Primer on Flat Rolling Primer on Flat Rolling Second Edition

John G. Lenard

Department of Mechanical and Mechatronics Engineering University of Waterloo Waterloo, Ontario, Canada

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Elsevier The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK 225 Wyman Street, Waltham, MA 02451, USA First edition 2007 Second edition 2014 Copyright © 2014, 2007 Elsevier Ltd. All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangement with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-099418-5

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For Elisa-Chaya, whose smile makes my day Preface to the Second Edition

In the first edition of this book, I wrote that the process of flat rolling has not changed for centuries, not since Leonardo da Vinci rolled soft metals on his hand-driven mill. While the process hasn’t changed in the last few years, either, there have been numerous, important innovations. The first edition was intended to present the basic ideas of flat rolling and that objective also has not changed in the second edition. Several chapters have been added, however, expanding on the fundamental concepts. In Chapter 1, various rolling processes À not only flat rolling, exclusively À have been described. In a new Chapter 2, Don Adair, the manager of Flat Rolling Operations at Quad Engineering of Toronto and Edwin B. Intong, former technical manager for Electrical & Automation, US Steel Engineers and Consultants, present a history of the development of different generations of hot strip mills. The changes are highlighted and their importance is pointed out. In an interesting section, the innovations, introduced in recent times, are listed. Another new addition, Chapter 3, prepared by Dr J.B. Tiley, hot rolling consultant, discusses current trends of roll design, focusing on the development of new materials for work rolls and back-up rolls with the objectives of reducing roll wear and enhancing roll life. Chapter 4, unchanged, describes the basic ideas of flat rolling. Chapter 5 deals with mathematical modelling of the process. Section 3.8 in the first edition, dealing with Neural Networks, has been removed. While the use of neural networks created some excitement some years ago, this didn’t continue, as engineers prefer to use statistical methods for control and predictions. Chapter 6, Advanced Finite Element Modelling is new. It is the result of the cooperative research of Dr G. Krallics, Z. Be´zi, M. Szucs˝ of the University of Miskolc, Miskolc, Hungary and the present author. It is acknowledged that finite element modelling of flat rolling has often been dealt with in the literature. Details of some of these models are given and a new model that accounts for all three components of the flat rolling system À the roll, the rolled strip and its interface À is introduced. Its predictive ability is presented by comparing its computations to experimental results. Chapter 7, Simulation and Reduction of Local Buckles in Cold Rolling, goes somewhat beyond the basics. It was prepared by Dr Yuli Liu, chief process engineer, manager of Process Engineering and Development of Quad Engineering Inc. An advanced mathematical model, consisting of several modules, is given in some detail. The chapter is of some importance as it deals with problems of shape and dimensional inaccuracies in the rolling process. However, using the model is neither easy nor immediate. A potential user must refer to the original publications. xii Preface to the Second Edition

Chapter 8 is unchanged. A new section, Nanotribology, is now included in Chapter 9, Tribology. The topic is briefly described. Chapter 10 is unchanged, as well. Most of the first edition dealt with steels. In the second edition, a presentation of hot rolling of aluminium alloys (Chapter 11) broadens the scope of the book. This portion has been prepared by Dr Mary Wells of the University of Waterloo. The rest of the first edition is repeated. None of Chapters 12À15 has been changed. A well-acknowledged chapter in the first edition presented problems and their solutions; this chapter has been repeated unchanged.

Advice for Instructors

There are several topics mentioned in this book, the thorough understanding of which needs a broad and varied background. The instructor should be aware of the preparation of the audience and make sure that the following subjects are understood well before starting on the presentations of the book’s contents. A brief quiz during the first lecture and the discussion of the results are often helpful in finding out what needs to be reviewed. In the present writer’s experience with rolling mill engineers, this background may have been there in the listeners’ college or university days but if such knowledge has not been used daily for some considerable time, gaps are certain to exist. It is strongly recommended that at least the first six lectures be devoted to a review of the following. The ideas involved with the strength of materials should be mastered first. These include the theory of elasticity and the analysis of stress and strain; the idea of equilibrium, static and dynamic. Principal directions, principal stresses and strains also need to be appreciated. Boundary conditions, surface and body forces should be clarified and it may be helpful to assign, and then discuss in class, some esoteric examples such as the free-body diagram of a tooth while it is being extracted or the forces and torques acting on a rail car wheel in motion. Identifying and sketching the loads on a bullet in flight would also pose a challenge. If these are well understood, their application during the course should become easy. The difference between engineering and true stresses and strains should be made clear. Strain rates and the conditions under which they remain constant in a test need to be mentioned. The theory of plasticity is used throughout the book without developing the basic ideas. ElasticÀplastic boundaries, the yield and the flow criteria, the associated flow rules, the constancy of volume and the compatibility equations should be presented as part of the review. The stress and strain tensors should be mentioned in addition to the tensor invariants. Basic ideas from the field of metallurgy are needed. The grain structure of metals, the carbon equilibrium diagram, the hardening and restoration mechanisms and the hot and cold response of metals to loading are all used in many of the developments in the course. It would be helpful for the students to have actually mounted, polished and etched a piece of metal for metallographical examination. Preface to the Second Edition xiii

Some time should be devoted to a discussion of Tribology as well. Viscosity, Reynold’s equation, lubricant and emulsion chemistry are all necessary here. As a last comment to the instructors, nothing replaces the actual hands-on experimentation. Having a well-instrumented rolling mill and conducting some carefully designed experiments would lead to immeasurable benefits. Some care needs to be exercised in assigning the problems from Chapter 11. Many of them are fairly straightforward and require the application of the ideas presented in the text. Many of them, however, require extensive reading and may well lead to some frustration. A discussion of the solution in class is often highly appreciated. Seminars or class discussions are suggested when dealing with Chapters 7À10. These may require advance preparation so the discussions would not become pro- fessorial presentations. State-of-the-art reviews have been found helpful. John G. Lenard, 2013 Preface to the First Edition

I have been dealing with problems of the flat rolling process for the last 30 years. This included mathematical modelling, experimentation, consulting, publishing in technical journals and presenting my research at conferences and in industry, as well as lecturing on the topic at levels, appropriate for second- and third-year undergraduate students, graduate students and practicing engineers and technologists of aluminium and steel companies. The present book is a compilation of my experience, prepared for use by practitioners who work with metal rolling and who want to know about the “whys”, the “whats” and the interdependence of the material and process parameters of the rolling process. The book may also be useful for graduate students researching flat rolling. My interest in the process began while I spent a year at Stelco Research as an NSERC Senior Industrial Fellow, shortly after starting my academic career. I became aware of the tremendous complexity underlying the seemingly very simple process of metal rolling. I realized that while the process of flat rolling À that of two cylinders rotating in opposite directions and reducing the thickness of a strip as it passes in between them À has not changed for centuries, its current sophistication places it at the top of the “high-tech” activities. On return to academia and as soon as research funds allowed, I designed and built a simple two-high experimental rolling mill and instrumented it to measure the important variables. The mill has been in use ever since to roll various metals À mostly aluminium and steel alloys À under a large variety of conditions. These conditions included dry and lubricated passes, use of neat oils and emulsions, high, low and intermediate temperatures, heated and non- heated rolls and speeds and reductions as high and low as the mill allowed. During these experiments my students and I used smooth and rough roll surfaces, prepared by grinding or sand blasting. In each of the tests the roll separating forces, the roll torques, the entry and exit thickness, the rolling speed, the forward slip, the entry and exit temperatures of the strip, the roll’s surface temperature, the amount of the lubricant, the flow rate and the temperature of the emulsion, the droplet size in the emulsion, the change of the width and the reduction of the strips were measured. In addition to the experiments performed by myself, by academic visitors from China, Egypt, Germany, Hungary, India, Israel, Japan, Poland and South Korea, and by my graduate students, twice each year my undergraduate classes, typically 80À100 students strong, performed flat rolling tests, providing me with a very respectable collection of data. xvi Preface to the First Edition

Mathematical modelling of the process proceeded parallel to the experimental studies. The attention was on establishing the predictive abilities of the available models of the flat rolling process. The assumptions made in the derivation of the traditional 1D models were critically examined and were improved on by developing an advanced 1D model which makes use of as few arbitrary assumptions as possible. The use of finite-element models was also explored in cooperation with Prof. Pietrzyk (University of Mining and Metallurgy, Krakow, Poland) and his colleagues and students. During my academic career I offered, once or twice a year, a specialist course on rolling, designed for technologists and engineers who work in the metal rolling industry. The educational level of the audience varied broadly from those who had completed high school to those with doctoral degrees. Each year I found two unchanging phenomena. The first was the shaky background my listeners pos- sessed, essentially regardless of their education. When asked about the difference between engineering strains and true strains, the difference between the plane- stress and plane-strain conditions, the difference between static and dynamic recrystallization and so on, the large majority of them betrayed serious ignorance. The second was the lack of a textbook that included all I needed to develop the ideas in the course. The present book, resulting from the notes I used in these courses, attempts to compile, present and explain the disparate components needed for a clear understanding of the topic. The book contains 11 chapters. The first 10 of these deal with various aspects of the flat rolling process and the 11th presents a set of assignments and incomplete solutions, formulated to test the understanding of the reader of the material presented. Each chapter ends with a set of conclusions. The flat rolling process is defined in Chapter 1, the Introduction. The objectives are to give a very brief overview of the process. Details of the hot rolling process, using hot strip mills, are given. Continuous casting is described. The cold rolling process and cold mill configurations are presented next. A general discussion of the rolling process is presented in Chapter 2. The components of a metal rolling system are defined. Reference is made to the rolling mill, designed by Leonardo da Vinci, and the scale model, built following his drawings. A description of the physical and metallurgical events during the process is given, including the events as the strip to be rolled is ready to enter the roll gap, as it is partially reduced and as the process becomes one of steady state. The independent variables of the system À the mill, the rolled metal and their interface À are listed. The minimum value of the coefficient of friction, necessary to commence the rolling process, is given. Some of the simplifying assumptions that are usually made in mathematical models of the process of flat rolling are critically discussed: these include the idea of “plane-strain plastic flow” and “homogeneous compression of the strip”. Microstructures of a fully recrystallized Nb steel, an AISI 1008 steel and a cold-rolled low carbon steel are presented. Mathematical modelling of the rolling process is the topic of Chapter 3. Traditional and more advanced models are discussed in terms of their capabilities as far as their predictions are concerned. Models for both mechanical and metallurgical Preface to the First Edition xvii events are included. The chapter ends with the identification of three parameters necessary for efficient, accurate and consistent modelling: the coefficients of heat transfer and friction and the resistance of the material to deformation. Chapters 4 and 5 treat these in turn: material behaviour and tribology, respectively. In both, the emphasis is on how the concepts are to be used when combined with the models, presented in the previous chapter. The objectives in preparing Chapter 6 are somewhat different. The chapter is entitled “Sensitivity Studies” and in spite of some examination of the sensitivity of the predictions in previous chapters, some more calculations and applications are added. Temper rolling is considered in Chapter 7. The differences between the usual flat rolling process and temper rolling are pointed out. Several mathematical models are given and the assumptions made in their development are discussed. The components that should make up a complete model of the process are listed. The tenor of the book changes at that point. In each of Chapters 8, 9 and 10 À accumulative roll bonding, flexible rolling and cold roll bonding, respectively À a review of the literature is followed by the detailed descriptions of experimental work. Chapter 11 contains two sections. In the first, problems are listed for each of the chapters. Some of these require the direct application of the expressions and the formulas presented in the book. Some answers require Internet searches. Some require development of computer programs. Some are suggested topics for seminars or class discussions. In the second part the solutions are given. Again, this is done in a variety of ways: in some cases detailed solutions are given, while in some others only the numerical answers are indicated. As well, in some instances, only a set of hints and recommended approaches are suggested. I would like to acknowledge the contributions of my undergraduate and graduate students without whom my research would not have progressed. Also, I would like to thank the visiting scientists with whom cooperation was always most enjoyable. This book couldn’t have been produced without the active encouragement of my wife Harriet and my daughter Patti. I am deeply grateful for their continuing support. John G. Lenard, 2007 Contributors

G Krallics University of Miskolc, Institute for Materials Science, Hungary

Edwin B. Intong Former Technical Manager for Electrical & Automation, US Steel Engineers and Consultants

Yuli Liu Manager, Process Engineering and Development, Quad Engineering Inc., Toronto, Ontario, Canada

John Tiley Hot Rolling Consultant

Donald Richard Adair Flat Rolling Manager, Quad Engineering Inc, Toronto, Ontario, Canada

M. Wells Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, Ontario, Canada Acknowledgements

The cooperation by Prof. Wells, Dr. Krallics, Dr. Liu, Dr. Tiley, Mr. Adair and Mr. Intong is greatly appreciated. 1 Introduction

1.1 The Flat Rolling Process

The mechanical objective of the flat rolling process is simple. It is to reduce the thickness of the work piece from the initial thickness to a pre-determined final thickness. This is accomplished on a rolling mill, in which two work rolls, rotating in opposite directions, draw the strip or plate to be rolled into the roll gap and force it through to the exit, causing the required reduction of the thickness. As these events progress, the material’s mechanical attributes change. These in turn cause changes to the metallurgical attributes of the metal, which, arguably are of more importance as far as the product is concerned. A schematic, three-dimensional diagram of the back-up rolls and the work rolls is shown in Figure 1.1 where a single- stand, four-high mill is depicted; this may be a single-stand roughing mill. Figure 1.1 shows the back-up rolls, the much smaller work rolls, the strip being rolled and the roll separating forces acting on the journals of the back-up roll bearings, keeping the centre-to-centre distance of the bearings as constant as possible1. As will be demonstrated in Chapter 10, the energy requirements of the process may be decreased when small diameter work rolls are used. The drawback of that step is the reduced strength of the work roll which necessitates the use of the massive back-up rolls to minimize the deflections of the work roll.

1.1.1 Hot, Cold and Warm Rolling While the rolling process may be performed at temperatures above half of the melting point of the metal, termed hot rolling, or below that temperature, in which case one deals with cold rolling, the division into these two categories should not be considered as being cast in stone. There is a temperature range, beginning below and ending above the dividing line between hot and cold rolling, within which the process is termed warm rolling and in some specific instances and for some materials this is the preferred process to follow. These processes lead to mechanical and metallurgical changes of the attributes of the work piece, which are not possible to achieve in either the cold or the hot flat rolling regimes.

1Mill stretch will be discussed in Chapter 5.

1.2 The Hot Rolling Process

Hot rolling of metals is usually carried out in an integrated steel mill, on a “Hot Strip Mill”, or since some changes were introduced in the last couple of decades, on mini-mills2. Both have advantages and disadvantages, of course, such as capital costs, flexibility, quality of the product and danger to the environment. A schematic diagram of a traditional hot strip mill is depicted in Figure 1.2, showing the major components. There are several basic components in the traditional hot strip rolling mill. In what follows, these are discussed briefly3.

Roll force

Back-up roll Work roll

Work piece

Figure 1.1 A schematic diagram of a single-stand, four-high set of rolls.

Edge rollers Pyrometers Descaler

Reheating Transfer Runout table furnaces table Finishing mill and Coilers Roughing Flying Cooling banks shear mill X-ray

Figure 1.2 The schematic diagram of a traditional hot strip mill.

2See Section 1.4 for a brief discussion of mini-mills. Also, see Section 2.8 for a discussion of compact strip production. 3See Chapter 2 where the development of various configurations of hot strip mills is described. Introduction 3

1.2.1 Reheating Furnace The reheating furnace constitutes the first stop of the slab after its delivery from the slab yard. The slab is heated up to 1200À1250C in the furnace to remove the cast dendrite structures and dissolve most of the alloying elements. The decisions to be made in running the reheat furnace in an optimal fashion concern the temperature and the environment within. If the temperature is higher than necessary, more chemical components will enter into solid solution but the costs associated with the operation become very high and the thickness of the layer of the primary scale will grow. If the temperature is too low, not all alloying elements will enter into solid solution, affecting the metallurgical development of the product, and the likelihood of hard precipitates remaining in the metal increases. As well, thinner layers of scale will form, a fairly significant advantage. A judicial compromise is necessary here and is usually based on financial consideration. The cost savings associated with a one-degree reduction of the temperature within the furnace can be calculated without too many difficulties; the changes to the formation of solid solutions may be estimated but the annual savings may well be significant. Primary scales of several millimetres thickness form on the slab’s surface in the reheat furnace. The thickness of the scale may be reduced by providing a protective environment within the furnace, albeit at some increased cost. As the furnace doors open and the hot slab slides down on the skids to the conveyor table, the instant chilling, caused by the water-cooled skids, causes marks that are often noticeable on the finished product. As well, fast cooling of the surfaces and especially of the edges is also immediately noticeable, indicating a non-uniform distribution of the temperature within the slab and leading to possibly non-homogeneous dimensional, mechanical and metallurgical attributes.

1.2.2 Rough Rolling Before the rolling process begins, the scale is removed by a high-pressure water spray and/or scale breakers. The slab is then rolled in the roughing stands in which the thickness of the slab is reduced from approximately 200À300 mm to about 50 mm in several passes, typically four or five. The speeds in the rougher vary from about 1 m/s to about 5 m/s. In the roughing process the width increases in each pass and is controlled by vertical edge rollers. The vertical edgers compress and deform the slab somewhat, causing some thickening which is corrected in the subsequent passes. A large variety of roughing mill configurations is possible, from single-stand reversing mills to multi-stand, one-directional mills, referred to as roughing trains. These usually have a scale breaker as the first stand where the mill deforms the slab sufficiently just to loosen the scale, which is then removed by the high-pressure water jets. Roughing scale breakers are usually vertical edgers, capable of reducing the width of the slab by up to 5À10 cm and causing stresses at the steel surface-scale layer interface which then separate the scales. Roll diameters are 4 Primer on Flat Rolling near 1000 mm. The rolls are usually made of cast steel or tool steel4. Roughing stands are either of the two- or four-high configurations. At the end of the rough rolling process, the strip is sent to the finishing mill along the transfer table where it is referred to as the “transfer bar”. The temperature of the slab in the roughing stands is high enough so that the transfer bar is fully recrystallized, containing strain free, equiaxed grains. In general, though, the grain structure at the end of the rough rolling process seems to have little influence on the structure by the time the strip has passed through several stands of the finishing mill.

1.2.3 Coil Box Not shown in Figure 1.2 is a device À an invention by the Steel Company of Canada and first installed in the early 1970s in Stelco’s Hilton Works À called the coil box5, placed between the roughing mill and the finishing train, in place of the transfer table. Since that time, several integrated steel companies have installed the coil box in their hot strip mills. A photograph of the coil box is shown in Figure 1.3. When the words “Coil Box” are entered into Google6, a plethora of information is found, including the possibility of watching a video of the coil box in motion. A detailed description of the events when the steel arrives to the coil box and when it is within the coil box are also available on-line. The transfer bars, exiting from the roughing stand are formed into coils at the coil box, a patented design of the Steel Company of Canada. The coil box consists of two entry rolls, three bending rolls, a forming roll, two sets of cradle rolls, coil stabilizers, peeler, transfer arm and pinch rolls. The adoption of a coil box configuration has several advantages:

G it reduces the overall length of the mill line; G it increases the productivity; G it enlarges the strip width and the length to be rolled and G it eliminates the thermal rundown along the strip length when compared to the conventional HSM. Thus, uniform temperature and constant rolling speed conditions are maintained. On uncoiling from the coil box, the transfer bars are end-cut, processed through high-pressure descaling sprays, and then they are ready to enter the finishing stands. With the introduction of advanced high-strength steels such as Hot Roll Dual Phase steels7, the benefits of the coil box are even more significant in providing uniform mechanical properties throughout the length of the coil8.

4See Chapter 3 for a description of roll materials. 5See Chapter 2 for a description of the coil box. 6It is acknowledged that the contents of websites on the Internet are changed and updated regularly. 7To be discussed in Chapter 8. 8The first coil box was installed in the Hilton Works shortly before the writer spent a year as a Senior Industrial Fellow at the Research Department of the Steel Company of Canada. At that time the information concerning the coil box was proprietary and so carefully guarded that no permission was obtained to read any of the reports written on the performance or the analysis of the equipment. Introduction 5

Figure 1.3 The coil box. Source: Courtesy The Steel Company of Canada.

1.2.4 Finish Rolling When the transfer bar, now coiled up in the coil box, reaches the appropriate temperature, it is uncoiled and is ready to enter the last several stands of the strip mill, the finishing train. The crop shear prepares the leading edge for entry and the transfer bar enters the first stand, assisted by edge rollers. Its velocity is in the range of 2.5À5 m/s9. The finishing train in the strip mill is traditionally composed of five to seven tandem stands. The roll configuration is usually four-high, employing large diameter back-up rolls and smaller diameter work rolls. The entry of the strip into the first stand is carefully controlled and is initiated when the temperature is deemed appropriate, according to the draft schedule, which is prepared using sophisticated off-line mathematical models. These determine the reductions and the speeds at each mill stand as well as predicting the resulting mechanical and metallurgical attributes of the finished product. After entry into the first stand, the strip is continuously rolled in the finishing mill. At the entry to the finishing mill, the temperature of the strip is measured and at the exit, both temperature and thickness are measured; the thickness at the exit from each intermediate stand is estimated

9When the present author was working in the Research Department of the Steel Company of Canada, some consideration was given to increasing the entry speed of the transfer bar into the first stand of the finishing mill. The project was abandoned when the possibility of the steel becoming airborne was realized. 6 Primer on Flat Rolling using mass conservation10. In some modern mills there are several optical pyrometers placed along the finishing train. The Automatic Gauge Control (AGC) system uses the feedback signals from several transducers to control the exit thickness of the strip. The finishing temperature may also be controlled by changing the rolling speed. However, only small variations of the rolling speed are possible without causing tearing, if the speed of the subsequent mill stand is too high, or buckling, referred to as “cobble” of the strip11, when the speed there is too low. On some newer and more modern strip mills, interstand cooling and/or heating devices have been installed, which minimize the temperature variation across the rolled strip and thereby increase the homogeneity and the quality of the product. As the thickness is reduced the speed must increase, as demanded by mass conservation, and the speeds in the last stand may be as high as 10À20 m/s. The rolls of the finishing mill are cooled by water jets strategically placed around the rolls. Without cooling, the surface temperature of the work rolls would rise to unacceptable levels. It has been estimated that when in contact with the hot strip, the roll surface temperatures could rise to as high as 500C at a very fast rate. Of course, the roll surface would cool during its journey as it is turning around and is subjected to water cooling, but the thermal fatigue it experiences accelerates roll wear and is, in fact, one of the major contributors to it. It is possible to measure roll surface temperatures by thermocouples embedded in the roll, with their tips positioned close to the surface12. A mathematical model would then be necessary to extrapolate the temperatures to the surface. There are usually scale breakers before the first stand of the finishing train, consisting of one or two sets of pinch rolls, exerting only enough pressure on the strip to break off the scale. The strip exits from the finishing train at a thickness of 1À4 mm. The Hylsa steel mill in Monterrey, Mexico produces hot rolled strip of 0.91 mm thickness. Bobig and Stella (2004) describe the semi-endless rolling and ferritic rolling processes. These, introduced in the thin slab rolling plant EZZ Flat Steel in Egypt, produce 0.8 mm-thick coils. The ferritic rolling leads to reduced scale growth and lower roll wear. During the last decade the materials used for the rolls on the hot strip mill were changed from chill cast to tool steels, reducing roll wear in a most significant manner13. There have also been reports of significant changes of the coefficient of friction in the roll gap after the switch of roll materials. Tool steels rolls, once implemented correctly, do provide benefits that offset their higher costs. The impact of lubricant interactions with these new roll chemistries has not been fully explored (Nelson, 2006).

10The lack of pyrometers along the finishing train often causes difficulties when the events at the mill stands are modelled. 11See Chapter 7 for a mathematical model, simulating the formation of buckles. 12This would not, of course, be permitted in a production mill. Results on the rise of the surface temperature of the roll, obtained using eight thermocouples embedded in the work roll of a laboratory mill, are presented in Chapter 9. 13Roll wear will be discussed in Chapter 3 and in Chapter 9. Roll materials are described in Chapter 3. Introduction 7

1.2.5 Cooling After exiting the finishing mill, the strip, at a temperature of 800À900C, is cooled further under controlled conditions by a water curtain on the run-out table. The run-out table may be as long as 150À200 m. Cooling water is sprayed on the top of the steel at a flow rate of 20,000À50,000 gpm; and on the bottom surface at 5000À20,000 gpm (1 gpm 5 4.55 l/min). The purpose of cooling is, of course, to reduce the temperature for coiling and transportation, but also to allow faster cooling of the finished product, resulting in higher strength. The cooling process plays a major role in the thermalÀmechanical schedule, designed to affect the microstructure of the product.

1.2.6 Coiling At the exit of the run-out table, the temperature of the strip is measured and the strip is coiled by the coiler. After further cooling, the steel coils are ready for shipping.

1.2.7 The Hot Strip Mill A photograph of a hot strip mill of Dofasco Inc. is shown in Figure 1.4. A pair of work rolls is visible, stored in the foreground of the figure and ready to be placed in the stands14.

1.3 Continuous Casting

Irwing (1993) describes the history of the development of continuous casting and identifies Mannesmann AG, where a production plant went into operation in 1950. A continuous casting plant was installed at Barrow Steel in Great Britain in 1951. The essential idea of the process is simple: molten steel is poured into a water- cooled, oscillating mould. The cooled copper wall of the mould solidifies the outer layer of the steel and as the steel is moving vertically downward, the solidified skin thickens. As the steel leaves the mould, it is cooled further by water sprays. The solidifying steel is supported by rollers, which prevent outward bulging. The continuous casting process replaced the ingot casting several decades ago and succeeded in increasing productivity. The complete continuous casting process is shown in Figure 1.5. The figure shows the ladle into which the molten steel is poured. From the ladle the steel is metered into the tundish and from there it enters the water-cooled, oscillating mould. As the steel strand exits the mould, it is solidifying further; an indication of the solidification front is also shown in Figure 1.5.

14The rolls are changed at regular intervals in the hot strip mill. The change takes place very fast, such that the mill need not be shut down. 8 Primer on Flat Rolling

Figure 1.4 The seven-stand finishing mill of Dofasco Inc. Source: Courtesy Dofasco Inc.

Using the withdrawal rolls and the bending rolls, the now solid but still very hot strand is straightened and cut to pre-determined sizes by the cut-off torch. There are two possible subsequent activities at this point. The slabs may be allowed to cool and are then stored in the slab yard, retrieved as needed by customers and reheated in the reheat furnaces and rolled, in the hot strip mill, as depicted in Figure 1.3. Alternatively, they may be rolled directly, as shown in Figure 1.6.

1.4 Mini-Mills (See Also Chapter 2)

The American Iron and Steel Institute’s website gives the following definition for mini-mills:

Normally defined as steel mills that melt scrap metal to produce commodity products. Although the mini-mills are subject to the same steel processing requirements after the caster as the integrated steel companies, they differ greatly in regard to their minimum efficient size, labour relations, product markets, and man- agement style. Introduction 9

Ladle

Molten steel Tundish Mold flux Submerged entry nozzle Water-cooled mold Molten steel Solidified steel Guide rolls Water spray

Cooling chamber

Withdrawal rolls

Bending rolls Slab straightening rolls Cutoff torch Continuous slab Slab

Figure 1.5 Continuous slab casting. Source: Groover (2002); reproduced with permission.

Entry from

the caster Tunnel Descaler Five-stand hot mill Spray zone Fine zone Trim zone Coiler

Figure 1.6 Continuous casting and direct rolling. Source: Following Pleschiutschnigg et al. (2004).

Currently in the United States 52% of the steel is rolled by 20 integrated steel mills and 48% by more than 100 mini-mills. The integrated mills roll approximately 400 tons/h while the mini-mills are capable of 100 tons/h. Information is also available from Wikipedia, a web-based encyclopaedia. It identifies mini-mills as secondary steel producers. Also, it mentions NUCOR as one of the world’s largest steel producers, which uses mini-mills exclusively. A very impressive number (79%) of mini-mill customers expressed satisfaction with their suppliers15.

152001 Customer Satisfaction Report, Jacobson & Associates. 10 Primer on Flat Rolling

The website www.environmentaldefense.org gives information regarding the recycling activities of mini-mills, stating that they conserve 1.25 tons of iron ore, 0.5 tons of coal and 40 lbs of limestone for every ton of steel recycled.

1.5 The Cold Rolling Process

The layers of scales are removed from the surfaces of the strips by pickling, usually in hydrochloric acid. This is followed by further reduction of the thickness, produced by cold rolling. Essentially there are three major objectives in this step: to reduce the thickness further, to increase the rolled metals’ strength by strain hardening and to improve the dimensional consistency of the product. An additional objective may be to remove the yield point extension by temper rolling, in which a small reduction, typically 0.5À5%, only is used16.

1.5.1 Cold Rolling Mill Configurations A large variation of configurations is possible in this process. An example of a modern cold rolling mill, for aluminium, is shown in Figure 1.7. The mill is six- high, having two small diameter work rolls of 470 mm diameter and two sets of back-up rolls. The diameter of the intermediate back-up roll is 510 mm and the third back-up roll is of 1300 mm diameter. The mill is capable of producing strips of 0.08 mm thickness at speeds up to 1800 m/min. Mill types, design details and configurations are so numerous that it is impossible to list them all in a brief set of notes. Mill frames, bearings and chucks, screw- down arrangements, loopers, control systems, number of stands, drive systems, spindles, lubricant or emulsion delivery, roll cooling, roll bending devices, shears and coilers may have practically infinite variations in design. Roll materials may also vary, and as recent literature indicates, the chill cast or high chrome rolls are being replaced by tool steel rolls. In what follows, only a set of figures indicating various roll arrangements is presented. Figure 1.8 shows the simplest two-high version in which two work rolls of fairly large diameters are used. The simplicity À low number of components À is out- weighed by the disadvantage of the need for massive rolls to minimize roll bending. A more advanced and significantly more rigid arrangement is the six-high configuration, in which the bending of the work rolls is reduced substantially by the large back-up rolls. Also, advantage is taken of the lower energy requirements, needed when the work rolls are of smaller diameters. The accuracy and consistency of the strip dimensions increase as the number of back-up rolls increases, resulting in a significant reduction of the deflections of the small work-rolls. The stiffness of the complete rolling mill also increases. Figure 1.9 shows a photograph of a

16Temper rolling is discussed in Chapter 12. Introduction 11

Figure 1.7 A schematic diagram of a modern cold rolling mill for aluminium (Hishikawa et al., 1990).

Two - high mill Six-high mill

Figure 1.8 A two- and a six-high mill. 12 Primer on Flat Rolling

Figure 1.9 A 20-high mill, for rolling copper and copper alloys, built by SUNDWIG GmbH.

20-high mill, built by SUNDWIG GmbH. The progressively increasing roll diameters, starting with the very small work rolls, are clearly observable. Bill and Scriven (1979) describe the details of the Sendzimir mill À which is used for both hot and cold rolling À and show various designs and configurations. They describe the advantages and the disadvantages of using small diameter work rolls, and the history of how engineers attempted to overcome this problem. Tadeusz Sendzimir, a Polish engineer and inventor, designed the clus- ter mill, which, named after him, was built as an experimental rolling mill in 1931 in Du¨sseldorf, Germany. In one of the designs, a type 1-2-3-4 arrangement shown in Figure 1.10, similar to the 20-high mill, illustrated above, the work rolls are driven through friction contact. This mill, and the other versions of it, are capable of producing very high reduction in one pass and can roll a strip to very low thickness. Backofen (1972) writes that the work roll may well have a diameter under 1v (25.4 mm) and the exit thickness may be as low as one- thousandth of an inch (0.025 mm). Further, since the small work rolls flatten less, they can continue to roll metal even after significant strain hardening with no need for intermediate annealing. The work rolls are often made of tungsten carbide, resulting in much longer roll life and producing a mirror finish on the rolled surfaces. The ridges, sometimes created by the small work rolls, are smoothed by subsequent operations. The Platzer planetary mill, shown in Figure 1.11, is also capable of very high reductions. In some of the versions, the mill has two back-up beams which are stationary. Around these are the intermediate and the work rolls. Feed rolls force the strip into the roll gap. The work roll diameters range from a low of 75 to 225 mm, Introduction 13

Figure 1.10 Sendzimir mill (Bill and Scriven, 1978).

depending on the width, much larger than in the Sendzimir mill of Figure 1.10. Fink and Buch (1979) indicate that 98% reductions are achievable on the Platzer mill, in one pass. It is interesting to note that the small work rolls rotate in a direction opposite the rolling direction. The number of roll contacts may be as high as 40À60 s21.

1.6 The Warm Rolling Process

The temperature range for this process is not defined very closely; it starts somewhat below half of the homologous temperature17 and ends somewhat above that. In the process both the strain and the rate of strain affect the mechanical and metallurgical attributes of the rolled metal and in process design these need to be accounted for carefully. The energy requirements are, of course, higher than those for hot rolling but lower than for cold rolling. The strength of the resulting product is higher than what can be achieved by hot rolling. While there is an accumulation of scales of the surfaces, the amount is significantly less than in the hot rolling process. The ferrite rolling, mentioned above, may be considered a warm rolling process, although this suggestion may be somewhat controversial.

17The homologous temperature range is defined such that one of the end points is absolute zero while the other is the melting temperature of the particular metal. Figure 1.11 The Platzer planetary mill (Fink and Buch, 1979). Introduction 15

1.7 Further Reading

A large number of books dealing with the rolling process are available. Among these the excellent books of Roberts Cold Rolling of Steel Hot Rolling of Steel and Flat Processing of Steel (Roberts, 1978, 1983, 1988), stand out. These are emi- nently readable, giving the history of the processes, detailed description of the equipment and the mathematical treatment. Rolling of shapes as well as flats is considered. Rolling of metals is considered exclusively by Underwood (1950), Starling (1962), Larke (1965), Tarnovskii et al. (1965), Tselikov (1967), Wusatowski (1969), Pietrzyk and Lenard (1991), Ginzburg (1993) and Lenard et al. (1999). Books dealing with the theory of plasticity or metal forming have chapters devoted to the rolling of metals. These include the books of Hill (1950), Hoffman and Sachs (1953), Johnson and Mellor (1962), Avitzur (1968), Backofen (1972), Rowe (1977), Lubliner (1990), Mielnik (1991), Hosford and Caddell (1983) and Wagoner and Chenot (1996). It may be necessary to review the background to plastic forming of metals. The reader may then refer to textbooks dealing with the mathematical theory of plasticity, theory of elasticity as well as continuum mechanics. Perusing books dealing with the metallurgical phenomena of hot and cold metal forming may also be useful. The list of technical publications dealing with various aspects of the rolling process is prohibitively long to be included here.

1.8 Conclusion

The concerns of the present book, strips and plates, were defined according to their geometry, such that the ratio of their width to thickness is much larger than unity. The flat rolling process, capable of producing strips and plates, was described in general terms. The integrated steel mill and hot strip mill, including its components, were described in some detail. Hot, warm and cold rolling were mentioned and the temperature ranges for each were given. A brief presentation of some mill configurations was also given, including two-, four- and six-high arrangements. The Steckel mill, the Sendzimir mill and the planetary mill were discussed, accompanied by several illustrations. Mini-mills were presented and some comparisons of their capabilities to integrated steel mills were demonstrated. Material for further reading was also included, classified into two sections. In one, texts dealing with a general treatment of plastic deformation of metals are listed. These include the necessary theory of plasticity in addition to the application of the theories to the analysis bulk and sheet metal forming problems. The second category includes specialist books, dealing with the process of rolling. 2 History of Hot Strip Mills1

2.1 Hot Strip Mill Evolution

Hot strip mills have evolved in various steps from original tinplate and sheet rolling mills. The development of semi-continuous and continuous multi-stand mills followed, producing small-diameter and eventually large-diameter steel coil products. Low-, medium- and high-carbon steels, high-strength low alloy steels, X grade pipe steels, silicon steels and stainless steels were rolled. The early hot strip mills were first constructed in 1924. They were of low capacity, producing 300,000À900,000 short tons per year with low specific coil weights. Shortly after the advent of the early hot strip mills, reduced cost and lower capacity single- stand reversing hot strip mills, known as Steckel mills, were developed and placed in operation. These mills employed a heated coiling furnace on each side of the mill stand. Higher capacity Steckel mills were also developed. These included an upstream reversing rougher for slab reduction to transfer bar for finishing in the Steckel mill. By 1961, the early hot strip mills had been upgraded or replaced by larger capacity semi-continuous and continuous Generation I hot strip mills with capacities of about 400,000À3,000,000 short tons per year. These mills employed relatively small slab reheating furnaces and continued to produce low specific coil weights. Generation I mills also utilized a reversing rougher to roll discrete plates which were removed by plate transfer before the finishing mill. In some of these cases the reversing rougher was wider than the finishing mill in order to produce the desired maximum plate width. Some other Generation I mills utilized the finishing mill to roll discrete plates which were removed by a plate transfer either before or after the down coilers. The practice of rolling discrete plates on some of the Generation I mills was discontinued in most Generation II mills, many of which produced coiled plate products. Prior to the development of continuous casting of slabs, first introduced in the 1950s, cast ingots were reheated in soaking pits and rolled on slabbing mills to produce slabs for reheating and rolling to a finished strip on the hot strip mills. Often the maximum strip width to be produced was greater than the maximum slab width that could be provided by the slabbing mill. In these cases the first roughing stand of the hot strip mill was a broadside mill with entry and delivery slab turning tables or slab lift and turn mechanisms which were used to rotate the slabs 90.

1Contributed by Donald R. Adair, flat rolling manager, Quad Engineering Inc. and Edwin B. Intong, former technical manager for Electrical & Automation, US Steel Engineers and Consultants.

The broadside mill then cross-rolled the slab in a single pass to increase the width, as required. Eventually the development of continuous slab casting eliminated the use of slabbing mills except for a few specialty steel mills. Most hot strip mills today roll only continuous cast slabs. For energy conservation and increased slab reheating capacity, continuous slab caster hot strip mill installations ideally are arranged to permit the direct charging of hot slabs into the hot strip mill reheat furnaces. From 1961 through 1970, 11 new Generation II hot strip mills were built in the United States (Ess, 1970). These mills were of heavier design, higher power and higher speed with increased annual capacities of 2,000,000À3,800,000 short tons per year and produced larger coil unit weights of 16.8À22.3 kg/mm. Many other Generation II type mills have been built throughout the world from 1961 to date. The finished strip thickness ranged from 1.2 to 12.7 mm for the initial Generation II mills and from 1.2 to 25.4 mm for upgraded and newer Generation II mills. In actual practice the Generation II mills don’t roll much below 1.7 mm due to strip threading and shape problems. In the 1970s, the Steel Company of Canada (Stelco) developed the hot strip mill coil box (See Figure 1.3; also the web-site www.hatch.ca/technologies/Coilbox/ 80FLDR002_Coilbox-en.pdf) and the concept of Generation III hot strip mills by installing a prototype coil box in the Stelco 56v hot strip mill in Hamilton, Ontario, Canada. The transfer bar from the reversing rougher was coiled in the coil box and uncoiled into the finishing mill, thus retaining the transfer bar heat for more uniform finish mill strip entry temperature, resulting in improved strip quality and reduced motor power during finish mill rolling. It was also recognized that the coil box concept would permit larger coil unit weights to be produced on shorter new mills or increased coil unit weights to be produced on existing space-limited older mills. Since 1978, when the first commercial coil box was installed in a new hot strip mill at John Lysaght in Australia, many Generation III coil box hot strip mills, including new or upgraded existing mills, have been operating throughout the world. Finished strip thickness ranges from 1.0 to 25.4 mm for coil box hot strip mills. In the 1980s, the compact strip production (CSP) type of hot strip mill was developed by SMS consisting of an electric arc furnace and a thin slab caster which produced a slab thickness of 50 mm and a direct coupled hot strip mill for rolling the thin slab to finished coils. Since 1989 when the first commercial CSP plant was started up at Nucor in Crawfordsville, IN, there have been many CSP hot strip mills operating throughout the world. Mannesmann Demag (now part of SMS), Danieli and Sumitomo have also developed thin slab caster-direct rolling hot strip mills. Mitsubishi-Hitachi has also provided thin slab caster hot strip mills in cooperation with Mannesmann Demag and using the Sumitomo thin slab casting technology. In addition to thin slab caster hot strip mills in which the slabs are parted for rolling to coils, Arvedi Cremoni/Siemens VAI has developed an endless thin slab casting and rolling process, ESP (endless strip production), wherein the cast slab is directly rolled to finished strip, which is then parted prior to coiling. Finished strip thickness ranged from 0.8 to 25.4 mm for the thin slab caster type hot strip mills. History of Hot Strip Mills 19

2.2 Early Hot Strip Mills

By 1940, the early hot strip mills of wide strip category, considered to be 914 mm or wider, included five mills located outside the United States (a 1676-mm mill in the Soviet Union, a 1270-mm mill in Germany, a 1092-mm mill in Japan, and 1422-mm and 1372-mm mills in England) and 28 mills located in the United States (Ess, 1941). The US mills were equipped with five, six or seven finish stands and except for a few semi-continuous mills which had single reversing roughing mills, most were of the fully continuous type with four or five roughing stands2. The nominal annual capacities of the US mills totalled 17,616,000 short tons. The nominal capacities were considered by American Iron and Steel Engineers (AISE) at that time to be conservative and the actual total annual capacity was estimated to be at least 22,500,000 short tons. The finish mill work-roll and back-up roll diameters of the US mills were relatively small, ranging from 546 to 699 mm for the work rolls and 1118 to 1372 mm for the back-up rolls. The mills were limited to producing relatively small specific coil weights, ranging from 5.8 to 11.6 kg/mm, except in the case of Weirton’s mill with 14.8 kg/mm. The early US mills are listed in the Appendix, including year built, mill width, maximum coil unit weight (PIW) and nominal annual capacities in short tons (TPY) totalling 17,616,000 TPY.

2.3 Early Steckel Mills

In 1925, the inventor Abraham P. Steckel in Youngstown, OH, first introduced the steel industry to the concept of a single-stand reversing hot rolling mill with entry and delivery coiling furnaces to reduce steel slabs into hot rolled coils. The first trial Steckel mill was 914 mm wide and began operation in 1931 at the Youngstown Sheet and Tube Co. in Briar Hill, OH. This mill was relocated to Indiana Harbor, Indiana, where it operated until 1935. It was then moved to Dominion Foundry and Steel Company (Dofasco) in Hamilton, Ontario, Canada, where it operated for 20 years before being replaced by a 1676-mm-wide Steckel mill. Some of the early Steckel mills are listed in the Appendix, including the year built, mill width and four- or two high type of mill stand.

2.4 Generation I Hot Strip Mills (USA)

By 1961, there were 30 hot strip mills operating in the United States of wide strip category (Ess, 1970). These mills were 914 mm or wider that had replaced, upgraded or added to the early hot strip mills and as the immediate forerunners to the Generation II hot strip mills, generally classified as Generation I hot strip mills.

233 Data Handbook, published by 33 Magazine in 1970, Newark, NJ. 33 Magazine is not published any longer. 20 Primer on Flat Rolling

The Generation I mills were a mixture of the semi-continuous and continuous types. The semi-continuous mills had single reversing roughing stands and the continuous mills had four or five roughing stands. All had five or six finishing stands (33 Magazine, 1970). The nominal annual capacities of the 30 Generation I mills were much higher than the early hot strip mills, totalling 55,809,000 short tons (Ess, 1970). The finish mill work-roll and back-up roll diameters of the Generation I mills ranged from 546 to 762 mm for the work rolls and 1118 to 1499 mm for the back- up rolls, not too different from the dimensions of the rolls of the early hot strip mills. The coil unit weights were also similar to the early hot strip mills in the range of 5.8À12.0 kg/mm. The total finish mill power ranged from 17,158 to 21,261 kW. The maximum delivery speed ranged from 595 to 710 m/min. The finish mill power indexes K, the total finish mill kilowatt per millimetre of mill width per 1000 m/min maximum finish mill speed, ranged from 9.5 to 23.0. The US Generation I hot strip mills are listed in the Appendix, including year built (year rebuilt), mill width, maximum coil unit weight (PIW) and nominal annual capacities in short tons (TPY) totalling 55,809,000 TPY.

2.5 Generation II Hot Strip Mills (USA)

Between 1961 and 1970, 11 new hot strip mills were built in the United States (Ess, 1970) and a 12th new mill was added in 1974. These mills resulted from customer requirements for closer strip tolerances which required larger, stiffer mill stands, and for larger coils also desired by the steel producers for increased operating efficiency. The work-roll and back-up roll diameters ranged from 635 to 724 mm and 1372 to1575 mm, respectively (Ess, 1970). The larger roll sizes and mill housing sections were designed to accommodate increased rolling forces and torques. For improved quality and increased productivity, most of the Generation II Mills included newly developed quick work-roll change systems instead of slower, less safe crane-operated porter bar work-roll changing. The US Generation II hot strip mills were relatively wide with widths ranging from 2032 to 2184 mm except for the first mill and the last mill which were 1473 and 1727 mm wide, respectively. The mills were equipped with six or seven finish stands. Two of the mills were semi-continuous with single reversing roughing stands and one or two non-reversing roughing stands. The remaining 10 mills were fully continuous with 4À6 roughing stands (Ess, 1970). For improved thickness tolerance using uniform temperature control and increased production rates using higher rolling speeds, the finish stands of these mills were more highly powered than the earlier mills in order to incorporate the new development of “zoom rolling”. With zoom rolling technology, the strip head end is threaded from the last finish stand to the coiler at a constant speed up to about 610 m/min, and then the finishing mill is accelerated to higher rolling speeds History of Hot Strip Mills 21 as permitted by the finish mill speed cone and power up to about 1220 m/min. Prior to the development of zoom rolling, the entire coil was rolled at the particular product coiler thread speed, which contributed to limited production capacities of the earlier mills. In these mills the maximum slab dimensions ranged from 203 to 305 mm thickness, 1346 to 2032 mm width and 8.5 to 12.2 m length with maximum weights of 18À47 short tons. The minimum slab widths ranged from 356 to 711 mm. The finished strip thickness ranged from 1.2 to 12.7 mm and the width from 356 to 2032 mm. The total finish mill power ranged from 23,364 to 62,664 kW and the maximum delivery speed ranged from 738 to 1280 m/min (Ess, 1970). The finish mill power indexes K ranged from 14.2 to 23.5. The original annual rated capacities of the first 11 Generation II mills were approximately 2,000,000 to 3,800,000 short tons and the coil unit weights were increased from those of the earlier mills to 16.8À22.3 kg/mm (Ess, 1970). The original annual rated capacity of the 12th Generation II mill is assumed to have been 2,400,000 short tons and the maximum coil unit weight of 18.6 kg/mm. The original annual rated capacities for the 12 US Generation II mills totalled 34,700,000 short tons. Most of these mills have been significantly upgraded with annual capacities increased to as high as 6,300,000 short tons, as in the case of US Steel, in Gary, IN, one of the world’s highest producing hot strip mills. The current annual rated capacities of the US Generation II mills range from 1,500,000 to 6,300,000 short tons and total 45,000,000 short tons3 excluding J&L Cleveland, which has been removed. The 12 Generation II hot strip mills are listed in the Appendix.

2.6 Other Generation I and II Hot Strip Mills

Between 1961 and 1979, many other Generation II type hot strip mills and a few Generation I type mills were built outside the United States. The Generation II mills were equipped with six to nine finish stands (including future provisions). The mills were semi-continuous with one to four roughing stands (known as 3/4 continuous mills, where there is one reversing rougher and three non-reversing roughing stands, followed by the finishing stands) or fully continuous with four to seven roughing stands. Of interest regarding the semi-continuous mills is the Iscor mill in South Africa which was originally provided with a single reversing rougher. Provisions for the addition of a second tandem reversing rougher were also made and this was subsequently installed. Further, the Erdemir mill in Turkey was designed for future conversion to a continuous mill. It was originally provided with a reversing rougher preceded by a non-reversing horizontal scale breaker. The latter was subsequently replaced with a second reversing rougher, therefore providing

3Hot Strip Mill Roundup, Association for Iron and Steel Technology, November 2012; Warrendale, PA. 22 Primer on Flat Rolling two independent reversing roughing mills in lieu of conversion to a continuous mill. The total finish mill power for these Generation II mills ranged from 22,380 to 101,531 kW. The maximum delivery speed ranged from 694 to 1851 m/min and the finish mill power indexes K ranged from 15.0 to 25.9. Other Generation I and II hot strip mills are listed in the Appendix, including year built, mill width, maximum coil unit weight (PIW), and Generation I or II type. Mills with maximum coil unit weight capability in excess of 1500 PIW are indicated as Generation II Super.

2.7 Generation III À Coil Box Hot Strip Mills

In the 1970s, Stelco of Canada developed the hot coil box designed for installation in hot strip mills ahead of the finish mill for coiling the transfer bar from the roughing mill and uncoiling it into the finishing mill. The coil box reduced the distance between the roughing mill and the finishing mill while retaining the transfer bar heat during finish mill rolling. In this way the entire transfer bar would enter the finish mill at approximately the same temperature and the entire strip could be rolled at thread speed without acceleration thereby reducing finish mill power requirements and improving finished strip thickness tolerance and metallurgical uniformity. In 1978, the first commercial coil box was installed in a new hot strip mill at John Lysaght in Australia. Stelco then sold numerous licenses to other steel producers for the application of coil boxes to existing or new hot strip mills. Stelco eventually transferred the coil box technology to Hatch Steel Technologies of Canada, who continue to provide coil box technology throughout the world. The original coil box concept, which permitted large unit weight coils to be rolled without zoom rolling, generally resulted in somewhat lower mill capacities in the area of 2À2.5 million short tons per year as compared to the Generation II mill capacities. Some of the newer coil box mills of today, such as Colakoglu in Turkey, are equipped with a very high-powered reversing roughing mill and finishing mill designed for higher speed zoom rolling from the coil box. Inter-stand water cooling is employed for strip temperature control, increasing the coil box mill production capacities while at the same time retaining the coil box advantage of a shorter mill layout. Most of the worldwide coil box hot strip mills are listed in the Appendix, including year built, mill width, new or existing mill, and annual capacities in metric tons4.

2.8 Thin Slab Hot Strip Mills

In the early 1980s, SMS Siemag (SMS) developed a pilot plant for a new type of hot strip mill process known as CSP in which thin slabs of approximately

4Coil Box Users, 1978À2007, by Hatch Steel Technologies, 2012. History of Hot Strip Mills 23

2v (50 mm) thickness are continuously cast and directly rolled in a rolling mill, which includes tunnel furnace(s), multiple rolling stands, strip cooling and down coilers. The CSP hot strip mills provided by SMS, beginning with Nucor at Crawfordsville, IN, in 1989, are listed in the appendix, including year built, maximum strip width and annual capacities in metric tons5. Other thin slab casting and direct rolling hot strip mill processes have been developed, including the ISP (in line strip production process) by Mannesmann Demag (now part of SMS); FTSC (flexible thin slab casting) by Danieli; QSP (quick ship program) by Sumitomo and ESP by Arvedi Cremoni/Siemens VAI. The Mannesmann Demag and some of the Danieli thin slab caster hot strip mills are listed in the Appendix, including year built, maximum strip width and annual capacities in metric tons. Thin slab caster hot strip mills referenced by Mitsubishi-Hitachi, some of which were provided with Mannesmann Demag ISP or Sumitomo QSP thin slab casting technology, are listed in the Appendix, including year built, maximum strip width and annual capacities in metric tons6. The Arvedi Cremoni/Siemens ESP thin caster Hot strip mill (HSM) is listed in the Appendix including year built, maximum strip width and annual capacity in metric tons. The introduction of the thin slab hot strip mills has relegated Generation I, II and III hot strip mills to be referred to as conventional slab hot strip mills.

2.9 Newer Generation II Hot Strip Mills

From 1980 to date, newer Generation II hot strip mills were built almost exclusively as semi-continuous mills, essentially eliminating the previous classical and costly continuous roughing mill configuration. The 1727-mm Dofasco hot strip mill, built in 1982, is unique in that it includes a single reversing mill that is actually a 1372-mm two-high universal slabbing mill. It initially rolled ingots from soaking pits to finish mill transfer bar when the mill started up in 1982 with only five finish stands. Subsequently Dofasco installed slab casters eliminating ingot rolling and the current mill annual production is 4,600,000 short tons with the slabbing mill and seven finish stands. Several of these newer semi-continuous mills include slab sizing presses to reduce slab caster width change requirements and all include single or multiple roughing stands which in some cases have sufficient capacity to achieve yearly ton- nages equivalent to or greater than the older continuous roughing train Generation II mills. Some of the newer Generation II hot strip mills are listed in the Appendix, including year built, mill width and annual capacities in metric tons7.

5Siemens AG, Compact Strip Production, 2010. 6Mitsubishi-Hitachi Machinery Inc., Hot Strip Mills, 1954À2010. 7SMS Siemag, Hot Rolling Mills Division, Reference List of Conventional Hot Strip Mills, 2010. 24 Primer on Flat Rolling

2.10 Modern Steckel Mills

The development of modern Steckel mills has included many items of improved design including more effective descaling, hydraulic automatic gauge control (HAGC), work-roll bending control, larger coiling furnaces and coiling drums, quick work-roll change, retractable furnace tables, higher speed rolling, combination strip and discrete plate, coiled plate Steckel mills and twin-stand Steckel mills. A number of the more modern Steckel mills throughout the world are listed in the Appendix, including year built, maximum strip width and annual capacities in metric tons. Twin-stand Steckel mills are indicated by an asterisk8.

2.11 Hot Mill Electrical Systems

2.11.1 Power System

Integrated steel plants usually provide a part of their electric power requirements with their own generating equipment, the power being produced using by-product fuels. Due to the large amount of power required by the mills and the concentration of this load, electrical power may be furnished by plant generation, by purchase from outside utilities or by a combination of the two. Where the plant generates its own power, it is usually that of the plant distribution system which then feeds the various load centres. With purchased power, incoming lines are often at the high transmission voltage which is stepped down to a suitable plant distribution voltage in a nearby outdoor station. This reduced voltage may tie into the plant general power system, forming a combined system. The basic connected electrical load for hot rolling mills varies widely, ranging from 32 to 70 MVA, depending on the size of the mill and the extent of operations. In the conventional layout of the hot strip mill, the motor room is usually parallel to the mill and lies between the hot mill building and the slab yard. The main mill drive motors are located in this area together with their respective gear reducers, switching and control equipment, motor generator sets, etc.

2.11.2 Motors and Drive Systems

The original prime movers used in rolling mills were reciprocating steam engines, designed for low pressure steam. Later, gas engines using blast furnace gas as fuel were developed and were used as prime movers for electric generators to drive the mills. Nowadays, virtually all mill stands and auxiliary equipment are powered by electric motors. Where the stand can operate at constant speed in one direction, the alternating current motor is generally used. For variable speed and reversible drives, the more expensive direct current motor is commonly employed. Since the early nineteenth century, the desired range of speed has increased and closer speed

8SMS Siemag, Hot Rolling Mills Division, Reference List of Steckel Mills, 2010. History of Hot Strip Mills 25 regulation has been required. This was responsible for the adoption of variable speed direct current motors for the main drives. Motor selection is based on the heaviest schedules after investigating the requirements of the various schedules. Two factors of importance in motor selection are the peak loads imposed and the motor heating or root-mean-square loads. The motors driving the roughing mill stands are usually of equal power. The motors driving the first five or six stands of the finishing mill are of equal power while that at the last stand is about 370À740 kW less. In the case of a continuous hot strip mill, due to the fact that the slab is in one roughing stand at a time as it passes through the mill, the motors driving these stands may be of constant speed. These motors are generally of the wound rotor induction type, controlled by slip regulators with torque motors to reduce power peaks. Since the duration of passes in a roughing stand is short and the time that the steel is in the rolls is only a small percentage of the total time, induction motors with flywheels make an ideal application. Synchronous motors offer the advantage of power factor corrections, simplification of control, low-cost and high electrical efficiency. As the strip passes through the finishing train, it is in all finishing stands at the same time and the finishing drives must be of variable speed. The mill speeds are determined from the rolling schedules and the speeds determine the related values of reduction gear ratios and motor speeds. For DC motors the speed is controlled by the potential applied to the armature until the base speed is attained. To achieve higher speeds, usually up to two or three times the base speed, the field current is decreased. The motor speed is accurately regulated by means of a speed regulator system using reference either from operator settings or from a computer system. AC type motors include synchronous and wound rotor induction motors, the former usually being less expensive if an exciter is not required. The synchronous motor has the stator wound with core and coils in the same manner as the squirrel cage and wound rotor type. Apart from being less expensive, a synchronous motor has the added advantages of high efficiency and the ability to correct the power factor. In the late 1960s, AC drive technology progressed to the point that AC motors could be controlled by a combination of power converters (e.g. cyclo converters with use of thyristors) or the newly developed solid-state power transistors. The latter benefitted with technologies, such as GTO, IGBT and IEGT power devices9. These allowed the AC power to be inverted to DC and then converted back to AC to drive the AC motors. Such technological advancement allowed the stringent regulation of the AC motors, not possible before. In modern mills, the drive of choice is AC motors with DC systems relegated to upgrades of existing mills having such drives.

9GTO, gate turn-off thyristor; IGBT, insulated gate bipolar transistor; IEGT, injection-enhanced gate transistor. 26 Primer on Flat Rolling

2.11.3 Computer Control

The automation of hot rolling mills has matured such that the control system has been organized in hierarchical levels. At the topmost is what is known as level 3, which has the production planning and control function. This level is responsible for setting the optimized production for production orders and feeds the raw material and target product information to the next lower level of control. Level 3 also collects product information resulting from the rolling process and saves it in data- bases for future retrieval. The next lower level of control, level 2, is known as process automation, where each product is rolled and tracked individually in the plant. This function also contains mathematical models which calculate the optimum rolling set-ups for the mill (e.g. gap, speed, force, temperatures). The next lower level of control, level 1, is the basic automation which controls the basic sequences and automation functions of the individual equipment in the hot mill. These may include dedicated technological functions, such as automatic gage control, width control, temperature control, in the laminar flow system and others. The lowest level, level 0, includes the basic drive equipment regulators and controls.

2.12 Hot Strip Mill Innovations

Over the years many important hot strip mill innovations have been developed, a selection of which is listed below. Walking beam slab reheat furnaces in place of pusher furnaces for improved slab heating and reduced slab skid marks Furnace slab extractors in place of furnace dropout chutes for reduced equipment damage and slab defects Slab descale boxes with higher pressure headers in place of vertical and horizontal scale breakers for removal of furnace scale Slab sizing press (slab squeezer) to reduce slab caster width change requirements Vertical motor-driven roughing mill edgers in place of horizontal motor bevel gear- driven edgers for reduced maintenance Edger rolls with box pass to contain slab corner bulging for reduction of slab width spread Hydraulic automatic width control (AWC) edger adjustment in place of fixed edger screws for more uniform slab width control Two-stand tandem reversing rougher for increased roughing capacity with reduced mill length Direct motor table roller drives in place of bevel gear table roller group drives for reduced maintenance Delay table heat retention covers to reduce transfer bar temperature run-down History of Hot Strip Mills 27

Hot coil box addition to eliminate transfer bar temperature run-down, reduce finish mill power requirements, increase coil unit weight on existing mills or reduce mill length on new mills Mandrel-less coil box to eliminate coil transfer mandrel temperature effect High-pressure finishing mill entry descaling units to replace two high roll finish scale breakers M stand (named after its original developer, Nippon Steel, Muroran Works) addition in front of an existing small crop shear for thicker transfer bar up to 60 mm from the roughing mill, in order to increase coil unit weight, increase reversing mill capacity, reduce transfer bar temperature run-down and reduce finish mill power and zoom rolling requirements Heavy crop shear and F0 stand addition close coupled in front of finish stand F1 for thicker transfer bar up to 60 mm from the roughing mill in order to increase slab/coil unit weight, increase reversing mill capacity, reduce transfer bar temperature run-down and reduce finish mill power and zoom rolling requirements Finish mill quick work-roll change instead of porter bar for shorter and safer work-roll change for increased productivity and improved product quality Direct transfer of finish mill work rolls to and from the roll shop to reduce roll handling by overhead crane and roll transfer car Finish mill gear spindles and universal joint spindles to replace slipper spindles for reduced maintenance and smoother, safer operation Finish mill rolling lubrication application for reduced work-roll wear and reduced rolling power requirements Finish mill work-roll bending for work-roll crown and strip shape control Finish mill HAGC for more uniform thickness control Finish mill zoom (accelerated) rolling for reduced transfer bar temperature run-down and higher mill production Conventional slab hot strip mill-finish mill endless rolling for reduced head end threading, improved yield and higher productivity Thin slab hot strip mills Thin slab endless rolling hot strip mills Hydraulic finish mill loopers to replace motor-operated loopers for faster looper response for more uniform strip width control and reduced strip necking Finish mill continuous variable crown (CVC) work-roll shifting control for improved strip shape Finish mill work-roll shifting for reduced roll wear and extended rolling campaign Work-roll pair cross-rolling for improved strip shape On-line work-roll grinding for extended rolling campaign Six-high finish stands for reduced strip edge drop and increased roll bending capacity Tool steel, high-chrome iron and high-speed steel finish mill work rolls for reduced roll wear and extended roll campaigns Finish mill long stroke HAGC cylinders to eliminate mill screw-down maintenance requirements Finish mill inter-stand laminar strip cooling to permit higher speed zoom rolling on Generation II and Generation III mills for higher production Finish mill fume suppression sprays to eliminate the need for fume exhaust systems Finish mill delivery profilometers (fixed and traversing X-ray gauges) for on-line strip profile measurement 28 Primer on Flat Rolling

Finish mill delivery shapemeters Finish mill delivery multi-functional temperature, thickness, shape, profile and width gauges Finish mill delivery in line strip surface inspection Individual motor-driven close-centred run-out table rollers without aprons for thin-gauge higher speed coiler threading Water curtain strip cooling headers for increased strip cooling efficiency Three-stage strip cooling system for production of dual phase and transformation induced plasticity (TRIP) steels Servo valve position-controlled hydraulic cylinders for down coiler entry-side guards, pinch rolls and wrapper rolls jump control in place of air cylinders for improved strip coiling quality Down coiler double step mandrel expansion to tighten coil inner wraps Down coiler mandrel improvements for longer mandrel campaigns Eye horizontal coil handling to eliminate coil eye vertical chain conveyor maintenance and coil edge damage.

2.13 Revamped Hot Strip Mills

Many hot strip mills have been upgraded in order to increase mill productivity, yield and quality, and to reduce maintenance. An example of a US Generation II hot strip mill to be significantly upgraded is the US Steel 84v continuous hot strip mill in Gary, IN. This mill was originally built in 1967 with an estimated annual capacity of 3.5 million short tons. The mill has undergone productivity and quality improvements including new computer control, digital drives and automation, CVC roll shifting, work-roll bending, HAGC, in line strip inspection, hydraulic down coilers and other improvements that have raised the current annual capacity to 6.3 million short tons. In the case of Generation I mills, some have been modified to a new configuration in order to increase the coil unit weight capability, improve strip quality, expand product range and increase mill production. A few examples of modified Generation I Hot strip mills are listed below. British Steel, Port Talbot, UK: 2032 mm fully continuous mill started up in 1953, capable of producing 9 kg/mm coils, was upgraded in place over the period of 1984À1986 including new furnaces, new single reversing rougher, new coil box, repowering for existing six stand finishing mill, new finish stand F7 and new down coilers capable of producing 18 kg/mm coils. Nippon Steel, Muroran Works, Japan: 1422 mm semi-continuous mill started up in 1957, capable of producing 7 kg/mm coils with an annual capacity of 790,000 metric tons, was upgraded in place over the period of 1965À1979 including new furnaces; new heavy reduction M stand with entry edger for introducing the 60 mm maximum transfer bar into existing crop shear; repowered, upgraded, higher speed six finishing stands and repowering of existing two down coilers capable of producing 21.4 kg/mm coils with annual capacity of 3,000,000 metric tons. History of Hot Strip Mills 29

Pittsburgh Steel, Allenport, PA: 1676 mm semi-continuous mill started up in 1953, capable of producing 9 kg/mm coils with a yearly capacity of 900,000 short tons, was disman- tled, rebuilt, modernized and relocated over the period of 1989À1993 by National Steel Corporation (NSC) in Iligan City, Philippines with the addition of a new furnace, heavy reversing rougher with a new edger, new heavy crop shear for 60 mm maximum transfer bar into the F0 stand (original Allenport reversing rougher) close coupled in tandem with modernized and repowered six existing finishing stands and new down coiler capable of producing 18 kg/mm coils with annual mill capacity of 2,500,000 short tons with addition of second furnace and down coiler. 3 Roll Design1

3.1 Introduction

The stock surface, under rolling conditions, is a mirror image of the roll surface, which erodes with time because the contact area between the roll and stock suffers wear. This erosion depends on a large number of factors, including the loads, and hence the stresses, they experience. As well, it depends on the roll grade, the chemical composition and the microstructure. Loads in an industrial rolling mill are not clearly and precisely defined. There are mathematical models to calculate the maximum stress in rolls, determined by design limits for maximum separation force, torque, Hertzian pressure, etc. These criteria are valid only under normal rolling conditions À meaning no non-standard events occur, such as cobbles2 (see also Chapter 7 for a mathematical model of the mechanisms of the formation of cobbles). They also change continuously with progressive wear in the contact zone. During normal rolling conditions, for example a stable operating period, mills experience changes in the rolling conditions. After a roll change or a mill stop, rolls need some time to return to stable thermal conditions. Sometimes there are severe rolling accidents, due to faults by operators or weak rolled materials with internal defects, or because of other problems in a mill such as a power cut, mechanical problems or issues in the water cooling system. While problems of this kind can never be predicted, they have a detrimental effect on all rolling schedules, including stresses in the roll. During abnormal rolling conditions, for example a cobble or torn-off tail end of the strip, roll damage often occurs with consequences for the mill and the rolled product. The rolls experience local over-stressing, due to double or even triple strip thickness in the roll bite, which they are not capable of resisting. They must be checked for damage before returning to service.

3.2 General Overview

While casting and forging are old technologies going back more than 3000 years, rolling assumed major importance in the industrialized world only during the nineteenth century. Initially, steel was rolled to profiles (rails, beams, channels,

1Contributed by Dr. John B. Tiley, hot rolling consultant. 2Cobbles are sudden changes in the thickness of the rolled strip. They are mechanical marks formed during localized overloads where the resultant stress imparted exceeds the yield strength of the rolled material.

Primer on Flat Rolling. DOI: http://dx.doi.org/10.1016/B978-0-08-099418-5.00003-2 © 2014 Elsevier Ltd. All rights reserved. 32 Primer on Flat Rolling rounds), but since about 1930 flat products (sheet and strip) have become increas- ingly dominant. Profiles and flats are hot rolled (the latter to a minimum size). Thin, flat products are finished by cold rolling for various reasons À for example, to achieve a better shape and profile, or because of mechanical properties or surface conditions. In the recent past (,20 years), rolling technology has improved and changed dramatically. Rolls have always remained the critical part of rolling mills. The development of roll qualities and roll-making technology followed the development of rolling technology. Product dimensional and surface quality requirements continued to tighten as the resolution of the measuring instruments improved. Rolling mill industries followed the evolution of the automotive industry. The demand for rolled products continued to grow. While the need for rolls decreased due to improved rolling technology and better roll qualities, the market situation for rolls is continuously undergoing changes.

3.3 Historical Development of Rolls for Rolling Mills

The quality of the strip and the productivity of the hot rolling mills are two of the most important concerns in the steelmaking plants. The quality of the strip is evaluated mainly by means of its shape, roughness and dimensional tolerances, all of them depending strongly on the shape, profile and surface quality of the work roll. The productivity of the rolling mill is directly related to the length of campaigns of the rolls, the essential aim being to maintain, over time, the initial surface roughness, shape and strip profile, which are affected by wear. Saving costs with minimum roll grinding is also important, since rolls are responsible for 5À15% of overall production costs. In most cases, work rolls for the finishing stands of hot rolling mills are cast-composite components made of an outer shell of wear-resistant material and a core of ductile iron or steel. For the work rolls of the early finishing train (stands 1 to 4), the development of materials for the outer shell has enjoyed rapid advances beginning in the early 1980s, culmi- nating in the application of cast alloys of the FeÀCÀCrÀWÀMoÀV system which gradually replaced high-chromium cast iron and Ni-hard cast iron with better performance (Hashimoto et al., 1995; Sano et al., 1992; Savage et al., 1996). These alloys are generically termed “high-speed” steels or multi-component white cast iron (Ziehenberger and Windhager, 2007; Garza-Montes-de-Oca, 2011). The idea of using these alloys for manufacturing work rolls for hot strip mills resulted from an insight into the requirements involved in this type of application: fundamentally, the capacity to retain a high level of hardness even when experiencing high temperatures, and also strong wear resistance. Thus, although alloys have been specifically designed to meet the operational conditions of each hot-strip mill plant, their chemical compositions generally fall into the following ranges: 1.5À2.5% C; up to 6% W; up to 6% Mo; 3À8% Cr and 4À10% V. In the nineteenth century, unalloyed grey iron, modified only by various carbon equivalents and different cooling rates (grey iron chill moulds or sand moulds), and Roll Design 33 forged steel were used for rolls. The cast iron grades varied from “mild-hard” to “half-hard” to “clear chill”, where the roll barrel showed a white iron layer surface (free of graphite) and grey iron core, due to reduced cooling rate. This type of roll was used for flat rolling without any roll cooling in “sheet mills”. In the twentieth century, cast steel rolls were developed with a carbon content of up to 2.4%, with and without graphite, and are still produced today. Around 1930, Indefinite Chill Double Poured (ICDP) rolls were invented for hot rolling, especially for work rolls in the finishing train of hot strip mills. These were also used for many other applications such as roughing stands of hot strip mills and work rolls in plate mills. This grade was to become the world standard for many years with very limited variation. For some time, no other material could replace this grade for several applications. In the late 1990s, ICDP rolls enhanced with carbides improved roll performance and started a new phase for this grade which is still successfully in use today in work rolls for early finishing stands and for plate mills. Around 1950, nodular iron was invented and introduced into roll manufacturing, unalloyed as well as frequently alloyed using Cr, Ni and Mo, giving good wear resistance and strength at the same time. The use of “high chromium iron” (2À3% C, 15À20% Cr) and later on “high chromium steel” (1À2% C, 10À15% Cr) brought new materials, with high wear resistance and “forgiveness”, into rolls. This was a major step towards greater productivity in the industry. In 1985, High Strength Tool Steel3 (HSS) materials were introduced into rolls and evolved to “Semi-Tool-Steel” grades. After initial problems, these changes brought new opportunities for better roll performance. After the introduction of new grades to the mills it was often necessary to change rolling conditions. All of these roll grades are currently used for flat and long products. Forged steel rolls were also improved for cold rolling to give higher hardness penetration after heat treatment by increasing the content of alloying elements. The chromium content was increased from 2% to 5% and induction hardening was used to achieve the desired surface hardness. Chromium plating of work rolls after grinding and shot blasting also helped maintain the necessary surface roughness. In reality, rolls are tools for metal forming. Therefore the development of suitable roll materials goes hand in hand with the development of other cutting and non-cutting tools in metal industries. Rolls are relatively large tools with an extended life, but ultimately, they are only tools.

3.4 Roll Wear

Work roll wear is a complex process characterized by the simultaneous operation of several surface degradation phenomena. The essential target of the rolling mill plant is to keep the shape, profile and surface roughness as close as possible to the initial

3High-Speed Tool Steels are FeÀCÀX multi-component alloy systems where X represents chromium, tungsten, molybdenum, vanadium or cobalt. 34 Primer on Flat Rolling ones that existed at the last roll change. The better performance of the HSS rolls, in comparison to the forerunner work roll materials, is related to its microstructure characteristics: the presence of great amounts of very hard (2800À3000 HV), fine and discontinuous metallic carbide, eutectic carbides and a matrix hardened by secondary precipitated carbides. The microstructure of the high-chromium cast iron, for instance, consists of the softer M7C3 eutectic carbide (1100À1800 HV) and a less high-temperature-resistant matrix (Maratray, 1970; Boccalini and Goldenstein, 2001; Lee, 1997; Webber, 2011). The degradation of the work rolls for the early finishing stands involves, at least, abrasion, oxidation, adhesion and thermal fatigue (Park et al., 1999; Spuzic, 1996; Ryu et al., 1992). Thermal fatigue results from stresses developed by cyclic heating and cooling of a very thin boundary layer close to the work roll surface, no thicker than 1% of the work roll radius (Chang, 1999). This layer is repeatedly heated by the hot strip, the work of plastic deformation and the roll/strip friction in the roll bite and cooled by water during the remaining portion of its rotation. The boundary layer is thus subjected to compressive stresses during the heating cycle, since its thermal expansion is constrained by the bulk roll which remains at approximately constant temperatures during the operation. If the compressive stresses are high enough to plastically deform the layer (softened by the high temperature), residual tensile stresses in excess of the rupture strength may develop during the cooling cycle and cracking will take place (Debarbadillo and Trozzi, 1981). The primary crack pattern is related to thermal and mechanical stresses imposed on the roll and not to the microstructure of the shell material. Eutectic carbides4 play a decisive role in the nucleation and propagation of the secondary cracks (Chang, 1999; Caithness et al., 1999). Thermal fatigue experiments show that secondary cracks nucleate at eutectic carbides, caused by the stress concentration induced by the difference between the thermal coefficients of expansion of the carbide and the matrix, and propagate along the carbide/matrix interface (Lee et al., 1997; Wisniewski et al., 1991). Since the presence of eutectic carbides, and thus crack nucleation, is unavoidable, improving thermal fatigue resistance requires their refining and homogeneous distribution so as to avoid the formation of easy crack propagation paths, like inter-dendritic or inter-cellular coarse M7C3 or M2C carbides. The combination of thermal fatigue and mechanical stresses, inherent to the rolling process, progressively extends and branches the cracking network through the sub-surface thickness. This process may lead to a catastrophic deterioration in which large segments of the roll surface, containing the oxide layer built up during rolling together with portions of the roll material, are peeled off. The peeling, known as “banding”, leaves a roughened roll surface, unsuitable for further rolling. Adhesion is a consequence of the micro-welding regions of the strip metal into roll metal in the sticking zone of the roll gap, mostly where there is no relative motion between the strip and roll surfaces (Hashimoto, 1995). Resultant wearing

4Eutectic carbides are carbides formed during freezing in ferrous alloys. Roll Design 35 takes place when the interfaces in contact are made to slide and the micro-welded regions must separate, hot shearing the roll material (Werquin et al., 1990; Ludema, 1992; Lanteri et al., 1998). The formation of large pores in the surface of the rolls, commonly named “comet tails”, is attributed to the intense occurrence of adhesion (Werquin et al., 1990). Adhesion resistance of the roll materials is improved by increasing the volume fraction of eutectic carbides. Werquin et al. (1990) explained this behaviour, indicating that the hot hardness of the eutectic carbides is higher than that of the matrix, allowing for higher hot shear strength. Hence, within the same concept, he suggested that adhesion resistance can be further improved through increasing the hot hardness of the matrix by means of secondary hardening heat treatment. It is noted that adhesion is primarily controlled by the physico-chemical interaction between the materials of the roll and the strip.

3.5 Friction and Wear

Work rolls in hot strip mills wear due to friction between the roll surface and hot strip. (Back-up rolls wear as well, but under pressure with elastic deformation, the friction is much less, except for the high work roll roughness in the last stand of a sheet rolling tandem cold mill). The wear is uneven from one end of the barrel to the other. A shining, dark oxide layer always covers the roll surface, from the beginning to the end of a campaign, as long as there is no banding/peeling. Friction and wear occur whenever materials move and slide relative to each other. In rolling mills, wear takes place mainly at the areas of highest friction; that is between a roll and the rolled material. (There is, of course, also wear in mills on other parts with almost no friction. For example, the barrel of back-up rolls in four high hot mills is subject to wear as well, but the amount is not as high). Wear normally is not equally distributed on the barrel from one end to the other because strip conditions vary over the strip width and the edges of rolls are never in contact with rolled material at all. In hot rolling mills the roll surface is also influenced by changing temperatures during each rotation, which may create fire-cracks, influencing friction and wear. A third factor of the impact on wear is roll/strip cooling. Cooling agents may be water or a water/emulsion mixture. These may be clean or not and may contain lubricants and chemical additives, potentially causing damage to the roll surface. Wear and fire-cracks increase progressively during the rolling campaign. After each campaign, rolls are redressed and reground to re-establish the correct original shape and surface roughness. In this process the diameter of a roll is reduced by grinding until all visible signs of wear and fire-cracks have been removed, often under the control of eddy current testing equipment, which is required for HSS rolls. Additionally, some amount of stock may be removed for safety reasons, to reduce the risk of problems due to undetected sub-surface damage. 36 Primer on Flat Rolling

Bakelite

Scale layer

Substrate

Figure 3.1 Cross-section of oxide flake, pressed into the substrate (2003 magnification). Source: Reproduced from Tiley (2006).

If the roll surface is damaged from a mill accident, for example, by local over- stressing or fire-cracks, then these have to be eliminated immediately by further grinding, otherwise the roll will fail catastrophically. This is a new discipline which operators need to follow to be successful when converting to HSS work rolls (Figure 3.1). At the end of roll life the total rolled material (in terms of tons or length of product) is related to total stock-removal for normal rolling, excluding stock- removal for accidents. This results in a figure for productive or effective roll performance. If stock-removal for accidents is included, then this figure is called “total roll performance”. These figures are the most important ones for evaluating the performance of the roll, allowing a comparison with other grades, and in making a distinction between various roll suppliers. During the last 50 years different roll grades were used for work rolls of hot strip mills and for some time up to four different grades were used simultaneously in different stands. All these grades weremanufacturedinthesamerangeofhard- ness. However, wear performance differed widely. With the realization that roll wear decreases when extremely hard carbides are used, the development of carbide-enhanced ICDP rolls followed this strategy using carbides of higher hardness in a strong martensitic structure. The performance of these ICDP rolls was improved in many mills, by a minimum of 20%, maximum beyond 100%. Other grades will likely follow the same direction, bearing in mind that wear resistance is just one required property. For the same roll grade, the little increase in wear resistance that might theoretically exist is often over-compensated for by the risk of unproductive loss of roll Roll Design 37

Figure 3.2 Progressive build-up of the oxide layer(s) on an HSS work roll. Source: Reproduced from Kerr (1999); with permission. life following rolling accidents. By increasing hardness, the ductility of the material is reduced and for all materials there is an optimum of useful hardness. Unfortunately it is not easy to find general rules for the best wear-resistant materials for all applications because the wear conditions vary too widely (Figure 3.2). 4 Flat Rolling A General Discussion

4.1 The Flat Rolling Process

The essential concept of the flat rolling process is simple and it has been in use for centuries to produce sheets and strips, or in other words, flat products. Leonardo da Vinci employed it to roll lead, utilizing a hand-cranked mill, depicted in Roberts’ book, Cold Rolling of Steel (1978). The Leonardo museum, located in the medieval Castello Guidi, built between 1100 and 1200 AD in the city of Vinci about 30 km from Firenze, contains some interesting examples of Leonardo’s plans for a rolling mill, shown in the website of the museum (http://www.leonet.it/comuni/ vinci/). Two figures, reproduced from that website, are given below. Figure 4.1 is a page of Leonardo’s plans on which the handwriting is, unfortunately, indecipher- able. The scale model, built according to these plans and shown in Figure 4.2,is able to roll a sheet of tin 30 cm wide. The basic idea for the production of flat pieces of materials by rolling has not changed since the process was introduced. Dimensions, materials, precision, speed, the mechanical and metallurgical quality of the product and most importantly the mathematical analysis and the control of the process have evolved; however, and as a result, the flat rolling process may truly be considered one of the most successful “high-tech” processes, since for modern, efficient and productive applications, the theories and practice of metallurgy, mechanics, mechatronics, surface engineering, automatic control, continuum mechanics, mathematical modelling, heat transfer, fluid mechanics, chemical engineering and chemistry, tribology and, encompassing all, computer science are absolutely necessary.

4.1.1 Hot, Cold and Warm Rolling The rolling process may, of course, be performed at low and high temperatures, in the cold rolling mill or in the hot strip mill, respectively, as already mentioned in Chapter 1. The formal distinction between what is low and what is high temperature, and in consequence, what are the cold and hot rolling processes, is made by considering the homologous temperature, in which the low end is at absolute zero and the high end is at the melting point of the metal to be rolled, Tm. When the process is performed at a temperature below 0.5Tm, it is usually termed cold rolling, while above that limit, hot rolling occurs. In addition to the above strict definitions of hot and cold rolling, there is the warm rolling process as well. The temperature range for this phase is not defined very precisely but it starts somewhat below

Figure 4.1 Leonardo’s plans for a simple rolling mill.

Figure 4.2 The scale model of Leonardo’s mill.

0.5Tm and changes to hot rolling at some temperature above that. Each of these processes has advantages and disadvantages, of course. At the high temperatures at which hot rolling is performed the metal is softer so less power may be needed for a particular reduction. Further, understanding the effects of the process parameters Flat Rolling A General Discussion 41 of the rolling process on the mechanical and metallurgical attributes allows the development of metals with specific, engineered properties; the process is termed thermal-mechanical treatment. The disadvantage of rolling at high temperatures concerns the development of a layer of scale on the surface and its effect on the process and the quality of the resulting product. All of these need to be clearly understood, such that they may be controlled with confidence. Cold rolling follows the pickling process in which the layer of scale is removed. Here the control of dimensional consistency and surface quality is the most important objective. Strict thickness and width tolerances must be maintained for the product to be commercially acceptable. During warm rolling some of the disadvantages of the hot rolling process are minimized as scale formation is less intense. The energy requirements increase, however, as the metal’s resistance to deformation is now higher.

4.1.2 Mathematical Modelling Use of sophisticated on-line and off-line mathematical models allows these activities to proceed. A number of models have been developed, some simple and some making use of the availability of the finite-element method. Among the latter, the model developed by the American Iron and Steel Institute (Hot Strip Mill Model, HSMM) stands out. A quotation from the AISI website is given below:

HSMM is one of several commercially licensed technologies developed under AISIs advanced process control programme, a collaborative effort among steelmakers and the U.S. Department of Energy to create breakout steel technologies. HSMM simulates the steel hot rolling process for a variety of steel grades and products, and forecasts final microstructure and properties, allowing the user to achieve a deeper insight into operations while optimizing product properties. Prior to its commercial release, it was used for several years by steel companies that helped develop the technology.

In each of these models the ideas of equilibrium, material behaviour and tribology are used to describe the physical phenomena. The first of these three is based on Newton’s laws. The latter two require experimentation and the translation of the experimental data into mathematical expressions for use in the models. The traditional and simplest approach when mathematical analyses of metal forming processes are considered is to allow the material’s resistance to deformation to be exclusively strain-dependent in the cold forming regime and to be exclusively strain-rate-dependent in the hot range. In the warm forming range the metal’s strength is usually both strain- and strain-rate-dependent. It is acknowledged, of course, that these are much simplified and in Chapter 8, dealing with the attributes of the metals, other more inclusive and more sophisticated material models are presented. In advanced mathematical treatment, by finite difference or finite-element techniques, the metals’ constitutive relation should be described in terms of several independent variables, including, at the very least, the strain, strain rate, temperature and 42 Primer on Flat Rolling metallurgical parameters, one of which is the ZenerHollomon parameter, used extensively (see Chapter 4 on material attributes for the definition). In some instances the metal’s chemical composition is also included in the equations. The tribological events at the contact of the work roll and the rolled metal are also to be described in terms of parameters and variables.

4.1.3 The Independent and Dependent Variables The discussion above leads to the consideration of both the dependent and independent variables of the flat rolling process. It is first necessary to identify the boundaries of the domain under consideration. For the arguments presented in what follows, these define a single stand of a two- or four-high mill, such that they are used in most laboratories during the development of the data on forces, torques, energy requirements and the resulting microstructures. Under industrial conditions, a full-size mill stand, regardless of being the roughing mill or the finishing train, is being considered. It is a metal rolling system that is, fact, being defined and, in the same sense as any metal forming system, is divided into three essentially independent but interconnected components: the rolling mill, the rolled metal and their interface. The significant dependent variables depend on what the objectives of the industry, of the engineers operating the rolling process, of the researcher developing a mathematical model or of the customers happen to be. There may be three separate but interdependent objectives. One of these is the design of the rolling mill. Another objective is the design of the rolled strip while the third potential objective is the design of the interface between the rolls and the rolled strip. The essential components of the rolling mill are the work rolls and the back-up rolls, the bearings, the mill frame, the drive spindles, the inter-stand tensioning devices, the heating and cooling equipment, the lubricant delivery apparatus and the driving motor. Their attributes, which affect the rolling process, include their dimensions in addition to roll material, crowning, roll surface hardness, roll roughness and its orientation, mill frame load-carrying area, mill stiffness and hence, mill stretch. In choosing the driving motor, its power and speed are to be determined. Mill dynamics is also a significant contributor to mill performance and is a function of all of the above in addition to the process variables, such as the reduction, the speed and the dimensional consistency of the as-received metal. All of these affect the quality of the rolled metal. All may be considered as the independent variables. The variables associated with the rolled metal include its mechanical, surface, metallurgical and tribological attributes: yield strength, tensile strength, strain and strain rate sensitivity, bulk and surface hardness, ductility and formability, fatigue resistance, chemical composition, weldability, grain size and distribution, precipitates, surface roughness and roughness orientation. Since the transfer of thermal and mechanical energy is accomplished at the interface of the work roll and the rolled strip, and the efficiency of that transfer is Flat Rolling A General Discussion 43 one of the most critical parameters of the process, the attributes of the contact are, arguably, the most important when the quality dimensional accuracy, consistency and uniformity of the surface parameters of the product is designed. The surface roughness of the rolls and their directions have already been mentioned. The attributes of the lubricant are to be considered here and among these are the viscosity and its temperature and pressure sensitivity, density, chemical composition and droplet dimensions if an emulsion is used. While environmental friendliness of the lubricant and the manner of its disposition after use are other, most important, considerations, they do not affect the quality of the product. While all of these variables should be considered when the rolling process is designed and/or analysed, they rarely, if ever, are. Engineers simplify the task and consider only what is absolutely necessary. The rolling mill, its capabilities, its driving system and its peculiarities are, of course, given and will be changed only when forced by industrial competition or the development of processes that lead to increased productivity and/or reduced costs. The metal to be rolled must, in usual circumstances, be one in the product mix offered by the particular company. In unusual circumstances, if the customer requires a chemical composition or mechanical and metallurgical attributes different from those available, either the request would be refused and another metal among the company’s products similar to the customer’s prescription would be offered or the costs of the development of the new metal would form a major part of the complete process costs. This second possibility would, of course, be prohibitively expensive. The choice of the lubricant, its volume flow and the lubrication system are often considered to be no more than a maintenance issue and in the opinion of the present writer, it is an uninformed and a highly mistaken view. The most significant independent variables usually considered in the analyses of the flat rolling process are listed below, classified according to the three components of the metal rolling system. These will be dealt with in what follows, in some considerable detail.

The rolling mill Roll diameter and length Roll material Roll surface roughness Roll toughness and hardness The metal Chemical composition Prior history grain size, precipitates Constitutive relation Initial surface roughness The interface Lubricant attributes chemical compositions, viscosity, temperature and pressure sensitivity, density; droplet size if an emulsion is used The rolling process Reduction Speed Temperature 44 Primer on Flat Rolling

4.2 The Physical Events Before, During and After the Pass

These events have been mentioned by Lenard et al. (1999). In the discussion that follows the earlier presentation has been enlarged and some more important ideas have been included. The first consideration is that in the presentation, the shapes of the work roll and the rolled strip are considered to be highly idealized. The work roll’s cross section is taken to be a perfect circle at the start and that makes the roll a perfect cylinder. The line connecting the roll centres is taken to be perpendicular to the direction of rolling and it remains so during the pass. The distance between the roll centres is considered to remain unchanged in most analyses, ignoring mill stretch1. The rotational speed of the work rolls is and remains constant, even after the loads on it increase and the inevitable slowdown under high torque loads is ignored. The strip to be rolled is straight, its sides make right angles to one another and the thickness and the width are uniform as is the surface roughness. The following events occur when the process of rolling a flat piece of metal starts, continues and is completed. As mentioned above, the work rolls are considered to be rotating at a constant angular velocity. The strip or the slab is moved towards the entry and is made to contact the rotating work rolls, either by vertical edge rollers or by a conveyor system or both. When contact is made, the leading edge of the strip enters the deformation zone because of the friction forces exerted by the work rolls on it and it is not difficult to show that the minimum coefficient of friction, necessary for successful entry is given by:

μ 5 φ ð : Þ min tan 1 4 1 φ where 1 is the bite angle. In most analytical accounts of the entry, the first contact is assumed to be along a straight line, across the whole length of contact with the work roll. In reality, there must be a significant amount of deformation of both the edges of the strip and the roll, suggested by common sense as well as by the loud noise always heard on entry to the roll gap of an industrial mill. The leading edge of the strip must be thickened somewhat on contact and the roll must be flattened, but the present author is familiar with surprisingly very little research concerning the geometrical changes at the instant of entry. In one of the early attempts Kobasa and Schultz (1968) used high-speed pho- tography which allowed some, albeit limited, visualization of the entry conditions and the length of contact in the rolling process. The published photographs do not allow a clear, close look at the deformation at the initial contact, however. The stresses in the metal increase, and the limit of elasticity of the strip is reached soon after entry, followed by permanent, plastic deformation. One usually assumes that the flat strip rolling process can be described in terms of one independent variable, taken in the direction of rolling, and that the stresses do not vary across the strip thickness. This assumption in turn is based on the usual homogeneous compression

1Ignoring the mill stretch in the control software would, of course, lead to significant errors; mill stretch is accounted for in the set-up programs used for rolling mills. Flat Rolling A General Discussion 45 assumption and the often-employed statement that “planes remain planes”. With these assumptions, the elasticplastic boundary becomes a plane perpendicular to the direction of rolling and on that plane the criterion of yielding is satisfied first. When the process is treated as a two-dimensional problem, as in many finite-element analyses, the elasticplastic boundary may be quite different from that described above. When strip rolling is considered and the roll diameter/strip thickness ratio is large in industrial settings this ratio varies from about 2530 at the first stand of the finishing train of the hot strip mill to as high as 4001000 in the last stand the one-dimensional treatment is perfectly adequate, and yielding and the beginning of plastic flow are taken to occur on a plane, parallel to the line connecting the roll centres, as just described. The permanent deformation regime then remains in existence through most of the roll gap region, followed by the elastic unloading regime which starts when the converging channel of the roll gap begins to diverge. Often, the yielding process is simplified further, and rigid plastic material behaviour is assumed to exist, ignoring the elastic deformation completely. With this assumption, the rolled metal is taken to satisfy the yield criterion and to begin full plastic flow as soon as it enters the roll gap. The assumption of rigid plastic behaviour is usually acceptable when hot rolling is analysed. It has often been shown that treating the cold rolling process requires the use of elasticplastic material models. These events are illustrated in Figure 4.3AC, which show a schematic diagram of a two-high mill and a strip ready to be rolled, rolled partway through and rolled continuously, in a steady-state condition. As well, the free-body diagram of the roll is indicated, showing the pressures, forces and torques acting on it. The conditions shown describe either a laboratory situation where no front and back tensions exist or a single stand reversing mill, such as a roughing mill. Three stages of the rolling process are shown in Figure 4.3AC, respectively. In Figure 4.3A the strip is about to make contact. The strip velocity at this point is dependent on the edge rollers or the conveyor system but it is usually significantly less than the surface velocity of the work rolls. If the coefficient of friction is larger than the tangent of the bite angle, as indicated by Eq. (4.1), a relationship that is often used to determine the minimum friction necessary to start the rolling process, the strip enters the deformation zone. In a laboratory mill, the usual practice is to carefully and lightly push the strip2, placed on the delivery table, towards the work rolls and allow the friction forces to cause entry. Under certain circumstances, when viscous lubricants are used or the roll speed is high, it is necessary to mill a shallow taper on the leading edge of the strip to facilitate the bite. Care must be exercised in this case concerning the placement of the lubricant. In an experiment in the writer’s laboratory, a highly viscous lubricant was used and to ensure bite, the tapered leading edge was left dry. Entry was achieved but when the rolls encountered the portion covered with the oil, the strip tore into two, in a violent fashion; sudden changes of the tribological conditions along the length of the strip to be rolled should be avoided.

2Caution is highly recommended here. The strip should never be pushed by hand but by a long piece of wood or another strip; this practice will eliminate the danger of rolling a few of the operator’s fingers. 46 Primer on Flat Rolling

Figure 4.3 (A) Schematic diagram of the strip’s entry into the roll gap Entry is imminent Work roll (Lenard et al. (1999) reproduced with permission). (B) The strip is Force of friction partially in the deformation zone (Lenard et al. (1999) reproduced with permission). (C) Free-body Strip diagram of the work roll. Roll flattening Source: Lenard et al. (1999) Thickening of reproduced with permission. the strip

(A)

Roll separating force

The strip is partially in the roll gap Roll torque

Elastic region Plastic region

(B) Elastic–plastic interface

Roll separating force

Shear stresses Roll torque on the roll surface

Roll pressure (C) distribution

In a hot strip mill, vertical edge rolls force the strip into the roll gap in the first stand and the momentum of the strip exiting from there carries it into the deformation zone of the next stand. In either case, entry creates some longitudinal compression of the strip and there will be some initial thickening as well, more than in cold rolling. This is accompanied by local, elastic deformation of the work rolls, indicating that the usual simplification about the entry point located where the perfectly straight edge of the strip encounters the undeformed, perfectly cylindrical roll does not represent reality very well. In a cold mill the strip is often threaded through the stationary mill and attached to the coiler on the exit side. The rolls Flat Rolling A General Discussion 47 then close and gradually reach the pre-determined roll gap and the mill is started, so no bite is required. Figure 4.3B shows the strip about halfway through the deformation zone. As previously mentioned, the unloaded metal first experiences elastic deformation, and when and where the yield criterion is first satisfied, plastic flow is observed. These two regimes are separated by an elasticplastic boundary, the shape and the location of which should be determined by the mathematical analysis of the rolling process. In the elastic region, the theory of elasticity governs the deformation of the metal. In the permanent deformation region, the criterion of yielding, the appropriate associated flow rule, the condition of incompressibility and the appropriate compatibility conditions describe the situation in a satisfactory manner. The rolls are further deformed they bend and flatten. The magnitude of the roll stresses should not exceed the yield strength of the roll material. The theory of elasticity is to be used to determine the roll distortion and the corresponding changes of the length of contact. In Figure 4.3C, the leading edge of the rolled metal has exited and the rolling process is continuing, essentially as a steady-state event. The figure shows the pressures, the forces and the torques acting on the roll and on the strip. These include the roll pressure distribution and the interfacial shear stress, the integrals of which over the contact length lead to the roll separating force and the roll torque. These are the dependent variables the mathematical models are designed to determine. If front and back tensions are present, as would be the case under industrial conditions in the finishing mill stands, their effect on the longitudinal stresses at the entry and exit should be included in the definitions of the boundary conditions. The roll separating force and the roll torque may be used to study the metal flow in the roll gap. As well, they may be used to design the rolling mill itself. Knowledge of the magnitude of the roll separating force is needed to size the mill frame, the roll neck bearings and the roll dimensions, including roll crowning and roll flattening. The roll torque is necessary to establish the dimensions of the spindles, the power required of the driving motor and the couplings. The surface velocities of the roll and the strip should also be considered. It may be assumed that the driving motor is of the constant torque variety and that the rolls rotate at a constant angular velocity, even though there may be some slowdown under high loads. The strip usually enters the roll gap at a surface velocity less than that of the roll. The friction force always points in the direction of the relative motion, and on the entering strip it acts to aid its movement. As the compression of the strip proceeds, its velocity increases3 and approaches that of the roll’s surface. When the two velocities are equal, the no-slip region is reached, often referred to as the neutral point4. At that location, the strip and the roll move together, and their relative velocity vanishes. If the neutral point is between the entry and the exit, the strip experiences further compression beyond it, and its surface velocity surpasses

3Recall that the assumption of plane-strain flow implies no width changes. Hence, incompressibility implies that the sum of thickness and the length strains should vanish. 4The ideas of the “neutral point” and the “neutral region” will be discussed in more detail in Chapter 12. 48 Primer on Flat Rolling that of the roll. Several researchers suggest that reference should be made to a neutral region instead of a neutral point, hypothesizing that the no-slip condition extends over some distance. In that region, between the neutral point and the exit, the friction force on the strip has changed direction and is now retarding its motion. The site between the entry and the neutral point is often referred to as the region of backwards slip. The location between the neutral point and the exit is called the region of forward slip. The forces shown are the external loads acting on the work rolls. The rolls are in equilibrium, of course, and the surface forces at the contact must be balanced by other, external forces. These originate at the bearings that exert the forces on the rolls to keep them in equilibrium, in a relatively stationary position. There are two types of loads at the roll bearings. One is a vertical force, minimizing the possibility of the roll moving upwards, called the roll separating force. The other is a turning moment, originating from the drive spindle, referred to as the roll torque. In a two-high mill, these are the loads acting on the work roll, balancing the effects of the loads originating at the interface: the pressure of the strip on the roll and the interfacial frictional forces. If a four-high configuration is studied, the forces normal and shear at the back-up roll and the work roll contact need to be included as part of the free- body diagram. The picture changes somewhat when front and back tensions are also considered, as would have to be done to account for the effects of the preceding and the subsequent mill stands and the effects of the loopers these are devices between the mill stands that keep some tensile forces in the strips. These forces act in the direction of rolling, of course, and would have an effect on the magnitudes of both the roll separating forces and the roll torques. It is possible and simple to include the effect of inter-stand tensions in the mathematical models of the process. Knowledge of the roll separating force and torque is necessary for three possible purposes: 1. to design the mill its frame, bearings, drive systems, lubricants and their delivery; 2. to determine the dimensions and the properties of the rolled metal; 3. to allow the development of control systems for on-line control of the process.

4.2.1 Some Assumptions and Simplifications In dealing with the process of flat rolling, it is advantageous to consider two assumptions frequently made when mathematical models are developed. The first is to acknowledge the almost true fact that the width of the flat product is practically unchanged: the plane-strain flow phenomenon. The second, again almost true, which allows the use of ordinary differential equations in the models, is the planes remain planes simplification.

4.2.1.1 Plane-Strain Flow The flat rolling process is usually taken to be essentially two-dimensional in the sense that the width of the product does not change much during the pass when Flat Rolling A General Discussion 49 compared to thickness and length changes and this makes the assumption of plane- strain plastic flow5 quite realistic. When one is to study roll bending and the attendant changes of the shape of the rolled metal in addition to the changes of the field variables across the width stress, strain, strain rate, temperature, grain distributions a change from the 2D mathematical formulation to 3D is unavoidable. Of course, the width changes in the flat rolling process and this and its effect on the resulting product have been considered in numerous publications. As long as the width to thickness ratio is over 10, however, this change is not taken to be very significant. In rolling experiments, using strips of about 1 mm thickness and 1025 mm width, the strain in the width direction is rarely over 23%.

4.2.1.2 Homogeneous Compression A discussion of the homogeneous compression assumption is also necessary here. This phenomenon has been studied experimentally by visio-plasticity methods in addition to observing the deformation of pins inserted into the rolled metal. Figure 4.4 shows in part (A) that the originally straight lines bend, while in (B) they do not and the original planes remain planes. In the second case, the compression of the strip during the rolling pass is referred to as “homogeneous compression”. Schey (2000) differentiates between the two possibilities, depending on the magnitude of the ratio of the average strip thicknesspffiffiffiffiffiffiffiffiffiffiffi in the pass, have 5 0:5ðhentry 1 hexitÞ, and the length of the contact, L 5 R0Δh, where R0 is the radius of the flattened but still circular work roll (this idea will be discussed later, in Chapter 5, dealing with mathematical modelling of the process) and Δh 5 hentry 2 hexit, of course. When have=L is larger than unity the deformation in inhomogeneous and the originally straight planes bend as shown in Figure 4.4A. When the ratio is under unity, the effects of friction on the rolling forces and torques are significant and the homogeneous compression assumption may be made with confidence. When strip rolling is discussed, whether hot or cold, the “planes remain planes” assumption is very close to reality, with one possible exception. This concerns metal flow in the first few passes of the slab through the roughing train of a hot strip mill where the strip thickness is in the order of 200300 mm and the work rolls maybe 1 m or more in diameter, leading to a roll diameter/strip thickness ratio in the order of 35. In the finishing train this ratio increases by at least an order of magnitude and the plane-strain assumption becomes acceptable. Venter and Adb- Rabbo (1980) examined the effect of Orowan’s (1943) inhomogeneity parameter on the stress distribution in the rolled metal. They concluded that the effect is more significant when sticking friction is considered to exist, compared to sliding fric- tion6. The distributions of the roll pressure, with or without the inhomogeneity parameter differed by about 10%.

5The deformation is deemed “plane-strain” when the strains in two directions are very much larger than that in the third direction. 6While the sticking friction has been assumed to exist in hot rolling in past analyses, recent studies indicate that it rarely occurs in the flat rolling process; use of lubricants reduce the coefficient of friction. 50 Primer on Flat Rolling

(A) (B)

Figure 4.4 (A) Non-homogeneous compression. (B) Homogeneous compression.

Some further consideration of the term “the width doesn’t change by much” is necessary here, in light of a recent publication by Sheppard and Duan (2002) who used FORGE3s V3, a three-dimensional, implicit, thermomechanically coupled, commercially available finite-element programme to analyse spread during hot rolling of aluminium slabs. While the authors’ predictions correspond to experimental and industrial data very well, the slabs they examined cannot be considered to behave according to the plane-strain assumption. In their study the slabs measure 25 mm width and 25 mm entry thickness, rolled using a roll diameter of 250 mm. In the industrial example, the measurements are 1129 mm width and 228 mm entry thickness. The roll diameter is 678 mm. In both cases lateral spread, measured and calculated, is shown to be considerable. When one considers strip rolling, however, in which case the roll diameter to entry thickness ratio is large in comparison to unity in addition to the width/thickness ratio also being large, homogeneous compression planes remaining planes during the pass as well as the assumption of plane-strain flow are quite close to the actual events. In what follows, both assumptions will be made without any further reference. Further simplifications and assumptions will be detailed and discussed in Chapter 5, dealing with the details of mathematical modelling of the flat rolling process.

4.3 The Metallurgical Events Before and After the Rolling Process

The rolling process begins by continuous casting7, or if an older, not modernized steel plant is considered, by ingot casting. In the most modern mills continuous casting is followed directly by hot rolling. In all of these cases the pre-rolling

7See Figure 1.5, Chapter 1. Flat Rolling A General Discussion 51

Figure 4.5 (A) The microstructure of an Nb-carrying steel fully recrystallized after 55% deformation in five passes at 11001070C. (B) The same steel, subjected to the same deformation pattern but at a lower temperature of 1000960C, shows significant grain elongation. The magnification is 1003. Source: Cuddy (1981). structure consists of dendrites which are subsequently removed in the reheat furnaces in which most of the alloying elements enter into solid solution. It may be assumed, then, that at the start of the rough rolling process the sample is in the austenite range and that it has been fully annealed and recrystallized before entry into the roughing mill stands8. The structure is made up of strain free, equiaxed grains. The steel is reduced in the rougher, in several steps, all performed at relatively high temperatures and not excessive rates of strain and it then passes on to the finishing train. The grain structure at this stage depends on the pass schedule in the rougher, but, as has been mentioned above, the influence of the metallurgical structure prior to entry into the finishing train has little influence on the final attributes. Two typical examples of the steel’s structure are shown in Figure 4.5A and B, reproduced from the publication of Cuddy (1981). The figures show the microstructures obtained by subjecting the samples to several, sequential plane-strain compression tests9. The chemical composition of the microalloyed steel was also given; it contained 0.057% C, 1.44% Mn and 0.112% Nb. The steel was reheated to 1200C and deformed by 55% in five passes. Figure 4.5A shows a fully recrystallized structure obtained at a deformation temperature of 11001070C. The test shown in Figure 4.5B, which was conducted at a lower deformation temperature of 1000960C, indicates flattened grains and, as a result, some strain hardening.

8It is difficult to prove the validity of this assumption as it is impossible to interrupt the rolling process to remove a piece of the hot steel for metallography. Some of the micrographs that are shown have been obtained from various laboratory simulations. 9Use of sequential, multi-stage hot compression tests in simulating the multi-pass rolling process will be discussed in Chapter 8. 52 Primer on Flat Rolling

Figure 4.6 The structure of an AISI 1008 steel, finish rolled, coiled and then hot rolled from a thickness of 3 mm, reduced by 10%. The magnification is 2503. Source: ASM Handbook (1985) reproduced with permission.

Following rough rolling, the transfer bar enters the finishing train where the microstructure undergoes further changes, again depending on the draft schedule which is usually prepared off-line, using mathematical models that are able to predict the expected metallurgical and mechanical attributes. There are prohibitively many possibilities to consider in one book so only a typical structure is shown in Figure 4.6, reproduced from the ASM Handbook (1985). The structure of a capped AISI 1008 steel is shown at a magnification of 250. The steel was finish rolled, coiled, then hot rolled from a thickness of 3 mm and reduced by 10%. The steel was then cooled in air, resulting in the fully ferritic microstructure. The next step that follows is the cold rolling process after the hot rolled, scaled surface is cleaned by pickling in hydrochloric acid. Several passes reduce the thickness further. The effects of progressively higher reductions are shown in Figure 4.7, demonstrating the resulting grain elongation.

4.4 Limitations of the Flat Rolling Process

There are several limits that designers of the draft schedule of flat rolling must consider. One of these, the minimum coefficient of friction necessary to initiate the process, has been mentioned above, see Eq. (4.1). Other limitations of the process include the minimum rollable thickness, alligatoring and edge-cracking. The first of these appears to be caused by the creation of a hydrostatic state of stress in the deformation zone. The latter two are also the consequence of the stress distribution, specifically the tensile stresses associated with the elongation of the rolled samples. Flat Rolling A General Discussion 53

Figure 4.7 Microstructure of a cold rolled, low-carbon steel sheet showing ferrite grains at (A) 30%, (B) 50%, (C) 70% and (D) 90% cold reduction. The magnification is 5003. Source: Benscoter and Bramfitt (2004) reproduced with permission.

4.4.1 The Minimum Rollable Thickness This phenomenon10 is observed to occur when a thin, hard strip is to be reduced in a single rolling pass using large-diameter rolls. In order to increase the reduction, the work rolls are progressively brought closer and closer in an attempt to reduce the roll gap. As the reduction is increased, the compression on the strip is also increasing and the work rolls deform more and more. After a certain gap dimension is reached, no further reductions of the thickness of the strip are possible; the minimum rollable thickness has been reached. A hydrostatic state of stress is supposed to have been built up within the strip in the deformation zone. Recalling that the material undergoing permanent plastic deformation retains its volume, no further change of the dimensions of the metal is possible. If the work rolls are forced to close still further, they flatten more, the mill frame stretches further and the minimum rollable thickness cannot be reduced any more. Further attempts are likely to

10The minimum thickness problem will be mentioned again in Chapter 12. 54 Primer on Flat Rolling cause damage to the mill. This thickness is a function of the material attributes of the metal as well as the elastic attributes of the work roll and of the mill frame. Early researchers estimated the magnitude of the minimum obtainable thickness in a rolling pass. Stone (1953) presented the formula

3:58Dμσ h 5 fm ð4:2Þ min E where the roll diameter is D, its elastic modulus is E, and σfm is the resistance of the rolled material to reduction (see Eq. (5.2)). Tong and Sachs (1957) also predict that the minimum rollable thickness is proportional to the same parameters as in Eq. (4.2). Johnson and Bentall (1969) hypothesize that the minimum rollable thickness does not actually exist in practice. Domanti et al. (1994) write that rolled thickness, beyond those predicted, was achieved in foil rolling mills. Nevertheless, the minimum rollable thickness is a real, actual limitation of the rolling process and its existence has been demonstrated in several instances. Researchers, using small scale laboratory rolling mills, are cautioned against attempting to demonstrate the existence of the minimum thickness. It is possible to force the work rolls together more and more, of course, but the chances of creating permanent damage to the mill and the attendant costs of replacing the cracked rolls are both usually prohibitively high.

4.4.2 Alligatoring and Edge-Cracking The rolled strip’s length grows while it is being reduced and the tensile strains in the direction of rolling often limit the reductions possible in a single pass. The stress distribution in the deformation zone may cause either alligatoring or edge- cracking. These were purposefully created while hot rolling aluminium strips with tapered edges (Duly et al., 1998), in order to examine the workability of the alloys. In each pass the work rolls were covered with a light coating of mineral seal oil. Severe edge-cracking is shown in Figure 4.8, rolled at a temperature of 505C. Edge-cracking and alligatoring are demonstrated in Figure 4.9.

Figure 4.8 Edge-cracking of an aluminium alloy, hot rolled at 505C to a strain of 0.6. Source: Duly et al. (1998). Flat Rolling A General Discussion 55

Figure 4.9 Alligatoring of an aluminium alloy, hot rolled at 497C to a strain of 0.56.

Workability and the limits of the process during hot rolling of steel and aluminium were considered in some detail by Lenard (2003).

4.5 Conclusion

A brief, general presentation of the flat rolling process was given. Two assumptions the “planes remain planes” and “homogeneous compression” necessary for the understanding of the flat rolling process were critically examined. The physical and the metallurgical events experienced by the steel were discussed. These included the examination of the free-body diagram of the work roll, in three conditions: the strip is ready to enter the roll gap; it is partially through; and steady-state rolling has been reached. As far as the metallurgical phenomena are concerned, several micrographs were presented, each showing the microstructure of the rolled strips, undergoing various rolling schedules. The limits of the process were presented. 5 Mathematical and Physical Modelling of the Flat Rolling Process

The essential, basic ideas in mathematical modelling of the flat rolling process are presented first. Empirical and one-dimensional (1D) models, applicable for strip rolling, are described and their predictive capabilities are demonstrated. Extremum principles specifically the upper bound theorem are considered. The need whether or not to include the effect of inertia forces in 1D models is discussed. A model, employing the friction factor instead of the coefficient of friction, is derived and its predictive abilities are examined in detail. The development of the microstructure as a result of the restoration and hardening phenomena during hot rolling and its effect on the resulting mechanical attributes are given. Thermalmechanical treatment is briefly discussed and the physical simulation of the flat rolling process is also included. In the last section, several phenomena, often ignored in mathematical modelling of the process, are given. These include the forward slip, mill stretch, roll bending, the lever arm and the effects of cumulative strain hardening. An approach that considers the difficulties associated with determining the relevant values of the coefficient of friction and the metals’ resistance to deformation for use in modelling is suggested.

5.1 A Discussion of Mathematical Modelling

Mathematical models of the flat rolling process are numerous and are easily available in the technical literature. The publications date from the early days of the twentieth century to the present. Their complexity, mathematical rigour, predictive ability and ease of use vary broadly. In what follows, models applicable to strip and plate rolling only will be presented, such that the large-roll diameter to strip thickness ratios allow the application of the “planes remain planes” assumption, implying that nearly perfect homogeneous compression is present in the deforming metal. This step and the additional assumption of the plane-strain plastic flow con- dition1 ensure that there will be only one independent variable in the equations: the

1These two assumptions have been discussed in Chapter 4.

Primer on Flat Rolling. DOI: http://dx.doi.org/10.1016/B978-0-08-099418-5.00005-6 © 2014 Elsevier Ltd. All rights reserved. 58 Primer on Flat Rolling distance along the direction of rolling or the angular variable around the roll. Thus, ordinary differential equations will be obtained, the integration of which is considerably less difficult than that of partial differential equations that would be obtained without the two assumptions. The available models can be listed according to the objectives the authors had while devising them. They are applicable equally well to hot, warm or cold rolling. These objectives may include the following:

G a simple, fast calculation of the roll separating forces; G in addition to the roll separating force, the roll torque, the temperature rise and the required power are to be calculated; G in further addition to the above, the determination of the metallurgical parameters and the material attributes as a result of the hot and cold rolling are to be determined. A more extensive list of the use of mathematical models of the rolling process is given by Hodgson et al. (1993). The authors add set-up and on-line control of the rolling mills and the rolling process to the use of the models, in addition to the following (the list is quoted directly from the reference):

G Minimize mill trials for product and process development. G Evaluate the impact of different mill configurations and new hardware on the process and the work piece. G Predict variables which cannot be easily measured (e.g. bulk temperature, temperature distribution, austenite grain size, post-cooling mechanical and metallurgical attributes). G Perform sensitivity analyses to determine which process variables should be measured and controlled to achieve the required quality, or final properties, of the product G Aid hardware design. G Further understand the physical process. Another comment needs to be mentioned in the context of using the models for predictions of rolling load, etc. The present author has been involved in the study of 1D models of the flat rolling process for quite some time. The studies involved experimentation as well as modelling and the predictive abilities of several 1D models were investigated. When the research studies began to appear in the technical literature, using finite-element models to investigate the flat rolling process, the following suggestion was made to several authors: experimental data would be provided and let us all compare our predictions. None took up the challenge. One comment was received: “Our analyses are performed to get insight into the mechanics of the process, not for predictions”. In what follows, some of the basic, classical 1D models2 are reviewed in addition to some of the more recent efforts. While the following list is not complete, it gives the most popular and well-known formulations. A model which includes an account of the variation of frictional effects along the roll/strip contact is also described, employing the friction factor instead of the coefficient of friction. Upper bound analysis of the process is discussed. The development of the metallurgical structure of

2It is recognized that these models were published quite some time ago, yet they often form the bases of existing on-line models. Mathematical and Physical Modelling of the Flat Rolling Process 59 the rolled strips is then reviewed and empirical relations, allowing the calculation of these parameters, are listed. As well, relations that predict the attributes of the material after the rolling process are given. Further, the predictive abilities of the models are presented and compared to each other and to experimental data. Each of these models will be developed in more detail in subsequent sections, classified as follows. The empirical models. An example of these is presented by Schey (2000). These can be used with considerable ease. Manual calculations, spreadsheets or simple computer programs are sufficient while calculating the roll separating force. The major objective of the models is just that: a simple and fast but reasonably accurate prediction of the roll separating force. The roll torque, the power and the temperature rise may also be obtained, but their accuracy is usually not quite as good as that of the force, no doubt because of the assumptions made in their determination; The 1D models. These are capable of predicting the roll separating forces as well as the roll torques quite well. The traditional models of these types are based on the classical Orowan approach, including the idea of the “friction hill” (Orowan, 1943) and its simplifications. For cold rolling the Bland and Ford (1948) technique and for hot-rolling Sims model (Sims, 1954) are often used in the steel industry, usually as a first approximation, often followed by adjusting the predictions to data taken on a particular rolling mill. Alternatively, the Cook and McCrum tables (1958), based on the 1D Sims model, may be employed. The predictive ability here is enhanced by accounting for the flattening of the work roll under the action of the roll pressure. The well-known Hitchcock formula (1935) is used in these models to estimate the magnitude of the radius of the flattened but still circular work roll while in a more refined 1D version (Roychoudhury and Lenard, 1984) the elastic deformation of the roll is analysed, using the 2D theory of elasticity. Interfacial frictional phenomena are modelled in two ways: mostly using the coefficient of friction and sometimes the friction factor3. The objectives here are similar to those above: that of the calculation of the roll separating force and the roll torque. The models can also be used to estimate the dimensions of some components of the rolling mill, such as the cross-sectional areas of the load carrying columns of the mill frame, the dimensions of the bearings, the drive spindles and possibly the power of the driving motor4. In addition to the above, the empirical relations, describing the evolution of the metallurgical structure (the amount of static, dynamic and metadynamic recrystallization, recovery, precipitation, retained strain and volume fraction of ferrite, as well as the mechanical and metallurgical attributes after hot forming and cooling) during and after hot rolling may be added to the 1D models. These equations are based on the studies of Sellars (1979, 1990), Roberts et al. (1983), Laasraoui and Jonas (1991a, b), Choquet et al. (1990), Hodgson and Gibbs (1992), Yada (1987), Beynon and Sellars (1992), Sakai (1995), Kuziak et al. (1997) and Devadas et al. (1991). The predictive abilities of the above relations,

3See Chapter 9 for the definitions. 4Note that the driving motor must have sufficient power to overcome friction losses and to compensate for the significantly less than 100% efficiency of the drive train. See Eqs. (9.8)(9.11). 60 Primer on Flat Rolling while combined with 2D finite-element mechanical models, were reviewed by Lenard, Pietrzyk and Cser (1999). A discussion of modelling the rolling process has been given in Chapter 16 of the Handbook of Workability and Process Design (2003). Extremum theorems.A model based on the extremum principles which, using the upper bound theorem, gives a conservative estimate of the power necessary for the rolling process. Finite-element models. These are discussed in some detail in Chapter 6, where an advanced version is presented. In addition to the care in the formulation of the models, the predictive ability of all of them depends, in a very significant manner, on the appropriate description of the rolled metal’s resistance to deformation and on the way the frictional resistance at the contact surfaces is expressed. Both of these phenomena are considered in detail in subsequent chapters. Material and metallurgical attributes are the topic of Chapter 8 while tribology is treated in Chapter 9. Caution is necessary when the choice of the most appropriate model in a particular set of circumstances is made. Often there is the tendency among researchers to select an advanced model and expect superior predictive capabilities. This step usually results in disappointment. The guiding principle should always be to make the complexity of the model match the complexity of the process and especially that of the objectives. Further, the mathematical rigour of all components, such as the material and the friction models, should match the rigour of the mechanical and the metallurgical formulations. A few comments, concerning the propensity of researchers to comment on the predictive capabilities of their models, are appropriate here. The usual tendency, when the predictions are compared to a few experimental results and the numbers compare well, is to proclaim that the model represents physical events very well. There are two concepts to consider, however, before good predictive ability is to be acknowledged. These are accuracy and consistency. A model that is accurate only sometimes and for which an error analysis has not been considered5 in the study is essentially useless. A model whose predictions may not be most accurate but are consistent, as demonstrated by the low standard deviation of the difference between calculated and measured data, is always useful as it can always be adjusted, a practice often followed in industry. While mathematical models of the flat rolling process have been published regularly in the technical literature, a complete list of all of them is much too large to be included in the present volume. Several conferences have been held in the recent past, entitled “Modelling of Metal Rolling Processes”, and the issues involved with all aspects of the rolling process have been discussed at these gatherings. An interesting review of on-line and off-line mathematical models for flat rolling was recently published by Yuen (2003). He examined the models that account for the flattening of the work rolls as well as those that don’t include it. He also

5It is indeed rare to see the analysis of the magnitudes of possible errors in the mathematical models. Mathematical and Physical Modelling of the Flat Rolling Process 61 discussed the models available for foil and temper rolling. He concluded that more sophisticated models are expected to be adopted for on-line applications and added that there is an “urgent need for robust algorithms in order to implement these superior models”. Mathematical models of the rolling process are now also available commercially. One of the outstanding ones, mentioned already in Chapter 4, has been made available by the American Iron and Steel Institute, the details of which can be obtained from the website www.integpg.com/Products/HSMM.asp. A large collection of software for simulation and process control can also be found at the website www.mefos.se/simulati-vb.htm.

5.2 A Simple Model

A simple model, fast enough for on-line calculations of the roll separating force, has been presented by Schey in his text Introduction to Manufacturing Processes, 3rd edition (Schey, 2000). The model expresses the roll separating force per unit width in terms of the averagepffiffiffiffiffiffiffiffiffiffiffi flow strength of the rolled metal in the pass, the projected contact length, L 5 R0Δh, a multiplier, identified by Schey as the pressure intensification factor, Qp, to accountpffiffiffiffiffi for the shape factor and friction, and a correction for the plane-strain flow, ð2= 3Þ 1:15, in the roll gap. The radius of the work roll, flattened by the loads on it, is designated by R0 (see below for Hitchcock’s equation, Eq. (5.3)). For the case when homogeneous compression of the strip may be assumed and frictional effects are significant, e.g. L/have . 1, and have is the average strip thickness, the model is written

Pr 5 1:15QpσfmL ð5:1Þ where the mean flow strength of the metal, σfm, is obtained by integrating the stressstrain relation over the total strain, experienced by the rolled strip:

ðε 1 max σ 5 σðεÞ ε ð : Þ fm ε d 5 2 max 0

The radius of the flattened roll, R0, is obtainable from Hitchcock’s relation (Hitchcock, 1935) 2 2 0 16ð1 v Þ R 5 R 1 1 Pr ð5:3Þ πEðhentry 2 hexitÞ

Hitchcock’s equation and the assumptions on which it is based have been controversial ever since it was published quite some time ago. While the critiques are valid the roll doesn’t remain circular in the contact zone and the roll pressure distribution is not elliptical Eq. (5.3) still enjoys widespread use. 62 Primer on Flat Rolling

Roberts (1978) examines the validity of Hitchcock’s equation6 and concludes that

the generally accepted Hitchcock equation, even considering elastic strip flattening, is not adequate to predict the length of the arc of contact between roll and strip.

Schey (2000) also presents an approach to deal with the rolling of thick plates where the assumption of homogeneous compression is not valid any longer. In these cases the shape factor is less than unity, L=have # 1. The plastic deformation is affected less by friction and a different pressure intensification factor is to be used7. The simplicity of the model of Eq. (5.1) is evident when one considers its relation to simple compression, akin to open die forging, expressing the force needed by the product of the mean flow strength and the projected contact length. This is then adapted to that of flat rolling by the application of the multiplier and the correction for plane-strain flow. In a rolling pass the total true strain is εmax 5 lnðhentry=hexitÞ and in cold rolling the stressstrain relation is usually taken as σðεÞ 5 Kεn or σðεÞ 5 Yð11 BεÞn1 where K, n, Y, B and n1 are material constants. Other formulations for the metal’s resistance to deformation are also possible of course, and some of these will be presented in Chapter 8. When hot rolling is analysed, the mean flow strength is expressed in terms of the average rate of strain, ε_ave:

m σfm 5 Cðε_aveÞ ð5:4Þ where the average strain rate is given, in terms of the roll surface velocity, v, the projected contact length and the strain by

v hentry ε_ave 5 ln ð5:5Þ L hexit

The material parameters C, m, K, n may be taken from the data of Altan and Boulger (1973) for a large number of steels and non-ferrous metals or may be determined in a testing program. Choosing more complex constitutive relations requires the use of non-linear regression analysis to determine the material constants, such as Y, B and n1. Care is to be taken when previously published material models are considered for use. Karagiozis and Lenard (1987) compared the predictive capabilities of several published constitutive relations, all claimed by their authors to be valid for low-carbon steels (see Figure 8.16). The recommendation therefore is, as follows: if there is any doubt about the accuracy of the material model that describes the resistance to deformation of the metal to be rolled, independent testing for the strength is necessary8.

6See also Section 5.4.1 where roll deformation is discussed in more detail. 7This problem is not dealt with in the present manuscript. If interested, refer to the original reference (Schey, 2000). 8Testing techniques are described in Chapter 8. Mathematical and Physical Modelling of the Flat Rolling Process 63

Finally, the multiplier Qp, the pressure intensification factor, is obtained from Figure 5.1 in terms of the coefficient of friction and the shape factor L/have, where have is the average of the entry and exit thickness. The torque to drive both rolls per unit width is then expressed, assuming that the roll force acts halfway between the entry and the exit:

M 5 PrL ð5:6Þ

The lever arm, the distance by which the roll separating force is to be multiplied to determine the roll torque, is thus defined as the projected contact length9. The power to drive the mill is determined using the torque, from Eq. (5.6), and the roll velocity. The relation gives the power in watts, provided the contact length is in m, the velocity in m/s, the roll radius in m, and the units of the width, w, match those of the roll separating force/unit width: v P 5 P wL ð5:7Þ r R0 Note that Eq. (5.7) gives only the power for plastic forming of the strip and thus, it is not to be confused with the power needed to drive the rolling mill, which is significantly larger. In order to develop the specifications for the power of the driving motor of the mill, friction losses and the efficiency of all drive-train components need to be considered. Rowe (1977) defines the overall power requirement in terms of these parameters, in the form

1 5 ð 1 Þð: Þ Ptotal η η 2P 4Pn 5 8 m t

6 Figure 5.1 The pressure intensification factor Qp. Source: Schey (2000); reproduced with 5 permission. Sticking μ = 02 4 0.15 f σ / p

p 3 =

p 0.1 Q 2 0.05 0 1

0 04812 16 20 L/h

9See Section 5.11.5 where the lever arm is discussed in some more detail. 64 Primer on Flat Rolling

η η where m is the efficiency of the driving motor and t is the efficiency of the transmission, including all of its components. Equation (5.8) supplies the power required for plastic deformation and the friction losses in the four roll-neck bearings (Pn) μ φ_ μ are given by nPrw d , where n is the coefficient of friction in the bearings, d is the bearing diameter and φ_ is the angular velocity of the roll. Roberts (1978) includes the resistance of the material in an equation that can also be used to predict the mill power: 1 ðσ 2 σ Þð1 2 rÞ 5 σ 1 entry exit 1 ð : Þ Ptotal η hentrywv cr 2 : Pn 5 9 m 1 0 5r

2 where σc is the dynamic, constrained yield strength of the rolled strip in N/m , w is the width in m, v is the velocity in m/s and r is the reduction in fractions; the power then will be obtained in W. The yield strength at room temperature, in lb/in2, of the “softer” strip is given in terms of the reduction as10

σ 5 40000 1 1773r 2 29:2r2 1 0:195r3 ð5:10Þ where the reduction, r, is given again as a fraction. The constrained yield strength is then taken to depend on the rate of strain: σ 5 : σ 1 ε_ : c 1 155ð 4460 log10 1000 Þð5 11Þ

Roberts (1978) presents a set of calculations of the overall power required to cold roll hard and soft low-carbon steels through a six-stand rolling mill. The predicted and measured input motor powers varied by at most 20% and were under 10% in most cases. In the calculations, the efficiency of the driving motor was taken from a low of 76.788.7%, fairly reasonable values. In light of the successful predictive ability, the use of Eq. (5.9) is recommended. As an alternative, the upper bound method may be used to calculate the power necessary to roll the strip; as is well known, the upper bound method gives a conservative estimate of the power for plastic deformation11. In an unpublished study, a simple experiment to estimate the power losses due to friction and drive-train inefficiency was conducted in the present writer’s laboratory. The work rolls were compressed to a certain magnitude of the roll separating force with no strip in between. The mill was turned on and the torque to drive the mill only was measured. The power thus obtained was in the order of 30% of the power when a strip was reduced in a similar fashion. The rise of the temperature of the strip in the pass due to plastic work may be estimated by P ΔT 5 ð5:12Þ gain mass flow 3 specific heat

10The conversion to SI units is 1 lb/in2 5 6.89476 3 1023 MPa. 11The upper bound approach is treated later in this Chapter; see Section 5.8.1. Mathematical and Physical Modelling of the Flat Rolling Process 65 where the power is to be expressed in J/s, the mass flow is to be in kg/s and the specific heat of the metal (cp) is to be in J/kg C. Equation (5.12) may be written in terms of the roll force, the geometry and the density in the form

0 PrL=R ΔTgain 5 1000 ð5:13Þ ρcphave

Note that while neither the speed of rolling nor the width of the strip appears in the equation, the strain rate would increase with increasing speeds and that would affect the magnitude of the roll separating force, and hence, the power. Note further that an error was knowingly committed in estimating the mass flow: the roll surface speed and the average strip thickness were used instead of the thickness at the no- slip point, which should have been used. While the location of the neutral point may be estimated, it is not known precisely, so the small error, no more than 10%, may be forgivable. Further, care must be exercised in the use of units. The roll separating force is to be in N/m; the contact length is to be in m; the roll radius is to be in m; the density is to be in kg/m3; the specific heat is to be in J/kg C; and the average strip thickness is to be in m. Roberts (1983) also gives a useful expression to estimate the temperature rise of the strip in the pass, which simply takes the work done/unit volume and assumes that all of that is converted to heat. The temperature increase, due to reduction r, may then be calculated by

σfm 1 ΔTgain 5 ln ð5:14Þ ρcp 1 2 r

A numerical experiment illustrates the magnitude of the predicted rise of the temperature of a hot-rolled steel strip. For the example, consider a 30% reduction of an initially 10 mm thick strip, using 500 mm radius work rolls which rotate at 50 rpm. Assume that the coefficient of friction is 0.2, a reasonable magnitude when some lubrication is used. Let the density be 7570 kg/m3,takethespecific heat of the steel to equal 650 J/kg K and let the average flow strength in the pass be 150 MPa. The temperature rise is now predicted to be 20C, by Eq. (5.12).The process and material parameters change drastically in the last stand of the finishing mill. Let the entry thickness there be 2 mm and the reduction to be 50%, so the final strip thickness will be 1 mm. The temperature of the strip is lower now, so the average flow strength is 250 MPa; this number includes the effect of the strain rate, caused by the increased rolling speed. The temperature rise is now much higher, calculated to be 135C. The temperature loss in the pass, due to conduction only is obtained as suggested by Seredynski (1973) in terms of the pass parameters, the heat transfer coefficient12, the density and the specific heat of the rolled steel. Seredynski’s formula is

12The heat transfer coefficient will be discussed in detail in Chapter 9, Tribology. 66 Primer on Flat Rolling rffiffiffiffiffiffiffiffiffiffiffiffiffi r 21 ΔTloss 5 60α ðTstrip 2 TrollÞ½ð12rÞπρcpN ð5:15Þ hentryR where α is the heat transfer coefficient at the roll/strip interface (Seredynski gives its 2 value as 44 kW/m K); r is the reduction in fractions, hentry is the entry thickness, R is the original, undeformed roll radius, Tstrip and Troll are the temperatures of the strip and 3 the roll, respectively, ρ is the density (7570 kg/m ), cp is the specific heat (650 J/kg K) and N is the roll rpm13. The temperature loss in the above two examples may be estimated now, using Eq. (5.15). Most of the numbers are known except one: the temperature of the roll. Roberts (1983) shows the experimental results of Stevens et al. (1971) who used thermocouples embedded in a full-scale work roll to monitor the rise of the temperature of the surface. The results indicate that the roll surface temperature may rise by as much as 500C14. With these numbers, the strip entering the roll gap at 1000C may cool by as much as 19C. In the second example, the loss of temperature is estimated to be 15.6C, less than before because of the shorter contact time. The final temperature of the strip after rolling will be the algebraic sum of these two values15. Roberts (1983) also presents the analysis of Stevens et al. (1971) to estimate the rise of the surface temperature of the roll, in terms of the bulk temperature of the roll, the time of contact and the thermal properties of the roll material: its thermal conductivity, its thermal diffusivity and the conductance. The calculations presented show that the rise of the surface temperature of the roll is somewhat less than those of the experiments of Stevens et al. (1971). The rise of the roll’s surface temperature may be estimated by the relation developed by Stevens et al. (1971). The equation relates the roll’s surface temperature (Troll), the roll’s temperature some distance below the surface (T0), the strip’s temperature at the entry (Tstrip) to the time of contact (t), the density and to several thermal parameters of the roll material. The formula is written in the form (Roberts, 1983) rffiffiffiffiffiffiffiffiffi T 2 T t roll 0 5 α ð5:16Þ Tstrip 2 T0 kρcp where α is the heat transfer coefficient at the roll/strip interface in W/m2 Kandk is the thermal conductivity of the roll material in W/m K. Roberts (1983) writes that the magnitude of T0 used in the calculations is not a critical variable. Typical calculations may be performed to appreciate the validity of the assumptions made above concerning the rise of the temperature of the roll’s surface. The thermal

13The numbers are taken from Roberts (1983). 14The experiments of Tiley and Lenard (2003) on an experimental mill indicate that the roll’s surface temperature may rise by as much as 200C. 15Note that in the example only two phenomena were considered: temperature rise due to plastic work done and temperature loss due to conduction. A more advanced thermal treatment needs to consider the temperature changes associated with radiation, convection and the metallurgical events. Mathematical and Physical Modelling of the Flat Rolling Process 67 conductivity, in W/m K, is dependent on the temperature, as indicated by Pietrzyk and Lenard (1991): 22:025T k 5 23:16 1 51:96 exp ð5:17Þ 1000 where T is in K. When the time of contact is 0.01 s, and the heat transfer coefficient, the specific heat and the density are as in the example above, the conductivity is calculated to be 28 W/m K by Eq. (5.17), the strip is at a temperature of 900C, the roll’s bulk temperature is 100C, and the roll’s surface is predicted to rise by 400C, close to the measurements of Stevens et al. (1971).

5.3 1D Models

5.3.1 The Classical Orowan Model The 1D models are all based on the equilibrium method in which a slab of the deforming material is isolated and a balance of all external forces acting on it is used to develop a differential equation of equilibrium16. Since the original treatment, published by Orowan (1943), is often considered to be the industry standard and other models’ predictions are usually compared to its calculations, it is worthwhile to review it in some detail. A detailed review and a thorough critical discussion of the method have also been given by Alexander (1972)17, who published a computer program in FORTRAN to analyse the flat rolling process. The model is based on the static equilibrium of the forces in a slab of metal undergoing plastic deformation between the rolls; see Figure 5.2. The forces due to the roll pressure, distributed along the contact arc, the interfacial shear stress and the stresses in the longitudinal and the transverse directions, form the force system, the equilibrium of which in the direction of rolling leads to the basic equation of balance. The assumption that planes remain planes allows this relation to be a 1D, ordinary differential equation of equilibrium in terms of the dependent variables: the roll pressure p, the strip thickness h, the radius of the deformed roll R0, the interfacial shear stress τ, the stress in the direction of rolling σx and the independent variable x, indicating the distance in the direction of rolling, measured from the line connecting the roll centres:

dðσ hÞ dh x 1 p 7 2μp 5 0 ð5:18Þ dx dx

16If inertia forces are expected to be significant contributors to the stresses, equations of motion need to be developed, equating the sum of all forces to the product of the mass and the acceleration. This concept is dealt with in Section 5.5. 17Note that Alexander indicated the existence of compressive stresses in the direction of rolling, acting on the isolated slab. In Figure 5.2 these stresses are shown as tensile and the boundary conditions are expected to determine if they are tensile or compressive. 68 Primer on Flat Rolling

y φ R'

pR' dφ Work roll τR' dφ

Rolled strip h+dhh (σ+dσ)(h+dh) σh x hexit h dx entry Slab

Figure 5.2 The schematic diagram of the rolled strip and the roll, showing the forces acting on a slab of the deforming material. where the 7 sign indicates that the equation above describes the conditions of equilibrium between the neutral point and the entry (when using the negative sign), as well as between the neutral point and the exit (when using the positive sign). In fact, Eq. (5.18) is composed of two independent, ordinary, first-order differential equations, containing four dependent variables, σx; p and h, all of which depend on R0 in turn in addition to the coefficient of friction. The interfacial shear stress has already been replaced by the product of the coefficient of friction and the normal pressure in Eq. (5.18), as suggested by the CoulombAmonton formulation. The necessary additional independent equations are obtained from the theory of plasticity and the geometry of the deformation zone. These include the HuberMises criterion of plastic flow, relating the stress components in the direction of rolling and perpendicular to it to the metal’s flow strength. With the assumption of plane-strain plastic flow, the criterion becomes:

σx 1 p 5 2k ð5:19Þ where k designates the metal’s flow strength in pure shear. The other variable, the strip thickness, can be obtained from geometry:

x2 h 5 h 1 2R0ð1 2 cos φÞh 1 ð5:20Þ 2 2 R0 The approximate formula is valid as long as the angles are much smaller than unity, true in the case of thin plate and strip rolling. The radius of the flattened roll is obtained using the original Hitchcock equation, given above; see Eq. (5.3). In order to integrate Eq. (5.18), the metal’s resistance to deformation is to be described and the interfacial shear stress needs to be given, usually as a function of the coefficient of friction and the roll pressure

τ 5 μp ð5:21Þ Mathematical and Physical Modelling of the Flat Rolling Process 69 as was done already in Eq. (5.18). Substituting Eq. (5.19) into Eq. (5.18) leads to

dp p 2k dh dð2kÞ 6 2μ 5 1 ð5:22Þ dx h h dx dx which, with the use of Eq. (5.20) and an expression for 2k see Eqs. (5.23) and (5.24) is ready to be integrated. The computation to determine the roll separating force and the roll torque begins with the integration of the equilibrium equations for the roll pressure. Starting at 5 σ 2 2 τ φ entry, using the appropriate boundary conditions [pentry entry 2kentry tan 1], φ 2 where 1 is the roll gap angle and the ve sign of the coefficient of the friction term, integration leads to a curve for the roll pressure. The next step is integration from the exit, and again, using the appropriate boundary condition there [pexit 5 2kexit 2 σexit] and now the 1 sign leads to another curve for the pressure distribution. Two curves thus produced give the pressures exerted by the rolled strip on the roll, referred to as the friction hill. (Note that the subscripts “entry” and “exit” in the parentheses refer to the values of the designated parameters at those locations. The terms σentry and σexit indicate the front and the back tensions, respectively. In most laboratory mills or a single-stand roughing mill, these are not applied.) The location of the intersection of the curves is defined as that of the neutral point, at which the roll surface velocity and that of the strip are equal and no relative movement between them takes place. Further integration of the roll pressure distribution over the contact, from entry to the exit, leads to the roll separating force. Integration of the product of the roll radius and shear forces from the entry to the exit leads to the roll torque. The necessity of accounting for the flattening of the work roll makes an iterative solution unavoidable. In the first set of calculations, rigid rolls are assumed to exist, i.e. R 5 R0. In the second iteration, the roll separating force, that has just been determined, is used to calculate the flattening of the roll employing Hitchcock’s relation (see Eq. (5.3)) and, using the radius of the flattened roll, a new roll force is obtained. The iteration is stopped when a pre- determined tolerance level on the roll force is satisfied. Corrections for the contribution of the elastic entry and exit regions can also be included in Orowan’s model; for the details, see Alexander (1972). Equation (5.18) may be used to analyse the cold, warm or hot flat rolling processes, the difference being the manner of the description of the term 2k, the metal’s resistance to deformation. If cold rolling is considered, one may follow Alexander (1972) and use the relation 2 2 h n1 2k 5 pffiffiffi Y 11 pffiffiffi Bln entry ð5:23Þ 3 3 h pffiffiffi where the 2= 3 multiplier corrects the stressstrain relation, obtained in a uniaxial tension or compression test, to be applicable for the analysis of the plane-strain flow 70 Primer on Flat Rolling problem of flat rolling. If hot rolling is to be studied, the resistance to deformation needs to be expressed in terms of the strain rate, at the least. A form often used is

2 2k 5 pffiffiffi Cε_m ð5:24Þ 3 where C and m need to be determined in independent tests. In a more advanced approach, the equation should include several more parameters. These will be discussed further in Chapter 8.

5.3.2 Sims’ Model Sims (1954) takes advantage of the fact that the angles in the roll gap are small when compared to unity, leading to the approximations sin φ tan φ φ and 1 2 cos φ φ2=2. He also assumes that the product of the interfacial shear stress and the angular variable is negligible when compared to other terms and that sticking friction, i.e. τ 5 k, is present in the contact between the roll and the strip18. These simplifications, in addition to assuming that the material of the rolled metal is characterized as rigid-ideally plastic, allow for a closed-form integration of the equation of equilibrium, and the roll separating force per unit width is obtained as

Pr 5 2kLQp ð5:25Þ an equation that is similar to that of Schey, see Eq. (5.1).InEq. (5.25) the term 2k stands for the yield strength of the metal,pffiffiffiffiffiffiffiffiffiffiffiffi obtained in plane-strain compression. The 0 contact length is as given above, L 5 R Δh, and the multiplier Qp is dependent on the ratio of the radius of the flattened roll, the exit thickness of the rolled strip and the thickness of the strip at the neutral point, Y: " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π 2 π 2 0 1 r 21 r 1 r R Y Qp 5 tan 2 2 ln 2 r 1 2 r 4 r hexit hexit : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ð5 26Þ 1 1 2 r R0 1 1 ln 2 r hexit 1 2 r where the thickness of the strip at the neutral point is found by equating the φ magnitudes of the roll pressures there. The location of the neutral point, n,is obtained from rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi π 0 0 0 2 5 R 21 R φ 2 R 21 r : lnð1 rÞ 2 tan n tan ð5 27Þ 4 hexit hexit hexit 1 2 r

18Note that the sticking friction assumption is not appropriate, even in hot rolling. The coefficient of friction, as a function of the temperature, will be discussed in Chapter 9. Mathematical and Physical Modelling of the Flat Rolling Process 71

In the above relations r stands for the reduction in the pass, expressed as a fraction. Because of its simplicity, Sims’ model often forms the basis of on-line roll force models for hot rolling in the steel rolling industry, although it is adapted for the particular mill on which it is used. The mathematical model of Caglayan and Beynon (1993), called SLIMMER, makes use of Sims’ approach and combines it with several relationships that describe the microstructural evolution of the rolled metal. The model developed by Svietlichnyy and Pietrzyk (1999) for on-line control of hot plate rolling also uses Sims’ model to calculate the roll separating forces.

5.3.3 Bland and Ford’s Model In addition to the small angle assumption, Bland and Ford (1948) assume that the roll pressure equals the stress in the vertical direction and since the difference between them is a function of the cosine of very small angles, the error is not large, especially in cold rolling where roll diameters are usually much larger than the thickness of the strip. As with Sims’ model, this allows a closed-form solution to be obtained. The roll force is then expressed: () ðφ ðφ n 1 ÂÃ 0 h h Pr 5 2kR expðμHÞdφ 1 exp μðHentry 2 HÞ dφ ð5:28Þ h φ h 0 exit n entry where H is given by rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi R0 R0 H 5 2 tan21 φ ð5:29Þ hexit hexit

The location of the neutral point is calculated by rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi h H h φ 5 exittan n exit ð5:30Þ n R0 2 R0 and the term Hn is determined by the formula

Hentry 1 hentry Hn 5 2 ln ð5:31Þ 2 2μ hexit where the subscripts indicate the conditions at the entry orp atffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the exit. The angular φ 5 2 = 0 distance at the entry, the roll bite, is obtained by 1 ðhentry hexitÞ R .The Bland and Ford model is often used in the rolling industry in the analysis and the control of the cold rolling process. Puchi-Cabrera (2001) used the Bland and Ford approach in a different way. He considered cold rolling of an aluminium alloy, from a thickness of 6 mm to a final thickness of 0.012 mm. The industrial practice is to roll the alloy in three stages. 72 Primer on Flat Rolling

In the first stage the work piece is reduced from 6 to 0.68 mm in four passes. In the second stage the alloy is annealed and in the third stage it is reduced, in several passes, to the 0.012 mm thickness. In each of the rolling passes the reduction, and hence the rolling load, is reduced and the mill’s full capability is not utilized. The author considered the effects of maintaining a constant load during each of the multi-stage reductions and concluded that this may create clear advantages in terms of productivity, product quality and roll life.

5.4 Refinements of the Orowan Model

Introducing the equations of elasticity to analyse the elastic entry and exit regions, as well as the deformation of the work roll, leads to a somewhat more fundamental model of the flat rolling process (Roychoudhury and Lenard, 1984). The model is still based on the equilibrium method the Orowan approach and it is applicable when the roll radius to strip thickness ratios are much larger than unity, allowing for the assumption of homogeneous compression in the roll gap. The differences between this model and the Orowan approach are as follows:

G Hitchcock’s equation is replaced by a 2D elastic analysis of roll deformation. The rolls are assumed to be solid cylinders initially, which deform under the action of non-symmetrical normal and shear stresses during the pass. Two-, four- or six-high roll arrangements can be treated by the analysis, depending on how keeping the work rolls in balance is described mathematically. The 2D theory of elasticity, coupled with the elasticplastic 1D treatment of the rolled strip, is used to determine the contour of the deformed roll. G The elastic loading and unloading regions in the rolled strip at the entry and exit, respectively, are analysed using the 1D theory of elasticity. The locations of the elastic/plastic interfaces at both locations then become parts of the unknowns and are determined during the solution process by using the HuberMises criterion of plastic flow. G The equation of equilibrium is written using the variable in the direction of rolling as the independent variable. The roll gap is divided into a finite number of slabs each of which is assumed to be either elastic or ideally plastic. As the metal is deformed and strain hard- ens during the rolling process, the flow strength of each slab is changed accordingly. As well, the roll pressure and the interfacial shear stresses are expressed in terms of Fourier series. A closed-form solution for each slab is thus obtained. Assembling the slabs is accomplished by enforcing horizontal equilibrium, leading to the complete solution for the pressure distribution and hence to the roll separating force and the roll torque. G The roll pressures and the interfacial shear stress distributions thus obtained are then used to calculate the contour of the deformed roll, using the Fourier series and the biharmonic equation. G The current shape of the flattened work roll is used to determine the roll pressure, interfacial shear stress, the roll separating force and the roll torque, which are employed to re- calculate the roll contour. G The iteration is continued until satisfactory convergence of the roll force is reached. Since the details of the model have been published (Roychoudhury and Lenard, 1984), only a brief exposition is given below. The schematic diagram from which the equation of equilibrium is derived, as shown in Figure 5.3, differs from the one Mathematical and Physical Modelling of the Flat Rolling Process 73

y x

y = f(x) R´ C

p μp h h 1 1e h h2e T1 T2 Elastic compression Elastic recovery

Figure 5.3 Schematic diagram of the rolled metal and the roll, used in the model of Roychoudhury and Lenard (1984). used by Orowan (1943) and Alexander (1972) in that the roll contour is taken to be an unknown function y 5 f(x), to be determined as part of the computations. The balance of the forces of a slab of the rolled metal is now derived using the direction of rolling as the independent variable, leading to d dy dy hp2 2k 7 τ 5 2 p 6 τ ð5:32Þ dx dx dx and as before, the positive algebraic sign in front of the interfacial shear stress indicates the region between the neutral point and the exit and the lower sign designates the region between the neutral point and the entry. The thickness of the strip, using C to designate half the distance between the two centres of the two roll-neck bearings, is

h 5 2ðy 1 CÞð5:33Þ

Using the coordinate system in Figure 5.3, the term y in Eq. (5.33) becomes a negative number. Expressing the roll contour of one particular slab as a straight line, defined by

y 5 ax 1 b ð5:34Þ and, as mentioned above, assuming that each slab is either elastic or is made of an ideally plastic metal simplifies Eq. (5.32), and closed-form integration, slab by 74 Primer on Flat Rolling slab, is now possible. The constants of integration are determined by assembling the slabs such that horizontal equilibrium is assured. The elastic regions at the entry and exit are also explicitly accounted for in the model. The equation of equilibrium, Eq. (5.32), is valid in those regions as well. Combining them with the 1D plane-strain form of Hooke’s law leads to the stress distributions in the elastic loading and recovery regions. Using the HuberMises yield criterion, and matching the elastic and the plastic stress distribution, the locations of the elasticplastic boundaries are determined. The analysis requires the explicit determination of the constants a and b, defining the roll contour at each slab. By expressing the roll pressures and the interfacial shear stress distributions in terms of Fourier series and analysing roll flattening following Michell’s 2D elastic treatment (Michell, 1900), the roll separating forces are obtained as " ð ð xn dy xexit dy Pr 5 p 1 1 μ dx 1 1 1 μ dx xentry dx xn dx : ð ð # ð5 35Þ xn dy xexit dy 1 μ 2 dx 1 μ 1 dx xentry dx xn dx and the roll torques by ð ð xn dy dy xexit dy dy M=25p x2y 1μ y1xy dx2p x2y 2μ y1x dx xentry dx dx xn dx dx ð5:36Þ

5.4.1 The Deformation of the Work Roll The critique of Roberts (1978) concerning the use of Hitchcock’s formula to calculate the radius of the flattened but still circular work rolls is well accepted. The elastic flattening of the rolls has been treated by Jortner et al. (1959). The authors considered the effect of a force on the deflection of a solid cylinder and showed that the rolls don’t, in fact, remain circular in the deformation zone. Non- circular roll profiles have also been developed by Grimble (1976) and Grimble et al. (1978). The problem of the deformation of the work roll is treated here by assuming that the work roll is a solid cylinder, subjected to non-symmetrical loads. The loading diagram is shown in Figure 5.4 (Roychoudhury and Lenard, 1984) where the roll pressure is designated by pðϕÞ and the interfacial shear stress by τðϕÞ. The roll is kept in equilibrium in one of two ways. If there is a back-up roll, the pressure between the two rolls will keep the work roll in its place. If a two-high mill is considered, the roll centre is taken to be stationary, achieved by letting 2ξ 5 π, where 2ξ is the extent of the pressure distribution of the now imaginary back-up roll. Mathematical and Physical Modelling of the Flat Rolling Process 75

∞ φ = + φ + φ Rb ( ) R0 ∑[Rancos(n ) Rbnsin(n )] n=1

ξ ξ

β β E a a 1 1 ⎛πφ ⎞ τ ()φ = G sin⎜ ⎟ F ⎝ 2β ⎠ G

⎛πφ ⎞ p(φ) = −E − F cos⎜ ⎟ ⎝ 2β ⎠

Figure 5.4 The loading diagram of the work roll, showing the roll pressure and the interfacial shear stress distributions in addition to the forces that keep the roll in equilibrium (Roychoudhury and Lenard, 1984).

The stress distribution in any problem of linear elasticity should satisfy the biharmonic equation, which in the 2D cylindrical coordinates are δ 1 δ 1 δ2 δ2φ 1 δφ 1 δ2φ 1 1 1 1 5 0 ð5:37Þ δr2 r δr r2 δϕ2 δr2 r δr r2 δϕ2 where the stress components are defined in terms of the Airy stress function, φ,as

1 δφ 1 δ2φ σ 5 1 ð5:38Þ r r δr r2 δϕ2

δ2φ σϕ 5 ð5:39Þ δr2 76 Primer on Flat Rolling and δ 1 δφ τ ϕ 52 ð5:40Þ r δr r δϕ

Following Michell (1900), the stress and the strain distributions may be calculated using biharmonic functions

XN φ 5 c r2 1 d r3 sin ϕ 1 d r3 cos ϕ 1 ða rn 1 b rn12ÞsinðnϕÞ 0 1 2 1n 1n : n52 ð5 41Þ n n12 1ða2nr 1 b2nr ÞcosðnϕÞ where the constants a1n; a2n; b1n; b2n; c0; d1 and d2 need to be determined such that the stress boundary conditions at r 5 R are satisfied:

σr 5 pðϕÞ and τrϕ 5 τðϕÞð5:42Þ

The coefficients in Eq. (5.41) can be determined by representing the normal and the shear loading on the roll’s surface in terms of Fourier series

XN pðϕÞ 5 pa0 1 ½pan cosðnϕÞ 1 pbn sinðnϕÞ ð5:43Þ n51 and

XN qðϕÞ 5 qa0 1 ½qan cosðnϕÞ 1 qbn sinðnϕÞ ð5:44Þ n51 where the coefficients may be determined by the Euler formulas19. The roll flattening, thus determined, was tested in a simple experiment. The side of the 125 mm radius work roll was fitted by two strain gauges and the strains during rolling of commercially pure aluminium alloys were measured20. These were compared to the calculated strains. The results are shown in Figure 5.5, plotting the radial strains against the angular distance around the work roll, using the data from the strain gauge near the edge, at 120 mm from the roll’s centre. It is observed that the predicted strains by the 2D elastic analysis compare well to the measurements.

5.5 The Effect of the Inertia Force

The metal to be rolled enters the roll gap at some velocity, which is usually lower than the surface velocity of the roll. As the thickness is reduced, the width remains

19The detailed development of the 2D analysis of Michell (1900) is given by Pietrzyk and Lenard (1991). 20The 50 mm wide, 2 mm thick aluminium strips were reduced by 5% in the tests. Mathematical and Physical Modelling of the Flat Rolling Process 77

–1000 Analytical –800 Experimental

–600 6 ×10

r –400 ε Roll separating force 1800N r R –200 / = 0.957

0 100 0 πππ π π π π 3 5 3 7 π 8482 848

Figure 5.5 The calculated and measured radial strains of the work roll (Roychoudhury and Lenard, 1984). unchanged and the length grows; the metal accelerates and exits from the roll gap at a velocity larger than that of the roll under most circumstances. Hence, there exists a force due to this acceleration and its effect on the rolling variables needs to be established. While 1D models usually ignore this contribution, the finite- element models usually include it in their analyses. In what follows the validity of these approaches will be discussed and the potential effect of the mass 3 acceleration term on the roll force, etc. will be given in numerical terms.

5.5.1 The Equation of Motion The effects of the inertia forces on the rolling process have rarely been analysed explicitly. In what follows, this effect will be considered in some detail. Equating the forces acting on a slab of the material in the roll gap to the product of the mass of the slab and its acceleration, and using the distance in the direction of rolling as the independent variable, leads to

dðσ hÞ dh ma x 1 p 7 2μp 5 ð5:45Þ dx dx dx where the mass/unit width is given by m 5 h dx ρ, the density is designated by ρ and the acceleration is a. TheleftsideofEq. (5.45) is, of course, identical to that of Eq. (5.18). In order to develop a relation for the acceleration of the slab, use is made of mass conservation, which requires that dðvhÞ 5 0, leading to a 5 dv=dt 52ðv=hÞðdh=dtÞ. The time derivative of the strip’s thickness may be 2 0 obtained from the simplified version of Eq. (5.20), h 5 h2 1 x =R , written in terms of x, the variable along the direction of rolling, in the form dh=dt 5 2xv=R0, where the strip velocity, v, in terms of the roll’s surface velocity and the thickness at the neutral 78 Primer on Flat Rolling point, is given as v 5 vrhn=h. Substituting the above into Eq. (5.45) along with the HuberMises criterion of plastic flow yields the differential equation of motion:

dp p 2k dh dð2kÞ 2xρ ðv h Þ2 6 2μ 5 1 1 r n ð5:46Þ dx h h dx dx R0 h3

It is now possible to estimate the orders of the magnitudes of the terms of Eq. (5.46). The magnitude of the roll pressure is in the order of several hundred MPa. The magnitude of the last term of the equation, for any reasonable set of rolling parameters, is less than 1% of that.

5.5.2 A Numerical Approach In another, simpler approach, the inertia force acting on the whole of the mass in the deformation zone can be determined. The acceleration is then given by 5 2 =Δ a ðvexit ventryÞ t and thep timeffiffiffiffiffiffiffiffiffiffiffi taken for a cross-sectional plane to travel from Δ 5 0Δ = the entry top theffiffiffiffiffiffiffiffiffiffiffiffi exit is t R h vr. The mass of the metal in the roll gap is 0 m 5 ρwhave R Δh so the inertia force is

FI 5 ρwhavevrðvexit 2 ventryÞð5:47Þ

For the inertia force to be a significant contributor in the analysis of permanent deformation, the stress it creates over the average cross-section of the rolled metal should be similar in magnitude to the yield strength. From Eq. (5.47) equate the stress created by the inertia force to the yield strength:

FI σyield 5 5 ρvrðvexit 2 ventryÞð5:48Þ whave

For any realistic set of numbers, the difference between the exit and entry velocities becomes unrealistically high, underscoring the conclusions drawn above: the contribution of the inertia force may be safely ignored. To get a numerical estimate, take a steel whose density is 7850 kg/m3 and let a 1 m diameter roll have a rotational speed of 100 rpm, leading to a roll surface velocity of 5.24 m/s. Let the entry thickness be 5 mm, at a velocity of 5 m/s. The exit velocity is then, from mass conservation, 12.5 m/s. Substituting these numbers in the right side of Eq. (5.48) leads to a stress, due to inertia effects alone, of 0.31 MPa, clearly negligible in comparison to the magnitudes of all other stress components.

5.6 The Predictive Ability of the Mathematical Models

The decision to be made when choosing a mathematical model to analyse the flat rolling process is not an easy one. In what follows, it is assumed that the objective of the analysis is to predict only some of the rolling variables, namely the roll Mathematical and Physical Modelling of the Flat Rolling Process 79 separating force. The predictive abilities of some of the models discussed above will be presented and critically discussed. The experimental data, developed by McConnell and Lenard (2000), will be used. In that project, low-carbon steels were rolled at various rolling speeds and to various reductions, using low viscosity oils for lubrication. The roll separating forces and the roll torques were measured. In the calculations that follow, the roll separating forces are considered and the predictive abilities of three models those given by Schey (2000), Bland and Ford (1948) and Roychoudhury and Lenard (1984) are compared. The results of the comparison are shown in Figure 5.6 in terms of the ratios of the measured and the calculated roll forces for each of the methods of calculation, as functions of the rotational speed of the roll. The reductions vary from a low of 14% up to 50% in the rolling passes. The data given in the figure need to be discussed very carefully and in some detail. The essential information for modelling includes the material’s resistance to deformation and the coefficient of friction. The former was given by McConnell and Lenard (2000) for the steel used here, obtained in traditional uniaxial tension : tests, as σ 5 150ð11234εÞ0 251 MPa. The latter was determined by inverse calculations, matching the measured separating forces to those calculated by the approach of Roychoudhury and Lenard (1984). This is evident in Figure 5.6, as the triangles of Roychoudhury and Lenard are always very close to unity, as expected, of course. The magnitude of the coefficient of friction, thus obtained, was then used in the two other methods of calculations. The diamonds of the Bland and Ford (1948) approach are approximately 20% over unity. The crosses of the Schey (2000) technique are not very consistent. It needs to be pointed out that when judging a model for its predictive ability, consistency is much more important than accuracy, since predictions with low standard deviation can always be adjusted by the use of carefully determined factors.

1.50 Figure 5.6 Comparison of the predictive capabilities of three simple models for cold rolling of low-carbon steel strips. 1.25 calculated F / 1.00 measured F 0.75 Schey's method Bland and Ford's method Roychoudhury and Lenard's method 0.50 0 1000 2000 3000 Roll speed (mm/s) 80 Primer on Flat Rolling

Both of these approaches Bland and Ford’s and Schey’s could have been used to determine the coefficient of friction, in an inverse manner, of course. Both would have yielded values for μ that would vary broadly and would be quite different from those, used in Figure 5.6, indicating that the inverse method for the determination of the coefficient may not be the most suitable approach. Instead, independent experiments, to be discussed in Chapter 9, are recommended.

5.7 The Friction Factor in the Flat Rolling Process

The referee of a manuscript of the present author and his student (Lenard and Barbulovic-Nad, 2002) questioned the use of the coefficient of friction in bulk forming processes, stating correctly that at high normal pressures the physical meaning of μ is lost. The rebuttal, which was accepted by the editor, is quoted below:

It is realized that the coefficient of friction obtained by inverse modelling, while it may be close to the actual value, is in fact an effective one. Further, while application of the traditional definition of the coefficient, as the ratio of the tangential to the normal forces, in metal forming operation has been questioned, it still remains a parameter in widespread use. As shown by Schey (2000), μ reaches a maximum as the normal stresses increase. This condition, while it may be reached during dry contact, is not likely to be observed when forming occurs in the boundary or mixed lubrication regimes. Azarkhin and Richmond (1992) also show that the friction factor will be less than unity, even when adhesion is the main cause of frictional resistance.

Nevertheless, the comments of the referee were taken seriously and they gave the impetus to develop a model of flat rolling, using the friction factor, instead of the coefficient of friction. Pashley et al. (1984) examined the three most significant factors that contribute to surface interactions involving adhesion: the area of real contact, the interfacial bond strength and the mechanical properties of the interface. They used a tungsten tip and a nickel flat, the tungsten being nearly 10 times harder than the nickel. When the surfaces were clean, the junction failed at a stress level roughly equal to the yield strength of the metal. Li and Kobayashi (1982) included the effect of the relative velocity of the sliding surfaces in their formulation of the frictional model. A similar model is used in Elroll21, a finite-element software developed by Pietrzyk (1982) in which the μ coefficient of friction is defined in terms of a constant value, 0, the relative velocity of the roll and the strip, Δv,andaconstant,a, which is chosen to be 1023, in the form

μ 2 Δv 5 21 ð : Þ μ π tan 5 49 0 a

21Elroll, a finite-element program that analyses the flat rolling process has been developed in the Department of Modelling and Information Technology AGH in Krako´w, Poland. The distributor of the software may be reached by e-mail: [email protected]. Mathematical and Physical Modelling of the Flat Rolling Process 81

Most commercially available finite-element software packages allow the user to choose the manner in which friction is to be modelled. A random search on the Internet yielded a 1994 newsletter from the MARC Corporation, giving an equation for the friction force:

2 Δv f 5 μf tan21 ð5:50Þ t n π C where μ is the coefficient of friction, fn is the normal force and C is a constant. Another relationship for the interfacial shear stress in terms of the relative velocity of the roll and the rolled strip was given by Gratacos et al. (1992):

σ Δv jτj 5 m pfmffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5:51Þ 3 Δv2 2 K2 where K is identified as the “regularization parameter for the friction law”, given later as a very small number, in the order of 0.001. An interesting expression was presented by Nadai (1939) for the interfacial shear stress as a function of the relative velocity of the strip and the roll, the lubricant’s viscosity and the thickness of the oil film:

ηðv 2 v Þ τ 5 strip r ð5:52Þ hfilm;ave where the thickness of the oil film is to be the average over the rolling pass. A third possibility in modelling friction is presented by Carter (1994) by relating the fractional shear strength of the contacting interface to the normal component of the deviatoric stress, through a “constant of proportionality”, identified as “much like the coefficient of friction”. Carter also states, unfortunately without referencing the information, that in simple compression “the fracture strength of the junction is close to the shear strength of the softer material”. Regardless of the manner in which friction is to be modelled, some difficulties, uncertainties and unknowns will always remain. In what follows, the friction factor is used in developing a 1D model of the flat rolling process, partially as the result of the comment of a reviewer of one of the present author’s recent manuscripts, questioning the use of the coefficient of friction in the flat rolling process. Aiding in making the decision to use the friction factor, in spite of the lack of precise knowledge of the magnitude of “k”, the shear strength of the interface, are three factors. One is consideration of the pressure sensitivity of lubricants, which, for an SAE 10 W oil, is given as 0.0229 MPa21 by Booser (1984) who also gives the viscosity as 32.6 mm2/s. If the roll pressure is 800 MPa, not an unreasonable magnitude when cold rolling steel, the Barus equation (see Eq. (9.48)) gives the viscosity at that pressure as 2.9 3 109 mm2/s. This number, while possibly unrealistically high, indicates that shearing the lubricant at that pressure may require as much of an effort as shearing the metal. The other is the comment, referred to above, concerning the level of 82 Primer on Flat Rolling stress at which the tungstennickel junction failed (Pashley et al., 1984) and the third, also mentioned above, is the conclusion that ploughing was the major frictional mechanism (Lenard, 2004; Dick and Lenard, 2004) when a sand-blasted roll was used. Carter (1994) and Montmitonnet et al. (2000) reinforce this last factor by indicating that ploughing may be as important as adhesion in understanding frictional resistance. The objectives are then:

G to develop a 1D model of the flat rolling process using the friction factor; G to determine the dependence of the friction factor on speed and the reduction by using data, developed earlier on cold rolling of steel strips (McConnell and Lenard, 2000); G to test the predictive capability of the model by comparing the predictions to experimental data; G to develop a correlation of the coefficient of friction and the friction factor.

5.7.1 The Mathematical Model The choice of the level of complexity in mathematical modelling of the flat rolling process depends on the objectives of the researchers. The comments in the last para- graph of the concluding chapter of Pietrzyk and Lenard (1991) are still valid: if the aim is to analyse mechanical events in strip rolling, where the roll diameter/strip thickness ratio is large, a 1D treatment is sufficient. This is followed in the present analysis in which the usual simplifying assumptions of previous workers are also employed. These include the assumptions of rigid rolls, homogeneous compression, a rigid-plastic material which remains isotropic and homogeneous as the rolling process continues. The angles are taken as small when compared to unity. As well, inertia forces are small in comparison to other forces and are therefore ignored. The usual, 1D schematic diagram of the flat rolling process is used and the balance of forces in the direction of rolling on a slab of the rolled metal then leads to the well-known relation

dðσ hÞ dh x 1 p 2 2τ 5 0 ð5:53Þ dx dx where σx is the stress in the direction of rolling and p is the roll pressure. In simplifying Eq. (5.53), the HuberMises flow criterion, σx 1 p 5 2k, is used, 2 the strip thickness is taken in terms of the independent variable, h D hexit 1 x =R, and the interfacial shear stress is defined by the friction factor, τ 5 mk. The shear strength of the softer material, the rolled steel, is taken as k and as usual, 0 # m # 1. A first-order ordinary differential equation for the roll pressure is then obtained:

dp 2k 5 ð 2 Þð: Þ 2 2x mR 5 54 dx hexitR 1 x where the friction factor, m, is to be expressed in terms of the significant variables and parameters. Equations (5.49)(5.52) expressed the coefficient of friction or the friction stress in various forms, as functions of the relative velocity of the strip with Mathematical and Physical Modelling of the Flat Rolling Process 83 respect to the roll, Δv, acknowledging the well-known speed dependence of the frictional resistance, in addition to the viscosity, the thickness of the oil film and several constants. Following these but also recognizing the dependence of friction on the normal stresses, the friction factor is now written as dependent on both load and speed:

5 2 2 2 1 21 Δ = : m aðx xnÞp b tan ð v qÞð5 55Þ where a and b are constants to be determined and q is a constant, taken arbitrarily to be 0.1. The relative velocity is given in terms of the location of the neutral point, xn, and the surface velocity of the roll, vr

x2 2 x2 Δ 5 n ð : Þ v vr 2 5 56 hexitR 1 x allowing the friction factor to vary from the point of entry to exit. Since the numer- ator in Eq. (5.56) changes algebraic sign as x varies, the friction factor also changes sign at the neutral point. At this stage of the calculations, the constants a and b in Eq. (5.55) and the location of the neutral point, xn, are not known. The computations start by integrating Eq. (5.54), using a RungeKutta approach, for the roll pressure, starting at the entry with the appropriate boundary condition, and using assumed values for all three unknowns, a, b and xn. The boundary condition at the exit is satisfied by adjusting the location of the neutral point. Integral of the roll pressure distribution, thus obtained, over the contact length, is the roll separating force. By adjusting the constants a and b in Eq. (5.55) for the friction factor, repeating the integration, the calculated and the measured roll separating forces are compared and when satisfactory convergence is reached, the constants a and b and the location of the neutral point are deemed to have been determined. At this point uniqueness of the predictions is not considered. The friction factor, thus determined, varies from a negative value at the entry to the neutral point where it reaches zero. Beyond that the factor becomes positive. Its average value, mave, is indicative of frictional resistance. The roll torque is determined using the power, W_ , required to roll the metal. The power is obtained as the sum of the power for internal deformation and friction: ð ð 2σ W_ 5 pfmffiffiffi ε_ dV 1 2 τ Δv dS ð5:57Þ 3 V S where the friction stress is as given above (τ 5 mk) and the torque is then, for both rolls,

RW_ M 5 ð5:58Þ vr

The closeness of the calculated and measured roll torques indicates that both constants a and b have been determined correctly. 84 Primer on Flat Rolling

5.7.2 Calculations Using the Model The predictive abilities of the model are tested in two instances. First, the roll separating forces are compared to those measured in earlier cold rolling experiments, followed by comparing the calculated and measured roll pressure distributions.

5.7.2.1 Cold Rolling of Steel Selected portions of the data, obtained by McConnell and Lenard (2000), are used to determine the friction factor. In that publication low-carbon steel strips, having a : true stresstrue strain relation of σ 5 150ð11234εÞ0 251 MPa and measuring 1 3 23 3 300 mm3, have been cold rolled, using lubricants, containing various additives and having broadly varying viscosities. The rolls were made of D2 tool steel, hardened to Rc 5 63 and were of 249.8 mm diameter. Their surface roughness was Ra 5 0.2 μm. The objective was to determine the coefficient of friction by inverse calculations and by the use of Hill’s formula. The data, used in the present study, involves a lubricant with a kinematic viscosity of 19.83 mm2/s and a density of 861.6 kg/m3. The dependence of the average friction factor on the surface velocity of the roll and on the reduction is shown in Figure 5.7. As expected from previous studies, the friction factor decreases with both increasing rolling speed and reduction, affected by the same mechanisms that affected the coefficient of friction. As has been pointed out in several instances, as the speed increases, more oil is drawn into the contact, leading to lower friction; and as the loads increase, the viscosity increases, also leading to lower friction, at least in the boundary and in the mixed lubricating regimes.

1.2 Figure 5.7 The friction factor as m v ave = –1.607(red) – 0.00013 r +1.256 a function of the reduction and the roll surface velocity. 1.0 Reduction 0.8 14%

ave 0.6 m

20% 0.4 35% 46% 0.2 Cold rolling steel Viscosity = 19.83 mm2 /s 0.0 0 1000 2000 3000 Roll speed (mm/s) Mathematical and Physical Modelling of the Flat Rolling Process 85

Statistical modelling, using non-linear regression analysis, gives the dependence of the friction factor on the two process parameters, load and speed. The relation is shown in Eq. (5.59):

mave 521:607ðredÞ 2 0:00013 vr 1 1:256 ð5:59Þ where “red” is the reduction in decimals. The predictive ability of Eq. (5.59) was tested on data, not used in its determination. The data points from McConnell and Lenard (2000) were taken using a different lubricant whose viscosity was similar to the one used in developing Eq. (5.59), its value being 20.03 mm2/s. The roll surface velocity was 2308 mm/s. The results are given in Table 5.1. In Column 1 of the table the roll forces, as measured, are given. In Column 2, the forces, as calculated by the model, are shown while in Columns 3 and 4, the torques are indicated. Column 5 lists the average friction factors that resulted in the calculated forces and torques. The friction factors, as predicted by Eq. (5.59), are shown in Column 6, demonstrating that within the range of the process parameters of the experiments, the equation predicts the friction factor reasonably well. As mentioned, the calculations proceed until the measured and calculated roll separating forces and roll torques are close, to within a pre-specified tolerance. The accuracy of the computations is shown in Figure 5.8, which gives the ratios of the measured and estimated loads on the mill against the number of tests. All speeds, from 261 to 2341 mm/s, and all reductions, from 14% to 46%, are included in the figure. In general, one may conclude that a reasonable accuracy has been reached. At lower reductions the differences between the experimental data and the calculations are larger, due to the deviation from homogenous compression. The numbers fall to near unity as the loads increase. McConnell and Lenard (2000) determined the coefficient of friction using Hill’s equation (see Eq. (9.26)). These values are compared to the friction factor in Figure 5.9. A linear relationship is evident and the equation

5 : μ 1 : : mave 4 425 Hill 0 01 ð5 60Þ relates the two descriptions of frictional events in the roll gap. A relationship between mave and μ, for use in forging, has been suggested by Kudo (1960) in the form

Table 5.1 A Comparison of the Predictions of the Model and That of Eq. (5.59)

Roll Force, Roll Force Roll Torque, Roll Torque mave,by mave from Measured by the Measured by the the Model Eq. (5.59) (N/mm) Model (Nm/mm) Model

6086 3927 39.39 37.38 0.214 0.17 8223 8239 45.48 45.03 0.302 0.301 7782 46.29 46.29 44.01 0.403 0.417 6871 6833 41.18 41.4 0.303 0.31 86 Primer on Flat Rolling

1.6 Figure 5.8 The accuracy of the computations: the ratio of the Cold rolling of steel measured and calculated roll force Roll speed = 261 – 2341 mm/s Reduction = 14 – 46% and torque. 1.4 Torque Load

1.2

1.0

Measured/estimated load and torque Reduction 14%26% 35% 46% 0.8 0 5 10 15 20 25 Number of experiments

1.2 Figure 5.9 The friction factor Cold rolling of steel versus the coefficient of friction Roll speed = 261 – 2341 mm/s by Hill’s formula. Reduction = 14 – 46%

) 0.8 m

Friction factor ( factor Friction 0.4

m = 4.425 μ + 0.01

0.0 0.0 0.1 0.2 0.3 Coefficient of friction ( μ)

pffiffiffi m = 3 μ 5 ave ð5:61Þ pave=σ0

Using the data of the flat rolling tests leads to the conclusion that the values of mave, predicted by Eqs. (5.59) and (5.60), are close at low speeds, underscoring the importance of the speed in defining either the coefficient of friction or the friction factor. Mathematical and Physical Modelling of the Flat Rolling Process 87

1.0 Figure 5.10 The variation of the friction factor along the roll/strip Cold rolling of steel Roll speed = 2341 mm/s contact.

0.5

0.0 Reduction

Friction factor Friction 46% 35% –0.5 26%

14%

–1.0 0246810 Distance from exit (mm)

The variation of the friction factor over the contact length is shown in Figure 5.10, at a roll surface velocity of 2341 mm/s and reductions ranging from 14% to 46%. It is observed that the positive values at the exit and the negative values at the entry are quite similar in magnitude, indicating that the surfaces in contact have been well lubricated. High values of the friction factor at the exit would imply that the lubricant has not been carried through the location of maximum pressure.

5.7.2.2 Distribution of the Roll Pressure at the Contact While integrating the friction hill over the contact gives realistic magnitudes of the roll separating forces, its shape has been shown to be unrealistic in several publications, starting with the work of Siebel and Lueg (1933). The friction hill is the result of the 1D models’ traditional approach, in which the intersection of the curves, obtained by integrating the equations of equilibrium from the entry and from the exit, is taken as the location of the maximum pressure and of the neutral point. In the present work the shooting approach is followed in which integration proceeds from the exit and satisfaction of the boundary condition at the entry indicates success. The roll pressure distribution, thus obtained, is different from the sharp cusp of the usual 1D models. Comparisons of the predicted roll pressures to experimental data are given in Figures 5.11 and 5.12.InFigure 5.11, the measurements of Lu et al. (2002) are used. Employing pins and transducers in the work roll, the authors rolled low- carbon steel strips at 1000C and reported on the distribution of the roll pressures and the interfacial shear stresses over the contact zone. One of these experiments is used here and the measured and calculated interfacial stresses are shown in Figure 5.11. The test was conducted at 33 rpm and the 20 mm thick slab was reduced by 20%. The roll separating force is read off Fig. 3 of Lu et al. (2002) as 88 Primer on Flat Rolling

150 Figure 5.11 Comparison of the Low carbon steel roll pressures, as measured by 100 Friction stress Rolled at 1000°C Lu et al. (2002) and calculated 35 rpm (412 mm/s) by the present model: hot rolling 20% reduction 50 of steel.

–50

–100 Roll pressure –150

–200 Roll pressure and friction stress (MPa) Lu et al. (2002) Present model –250 0 5 10 15 20 25 Distance from exit (mm)

250 Figure 5.12 Comparison of the 1100 H 14 Al roll pressures, as measured by 200 Malinowski Rolled at 100°C et al. (1993) Malinowski et al. (1993) and 12 rpm (157 mm/s) 150 Present model calculated by the present model: 39.3% reduction 100 warm rolling of aluminium. 50 0

–50 Friction stress –100 –150 Roll pressure –200

Roll pressure and friction stress (MPa) –250 –300 0 4 8 12 16 20 Distance from exit (mm)

3000 N/mm and the roll torque as 66 Nm/mm. (Note that this figure is for a roll speed of 40 rpm, and no roll force data are given for a speed of 33 rpm.) The average flow strength of the steel, at the temperature and the strain rate used, is obtained by Shida’s (1969) relations (see Eqs. (8.10)(8.16)) as 124.74 MPa. Figure 5.11 indicates that the predicted pressures and shear stresses match the measurements quite well, indicating that use of the friction factor, as a variable in the contact zone, is quite realistic. Mathematical and Physical Modelling of the Flat Rolling Process 89

Roll pressure and interfacial shear stress distributions, obtained during warm rolling of 1100-H14 aluminium alloy strips, have been presented by Malinowski et al. (1993). A comparison of the predictions of the present model to the measurements by Malinowski et al. (1993) is shown in Figure 5.12. The 6.28 mm thick strip has been reduced by 39.3% at 100C at a roll speed of 12 rpm. The average flow strength of the metal is taken as 163 MPa in the calculations. The roll separating force was measured to be 3240 N/mm. The model calculated it to be 3293 N/mm. Examination of the two figures leads to the conclusion that allowing the friction factor to vary from the entry to the exit in the roll gap leads to realistic calculations of the roll pressure distribution.

5.8 Extremum Principles

Arguably, the most powerful of the approximate techniques available to analyse metal forming processes are the extremum principles, specifically the upper bound and lower bound theories. Both theories are formulated to estimate the power required for plastic forming. The upper bound theorem can be shown to predict the power that is always more than necessary. The lower bound is designed to lead to a power that is less than needed. Hence, since the upper bound theorem is the more conservative and the more useful of the two, it will be described in some detail. In spite of the widespread use of these theories in the treatment of problems of metal forming, the upper bound approach has rarely been used to treat the process of flat rolling.

5.8.1 The Upper Bound Theorem The upper bound theorem is described by Avitzur (1968) as follows: Among all kinematically admissible strain rate fields the actual one minimizes the expression ð rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ð σ 2 fm 1 J 5 pffiffiffi ε_ijε_ij dV 1 τ Δv dS 2 Tivi dS ð5:62Þ 3 V 2 SΓ Si

A strain rate field derived from a kinematically admissible velocity field is kinematically admissible. In Eq. (5.62), J is the externally supplied power; the first integral represents the power for internal deformation over the volume of the body (V), the second evaluates power due to shearing over surfaces of velocity discontinuities (SΓ) and the last term accounts for power supplied by body tractions over the surface, designated by Si. There are several concepts, mentioned above, that require careful definition. The term “kinematically admissible velocity field” implies the requirement that the velocity field must satisfy constancy of volume and all boundary conditions. The concept of velocity discontinuities is also mentioned above. As Avitzur (1968) explains, the velocity field within a deforming body need not be continuous. As shown in Figure 5.13, it is permissible to divide a body into several zones, in each of which a different set of velocities may exist. The boundary, at which the velocity 90 Primer on Flat Rolling

Discontinuity v(N) Figure 5.13 Velocity discontinuity within 1 a metal forming system. Zone 2 v1 v(N) = v(N) v(T) 1 2 1 (N) v 2 (T) v(T) ≠ v(T) v 2 1 2 v Zone 1 2 may be discontinuous, is indicated in the figure; this boundary may be located at the die/metal contact or it may be within the deforming metal. When the flat rolling process is analysed, the roll/strip contact surface is considered one of these surfaces of discontinuity. In Figure 5.13, two zones are identified, Zone 1 and Zone 2. The velocity of a N material point in Zone 1 is v1; its component normal to the discontinuity is v1 and T the component parallel to the surface of discontinuity is v1 . In Zone 2 the velocity is N v2; its component normal to the discontinuity is v2 and the component parallel to the T surface of discontinuity is v2 . As shown in the figure, continuum mechanics requires only that the velocity components normal to the surface be continuous. The tangential components need not be equal on the two sides of the surface of discontinuity, giving rise to a region of high shearing stresses. Using these concepts and the assumption that the velocity of the rolled metal in the deformation zone moves towards the intersection of lines, tangential to the rigid rolls at the entry, the upper bound on the power, required for plastic deformation of the rolled metal, becomes (Avitzur, 1968) 2 sffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 2 σ 2 4 hentry 1 hexit hentry entry exit J 5 pffiffiffi σfmvhexit ln 1 2 1 1 pffiffiffi 3 hexit 4 R hexit ð2= 3Þσ0 : 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 ð5 63Þ m h h 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ entry 2 1 2 tan21 entry 2 1A5 hexit=R hexit hexit

At the boundaries separating these zones only the normal velocity component must be continuous; the tangential component in one zone may be different than the corresponding component on the other side of the separating surface. This velocity discontinuity, of magnitude

Δ 5 ðtÞ 2 ðtÞ : v vI vII ð5 64Þ will create shearing stresses along the “surface of discontinuity” SΓ in Eq. (5.62) whose magnitude is given by mσ τ 5 pffiffiffi0 ð5:65Þ 3 where the magnitude of the friction factor m may be in the range 0 # m # 1. Mathematical and Physical Modelling of the Flat Rolling Process 91

The roll torque, for both rolls, may be obtained from the power as

R M 5 J ð5:66Þ U_

Tirosh et al. (1982) applied Avitzur’s (1968) upper bound approach to analyse cold rolling of viscoplastic materials at high speeds. The authors focused their attention on the effect of the speed, the inertia and the temperature dependence of the material’s resistance to deformation on the roll separating forces and on the roll torques. The Bingham material model22 was taken as the constitutive relation, defining the stress deviator tensor components in terms of the metals’ viscosity as

2η Sij 5 pffiffiffiffiffi ε_ij ð5:67Þ 1 2 ðk= J2Þ where the dynamic viscosity of the material is designated by η, k stands for the yield strength of the material in pure shear, ε_ij are the components of the strain rate tensor and J2 is the second invariant of the of the stress deviator tensor, defined as

1 J 5 S S ð5:68Þ 2 2 ij ij

For completeness, recall that the deviator stress tensor components are defined in terms of the stress components by the relation

1 S 5 σ 2 σ δ ð5:69Þ ij ij 3 kk ij where δij is Kronecker’s delta. In deriving the velocity field, the authors assumed that the arc of contact may be replaced by straight lines. The flow pattern then becomes a radially converging flow, leading to the statement that the resulting stress field is “unavoidably approximate in nature”. The coefficient of friction is taken to depend on the speed of the rolled strip at entry. They use the two relations given by Sims and Arthur (1952):

μ 5 0:08 expð20:51v0Þð5:70Þ for

0 # v0 # 0:25 m=s

22It is rare to see the Bingham model used in problems dealing with plastic forming of metals. One of the exceptions is the work of Haddow on the compression of a disk; see Haddow, J.B. 1965. On the compression of a thin disk. Int. J. Mech. Sci. 7, 657660. 92 Primer on Flat Rolling and

μ 5 23 20:038 : 10 expð4v0 Þð5 71Þ for

0:25 # v0 # 1:5m=s

The predicted roll separating force and torque values compared very well to the experiments of Shida and Awazuhara (1973) on cold rolling of steel strips. Further, increasing speeds were found to cause increasing compressive loads on the rolls and increasing tensile stresses within the strip, both of which were most likely caused by the strain rate dependence of the rolled metal, as predicted by the Bingham material model.

5.9 Comparison of the Predicted Powers

Several mathematical models have been introduced in this chapter: an empirical model, 1D models and a model based on the upper bound theorem. Two formulas have also been given to estimate the power required to reduce a strip of metal in a rolling mill; see Eqs. (5.8) and (5.9). Their predictions can now be compared and discussed in light of their assumptions. Consider the hot rolling of a low-carbon steel strip in a single stand roughing mill. Let the entry thickness of the slab be 20 mm and its width to be 2000 mm. Its resistance to deformation is taken to be 150 MPa. It is reduced by 30% using work rolls of 800 mm diameter, rotating at 50 rpm. The coefficient of friction is taken to be 0.2. The estimated power for plastic deformation, by Eq. (5.7), is obtained as 6115 kW. With four bearings of 400 mm diameter and a coefficient of friction of 0.01 in the bearings, the losses there are estimated to be 500 kW. Assuming further that the efficiencies of the motor and the transmission are both 0.9, the total power to drive the mill is obtained, by Eq. (5.8), as 8170 kW. Using the same numbers and Eq. (5.9), the total power is estimated to be 5350 kW. The upper bound theorem, designed to give conservative estimates, also allows the prediction of the power, required for plastic deformation of the strip. Equation (5.63), with the friction factor equal to 0.8, yields 3200 kW. Calculations using the refined 1D model (see Section 5.4) leads to a torque for both rolls of 524 Nm/mm, which when used to compute the power needed to reduce the strip, gives 5483 kW. Adding on the power losses in the four bearing and using 0.9 for the efficiency of the motor as well as the transmissions, one obtains 7262 kW. Based on these calculations23,itis recommended to use either the refined 1D model or the empirical model. A conservative number for the motor power is likely to result.

23It is to be noted that only one set of data was used in the exercise. Statistical analysis is necessary to prove the consistency of the predictions of any of the models. Mathematical and Physical Modelling of the Flat Rolling Process 93

The second number appears to be closer to reality, based on Robert’s calculations, mentioned above.

5.10 The Development of the Mechanical Attributes of the Rolled Strip

A detailed exposition of state-of-the-art of the evolution of the microstructure and the resulting mechanical attributes after hot flat rolling have been presented by Lenard et al. (1999). Carbon and alloy steels were included in the discussion and the predictions of the metallurgical model have been substantiated by comparing them to results obtained in the laboratory and in industry. In what follows, a revised and updated version, dealing with carbon steels, is described. Numerical examples are also given. The development of the draft schedule of the hot or the cold controlled rolling processes is usually performed off-line, using sophisticated mathematical models, which are composed of mechanical, thermal and metallurgical parts. The objective of the process is the creation of steel with small, uniform ferrite grains and as the HallPetch equation demonstrates, this will increase the strength of the rolled metal. There are a limited number of parameters whose magnitudes may be chosen relatively freely, although several of them are connected through mass conservation. The parameters include:

G the starting temperature; G the strain per pass; G the strain rate per pass (within strict limits); G the interstand tensions G the interstand and the pre-coiler cooling rates. The engineer must also consider the given parameters which cannot be altered: these involve the rolling mill and its capabilities. At this point the chemical composition of the steel is also given and the designer of the draft schedule must keep that in mind. The thickness of the scale on the surface may be controlled to some extent by the scale breakers. The interfacial frictional forces may also be controlled, also only to some extent, by the careful design of lubricant and coolant delivery. Lenard and Pietrzyk (1993) showed in a numerical experiment that while austenite grain size of a low-carbon steel is not affected by the coefficient of friction, the higher the surface heat transfer coefficient, the lower the grain diameter near the surface of the rolled steel, as expected by the higher surface cooling rates. There is some evidence that the metallurgical structure at the end of the rough rolling process doesn’t affect the subsequent events in any significant manner so the concern here is with the design of the passes on the finishing train and the cooling banks. The metallurgical events that affect the eventual attributes of the rolled metal are the hardening and the restoration processes. The hardening processes include strain, strain rate and precipitation hardening; the restoration processes include 94 Primer on Flat Rolling recovery and recrystallization, static, dynamic or metadynamic. These, in turn, are affected by the mechanical and thermal events. The three critical temperatures need to be known: 1. The precipitation start and stop temperature 2. The recrystallization start and stop temperature 3. The transformation start and stop temperature. In Lenard et al. (1999), the analysis of the thermal and mechanical events was accomplished using a finite-element model of the rolling process. Carbon and alloy steels were examined. A 2D model was used to determine the distributions of the mechanical and the thermal variables during and after rolling, and these data were then used to calculate the resulting grain sizes, amount of recrystallization and the mechanical properties after cooling. In the present treatment, the 2D model is replaced by one of the 1D models described above, and the output of this is then employed as the initial condition to determine the development of the post-rolling attributes. Since carbon steels only are considered here which experience no significant precipitation, the two mechanisms of interest are the processes of recrystallization and transformation. Their progress depends on the temperature and time available, in addition to the strain and the rate of strain in the pass. Sellars (1990) summarized the importance of modelling of the evolution of the microstructure:

G for a given composition of alloy, the high-temperature flow stress is influenced to a large extent by the microstructure. Proper prediction of the rolling force is possible only if the relevant microstructure is known and the microstructure present at the end of the rolling and cooling operations controls the product properties. Austenite, a face-centred-cubic structure (FCC), is formed after solidifying. It is designated by γ. On further cooling to the Ar3 temperature, the ferrite (α)grains appear and the steel reaches the two-phase region. The structure of the ferrite is body-centred-cubic (BCC). As the temperature drops to the Ar1 temperature, the transformation stops and the steel has become fully ferritic. Depending on the carbon content and cooling rates, other phases such as pearlite or bainite may appear as well. The two temperatures, Ar3 and Ar1, are affected by the chemical composition, pre-strain, cooling rate and initial austenite grain size; see Hwu and Lenard (1998). The costs associated with industrial trials are prohibitively high. The trials are expensive and difficult to control and monitor, and are necessarily constrained by the capabilities of existing plants. Laboratory simulation tests are unable to reproduce all conditions of industrial hot rolling24. Both the torsion and compression tests have limitations on the attainable strain rates, particularly in relation to strip or rod rolling. Further limitations are evident in torsion testing, in which the sample also develops different textures from those in flat rolling. On the other hand, the plane-strain and axisymmetrical compression tests cannot achieve the total strains of complete industrial rolling schedules. Hence, the use of off-line models which, in

24In spite of this limitation, laboratory simulation of the multi-stage hot rolling process yielded extremely useful results. Mathematical and Physical Modelling of the Flat Rolling Process 95 spite of the critique above, have been obtained from laboratory simulation tests is very useful, especially if the consistency and the accuracy of their predictions can be demonstrated. It must be realized at this point that the predictive abilities of these models have been substantiated by comparing their predictions to a selected number of measurements. Statistical analysis of the predictions, while necessary, is not widespread.

5.10.1 ThermalMechanical Treatment The two major objectives of the hot-rolling process are to control the dimensions of the product and to affect the attributes the metal will possess on cooling. For most commercial products in the steel industry, their external shapes are the result of hot deformation, such as hot rolling, while the necessary mechanical properties are obtained by alloying elements and heat treatment after deformation. However, metallurgical changes caused by hot deformation may result in additional beneficial effects on the mechanical properties of steels and sometimes can eliminate heat treatment after deformation. Thermomechanical processing is a technique to combine shaping and heat treating of steel. Controlled rolling is a typical example of thermomechanical processing in which the austenite is conditioned to produce a fine ferrite grain size. The development of controlled rolling approaches, used for carbon steels, is shown in Figure 5.14. There are four different techniques in the figure, in which “R” refers to rolling in the roughing mill and “F” indicates the finishing mill. In the first method, both roughing and finishing are completed at a temperature at which the steel is fully recrystallized. The resulting product will emerge as soft and ductile. In the second, the finish rolling process is interrupted and the steel is allowed to cool but the rolling process is still completed in the full recrystallization region. The result is a steel that is somewhat harder than in the first process. In both the third and the fourth strategies, the processing temperatures are further

1250 1250 R R 1150 γ R 1050 F 1100 $

C) R ° 1000 Recryst. F 950 850 100°C[Nb] γ 900 $ γ F F Unrecryst Unrecryst A 800 r γ+α γ+α 3

Temperature ( Temperature Ar A l r1 α+θ α+θ Holding Holding Holding

Time

Figure 5.14 Controlled rolling strategies. 96 Primer on Flat Rolling decreased and rolling is completed such that the steel is only partially recrystallized or, in the last step, finish rolling is performed in the two-phase region. Hodgson and Barnett (2000) review the practice of thermomechanical processing of steels. They list the processes in use in industry, classifying them as those carried out during the deformation process and those performed during the cooling phase, after deformation. These processes are:

G Conventional controlled rolling to improve strength and toughness G Recrystallization controlled rolling to achieve fine grains by affecting austenite grain growth and higher strength by precipitation hardening G Accelerated cooling, direct quenching, quench and self-tempering to affect the transformation mechanisms G Warm forming to affect the ferrite phase G Intercritical rolling of the austeniteferrite structure to increase the strength and toughness. The authors also write about how thermomechanical processes may develop in the future. They identify the processes that produce ultrafine grains by heavy plastic deformation (see Chapter 13) and by the application of magnetic fields. In another process (Hodgson et al., 1998), austenite grains are coarsened prior to deformation, followed by a small reduction and cooling. The result is a composite strip with ultrafine grains in the surface layers, possessing markedly increased strength.

5.10.1.1 Controlled-Rolling of CMn Steels Notch ductility and yield strength can both be improved by α grain refinement. Among other techniques for grain refinement, European mills utilized controlled low-temperature hot rolling in order to refine the α grains and to increase the toughness. The following features were generally applied in this controlled rolling process: Interrupting the hot-rolling operation when the slab had been reduced to the prescribed thickness, e.g. 1.65 times the final thickness. Recommencing hot-working when the slab has reached a prescribed temperature and finishing at temperatures in the austenite (γ) range, above the Ar3 but lower than the conventional finishing temperatures, e.g. down to 800C. The low-temperature finish rolling practice refines the γ grains, hence the transformed α grains. A considerable additional grain refinement can be achieved by rolling in the non-recrystallized γ region, where deformation bands increase nucleation sites for α grains. However, the temperature range for non-recrystallized austenite in CMn steels is relatively narrow, and this mechanism for grain refinement cannot be effectively utilized, due to the risk of getting into the two-phase region deformation.

5.10.1.2 Dynamic and Metadynamic Recrystallization Controlled Rolling In rod and bar rolling, using high strain rates (1001000 s21), short interpass times (between a few tens of milliseconds to a few hundreds of milliseconds), and large strains per pass (0.40.6) dynamic recrystallization has been found to occur. It has been proposed that under appropriate conditions, dynamic recrystallization also Mathematical and Physical Modelling of the Flat Rolling Process 97 occurs during strip rolling of niobium HSLA steels. The occurrence of dynamic recrystallization during simulated strip rolling of HSLA steels has been cited by several other authors. The results of an analysis of the events during strip rolling also indicated that dynamic recrystallization is happening during rolling of Nb. Dynamic recrystallization affects rolling loads and is reported to produce considerably finer ferrite grains (B3 μm) than those transformed from pancaked austenite (B7 μm). However, there are concerns regarding the validity and applicability of the results obtained in all of the above studies to the real mill practice, primarily due to the low strain rates employed in the experiments. Conventional controlled rolling relies on static recrystallization in the early stages of finish rolling to refine the austenite and pancaking of the austenite in the last stages to enhance ferrite grain nucleation during transformation. In contrast, dynamic recrystallization favours higher reductions in the first few stands to exceed the critical strain for the onset of dynamic recrystallization. Dynamic recrystallization controlled rolling leads to greater ferrite grain refinement through austenite grain refinement. Another advantage of the initiation of dynamic recrystallization during rolling is a marked reduction in roll forces and torques, which in turn trans- lates to savings in energy consumption and less roll wear. Also, the gauge accuracy will be enhanced due to the lower reductions required in the last stands. The only justification that the author found in the literature regarding the better grain refinement through dynamic recrystallization is the higher nucleation rate and formation of “necklace structure” during deformation. High density of grain nucleation already incorporated into the matrix expedites the post-dynamic recrystallization compared to static recrystallization, as no incubation time is required.

5.10.1.3 Effects of Recrystallization Type on the Grain Size Many different authors have attempted to develop models predicting grain sizes produced by static, dynamic and metadynamic recrystallization for different materials. The general observation, common in all these models, is that statically recrystallized grain size is a function of initial grain size, temperature and amount of strain, while dynamically and metadynamically recrystallized grain sizes are only a function of ZenerHollomon parameter, i.e. temperature and strain rate, in an inverse power law form. This indicates that increasing strain rate and decreasing rolling temperature lead to more grain refinement provided dynamic and metadynamic recrystallization are in place. Another common understanding is that rolling schedules with dynamic and metadynamic recrystallization produce finer final grain sizes compared to schedules with only static recrystallization. This idea is appeal- ing to the steel manufacturers to achieve further grain refinement.

5.10.1.4 Controversies Regarding the Type of Recrystallization in Strip Rolling The occurrence of dynamic recrystallization by strain accumulation during industrial hot strip rolling schedules has been questioned. It has been argued that the 98 Primer on Flat Rolling kinetics of static recrystallization approaches those of dynamic recrystallization as the strain increases. In addition, interpass times are generally much greater than deformation times. Hence, softening of the material during strip rolling may be due to enhanced static recrystallization. This controversy, in spite of its practical importance in terms of final mechanical properties and mill set-up, still remains. The physical proof of the possibility of dynamic recrystallization during strip rolling is notoriously difficult, since it requires extremely fast quenching of steel during deformation to freeze the structure and look for dynamically created grain nuclei. Most of the mill engineers do not believe in the possibility of dynamic recrystallization in any kind of steel during strip rolling. This belief has been reinforced by the fact that the possibility of dynamic recrystallization has not been taken into account in the conventional strip mill set-up and in control modules developed by General Electric and Westinghouse. In these control modules, which are in use in North America, it is assumed that the steel repeatedly goes through only work hardening during deformation and static softening during interpass times. This assumption may lead to erroneous roll force prediction if the steel actually softens in one or more stands instead of hardening. There are few examples of phenomenological models being in use in hot rolling. This is due to the complexity of the process and the lack of true phenomenological models to be able to quantitatively describe the microstructural changes occurring during rolling. Historically, there has been widespread application of empirical and semi-empirical models, based on simple statistical regressions of large data sets of plant measurements. The accuracy of these models depends on the availability of a general form of mathematical equations which can fit the data well. Data are usually non-linear and interrelated, which limits the applicability and accuracy of regression models. It is expected that the next decade will see extensive use of both the semi-empirical and heuristic models within the steel industry.

5.10.2 Conventional Microstructure Evolution Models Mathematical models of the evolution of the microstructure have been published in the technical literature. Sellars (1990), Roberts et al. (1983), Laasraoui and Jonas (1991a, b), Choquet et al. (1990), Hodgson and Gibbs (1992), Yada (1987), Beynon and Sellars (1992), Sakai (1995), Kuziak et al. (1997) and Devadas et al. (1991) presented various closed-form equations, describing the processes of recrystallization and grain growth. The restoration processes are time dependent and since in industrial hot rolling the strain rates are high, there is not enough time to trigger dynamic restoration of the work-hardened material; note that the time available is determined by the ratio of the strain and the strain rate, t 5 εpass=ε_pass. To demonstrate the validity of this statement, take a typical set of numbers in the first stand of a hot strip mill. Let the entry thickness be 15 mm and consider a fairly high, 50% reduction at a roll speed of 30 rpm. Take the diameter of the work roll as 500 mm. The average strain rate in the pass is then estimated to be 12 s21, the true strain is 0.69 and hence the pass takes place in about 55 ms, indicating that concurrent static recovery, accompanied Mathematical and Physical Modelling of the Flat Rolling Process 99 by static recrystallization, usually occurs after deformation. Both static recovery and recrystallization have been observed in austenite, although the extent of the former is rather limited. Some caution is introduced at this point. Biglou et al. (1995) considered industrial hot-rolling schedules of Nb-bearing microalloyed steels. Torsion testing was used to simulate the finish rolling schedules and some softening, attributed to metadynamic recrystallization, was found in the third stand. As well, accumulating strains have been thought to contribute to dynamic recrystallization.

5.10.2.1 Static Changes of the Microstructure The first step is to attempt to control the temperature at the entry to the first stand of the finishing mill. The success of this attempt is limited by the temperature of the just-hot-rolled strip, called the transfer bar, which is most likely waiting to exit the coil box. The temperature of the strip entering the coil box is controlled by the reheat furnace, held at 12001250C, and by the heat gains and losses in the roughing passes. The difference between the head and the tail temperatures is minimized while the steel is coiled up in the coil box within which the cooling rate is quite slow. The entry temperature therefore will depend on all of the above: the temperature in the reheat furnace, the gains and losses during rough rolling and the time spent in the coil box. The temperature above which recrystallization will occur is given by Boratto et al. (1988): pffiffiffiffiffiffiffiffiffi T 5 887 1 464 ½C 1 ð6445 ½Nb 2 644 Nb NRX pffiffiffiffiffiffiffi ð5:72Þ 1 ð732 ½V 2 230 ½VÞ 1 890 ½Ti 1 363 ½Al 2 357 ½Si so a low-carbon steel, containing 0.05% C, will recrystallize above 910C. The higher the carbon content, the higher the temperature above which recrystallization will be present. The entry temperature into the first stand is usually higher than 900C, unless ferrite rolling is contemplated. The extent of static recovery, defined as a softening process in which the decrease of density and the change in the distribution of the dislocations after hot deformation or during annealing are the operating mechanisms, is rather limited in hot-rolling processes. There is a general consensus that the maximum amount of softening during holding, attributable to recovery, is approximately 20%. The hot deformation of austenite at strains typically encountered in plate or strip rolling processes leads to significant work hardening, which is usually not removed by either dynamic softening processes or static recovery. This hardening creates a high driving force for static softening processes. The mechanism of these processes is explained clearly by Hodgson et al. (1993). Some of his observations are presented briefly below:

G A minimum amount of deformation (critical strain) is necessary before static recrystallization can take place. G The lower the degree of deformation, the higher the temperature required to initiate static recrystallization. 100 Primer on Flat Rolling

G The final grain size depends on the degree of deformation and to a lesser extent, on the annealing temperature. G The larger the original grain size, the slower the rate of recrystallization. During conventional preheating at high temperatures, incomplete recrystallization can take place at an early stage of the rolling process when small reductions are applied. The accumulation of strains then leads to full recrystallization in subsequent passes and, in consequence, the effect of the initial conditions on the down- stream final microstructure is very small and is usually neglected. The recrystallized volume fraction X is determined by the JohnsonMehl AvramiKolmogorov equation as a function of the holding time after deformation: "# t k X 5 1 2 exp A ð5:73Þ tX where t is the holding time, tX is the time for a given volume fraction X to recrystallize, A 5 ln(X) and k is the Avrami exponent. The majority of microstructure evolution models has been developed for X 5 0.5, indicating that tX in Eq. (5.73) represents the time for 50% recrystallization and the constant A 520.695. The most commonly used form describing the time for 50% recrystallization (t0.5X)is p q r s QRX t : 5 Bε D Z ε_ exp ð5:74Þ 0 5X RT where ε is the strain, D is the grain size prior to deformation in μm, Z is the 25 ZenerHollomon parameter , ε_ is the strain rate, QRX is the apparent activation energy for recrystallization, R is the gas constant and T is the absolute temperature. 219 Sellars (1990) gives B 5 2.5 3 10 , p 524, q 5 2, QRX 5 300,000 J/mole and the Avrami exponent, k 5 1.7. The exponents of the ZenerHollomon parameter and the strain rate are indicated to equal zero. Equation (5.74) implies that the time for 50% recrystallization decreases with increasing strains and grows with the grain size. The time required for 50% recrystallization is given as a function of the temperature of the pass in Figure 5.15, for a set of realistic strains and pre-pass grain sizes. It is clear that the steel will recrystallize quite fast at higher strains and at higher temperatures. The recrystallized grain size is reportedly sensitive to the temperature. The most commonly used form of the equation (Sellars, 1990) describes the dependence of the grain size after recrystallization (Dr) on the strain, the strain rate, the prior austenite grain size, the apparent activation energy and the temperature:

2Q D 5 C 1 C εmε_nDl exp d ð5:75Þ r 1 2 RT

Sellars (1990) gives the magnitudes of the constants and the exponents in Eq. (5.75) as follows: C1 5 0; C2 5 0.5; m 521; n 5 0; l 5 0.67; Qd 5 0, indicating that the

ÀÁ 25The ZenerHollomon parameter is defined as Z 5 ε_ exp Q=RT . Mathematical and Physical Modelling of the Flat Rolling Process 101

1E+5 Figure 5.15 The time required Strain Dγ (μm) for 50% recrystallization as a 1E+4 0.1 50 function of the temperature, the 0.1 100 strain and the initial grain size. 1E+3 0.1 150 0.5 50 1E+2 0.5 100 0.5 150 1E+1

1E+0

1E–1

1E–2 Sellars (1990) Time for 50% recrystallization (s) Time for 1E–3

1E–4 600 800 1000 1200 1400 1600 Temperature (°C) strain and the prior austenite grain size are the most significant variables. Roberts et al. (1983) provides somewhat different magnitudes. He gives C1 5 6.2; C2 5 55.7; m 520.65; n 5 0; l 5 0.5; Qd 5 35 000 J/mole. Note that Sellars’ equation excludes the dependence of the recrystallized grain size on the temperature but Roberts’ accounts for it. Both researchers indicate that the grain size after recrystallization is independent of the rate of strain. Choquet et al. (1990) and Hodgson and Gibbs (1992) also gave various magnitudes for the coefficients and the exponents. Laasraoui and Jonas (1991a,b) offer another relation for the recrystallized grain size in a CMn steel in terms of the strain and the pre-deformation austenite grain size, similar to that of Sellars (1990) with somewhat different exponent for the strain:

0:67 20:67 Dr 5 0:5D ε ð5:76Þ

The equations predict different grain sizes for the same initial conditions. Assuming an initial grain size of 50 μm, a strain of 0.30 and at a temperature of 800C, Sellars and Roberts predict a grain size of 23 μm, while Laasraoui and Jonas predict 15 μm. Increasing the initial size to 150 μmgives48μm by Sellars, 36 μm by Roberts and 32 μm by Laasraoui and Jonas. It is difficult to recommend one of these relations for use without some more data, preferably analysed statistically. The time for the completion of recrystallization is calculated from the Avrami equation for the recrystallized volume fraction, X (Eq. (5.73)). The constant A is taken to correspond to 50% recrystallization. In that case X 5 0.5, both t and tX are t0.5X and A 5 lnð1 2 0:5Þ 520:693. The time for X% of recrystallization is thus

1 lnð12XÞ k t 5 t ð5:77Þ A X 102 Primer on Flat Rolling

and the time for 95% recrystallization, when tX 5 t0.50 is

1 k lnð0:05Þ 1 t : 5 t : 5 4:3219k t : ð5:78Þ 0 95 lnð0:5Þ 0 50 0 50

Situations when partial recrystallization takes place during interpass times are common in the industrial rolling processes. Beynon and Sellars (1992) present an equation to calculate the grain size at the entry to the next pass:

ð4=3Þ 2 Dp 5 DrX 1 Dð12XÞ ð5:79Þ where D represents the grain size prior to deformation, Dp is the recrystallized grain size and X is the recrystallized volume fraction. Considering some of the grain sizes used above (800 C, D 5 50 μm, Dr 5 23 μm and 75% recrystallization), the average grain size of the rolled strip entering the next stand is predicted to be nearly 19 μm.

5.10.2.2 Dynamic Softening All softening processes that take place during plastic deformation are referred to as dynamic ones. These include dynamic recovery and dynamic recrystallization. The conventional models of dynamic recrystallization involve equations describing the critical strain, kinetics of dynamic recrystallization and the grain size after dynamic recrystallization. The critical strain at which dynamic recrystallization starts is given in terms of the ZenerHollomon parameter, the grain size and several constants:

p q εc 5 AZ D ð5:80Þ

Sellars (1990), considering a CMn steel, defines A 5 4.9 3 1024, p 5 0.15, q 5 0.5 24 and QDRX 5 312,000 J/mole. Laasraoui and Jonas (1991a) give A 5 9.82 3 10 , p 5 0.13, q 5 0 and QDRX 5 312,000 J/mole for a similar steel. A check of Sellars’ predictions of the critical strain is possible by considering the true stresstrue strain curve for a 0.05% C steel at 975C at a strain rate of 1.4 3 1023 s21, presented by Jonas and Sakai (1984). Reading the critical strain off the curve, one obtains εc 0:14. The grain size is taken as 65 μm and the predicted critical strain is then found to be 0.15. The equation describing dynamically recrystallized volume fraction is "# k ε2εc XDRX 5 1 2 exp B ð5:81Þ εp

where εp is the strain at the peak stress, usually calculated as εp 5 Cεc. Hodgson et al. (1993) give the coefficients in Eq. (5.81) for CMn steels as B 5 0.8, k 5 1.4 and C 5 1.25. The strain for 50% recrystallization is calculated as Mathematical and Physical Modelling of the Flat Rolling Process 103 23 0:28 0:05 51880 ε : 5 1:144 3 10 D ε_ exp ð5:82Þ 0 5X RT

The equation describing the grain size after dynamic recrystallization is

r DDRX 5 BZ ð5:83Þ

Sellars (1980) provides the coefficients in Eq. (5.83) for CMn steels as B 5 1.8 3 103 and r 520.15. The apparent activation energy is, as given above, 312,000 J/mole.

5.10.2.3 Metadynamic Recrystallization When dynamic recrystallization starts during the deformation and the recrystallized nuclei continue to grow after the deformation ends, the phenomenon is identified as metadynamic recrystallization. The equation describing the time for 50% metadynamic recrystallization is s Q t : 5 A Z exp ð5:84Þ 0 5 1 RT

The constants in Eq. (5.84) for carbonmanganese steels are given by Hodgson et al. (1993): A1 5 1.12, s 520.8, Qd (the activation energy of deformation in the ZenerHollomon parameter) 5 312,000 J/mole and Q 5 300,000 J/mole. The metadynamic grain size is (Hodgson et al., 1993)

u DMD 5 AZ ð5:85Þ where A 5 2.6 3 104 and u 520.25. The grain size during metadynamic recrystallization is calculated as the weighted average of the contributing grains:

DðtÞ 5 DDRX 1 ðDMD 2 DDRXÞXMD ð5:86Þ

In Eq. (5.86), XMD is the volume fraction after metadynamic recrystallization, calculated from the Avrami equation with k 5 1.5.

5.10.2.4 Grain Growth Following complete static or metadynamic recrystallization, the equiaxed austenite microstructure coarsens by grain growth. Nanba et al. (1992) and Hodgson and Gibbs (1992) presented the equation for CMn steels: 2 Q DðtÞn 5 Dn 1 k t exp g ð5:87Þ RX g RT 104 Primer on Flat Rolling

where DRX is the fully recrystallized grain size, t is the time after complete recrystallization, Qg 5 66,600 J/mole is the apparent activation energy for grain growth, 12 and n 5 2 and kg 5 4.27 3 10 are constants. The equation predicts a linear dependence of the grain growth on the time.

5.10.3 Properties at Room Temperatures Empirical relations, leading to the mechanical attributes of the rolled product, have also been developed and in what follows, these are reviewed in some detail. At the Ar3 temperature, given by

Ar3 5 910 2 310ðCÞ 2 80ðMnÞ 2 20ðCuÞ 2 15ðCrÞ 2 80ðMoÞ 1 0:35ðt 2 8Þ ð5:88Þ the austenite grains begin their transformation to ferrite grains. Equation (5.88) was developed for plate rolling and t represents the thickness of the plate.

5.10.3.1 Ferrite Grain Size The ferrite grain sizes may be estimated by the relation of Sellars and Beynon (1984):

5 2 : ε1=2 3 : 1 21=2 1 2 2 : 3 22 : Dα ð1 0 45 r Þ f1 4 5Cr 22½1 expð 1 5 10 DÞg ð5 89Þ where Dα is the ferrite grain size in μm, Cr is the cooling rate in K/s, D is the austenite grain size, also in μm and εr is the accumulated strain. When the cooling rate is taken as 20 K/s, the accumulated strain as 0.4 and the grain size as 50 μm, Eq. (5.89) predicts a ferrite grain size of 10 μm.

5.10.3.2 Lower Yield Stress

According to the HallPetch equation, the lower yield stress σy for a homogeneous microstructure is expressed as

20:5 σy 5 σ0 1 KyDα ð5:90Þ

where σ0 is the lattice friction stress, Ky is the grain boundary unlocking term for 23/2 high-angle grain boundaries, taken as 15.118.1 N mm and Dα is the ferrite grain size. Le Bon and Saint Martin (1976) presented a simple equation for the lower yield stress of carbon steels, in terms of the ferrite grain size:

20:5 σy 5 190 1 15:9ð0:001DαÞ ð5:91Þ Mathematical and Physical Modelling of the Flat Rolling Process 105

5.10.3.3 Tensile Strength Hodgson and Gibbs (1992) published a simple formula expressing the tensile strength of carbon steels with some alloying elements:

σu 5 164:9 1 634:7 ½C 1 53:6 ½Mn 1 99:7 ½Si 1 651:9 ½P ð : Þ 20:5 5 92 1 472:6 ½Ni 1 3339 ½N 1 11ð0:001DαÞ

The tensile strength of a carbon steel, containing 0.1% C and 0.6% Mn, and 10 μm ferrite grains, is estimated as 370 MPa. The yield strength of the steel is 349 MPa.

5.10.4 Physical Simulation In spite of the comments above regarding the limited abilities of physical simulation of thermalmechanical treatment, useful and detailed information can be obtained about the hot response of steels. While the number of publications in the field is too numerous to be reviewed here, many of the equations, given above, have been obtained as a result of simulation experiments: multi-stage compression and torsion tests have been found to be very useful. Some of the publications have been reviewed by Lenard et al. (1999); in one of these, Majta et al. (1996) performed multi-stage hot compression of a high-strength low-alloy steel, measured the yield strength after cooling and compared it very successfully to the measurements of a large number of researchers (Morrison et al., 1993; Coldren et al., 1981; Irvine and Baker, 1984). Several meetings have been devoted to the subject. One of the outstanding conferences was held in Pittsburgh in 1981, entitled “Thermomechanical Processing of Microalloyed Austenite”, edited by A.J. DeArdo, G.A. Ratz and P.J. Ray.

5.11 Miscellaneous Parameters and Relationships in the Flat Rolling Process

The mathematical models presented above take account of the contributions of the most significant variables and parameters. Several more phenomena are associated with the flat rolling process, however, and it is surprising that these aren’t usually included in the traditional analyses26. These are listed below; their definitions are given and simple formulae are presented for their evaluation.

5.11.1 The Forward Slip The relative velocities of the strip and the roll have been identified as having an effect on the rate of straining, lubrication, friction, scaling and the interfacial

26On-line and off-line models used in the rolling industry often include these parameters. 106 Primer on Flat Rolling forces. The forward slip, which is given in terms of the relative velocity, has, on occasion, been used to characterize tribological events. It is defined as

vexit 2 vroll Sf 5 ð5:93Þ vroll

In determining the exit velocity of the strip, one may use a variety of approaches. Optical techniques, which monitor the roll and the strip velocities, arguably offer the most accurate measurements. One often-used method is to mark the work roll surface using equally spaced, parallel lines, the separation of which is designated by lr. These lines make their impressions on the surface of the rolled strips and their distance on the strip, ls, may be measured using travelling microscopes. The forward slip can then be determined from these distances as

ls 2 lr Sf 5 ð5:94Þ lr

Using the idea of mass conservation or its equivalent, volume constancy, indicating that the volume of the rolled metal at any particular location is constant, it can be shown that the two formulas for the forward slip are identical. Researchers studying the development of surfaces as a result of flat rolling may well object to marking the roll surface as the lines may affect the interactions of the surfaces and the lubricants. The forward slip is often taken as a direct indication of frictional conditions in the roll gap. There are several formulas in the technical literature, connecting the coefficient of friction and the forward slip. These will be discussed in some more detail in Chapter 9.

5.11.2 Mill Stretch

When a certain exit thickness, hexit, is required, and the roll gap is to be set such that it is achieved, it is necessary to account for the extension of the mill frame, as well. The formula expressing the thickness that will result when the roll gap is set 0 to h1 is given below:

P h 5 h0 1 ð5:95Þ exit 1 S where P is the roll separating force in N and S is the mill stiffness, measured in N/mm. A typical value for the mill stiffness is 5 MN/mm; however, this would have to be ascertained for each particular mill.

5.11.3 Roll Bending Rowe presents a simple formula to estimate the maximum deflection at the centre of the work roll (Rowe, 1977), treating the roll as a simply supported beam, Mathematical and Physical Modelling of the Flat Rolling Process 107 loaded at its centre. The formula accounts for the deflections due to shear loading, as well:

Pl3 Pl Δ 5 1 0:2 ð5:96Þ EI AG where Δ is the maximum deflection of the roll at its centre in mm, P is the roll force in N and l is the length of the roll, bearing to bearing in mm. The elastic modulus is designated by E and taken as 200,000 MPa and the shear modulus by G, equal to 86,000 MPa. The cross-sectional area of the roll in mm2 is A and I is the moment of inertia27 of the roll’s cross-section in mm4. Roberts (1978) developed a more fundamental formula for the maximum deflection of the roll, based on the double integration method. The effects of both the normal and shear loads are included here as well:

PL2ð5L 1 24 Þ PL Δ 5 1 1 c 1 ð5:97Þ 6πED4 2πGD2 where L1 is the centre-to-centre distance of the bearings and c is half of the bearing length. The equation presented by Wusatowski (1969), similar to Eq. (5.97), includes several more geometrical parameters. He gives the roll deflection at the centre, including the effects of shear: 8 < 1 Â Δ 5 P 8L3 2 4L b2 1 b3 1 64c3ðD4=d4 2 1Þ :18:8ED4 1 1 9 ð : Þ = 5 98 1 ÂÃ 1 L 2 0:5b 1 2cðD2=d2 2 1 GD2π 1 ; where b is the width of the rolled strip. Simple calculations indicate interesting magnitudes of the deflection of a work roll at its centre. In a laboratory experiment, using a 250 mm diameter steel roll, with the bearings 400 mm apart, and reducing a 25 mm wide low-carbon steel strip by 50%, the roll separating force was measured to be 8000 N/mm. The roll deflection is then obtained as 0.337 mm, a little over 0.1% of the roll diameter, indicating no need for crowning. Similar calculation for an industrial case yields quite different numbers. Considering hot rolling of a 2000 mm wide low-carbon steel strip and reducing it with 1000 mm diameter rolls, leads to roll separating forces of 2434 MN, depending on the temperature and the reduction. Equation (5.96) now indicates that the roll deflection will be in the order of 26 mm. If no crowning is used, the cross-section of the rolled strip will not be satisfactory.

27A more suitable name for I is “second moment of the area”. 108 Primer on Flat Rolling

5.11.4 Cumulative Strain Hardening The cumulative effect of sequential straining on the resistance of the material to deformation is well understood. In what follows, a simple procedure to estimate this effect in multi-pass flat rolling is presented. In the example, a strip of steel is to be rolled in two consecutive passes. In the first pass its thickness at the entry is hentry and its thickness at the exit is hexit;1: hentry ε1 5 ln ð5:99Þ hexit;1 and the average flow strength is obtained by integrating the true stresstrue strain relation over the strain in the pass

ðε 1 1 σ 5 σðεÞ ε ð : Þ 1 ε d 5 100 1 0

In the second pass the entry thickness is hexit;1 and the exit thickness is hexit;2 so the strain in the second pass is hexit;1 ε2 5 ln ð5:101Þ hexit;2 and the total strain experienced by the strip so far hentry εtotal 5 ln ð5:102Þ hexit;2

The average flow strength in the second pass is then determined by integrating over the strain in the second pass:

ðε 1 total σ 5 σðεÞ ε ð : Þ 2 ε 2 ε d 5 103 total 1 ε1

The steps described above are illustrated by an example in which a low-carbon steel strip is reduced, first by a strain of 0.1, followed by another pass creating the same magnitude of the strain. The true stresstrue strain relation of the steel, in : MPa, is σ 5 100ð11182:02εÞ0 355 so the average flow strength in the first pass is obtained as 218 MPa and in the second, 326 MPa; Figure 5.16 shows the details.

5.11.5 The Lever Arm In the empirical model of the flat rolling process (see Section 5.2), the roll torque was calculated by assuming that the roll separating force acts halfway between the Mathematical and Physical Modelling of the Flat Rolling Process 109

400 Figure 5.16 The true 0.355 σ = 100(1+ 182.02ε) MPa stresstrue strain curve and the average flow strengths in the two passes. 300

200 σ 2 True stress (MPa) True σ 100 1

ε ε 1 cum 0 0.00 0.05 0.10 0.15 0.20 0.25 True strain entry and the exit, making the ratio of the torque for both rolls and the roll separating force the lever arm equal to the projected length of the contact, L. As mentioned above, while the predictions of the roll separating forces by the empirical model are reasonably accurate and consistent, those of the roll torque are not quite as good. The reason is found in the assumption of the magnitude of the lever arm. In an effort to develop a better appreciation of the lever arm, the data of McConnell and Lenard (2000) are employed once again. These include approximately 250 experiments. The ratios of the measured roll torques and the roll separating forces are calculated, yielding the actual lever arm, a. Then, the ratio of the projected contact length and the lever arm are calculated, resulting in

L 5 3:983 3 1025P 1 0:946 ð5:104Þ a r

The results are shown in Figure 5.17, where the ratio, L/a, is plotted versus the roll separating force. All data are included, with the roll speeds varying from a low of 260 mm/s to a high of 2400 mm/s and the reductions varying from a low of 12% to a high of 50%. The lever arm is obtained as 1 6 30%. Lundberg and Gustaffson (1993) estimate the lever arm in edge rolling to be close to unity.

5.12 How a Mathematical Model Should Be Used

For successful predictions of the rolling variables while using any of the available mathematical models, knowledge of the accurate magnitudes of the coefficient of friction or the friction factor and the metal’s resistance to deformation is absolutely necessary. While it is clear that without them the predictions become essentially 110 Primer on Flat Rolling

2.0 Figure 5.17 The dependence of the ratio of the contact length Contact length = 3.983 × 10–5 P +0.947 Lever arm r and the lever arm on the roll 1.6 separating force.

1.2

0.8 Increasing speeds Increasing reductions

Contact length/lever arm Contact length/lever Cold rolling of steel strips 0.4 Various lubricants Reductions from 12 – 50% Roll speeds from 260 – 2400 mm/s 0.0 0 2000 4000 6000 8000 10000 Roll separating force (N/mm) useless, their determination may cause almost insurmountable difficulties in many instances. The following steps are then recommended.

5.12.1 Establish the Magnitude of the Coefficient of Friction Conduct a carefully controlled set of rolling tests and measure the roll separating force as a function of the rolling speed and the reductions. If hot rolling is studied, the temperature also becomes one of the independent parameters and its effect also needs to be taken into account. Its measurement is not easy. Arguably the best approach may be to embed thermocouples in the strip to be rolled, even though the stress concentration this causes may affect the magnitude of the reduction. Optical pyrometers may be used instead at both the entry and the exit and the average of their readings may give the average surface temperature quite closely. Once the data are collected, using one of the models of the rolling process and employing the inverse method, determine the coefficient of friction such that the measured and the calculated roll forces agree. This should then be followed by using non- linear regression analysis to develop a relationship of the coefficient of friction as a function of the speed and the reduction and possibly the temperature. In further modelling, this equation may then be used with good confidence.

5.12.2 Establish the Metal’s Resistance to Deformation Use the plane-strain compression test to determine the material’s resistance to deformation. If a plane-strain press is not available, a uniaxial tension or compression test will be acceptable; if both are possible, choose the compression test. The experimental difficulties increase if one deals with hot deformation. In an ideal case, isothermal tests should be conducted. Mathematical and Physical Modelling of the Flat Rolling Process 111

Regardless of whether hot or cold rolling is considered, the effects of friction and temperature rise should be removed from the data. If experimental equipment, needed to determine the material’s strength, is not available, one has no choice but to rely on published data, the perils of which have been pointed out elsewhere. For hot rolling, Shida’s (1969) equations are recommended, but checking them against the data of Suzuki et al. (1968) would be helpful.

5.13 Conclusions

In this chapter mathematical models that describe the mechanical and the metallurgical phenomena during flat rolling of metals were discussed. Modelling of the rolling process of strips and thin plates were examined exclusively so a 1D treatment was considered to be satisfactory. The potential objectives of modelling were listed first. The models presented were classified according to their level of sophistication. These started with an empirical model and were followed by several well- known 1D models, including a 1D elasticplastic model that takes careful account of the elastic entry and exit regions as well as the elastic flattening of the work roll. In another 1D model the coefficient of friction was replaced by the friction factor which was allowed to vary along the contact region from the entry to the exit. Based on past experience, the factor was taken to depend on the roll pressure, the relative velocity of the roll and the strip and the distance along the contact. The roll pressure distribution was calculated by using the shooting method: numerical integration of the equation of equilibrium was started at the exit and the location of the no-slip point was adjusted to meet the boundary condition at the entry. The extremum theorem the upper bound formulation was also used to estimate the power needed to roll a strip. Following the mechanical models, the development of the microstructure during and after the rolling pass was described. Empirical relations that can be used to estimate the metallurgical parameters during rolling of low-carbon steels were listed. A few numerical examples indicated the predictions of the equations. The chapter was concluded by presenting several parameters and relationships in the rolling process which are not usually included in mathematical models: the forward slip, mill stretch, roll bending and the effect of cumulative strain hardening. Recommendations regarding the choice of a model to analyse the flat rolling process may now be made. As long as the roll diameter to thickness ratio is much larger than unity, a criterion that is satisfied well in strip and thin plate rolling, the “planes remain planes” assumption is valid and 1D analyses are satisfactory. The choice of the model depends on the objectives of the user. The guiding principle should be to use the simplest model that satisfies the need. Regardless of the model chosen, the accurate knowledge of three parameters is necessary: the coefficient of friction, the heat transfer coefficient and the metal’s resistance to deformation. 112 Primer on Flat Rolling

If the roll separating force only is needed, the empirical model of Section 5.2 is adequate. If the force and the torque are needed, a 1D model should be used. For somewhat more confidence in the predictions, the refined 1D model should be employed. If, in addition to the above, the temperature changes, roll flattening, required power and the metallurgical events are to be determined, the refined 1D model is recommended. See Section 5.4. If the distributions of the normal pressures and the interfacial shear stresses on the work roll are wanted, the coefficient of friction or the friction factor should be expressed as a variable from the entry to the exit of the roll gap. The shape of the coefficient of friction distribution may be based on existing data. If rolling of thicker plates is to be analysed, the recommendation is to use the finite- element technique. 6 An Advanced Finite Element Model of the Flat, Cold Rolling Process1

6.1 Introduction

As long as the mill frame is taken to be rigid, the three important components of the metal rolling system include the work rolls, the rolled strip and their interface. While a complete mathematical model of the cold, flat rolling process should account for the behaviour of all three, published studies often include several simplifications. The most significant of these concerns the modelling of the frictional resistance at the interface between the work roll and the rolled strip, the location where the transfer of energy is accomplished. While it is well known that the coefficient of friction and the friction factor depend, in a significant manner, on the relative velocity, the interfacial pressure, the viscosity and the surface roughness, a model describing the relationship of these parameters to the frictional events has still not been provided. In most of the literature the coefficient of friction is treated as a free parameter whose magnitude is determined by inverse calculations. While most analyses treat the rolled metal as isotropic (Dixit and Dixit, 1996), a rare exception is the work of the same authors (Dixit and Dixit, 1997). Hartley et al. (1979) commented that earlier finite element studies of metal forming have used unrealistic frictional conditions. Hence, the authors included a layer of elements at the interface whose properties controlled friction. They applied their model to ring compression and found excellent agreement of their predictions and experiments. Liu et al. (1985) analysed the elasticplastic cold rolling process employing the layer of elements at the surface of contact, following Hartley et al. (1979). In their analysis, the HuberMises flow criterion and the PrandtlReuss relationship with strain hardening were included. The work rolls were taken to be rigid. While the calculated roll pressure distributions were close to those measured by Al-Salehi et al. (1973) for low reduction, there were large differences when higher reductions were considered. Li and Kobayashi (1982) expressed the frictional shear stress at the roll/rolled metal contact as a function of the relative velocity. They analysed the problem of

1This chapter is based on “Lemezhengerle´sKı´se´rleti Vizsga´lata e´sVe´geselemes Modelleze´se” by Z. Be´zi, G. Kra´llics, M. Szucs˝ and J. Le´na´rd, published in Hungarian in Anyagme´rno¨ki Tudoma´ny, Vol. 37, 2012, pp. 2333; with permission.

Primer on Flat Rolling. DOI: http://dx.doi.org/10.1016/B978-0-08-099418-5.00006-8 © 2014 Elsevier Ltd. All rights reserved. 114 Primer on Flat Rolling cold rolling of rigid-plastic strips between rigid work rolls. They also compared their predicted roll pressure distributions to the data of Al-Salehi et al. (1973). As with the study of Liu et al. (1985), some of the comparisons were successful and some were not. Tieu and Liu (2004) used four pin-transducer combinations embedded in the work roll to monitor the variation of the interfacial normal and shear stresses while rolling aluminium alloy and carbon steel strips. They confirmed earlier studies (Siebel and Lueg, 1933; van Rooyen and Backofen, 1957; Banerji and Rice, 1972; Hum et al., 1996, among others), showing that the coefficient of friction varies from the entry to the exit. As well, they showed that while the use of the “friction hill” in the calculations of the roll pressure distributions offers a useful shortcut in analyses, actual roll pressures possess rounded tops, not saddle points. They indicated that the forward slip increases as the reduction is increased. Further, they demonstrated that both the coefficient of friction and the forward slip drop as the relative velocity between the roll and the strip increases. Jiang et al., (2003) analysed cold, flat rolling of steel and copper strips. They employed a three-dimensional, rigid-plastic finite element model and used a frictional shear stress as suggested by Kobayashi et al. (1989). The shear stress therefore depended on a friction factor, the relative velocity, the yield stress of the rolled metal, a coefficient of friction and two positive constants, defined for both the forward and the backward slip zones. The authors write “... a major handicap to produce an accurate and reliable model ... is the lack of a well-defined friction boundary condition ...”, a statement with which the present author agree whole- heartedly. It is regretted therefore that no specific information regarding the friction model used was provided. There was no reference to the deformation of the work rolls, so it may be inferred that they were taken to be rigid. The predictions of the model compared well to measured roll pressures, the origins of which, however, were not identified. In more recent studies Dvorkin et al. (1997) and Jiang et al. (2004) analysed cold rolling of rigid-plastic strips. In contrast, Gudur and Dixit (2008) considered the behaviour of elasticplastic metals. The effects of both rigid and elastic rolls were studied by Shangwu et al. (1999). The least well-understood phenomena are still the frictional conditions at the interface. In the present work, the coefficient of friction is taken to depend on the relative velocity of the roll and the rolled strip. Muniz (2007) used what is arguably the most complete finite element model of the flat rolling process. He took account of elastic work rolls, elasticplastic rolled strips and a coefficient of friction which was dependent on the relative velocity of the roll and the rolled strip. He presented the distributions of the parameters stress, strain, rate of strain, temperature in the rolled strips and in the rolls. Beyond a statement that the results confirm others’ studies, no experimental substantiation was given. As observed from the review of the literature, a complete model that accounts for the behaviour of all three of the rolling system’s components and is fully An Advanced Finite Element Model of the Flat, Cold Rolling Process 115 substantiated by comparing its predictions to experimental data, is not yet available. Hence, the objective of the present chapter is to include all three components of the rolling system in a finite element model. The elastic deformation of the work roll, the elasticplastic rolled strip and the velocity-dependent coefficient of friction are to be modelled. The magnitude of the average coefficient of friction is determined by an inverse approach, such that the computed roll separating force, the roll torque and the forward slip match the experimental values. Following the validation of the predictive accuracy of the model, the distributions of the roll pressures, interfacial shear stresses and the equivalent strains in addition to the changes of the radius of the deformed roll are documented.

6.2 Modelling the Flat Rolling Process

Pa´czelt writes that the most important aspect of the formulation of a mathematical model is the close correlation of the equations of continuum mechanics and the physical events (Pa´czelt, 1999). In the finite element approach use of the virtual work hypothesis leads to this correlation. Starting with the known initial conditions at time t, and using the Updated Lagrange approach, iteration leads to the state at time t 1 Δt (Dixit and Dixit, 2008): ð t1Δtσ δ t1Δtε t1Δt 5 t1Δt : ij ð ijÞd V R ð6 1Þ t1ΔtV

t1Δt t1Δtσ where R is the virtual work of external forces, ij is the Cauchy stress δ t1Δtε tensor at time t and ð ijÞ is the variation of the virtual deformation. Since the solution at time t 1 Δt is unknown, Eq. (6.1) is transformed to reflect events at time t: ð t1Δt δ t1Δt t 5 t1Δt : tSij ð teijÞd V R ð6 2Þ tV

t1Δt δ t1Δt where tSij is the second PiolaKirchhoff stress tensor and ð teijÞ is the GreenLagrange virtual deformation. The equation of motion is now expressed following Bathe (1996): ð ð ð Δ δ Δε t 1 tσ δ Δη t 1 tσ δ Δε t 5 t1Δt : t Sij ðt ijÞd V ij ðt ijÞd V ij ðt ijÞd V R ð6 3Þ tV tV tV

Employing the constitutive relations

σ 5 EP L : d ij Cijkl dekl ð6 4Þ 116 Primer on Flat Rolling the linear form of the equations of motion is then written as (Bathe, 1996) ð ð ð t EP Δ δ Δ t 1 tσ δ Δη t 1 tσ δ Δ t 5 t1Δt Cijkl t ekl ðt eijÞd V ij ðt ijÞ d V ij ðt eijÞd V R tV tV tV ð6:5Þ EP tσ where Cijkl is the constitutive tensor and ij is the Cauchy deformation tensor at η time t. The terms eij and ij denote the linear and the non-linear changes in the deformation, as referred to the configuration at time t:

1 1 Δe 5 ð Δu ; 1 Δu ; Þ; Δη 5 ð Δu ; 1 Δu ; Þð6:6Þ t ij 2 t i j t j i t ij 2 t k i t k j

The Updated Lagrange method takes detailed account of all non-linear effects, including the material’s non-linearity and large deformations. The virtual work hypothesis then leads to the non-linear discretized algebraic system of equations

t Δ 1 t 5 t1Δt : ½Ktf ug ff g fFgð6 7Þ which is solved by the NewtonRaphsoniteration,stepbystep(MSC-MARC, 2011). In the present study, the MSC-MARC finite element software is used to analyse the cold rolling process. The material is taken to be elasticplastic. In the first step, the work rolls are treated as rigid. In the second step, the computations are repeated, and the elastic deformation of the rolls is accounted for. Figure 6.1 shows

Figure 6.1 Discretization of the roll. An Advanced Finite Element Model of the Flat, Cold Rolling Process 117 the isoparametric elements (QUAD 4/11) and the refinement of the net near the contact region. A total of 4785 four-node elements are used. The deformation of the rolled strip is assumed to be two-dimensional. The rolling process is taken to be symmetrical, leading to significant simplifications. The rolled material remains isotropic and it obeys the HuberMises yield criterion. Young’s modulus is 69 GPa and Poisson’s ratio is 0.3. The constitutive relation in the plastic region is

0:143 kf 5 270ð1177εÞ ð6:8Þ where kf is the flow strength of the rolled metal and ε is the effective strain. Young’s modulus of the roll material is 210 GPa and the Poisson’s ratio is 0.3. As discussed in Section 5.7, the coefficient of friction in the contact region is given in terms of the relative velocity between the roll and the rolled strip:

2 Δν μ 5 μ arctan ð6:9Þ 0 π C μ where 0 is the maximum Coulomb coefficient of friction and the relative speed between the roll and the rolled strip is Δν 5 νh 2 νt. The subscript h refers to the roll and t indicates the rolled strip. C is a parameter, taken to be C 5 vh=20. Note that Eq. (6.9) allows for the change of algebraic sign of the coefficient of friction at the neutral point.

6.3 Experiments

Cold rolling experiments were conducted on a STANAT experimental rolling mill. The mill was driven by a 12 kW DC motor, through a four-speed gearbox. The top surface speed of the rolls was 1100 mm/s. The surfaces of the 150 mm diameter, 203 mm long tool steel rolls were sandblasted, creating random roughness of Ra 5 0.3 μm. Load cells, placed over the bearing blocks of the top roll, monitored the roll separating forces. The roll torques were measured by transducers placed in the drive spindles. The signals from two photodiodes at the exit led to the exit speed of the rolled strip, and hence to the forward slip. The rotational speed of the top roll was monitored by a tachometer. All data were collected using a DASH 16A/D board and Labview. The 6061 T6 aluminium strips were 1.0 mm thick, 25 mm wide and 300 mm long. They contained 1% Mg, 0.6% Si, 0.3% Cu and 0.2% Cr. Their initial surface roughness, both along and across the strips was Ra 5 0.2 μm. Prior to the tests, all strips were cleaned using acetone. Mineral seal oil was used as the lubricant, 10 drops on each surface, spread using a roller. Its kinematic viscosity at 40C was 4.4 mm2/s and at 100C, 1.53 mm2/s. Its density was 850 kg/m3 at 40C (Lenard, 2004). 118 Primer on Flat Rolling

6.4 Results

The details of the experiments, used in the calculations, are given in Table 6.1. The reduction and the rolling speed are given in the first two columns. The coefficient of friction, as calculated by Hill’s formula (see Eq. (9.26)), is shown in the third column, while the forward slip, the roll separating force and the roll torque for both rolls follow. The first set of results of the calculations is shown in Figures 6.2 and 6.3.In both figures the distance along the roll gap is given in the abscissa while the roll pressures and the interfacial shear stresses are indicated on the ordinate. As expected, increasing reductions result in higher roll pressures and interfacial shear stresses. As in the one-dimensional models, the location of the maximum roll pressure coincides with the location of the change of the algebraic sign of the shear stress. Accounting for elastic deformation of the rolls leads to larger length of contact, resulting in significantly larger roll pressures on the rigid rolls. It is clear that accounting for the elastic deformation of the rolls leads to larger contact length and zones of deformation. The effect of the speed of rolling is also observed in Figures 6.2 and 6.3. As the speed increases, the pressures drop, as do the interfacial shear stresses caused by the improving tribological situation at the contact. Recall that increasing the speed of lubricated rolling consistently resulted in lower coefficients of friction; see Figures 5.28 and 5.29 in Section 5.5.2. The distribution of the total equivalent plastic strain in the rolled strip, undergoing 60% reduction at a speed of 0.796 m/s, is shown in Figure 6.4. The grey area indicates the roll. The changes of the roll radius as a function of the reduction and the rolling speed are shown in Figure 6.5. The distributions of the roll pressures are also

Table 6.1 Experimental Details Used in the Calculations

Pass Details Experimental Results

Reduction Roll Speed Coefficient of Forward Roll Force Roll Torque (%) (m/s) Friction (Hill) Slip (%) (N/mm) (N m/mm)

17.86 0.376 0.246 4.161 2750.89 10.88 19.39 0.184 0.204 3.446 2720.13 10.55 17.84 0.875 0.219 3.577 2613.71 8.96 43.04 1.036 0.143 21.262 4615.95 25.84 43.49 0.312 0.135 21.704 4484.31 26.19 44.87 0.833 0.108 1.821 4067.00 25.38 60.19 1.753 0.164 9.919 7173.67 44.10 60.04 0.349 0.150 10.666 6663.80 43.04 62.35 0.796 0.129 3.764 6151.85 41.30 An Advanced Finite Element Model of the Flat, Cold Rolling Process 119

700

600

500

400 p_m r = 17% v = 0.184 m/s τ_m p_r 300 τ_r p_m v = 0.875 m/s Stress (Mpa) 200 τ_m p_r τ 100 _r

–100 0123 456 x 1 (mm)

Figure 6.2 The roll pressures (p) and the interfacial shear stresses (τ) at 17% reduction and various rolling speeds. The index “m” refers to rigid rolls while “r” indicates elastic rolls.

1400 p_m r = 60% v = 0.175 m/s τ_m 1200 p_r τ_r 1000 p_m v = 0.796 m/s τ_m p_r 800 τ_r

600

Stress (Mpa) 400

200

–200 0246810 x 1 (mm)

Figure 6.3 The roll pressures (p) and the interfacial shear stresses (τ) at 60% reduction and various rolling speeds. The index “m” refers to rigid rolls while “r” indicates elastic rolls. 120 Primer on Flat Rolling

1.08 0.94 0.8 0.67 0.54 0.4 0.27 0.13

Figure 6.4 The plastic deformation of the strip, showing the equivalent strain distribution. The grey area indicates the roll.

r v = 0.184 m/s p = 519 MPa 0.08 = 17% max 0.375 522 0.875 556 r = 44% v = 0.104 560 0.312 635 0.06 0.833 675 r = 60% v = 0.175 856 0.349 986 0.796 1088 0.04 (mm) R Δ

0.02

0.00 010203040 50 60 Contact length (mm)

Figure 6.5 The changes of the roll radius and the length of contact at various speeds and reductions.

indicated in the figure and it is evident that they occur at the largest changes of the radii. The iso-stress distribution on the work roll is shown in Figure 6.6. It is noted that the most highly stressed location is not at the contact zone but is located a few millimetres from the surface. Figure 6.7 indicates the changes of the stress components as a function of time. While the calculations were performed for a reduction of 60.04%, similar results have been obtained in other cases, as well.

6.5 Comparison of the Experimental and Numerical Results

Tables 6.2 and 6.3 list the comparison of the measured and the computed data. The results of the calculations are given in Table 6.2. In each case the coefficient An Advanced Finite Element Model of the Flat, Cold Rolling Process 121

Figure 6.6 Iso-stress distribution in the work roll.

171 228 285 341 114 398 455 58 507

(MPa)

600 σ e 400 σ 11 σ 200 22 σ 12 0

–200 Stress (MPa) –400

–600

–800 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Time (s)

Figure 6.7 The maximum stress components in the work roll as function of the time of contact. 122 Primer on Flat Rolling

Table 6.2 Results of the Calculations, Using the Finite Element Method

Rigid Roll Elastic Roll

Forward Roll Roll Torque Coefficient Forward Roll Roll Torque Coefficient Slip (%) Force (N m/mm) of Friction Slip (%) Force (N m/mm) of Friction (N/mm) (N/mm)

3.62 2751.34 10.74 0.242 2.80 2754.03 9.96 0.158 3.32 2721.34 11.11 0.170 1.99 2726.32 10.39 0.105 3.25 2616.91 10.22 0.190 2.36 2615.19 9.51 0.112 5.59 4611.84 28.24 0.128 4.78 4650.39 26.73 0.105 4.76 4466.21 27.62 0.115 3.83 4481.28 26.17 0.094 1.69 4063.77 25.92 0.083 2.01 4146.33 25.18 0.071 10.77 7121.51 49.39 0.150 8.98 7157.77 45.49 0.127 9.48 6581.45 45.71 0.137 7.86 6677.90 43.04 0.116 7.77 6107.88 43.85 0.117 5.93 6143.75 41.26 0.102

Table 6.3 Comparison of the Measured and the Computed Data

Rigid Roll Elastic Roll

Roll Force Roll Torque Forward Slip Roll Force Roll Torque Forward Slip Difference Difference Difference Difference Difference Difference (%) (%) (%) (%) (%) (%)

0.016 2 1.287 2 0.541 0.114 2 8.456 2 1.361 0.044 5.308 2 0.126 0.228 2 1.517 2 1.456 0.122 14.063 2 0.327 0.057 6.138 2 1.217 2 0.089 9.288 6.852 0.746 3.444 6.042 2 0.404 5.460 6.464 2 0.068 2 0.076 5.534 2 0.079 2.128 1.508 1.951 2 0.788 1.828 2 0.727 11.995 0.851 2 0.222 3.152 2 0.939 2 1.236 6.204 2 1.186 0.212 0.000 2 2.806 2 0.715 6.174 4.006 2 0.132 2 0.097 2.166

of friction was chosen such that the difference between the measured and the computed roll separating forces is minimized. It is noted that when the elastic rolls are used the coefficients are significantly less than the ones obtained with the rigid rolls. The percentage differences between the finite element method results and the experimental data are given in Table 6.3. As expected, the roll forces are computed well when either rigid or elastic rolls are used. The roll torque differences are much less when the elastic deformations of the work rolls are accounted for. An Advanced Finite Element Model of the Flat, Cold Rolling Process 123

6.6 Conclusion

The parameters of the cold strip rolling process were measured and calculated, using an advanced finite element model. Elasticplastic deformation of the rolled strip was taken into account. Two approaches were followed: in the first, the roll was assumed to remain rigid, while in the second, its elastic deformation was considered. The coefficient of friction was expressed as a function of the relative velocity between the roll and the rolled strip. In each calculation it was chosen such that the difference between the measurements and the calculations was minimized. The coefficient of friction dropped as the rolling speed was increased. As well, while the roll forces were determined accurately by using either rigid or elastic rolls, the accuracy of the roll torque computations increased when the elastic deformation of the rolls and the elasticplastic deformation of the strip were accounted for.

6.7 Acknowledgements

The financial assistance of The Centre of Excellence for the Development of the Quality of Higher Education in the University of Miskolc, TA´ MOP-4.2.1.B-10/2/KONV-2010-0001 project, is gratefully acknowledged. 7 Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling1

7.1 Introduction

Local buckles are shape defects resulting from the local increase in strip elongation along a line at any position across the strip width. They can be caused by three irregular factors in a cold mill. These are 1. The hot rolled coil ridges or high spots, in which case the elongation will increase locally at the position of the ridge during cold rolling and local buckles, also called ridge- buckles, may occur on the cold rolled strip. 2. The feed stock local yield stress drop, caused during casting and hot rolling when the microstructure variation across the width resulted in areas that contain coarse grains having a lower yield stress which will elongate more during cold rolling, thus producing local buckles. 3. The work roll crown ridges in the last stand which may be induced by clogging of the roll cooling nozzles or improper maintenance of the cold mill work rolls. In what follows, a mathematical model to simulate strip shape and the local buckle’s formation under the above-mentioned three irregular conditions in the cold rolling process is described. The shape model includes five underlying models as follows: The first model is a strain rate based finite difference 3D strip deformation model The second model calculates the roll stack deflection and work roll 3D flattening The third model is a work roll thermal crown model The fourth model is an analytical model to determine the tension distribution between stands and after the last stand The fifth model is a local buckling model to predict the local buckling threshold and the shape of the local buckles. By combining the above five models, the software is capable of simulating the formation of local buckles and calculating their limiting values. This chapter intro- duces the models and the simulation results.

1Contributed by Dr. Yuli Liu, chief process engineer and manager of Process Engineering and Development, Quad Engineering Inc. This chapter is a combination of the following publications: Liu et al. (2005, 2007a,b, 2011a,b, 2012). The models have also been presented in Vladimir, B. (Ed.), 2009. Flat-Rolled Steel Processes. CRC Press, Ginzburg.

7.2 Strain Rate Based Strip 3D Deformation Model

7.2.1 Analysis Model of Deformation Zone Figure 7.1 shows the strip deformation zone and the stresses acting on an element in the roll bite. The nomenclatures of geometrical dimensions and stresses are also shown in the figure. 7.2.2 Strip Thickness Distribution in the Roll Bite The strip thickness profile in the roll bite along the rolling direction is assumed to be parabolic, given by x 2 hðx; yÞ 5 h2ðyÞ 1 ½h1ðyÞ 2 h2ðyÞ ð7:1Þ ld where

ld 5 contact length between the roll and strip; h1(y) 5 strip entry thickness profile in the width direction; h2(y) 5 strip exit thickness profile in the width direction. The entry thickness profile in the first stand is equal to the hot rolled band profile h0(y). Under normal rolling conditions, the hot band profile is fitted with the following function: y 2 y 4 h ðyÞ 5 a 1 a 1 a ð7:2Þ 0 0 2 b=2 4 b=2

z x p τ y

σ σ σ σ σ σ x +d x σx σ y +d y y 1 2 y OOy h x h (y) h +dh dx n 1 h (y) x 2 l d

y τxy +dτxy

σy +dσy τ σ +dσ y x x σx dy τ xy y σy

x O dx b x b+Δb

Figure 7.1 Sketch of strip 3D deformation model. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 127 where

a0, a2, a4 5 fitting coefficients; b 5 the width of the strip. If the hot band has a ridge, another parabolic term is added to the above function within the ridge width to form the ridged profile: 8 ! <> 2 y2yr h0ðyÞ 1 Δhr 1 2 yr 2 wr # y # yr 1 wr h ðyÞ 5 wr ð7:3Þ 0r :> h0ðyÞ y , yr 2 wr or y . yr 1 wr where

h0r(y) 5 hot band profile with a ridge; Δhr 5 maximum height of the ridge; yr 5 y coordinate of the ridge centre; wr 5 half width of the ridge.

7.2.3 Strain Rate and Velocity Field Model It is assumed that the ratio of transverse strain rate to longitudinal strain rate, β,is constant along the rolling direction

ξ βð Þ 5 y ð : Þ y ξ 7 4 x

With this assumption and applying the constant volume principle, the strain rate field (ξx, ξy, ξz, ηxy) and the velocity field (vx, vy) in the roll bite become

1 @h ξ v ð7:5Þ z h @x x β ξ 52 ξ ð7:6Þ y 1 1 β z

1 ξ 52 ξ ð7:7Þ x 1 1 β z

@v @v η 5 x 2 y ð7:8Þ xy @y @x

ð1=11βÞ vx 5 vnðhn=hÞ ð7:9Þ ð y β 1 @h 5 ð : Þ vy 1 β @ vx dy 7 10 0 1 h x 128 Primer on Flat Rolling

where hn and vn are thickness and longitudinal velocity distributions at the neutral plane, respectively.

7.2.4 Yield Criterion and Plastic Flow Equation Considering the yield stress variation along width direction, the HuberMises yield criterion and the LevyMises plastic flow equations can be expressed by the stress field (σx, σy, σz, τxy) as follows:

2 2 2 2 ðσx 2σyÞ 1 ðσy 2σzÞ 1 ðσz 2σxÞ 1 6τxy 5 6½kðyÞ ð7:11Þ and

ξ ξ ξ η H x 5 y 5 z 5 xy 5 ð7:12Þ σx 2 σm σy 2 σm σy 2 σm τxy kðyÞ where

1 σ 5 ðσ 1 σ 1 σ Þ m 3 x y z H 5 effective shear strain rate; k(y) 5 yield stress in shear, a function of the width direction in the case of local yield stress variation:

8 ! <> 2 y2yr k0 1 Δk 1 2 yr 2 wr # y # yr 1 wr kðyÞ 5 wr ð7:13Þ :> k0 y , yr 2 wr or y . yr 1 wr where

k0 5 average yield stress in shear in the roll bite, considered to be a constant value; Δk 5 maximum value of local yield stress increase (positive) or drop (negative). Combining the yield condition, the plastic flow equations and the strain rate equations leads to the stressstrain rate relationships 8 ð Þ > σ 5 σ 1 2k y ξ 1 ξ > y x ð2 y zÞ > H > < 2kðyÞ σ 5 σ 1 ðξ 1 2ξ Þ : > z x H y z ð7 14Þ > > > kðyÞ :> τxy 5 η H xy Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 129

7.2.5 Surface Friction Model

The surface friction τf is divided into components in the longitudinal and transverse directions, τx; τy:

vx 2 vn vsx τx 5 τf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 τf ð7:15Þ 2 2 1 2 vs ðvx vnÞ vy

vy vy τy 5 τf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 τf ð7:16Þ 2 2 1 2 vs ðvx vnÞ vy

( μp μp # kðyÞ τf 5 ð7:17Þ kðyÞ μp . kðyÞ where p 5 specific roll force; μ 5 friction coefficient; vs 5 relative speed of the roll and the strip surfaces; vsx 5 longitudinal component of the relative speed of the roll and the strip surfaces.

7.2.6 Longitudinal Equilibrium Equation Using slab analysis, the longitudinal equilibrium equation is (Tozawa, 1984):

@ðσ hÞ @h x 1 p 2 2τ 5 0 ð7:18Þ @x @x x

Introducing the stressstrain rate relationships and the surface friction equations into the longitudinal equilibrium equation and making further simplifications, the following equations are obtained: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @σ μ ð Þ μ @ H2 2 η2 x 1 2 vsx σ 2 k y 2 vsx 2 h xy 1 β 5 μ # x 2ð2 Þ 0 p kðyÞ @x hvs hH vs @x 1 1 β 1 β ð7:19Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @σ @ ð Þ H2 2 η2 x 2 vsx 1 h k y xy 1 β 5 μ . : 2k 2ð2 Þ 0 p kðyÞð7 20Þ @x hvs @x Hh 1 1 β 1 β

The above equations are solved numerically. 130 Primer on Flat Rolling

7.2.7 Entry and Exit Tension Stress Models The entry and exit tension stress models developed by Liu et al. (2005) are employed. The entry tension stress is then ! E v E v ðh =h Þð1=11βÞ σ 5 σ 1 s x1 2 1 1 σ 5 σ 1 s n n 1 2 1 1 σ 1 1 2 υ2 0 1 2 υ2 0 1 s vx1 1 s vx1 ð7:21Þ where

σ1 5 average entry tension stress; vx1 5 entry velocity distribution due to plastic deformation in the roll bite; vx1 5 average entry velocity; σ0 5 residual stress of the incoming strip; Es 5 elastic modulus of the strip; 2 υs 5 Poisson’s ratio of the strip . The exit tension stress is "# E v E v ðh =h Þð1=11βÞ σ 5 σ 1 s 1 2 x2 5 σ 1 s 1 2 n n 2 ð7:22Þ 2 2 2 υ2 2 2 υ2 1 s vx2 1 s vx2 where

σ2 5 average exit tension stress; vx2 5 exit velocity distribution due to plastic deformation in the roll bite; vx2 5 average exit velocity.

7.2.8 Transverse Equilibrium Equation The finite difference form of the transverse equilibrium equation is used to avoid the discontinuities of the partial derivatives in the y direction: Δσ Δτ h y 2 h xy 2 2τ 5 0 ð7:23Þ Δy Δx y

To get better convergence, a simplified form of the transverse equilibrium equation is adopted here based on the method of weighted residuals to get an approximate solution (Ishikawa, 1987): ð ld Δσ Δτ y 2 xy 2 τ 5 ð : Þ h Δ h Δ 2 y dx 0 7 24 0 y x

2Note that the symbol for Poisson’s ration here is υ; it is used to avoid confusion with the symbol for the velocities, ν. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 131 with the boundary condition at the strip edge y 5 b/2 ð ld hσy dx 5 0 ð7:25Þ 0

7.2.9 Numerical Scheme The above equations are solved using finite difference methods. The flow chart of the calculation procedure is shown in Figure 7.2.

Assume neutral plane profile

Mesh the deformation zone

Calculate thickness distribution and its derivatives

Assume lateral spread ratio

Calculate strain rate field and velocity field

Calculate stress field from entry to exit

Calculate residual values of transverse equilibrium equations Modify spread ratio

N Equilibrium equation satisfied? Modify neutral plane profile Y

N Longitudinal stress profile = tension profile at exit?

Y Calculate roll force distribution and lateral spread

Out

Figure 7.2 Flow chart of strip 3D deformation model. 132 Primer on Flat Rolling

The stress fields are calculated from the entry to the exit by solving the longitudinal equilibrium equations (7.19) and (7.20) with the initial values determined by Eq. (7.21). There are two main iterative loops in the calculation procedure. The inner loop calculates the lateral spread ratio by satisfying the transverse equilibrium equation (7.24) and the strip edge boundary condition (7.25). The outer loop determines the neutral plane profile by matching the exit tension stress calculated by the longitudinal equilibrium equation with the tension stress determined by the exit tension model (7.22).

7.3 Work Roll Thermal Crown Model

An axisymmetric 2D finite difference model is developed to calculate the work roll temperature field and thermal crown. Since there are several different heat transfer zones along the circumferential direction, the weighted average of the heat fluxes during one revolution is used in the calculation. The heat transfer coefficient formulations by Ginzburg (1997), Tseng and Wang (1996), Steden and Tellman (1987) and Devadas and Samarasekera (1986) are adopted to calculate the heat fluxes in different zones. The roll cooling nozzles are controlled independently; therefore, the roll thermal crown ridge effect can be simulated by turning off one or two nozzles. The governing equation and solving procedure used by Ginzburg (1997) are followed. Extensive measurements of work roll temperature field and work roll thermal crown were carried out at Dofasco’s CPCM3. The work roll thermal crown model was tuned and verified using the data collected during the measurements.

7.4 Roll Stack Deformation Model

The roll stack deformation model for a 4-high mill, considering possible work roll crown ridge and kiss rolling condition4, is shown in Figure 7.3.

7.4.1 Roll Separating Forces The roll separating forces at the drive side and operator side may be different. They are obtained by balancing the force and moment of the roll system:

PL 5 P1 1 Pk 1 FL 1 FR 2 PR ð7:26Þ " 1 Xm P 5 l ðP 1 P 1 F 1 F Þ 1 p Δy y R ð 1 Þ b 1 k R L cj j j Lb 2lb j51 0 1 0 13 ð7:27Þ L 2 L L 2 L 2 F @l 1 w bA 1 F @l 1 b wA5 L w 2 R w 2

3CPCM 5 coupled pickle line and tandem cold mill. 4“Kiss” indicates that the work rolls are in contact. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 133

Z Pr Pl lb Lb lb

O Db Y

Contact length

o D w Y p P P k k

b Fr Fl lwLw lw

Figure 7.3 Roll stack deformation model. where

PL 5 roll separating force at the left side (drive side); P1 5 total rolling force; Fl 5 work roll bending force on the left side (drive side); Fr 5 work roll bending force on the right side (operator side); PR 5 roll separating force on the right side (operator side); Pk 5 total contact force between the top and the bottom work rolls; Lb 5 back-up roll barrel length; Lw 5 work roll barrel length; lb 5 back-up roll neck length; lw 5 work roll neck length; pc 5 combined work roll contact pressure and rolling pressure; 8 < P rolling zone 5 pc : pk work roll to work roll contact zone 0 non-contact zone

P 5 roll pressure per unit width; pk 5 work roll contact pressure per unit width; yj 5 coordinate of the jth element in the y direction (roll axial direction); Δyj 5 the length of the jth element; m 5 number of elements. 134 Primer on Flat Rolling

7.4.2 Roll Equilibrium Equations Two independent roll equilibrium equations are used to determine the inter-roll pressure distribution:

Xm qjΔyj 2 PL 2 PR 5 0 ð7:28Þ j51

Xm qjΔyjyj 2 lbðPR 2 PLÞ 2 PRLb 5 0 ð7:29Þ j51 where q 5 the contact pressure between the back-up roll and the work roll.

7.4.3 Roll Deflection Equations

With the formulation of influence functions, the back-up roll deflection Zb is expressed as

Xm Xm yi Zbi 5 αbijqjΔyj 1 αbLPL 1 αbRPR 1 αΔbjqjΔyj 5 Lb 5 j 1 j 1 : Xm ð7 30Þ 1 αΔbjqjΔyj ði 5 1; 2; ...; mÞ j51 and the work roll deflection Zw is expressed as

Xm Xm Zwi 5 αwijpcjΔyj 2 αwijqjΔyj 2 αFLFL 2 αFRFR j51 j51 ð7:31Þ yi 1 Δwz 1 Kwz ði 5 1; 2; ...; mÞ Lc where

Lc 5 contact length between rolls; Δwz 5 work roll rigid skewing parameter; Kwz 5 work roll rigid movement; α 5 influence functions for different cases; i 5 suffix counter, denoting the position of an element in the axial direction.

7.4.4 Roll Deformation Compatibility Equation The deformation compatibility between the back-up roll and the work roll is

Zbi 1 δbwi 5 Zwi 1 CRbi 1 CRwi ði 5 1; 2; ...; mÞð7:32Þ Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 135 where

CRb, CRw 5 back-up and work roll crown radii, relative to the middle of the roll barrel; δbw 5 contact deformation between the work roll and the back-up roll (Lian and Liu, 1995). Roll crown profile consists of roll initial crown and roll thermal crown. If there is no crown ridge, the roll initial crown profile is assumed to be parabolic. Another parabolic term is added to the roll initial crown profile if a roll crown ridge exists.

8 ! <> 2 y2yr CwðyÞ 1 ΔCwr 1 2 yr 2 wr # y # yr 1 wr C ðyÞ 5 wr ð7:33Þ Rw :> CwðyÞ y , yr 2 wr or y . yr 1 wr where

CRw(y) 5 work roll initial crown include a ridge; Cw(y) 5 parabolic work roll initial crown; ΔCwr 5 maximum height of the work roll crown ridge.

7.4.5 Roll Gap Profile The roll gap profile is

u u d d hi 5 hz 1 ðZ 2 Z Þ 1 ðZ 2 Z Þ 1 2ðδ i 2 δ zÞ wi wz wz wz ww ww ð7:34Þ 2 2ðCRwi 2 CRwzÞði 5 1; 2; ...; mÞ where

hz 5 exit thickness at the centre; u 5 the top work roll; d 5 the bottom work roll; z 5 centre of the strip; δww 5 work roll flattening (Ishikawa, 1987).

7.4.6 Calculation Procedure The kernel part of the roll deformation calculation solves the equation system consisting of roll deformation compatibility and roll equilibrium equations. The flow chart of the calculation procedure is shown in Figure 7.4. Considering the inter-roll pressure peaks due to the work roll crown ridge, iteration to determine the inter-roll pressure distribution is used in the program. This also determines the inter-roll contact length at the same time. Another iteration loop calculates the kiss pressure, if kiss rolling occurs. 136 Primer on Flat Rolling

Assume kiss pressure = 0

Calculate combined work roll pressure

Assume WR & BUR contact length & pressure distribution

Assemble compatibility & equilibrium equations

Modify contact length, pressure distribution Solve compatibility & equilibrium equations

Pressure N distribution converge?

Y Calculate work roll deflection and flattening

Calculate work roll gap profile

Y N Work roll gap> = 0?

Kiss pressure = 0 Iteratively calculate kiss pressure

N Kiss pressure converge? Modify kiss pressure

Out

Figure 7.4 Flow chart of roll stack deformation model.

7.5 Stress Unloading Model

The analytical stress unloading model of Yukawa et al. (1987) is adopted to calculate the residual stress. The residual stress σxr is

Ð = b2 2 σ 2 υ σ 1 σ 0 f x2 sð y2 z2Þgh2 dy σxr 5 σx2 2 υsðσy2 1 σz2Þ 2 Ð ð7:35Þ b2=2 0 h2 dy Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 137 where

h2 5 strip thickness at the exit of the roll bite; b2 5 strip width at the exit of the roll bite.

7.6 Local Buckling Threshold Model

The local buckling threshold model is derived based on conventional thin plate buckling theory with a discrete method (Liu et al., 2011a). Applying the governing equation of thin plate bending (Timoshenko and Gere, 1989) to a thin plate with thickness h and width b (h«b), subject to a residual stress distribution with a compressive force per unit width Nx 52hσx0 and varying with y across the width, leads to

@2w @4w @4w N @2w 1 2 1 52 x ð7:36Þ @x4 @x2@y2 @y4 D @x2 where w 5 vertical deflection of the plate; D 5 flexural rigidity of the plate given by

Eh3 D 5 ð7:37Þ 12ð1 2 υ2Þ

E 5 Young’s modulus and υ is Poisson’s ratio. Introducing the non-dimensional compressive stress

N b2 k 5 x ð7:38Þ π2D the governing equation becomes

@4w @4w @4w gkπ2 @2w 1 2 1 52 ð7:39Þ @x4 @x2@y2 @y4 b2 @x2 where g is a multiplier of the force Nx. In order to deal with an arbitrary distribution of residual stresses, the discrete method is used by dividing the plate into 2m elements along the width (Bush et al., 2001). For each element, a solution is sought of the form

nπx w ðx; yÞ 5 sin f ðyÞði 5 1; 2; ...; 2mÞð7:40Þ i a i where n is the number and a is the length of half-waves of the deflected curve. 138 Primer on Flat Rolling

Substituting Eq. (7.40) into Eq. (7.39) results in d4f nπ 2 d2f nπ 4 π2 nπ 2 i 2 2 i 1 2 gk f 5 0 ði 5 1; 2; ...; 2mÞð7:41Þ dy4 a dy2 a i b2 a i

2 2 Letting A 5 ðnπ=aÞ and K 5 gkiðπ =b Þ, Eq. (7.41) is rewritten as

d4f d2f ÂÃ i 2 2A2 i 1 A4 2 KA2 f 5 0 ði 5 1; 2; ...; 2mÞð7:42Þ dy4 dy2 i

Assuming that the residual stress ki is constant within one element, the solution to Eq. (7.42) becomes (Timoshenko and Gere, 1989)

2α α 5 iy 1 iy 1 β 1 β 5 ; ; ...; : fiðyÞ Ci1e Ci2e Ci3 cosð iyÞ Ci4 sinð iyÞði 1 2 2mÞð7 43Þ

α β where i and i are real and positive and are given by pffiffiffiffi 2 1=2 αi 5 ½A 1A K ði 5 1; 2; ...; 2mÞð7:44Þ pffiffiffiffi β 5 2 2 1 1=2 5 ; ; ...; : i ½ A A K ði 1 2 2mÞð7 45Þ

The solution function (7.43) applies to each element. Each solution function contains four arbitrary constants Cij ðj 5 1; 2; 3; 4Þ, thus, the general solution across the width of the strip is represented by 2m solution functions, involving 8m arbitrary constants. The continuity condition is imposed on f ; df=dy; d2f=dy2; d3f=dy3 between the elements. There are two boundary conditions at the left and right sides of the plate. The total conditions are 8m. The resulting 8m linear equations are homogeneous. In matrix form they are

X8m MIJ CJ 5 0 ðI 5 1; ...; 8mÞð7:46Þ J51

The coefficient matrix M is derived based on the continuity conditions and boundary conditions. Buckling occurs when a non-zero solution for the coefficients CJ is possible, and the condition for this is that the determinant of the coefficient matrix MIJ vanishes,

jMIJj 5 0 ð7:47Þ

For any chosen non-dimensional residual stress ki, the coefficient matrix M is a function of the multiplier g. The values of g such that the determinant is zero are obtained using the bisection method. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 139

Once the multiplier g is calculated, the critical residual stress σc is obtained from the maximum value of the input residual stress σmax:

σc 5 gσmax ð7:48Þ

Even though the buckling threshold can be calculated using the above model, the local buckling amplitude of the shape of post-buckling cannot be determined since Eq. (7.46) are homogeneous and have multiple solutions. To calculate the local buckling shape, the energy method in large deflection buckling theory is used.

7.7 Local Buckling Shape Model

The function describing the strip shape after local buckling is assumed to have the following format: 8 2 3 2 = # , > 0 b 2 y c < π 2 π ; 5 4 2 2 ðy cÞ5 x # # 1 : wðx yÞ > f 1 cos sin c y c Lm ð7 49Þ :> Lm Ln 0 c 1 Lm , y # b=2 where f is the nominal amplitude of the buckles to be determined. The widths in the x and y directions are denoted by Lm and Ln. The parameter c indicates the edge of the buckle. Introducing Y 5 y 2 c; X 5 x; α 5 2π=Lm; β 5 π=Ln leads to Y0 52b=2 2 c and Yb 5 b=2 2 c and Eq. (7.49) is rewritten as a function of Xand Y 8 < 0 Y0 # Y , 0 ; 5 2 α β # # : wðX YÞ : f ð1 cos YÞsin X 0 Y Lm ð7 50Þ 0 Lm , Y # Yb

The bending strain energy Vw and the neutral plane strain energy Vc of the strip are calculated following Lian et al. (1987): ()"# ð ð 2 2 D Ln Yb @2w @2w @2w @2w @2w V 5 1 2 2ð1 2 vÞ 2 dX dY w @ 2 @ 2 @ 2 @ 2 @ @ 2 0 Y0 X Y X Y X Y ð7:51Þ

! ð ð 2 Eh Ln Yb @u 1 @w 2 5 ε 1 1 ð : Þ Vc 0 @ @ dX dY 7 52 2 0 Y0 X 2 X

where ε0 is the initial residual strain and u is the displacement function in the longitudinal direction. 140 Primer on Flat Rolling

The governing equation of the neutral plane displacement u is obtained from the equilibrium equation and the stressstrain relationship (Lian et al., 1987)

@2u @w @2w 1 5 0 ð7:53Þ @X2 @X @X2

Solving Eq. (7.53) and considering the boundary conditions results in

1 u 52 f 2βð12cos αYÞ2 sin 2βX 1 ε X ð7:54Þ 8 1 where ε1 is the shortened strain of the neutral plane after the strip lost stability. The total strain energy V of a strip of length L is

L V 5 ðVw 1 VcÞð7:55Þ Ln

Integrating Eqs. (7.51) and (7.52) and substituting them into Eq. (7.55): πDf 2LÂÃEhL 3πf 2β2ε 35πf 4β4 V 5 α4 12α2β2 13β4 1 f 2π2β2S 1S 1bε2 1 1 1 4α 2 0 1 1 2α 64α ð7:56Þ where ð Yb X2m 5 ε2 5 ε2 Δ S1 0 dY 0i b Y0 i51 ð Xim 1 Lm 1 S 5 ε ð12cos αYÞ2dY 5 ε ð12cos αY Þ2Δb 0 2π2 0 2π2 0i i 0 i5i0 and the initial residual strain is ε0i 5 σx0i=E ði 5 1; 2; ...; mÞ. Based on the energy principle, the buckled strip reaches its stable state when the total strain energy V is the minimum value:

@V @V @V @V @V 5 0; 5 0; 5 0; 5 0 and 5 0 @ε1 @f @Ln @Lm @c

From these conditions, the following equations are derived

3πf 2β2 ε 52 ð7:57Þ 1 4αb Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 141

ðD=EhÞðα4 1 2α2β2 1 3β4Þ 1 2παβ2S f 2 5 0 ð7:58Þ ðβ4=4Þðð9π=αbÞ 2 ð35=4ÞÞ pffiffiffi 4 3 L 5 L ð7:59Þ n 2 m h2 3 24L4 4L2 @S 3L2ε 35f 2 2 n 2 n 1 2L2 0 1 n 1 1 5 0 ð7:60Þ 2 2 4 2 n @ π2 12ð1 v Þ 2 Lm Lm Lm 2 64

@S 2L2 0 5 0 ð7:61Þ n @c where

@S Δb Xi1 2πðy 2 cÞ 0 52 ε ð 2 Þ i 2 0i yi c sin @Lm 2πL Lm m i5i0

Xi1 @S0 Δb 2πðyi 2 cÞ 52 ε0i sin @c 2πLm Lm i5i0

Solving Eqs. (7.57)(7.61), the parameters f, ε1, Lm, Ln are obtained. In order to solve the above equations, the minimization method of Press is used (Press, 1988). The cost function for the minimization method is defined as

; 5 2 1 2 : FðLm cÞ Z1 Z2 ð7 62Þ where 2 3 h 3 24L4 4L2 @S 3L2ε 35f 2 Z 5 2 4 2 n 2 n5 1 2L2 0 1 n 1 1 1 2 2 4 2 n @ π2 12ð1 v Þ 2 Lm Lm Lm 2 64 @S Z 5 2L2 0 2 n @c

Once the parameters are calculated, the buckling shape is obtained, using Eq. (7.49).

7.8 Flow Chart of the Main Program

Combining the above models, a program that simulates the local shape phenomena in a tandem cold mill is formed. The program is designed in a way that a multi-coil line-up can be simulated in one run. The main flow chart of the local shape simulation program is shown in Figure 7.5. 142 Primer on Flat Rolling

Run shape simulation

Get simulation option, number of colis, number of stands

Coil number = 1

Get coil data

Stand number = 1

Get stand data

Call thermal model

Roll profile = initial + thermal Coil number ++

Assume strip exit profile

Call 3D strip deformation model

Call roll deformation model Modify exit profile

N Exit profile converge? Stand number ++

Call stress unloading model

N Last stand? Pass residual stress and exit profile to next stand

Y Call buckling threshold model

Y Buckling? Call buckling shape model

Output results

N Last coil?

Y Return to GUI

Figure 7.5 Main flow chart of the shape simulation program. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 143

7.9 Model Tuning and Verification

The model was tuned and verified under no local buckle conditions using the data collected from a CPCM to establish the base cases for the simulation. The shape meter readings of 20 coils in two line-ups were used for the model tuning and verification. The shape readings in I-units5 were translated into residual stress and tension stress distributions acting on the shape meter. For each coil, the average value of 200 readings in a stable rolling period was used as the shape reading of the coil. The corresponding process parameters for each coil were then extracted from the data- base of the mill. A tuning factor was added to modify the work roll crown to compensate for effects not considered in the model, such as roll horizontal deflection, effects of roll wear and errors in roll grinding. This tuning factor was determined by fitting the shape readings of one coil for each line-up. With a fixed tuning factor, the shape readings of other coils in the line-up were used to verify the model. The model provided a consistent shape prediction for all the coils in each line-up. The shapes of sample coils of the line-ups are shown in Figures 7.6 and 7.7. Although there are still discrepancies, the consistency of the overall match of the measured and predicted shape proves the validity of the model. The buckling models were verified separately using theoretical results and the FEA method (Liu et al., 2011b).

7.10 User Interface

A graphic user interface has also been developed to integrate the sub-models and to manage the simulation options and the input and output data, and then graphically

120 Predicted Measured 100

20 Tension stress distribution (MPa) stress distribution Tension 0 0 100 200 300 400 500 600 Distance to strip centre (mm)

Figure 7.6 Sample comparison of first line-up.

5The I-unit is a measure of strip waviness. 144 Primer on Flat Rolling

70 Predicted 60 Measured

10 Tension stress distribution (MPa) stress distribution Tension 0 0 100 200 300 400 500 600 Distance to strip centre (mm)

Figure 7.7 Sample comparison of second line-up. display the simulation results. The simulation results are output to data files which can be further processed by the user. The major results currently selected for graphical display in the user interface include 3D longitudinal stress, 3D transverse stress, 3D vertical stress, 3D shear stress, lateral spread ratio, neutral plane profile, exit velocity profile, entry tension stress profile, exit tension stress profile, residual stress distribution, inter-roll contact pressure distribution, exit thickness profile, roll thermal profile, roll force distribution and local buckling shape. Since the local shape defects are the main focus of this simulation program, three abnormal rolling conditions are listed as the main simulation options. These abnormal rolling conditions are simulated and discussed in subsequent sections.

7.11 Base Case for Local Shape Defect Simulation

To simulate the local shape defects from various sources, an actual rolling case is selected as the base in which no local shape defect was produced. The various causes of local shape defects are then added to the base case to examine their effects. In the calculations, a 5-stand mill was considered. The work roll diameters ranged from 495 to 516 mm while the back-up roll diameters varied from 1400 to 1511 mm. The thermal conductivitywastakentobe0.06W/mmCandthe thermal expansion coefficient was 0.000013 mm/C.Thespecificheatwas 434 J/kg C. The entry thickness of the strip into the first stand was 2.442 mm and into the last stand, 0.417 mm. The coefficient of friction varied from a high 0.05 in the first stand to 0.028 in the last stand. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 145

7.12 Effects of Entry Strip Profile Ridge

The feed stock of cold strip mills are coils rolled by hot strip mills. If abnormal conditions, such as excessive roll wear, occur in the hot strip mill, thickness profile ridges are produced on the hot rolled strip. When the hot rolled strip with ridges is subsequently fed into a cold mill, local buckles are produced in the cold rolled strip. The extra material forming the thickness ridge may be divided into three portions in subsequent cold rolling. A portion of the material will remain in the form of a thickness ridge in the cold rolled strip due to a local roll flattening increase, caused by the higher local roll force when rolling the thickness ridge. The second portion of the material will produce a local elongation increase, causing a local strip compression increase that may induce local buckles. The third portion of the material will spread laterally, which actually has the effect of attenuating the local buckles and the thickness ridges of the cold rolled strip. The percentage of each portion depends on the parameters of the rolls, products and the rolling process and may vary over a wide range. This shape simulation program is capable of quantitatively determining what happens when rolling a thickness profile ridge. Figures 7.87.16 show the major simulation results when cold rolling a coil with two thickness ridges in a five-stand fully continuous tandem mill. The height of the thickness ridges is assumed to be 2% of the entry strip thickness, 0.051 mm. The thickness ridges are assumed to be parabolically distributed over the 100 mm width and symmetrical about the centre line of the strip. The strip thickness profiles after each stand are shown in Figure 7.8. Even though the height of the ridges is continuously reduced from stand to stand, the ridges could not be totally eliminated. There are still thickness ridges, about 0.004 mm in height, left in the final product. The additional reductions to the thickness ridges will cause peaks in the roll force distribution as shown in Figure 7.9. Figure 7.10 shows the 3D stress distributions in the roll bite when rolling a thickness ridge. It is clear that the thickness ridges cause reduced normal stresses

2 Stand 1 Stand 2 Stand 3 Stand 4 Stand 5

1.5

0.5 Exit thickness profile (mm) Exit thickness

0 0 400 800 1200 1600 Distance to roll end (mm)

Figure 7.8 Exit thickness profile. 146 Primer on Flat Rolling

16,000 Figure 7.9 Roll force transverse distribution.

12,000

8000

Stand 1 4000 Stand 2 Stand 3 Roll force distribution (N/mm) distribution Roll force Stand 4 Stand 5 0 0 400 800 1200 1600 Distance to roll end (mm)

Figure 7.10 3D stress distribution in roll bite.

370 270 170 70 –30 –130 0 3 distribution (MPa) distribution 0 Longitudinal stress 200 6 9 Half width (mm)400 12 600 15 Contact arc (mm) 800 18

–300

–600

(MPa) 0 0 3 200 6 9 Vertical stress distribution Vertical Half width (mm)400 12 600 15 Contact arc (mm) 800 18 Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 147

0.06 Stand 1 0.05 Stand 2 Stand 3 0.04 Stand 4 Stand 5 0.03

0.02

0.01 Lateral spread ratio Lateral

–0.01 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.11 Lateral spread ratio distribution.

3 Stand 1 Stand 2 2 Stand 3 Stand 4 Neutral plane profile (mm) Neutral Stand 5 1 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.12 Neutral plane profile. locally in all three directions. The ridges also induce the local shear stress variation due to lateral spread and speed variation. The thickness ridge will also induce lateral spread around it. However, the magnitude of the lateral spread ratio, which is defined as the ratio between lateral strain rate and longitudinal strain rate, is much smaller in the region of the thickness ridge than that in the strip edge. The lateral spread ratio in the region of the thickness ridge also decreases from the first stand to the last stand as shown in Figure 7.11. The lateral spread ratio at the strip edge, however, may not always decrease from the first to the last stand. Because of the thickness ridge and lateral spread at the edge, the neutral planes are not “planes” anymore. They are curved by the material flow variation and 148 Primer on Flat Rolling lateral spread as shown in Figure 7.12. The curved portions are mainly at the region of the ridge and the strip edge. The relative exit speed profile, which is defined as the ratio of the strip longitudinal speed at the roll bite exit to the roll linear speed, is also curved around the ridge and the strip edge as shown in Figure 7.13. The entry tension stress profile, which mainly depends on the residual stress formed in the previous stand and the entry speed profile of the current stand, is also influenced by the thickness ridge (Figure 7.14). The exit tension stress drops around the ridge area are evident as shown in Figure 7.15.

1.06

1.05

1.04

Stand 1 1.03 Stand 2 Stand 3 Relative exit velocity profile velocity exit Relative Stand 4 Stand 5 1.02 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.13 Relative exit velocity profile.

Stand 1 400 Stand 2 Stand 3 Stand 4 Stand 5 200

0 Entry tension profile (MPa)

–200 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.14 Entry tension stress profile. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 149

Stand 1 400 Stand 2 Stand 3 Stand 4 Stand 5 200

0 Exit tension profile (MPa)

–200 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.15 Exit tension stress profile.

200 Stand 1 Stand 2 Stand 3 100 Stand 4 Stand 5

Residual stress profile (MPa) –100

0 200 400 600 800 Distance to strip centre (mm)

Figure 7.16 Residual stress profile.

However, the amount of tension stress drop decreases substantially from the first stand to the last stand. The exit tension stress drop in stand 5 is only about a quarter of that in stand 1. A similar phenomenon also occurs in the residual stress profile as shown in Figure 7.16. The interaction among reduction, lateral spread, tension, residual stress, roll bending and flattening of all stands determines the final residual stress due to thickness ridge rolling, which will dictate the local shape of the product. One feature that distinguishes the local hot band ridge defects from other causes, such as yield stress drop or roll crown ridge, is that both local thickening and local buckles exist at the same position as shown in Figures 7.8 and 7.16. 150 Primer on Flat Rolling

7.13 Effect of Local Yield Stress Drop

There are abnormal metallurgical or physical conditions that may cause non- uniform yield stress across the strip width. For example, microstructure changes occurring in the finishing mill as a result of temperature variation in various zones of the strip may result in yield stress variations in the hot band. If a narrow slice of the hot rolled strip is softer than the rest, local buckles may be induced as a result of cold rolling. To see the effects of local yield stress drop, a parabolic yield stress valley with 15% yield stress drop in a 100-mm-wide slice is assumed. Yield stress distributions for each stand are shown in Figure 7.17. The local yield stress drops will induce a roll force drop at corresponding posi- tions as shown in Figure 7.17. However, the percentage of roll force drop is only about 23%, which is much smaller than the 15% yield stress drop. The reason is that the local roll force drops reduce the local roll flattening, which then reduces the local roll gaps and increases the local reductions slightly, which in turn offsets part of the roll force drops (Figure 7.18). The slightly increased local reductions cause strip exit speeds to increase locally as shown in Figure 7.19, which in turn changes the neutral plane profiles as shown in Figure 7.20. The slightly increased local reductions also cause compression at the entry to and exit from the roll bites and produce local tension stress drops at the entry and exit sides as shown in Figures 7.21 and 7.22. Because of the local exit tension stress drops, there are local compressions in the residual stress profile after unloading as shown in Figure 7.23. The local compressions in the residual stress profile would produce local buckles if they exceeded certain limits. Contrary to the local shape defects produced by hot band ridges, in

500

400

300

200 Stand 1 Stand 2 100 Stand 3

Yield stress in shear profile (MPa) Yield Stand 4 Stand 5 0 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.17 Assumed yield stress profile. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 151

14,000 Figure 7.18 Roll force profile.

10,500

7000 Stand 1 Stand 2 3500 Stand 3 Stand 4 Roll force distribution (N/mm) distribution Roll force Stand 5

0 0 400 800 1200 1600 Distance to roll end (mm)

1.06 Figure 7.19 Relative exit velocity profile.

1.05

1.04

Stand 1 1.03 Stand 2

Relative exit velocity profile velocity exit Relative Stand 3 Stand 4 Stand 5 1.02 0 200 400 600 800 Distance to strip centre (mm)

6 Figure 7.20 Neutral plane profile. 5

3 Stand 1 Stand 2 2 Neutral plane profile (mm) Neutral Stand 3 Stand 4 Stand 5 1 0 200 400 600 800 Distance to strip centre (mm) 152 Primer on Flat Rolling

Figure 7.21 Entry tension Stand 1 400 Stand 2 stress profile. Stand 3 Stand 4 Stand 5 200

0 Entry tension profile (MPa)

–200 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.22 Exit tension stress Stand 1 400 Stand 2 profile. Stand 3 Stand 4 Stand 5 200

Exit tension profile (MPa) 0

–200 0 200 400 600 800 Distance to strip centre (mm)

200 Figure 7.23 Residual stress Stand 1 profile. Stand 2 Stand 3 100 Stand 4 Stand 5

–100 Residual stress profile (MPa)

0 200 400 600 800 Distance to strip centre (mm) Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 153

0.5

0.4

0.3 Exit thickness profile (mm) Exit thickness

0.2 0 400 800 1200 1600 Distance to roll end (mm)

Figure 7.24 Thickness profile after stand 5. which local thickening occurs at the location of the local buckles, local strip thinning would occur at the position of the buckles were they caused by local yield stress drops as shown in Figure 7.24.

7.14 Roll Cooling Nozzle Clog or Work Roll Crown Ridge Effect

Abnormal conditions in the cold mill may also cause local shape defects. If some roll cooling nozzles are clogged, a roll thermal ridge may be formed causing local buckles. As an example, assuming that two adjacent work roll cooling nozzles are clogged, a 0.007-mm work roll thermal crown ridge will be produced at thermal steady state conditions for the base case simulated. If the roll ridge occurs in the early stands, the potential local shape defects produced could be ironed flat in subsequent stands before the buckles occur. However, if the work roll thermal ridge is in the last stand, local buckles may be produced in the final product. Figure 7.25 shows the roll force distributions, assuming that four roll cooling nozzles are clogged at stand 5, two in each half of strip width and symmetric to the strip centre line. Maximum 0.007-mm roll thermal crown ridges are built up at the thermal steady state conditions of stand 5, which will produce the roll force peaks in stand 5 as shown in Figure 7.25. Local reduction increases induced by the roll force peaks will cause local drop of the entry tension stress profile, as shown in Figure 7.26, and a local drop in the exit tension stress profile, as shown in Figure 7.27. The local drop in exit tension stress profile will cause a local compression in the residual stress profile after unloading as shown in Figure 7.28. 154 Primer on Flat Rolling

10,000

5000 Stand 1 Stand 2 Stand 3 Roll force distribution (N/mm) distribution Roll force Stand 4 Stand 5 0 0 400 800 1200 1600 Distance to roll end (mm)

Figure 7.25 Roll force profile.

500 Stand 1 Stand 2 400 Stand 3 Stand 4 300 Stand 5

200

100

Entry tension profile (MPa) 0

–100 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.26 Entry tension stress profile.

7.15 Identification of Causes of Local Buckles

The effects of friction coefficient variations in the width direction and feed stock local residual stress are also simulated. In the practical variation range, they are unlikely to cause local shape buckles. Therefore, they are excluded from the possible causes of the local shape defects. Since the local residual stress induced by upstream stands can be ironed out, the working conditions of the upstream stands are excluded from the causes of the local buckles. The remaining possible causes of local shape defects are then the hot band ridge, the yield stress local drop and the abnormal rolls Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 155

500 Stand 1 400 Stand 2 Stand 3 Stand 4 300 Stand 5

200

100 Exit tension profile (MPa) 0

–100 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.27 Exit tension stress profile.

200 Stand 1 Stand 2 150 Stand 3 Stand 4 100 Stand 5

Residual stress profile (MPa) –50

–100 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.28 Residual stress profile. or operating conditions in the last stand. Determination of which of the three irregular factors in a cold mill causes the occurrence of the observed local buckles requires a method to identify their specific causes. This is accomplished using simulation by measurement and analysis of the thickness profile at buckled regions across the strip width. Thus, three causes of local buckles are identified: 1. Local increased strip thickness and local buckles at the same position indicating feed stock ridge as the cause. 2. Local decreased strip thickness and local buckles at the same position, thus indicating that the work roll cooling system and condition of the work rolls as well as the operating conditions of the last stand should be checked first for any abnormalities. 156 Primer on Flat Rolling

Measuring strip thickness across strip width

Local increased strip Local decreased strip thickness and local thickness and local buckles at same position buckles at same position

Feed stock Lower yield stress Abnormalities ridge of coarse grains in at last stand buckling region

Local yield Roll crown ridge stress drop at last stand

Figure 7.29 Flow chart of the procedure to identify causes of local buckles.

3. Local decreased strip thickness and local buckles at the same position without any abnormalities in the last stand, indicating local yield stress drop as the cause of the local buckles. This can be identified by checking the yield strength or microstructure across the strip width. Figure 7.29 shows the flow chart of the above described procedure to identify the causes of local buckles.

7.16 Predicting Limiting Values for Factors Causing Local Buckles

The local buckling simulation models are capable of calculating the limiting values of the factors causing local buckles to start occurring. These limiting values are important for quality control of cold rolled coils. A sample calculation was performed for a 0.275-mm-thick product with the local compression width of 110 mm. Assuming a parabolic local compression stress distribution, a critical local buckle stress for this case is 12.96 MPa. The limiting value of the feed stock ridge to induce this critical local compression residual stress in the final product is 0.046 mm as shown in Figures 7.30 and 7.31. Figure 7.30 shows the strip transverse thickness distribution at the entry side of each stand. Figure 7.31 shows the residual stress transverse distribution after each stand. From Figure 7.31, it is seen that while the local compression residual stress induced by feed stock ridge is quite large after the first pass, it decreases pass after pass. The local compression residual stress after the fifth pass is only about 18% of that after the first pass. Even though the critical local buckling stress decreases quickly as the strip is getting thinner, the likelihood of local buckles caused by feed stock ridge may not increase quickly since the local compression residual stress also decreases as the strip is getting thinner. This phenomenon is also the reason that the calculated limiting values for the feed stock ridge to induce local Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 157

3 Stand 1 Stand 2 Stand 3 Stand 4 Stand 5

1 Entry profile (mm) thickness

0 400 800 1200 1600 Distance to roll end (mm)

Figure 7.30 Thickness profile when the feed stock ridge is at its limiting value.

100 Stand 1 80 Stand 2 Stand 3 60 Stand 4 Stand 5 40

–20 Residual stress profile (MPa) –40

–60 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.31 Local compression residual stresses induced by the ridge at the limiting value. buckles are larger than the results of other calculations (Tieu et al., 2008). Larger entry thickness is another reason for the larger limiting value of the feed stock ridge. Nevertheless, since retained ridges will also cause build-up in the coil (Blazevic, 2002), the feed stock ridge may not be allowed even though it may be smaller than the limiting value for local buckles to occur. Therefore, the limiting values of feed stock ridge causing coil build-up need to be further investigated and compared with the limiting values of feed stock ridge causing local buckles. The smaller limiting value either from local buckling condition or coil build-up condition should be used as the limiting value for hot band ridges. 158 Primer on Flat Rolling

The limiting value of the local yield stress drop to induce this critical local compression residual stress is 12.08 MPa as shown in Figures 7.32 and 7.33. Figure 7.32 shows the yield stress in shear along the strip width at each pass. Figure 7.33 shows the residual stress distribution along the strip width after each pass. From Figure 7.33, it is seen that the local compression residual stresses are almost the same after each pass. This phenomenon indicates the likelihood that the local buckling, induced by the local yield stress drop, increases quickly pass after pass as the strip is getting thinner. Since the increased local elongation caused by the local yield stress drop may also cause coil build-up, the limiting values of the

450 Stand 1 Stand 2 Stand 3 Stand 4 Stand 5

400

350 Yield stress in shear (MPa) Yield 300

0 200 400 600 800 Distance to strip centre (mm)

Figure 7.32 Shear yield stress variation.

100 Stand 1 Stand 2 80 Stand 3 60 Stand 4 Stand 5 40

–20

Residual stress profile (MPa) –40

–60 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.33 Local compression residual stress induced by local yield stress drop at its limiting value. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 159 local yield stress drop causing coil build-up need to be further investigated. The smaller limiting value from local buckling condition or coil build-up condition should be used as the actual limiting value of the local yield stress drop. Comparing Figure 7.33 to Figure 7.31, it is concluded that the local buckles are more and more likely to be induced by the local yield stress drop than by the feed stock ridge as the strip is getting thinner. The limiting value of the last stand work roll diameter crown ridge to induce this critical local compressive residual stress is 0.010 mm as shown in Figures 7.34 and 7.35.

0.08 Stand 1 Stand 2 Stand 3 0.06 Stand 4 Stand 5

0.04

0.02 Initial work roll profile (mm)

0 0 400 800 1200 1600 Distance to roll end (mm)

Figure 7.34 Last stand work roll crown ridge at its limiting value.

100 Stand 1 80 Stand 2 Stand 3 60 Stand 4 Stand 5 40

–20 Residual stress profile (MPa) –40

–60 0 200 400 600 800 Distance to strip centre (mm)

Figure 7.35 Local compression residual stress induced by the last stand work roll crown ridge at its limiting value. 160 Primer on Flat Rolling

Figure 7.34 shows the work roll radius crown curve for each stand. Figure 7.35 shows the residual stress transverse distribution after each stand. The results show that the local compression residual stress is sensitive to the roll crown ridge. Small roll crown ridge at the last stand may induce local buckles. Since the increased local elongation caused by the roll crown ridge may also cause coil build-up, its limiting values need to be further investigated. The smaller limiting value from local buckling condition or coil build-up condition should be used as the actual limiting value of the roll crown ridge. A summary of the calculation conditions and the limiting values causing local buckles for this sample case is shown in Table 7.1. Both finished product thickness and local compression width have a very significant influence on the limiting values for the factors causing local buckles. Figure 7.36 shows the variation of limiting values of feed stock ridge height versus local compression width for three different strip thicknesses. It is seen that the limiting values of feed stock ridge height increase as the strip thickness increases and the local compression width decreases. Figure 7.37 shows the variation of the limiting values of the local yield stress drop versus local compression width for three different strip thicknesses. It is seen that the limiting values of the local yield stress drop increase as the strip thickness increases and the local compression width decreases.

Table 7.1 Summary of Conditions and Limiting Values

Feed Product Local Buckled Local Limiting Limiting Limiting Stock Thickness Compression Width Buckle Value of Value of Value of Thickness Width Critical Feed Local Yield Last Stand Stress Stock Stress Roll Crown Ridge Drop Ridge

2.442 mm 0.275 mm 110 mm 96.8 mm 12.96 MPa 0.046 mm 12.08 MPa 0.010 mm

0.16 h=0.25 h=0.275 h=0.30 0.12

0.08

0.04 Feed stock ridge height (mm) stock Feed

0 70 80 90 100 110 120 130 140 150 Ridge width (mm)

Figure 7.36 Limiting values of feed stock ridge height versus the local compression width. Flat Rolling Simulation and Reduction of Local Buckles in Cold Rolling 161

h=0.25 40 h=0.275 h=0.30

10 Yield stress drop (MPa) Yield

0 70 80 90 100 110 120 130 140 150 Ridge width (mm)

Figure 7.37 Limiting values of the local yield stress drop versus the local compression width.

h=0.25 0.03 h=0.275 h=0.30

0.02

0.01 Work roll ridge Work height (mm)

0 70 80 90 100 110 120 130 140 150 Ridge width (mm)

Figure 7.38 Limiting values of work roll crown ridge versus local compression width.

Figure 7.38 shows the variation of limiting values of the work roll crown ridge at the last stand versus the local compression width for three different strip thicknesses. It is seen that the limiting values of the work roll crown ridge increase as the strip thickness increases and the local compression width decreases.

7.17 Reduction of Local Buckles

Even though the cold mill itself may cause local buckles, the majority of the local buckles are caused by the feed stock from the hot mill (Melfo et al., 2006). Therefore, proper monitoring of the hot band profile and mechanical properties 162 Primer on Flat Rolling will be the first choice to reduce local buckles. Since hot band ridges are most likely related to uneven roll wear, the measures to cause uniform roll wear, such as on-line roll grinding and work roll axial shifting, are suggested for reducing local buckles. If all measures in the hot strip mills fail and ridges are produced on the hot band, flattening the ridges before cold rolling using a mill stand with very high local rigidity (Liu et al., 2011b, 2012) can be used to reduce the local buckles significantly. The uniformity of the metallurgy-related factors in the hot strip mills and casters should be properly maintained to avoid the local yield stress drop and to reduce local buckles. Finally, rolls, roll cooling systems and other operational conditions of the cold mill need to be carefully maintained to avoid the local shape defects produced solely by the cold mill. 8 Material Attributes

8.1 Introduction

The metal’s resistance to deformation is often referred to by several names. It may be called the flow stress, the flow strength or the constitutive relation; the bottom line is that a relationship of the metal’s strength to other, independent variables is being considered. The best identification is the term “resistance to deformation” since it describes a material property and its meaning is clear: it indicates how the material reacts when it is loaded and deformed by external forces. The need for understanding the intricacies of the material’s resistance to deformation has been indicated in Chapters 5 and 6. This includes two ideas: the appreciation of the physical and metallurgical capabilities and the response of the materials while in service as well as the development of mathematical models of the metals’ resistance to deformation. The former is necessary to provide insight and to aid in the design and the planning of the metal forming processes. The latter is critical for the success of the predictive models of the flat rolling process. Several steps need to be completed in order to reach these objectives. These are listed below:

G Determine the independent variables that are expected to affect the metal’s resistance to deformation. G Determine the metal’s resistance to deformation in an appropriate test, one that allows the variation of the independent variables over an appropriate range of magnitudes. G Develop through mathematical modelling (e.g. non-linear regression analysis, artificial intelligence or storing data in a multi-dimensional matrix of data) a true stresstrue strain, strain rate, temperature, etc. relation. In what follows, selected attributes of some of the steel and aluminium alloys used in the metal forming industry will be briefly reviewed and will be compared, with special attention paid to the automotive industry. Recently developed alloys will also be briefly introduced. Traditional testing techniques to determine the metals’ attributes will be discussed next; their advantages and disadvantages are given. This will be followed by a presentation of the mathematical description of some of the attributes. Metallurgical events will also be discussed and the grain structures accompanying hot or cold deformation processes will be presented. Most of the comments will concern steels, reflecting the experience of the writer. Hot rolling of aluminum is discussed in Chapter 11.

8.2 Recently Developed Steels

The traditional metals used in the metal forming industry include the alloys of steel, aluminium, copper and titanium. New alloys have been developed in the last several decades, mostly driven by the need of the automotive industry to reduce weights and gasoline consumption and thus reduce air pollution. This need created one of the most important current objectives of the carmakers, which is to develop the technology to produce lightweight components. The materials used in this regard must have attributes that include high strength and high ductility. The new ferrous metals being introduced include the interstitial free (IF) steels, bake hardenable steels, transformation-induced plasticity (TRIP) steels, the high-strength low-alloy (HSLA) steels, dual phase (DP) steels and martensitic and manganeseboron steels, having yield strengths that vary from a low of 200 up to 1250 MPa. The elongation of these steels decreases as the strength increases, from a high of nearly 40% to a low of 45%, affecting the design of the subsequent applications. A recent review (Ehrhardt et al., 2004) lists many of these steels and indicates that light construction steels with induced plasticity possess tensile strength in the order of 1000 MPa and remarkably high total elongation of 6070%. The website of the American Iron and Steel Industry also includes up-to-date information concerning the description and the processing of recently developed steel alloys. As given in the site, advanced high-strength steels (AHSSs) in use in the automotive industry include the DP steels the microstructure of which includes ferrite and up to 20% and 70% volume fraction of martensite. While the use of bainite helps to enhance the capability to resist stretching on a blanked edge, the ferrite phase leads to high ductility and creates high work hardening rates, which give the DP steels higher tensile strength than conventional steels. Further, TRIP steels are also used, the microstructure of which consists of a ferrite matrix containing a dispersion of hard second phases martensite and/or bainite in addition to retained austenite in volume fractions greater than 5%. During deformation, the hard second phases create a high work hardening rate while the retained austenite transforms to martensite, increasing the work hardening rate at higher strain levels. The complex phase (CP) steels consist of a very fine microstructure of ferrite and a higher volume fraction of hard phases that are further strengthened by fine precipitates. In the martensitic (MART) steels the austenite that exists during hot rolling or annealing is transformed almost entirely to martensite during quenching on the run-out table or in the cooling section of the annealing line. All AHSSs are produced by controlling the cooling rate from the austenite or austenite plus ferrite phase, either on the run-out table of the hot mill (for hot rolled products) or in the cooling section of the continuous annealing furnace (continuously annealed or hot dip-coated products). AHSS cooling patterns and resultant microstructures are schematically illustrated on the continuous cooling-transformation diagram, available for examination in the American Iron and Steel Institute (AISI) website. The cooling patterns are designed on the bases of mathematical models, which attempt to predict the structures and properties resulting from the processing technique. Material Attributes 165

Figure 8.1 Microstructure of an extra-deep-drawing ferritic steel 3003 . Source: Reproduced from http:// www.mittalsteel.com/.

Research is continuing in the development of twinning-induced plasticity steels and lightweight steels with induced plasticity (Gigacher et al., 2005)1.

8.2.1 Very Low Carbon Steels The structure of these steels is fully ferritic. A micrograph, reproduced from the website http://www.mittalsteel.com is shown in Figure 8.1. While the strength of these steels is very low, their very high formability makes them ideal candidates for parts that carry low loads but require high-strain carrying ability during the forming process. Automotive components and motor lamination steels are potential uses.

8.2.2 IF Steels These steels contain less than 0.003% C. The nitrogen level is also reduced during their preparation and the remaining carbon and nitrogen are tied up using small amounts of alloying elements, such as Ti or Nb. The steels are finish rolled above 950C. Their strengths are low but they possess very high formability, especially after annealing (138165 MPa yield strength, 4145% elongation). Their structure is very similar to that shown in Figure 8.1, above.

8.2.3 Bake-Hardening Steels The carbon content is even lower, 0.001% C. The steels harden during the paint- curing cycle, performed usually at 175C, for 30 min. The hardening is caused by the precipitation of carbonitrides. The as-received yield strength of the steel is 210310 MPa; after baking and a 2% pre-strain these rise to 280365 MPa. There is little change of the tensile strength but the dent resistance is increased. These

1I am grateful to Dr. G. Nadkarni, Mittal Steel, Southfield, MI, and to Dr. J. B. Tiley, hot rolling consultant, to bring these grades of steel to my attention. 166 Primer on Flat Rolling

Figure 8.2 BH steel. Source: Reproduced from http:// www.mittalsteel.com/.

grades are extensively used in automotive outer body panels. A typical microstructure of a bake-hardening (BH) steel is given in Figure 8.2.

8.2.4 TRIP Steels These steels are highly alloyed and have been heat treated to produce metastable (that is, not fully stable with respect to transformation) austenite plus martensite. When subjected to permanent deformation, the austenite experiences strain-induced transformation to martensite. A tempering process may follow the transformation. The steels are highly ductile and are strain rate sensitive. Their tensile strength can reach magnitudes as high as 800 MPa. They respond well to bake hardening and an extra 70 MPa strength is the result. These steels are one of the newest families of AHSSs currently under development for the automotive industry. The steels have a microstructure of soft ferrite grains with bainite and retained austenite. The hard martensite delays the onset of necking, resulting in a product with high total elongation, excellent formability and high crash energy absorption. In addition, TRIP steels also exhibit extremely high fatigue endurance limits, thereby providing excellent durability performance. The micrograph of Mittal’s TRIP steel is shown in Figure 8.3.

8.2.5 HSLA Steels The HSLA steels, often referred to as microalloyed steels, are low-carbon steels with the strength increased by small amounts of alloying elements such as niobium, vanadium, titanium, molybdenum or boron, singly or in combinations. Their tensile strength may reach 450 MPa and their ductility may be as high as 30%. Thermomechanical processing is used to affect their mechanical and metallurgical attributes. Arguably, one of the best collections of information concerning some of these alloys appears in the Proceedings of the International Conference on the Thermomechanical Processing of Microalloyed Austenite, held in Pittsburgh, in Material Attributes 167

Figure 8.3 TRIP steel. Source: Reproduced from http:// www.mittalsteel.com/.

Figure 8.4 Micrographs of a boron steel; (A) ε_ 5 1:7 3 1022 s21 and ε 5 0:37; (B) ε_ 5 1:7 3 1021 s21 and ε 5 0:46 (Maki et al., 1981).

1981. Micrographs of many of these steels under a large number of processing conditions have been published at that conference. In what follows, two examples are shown. Figure 8.4 shows two micrographs (Cuddy, 1981) of a 0.057% C, 1.44% Mn, 0.112 Nb steel, reheated to 1200C and reduced 55% in five passes. In (A) the steel, at 11001070C, is fully recrystallized, while in (B) at 1000960C, the grain elongation is pronounced.

8.2.6 DP Steels These are low-alloy steels, similar to the HSLA steels. Their tensile strength is somewhat higher, 550 MPa. The structure, shown in Figure 8.5, contains approximately 20% martensite in a ductile ferrite matrix. As written on Mittal’s website, DP steels are one of the important new AHSS products developed for the automotive industry. Their microstructure typically consists of a soft ferrite phase with dispersed islands of a hard martensite phase. The martensite phase is substantially stronger than the ferrite phase. 168 Primer on Flat Rolling

Figure 8.5 A typical DP steel. Source: Reproduced from http:// www.mittalsteel.com/.

60 Austenitic Deep-drawing steels stainless steels 50

40 BH steels TRIP steels Duplex stainless 30 steels

Elongation (%) 20 Aluminium DP steels HSLA steels 10

Magnesium 0 0 100 200 300 400 500 600 Yield strength (MPa)

Figure 8.6 A compilation of material attributes. Source: After Pleschiutschnigg et al. (2004).

A compilation of the mechanical attributes of several materials is shown in Figure 8.6, reproduced, following Pleschiutschnigg et al. (2004). The figure gives the elongation and the yield strength, measured at 0.2% offset, of deep-drawing steels, austenitic stainless steels, BH steels, TRIP steels, duplex stainless steels, DP steels and HSLA steels in addition to aluminium and magnesium. It is noted that the deep-drawing quality steels and the austenitic stainless steels offer the highest formability. BH and TRIP steels indicate similar elongation but the TRIP steels also provide much higher strength. Aluminium is less strong and less formable but much lighter than the ferrous metals. Its competitiveness needs to be based on its superior strength to weight ratio. Material Attributes 169

The processing route that results in the TRIP steels and the DP steels is also discussed by Pleschiutschnigg et al. (2004). The rolling process is similar for both metals, the difference being the cooling rate: faster for the DP steels and slower for the TRIP steels. The authors indicate that the controlled rolling process, not the chemical composition, has a dominating influence on the results.

8.3 Steel and Aluminium

The competition between aluminium and steel alloys for use in the automotive industry is intense. The website of the US Steel Company gives some of the data, indicating the advantages of the steel grades over aluminium. The information below was taken directly from the website xnet3.uss.com/auto/index.htm, in April, 20062. A formability chart in the website compares the formability of several steels and aluminium. The winner as far as formability is concerned is the IF steel, indicating up to 55% total elongation. Its tensile strength is low, however, at most nearly 350 MPa. The strongest steel is the martensitic type, as expected, possessing a tensile strength of almost 1700 MPa but at a total elongation of about 7%, subsequent plastic forming processes need to be designed with extreme care. Aluminium appears to be somewhere in the middle, with elongation varying from a low of 7% up to 3032% and a maximum tensile strength of 600 MPa. TRIP steels have a strength between 600 and 1250 MPa and elongations of 1840%. A very interesting compilation of stressstrain curves for several steel and aluminium alloys is also given in the United States Steel (USS) website. Obtained at fairly low (0.0005 s21) and somewhat higher rates of strain (9.8 s21), the figure indicates that the steel alloys’ strengths increase with increasing rates of strain while those of the aluminium don’t. Specifically, the maximum strength of the TRIP 590Y steel, at 9.8 s21, is near 750 MPa and at the lower rate of 0.0005 s21 it is 620 MPa. In the same strain rate range, the strength of the DP steel increases by about 100 MPa as the rate of strain is increased and that of the deep- drawing quality steel increases by about 150 MPa. The 5754-0 aluminium alloy indicates no rate sensitivity. Note, however, that there are several aluminium alloys whose strengths are in fact, strain rate sensitive; an example is the commercial purity aluminium alloy, 1100-H0. Of major importance, also observable from the figure of USS, is the total strain sustained by each of the metals, as this has a major impact on the design of subsequent forming processes and will hence affect productivity. The clear winner here is the deep-drawing quality steel, deformed at the low strain rate note that the 0.0005 s21 is almost creep deforming to a fracture strain of 45%. The fracture strain of the TRIP steel at high rates B36% is near that of the 5754-0 aluminium, strained at the lower rate. Most of the steels appear to be more formable than the single aluminium alloy.

2An attempt was made to reproduce some of the excellent figures from the US Steel website, identified above. Copying and pasting proved unsuccessful as the results became hazy when printed. 170 Primer on Flat Rolling

Further data, also from the US Steel website, compares the strengths and the cost indices of the two metals. A quotation discussing the information is reproduced below:

The figure... shows common metallurgical grades undergoing a pre-strain of 2% and typical automotive paint bake cycle on the left compared to their prospective cost index shown on the right. Pricing for steel grades is based on seven combined typical market sources and the ULSAB-AVC cost model. Aluminum pricing was gathered based on 2002 publications from the MIT Material Systems Lab and typical market information, such as the American Metal Market (20022003).

The cost index of the aluminium alloys is more than five times that of the steels. For example, the 6111-T4 alloys yield strength is given as approximately 240 MPa and its cost index is 5.9. This may be compared to that of the deep-drawing quality steel, DP 600. The yield strength of this alloy is the highest among those shown, at 580 MPa but its cost index is 1.1. The Society of Automotive Engineers (SAE) grade 3 steel is demonstrating a yield strength of 250 MPa and a cost index of one. Information regarding aluminium alloys is also easily available from the Internet. Alcan’s website indicates the formation of a spin-off company, Novelis Inc., formed in 2005 and now dealing with rolled products and sheet metal operations. Novelis.com lists the benefits of aluminium over that of other materials; the list below is copied directly from the website.

The benefits of using lightweight aluminum sheet in transport applications are clear:

Aluminum offers high potential for weight savings, thus reducing emissions through the life of the vehicle, improving fuel efficiency and also handling; The metal is easily and widely recycled, saving energy and raw materials; It has very good deformation characteristics and manufacturing properties; Aluminum will absorb the same amount of crash energy as steel, at a little more than half the weight; It has good corrosion resistance. Novelis’ product range for the transportation market includes: sheet for automotive vehicle structures and body panels; pre-painted and plain sheet for commercial vehicle applications such as dump bodies, cabs and trailer flooring; “shate” for ship hulls and decks, tippers, road tankers, etc; plain, heat-treated or painted slit strips and coils customized to the needs of automotive part suppliers; high specification foil (industrial finstock) and brazing sheet for heat exchangers; and foil for insulation applications.

8.4 The Independent Variables

The traditional approach in identifying the variables and parameters that affect the behaviour of metals is quite simple and works well in many cases. The usual Material Attributes 171 formulations indicate that in the cold deformation regime the resistance to deformation is assumed to be an exclusive function of the strain, that is σcold 5 f ðεÞ, and in the high-temperature region, at a particular temperature, the only independent variable is the rate of strain, or σhot 5 gðε_Þ. While both relations are used regularly in the analyses of metal forming processes, they represent a much too simplified view of how the metals behave. A list of independent variables that affect the material attributes is much longer. It may include the strain, strain rate, temperature and metallurgical parameters (e.g. the grain size, ZenerHollomon parameter, chemical composition, activation energy, precipitation potential, amount of recrystallization, volume fraction of various phases). Arguably, it may also include the dependence of the results on the testing technique as it is very rare to see a successful comparison of stressstrain curves of a metal, obtained in tension, compression and torsion. Of course, an equation that includes all of these variables would never be used by engineers, so, as always, a compromise is needed. As will be demonstrated below, adequate results are obtained when the resistance to deformation is given in terms of the strain, rate of strain, temperature and the activation energy:

σ 5 f ðε; ε_; T; QÞð8:1Þ where Q is the activation energy, to be discussed in more detail later in this chapter, and T is the temperature, usually expressed in Kelvin. There are two steps involved in determining an actual, useful and usable form of Eq. (8.1). The first step requires a systematic testing programme designed with the end-use of the resulting data in mind3. The multi-dimensional databank thus obtained is then employed to develop an appropriate mathematical model which describes the metal’s resistance to deformation. In what follows, these two steps are discussed in some detail.

8.5 Traditional Testing Techniques

The objectives of the tests are several and depend strongly on the objectives of the tester. The materials engineer wants to know how the sample of the metal will react to loads. The engineer dealing with metal forming wants to know how to use the results of the test in analysing a metal forming process in addition to deciding what test to use to enable that analysis. The objectives of the materials chemist and phys- icist also would have an effect on the choice of the test. While there are numerous experiments available for the determination of the metal’s resistance to deformation

3These may include examining the behaviour of the material or the design and analysis of a metal forming process. 172 Primer on Flat Rolling for use in planning, designing and analysing metal forming processes, three of them are used most often. They are as follows:

G the tension test G the torsion test G the compression test G the axially symmetrical sample G the plane sample (width . thickness). In each of these, constant strain rates and constant temperature conditions need to be established so the only variables to be monitored remain the force and the deformation. As well, the strains are to be high enough to allow a direct comparison of the metals’ behaviour in the tests with those required in the actual process.

8.5.1 Tension Tests These are the easiest and simplest to perform, using samples of cylindrical or rectangular cross-sections. The advantages are as follows:

G there are no frictional problems to be considered and G the tests are governed by American Society for Testing and Materials (ASTM) codes (ASTM Standards, E 8 and E 8 M) so inter-laboratory variability is minimized. The disadvantages indicate that tension testing is not the most suitable when the information gathered is to be used to study metal forming processes. They are as follows:

G only low strains are possible, at most 4050%; G the uniaxial nature of the stress distribution is lost when diffuse straining ends and localized straining begins, subjecting the necked down region to triaxial tension; G in order to keep the rate of strain constant, increasing the cross-head velocity during the test is necessary; and G while it is possible, it is difficult to perform the test in isothermal conditions. A schematic diagram of a tension testing set-up is shown in Figure 8.7, reproduced from Schey (2000). A universal testing machine is shown, along with a flat sample, the actuator that moves the cross head, a load cell, an extensometer and a recording device which plots, online, the force-deformation curve. In an up-to-date, modern variant of this set-up, the measurements would be collected using digital data acquisition and a stressstrain curve would then be plotted in real time. Attention needs to be paid to the manner in which the sample is attached to the cross head. As shown in the figure, there are holes drilled in the sample and the attachment ensures the application of the force in the direction of its longitudinal centreline. At the same time, the effect of the stress concentration at the holes is to be considered carefully and the sample should be designed such that fracture doesn’t occur there. In many commercially available tensile testers, jaws attached to the machine through spherical seats are used to hold the sample and these also ensure appropriate alignment while minimizing the problems that may be caused by stress concentration. Material Attributes 173

Actuator

Displacement transducer

Extensometer

Moving P crosshead x–y recorder y

Test specimen x x y

P Voltage Voltage ∝Δl ∝P

Load cell

Figure 8.7 A tension test. Source: Schey (2000), reproduced with permission.

The comment made above, concerning the difficulties associated with providing isothermal conditions, may be appreciated by examining the figure. There are several possibilities, none easy. An openable, split furnace may be used which would enclose most of the sample and the extensometer. This would necessitate the use of very expensive extensometers that are capable of performing within the high- temperature furnace. A modern variation would make use of an optical device, focused on a deforming section of the sample, through a viewing hole of the furnace wall. In either case, the jaws would also need to withstand the high temperatures. The load cell and the rest of the testing machine would have to be protected from temperature damage, probably by installing in-line heat exchangers. Another possibility to heat the sample is the use of induction heaters and a coil which would cover only the reduced, deforming portion of the sample; however, induction heaters are bulky and very expensive.

8.5.2 Compression Testing These may be performed on cylindrical or plane samples; for details, see ASTM Standards E 9. Figure 8.8, again reproduced from Schey (2000), shows a schematic of the compression test, using a cylindrical sample, (A) and the force-deformation curve that results, (B). Note that the curve is increasing exponentially, reflecting 174 Primer on Flat Rolling

P A 0 A 400 1

300 0 h Displacement

1 transducer h (Δh) 200 , kN P Load 100 cell (P)

0 24681012 Δh h h (= 0 – 1), mm (A)P (B)

Figure 8.8 (A) The compression test (B) and the resulting force-deformation curve. Source: Schey (2000), reproduced with permission. the growing contact area and the attendant increase of frictional resistance. The advantages of the compression tests are as follows:

G Larger strains are possible, typically 120140% when cylinders are compressed and up to 200% when plane samples are tested. G The state of stress is mostly compressive, as in bulk forming. The disadvantages are as follows:

G Frictional forces at the ram-sample interface grow as the test progresses and their effects must be controlled and removed from the data. G Tensile straining at the cylindrical surfaces or the edges of plane samples limits the level of straining (the circumferential strain may be calculated easily, making use of mass conservation). G The achievement of constant true strain rates during the tests requires careful feedback control, making the use of a cam-plastometer or in a modern setting, a computer- controlled servohydraulic testing system. G The distribution of the strains in the normal direction is not uniform. G When plane-strain compression is performed, isothermal conditions are difficult to achieve. It is relatively simple to conduct a compression test at high temperatures and to make sure that almost isothermal conditions exist within the furnace. An openable furnace is necessary with a fairly long heated length. The sample is to be compressed between flat platens, made of a material that retains its strength at the test temperature. For steels the platens are often made of silicon carbide. Various Inconel alloys may also be used. As well, it is important to place water-cooled heat exchangers between the compression platens and the rest of the testing system, that is the load cell and the actuator. The procedure followed is also of importance as is the location of the thermocouple, or the temperature measuring system that controls the furnace temperature. Material Attributes 175

Figure 8.9 The plane-strain compression test. Source: Schey (2000), reproduced with permission.

In the plane-strain compression test, shown schematically in Figure 8.9, a flat sample is compressed between two flat dies. As long as the shape factor in the plane-strain test is similar to the shape factor in a flat rolling process, the strain distributions in the deforming portions within the two processes are similar (Pietrzyk et al., 1993). This allows one to recommend that in order to develop a mathematical model of the resistance to deformation for use in a one-dimensional model of flat rolling, the plane-strain compression tests should be used. An experimental difficulty in developing isothermal data in the plane-strain compression test is immediately evident. Enclosing the complete apparatus in a furnace is not practical. Heating the sample only is possible but not easy since the uniformity of the temperature distribution is difficult to maintain. Resistance heating, as performed in the Gleeble machines, or induction heating may be best, though in the latter, placing the coils may cause further difficulties.

8.5.3 Torsion Testing This type of testing materials is the most suitable when the data are to be used to analyse large-strain processes, such as a slab would experience during its journey through a hot strip mill, while it is being reduced from a thickness of about 300 mm to a final thickness of about 12 mm. Finite strains of 400500% can be obtained easily, allowing the simulation of the complete history of hot rolling, including the phenomena at the roughing mill and the finishing train of hot strip mills. The advantages are as follows:

G very large strains are possible; G constant rate of strain is simple to achieve; and G no frictional problems exist. 176 Primer on Flat Rolling

The disadvantages are that

G the torsional stresses and strains vary over the cross-section and a considerable amount of analysis is necessary to extract the uniaxial normal stressstrain data; and G the variation in the time it takes for different locations of the cross-section to experience metallurgical phenomena, specifically dynamic recovery and recrystallization, may cause a non-homogeneous structure. It is essential to allow the length of the sample to change without restrictions as the torsional testing proceeds, as constraining the length would induce longitudinal stresses in addition to the shearing stresses.

8.6 Potential Problems Encountered During the Testing Process

The usual approach to determine the metal’s resistance to deformation in order to simulate the hot or the cold rolling processes is to conduct compression tests, using plane or axially symmetrical samples4. While the test procedures are well understood and many of them are controlled by well-known standards, two areas of potential difficulties still exist: that of friction and that of temperature control. In what follows, these difficulties are discussed.

8.6.1 Friction Control This problem is encountered in the compression testing process, whether using axially symmetrical or plane samples. As the samples are being flattened, the contact area grows and continuously increasing effort must be devoted to overcome the frictional resistance at the compression platens. Baragar and Crawley (1984) showed that frictional effects are not very pronounced when strains under approximately 0.7 are considered. Above that level of deformation, however, the increasing frictional effects must be removed from the force-deformation data in order to obtain uniaxial behaviour. When using cylindrical samples, this may be accomplished by adopting the relation: m d p 5 σf 1 1 pffiffiffi ð8:2Þ 3 3 h

where the uniaxial flow strength is σf,pis the interfacial normal pressure, m is the friction factor and d and h are the current diameter and height of the sample,

4This statement may cause an argument among material scientists, many of whom value the advantages provided by the torsion test more than the simplicity of the compression tests. Material Attributes 177

200 Figure 8.10 True stresstrue strain curves of ε (1/s) aNbV steel, at 950C, under three different 1 conditions (Wang, 1989). 160 2

120 3 (MPa) σ 80 1. Specimen with flat ends 2. Specimen with recessed ends 40 3. Correction of curve-1 for friction (m = 0.18)

Nb–V steel, 950ºC, ε = 0.051/s 0 0.0 0.4 0.8 1.2 1.6 ε respectively. The friction factor is best determined in the ring compression test5 (Male and DePierre, 1970). Avitzur (1968) quotes Kudo’s (1960) formula, connecting the coefficientpffiffiffi of Coulomb friction and the friction factor in the form μðpave=σf Þ 5 m= 3. In room-temperature testing it is possible to minimize frictional problems by using a double layer of Teflon tape over the flat ends of the sample. In high-temperature tests a glass powder alcohol emulsion may be employed. Removing the effects of friction while the data obtained from plane-strain compression testing are analysed is equally important. In what follows, an example of the use of the above formula, Eq. (8.2), is presented, considering the compression test performed on a NbV microalloyed steel. Samples of the steel, measuring 10 mm in diameter and 15 mm long, were compressed under nearly isothermal conditions, at a constant true strain rate of 0.05 s21. The temperature of the sample was 950C. Three tests were conducted, the results of which are shown in Figure 8.10 (Wang, 1989). In all three tests, glass powder in an alcohol emulsion (Deltaglaze 19) was used as the lubricant. The first experiment used a sample prepared with its ends machined flat and beyond a true strain of 0.8 the resulting stressstrain curve indicated a steep rise which, if no elevated temperatures were employed, may be confused with strain hardening. In the second test, the well-known Rastegaev (1940) technique was followed, indenting the ends of the sample to a depth of 0.1 mm and leaving a ridge of about the same dimension. The objective was to trap the lubricant at the ram/sample interface. The resulting curve still indicated some rise. (It is noted that researchers often employ very shallow, concentric or spiral grooves on the flat ends to achieve the same objectives. The present writer’s experience indicates that multiple grooves are

5The ring compression test will discussed in detail in Chapter 9, Tribology. 178 Primer on Flat Rolling more difficult to machine without offering any significantly increased benefits over recessed ends in the reduction of friction.) In the third attempt, the value of the friction factor was determined, under the same conditions in a ring-test, to be 0.18. The uniaxial flow strength was calculated by Eq. (8.2) and is shown in Figure 8.10, identified as curve #3. The curve demonstrates the steady-state behaviour expected of the steel at the test temperature and strain rate.

8.6.2 Temperature Control The need to control the temperature during the tests for strength is equally important at both low and high temperatures. While it is essential to do so, it is almost impossible to conduct a test for the stressstrain curve under true isothermal conditions. It is equally difficult to measure the temperature of the sample accurately during the experiment. Overcoming the first difficulty is most important when the tests to determine the material’s resistance to deformation are conducted. The second problem is of significance when the test results are reported and mathematical models for use in subsequent analyses are to be developed6.

8.6.2.1 Isothermal Conditions The usual procedure in conducting a test is to preheat the furnace and the compression rams to the desired temperature. This is followed by opening the furnace door, placing the sample on the bottom ram for a sufficient length of time to reach a steady state, bringing the top ram in contact with the sample and starting the compression process. The rams are usually of a larger diameter than the compression sample and are of significantly larger thermal mass. They are connected to the load cell and the actuator by water-cooled heat exchangers, and their lengths are considerable, even if the heated length of the furnace is not very long. Because of the heat exchangers, the rams’ temperature is not uniform along their lengths and typically they are lower than that of the furnace. The furnace temperature is usually monitored by a thermocouple whose bead is a few millimetres away from the inner surface of the furnace’s insulation. The control of the furnace temperature is achieved by monitoring the output of this thermocouple. The average temperature within the furnace is quite cer- tainly lower than the indicated value. When the furnace is opened to allow the placing of the sample on the ram, considerable cooling takes place. While time consuming, expensive and labour intensive, thermocouples should always be embedded in the sample and in the loading rams. The thermocouple on the sample should be used to control the temperature of the furnace. The test is to commence when the sample and the ram temperatures are very close, within a pre-determined tolerance. (A modern alternative, of course, is the use of optical pyrometers through spy-holes in the furnace walls, instead of thermocouples).

6When reporting the results of the tests, many writers are guilty of not describing the equipment and the procedure in minute detail. Both of these are necessary if the tests are to be duplicated. Material Attributes 179

240 Figure 8.11 The temperature dependence of the peak stress of several steels. 200 Source: Lenard et al. (1999), reproduced with permission.

160

120

Peak stress (MPa) 0.120% Ti, 0.07% C 0.035% Ti, 0.06% C 80 0.028% Nb, 0.13% C AISI 5140 0.05% Nb, 0.12% C 40 600 800 1000 1200 Temperature (ºC)

The thermocouple in the sample will also indicate the temperature rise due to work done on it. In reporting the results this rise should be accounted for. Realizing that the work done per unit volume is almost exactly equal to the area under the true stresstrue strain curve, the temperature rise may be estimated by: Ð σ dε ΔT 5 ð8:3Þ cpρ where the specific heat is designated by cp and the density by ρ, both of which are temperature dependent; see Touloukian and Buyco (1970). Corrections to develop the flow curve under isothermal conditions require the determination of the temperature as the sample is being compressed and inter- and extrapolation to compute the appropriate values of the stresses. In these calculations it is assumed that all work done is converted into heat, an assumption which is close enough though not quite correct.

8.6.2.2 Monitoring the Temperature The potential accuracy of the temperature measurements should be considered, as well. Manufacturers’ catalogues list the accuracy of a type K (chromelalumel) thermocouple as 60.5%, full scale7. If testing at 1000C is considered, this indicates a potential error of 10C. The effect of this error in the magnitude of the temperature needs to be understood in light of the temperature sensitivity of the resistance to deformation of steels. This varies over a large range, as shown in Figure 8.11 which indicates the dependence of the peak stresses of several steels on the temperature.

7Platinumrhodium thermocouples, type R, could be used, of course, and these are considerably more accurate (B0.1%) than the type K version. They are much more expensive, however. 180 Primer on Flat Rolling

In the worst-case scenario, consider the microalloyed steel, containing 0.028% Nb and 0.13% C. The graph shows a slope of 0.9 MPa/C and the 10C difference would then indicate an error in the strength of 9 MPa. As the steel’s strength at that temperature is about 130 MPa, the very small error in temperature measurements creates a very much more significant error of about 7% in the strength data. The implications of this 7% are evident when considering the sensitivity of the predicted roll separating forces and roll torques to variations of the material’s strengths. Several high-temperature furnaces are available with a spy-hole, allowing the use of optical pyrometers which, when focused on the sample, would monitor its temperature as the test is proceeding. This approach is preferred over the use of thermocouples which require a hole to be drilled into the sample to house the thermocouple. While the stress concentration around the hole and the embedded thermocouple is not expected to affect the material’s resistance to deformation in any significant manner, it is nevertheless an interruption and can be avoided by the use of optical devices. Further problems with the embedded thermocouples include the different strength of the bead and the sample in addition to the possible imperfect contact of the bead and the bottom of the hole. The latter may be eased somewhat, but not eliminated completely, by the use of high conductivity cement.

8.7 The Shape of StressStrain Curves

The stressstrain curves of metals differ greatly, depending on the temperature at which the test is performed. The two cases, low- and high-temperature behaviour, are treated below.

8.7.1 Low Temperatures Stressstrain curves of an AISI 1008 steel, obtained in uniaxial tension and at room temperature, are given in Figure 8.12. Two curves are shown. In the first, the steel was tested as received. In the second, the results of cold working are evident, indicating that the steel’s strength increased by approximately 40% as a result of one cold-rolling pass in a two-high rolling mill, causing 58.5% reduction. Both curves indicate strain hardening. The steel is not highly ductile. Even in the annealed condition, the fracture occurred at a strain of 0.18. The stressstrain curve following the 58.5% cold reduction exhibits the yield point extension.

8.7.2 High Temperatures The shape of a stressstrain curve, obtained in a test, conducted at high temperatures, differs significantly from that at low temperatures, showing the effects of metallurgical phenomena on the resistance of the material to deformation. The difference is illustrated in Figure 8.13, which shows the true stresstrue strain curves Material Attributes 181

400 Figure 8.12 The stressstrain curves of an AISI 1008 steel.

300

200 AISI 1008 steel, annealed

As received Tensile stress (MPa) 100 58.5% reduction, single-stage

0 0.00 0.04 0.08 0.12 0.16 0.20 Strain obtained in compression testing of a 0.1% C, 0.0877% Nb, 0.0795% V steel. During the compression test, a glass-alcohol emulsion was used to cover the ends of the samples to minimize interfacial friction. While the curves have not been corrected for temperature rise, this omission is not expected to cause significant errors, since the rates of strain were not excessive. The curves demonstrate the traditionally expected behaviour of steels at high temperatures. Two metallurgical mechanisms, affecting the shape of the curves, are to be considered: these are the hardening and the restoration processes. The latter includes dynamic recovery and dynamic recrystallization. Since all samples were annealed prior to the tests, it may be safely assumed that the steels were initially fully recrystallized and that the austenite grains were uniform in size and were equiaxed8. As soon as the compression process begins, hardening due to the pancaking of the grains begins and at a fairly small strain, say 35%, dynamic recovery also starts. A micrograph taken at that strain would indicate the flattened grains. The migration of the dislocations may also be observed but no changes to the grains, other than some flattening, are expected due to dynamic recovery. The loading is now continued and the hardening and the softening processes occur simultaneously. When the rate of softening exceeds that of hardening, the slope of the stressstrain curve begins to decrease. At a particular strain, identified as the critical strain, usually denoted by εc, another restoration process, that of dynamic recrystallization, is initiated and the slope of the stressstrain curve drops even more. A micrograph, taken just beyond the critical strain, would show the new, strain-free grains nucleating, usually at the grain boundaries. The process is still

8A perfectly equiaxed figure is the circle with its diameter constant. An equiaxed grain is usually hexag- onal; see, for example, Figure 8.1. 182 Primer on Flat Rolling

(A) 250 Figure 8.13 True ε (s–1) stresstrue strain curves of 2 a high NbV steel, obtained 200 1 under nearly isothermal conditions and at strain rates 0.1 of 0.0012s21. 150

0.01

100 0.001 True stress (MPa)

50 T = 900°C

0 0.00 0.40 0.80 1.20 True strain (B) 250

ε –1 200 (s ) 2 1 150 0.1 (MPa)

σ 0.01 100 0.001

50 T = 950°C

0 0.00 0.40 0.80 1.20 ε continuing and now all three metallurgical events are active at the same time. Further straining reaches the condition when the rate of hardening just equals the rate of softening and a plateau in the stressstrain curve is reached, identified as the peak strain, εp. The stress at that location is referred to as the peak stress and is usually designated by σp. Further loading causes the softening rate to exceed the hardening rate and the material’s resistance to deformation falls until a steady state, at a strain identified as εss, is reached. Beyond that strain the stressstrain curve becomes independent of the strain but is dependent on the strain rate and the temperature. Material Attributes 183

Grains elongate Figure 8.14 A schematic Dislocation density increases diagram of a stressstrain Subgrains are created curve at high temperatures. Initial grains disappear Source Dynamically recrystallized grains are equaxial : Reproduced from εp Lenard et al. (1999), with permission; some changes Steady-state flow were introduced.

εc Stress ε ss ε = constant T = constant

Strain

Figure 8.15 The progress of dynamic recrystallization in a 0.1% C, 0.04% Nb steel. (A) shows the structure before deformation; (B) shows no recrystallization; (C) shows partial recrystallization; and (D) shows complete recrystallization (Cuddy, 1981).

A schematic diagram of a true stresstrue strain curve, obtained at high temperatures, is shown in Figure 8.14 (reproduced from Lenard et al., 1999, with some changes), with all three strains, εc, εp and εss indicated. Another set of micrographs (Figure 8.15AD) indicates the progress of dynamic recrystallization in a 0.1% C, 0.04% Nb steel (Cuddy, 1981). In Figure 8.15A, the structure before testing is shown. The austenite grains are large, measuring 370 μm on average. The structure, after straining to 0.4 at a temperature of 900Cata 184 Primer on Flat Rolling strain rate of 0.017 s21, is shown in Figure 8.15B. The recrystallization process has not started yet. Deformation to a strain of 0.43 at a higher temperature of 1000C and a strain rate of 0.05 s21 caused partial dynamic recrystallization (Figure 8.15C). The recrystallization process was completed when the sample was subjected to a strain of 0.55 at 1100C and at a strain rate of 0.17 s21 (Figure 8.15D).

8.8 Mathematical Representation of StressStrain Data

At this stage of the study, the necessary data on the metal’s resistance to deformation the stress, strain, rate of strain, temperature and hence, the ZenerHollomon parameter, defined as Z 5 ε_ expðQd=RTÞare in hand. In the expression Qd represents the activation energy for plastic deformation, R is the universal gas constant, 8.314 J/m/K and T is the temperature in Kelvin. The next step is to develop a mathematical model for further use in analysing a metal forming process. The traditional approach is to make use of non-linear regression analysis and then fit the experimental data, as best as possible, to a pre-determined relation. Another possibility is to place the experimental data in a multi-dimensional databank and when the stress values are needed in an application, inter- or extrapolate for them, at the actual strain, rate of strain and temperature. Two fairly recently developed possibilities to determine the material attributes have been developed, but so far they have not been employed extensively. One of them uses artificial intelligence, specifically neural networks, to estimate and predict the metals’ behaviour. The other, parameter identification, is based on a combination of a finite-element simulation of a test in the present instance, that would be a rolling pass with the measurements of the overall parameters, such as the roll separating force or the roll torque (Gelin and Ghouati, 1994; Kusiak et al., 1995; Malinowski et al., 1995; Khoddam et al., 1996; Gavrus et al., 1995). The measurements of the process parameters are then compared with the predictions by the finite-element method. An error norm is defined as the vector of distances between the measured and calculated values. The minimization of the error norm is used to determine the unknown parameters in the constitutive law. In the statistical method an equation is always obtained which can be used with more or less ease in the subsequent steps of the analysis. There are two difficulties. The first problem concerns the just-developed “best-fit curve” which may not fit all data points equally well, and therefore some carry-on errors are unavoidable. The second problem is encountered when additional material data are developed. The non-linear regression analysis needs then to be repeated and a new relation that fits the new data as well as before must be obtained. The latter deficiency, that of repeating the statistical re-development of the empirical relation, is overcome by the ability of the neural networks to renew them- selves. The disadvantage, often claimed by engineers, is that an equation is not obtained. Material Attributes 185

8.8.1 Material Models: StressStrain Relations There is an infinite number of possibilities in formulating the constitutive relations, both at high and low temperatures. These equations are just that: they are chosen in an arbitrary manner to describe the metals’ observed behaviour. The choice of the form and the independent variables are up to the researcher. Some of the better known and accepted forms are given below.

8.8.1.1 Relations for Cold Rolling While the choice of the form for stressstrain relations is practically infinite, two equations have been used regularly by researchers. Both relate the metal’s strength to strain only in addition to material constants which may depend on the rate of strain. The first is σ 5 Kεn, where the constants, K and n can be determined for any particular stressstrain data, either by a least-squares minimization routine or by forcing the curve through two pairs of stress and strain values. Both approaches are acceptable. The other relation, more suitable for the analysis of metal forming and particularly for the rolling process, also relates the metal’s strength to the strain in the form σ 5 Yð11BεÞn, where the three material constants need to be determined by fitting to experimental data. This expression indicates that the metal possesses some strength at zero strain. In addition to the strain, the strengths of some metals (e.g. titanium) are also dependent on the rate of strain. A relationship that has been found useful in such cases is σ 5 Yð11BεÞnε_m, where the exponent m is the strain rate hardening coefficient. Again, non-linear regression is needed to determine the coefficients and the exponents.

8.8.1.2 Relations for Use in Hot Rolling Statistical methods One of the simplest expressions, often used in the analysis of hot rolling problems, relates the metal’s strength to the average rate of strain and two material constants, in the form σ 5 Cε_m; values for the constants have been given by Altan and Boulger (1973) for a selection of ferrous and non-ferrous metals. An often-quoted source for stressstrainstrain rate relations is the compilation of experimental data, Suzuki et al. (1968). Stressstrain curves for a large number of ferrous and non-ferrous metals have been given, at various temperatures and rates of strain. The chemical compositions of the metals have also been provided. Several somewhat more complex equations were listed by Lenard et al. (1999), some of which are repeated below. One of these, based on the hyperbolic sine function, is due to Hatta et al. (1985). The hyperbolic sine law gives the strain rate in the form: Q ε_ 5 c sinh ðασÞn exp 2 ð8:4Þ RT 186 Primer on Flat Rolling

Hatta et al. (1985) define the various terms in Eq. (8.4), for a 0.16% C steel, as

c 5 exp ½24:4 2 1:69 ln Cðs21Þð8:5Þ

n 5 exp ½1:63 2 0:0375 ln Cð8:6Þ

α 5 exp ½2 4:822 1 0:0616 ln Cðin MPa21Þð8:7Þ and

Q 5 exp ½5:566 2 0:0502 ln Cðin kJ=moleÞð8:8Þ

While Hatta et al. (1985) determined the activation energy by non-linear regression analysis, a somewhat more fundamental approach, making use of experimental data, is likely to lead to more physically realistic values. The recommendations are to follow these steps:

G Re-write Eq. (8.4) in a different form: ε_ 5 Aσn expð2 ðQ=RTÞÞ; G perform a number of stressstrain tests at several temperatures and rates of strain; G obtain the peak stresses and prepare a loglog plot of the peak stresses versus the temperatures; G at an arbitrary stress level, obtain from the plot two temperatures and the corresponding rates of strain; and G determine the activation energy from the slope Q Δðlnðε_Þ=Δð21=RTÞ. The activation energy, thus determined for a 0.1% C, 1.093% Mn, 0.088% Nb, 0.0795% V steel, was 483 kJ/mole (Lenard et al., 1999). In general, higher alloy content leads to larger values of the activation energy. It is noted that the strain doesn’t appear in Hatta’s relations, indicating that they are strictly applicable in the steady-state region. Wang and Lenard (1991) included the strain in the exponents of Eq. (8.4) while developing a high-temperature model for the deformation of a NbV steel. Another set of empirical relations have been presented by Shida (1969), giving the metal’s resistance to deformation as a function of the temperature, carbon content, strain and strain rate. These equations have been used successfully in a number of publications, concerned with hot rolling or hot forging of steels. The relations have been developed by Shida for carbon steels. It is expected that use of carbon equivalent instead of the carbon content may allow Shida’s formulae to be used for alloy steels, as well. The carbon equivalent may be calculated as a function of the alloy content of the steel from the relation9:

Ceq 5 C 1 Mn=6 1 ðCu 1 NiÞ=15 1 ðCr 1 Mo 1 VÞ=5 ð8:9Þ

9There are several formulae available for “carbon equivalent”, mostly developed for the study and modelling of welding processes. Material Attributes 187

The flow strength of the steel, in kg/mm2, is given by Shida, in terms of the carbon content in %C, the rate of strain and the temperature: ε_ m σ 5 σ f ð8:10Þ f 10

The terms in Eq. (8.10) are defined, depending on the temperature of deformation. For

C 1 0:41 T $ 0:95 ð8:11Þ C 1 0:32 5 0:01 σ 5 0:28 exp 2 ð8:12Þ f T C 1 0:05 and

m 5 ð20:019C 1 0:126ÞT 1 ð0:075C 2 0:05Þð8:13Þ

For temperatures below that defined by Eq. (8.11), C 1 0:32 0:01 σ 5 0:28 qðC; TÞ exp 2 ð8:14Þ f 0:19 ðC 1 0:41Þ C 1 0:05

C10:49 2 C 1 0:06 qðC; TÞ 5 30ðC 1 0:9Þ T20:95 1 ð8:15Þ C10:42 C 1 0:09 and

0:027 m 5 ð0:081C 2 0:154ÞT 2 0:019C 1 0:207 1 ð8:16Þ C 1 0:32

The remaining parameters are

f 5 1:3ð5εÞn 2 1:5ε

n 5 0:41 2 0:07C and

T 5 ðT 1 273Þ=1000 188 Primer on Flat Rolling

320 Figure 8.16 A comparison of the Altan and Boulger predictions of several empirical relations, 280 Suzuki et al. designed for high-temperature behaviour. Shida 240 Hatta 800°C 200

160

120 Stress (MPa)

80 1200°C

0 0 4 8 12 16 Strain rate (s–1)

In the above relations, T is the temperature inC and C is the carbon content in weight %. The true strain is denoted by ε. Shida gives the limits of applicability of his empirical relations10 as C , 1.2%; T in between 700C and 1200C; ε_ in between 0.1 and 100 s21 and ε , 70%. While these equations have been used successfully in many instances, some caution is needed in specific applications. The difficulties are illustrated in Figure 8.16 where the predictions of four previously published equations or measurements are compared, by plotting the predicted strength as a function of the rate of strain, at a temperature of 800C (close to the transformation temperature) and 1200C, where the steel is fully in the austenitic state. A carbon steel is chosen for the comparison. The curve denoted by the crosses is due to Altan and Boulger (1973). The equations for a steel, containing 0.15% C, are, at a temperature of 800C, σ 5 145:38ε_0:109 MPa and at a temperature of 1200C, σ 5 39:27ε_0:181 MPa. The steel, closest to this and whose stressstrain curves are given by Suzuki et al. (1968), contains 0.147% C. Non-linear regression analysis gave the equations of the curves, plotted in Figure 8.16 (denoted by the diamonds), at 800C, σ 5 182:34ε0:1039 MPa and at 1200C, σ 5 59:06ε0:1698 MPa. The parameters of the equation due Hatta et al. (1985) are given above; see Eqs. (8.4)(8.8). This curve is given by the upward triangles. Finally, the curve obtained using Shida’s relations for a steel containing 0.15% C is denoted by the squares. It is observed that all four curves give the expected trends. The differences of the predictions are quite large, though, reaching up to 35%.

10In several cases of empirical relations, developed to represent the metal’s resistance to deformation, the limits of applicability are not given a major omission. Material Attributes 189

50 Figure 8.17 The predictions of the flow strength of a commercially pure –1 Temperature ε = 7.58 s aluminium alloy (Chun et al., 1999). 40 400°C

30 450°C

500°C

20 Training Testing True stress (MPa)

10 Neural network Experiment

0 0.00 0.20 0.40 0.60 0.80 1.00 True strain

Developing a databank of data Use of a multi-dimensional databank was explored by Lenard et al. (1987). Stress values at particular strain, strain rate and temperature were stored and retrieved as needed in the analysis of the flat rolling process. The results compared favourably with data obtained by other approaches. While it is believed that using a large databank removes the need to develop arbitrary empirical relations and that it would remove one error-prone step from the analysis of the rolling process, no extensive use of the approach is evident in the technical literature.

Artificial neural networks The predictive capabilities of the method are demonstrated by considering the hot compression testing of an aluminium alloy (Chun et al., 1999). Cylindrical samples of the Al 1100-H14 alloy of 20 mm diameter and 30 mm height have been used to determine the metal’s resistance to deformation. The specimens have been machined from plates with the longitudinal direction parallel to the rolling direction. The flat ends of each specimen were machined to a depth of 0.1B0.2 mm to retain the lubricant, boron nitride. A type K thermocouple in an INCONEL shield, with outside diameter of 1.54 and 0.26 mm thermocouple wires, was embedded centrally in each specimen. The compression tests were carried out on a servohydraulic testing system, at a true constant strain rate of 7.58 s21 and at sample temperatures of 400C, 450C and 500C. The results are shown in Figure 8.17. The network was trained using the data at the temperatures of 400C and 500C. Testing the network was performed by comparing the predictions to measurements obtained at 450C. The good predictive ability of the network is evident.

Parameter identification The parameter identification method has been developed in the last decade and applied to problems of metal forming (Gavrus et al., 1995; Boyer and Massoni, 190 Primer on Flat Rolling

Figure 8.18 The stressstrain 350°C 150 curves of an aluminium alloy, measured and compared to the calculations by the parameter 100 425°C identification technique (Lenard et al., 1999). 500°C

Stress (MPa) 50 Width of the platen 5 mm Experiment Thickness of the sample 10 mm Calculations Strain rate 1 s–1 0 0.0 0.2 0.4 0.6 0.8 1.0 Strain

2001; Gelin and Ghouati, 1999; Gelin and Ghouati, 1994; Kusiak et al., 1995). The details of the technique have been reviewed by Lenard et al. (1999). Pietrzyk (2001) used the technique to determine and predict the high-temperature behaviour of a low-carbon steel and a 304 stainless steel; the predictions and the measurements compared very well. The ability of the method to predict the hot stressstrain curves of a harder aluminium alloy, when subjectedtoplane-strain compression, is evident in Figure 8.18 (Lenard et al., 1999). A description of the experimental procedure and theresultsoftheanalysisweregivenbyPietrzykand Tibbals (1995). The experiments have been carried out at temperatures of 350C, 420C and 500C and at a strain rate of 1 s21. The initial thickness of the aluminium alloy samples was 10 mm. The width of the platens and of the sample was 15 mm.

8.9 Choosing a StressStrain Relation for Use in Modelling the Rolling Process

It is clear at this point that to satisfy the demands of its users the success of a mathematical model of the flat rolling process depends on how well the tribological and material attributes are treated. Tribology is to be discussed in the next chapter. The choice of a stressstrain equation will also contribute to success or failure and extreme caution is advised when that choice is made. While the researchers have several possible avenues to follow, one of the approaches, given below, will likely be chosen: 1. Conduct independent testing to determine the metal’s mechanical attributes and use non- linear regression analysis to develop a model for later use. 2. Search the existing technical literature for information on the attributes. It is strongly advised, however, that if at all possible, the first approach in the list should be followed. The reasons are clearly demonstrated in Figure 8.16. Material Attributes 191

8.10 Summary

The metals’ resistance to deformation was discussed in the chapter. First, the recently developed steels were presented and their micrographs were demonstrated. This was followed by a presentation of the perennial battle for supremacy between the steels and aluminium alloys. The most prevalent, traditional techniques available to test the metal’s response to loading were given, including tension, compression and the torsion tests. Their advantages and disadvantages were listed. Approaches towards the treatment of constitutive data were presented. The mathematical models, arrived at by statistical techniques, parameter identification and artificial intelligence methods, designed to describe the behaviour of the materials, were also included. A simple approach to determine the activation energy was mentioned. Recommendations concerning the determination of the metal’s resistance to deformation were given. 9 Tribology

9.1 Tribology A General Discussion

It is appropriate here to start with a quotation from Roberts (1997). He writes,

Of all the variables associated with rolling, none is more important than friction in the roll bite. Friction in rolling, as in many other mechanical processes can be a best friend or a mortal enemy, and its control within an optimum range for each process is essential.

While Roberts wrote about friction, it is further appropriate and even necessary to replace that term with “tribology”. The study of surfaces in relative motion, in contact and under pressure that of tribology is a very broad subject. It has been studied by scientists and applied by engineers for thousands of years. The points of view of its practitioners are equally wide, encompassing the disciplines of tribochemistry, tribophysics, chemistry, chemical engineering, nanotribology, surface analysis, surface engineering, fluid mechanics, heat transfer, mathematics and mechanical engineering and the list is far from complete. Attending a large, comprehensive conference, entitled “Tribology”, requires careful choice of the lectures to be attended and may easily lead to information overload. In the present context, that is, flat rolling of metals, the focus is on three interconnected phenomena: friction, lubrication and heat transfer at the contact surfaces. These, in turn, create roll wear, and in the metal rolling industry the costs associated with wear problems account for nearly 10% of total production costs. The cost of inappropriate understanding or application of tribological principles has been estimated to be as high as 6% of the GDP in the United States (Rabinowicz, 1982). Interesting data have been given in a recently published book by Stachowiak and Batchelor (2005) concerning the same topic. They quote the Jost report (1966) which estimated that the correct application of the basic principles of tribology would save the UK economy d515 million per annum. A report by Dake et al. (1986) indicated that in the United States about 11% of the total annual energy could be saved in the areas of transportation, turbo machinery, power generation and industrial processes.

A recent search on the Internet, using the word “Tribology”, yielded 237,000 results1. It is, of course, not realistic to check and evaluate all of these. One that appeared of potential interest was the Virtual Tribology Institute, a group of European organizations that deal with all aspects of the subject. Members are located in several European countries while the manager of the Institute is located in Belgium. Also found by the search is the Center for Tribology, identifying itself as the largest tribology testing laboratory in the world, located in Campbell, CA. Under “useful links” the site gives a list of universities where research on tribology is performed. Unfortunately, however, when checking the title “Northern American Universities”, schools in the United States only are mentioned, while Mexican and Canadian places are omitted. Probably and arguably the most complete website is the one provided by the University of Sheffield, checked in early 2007. The four sub-topics listed on the site are Research, Teaching, Tools and Information and Consultancy. Clicking on Tools and Information, a plethora of very useful items is found. The list of books dealing with the topic of tribology is likely the most complete available. The list of journals, periodicals and online resources are also most impressive. The list of “Tribologists Around the World” is very useful when one wants to know who is dealing with what in the field. It is evident that while the interactions of the components, parameters and variables of the field of tribology are beyond the capabilities of a single discipline, listing them is still valuable. For the most complete listing of the attributes of an interface, in addition to their interactions, the reader is referred to the table shown in Fig. 3.2 of “Tribology in Metalworking” by Schey (1983). The table was first presented at a conference in 1980, but to the best of the present writer’s knowledge, no better or more complete compilation has been given since. The three components of a metal working system are identified as the die, the work piece and the interface between the two, which includes the lubricant. The table is reproduced as Figure 9.1 and an examination of the interconnection of the parameters indicates the complexities of the process. As mentioned above, the phenomena of friction, lubrication and heat transfer will be discussed in turn in what follows, followed by a brief look at their combined effect: roll wear.

9.2 Friction

9.2.1 Real Surfaces An enlarged view of the cross-section of two surfaces is presented by Schey (1983) and the figure is reproduced here as Figure 9.2, clearly demonstrating the validity of the comment written by Batchelor and Stachowiak (1995), stating that surfaces

1The search was done in 2006. Tribology 195

Die Lubricant Workpiece Macrogeometry Rheology Macrogeometry MicrogeometryShear strength Microgeometry Roughness Temp. dependence Roughness Directionality Pressure dependence Directionality Mechanical properties Shear-rate dependence Mechanical properties Elastic Composition Elastic Plastic Bulk Plastic Ductile Carrier Ductile Fatigue Surface Fatigue CompositionBoundary and E.P. Composition Bulk Bulk Surface Surface Reaction product Temperature Reaction product Adsorbed film Adsorbed film Phases Application Phases Distribution Supply Distribution Interface Resupply Interface Temperature Temperature Atmosphere Velocity Velocity Process Process Geometry Surface extension Macro Virgin surfaces Micro Temperature Speed Contact time Approach Sliding Reactions Pressure Lubricant transfer Distribution Heat transfer

Friction Lubrication Adhesion Hydrodynamic Plastohydrodynamic Wear Mixed film Die Boundary Workpiece Dry

Equipment Product quality Pressures Surface finish Hydrodynamic Forces Deformation pattern lubrication Power requirements Metallurgical changes Plastohydrodynamic Mechanical properties lubrication Residual stresses Boundary Fracture lubrication

Figure 9.1 The tribological system. Source: Schey (1983), reproduced with permission. are never clean. An adsorbed film and an oxide layer are always present in industry as well as in a laboratory. It is, of course, possible, albeit difficult, to provide a controlled atmosphere while testing tribological attributes and this is often done. The results thus produced are of interest since the ability to control the independent variables is increased in a very significant manner; however, the applicability of 196 Primer on Flat Rolling

Die Figure 9.2 An enlarged view of the cross-section of two surfaces. Hard phases Source: Schey (1983), reproduced Matrix with permission. Adsorbed film ~ 30 Å

Reaction Workpiece (Oxide) film 20–100 Å Bulk

Surface layer 1–5 μm Disturbed enriched/depleted such data to a real-life, industrial environment is highly questionable. The surfaces are also never perfectly smooth2.

9.2.2 The Areas of Contact Two concepts need to be defined before any further discussion of the mechanisms of friction may be presented. The apparent area of contact, A, is the first and it is defined by the overall, outer dimensions of the contact surface. The real area of contact, usually denoted by Ar, affects the frictional phenomena in a much more fundamental manner and is defined as the totality of the areas in contact at the asperity tips. When the two surfaces approach one another, contact is first made at those tips, see Figure 9.3. It is to be realized that the shapes, dimensions, locations and contacts of the asperity tips are completely random; hence, the interactions may, most appropriately, be termed chaotic (Batchelor and Stachowiak, 1995). As the normal force increases, the asperities flatten and the real area of contact increases. If there is sufficient time note that no more than a few milliseconds, or even microseconds, are needed adhesive bonds are created, as described by the adhesion theory of Bowden and Tabor (1950) which gives the requirements for the establishment of adhesion: that the surfaces should be clean and close enough for interatomic contact. As the normal force increases and the asperity tips are flattened further, new, clean surfaces are created and the real area of contact approaches the apparent area. The rate of approach depends on the resistance to

2A strong criticism is offered here of the often-repeated statement “the surface is perfectly smooth” to imply a lack of friction. This comment was written very recently in a calculus text to be used in an introductory course. Exactly the opposite is correct as the smooth surfaces provide a large real contact area and lead to high frictional resistance. Moving two perfectly smooth surfaces in contact, relative to one another, would be difficult as the resistance to overcome would be at the maximum. The proper terminology, if no friction is to be assumed, should be “the surfaces are perfectly lubricated”, or better still, just write “the frictional resistance is taken to be zero”. Tribology 197

Flattened portions

Two surfaces are First contact is at The asperities approaching the asperity tips flatten and bonds are created

Figure 9.3 Asperities, valleys and the contact of two rough surfaces. deformation and the formability of the asperities. Relative movement of the contacting bodies is then possible only by applying a shear force, large enough to separate the contacting, flattened and bonded asperities.

9.2.2.1 The Relationship of the Apparent and the True Areas of Contact The definitions of the two areas, given above, are quite clear. The interest at this point is how A and Ar are related and how the true area may approach the apparent area when circumstances change. The implication of this is also clear: as the asperities flatten while the loads are increased, the nature of the contact is changing and this will affect all of the interfacial phenomena. Schey (1983) classifies the reaction of asperities to the normal and shear stresses at the contact surfaces in the following manner:

G The average normal stress is below the flow strength of the work piece, creating elastic stresses within the bulk. The asperity tips are experiencing permanent deformation and the true area is approaching the apparent area. G In addition to the normal pressures, which are still below the flow strength, there is relative sliding in between the die and the work piece. G The normal pressures cause plastic flow of the bulk of the work piece. Bowden and Tabor (1964) discuss the events that occur when the asperity tips come into contact and there is a certain amount of normal force applied to the two bodies as they are approaching one another. While the average normal pressure may be well below the flow strength of either component, the stresses at the extremely small areas of the asperity tips will always be high enough to create permanent deformation. The tips will then deform and the load carried (W) will determine the true area of contact in terms of the flow strength of either body as:

W Ar 5 ð9:1Þ σfm where σfm, the flow strength, may well increase as the deformation is proceeding due to strain and strain rate hardening. The magnitude of the tangential force, (F), 198 Primer on Flat Rolling required to move one of the bodies with respect to the other, depends on the shear strength (τ) of the junction. That force may be a combination of adhesive and ploughing forces, depending on the nature of the contacting surfaces:

W F 5 Arτ 5 τ ð9:2Þ σfm

A more recent examination of the response of asperity tips to loads is reviewed by Stachowiak and Batchelor (2005). They quote the studies of Whitehouse and Archard (1970) and Onions and Archard (1973) which indicate that a large propor- tion of the contact between asperities is elastic under normal operating loads. They further mention, however, that an exception, that is, permanent asperity deformation, may occur at the contact surfaces in metal working processes. Since the normal pressures in bulk forming are significantly larger than in sheet metal forming processes, plastic deformation of the asperities there is an important contributor to surface phenomena. The first step is to determine if the contact is mostly elastic or is there a significant amount of plastic deformation of the tips of the asperities. This is accomplished by calculating the plasticity index, ψ. Stachowiak and Batchelor (2005) present three definitions for the plasticity index. They identify the first as due to Greenwood and Williamson (1966), followed by those of Whitehouse and Archard (1970) and by Bower and Johnson (1989). The three formulas give the index in terms of material and geometrical attributes. The easiest to use plasticity index is the one given by Bower and Johnson (1989) in the form: 0 ψ 5 E σκ 0:5 : s ð Þ ð9 3Þ ps

ψ σ where the plasticity index for repeated sliding is s, isthermssurfacerough- ness of the harder surface in m, κ is the curvature of the asperity tip in m21 and 3 ps is the shakedown pressure of the softer surface. If the stresses are below the shakedown pressure, the deformation is elastic; otherwise, plastic deformation is expected. The term E0 is the composite Young’s modulus for the two bodies in contact, a and b:

2 ν2 2 ν2 1 5 1 a 1 1 b : 0 ð9 4Þ E Ea Eb and ν is Poisson’s ratio.

3The shakedown pressure is a limit; when the magnitudes of stresses are below it, elastic deformation is present, while above it, plastic deformation occurs (Stachowiak and Batchelor, 2005). Tribology 199

The Whitehouse and Archard (1970) model defines the plasticity index as: E0 σ ψ 5 ð9:5Þ H β where H is the hardness of the deforming surface, and β is the correlation distance. When ψ , 0:6, elastic deformations are expected. When ψ . 1, most of the contacts experience plastic deformation. In the flat rolling process, especially where metals are concerned, significant plastic deformation of the asperities is certain to occur. To show that this is the case, consider the data of McConnell and Lenard (2000) where in each pass the average roll pressures are calculated to be in the range of 700900 MPa while the yield strength of the rolled metal is in the order of 250300 MPa; permanent deformation will be present. A relationship between the apparent area and the true area, when plastic deformation takes place has been given by Majumdar and Bhushan (1991) in the form:

W 5 KφA ð9:6Þ AE0 r

5 =σ φ 5 σ = 0 5 = σ where K H y, y E ,andAr Ar A and y is the yield strength. Re-substitution of these terms into Eq. (9.6) gives the true area of contact as a function of the load and the hardness of the deforming body, in a form similar to the Bowden and Tabor (1964) model, with the hardness replacing the flow strength:

W A 5 ð9:7Þ r H

Korzekwa et al. (1992), in stating the need for quantitative understanding of friction, present a model for the evolution of the contact area in a sheet undergoing a plastic forming process. A rate dependent material, subjected to large range of strains, is considered along with the effect of bulk deformation on asperity flattening, modelled as the indentation of a flat surface by a rigid punch. A viscoplastic finite-element model is used in the calculations of the changing true contact area as the deformation is continuing. While the results concentrate on low contact pressures which are appropriate in sheet metal forming operations, it is believed that with increasing loads the trends may not change markedly. The data are presented in the form of graphs, reproduced here as Figure 9.4, showing the changing contact area fraction, defined as the ratio of the half width of the rigid indenter to half the distance between the centres of the indenters, as a function of the bulk effective strain. The deformation of stainless steel 304L was considered. The results indicate that the true area of contact increases as the normal pressures and the asperity slopes increase. As well, the straining directions also have a significant effect on the growth of the true contact area. 200 Primer on Flat Rolling

(A) Figure 9.4 Contact area 0.8 fraction as a function of the A bulk strain for a range 0.6 of normal pressures (A), asperity slopes (B) and straining directions (C). 0.4 Source: Korzekwa et al. (1992). 0.2 P = 8 MPa θ = 2° P Contact area fraction Contact area fraction = 20 MPa φ = 2.678 P = 40 MPa 0 0 0.1 0.2 0.3 0.4 0.5 Bulk effective strain (B) 0.8 A

0.6

0.4

0.2 θ = 1° P = 20 MPa Contact area fraction Contact area fraction θ = 2° φ = 2.678 θ = 5° 0 0 0.1 0.2 0.3 0.4 0.5 Bulk effective strain

Sutcliffe and co-workers have considered the problems associated with asperity deformation and the true and the apparent contact areas in flat rolling of metals. Their work is innovative and at the writing of the present manuscript, the most up to date. Sutcliffe (2000) lists two factors that affect frictional conditions: the manner in which the contacting surface asperities conform to each other while in contact and the frictional mechanisms at those contacting areas and at the valleys in between. He writes that in considering the deformation of the asperities, the effects of bulk deformation and of the wavelength need to be taken into account, as well. He writes that when sub-surface deformation is accounted for the realistic approach, especially in bulk metal forming and in flat rolling the asperities are shown to flatten more. Sutcliffe (2002) presents the rate of change of the ratio of contact areas as a function of the bulk strain. When the roll roughness is in the direction of rolling:

dA W 5 ð9:8Þ dε tan θ Tribology 201

1.0 Figure 9.5 The change of the area of contact ratio with bulk strain. 0.9 Source: Sutcliffe (2002), reproduced with permission. 0.8 P/2K = 1.0 0.66 0.7 A

0.6 P/2K = 0.40

0.5

0.4 Theory

Area of contact ratio ratio contact of Area 0.3 P/2K = 1.0 0.2 Experiments P/2K = 0.66 P/2K = 0.40 0.1

0 0.05 0.10 0.15 ε Bulk strain x

and when it is in the transverse direction: dA 1 W 5 2 A ð9:9Þ dε 1 1 ε tan θ where the bulk strain is designated by ε, the flattening rate is W and the slope of the asperities is θ. Integration of these relations yields the area of contact ratio as a function of the bulk strain and the normal pressure, shown in Figure 9.5. Reasonable agreement of the predictions and the measurements is observed in Figure 9.5. Stancu-Niederkorn et al. (1993) list some of the experimental techniques available to determine the real contact area. They classify them in two categories: off-process and in-process inspections. Off-process approaches, which cannot measure the elastic deformation, include measuring the profile after deformation by inspection or by interferometry. The authors describe an experimental technique to measure the real contact area while the bulk of the work piece undergoes plastic deformation using ultrasound waves, the in-process inspection. Measurements were taken in free upsetting of steel samples and in closed-die upsetting, using copper specimens. Dry and lubricated conditions were examined. In the free upsetting tests, the real area of contact increased fast with increasing loads. In the closed-die upsetting, the real contact area reached about 95% of the apparent area at a normal load of 1100 MPa. Azushima (2000) used the finite-element method to analyse the deformation of the hills and 202 Primer on Flat Rolling valleys as a result of the pressure of the entrapped oil. He plotted the dependence of the contact area ratio on the reduction of the height of the asperities and found that without the oil the flattening is much more pronounced. The area ratio was found to remain relatively constant when the lubricant was entrapped in the valleys. Siegert et al. (1999) described the development of optical measurement techniques and computer workstation technology, using which they characterized the topography of sheet surfaces in three dimensions. The instrument is expected to be usable directly in the press shop.

9.2.3 Definitions of Frictional Resistance There are two traditional approaches to express the frictional phenomena in between two surfaces in contact, in relative movement and under pressure. In one of these, the coefficient of friction, as defined by Amonton and Coulomb and applied in most analyses of problems of metal forming, is given as the ratio of the interfacial shear stress to the normal pressure:

μ 5 τ=p ð9:10Þ

The friction factor, on the other hand, is given as the ratio of the interfacial shear stress and the yield strength in shear of the softer material in the contact:

m 5 τ=k where 0 # m # 1 ð9:11Þ

The existence of perfect lubrication is indicated when m 5 0 while m 5 1 points to sticking conditions. In developing mathematical models of bulk metal forming processes either coefficient may be used; however, both describe the interactions at the interface in a highly simplified manner and both involve some conceptual difficulties. Schey (1983) points out that the AmontonCoulomb definition becomes meaningless when the normal pressure is several times the flow strength of the metal. This is because the interfacial shear stress cannot rise beyond the yield strength in pure shear of the materials in contact and its ratio to the increasing normal pressure would continue to decrease and hence, the coefficient would also decrease. Mro´z and Stupkiewicz (1998) agree with Schey (1983), writing:

...the classical AmontonCoulomb model is not suitable for most metal forming processes...

The difficulty with the application of the friction factor is the lack of precise knowledge of the meaning of k, originally defined as the yield strength in pure shear of the softer material of the pair in contact, while ideally it should represent the strength of the interface. Again as pointed out by Schey (1983), the properties of the interface are not necessarily identical to the properties of the materials in contact. By examining Figure 9.2, which shows a realistic view of an interface, involving surface layers, oxides and adsorbed films, equating k of the interface to that of one of the materials may indeed be troublesome. Tribology 203

Wanheim (1973) was among the first researchers to write that the usual CoulombAmonton model doesn’t apply at the high normal pressures which exist in bulk forming processes. In those cases he suggests that the frictional stress should be taken as a function of the normal pressure, surface topography, length of sliding, viscosity and the compressibility of the lubricant. Wanheim and Bay (1974, 1978) propose a general friction model, using the above-mentioned ideas. In their model, Coulomb friction is taken to be valid at low normal pressure whereas the friction stress approaches a constant value at high normal pressures. The approach was applied successfully to model the pressure distribution in plate rolling and the cross shear plate rolling process (Zhang and Bay, 1997). A mixed friction model was also used by Tamano and Yanagimoto (1978) with Coulomb friction at low pressures and sticking friction at high pressures. Another approach, mostly used in finite-element modelling, is to introduce a “friction layer” in between the contacting surfaces. Montmitonnet et al. (2000) discuss the wear mechanisms and the differential hardness of the tool and the work piece that create a third body in between the die and the worked metal, identified as the transfer layer. Regardless of the manner in which friction is to be modelled, some difficulties, uncertainties and unknowns will always remain. In Chapter 5, the friction factor was used in developing a 1D model of the flat rolling process, partially as the result of the comment of a reviewer of one of the present author’s recent manuscripts, questioning the use of the coefficient of friction in the flat rolling process. Several factors need to be considered when deciding to use “k”, the friction factor. One is consideration of the pressure-sensitivity of lubricants4, which, for an Society of Automotive Engineering (SAE) 10 W oil, is given as 0.0229 MPa21 by Booser (1984), who also gives the viscosity as 32.6 mm2/s. If the roll pressure is 800 MPa, not an unreasonable magnitude when cold rolling steel, the Barus equation (see Eq. (9.48)) gives the viscosity at that pressure as 2.9 3 109 mm2/s. This number, while possibly unrealistically high, indicates that shearing the lubricant at that pressure may well require as much of an effort as shearing the metal. The other is the comment, referred to above, concerning the level of stress at which the tungstennickel junction failed (Pashley et al., 1984) and the third, also mentioned above, is the conclusion that ploughing was the major frictional mechanism (Lenard, 2004; Dick and Lenard, 2005) when a sand-blasted roll was used. Carter (1994) and Montmitonnet et al. (2000) reinforce this last factor by indicating that ploughing may be as important as adhesion in understanding frictional resistance. The last consideration concerns the experimental difficulties in determining either the coefficient of friction or the friction factor. The coefficient is never measured directly in the many available tests that have been published; instead, the normal and the tangential forces are measured and their ratio yields the magnitudes. The friction factor, however, can be obtained easily by the well-known ring test5.

4The sensitivity of the lubricant to pressure will be discussed in Section 9.4.1.2. 5The ring test will be discussed in Section 9.3.1.3. 204 Primer on Flat Rolling

9.2.4 The Mechanisms of Friction The mechanisms of interface contact have been discussed with the aid of a very well prepared figure by Batchelor and Stachowiak (1995), reproduced here as Figure 9.6. They identify adhesion, ploughing and viscous shear as the main contributors to frictional resistance. In bulk forming processes the former two are the most likely events to occur, as complete separation of the surfaces and thus full hydrodynamic conditions are rarely realized in practice. Montmitonnet et al. (2000) discuss surface interactions further and mention the possibility that particles will be detached from one of the contacting bodies, possibly resulting in micro- cutting. Further, a wave may be pushed along the surface, creating a bulge, or repeated contact may cause fatigue failure. The relative magnitudes of adhesion and ploughing have been examined by Mro´z and Stupkiewicz (1998). While developing a constitutive model for friction in metal forming processes, the authors indicate that friction forces include both adhesion and ploughing. They present a combined friction model, which simulates the interaction of the tool’s asperities with that of the work piece. In the mathematical model the effect of bulk plastic deformation is neglected; however, and as they write, experimental verification of the predictions is still required. Much depends on the angle of attack between the contacting surface asperities and on how the harder surface of the tool is prepared. Grinding, the traditional approach in preparing the rolls in the metal rolling industry, would produce relatively shallow angles while sand blasting would result in sharp asperities. Examining the effect of progressively rougher, sand-blasted rolls on the coefficient of friction and the resulting rolled surfaces, ploughing appeared to be the major component, contributing to frictional forces and overwhelming the effects of adhesion (Lenard, 2004; Dick and Lenard, 2005).

Asperity of harder surface or trapped wear particle Ploughing Viscous drag

Body 1 Motion Shearing of film material body 1 Body 2 motion Wave of material Film material Plastically deformed layer body 2 Adhesion Adhesive bonding Deformed asperity Body 1 Motion

Figure 9.6 The major mechanisms of friction. Source: Batchelor and Stachowiak (1995). Tribology 205

9.3 Determining the Coefficient of Friction or the Friction Factor

Since the Coulomb coefficient of friction is defined as a ratio of forces and the friction factor is defined as a ratio of stresses, neither can be measured directly. Several experimental approaches are available, however, to determine various experimental parameters and thereby deduce the magnitude of the coefficient or the factor.

9.3.1 Experimental Methods Several methods for measuring interfacial frictional forces during plastic deformation have been developed, some of which have been listed by Wang and Lenard (1992). A more comprehensive list, applicable to other metal forming processes, including bulk and sheet metal forming, has been presented by Schey (1983). Some of the more useful approaches are described below.

9.3.1.1 The Embedded Pin Transducer Technique Originally suggested by Siebel and Lueg (1933) and adapted by van Rooyen and Backofen (1960) and Al-Salehi et al. (1973), the method has been applied to measure interfacial conditions in cold flat rolling (Karagiozis and Lenard, 1985; Lim and Lenard, 1984), warm rolling (Lenard and Malinowski, 1993) and hot rolling of steels (Lu et al., 2002) and aluminium (Hum et al., 1996). Variations of this procedure have been presented by Lenard (1990, 1991) and Yoneyama and Hatamura (1987). Typical results, obtained by this approach, are shown in Figures 9.7 and 9.8 for warm rolling of aluminium (Lenard and Malinowski, 1993) and hot rolling of

250 Figure 9.7 Roll pressure and friction 1100 H 14 Al 200 stress during warm rolling of an rolled at 100°C aluminium strip, obtained with the 150 12 rpm (157 mm/s) 39.5% reduction use of pins and transducers 100 embedded in the work roll. 50 Source: Lenard and Malinowski 0 (1993).

–50 Friction stress –100 –150 Roll pressure –200

Roll pressure and friction stress (MPa) –250 –300 048121620 Distance from exit (mm) 206 Primer on Flat Rolling

150 Figure 9.8 Roll pressure and Low-carbon steel friction stress during hot rolling 100 Friction stress Rolled at 1000°C 35 rpm (412 mm/s) of steel, obtained with the use of 20% reduction pins and transducers embedded 50 in the work roll. 0 Source: Lu et al. (2002).

–50

–100 Roll pressure –150

Roll pressure and friction stress (MPa) –200

–250 048121620 Distance from exit (mm)

steel (Lu et al., 2002), respectively. It is evident that the friction hill, derived by the traditional, 1D models of the flat rolling process and employing constant coefficients of friction, leads to unrealistic distribution of the roll pressures in the form of the friction hill. As Figures 9.7 and 9.8 show, detailed information of the distributions of interfacial frictional shear stresses and the work roll pressures may be obtained by these methods, but the experimental setup and the data acquisition are elabo- rate and costly. Since the major criticism concerns the possibility of some metal particle or oxide intruding into the clearance between the pins and their housing and invalidating the data, it is necessary to substantiate the resulting coefficients of friction by independent means. This substantiation has been performed successfully in several instances (see, e.g.,Humetal.,1996).Inthatstudy,the coefficients of friction, determined by the pin-transducer technique, were used in a model of the rolling process. The model calculated the roll forces and the roll torques which compared very well to the measured values, demonstrating that the technique leads to reliable data. Another difficulty encountered, when the embedded pins are used, is the interruption of the surface of the roll at the pins. The magnitude of the effect of this interruption is, at the present, unknown, but, considering the above described substantiation of the measurements, it is not expected to be significant. The use of the pins and transducers was reviewed quite some time ago by Cole and Sansome (1968). The authors concluded that the approach can provide reliable data as long as care is taken in the design, manufacture and calibration of the apparatus. A cantilever, machined out of the roll such that its tip is in the contact zone and fitted with strain gauges, and its various refinements were presented by Banerji and Rice (1972) and Jeswiet (1991). Tribology 207

9.3.1.2 The Refusal Technique Januszkiewicz and Sulek (1988) used the “refusal technique” to monitor the coefficient of friction necessary to initiate entry of the strip into the roll gap in a study of the effects of contaminants on the lubricating properties of lubricants. This approach makes use of the minimum coefficient of friction, required to initiate the rolling process. Recalling Eq. (4.1), the coefficient needed to allow entry is dependent only on the bite angle. At small reductions the bite angle is small and the required coefficient is also small. This fact is employed in the rolling process in which progressively smaller reductions are attempted in each pass. The bite angle at which entry is first successful is then reported as the coefficient of friction.

9.3.1.3 The Ring Compression Test The most popular and most widely used technique to establish the friction factor, however, is the ring compression test (Kunogi, 1954; Male and Cockroft, 1964; Male and DePierre, 1970; DePierre and Gurney, 1974). In the test a ring of specific dimensions is compressed in between flat dies and the changes of its dimensions are related directly to the friction factor. Using calibration curves, the friction factor is obtained easily. The derivation of the calibration curves is well described by Avitzur (1968), who also presented a detailed set of calculations indicating how the curves are to be determined. The schematic diagram of the ring test and a typical calibration chart are shown in Figures 9.9 and 9.10. Bhattacharyya (1981) showed that under some circumstances the compressed rings develop tapering, with the top and bottom surfaces deforming in a different manner, most likely due to different tribological conditions on the two surfaces. The tapering disappeared when the samples were pre-compressed, and the true areas of the contact at the top and the bottom surfaces converged. Tan et al. (1998) used different ring geometries to study the ring compression process. Concave,

Figure 9.9 The ring compression test. P

P 208 Primer on Flat Rolling

80 Figure 9.10 A typical calibration chart for m the ring test. 0.8

0.4 40

(%) 0.2 D I Δ 0

0.1

0.05 –40

0 20406080 ΔH (%) rectangular and convex shaped cross-sections were employed. The results indicated that the influence of strain hardening on friction is complicated. Friction was affected by the normal pressure in a significant manner. Szyndler et al. (2003) compressed austenitic stainless steel rings at high temperatures, without lubrication, and used an inverse analysis to determine the friction factor. The factor’s dependence on the temperature was found to be well described by the relation m 5 3:511 3 1024T 2 0:01846, where the temperature is in C6, indicating that the friction factor increases with increasing temperatures.

9.3.2 Semi-Analytical Methods Numerous attempts to relate the coefficient of friction or the friction factor to various parameters have been presented in the literature, too many to be reviewed here. Only some, considered to be the more useful, are presented below.

9.3.2.1 Forward Slip Coefficient of Friction Relations Several formulae, connecting the forward slip to the average coefficient of friction, have been published in the technical literature. The predictive abilities of these relations have been studied (Lenard, 1992), and the results have been compared to data produced by the embedded-pin technique. While the conclusions indicated that the reviewed equations don’t work very well, they are presented below for completeness. It is recalled that the forward slip is given by:

vexit 2 vroll Sf 5 ð9:12Þ vroll

6The effect of the temperature on the coefficient of friction will be discussed in more detail in Sections 9.3.2.3 and 9.4.1.3. Tribology 209 where vexit stands for the exit velocity of the rolled strip and vroll designates the surface velocity of the work roll. Sims’ formula (1952) connects the forward slip to the reduction, r, the flattened 0 roll radius, R , the exit thickness, h2 and the coefficient of friction, μ: rffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 1 r 1 1 tan21 S 5 tan21 2 ln ð9:13Þ f 2 1 2 r 2a 1 2 r where rffiffiffiffiffi μ R0 a 5 : 1 2 r h2

φ Ekelund’s (1933) formula gives the coefficient of friction in terms of the bite angle, 1, the roll radius and the forward slip as:

ðφ =2Þ2 μ 5 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9:14Þ u 2S ðφ =2Þ 2 u f 1 t2R 2 1 h2

Bland and Ford (1948) use similar variables in giving the coefficient of friction as:

h 2 h μ 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi ð9:15Þ 0 0 2 R ðh1 2 h2Þ 2 4 SfR h2

Roberts (1978) includes the roll force (Pr) and the torque for one roll (M/2) in addition to the reduction and the roll radius to define the coefficient of friction:

M=2 μ 5 "#rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9:16Þ S ð1 2 rÞ P R0 1 2 2 f r r

Another relationship, due to Inhaber (1966), includes the roll force and the roll torque, φ in addition to the neutral angle ( n), the arc of contact, the maximum pressure, P2, and the pressures at entry and the exit, P1 and P0, respectively. Note that in the absence of external tensions P1 equals the yield strength of the strip at the entry and the pressure at exit, P0, equals the yield strength there. The equation is

= 2 1 M 2 5 0 φ 2 φ P2 P1 : Pr 2R ð 1 nÞ ð9 17Þ Rμ lnðP2=P1Þ 210 Primer on Flat Rolling where the neutral angle is given in terms of the forward slip pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi φ 5 = 0 : n Sf h2 R ð9 18Þ and the maximum pressure is calculated from c0 ln P0 2 c1 ln P1 1 P1 2 P0 P2 5 exp ð9:19Þ c0 2 c1 where 1 P M=2 c 5 r 2 ð9:20Þ 0 0 φ φ μ 2R n R n and

P 1 M=2R0μ c 5 r ð9:21Þ 1 0 α 2 φ 2R ð nÞ

In order to test the predictive abilities of the above listed formulae, the forward slip needs to be measured. This may be accomplished in several ways. One of the often used methods is to mark the roll surface with fine lines, parallel to the roll axis and placed uniformly around the roll. As the strip is rolled, these marks create impressions on the rolled surface. The forward slip may then be calculated by:

l 2 l S 5 1 ð9:22Þ f l where l1 is the average of the distances between the marks on the strip’s surface and l is the distance between the lines on the roll’s surface. It is expected that as long as the lines are created carefully, using sharp, hard tools, the resulting data on the forward slip is reasonably accurate. The predictions of Eqs. (9.13)(9.17) were compared to measurements of the coefficient friction and forward slip while rolling strips of commercially pure aluminium (Lenard, 1992). The coefficients of friction were determined by force transducers embedded in the work rolls and the forward slip was obtained by the use of lines on the roll surface and by Eq. (9.22). The results are shown in Figure 9.11, plotting the coefficient of friction versus the forward slip. As Figure 9.11 shows, all relations predict realistic values for the coefficient of friction. However, the trends are not predicted well. Ekelund (1933), Sims (1952), Roberts (1978) and Bland and Ford (1948) all predict an increasing trend, reaching a plateau and then dropping; the measurements indicate an upward exponential. It is concluded that most of the formulae are not successful in providing reliable Tribology 211

0.3 Figure 9.11 Comparison of the calculated and the measured 0.25 forward slip. 0.2 Source: Lenard (1992).

0.15 Measurements Ekelund 0.1 Sims

Coefficient of friction Roberts 0.05 Inhaber Ford 0 0123456789101112 Forward slip

and consistent predictions. Arguably the best, though not perfect, approach is due to Inhaber (1966). Marking the roll’s surface, however carefully, may affect the interfacial conditions and hence, the frictional forces, although, as concluded when the embedded pins were used, these effects may not be very large. If lubricants are also present, the need to distribute them over the contact may also be compromised somewhat since the marks will act to retain some of the oils. There are alternatives for researchers who don’t want to mark the roll’s surface and these make use of optical devices. McConnell and Lenard (2000) used two photodiodes, located a known distance apart, to monitor the exit velocity of the rolled strip. The time interval between the signals of the diodes allowed the determination of the exit velocity and this, in turn, allowed the use of the original definition of the forward slip in terms of the roll’s and the strip’s speeds. Li et al. (2003) used laser Doppler velocimetry to measure the relative velocities of the roll and the strip. It must be noted that measurements of the forward slip are often error prone. A simple examination of either Eq. (9.12) or (9.22) illustrates the difficulties. If, for example, the roll’s surface velocity is 900 mm/s and the strip’s exit velocity is 1000 mm/s, the forward slip is determined to be 0.11. If, however, the roll velocity is mistakenly measured to be 909 mm/s a 1% error the corresponding error is almost an order of magnitude higher. An interesting approach to determine the coefficient of friction using the forward and the backward slip was given by Silk and Li (1999). They use the original definition of the forward slip (see Eq. (9.12)), and the backward slip as the relative difference between the entry speed of the strip (v0) and speed of the roll:

v cos φ 2 v 5 r 1 0 ð : Þ Sb φ 9 23 vr cos 1

The authors make several assumptions which lead to simple expressions for the coefficient of friction in the forward slip zone, defined as the region between the 212 Primer on Flat Rolling

μ neutral point and the exit, f, and in the backward slip zone which is the region μ between the entry and the neutral point, b:

Δh=R0 μ 5 rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9:24Þ f Δ h 2 2Sf hexit 2 0 4 0 R 2R 2 hexit and

Δ = 0 μ 5 rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih R : b ð9 25Þ Δh h cos φ h 2 2 4 ð1 2 S Þ entry 1 2 exit R0 b 2R0 R0

These relations are subject to the following assumptions:

G Coulomb friction exists between the roll and the rolled metal; G the roll pressure is constant over the contact area; and G the angles are small compared to unity. In order to determine the coefficients, numerical values for the forward and the backward slips are necessary. Silk and Lee (1999) use information obtained from the instrumented loopers of the hot strip mill of Hoogovens7; their results refer to stands F2 and F3 of the Hoogovens hot strip mill. The magnitudes of the coefficients of friction vary from 0.15 to 0.23 in both stands, realistic values considering the efficient lubrication applied in the mill.

9.3.2.2 Empirical Equations Cold Rolling The three well-known formulas, connecting the coefficient of friction to the roll separating force, rely on matching the measured and calculated forces and choosing the coefficient of friction to allow that match. One of the often-used formulae, given by Hill, is quoted by Hoffman and Sachs (1953) in the form: P h pffiffiffiffiffiffiffiffiffiffiffir 2 1:08 1 1:02 1 2 exit σ R0Δh hentry μ 5 sffiffiffiffiffiffiffiffiffiffi ð9:26Þ h R0 1:79 1 2 exit hentry hentry

where Pr is the roll separating force per unit width, σ is the average plane-strain flow strength in the pass and R0 is the radius of the flattened roll. Roberts (1967) derived a relationship for the coefficient of friction in terms of 0 the roll separating force Pr, the radius of the flattened roll R , the reduction r, the

7Hoogovens is a steel producer in the Netherlands. Tribology 213

average of the tensile stresses at the entry and exit σ1, the average flow strength of the metal in the pass, σ and the entry thickness of the strip, hentry: rffiffiffiffiffiffiffiffiffiffi"#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 μ 5 hentry Prð1 rÞ 1 2 1 5r : 2 0 0 1 ð9 27Þ R r σ 2 σ1 R hentryr 4

Ekelund’s equation, given by Rowe (1977) in the form of the roll separating force in terms of material and geometrical parameters and the coefficient of friction, may be inverted to yield the coefficient of friction: Pr pffiffiffiffiffiffiffiffiffiffiffi 2 1 ðhentry 1 hexitÞ 1 1:2Δh σ R0Δh μ 5 pffiffiffiffiffiffiffiffiffiffiffi ð9:28Þ 1:6 R0Δh

A comparison of the predicted magnitudes of the coefficient by these formulae is shown in Figure 9.12, using data obtained while cold rolling steel strips lubricated with a light mineral seal oil (McConnell and Lenard, 2000). Two nominal reductions are considered. The first is for 15% and the second is for 50%, of originally 0.96 mm thick, 25 mm wide, AISI 1005 carbon steel strips. The metal’s uniaxial flow strength, : in MPa, is σ 5 150ð11234 εÞ0 251. The tests were repeated at progressively increasing roll surface velocities, from a low of 0.2 to 2.4 m/s. Care was taken to apply the same amount of lubricant in each test: 10 drops of the oil on each side of the strip, spread evenly. In Figure 9.12, the coefficient of friction is plotted versus the roll surface velocity, which does not appear in any of the above formulae in an explicit manner. However, the effect of increasing speed is felt by the roll force, which, as expected, is reduced as the relative velocity at the contact surface increases. Increasing velocity is expected to bring more lubricant to the entry to the contact zone. The dropping frictional

0.50 Figure 9.12 The coefficient of friction, as predicted by Hill’s, Roberts’ and 15% reduction Hill Roberts Ekelund’s formulae, for cold rolling 0.40 Ekelund of a low-carbon steel. Source: McConnell and Lenard (2000). 0.30

0.20 45% reduction Coefficient of friction

0.10

0.00 0 500 1000 1500 2000 2500 Roll surface speed (mm/s) 214 Primer on Flat Rolling resistance indicates the efficient entrainment of the lubricant and its distribution between the roll and the strip surfaces. No starvation of the contacting surfaces is observed. All three formulae give realistic, albeit somewhat high, numbers for the coefficient of friction and all predict the expected trend of decreasing coefficient with increasing velocity. As well, the coefficient of friction is indicated to decrease as the reductions increase, demonstrating the combined effects of the increasing number of contact points, the increasing temperature and the increasing normal pressures. The first two phenomena result in increasing frictional resistance with reduction. The third causes increasing viscosity and hence, decreasing friction and, as shown by the data, it has the dominant effect on the coefficient of friction. The magnitudes vary over a wide range, however, indicating that the mathematical model also influences the results in a significant manner. An analytical approach, to determine the coefficient of friction, has been presented by Li (1999). Two approaches were given, both subject to several, a priori, assumptions: homogeneous plane-strain compression is present; the coefficient of friction is constant in the arc of contact; the strip is rigid-plastic; the neutral plane is within the arc of contact and the rate of strain is low. In the first approach, the roll pressure distribution, as predicted by Bland and Ford (1948), is used, the rolling strain is determined and the minimum coefficient of friction for steady-state rolling without skidding is obtained as: ε 2 μ 5 a : c ð9 29Þ H0 where H0 is the entry thickness and a is defined as a tension parameter in terms of the front and back tensions (σt;f and σt;b, respectively) and the rolled strip’s strength at the entry and exit (σ0 and σ1,respectively): σ ; σ ; a 5 ln 1 2 t b 2 ln 1 2 t f ð9:30Þ σ0 σ1

The coefficient of friction is then obtained in an iterative manner, until the measured and calculated rolling strains agree. In the second approach the forward slip is used. A relationship is then derived, in terms of the coefficient of friction, μ, the thickness at the entry and the exit, h0 and h1, the bite angle, α and the forward slip, Sf:

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 1 μ α μ α μ α μ cot α cot 1 2 cot α2 2μ2 2 φ cot cot h1 h1 ð1 cot 1Þh0 h1 4 5 5ðS 11Þh ð9:31Þ 11μ cot α f 1

Using the measured forward slip and iterating, the coefficient of friction is calculated. The magnitudes of the calculated coefficients of friction are as measured or Tribology 215 predicted elsewhere. The results indicate that as the rolling strain increases, the coefficient also increases, a finding valid for cold rolling of aluminium but contra- dicting most experimental data, obtained while rolling steel, which indicate the increasing reductions result in dropping coefficients of friction. Beynon et al. (2000) studied friction and the formation of scales on the surfaces of hot rolled steels. They determined the coefficient of friction by using the forward and the backward slip data, measuring the speed of the roll and of the strip simultaneously and using the model of Li (1999), described previously. Their results indicate that mixed sliding/sticking conditions exist in the contact zone. Further, a neutral zone, rather than a neutral plane, is present there. Martin et al. (1999) analysed friction during finish rolling of steel strips. A combination of experimental data and a coupled thermomechanical model of the rolling process was used. Their conclusions are interesting in that they contradict existing beliefs: the frictional conditions exhibited weak correlation with rolling speed, temperature and reduction.

9.3.2.3 The Study of Tabary et al. (1994) A somewhat different approach was followed by Tabary et al. (1994) in determining the coefficient of friction during cold rolling of fairly soft, 1200 aluminium alloy strips. The authors mention the difficulties associated with the determination of the coefficient of friction in the roll bite. One of the difficulties is the changing direction of the friction force in the roll gap, aiding the movement of the strip until the no-slip region is reached and retarding its movement beyond, until the exit is reached. The location of the no-slip region may be manipulated by applying external tensions and that is the approach followed in this study. Using external tensions the neutral point is forced to be at the exit, causing the friction forces in the deformation zone to act only in one direction. The von Karman differential equation of equilibrium was then integrated with assumed values for the coefficient of friction and the inlet yield strength. Both of these were adjusted until the calculated and the measured roll forces matched and the boundary condition at the exit was satisfied, so the method is essentially one of the inverse analyses. A rare and most welcome section of Tabary et al.’s paper is the analysis of the errors in the reported values of the coefficient of friction. They also account for the μ contribution of the hydrodynamic action to the coefficient of friction, h, according to the relation: τ ηΔ μ 5 0 21 u : h sinh ð9 32Þ p τ0hs where p is the average roll pressure and η is the viscosity at p. The average relative speed is Δu, the smooth film thickness is hs and τ0 is the Eyring shear stress, estimated to be 2 MPa. The results indicate that the coefficient of friction is a strong function of the reduction and of the ratio of the smooth lubricant film thickness to the 216 Primer on Flat Rolling combined rms roughness (Λ). The coefficients increase with increasing reduction and drop with increasing Λ, varying from a high of approximately 0.08 to a low of 0.02.

9.3.2.4 Empirical Equations and Experimental Data Hot Rolling Formulae, specifically intended for use in the analyses of flat, hot rolling of steel, have also been published. Those given by Roberts (1983) and by Geleji, as quoted by Wusatowski (1969), are presented below. Roberts’ formula indicates that the coefficient of friction increases with the temperature. Geleji’s relations indicate the opposite trend. Roberts (1983) combined the data obtained from an experimental two-high mill, an 84-in. hot strip mill and a 132-in. hot strip mill, all rolling well- descaled strips, and used a simple mathematical model to calculate the frictional coefficient. Linear regression analysis then led to the relation:

μ 5 2:7 3 1024T 2 0:08 ð9:33Þ where T is the temperature of the work piece in F. Geleji’s formulae, given below, have also been obtained by the inverse method, matching the measured and calculated roll forces. For steel rolls the coefficient of friction is given by:

μ 5 1:05 2 0:0005T 2 0:056v ð9:34Þ where the temperature is T, given here in C and v is the rolling velocity in m/s. For double poured and cast rolls the relevant formula is

μ 5 0:94 2 0:0005T 2 0:056v ð9:35Þ and for ground steel rolls:

μ 5 0:82 2 0:0005T 2 0:056v ð9:36Þ

It is observed that Geleji’s relations, indicating decreasing frictional resistance with increasing temperature and rolling speed, confirm experimental trends. Note also, that the predictions of Roberts see Eq. (9.33) indicate the opposite trend, agreeing with the results of Szyndler et al. (2003), who obtained the friction factor as a function of the temperature during ring compression of stainless steel samples. Rowe (1977) also gives Ekelund’s formula for the coefficient of friction in hot rolling of steel:

μ 5 0:84 2 0:0004T ð9:37Þ where the temperature is to be in excess of 700C, again indicating that increasing temperatures lead to lower values of the coefficient of friction. Tribology 217

Underwood (1950) attributes another equation to Ekelund, similar to those above, giving the coefficient of friction as:

μ 5 1:05 2 0:0005T ð9:38Þ

Roberts (1977) presents a relationship for the coefficient of friction for a well- descaled strip of steel, in terms of the strip’s temperature T in F, obtained by fitting an empirical relationship to data, obtained by inverse calculations: 22:61 3 104 μ 5 2:77 3 104 exp 1 0:21 ð9:39Þ 459 1 T

A comparison of the predictions indicates that the relations may not be completely reliable in all instances. For example, using a steel work roll and a rolling a steel strip at a temperature of 1000C and at a velocity of 3 m/s, Roberts’ equation predicts a coefficient of friction of 0.415 while Geleji’s relation gives 0.382, indicating that while the numbers are close, the difference, almost 8%, is not insignificant. When 900C is considered, Roberts’ coefficient becomes 0.366 and Geleji’s increases to 0.432, creating a large difference. Ekelund’s predictions are 0.44 and 0.48, at 900C and 1000C, respectively. It is difficult to recommend any of these relations for use in modelling. Lenard and Barbulovic-Nad (2002) hot rolled low-carbon steel strips at entry temperatures varying from a low of 800C to 1100C, using an emulsion of Imperial Oil 8581 and distilled water, at a ratio of 1:1000. During heating, the strips were held in a furnace which was purged using oxygen-free nitrogen, allowing close control of the scale thickness. The roll separating forces, the roll torques, the roll speed and the entry and exit strip surface temperatures were measured in each pass. The coefficient of friction was obtained by inverse calculations, using the refined 1D model, presented in Section 5.4. Non-linear regression analysis led to the relationship:

: φ_ ΔT p 1 2 μ 520:183 2 0:636 roll exp 20:279 1 0:248 ð9:40Þ ε_ T σfm

φ_ where the parameters are the ratio of the roll speed ( roll in revolutions/s), to the strain rate of the rolled strip (ε_ in s21), the ratio of the surface temperature drop (ΔT in K) to the average temperature in the deformation zone (T in K) and the ratio of the average interfacial pressure (p in MPa) to the metal’s resistance to deformation (σfm in MPa). Some of the data of Lenard and Barbulovic-Nad (2002) are shown in Figure 9.13, giving the coefficient of friction, calculated by Hill’s formula, as a function of the average temperature of the strip in the deformation zone. The two graphs demonstrate the difficulties in attempting to arrive at some definite conclusions. At low speeds and low reductions the coefficient appears to drop with increasing temperatures while at higher reductions and speeds, the opposite is noted. 218 Primer on Flat Rolling

0.50 Figure 9.13 The dependence of Reduction and roll speed Hill’s coefficient of friction on the ~10%, ~80 mm/s temperature at low speeds and 0.40 ~40%, ~470 mm/s reductions and at high speeds and reductions. Source 0.30 : Barbulovic-Nad and Lenard (2002).

0.20 Coefficient of friction (Hill) 0.10

Low-carbon steel 0.00 700 800 900 1000 1100 Average temperature (°C)

The behaviour of the coefficient of friction is most likely affected by the thickness of the layer of scale. Since the pre-test heating process was the same, the scale thickness at entry was also the same in both sets of experiments. At lower speeds, however, there is sufficient time for the thickness of the layer of scale to grow during the pass, and as has been pointed out often, thicker scales lead to lower frictional resistance. Wusatowski’s data (1965) may be used to clarify some of the apparent contradictions. He determined the coefficients of friction applicable during industrial hot rolling of carbon steels, using the inverse method. While he showed that the coefficient is strongly dependent on the temperature, the dependence is not linear. The coefficient increased with the temperature from about 750C to 900C and after reaching a plateau there, it dropped when the temperature increased further. The temperatures where the change of slopes occurred also depended on the effective carbon content of the steels. It may be concluded that the strength of the layer of scale and of the adhesion between it and the parent metal also affect the coefficient of friction. The effect of the strength of the scale layer on surface interactions is clearly indicated by Li and Sellars (1999) who showed how the scale may break up and how the hot steel may extrude through the cracks and contact the roll surface. Hot rolling ferritic stainless steel strips while carefully controlling the development of surface scales, Jin et al. (2002) developed a relation for the coefficient of friction (as calculated by Hill’s formula, Eq. (9.26)):

: : p 0 4242 T 0 3197 μ 5 1:0728 2 0:9014 1 0:0016 tscale 2 0:0014ε_ σfm 1000 ð9:41Þ where T is the entry temperature in K, tscale is the thickness of the layer of scale at the entry in μm and ε_ is the strain rate. The results of Jin et al. (2002) are shown in Tribology 219

0.8 Figure 9.14 The coefficient of friction Scale thickness from 1.5–11 μm as a function of the temperature Roll speeds from 750–980 mm/s Reductions from 25–33% during hot rolling of ferritic stainless 0.6 steel strips. Source: Jin et al. (2002).

0.4

0.2

Coefficient of friction (Hill's formula) Hot rolling ferritic stainless steels

0.0 800 900 1000 1100 1200 Entry temperature (°C)

Figure 9.14, plotting the coefficient of friction as a function of the temperature at the entry to the roll gap. The figure includes data obtained at temperatures varying from 900C to 1100C, scale thickness before the pass from 1.5 to 11 μm and roll surface speeds of 750980 mm/s. In spite of the broad scatter, the downward trend of the coefficient of friction with increasing temperature is clearly present. While these results may be compared to that of Szyndler et al. (2003) who found the opposite (the friction factor, in their ring tests, increased with the temperature), the comparison may again be one of apples and oranges. Szyndler et al. (2003) used ring compression tests, an austenitic stainless steel and no lubrication and gave no special attention to scale formation; Jin et al. (2002) used rolling, a ferritic stainless steel, a light mineral steel oil as the lubricant and exerted careful control of the development of the layer of scale. Considering the often-mentioned interaction of all parameters in creating a particular magnitude of the coefficient of friction, the different trends, while not easily predictable, are not really surprising. The discussions presented above indicate that while the coefficient of friction is dependent on the temperature in a significant manner, relations that attempt to use only some of the independent variables inevitably lead to errors. The interactions of the variables and the parameters need to be understood before reliable functional connections are established.

9.3.2.5 Inverse Calculations A method often followed to determine the coefficient of friction relies on conducting experiments during which a parameter that depends on the coefficient in a well- known manner is measured. In the mathematical model of the process the coefficient of friction then becomes the only unknown and is determined such that the measured and the calculated parameters match. 220 Primer on Flat Rolling

0.20 Figure 9.15 The coefficient of friction, obtained by Hill’s formula and by inverse analysis, using a 0.16 1D model of the rolling process.

0.12

0.08

μ μ 1D = 0.594 Hill + 0.0165

Coefficient of friction – 1D model 0.04

0.00 0.0 0.1 0.2 0.3 0.4 Coefficient of friction – Hill's formula

This approach was followed by McConnell and Lenard (2000) while cold rolling low-carbon steel strips. The coefficient of friction was calculated by Hill’s equation (Eq. (9.26)) in addition to the use of the 1D model of Roychoudhury and Lenard (1984), referred to in Chapter 5 as the refined version of Orowan’s model8.The results are shown in Figure 9.15, plotting the 1D coefficient of friction on the ordinate and Hill’s coefficient on the abscissa. It appears that the two coefficients are linearly related with the 1D model’s results approximately 40% below those of Hill. The relation of the two coefficients, obtained by non-linear regression analysis, is given by Eq. (9.42):

μ 5 : μ 1 : : 1D 0 594 Hill 0 0165 ð9 42Þ

9.3.2.6 Negative Forward Slip It has been observed in several experiments (Shirizly and Lenard, 2000; Shirizly et al., 2002) that at higher rolling speeds and larger reductions the forward slip becomes negative, indicating that the surface velocity of the exiting strip is less than that of the work roll. In these cases there is neither a neutral point nor a neutral region. Realizing that the mathematical models, presented above, include the idea that the rolled strip exits at a velocity higher than the surface velocity of the roll and the no-slip location is between the entry and the exit, a new approach is needed to analyse the instances when the forward slip is negative.

8See Section 5.4, Refinements of Orowan’s model. Tribology 221

Avitzur’s upper bound formulation (Avitzur, 1968) is adopted in the present work. The power to reduce the strip can be obtained from the measured roll torque and the roll speed:

Torque 3 roll surface velocity Power 5 ð9:43Þ Roll radius and this is equated to the power obtained using the kinematically admissible velocity field of Avitzur (see Eq. (5.63), Chapter 5). The friction factor, which is the only unknown in Eq. (9.43), can then be determined. The coefficient of friction is evaluated using the relationship:

pffiffiffi m= 3 μ 5 ð9:44Þ pave=σfm where the average pressure is obtained from the roll separating force divided by the projected contact area. Using the experimentally obtained power and the exit velocity of the strip, the average coefficient of friction in the roll gap is obtained directly.

9.3.2.7 The Correlation of the Coefficient of Friction, Determined in the Laboratory and in Industry Munther (1997) conducted hot rolling experiments on a small laboratory mill, using low-carbon and high-strength low-alloy steel strips. In each test, the roll separating force, the roll torque, the reduction, the thickness of the layer of scale, the speed and the temperatures at the entry and at the exit were measured. The coefficient of friction values were determined by inverse calculations, using the 2D finite-element code, Elroll. In addition, mill logs were obtained from Dofasco Inc., giving all necessary data to allow the determination of the coefficient of friction, again by inverse calculations. The results are illustrated in Figure 9.16, plotting the coefficient of friction values against the dimensionless group Δhσfm=Pr; in the plot the coefficients of friction from the laboratory mill were corrected to allow for the effect of geometry according to the square root of the ratio of the respective work roll radii: rffiffiffiffiffiffiffiffiffi μ 5 μ Rlab: : lab;corr: ð9 45Þ Rind:

It is noted that while there is some scatter, both the laboratory and the industrial data fall on the same trend line. 222 Primer on Flat Rolling

0.6 Figure 9.16 A comparison of the corrected values of the coefficient of Data from industry friction from a laboratory mill and Data from the laboratory those from industry. Source: Munther (1997). 0.4

0.2 Coefficient of friction

Hot rolling of low carbon steels 800–1100°C 15–40% reduction

0.0 0.00 0.10 0.20 0.30 Δh σ fm P r

9.4 Lubrication

The objectives of using lubricants and emulsions in the flat rolling process include energy conservation, protection of the work roll surfaces, control of the coefficient of friction, control of the resulting surface parameters and cooling. Each of these depends on several interacting variables and arguably, one of the most important among these is the coefficient of friction. In what follows, the basic concepts of lubricated flat rolling are discussed.

9.4.1 The Lubricant Heshmat et al. (1995) reviewed modelling of friction, interface tribology and wear for powder-lubricated systems and for solid contacts. He stated that anything in between the contacting surfaces is a lubricant, be it a powder, a contaminant, a layer of scale or, in fact, oil. In the present context, the lubricant considered is oil, of course, either in the neat form or as an emulsion, usually in water.

9.4.1.1 The Viscosity The behaviour of the lubricant in the contact zone is affected by its viscosity, defined as the factor of proportionality between the shear stress within the oil, τ, and the shear strain rate, γ_:

τ 5 ηγ_ ð9:46Þ Tribology 223

The factor of proportionality thus defined is referred to as the dynamic viscosity and its units are, in the SI system, Pa s. The kinematic viscosity, μ, is obtained by dividing the dynamic viscosity by the density, ρ:

μ 5 η=ρ ð9:47Þ and if the density is in kg/m3, the units of the kinematic viscosity are m2/s9. The viscosity of Newtonian fluids is taken to be a constant. Non-constant viscosity indicates a non-Newtonian fluid. In what follows, Newtonian behaviour only will be considered.

9.4.1.2 The ViscosityPressure Relationship The effect of the pressure on the viscosity, and in turn on the coefficient of friction or the friction factor, is significant and it cannot be ignored. The Barus equation, used frequently, gives the viscosity at an elevated pressure, η, in terms of the viscosity at η γ atmospheric pressure, 0, the pressureviscosity coefficient, ,andthepressure,p: η 5 η γ : 0 expð pÞð9 48Þ

Equation (9.48) is simple to use, but the user should be cautious: Stachowiak and Batchelor (2005) quote Sargent (1983) who wrote that the Barus equation leads to errors when applied at pressures in excess of 500 MPa. These errors have also been mentioned by Cameron (1966), showing that at high temperatures and pressures the exponential law can overestimate the viscosity by a factor of 500. The warning is repeated by Szeri (1998) who, however, limits the applicability of the Barus equation to pressures of only 0.5 MPa. The fact that during flat rolling of steels the normal pressures will easily exceed 500 MPa has been mentioned above and for realistic estimates of the viscosity other relations need to be employed. Cameron (1966) presents an equation for the viscosity under higher pressures in the form:

η 5 η 1 n : 0ð1 CpÞ ð9 49Þ where C and n are constants. The exponent n is taken to equal 16 and C is expressed in the form:

5 a 2 ηðb21Þ : C 10 ð1 bÞ 0 ð9 50Þ where b 5 0.938 and a 52ð0:4 1 oF=400Þ, where the temperature of the oil is to be expressed in F, the viscosity is to be in centipoises and the pressure in psi.

9Unfortunately, much of the information on the viscosity is not given in these units. Instead, often the poise (dyne s/cm2) is used for the dynamic viscosity and the cS (centi Stoke) for the kinematic viscosity. To convert from the poise to Pa s, multiply it by 0.1. When dealing with the kinematic viscosity, use the following: 1 cS equals 1 mm2/s. 224 Primer on Flat Rolling

The relation is expected to be valid under 300F. Stachowiak and Batchelor (2005) 21 η give the value of C in Pa , in the form of a plot of C versus 0. The viscosity in the figure is in centipoise (cP) and the temperature of the lubricant is in C. Another relation, giving the dynamic viscosity at room temperature by Roelands, is presented by Stachowiak and Batchelor (2005):

η 5 η 1 : 1 : 3 29 Z 2 : expfðln 0 9 67Þ½ð1 5 1 10 pÞ 1g ð9 51Þ where γ Z 5 : : 3 29 η 1 : 5 1 10 ðln 0 9 67Þ

For mineral oils, the viscositypressure coefficient is given by Wooster and quoted by Stachowiak and Batchelor (2005) as:

γ 5 : 1 : η 3 28 : ð0 6 0 965 log10 0Þ 10 ð9 52Þ

10 wheretheviscosityatzeropressure,η0,isincP and the pressureviscosity coefficient is in Pa21. The pressureviscosity coefficients can also be calculated following the study of Wu et al. (1989), who quote the formula of So and Klaus (1980) in a slightly revised form, giving the coefficient as:

3:0627 24 5:1903 1:5976 γ 5 1:030 1 3:509ðlog μ Þ 1 2:412 3 10 m0 ðlog μ Þ 0 0 ð9:53Þ 2 : μ 3:0975ρ0:1162 3 387ðlog 0Þ in units of kPa21 3 105; the predictions are shown by the authors to be very accurate. While in Eq. (9.52) the coefficient is given as a function of the viscosity only, in Eq. (9.53) the density as well as the temperature also affect it in addition to the viscosity. The constant m0 is defined as the viscositytemperature property, given by the ASTM slope divided by 0.2. The slope is given by Briant et al. (1989) and the coefficient m0 is then obtained from: ÂÃ μ 1 : 2 μ 1 : log logð 0 0 7Þ log logð 0 7Þ m0 5 ð1=0:2Þ ð9:54Þ log T0 2 log T

The lubricant density is also dependent on the pressure. Szeri (1998) quotes the relationship of Dowson and Higginson (1977): 0:6 3 1029p ρ 5 ρ 1 1 ð9:55Þ 0 1 1 1:7 3 1029p where the pressure is in Pa.

10100 cP 5 0.1 Pa s. Tribology 225

100,000 Figure 9.17 The viscosity as predicted by the Barus, Cameron Barus equation and Roelands Eqs. (9.48), 10,000 Roelands equation (9.49) and (9.51), respectively. Cameron's equation

1000

100 Dynamic viscosity (Pa s) η = 0.153 Pa s 10 0 γ = 23.7 × 10–9 m2/N

1 0 100 200 300 400 500 Pressure (MPa)

A comparison of the dependence of the viscosity on the pressure, as predicted by the Barus, Cameron and Roelands equations, is given in Figure 9.17.

9.4.1.3 The ViscosityTemperature Relationship Stachowiak and Batchelor (2005) quote from the work of Crouch and Cameron (1961) who gave four different relationships for the temperature and the viscosity. The first is due to Reynolds in the form:

η 5 b expð2aTÞð9:56Þ expected to be accurate for a limited temperature range; one due to Slotte

η 5 a=ðb1TÞc ð9:57Þ which may be more useful than Reynolds’ formula. The Walther equation, which forms the basis of the ASTM viscositytemperature chart11, is given next as:

1 μ 1 a 5 bdTc ð9:58Þ

11The chart is part of ASTM method D 341-77. 226 Primer on Flat Rolling where μ is the kinematic viscosity in m2/s. When the ASTM chart is used, the constant d is taken to be 10 and the constant a is 0.6. The last relation is the Vogel equation, identified as the most accurate: b η 5 a exp ð9:59Þ T 2 c

In these equations a, b, c and d are constants. In these equations η is in Pa s, μ is in m2/s and T is in K.

9.4.1.4 The Combined Effect of the Temperature and the Pressure on the Viscosity η 5 η γ 2 β Most researchers use the form 0 expð p TÞ to estimate the combined effects of the pressure and the temperature in the viscosity; the pressureviscosity coefficient is γ in units that match the pressure, p,andβ is the temperatureviscosity coefficient, also in units that match that of the temperature. When the magnitudes of the viscosity at two temperatures are available, determination of the viscositytemperature coefficient is simple, as long as one assumes a linear variation between the temperatures. Sa and Wilson (1994) use a somewhat more complex relation for the viscosity η 5 η γ 2 β 2 δ δ in the form 0 expð p T TpÞ where is identified as a cross-coefficient. The establishment of accurate values of that coefficient may not be easy or straightforward.

9.4.2 The Lubrication Regimes Arguably the best and most widely known approach to characterizing the lubrication regimes is with the aid of the Stribeck diagram (see Figure 9.18) first

Figure 9.18 The Stribeck curve. Dry; metal-to-metal contact

Boundary; a few lubricant pockets

Mixed; more μ lubricant pockets

Hydrodynamic; complete separation

η Δv p Tribology 227 developed by Stribeck, a German railway engineer who studied friction in the journal bearings or railcar wheels12. The diagram plots the coefficient of friction against a dimensionless group of parameters, identified as the Sommerfeld number, used in the design of journal bearings; it is the product of the dynamic viscosity and the relative velocity, divided by the normal pressure (Faires, 1955). Schey (1983) writes, however, that the use of the term “Sommerfeld number” is somewhat incorrect. Knowing the magnitude of the coefficient of friction, the curve allows one to determine the extent of various lubricating regimes in a metal forming process. In the first portion, where the viscosity of the lubricant and the relative velocity of the contacting surfaces are low and the interfacial pressure is high, boundary lubrication is observed, in which metal-to-metal contact is predominant in addition to some lubricant-to-metal contact. The roughness of the resulting surfaces will approach that of the forming die, that is, the work roll. As oils of higher viscosity are introduced in the contact zone at higher relative speeds, the boundary regime changes as more lubricant is drawn in the contact zone and more lubricating pockets are created in the valleys between the asperities, and the “mixed mode” of lubrication, involving less metal-to-metal contact, is found. Moving further towards the right along the axis of the Sommerfeld number, the hydrodynamic regime is located, characterized by complete separation of the contacting surfaces. In this regime, the increase of the coefficient of friction is a result of increasing frictional resistance in the oil film, usually characterized as a Newtonian fluid, separating the surfaces. In this region, the product surface roughens after rolling because of the free plastic deformation of the grains near and at the surface. The nature of the lubrication regimes is often defined in terms of the thickness of the lubricant film and the combined roughness of the work roll and the rolled strip. The ratio

h λ 5 min ð9:60Þ σ

13 σ where hmin is theqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oil film thickness , is the rms roughness of the two surfaces, σ 5 2 1 2 given by Rq1 Rq2 and Rq1 and Rq2 are the rms surface roughness values of the two surfaces. When the oil film thickness to surface roughness ratio is less than unity, boundary lubrication is present. When 1 # λ # 3 mixed lubrication prevails while for a ratio over three, hydrodynamic conditions exist and the contacting surfaces are fully separated. In the flat rolling process, mixed or boundary lubrication regimes are usually present.

12Dowson (1979) identifies Gu¨mbel (1914), who plotted the curve now known as the Stribeck curve. 13The oil film thickness will be discussed in Section 9.4.7. 228 Primer on Flat Rolling

9.4.3 A Well-Lubricated Contact in Flat Rolling Lubricants and emulsions are used to optimize the frictional events in the rolling process in addition to controlling the quality and the temperature of the resulting surfaces. Effective lubrication is essential to control the tribological interactions between the work rolls and the work piece in the flat rolling process. The interactions include four phenomena, the requirements of all which need to be satisfied to create well-lubricated contacts. First, sufficient amounts of the lubricant or the emulsion must be made available at the entry to the roll bite; the emphasis here is on the word “sufficient amounts”, implying that too little or too much can be equally counterproductive. The lubricant must then be entrained, that is, the oil or its droplets must be captured and drawn into the contact zone between the work roll and the rolled strip. After successful entry the lubricant must be spread through the contact uniformly well so all of the surfaces are covered evenly. The lubricant must then travel through the deformation zone to the exit and should not be squeezed out at the sides. The current industrial practice during the rolling of steel or aluminium strips is to use oil-in-water (O/W) emulsions, which, if the above requirements are satisfied, have been shown to create good lubricating conditions and acceptable surfaces in addition to efficient cooling. In some isolated cases neat oils only are used. Delivering sufficient amount of the lubricant or the emulsion is dependent on the hardware and on the practice followed by the operators of a particular mill. In laboratory experiments two possible approaches are employed. The emulsion may be sprayed either on the work rolls, or on the top and bottom surfaces of the entering strip directly at the point of entry. Schmid and Wilson (1995) found that the film thickness was greatest when the emulsion nozzles were directed into the gap. The volume flow is usually carefully controlled and is kept constant. If neat oil is used, it is usually applied by a pipette and then spread over the surfaces by a clean roller. In either case it is necessary to measure the weight of the strip to be rolled before and after the application of the lubricant so the actual pre-rolling oil film thickness should be well known. In industry, the lubricant delivery systems vary from mill to mill. Roberts (1978, 1983) describes many of these in detail, for both hot and cold rolling. The other three events entrainment, uniform cover of the surface and travel through the contact zone depend on the interaction of process and material parameters, including the rolling speed, the reduction and the resistance of the rolled metal to deformation. As well, and arguably more important than the attributes already mentioned, the surface roughness parameters of the roll and the strip to be rolled affect the lubrication process in a most significant manner. Since the roll is very much harder than the strip, the latter’s asperities are expected to flatten shortly after entry, implying that the real contact area reaches its maximum fast. While under industrial conditions the surface of the work roll changes due to wear and roll pick-up; in a laboratory, where the volume of the rolled metal is much less, the roll’s surface roughness is not expected to change in any significant measure. Hence, the nature of the surface roughness of the work roll is considered to be Tribology 229 among the most important contributors to the success of the last three requirements of the lubricating system. These comments are equally valid whether neat oils or emulsions are used. Since the use of emulsions is increasing in the industry, the discussions in what follows will concern mostly them. An additional phenomenon needs to be considered when O/W emulsions are used: that of the behaviour of the droplets when they encounter the entry region of the roll-strip conjunction.

9.4.4 Neat Oils or Emulsions? It is well known that the use of neat oils results in significant reduction of the loads on the mill. In order to make an intelligent choice between using neat oils or O/W emulsions, an answer to the question is needed: How do emulsions affect the roll separating force and the roll torque? Cold rolling experiments on low-carbon steel strips were conducted to provide an answer (Shirizly and Lenard, 2000). The actions of four lubricants were compared, in addition to dry rolling and using water only, for their abilities to affect the roll forces and the torques on the mill and the frictional conditions. The lubricants included the SAE 10 and SAE 60 automotive oils, the SAE 10 base oil with 5% oleic acid added as a boundary additive and the SAE 10 base oil, emulsified, using water and polyoxyethylene lauryl alcohol as the emulsifier, 4% by volume. Oleic acid was chosen as the boundary additive since it was shown to react to pressure and temperature less than several other fatty oils (Schey, 1983). While the automotive lubricants were not formulated for use in the flat rolling process, their properties are well known, and that is the reason for their choice in this comparative study. Reid and Schey (1977) also used automotive oils in their study of full- film lubrication during rolling of aluminium alloys.

9.4.4.1 Roll Force and Roll Torque Typical roll separating force and roll torque data, as a function of the reduction and at various roll surface speeds (20 and 160 rpm, leading to surface velocities of 262 and 2094 mm/s) are shown in Figures 9.199.21, respectively. The lubricants and the emulsions used are also indicated in Figures 9.199.21. As expected, the roll forces and the torques increase as the reduction is increased, in a fairly linear fashion. Increasing the speed of rolling is expected to create more favourable lubricating conditions in the roll gap as more oil is drawn into the contact zone, at least in the tests where neat oils have been used. Contradictory to this phenomenon are the potential difficulties when emulsions are used. These are the time available for the oil particles to plate out14, which at higher speeds is much less in addition to more difficulties with droplet capture. While the data for 20 and 160 rpm were plotted separately, the lower forces at the higher speeds are clearly observable. Also, it was expected that dry conditions will require the largest forces and torques

14Droplet capture will be discussed in Section 9.4.5. 230 Primer on Flat Rolling

12 Figure 9.19 The roll separating force Lubricants at 20 rpm. SAE 10, oleic acid 10 SAE 10 SAE 10, emulsified SAE 60 8 Dry Water

Roll separating force (kN/mm) force Roll separating 2 Roll speed = 20 rpm

0 01020304050 Reduction (%)

12 Figure 9.20 The roll separating Lubricants force at 160 rpm. SAE 10, oleic acid 10 SAE 10 SAE 10, emulsified SAE 60 8 Dry Water

4 Roll separating force (kN/mm) force Roll separating 2 Roll speed = 160 rpm

0 0 1020304050 Reduction (%) to reduce the strip and as noted, this expectation was realized at both rolling speeds. The lubrication effect is much more pronounced at high speeds. At 20 rpm, while dry conditions produced larger forces and torques, these are of the same order of magnitude as those caused by some of the other lubricants. Also, the effect of water only on the forces and torques was surprising. At both speeds, use of water created favourable conditions as far as the loads on the mill were concerned. While the roll torques were not the lowest with water only, they were among the lowest. It appears that the lubricating effects of water are comparable to that of the lubricants. Tribology 231

50.0 Figure 9.21 The roll torque at 20 160 rpm Lubricants 20 and 160 rpm. SAE 10, oleic acid SAE 10 40.0 SAE 10, emulsified SAE 60 Dry 30.0 Water

20.0 Roll torque (Nm/mm)

10.0

0.0 0 1020304050 Reduction (%)

The roll force appears to be much more sensitive to the variation of the rolling velocity than the roll torque. There is a clear drop of the force as the speed is increased, of about 25% magnitude. The torques, however, are not affected by the change of speed in any significant measure. There is one observable, interesting trend in the roll force data: as the reduction is increased, the forces required to roll the steel become more affected by the lubricant type at the lower speed of 20 rpm. At the higher speed the opposite trend is present. At low reductions, the oil type has a noticeable effect on the forces. This effect is less evident as the reduction is increased. In general, however, no significant effect of the type of lubricant or emulsion on the forces and the torques is observed in the data. In prior studies, concerning cold rolling of commercially pure aluminium using neat oils (Lenard and Zhang, 1997), there was a clear drop of the forces and torques when the SAE 5 oil was replaced by the much more viscous SAE 30. The expectations in the present work, that of lower loads on the mill as the viscosity is increased, albeit smaller than with the aluminium, were not realized. The data of Lin et al. (1991) appear to support this observation. The authors used four lubricants with viscosity indexes varying from a low of 97 to a high of 115 and found that the loads on the mill were not affected in any significant measure.

9.4.4.2 The Coefficient of Friction Hill’s formula is used to determine the magnitudes of the coefficient of friction in the cold rolling process. The results are shown in Figures 9.22 and 9.23, for 20 and 160 rpm, respectively, where the coefficient of friction is plotted against the reduction, for all lubricants, dry conditions and water only. 232 Primer on Flat Rolling

0.5 Figure 9.22 The coefficient of Lubricants friction as a function of the SAE 10, oleic acid reduction at 20 rpm. SAE 10 0.4 SAE 10, emulsified SAE 60 Dry Water 0.3

0.2 Coefficient of friction

0.1

Roll speed = 20 rpm

0.0 0 204060 Reduction (%)

0.5 Figure 9.23 The coefficient of Lubricants friction as a function of the SAE 10, oleic acid reduction at 160 rpm. SAE 10 0.4 SAE 10, emulsified SAE 60 Dry Water 0.3

0.2 Coefficient of friction

0.1

Roll speed = 160 rpm 0.0 0 204060 Reduction (%)

In general and as expected, the highest frictional resistance is observed at low speeds and dry conditions. At 20 rpm, the lowest magnitudes for the coefficient of friction are produced by water only as the lubricant, confirming the trend noted above with the loads on the mill. No significant differences in frictional resistance are noted when any of the oils, neat or emulsified, are used. In all cases the coefficient of friction is reduced as the reduction is increased. Rolling in the dry condition resulted in frictional values that are among the highest, but surprisingly not the highest. Tribology 233

There is an unmistakable, albeit not very pronounced, dependence of the frictional resistance on the lubricant viscosity at the 160 rpm rolling speed. Of the four lubricants, the most viscous, SAE 60, appears to yield the lowest coefficient of friction and the highest values are obtained under no lubricating conditions. The magnitude of frictional resistance with the SAE 10, containing the oleic acid additive, is significantly lower than those rolled dry, as expected. The SAE 10, neat or emulsified, leads to friction values that are practically identical and not very much different from SAE 10 and the boundary additive. When using the four lubricants, the coefficient of friction reduced with increasing reduction. As well, the coefficient of friction was observed to decrease as the speed of rolling increased, under both dry and lubricated conditions. Based on these results, the recommendation is to use emulsions as often as possible.

9.4.5 O/W Emulsions Most metal working emulsions are O/W systems where oil is the dispersed phase and water the continuous phase. Emulsions are composed of three primary ingre- dients: the oily phase, the emulsifier and water. The emulsions used in rolling are composed of a water phase in which spherical micelles of oil, with diameters ranging from 1 to 10 μm, are dispersed. To keep these micelles from coalescing, an emulsifier, sometimes referred to as a surfactant, is used. Emulsifiers are composed of a molecular structure having two distinct ends. The hydrophilic (water loving) end is made of polar covalent bonds and is therefore soluble in water. The lipophilic (oil loving) end is soluble in natural and synthetic oils. When the emulsion is formed, the hydrophilic groups will orient towards the water phase and the lipophilic hydrocarbon chain will orient towards the oil phase.

9.4.5.1 Behaviour of the Droplets Kumar et al. (1997) wrote that the fundamental problem in the use of emulsions is the behaviour of the oil particles, the capture of which by the entering strip or by the roll surfaces is not yet fully understood. A study, some time before these comments, clarified the mechanisms of droplet capture. A transparent, translating plate, against which a stainless steel roll of 0.8 μm surface roughness was pressed, and a high-speed camera were used to study the droplets in O/W emulsions (Nakahara et al., 1988). Three types of droplets were identified. The first types penetrate the contact zone, called penetration droplets. Some droplets enter but don’t travel through to the exit and these are identified as the stay droplets. The remaining are the droplets that are rejected completely, called the reverse droplets. A lubricant feed rate of 1.2 cc/s and a normal load of 49 N were used in the tests. The relative velocity was varied, from 5 to 20 mm/s. As the relative velocity increased the number of penetration droplets decreased. Both the oil film thickness and the oil concentration at the entry, where 234 Primer on Flat Rolling an oil-rich pool was observed, were dependent on the emulsion concentration, also found by Zhu et al. (1994) and Kimura and Okada (1989). The lubrication mechanisms were considered to be velocity dependent. In the low-speed range the oily pool at the entry provided the lubrication, while at higher speeds “fine O/W emulsion” produced the oil films. The minimum size of the particles observed was 50 μm, however, significantly larger than those in most practical rolling processes. As well, the load and the relative speeds in practice are much larger than in the tests of Nakahara et al. (1989). The observations of Nakahara et al. (1989) imply that under some circumstances no oil particles will travel through the deformation zone and starvation at the exit may result. In order to get a better understanding of the lubrication conditions, Kumar et al. (1997) offered an explanation for the experimental observations (Nakahara et al., 1988) that particles very close to the tooling are rejected and play no part in the lubrication process. The authors tried to explain that kind of behaviour through a computational fluid dynamics model of a rigid particle in the inlet zone. Their theory applies to emulsion lubrication, slurry lubrication and wear involving solid particles, suspended in a liquid. The dimensionless results for different particle sizes were obtained for symmetric and unsymmetrical inlet zones. The results indicate that the segregation location of a particle is closer to the centre of the gap as the particle size increases. That suggests that larger particles are pushed into the back-flowing centre region and are rejected. Larger particles segregate closer to the roll surface than smaller ones. Small particles that are in the back-flowing regions will be rejected from the inlet zone and will have their clearance with the tooling increase; thus they will not be entrained later.

9.4.5.2 Entrainment of the Emulsion Plate-out, dynamic concentration and the mixture theory have been used to explain the supply and subsequent entrainment of the oil particles into the contact zone. In the plate-out process the particles adhere to and coat the surfaces (Schey, 1983). The droplets adsorb onto the surface and spread to the wetting angle. Several droplets eventually cover the complete surface and are available to enter the conjunction. The usual criticism of the plate-out theory concerns its potential inapplicability under industrial conditions. With an increase in speed, which may reach 2030 m/s, there may not be sufficient time for the plate-out process to occur and the film thickness at the entry will decrease or disappear entirely as a result of oil starvation. As the pressure in between the roll and the strip increases, the oil droplets are flattened and, because of their higher viscosity, are drawn into the inlet zone (Wilson et al., 1994). The concentration of the oil is therefore increasing at the inlet, leading to the dynamic concentration theory, which hypothesizes that the O/W emulsion inverts to become a water-in-oil emulsion as the pressure in the contact zone increases. Larger droplets are more likely to be entrained (Schmid, 1997). Once they have penetrated the contact zone to the point where the gap and the droplet size are of similar magnitude, they are irreversibly captured. While the mixture theory is the last observable lubrication regime at very high speeds (Schmid and Wilson, 1996), Tribology 235 it appears unsuitable under realistic rolling situations when the film thickness is much less than the droplet diameter (Schmid and Wilson, 1995). Yan and Kuroda’s (1997a) model showed that there is a velocity difference between two phases of the emulsion and that this causes a variable concentration of the oil phase in the lubricating film. At low entraining speeds the oil pool is formed in the inlet zone, so the film thickness is obtained primarily by the oil phase. At high entraining speeds, the increment of oil concentration becomes slow and both the oil and water phases are entrained into the contact zone. These results agree qualitatively with the experimental observations of Zhu et al. (1994). Zhu et al. (1994) showed a set of experimental results of the elastohydrodynamic lubrication (EHL) film thickness with O/W emulsion in a wide range of rolling speeds for different oil concentration and pH values. Experimental observations indicated that the phase inversion/oil pool formation mechanism around the inlet zone takes place only at very low speeds, which are most likely far below particle speed ranges for major industrial applications. In a follow-up paper, Yan and Kuroda (1997b) extended their previous work and discussed the variation of the oil concentration along the lubricant film. They concluded that the reason that EHL film thickness of an emulsion is of the same order of magnitude as that of the neat oil is because the general elastohydrodynamic film thickness of an emulsion is smaller than the droplet size, and the increase in the oil concentration makes it the same order as that of the neat oils. Schmid and Wilson (1996) reviewed the published research on the use of emulsions and attempted to reconcile the apparent contradictions of different researchers. They described and identified different lubrication mechanisms of O/W emulsions that are highly dependent on speed effects. Among the mechanisms of emulsion behaviour they mentioned the plate-out theory (introduced by Schey in 1983) and the dynamic concentration of oil and the mixture theory. The plate-out theory states that when an oil droplet is exposed to a metal surface, the anionic layer formed by the polar orientation of the emulsifier molecules induces the droplets to adsorb onto the surface. Once on the surface, the droplet spreads to the wetting angle. Therefore, the metallic surface exposed to the emulsion can cause the plate-out of the oil layer, with a resultant increase in the film thickness and a corresponding decrease in friction, greater than expected from the properties of the emulsion’s bulk. The weak point in the plate-out theory is that in many rolling applications, the rolling speed is relatively high and there is not sufficient time for the droplet to develop and spread to the wetting angle.

9.4.5.3 The Emulsion in the Contact Zone Schmid and Wilson (1995) indicated that the mechanism of lubrication with O/W emulsion is highly dependent on speed effects which influence the frictional conditions (boundary lubrication, mixed, hydrodynamic, etc.). Wilson and Chang (1994) introduced a simple model of mixed lubrication of bulk metal forming processes under low-speed conditions, where the inlet zone does not contribute significantly to hydrodynamic pressure generation. The model showed that relatively high 236 Primer on Flat Rolling hydrodynamic pressures can be generated in the work zone under conditions where it was previously considered that hydrodynamic effects were unimportant. The outlet film thickness predicted by the model was much larger than those predicted using full-film or high-speed mixed lubrication theories. An interesting observation of the emulsifier and droplet size effect on the film thickness is introduced by Kimura and Okada (1989). They published experimental results in which elastohydrodynamic and boundary lubrication properties of O/W emulsions of mineral oil are studied with a variety of nonionic surfactants as emulsifying agents. Kimura and Okada used a pendulum friction tester in order to determine the boundary coefficient of friction. In general, they found that the boundary lubrication effects of the emulsions seem to be similar to those of the surfactants. The coefficient of friction was higher for emulsions than for surfactant solutions. By the use of X-ray transmission techniques, Kimura and Okada measured the EHL film thickness. They showed that when the emulsifying agent concentration is high, or the particle diameter is small, the minimum film thickness increases with increasing particle diameters. In order to measure the contact angle on the oilsteel interface, 1 μl drop of base oil or distilled water was placed on the roll’s cylindrical surface and left for several minutes at room temperature. Photographs showed that both oil and water had small contact angles on the roll with emulsifying agent of polyoxyethylene alkyl ethers, whereas the water gave a contact angle larger than 90 when polyoxyethylene monooleate and sorbitan monooleate were used, showing marked hydrophobicity. Dubey et al. (2005) cold rolled a low-carbon steel, using O/W emulsions. They concluded that larger oil droplets created thicker oil films. They confirmed earlier data, indicating that increasing speeds reduce the coefficient of friction.

9.4.6 A Physical Model of the Contact of the Roll and the Strip Sutcliffe (2002) considers the lubrication mechanisms in between the work roll and the rolled strip. His figure is reproduced here as Figure 9.24. He shows the two significant contributors to frictional resistance and relative motion of the roll and the strip: contact at the asperities and contact at the lubricant-filled valleys. His

(A) (B) Roll Oil drawn Sliding of roll Oil-filled into inlet relative to strip valley due to ‘Contact’ area entraining Sliding Direction action Roll Strip Oil drawn out of pit due Strip to sliding action (MPHL)

Figure 9.24 Schematic diagram of the lubrication mechanisms in flat rolling (A) and details of the contact, showing asperities and the lubricant-filled valleys (B). Source: Sutcliffe (2002), reproduced with permission. Tribology 237 figure emphasizes the importance of the changes to the asperities as the rolling process is continuing as both contributions change when high normal and shear stresses act on the bodies in contact. The mechanism at the contact is usually referred to as micro-plasto-hydrodynamic lubrication. One important aspect of this mechanism is the oil, drawn out of the valleys due to the sliding action of the roll and the strip. An effective coefficient of friction maythenbedefined:

μ 5 μ 1 2 μ : A c ð1 AÞ v ð9 61Þ

μ where A is the ratio of the true to the apparent contact area, c is the coefficient of μ friction at the contacts and v is the coefficient of friction at the valleys. When the behaviour of the lubricant is assumed to be Newtonian, the frictional stress and μ η hence, v may be estimated in terms of the dynamic viscosity of the lubricant ð Þ and its strain rate ðγ_Þ as given by Eq. (9.46). For non-Newtonian lubricant behaviour, the Eyring model is more appropriate: ηγ_ 21 τ 5 τ0 sinh ð9:62Þ τ0 where τ0 is the stress at which the non-linearity starts. Under the mixed or boundary lubrication regimes the contribution to friction is mostly from the contact areas. Zhang (2005) points out, however, that dry contact may occur even under elastohydrodynamic conditions.

9.4.7 The Thickness of the Oil Film One of the parameters, considered to have a very significant effect on the forces of friction, is the thickness of the oil film between the contacting surfaces, which, in the present context, refers to the roll/strip contact. The results of an experiment, conducted using a flat-die apparatus15, indicate the dependence of the coefficient of friction on the volume of the oil (Kosanov et al., 2006). It is realized that the geometry of the test is significantly different from that of flat rolling and therefore the numbers quoted here are not relevant to the rolling process. The trends, however, are expected to be similar. Since industrial practice in sheet metal forming indicates that 22.5 g/m2 of lubricant on the sheet surface will ensure good tribological phenomena, that was the amount used initially in a set of experiments. This amount of oil creates a film thickness of approximately 23 μm in the area of contact in the test, and a film thickness

15This technique involves compressing a lubricated sheet of material between two flat dies and drawing the sheet through while monitoring the normal and the horizontal forces. Constant velocity in the longitudinal drawing direction and a constant normal load are maintained throughout the test. The coefficient of friction is then taken as the average of the draw force divided by half of the normal force. 238 Primer on Flat Rolling

0.20 Figure 9.25 The dependence of Rustilo S 40/2 oil; μ =40mm2/s the coefficient of friction R μ Ground CI dies, a = 0.18 m on the film thickness in the Hot dip galvanized steel flat-die test. 0.15 1 MPa normal pressure 50 mm/s draw speed Source: Kosanov et al. (2006). dies cleaned after each test dies are not cleaned

0.10

Coefficient of friction 0.05

0.00 0 4 8 12 16 Amount of oil (g/m2) ratio, λ 5 film thickness=effective surface roughness, of about 23, so a mixed lubrication regime would be expected. Loud metallic noises heard during the preliminary tests and some die damage, however, indicated the presence of mostly boundary lubrication. Weighing the sheets after the tests and comparing the weights to the pre- test values suggested that squeezing-out of the lubricant at the sides was most likely not the cause of the noise. Experiments were then conducted in which the amount of the oil was systematically increased. The results are shown in Figure 9.25, where the amount of the oil is given on the abscissa and the resulting coefficient of friction is shown on the ordinate. The graphs appear similar to the Stribeck curve, showing a possible approach to a hydrodynamic regime. Mixed tribological conditions appear to be present up to approximately 8 g/m2.

9.4.7.1 Measurement of the Thickness of the Oil Film One of the earlier attempts to measure the thickness of the oil film was by Whetzel and Rodman (1959). They dissolved the lubricant that stayed on the rolled strip after the pass in a solvent, evaporated the solvent and thus determined the volume of the remaining oil. They assumed that the surfaces of the strip were covered in a uniform manner. Another technique to determine the thickness of the oil film, that of the “oil-drop” approach, was probably introduced by Saeki and Hashimoto (1967). They measured the weight of the strip before rolling, added a drop of the lubricant, measured the weight after the pass and the area covered by the oil drop, leading to the thickness of the film. Azushima (1978) also used the “oil-drop” approach to determine the thickness of the oil film. He rolled 1-mm-thick stainless steel strips at speeds which varied from a low of 4 to 850 m/min and used three lubricants of 61, 30 and 4 cSt kinematic viscosities (at 38C), respectively. The Tribology 239 thickness of the oil film decreased with increasing reduction and increased with increasing rolling speeds. Sutcliffe (1990) measured both the average film thickness and the area of contact ratio, using two aluminium alloys and lead, lubricated with a mineral oil of 1.951 Pa s dynamic viscosity. Two rolls with a rough and a smooth finish were employed. He followed two methods. The first was that of Azushima (1978) and the oil-drop technique. In the other, he used the roughness data of the rolled strips, obtained while rolled with the smooth roll. Using the theory developed by Sutcliffe and Johnson (1990), he presented the results in terms of the ratio of the mean film thickness to the combined rms roughness of the rough rolls and the strip, plotted against mean film thickness to the combined rms roughness of the smooth rolls and the strip. The predictions and the measurements were remarkably close. Zhu et al. (1994) used either a steel ball or a steel roller, rotating against a glass disk. The surfaces of the disk and both the ball and the roller were prepared to be extremely smooth. The flow rate and the lubricant temperature were closely controlled during the experiments. Optical interferometry was used to establish the thickness of the lubricating films. Neat oil, pure water and six O/W emulsions were tested, with the viscosity, at 40C, changing from a low of 0.66 cSt (for water) to a high of 296.15 cSt for the emulsion, containing 40% oil. The droplet dimensions in the emulsions didn’t differ by much, varying from 0.44 to 0.55 μm. The range of speeds was remarkably broad, from a low of 0.001 m/s to a high of 20 m/s. Using the neat oil, the film thickness increased linearly with the speed. The film thickness increased, dropped and then increased again with all six emulsions when the speed was increased. Lo and Yang (2001) presented an analytical method to determine the thickness of the oil film in cold rolling. They assumed that the work roll surface is smooth and a mixed lubrication regime governs the contact between the roll and the rolled metal. Under these conditions the average oil film thickness is approximately the same as the average depth of the valleys on the rolled strip; hence, measurements of the surface roughness of the rolled strips lead to the thickness of the oil film. Trijssenaar (2002) states clearly that “the a priori assumption that the cold rolling lubrication film exists of pure oil, is not correct”. She used the oil-drop method to estimate the film thickness while lubricating the strip with an O/W emulsion. She concluded that the Wilson and Walowit (1972) equation see below, Section 9.4.7.2, Eq. (9.63) needs to be corrected for the conditions of her experiments, since it doesn’t account for the surface roughness. When the corrections were introduced, the data and the predictions agreed very well. A portable infrared analyser to measure the oil film thickness was discussed in Nordic Steel & Mining Review (1998), developed by Spectra-Physics Vision Tech.

9.4.7.2 Calculation of the Oil Film Thickness Neat Oils Wilson and Walowit (1972) developed a mathematical model to study the lubrication conditions in strip rolling under hydrodynamic conditions. They used 240 Primer on Flat Rolling several simplifying assumptions to allow the integration of Reynolds’ equation and obtained the often-used relation for the thickness of the oil film at the entry to the roll gap: η γ 1 3 0 ðventry vrollÞR h1 5 ð9:63Þ Lf1 2 exp½2γðσy 2 σentryÞg

where η0 is the dynamic viscosity at 38 C, in Pa s, and γ is the pressureviscosity coefficient in Pa21. The radius of the roll is designated by R in m, the roll surface velocity is vroll, the entry velocity of the strip is ventry, both in m/s and L stands for the projected contact length, also in m. The average flow strength in the pass is given by σy, and the tensile stress at the entry is σentry, in units that match those of γ. The relationship predicts that the film thickness at the entry will increase as the viscosity, the viscositypressure coefficient, the velocity and the roll radius increase. Assuming that the rolls and the strip are rigid at the inlet, the oil film thickness in the contact zone can also be determined as

x2 2 x2 h 5 h 1 1 ð9:64Þ 1 2R where x is the distance from the line connecting the roll centres and x1 is location of the entry to the deformation zone.

Emulsions The model that comes closest to reality is due to Schmid and Wilson (1995). The authors derive a simple equation for the inlet oil film thickness and claim that the predictions of the model are supported by experiments. They also make the statement that the experiments seem to suggest that the efficiency of oil droplet capture increases with increasing rolling speed. However, use of the model depends on certain assumptions whose validity is proven by comparing the predictions to measurements. Thus, the model is semi-empirical. The background of the study is the division of the entry zone to the roll gap into three regions: the supply region, the concentration region and the pressurization region. In the supply region the oil droplets are isolated from the surfaces and from each other. In the concentration region ambient pressure exists and the local concentration of the oil times the film thickness remains constant. In the pressurization region the water is trapped within the oil film and no further concentration is possible. The two assumptions that must be made involve a capture coefficient (C) and the oil concentration at which inversion of the emulsion occurs. The assumption is that inversion will occur at a concentration of 0.907. The model next calculates the oil film thickness in the case of an infinite pressurization region, using the expression developed by Wilson and Walowit (1972). The expression is

6μ Uγ h 5 0 ð9:65Þ 0w θ½1 2 expð2γσÞ Tribology 241

2 where μ0 is the oil viscosity, Ns/m , U is the average of the strip inlet and the roll surface velocities, γ is the pressureviscosity coefficient, θ is the inlet angle and σ is the plane-strain flow strength of the rolled metal. The emulsion availability is calculated next, using the assumed value of the capture coefficient:

Cφ d 5 s s ð : Þ A φ 9 66 ih0w

In the expression ds is the oil droplet size, φs is the oil concentration in the original emulsion and φi is the oil concentration at the inversion. The non-dimensional oil film thickness H 5 h0/h0w is obtained from

H2 ðA2 1 2AÞH 1 A2 5 0 ð9:67Þ

The thickness of the oil film at the entry to the pressurization region, h0, can now be determined.

9.5 Dependence of the Coefficient of Friction or the Roll Separating Force on the Independent Variables

During the last several decades thousands of rolling experiments were conducted in the writer’s laboratory. Steel and aluminium alloys were rolled with the independent variables being the rolling speed, the reduction, the roll diameter and its surface roughness, the lubricant and the temperature. The roll separating forces, roll torques, the temperatures, the roll speed and the resulting reductions were measured. The magnitudes of the coefficient of friction were calculated using Hill’s formula, Eq. (9.26), which is based on equating the measured and the calculated roll separating forces. In what follows, these data are presented to illustrate the dependence of the coefficient of friction or the roll separating force on some of the independent variables. It is noted, of course, that the coefficient of friction, thus determined, is not the actual value since, as mentioned above, a large number of variables, parameters and their interaction affect its magnitude. Hill’s numbers, however, serve well in a relative sense and are good for the comparison of the trends.

9.5.1 The Dependence of the Coefficient on the Reduction There appears to be a general agreement that the coefficient of friction decreases as the reduction increases. This agreement is confirmed by the results shown in Figure 9.26, obtained while rolling low-carbon steel strips, where the reduction is plotted on the abscissa. Two neat oils are included in Figure 9.26 as well as two rolling speeds. Under all conditions the coefficient of friction demonstrates a 242 Primer on Flat Rolling

0.40 Figure 9.26 The dependence of the coefficient of friction on the Lubricant viscosity reduction low-carbon steel 25.15 mm2/s strips. 0.30 5.95 mm2/s

0.20 Roll speed

~0.26 m/s

0.10 Coefficient of friction (Hill) ~2.40 m/s

Cold rolling low-carbon steel strips 0.00 0.00 0.20 0.40 0.60 0.80 Reduction

0.24 Figure 9.27 The dependence of Alloy the coefficient of friction on the 1100-H0 reduction aluminium alloys. 0.20 1100-H14 5052-H34

0.16

0.12 Coefficient of friction 0.08

0.04 0 5 10 15 20 25 Reduction (%) downward concave trend. The lubricant viscosity appears not to have a major effect on the coefficient but the rolling speed does (see Section 9.5.2). The behaviour of the coefficient of friction as dependent on the reduction, and hence, the normal pressure, changes when strips made of softer materials are rolled. This is demonstrated in Figure 9.27, where various aluminium alloys were rolled at room temperatures (Karagiozis and Lenard, 1985). The coefficients of friction in these experiments were determined by the pins and transducers embedded in the work roll (Lim and Lenard, 1984). As shown, the coefficient of friction increases with the reduction, indicating the faster rate of asperity flattening of the softer Tribology 243

0.40 Figure 9.28 The dependence of the coefficient of friction on the Lubricant viscosity rolling speed low-carbon 2 25.15 mm /s steel strips. 0.30 5.95 mm2/s

Reduction

0.20 ~15%

0.10 Coefficient of friction (Hill) ~45%

Cold rolling low-carbon steel strips 0.00 0 1000 2000 3000 4000 Roll speed (mm/s) metals, leading to a faster rate of increasing real area of contact, agreeing with the results of Tabary et al. (1994), reviewed above, see Section 9.3.2.3.

9.5.2 The Dependence of the Coefficient on the Speed Under most circumstances and provided that the conditions for a well-lubricated contact are met, increasing velocity results in dropping coefficients of friction. This is aided by two events: the increased amount of lubricant being drawn into the contact zone and the time-dependent nature of the formation of the adhesive bonds16. While Figure 9.26 already indicated the dependence of the coefficient of friction on the rolling speed, a more explicit figure underlines the issue, see Figure 9.28. Here, the speed is plotted on the abscissa. Again the same two lubricants are used as in Figure 9.26. Two reductions are indicated and in both cases the decreasing coefficient is evident. The situation may change when the conditions for efficient lubrication are not met. Increasing the roughness of the work roll and changing the roll material result in increasing coefficient of friction with increasing rolling speeds, see Figure 9.29 (Lenard, 2004). Figure 9.29 gives the coefficient of friction, obtained while rolling 6061-T6 aluminium alloy strips, reduced by approximately 55%. The roll speed is given on the abscissa and the results are given for three values of the roll roughness. At the highest roughness of Ra 5 2.4 μm, the coefficient of friction increases with the speed. Reich et al. (2001) examine the slopes of the forward slip-speed plots obtained while cold rolling 3004 aluminium alloy strips using an O/W emulsion. They conclude that increasing values of the forward slip with increasing rolling speeds may indicate

16The adhesion hypothesis (Bowden and Tabor, 1950) states that the resistance to relative motion is caused by adhesive bonds formed between the contacting asperity tips, which are an interatomic distance apart. 244 Primer on Flat Rolling

0.5 Figure 9.29 The coefficient of friction as a function of the Roll roughness (μm) 0.3 roll speed and the surface 0.4 1.1 roughness of the work roll; 2.4 6061-T6 aluminium alloy strips are rolled. 0.3 Source: Lenard (2004).

0.2

0.1 Coefficient of friction (Hill's formula) Nominal reduction = 55% 0.0 0.00 0.50 1.00 1.50 Roll speed (m/s) that the contact zone is starved of adequate amount of the emulsion. As demonstrated above (see Eqs. (9.13)(9.17)), increasing forward slip indicates increasing coefficient of friction and that is given in Figure 9.29, leading to the possibility of starvation in the contact zone.

9.5.3 The Dependence of the Coefficient on the Surface Roughness of the Roll Dick and Lenard (2005) conducted cold rolling experiments on low-carbon steel strips, using progressively rougher rolls in a STANAT two-high variable speed mill. The strips were lubricated by O/W emulsions, delivered at a rate of 3 l/m. Five kinds of roll surfaces were prepared. In the first instance, the rolls were ground in the traditional manner to a surface roughness of approximately Ra 5 0.3 μm in the direction around and along the roll. The next surfaces were prepared by sand blasting, expected to create a random roughness direction. Using Blasto-Lite glass beads BT-11 resulted in a surface roughness nearly identical to that of the ground rolls, Ra 5 0.35 μm. The next surface was prepared using larger glass beads of grit #24, creating a randomly oriented surface, approximately Ra 5 0.91 μm. Following this, using #60 Lionblast oxide grit resulted in surface roughness of 1.31 μm and using BEI Pecal EG 12, another oxide grit, created surface roughness of approximately 1.76 μm. Three lubricants, supplied by ImperialOil,wereusedinanO/Wemulsion. Walzoel M3 is a low-viscosity, high-VI oil with synthetic ester lubricity agents and phosphorus-containing antiwear agents. Its kinematic viscosity of 8.65 mm2/s at 40C and 2.34 mm2/s at 100C. Kutwell 40 is a medium-viscosity and medium-VI paraffinic oil with sodium sulphonate surfactant and antirust additives, and no lubricity ester or antiwear agents with a viscosity of 37 mm2/s Tribology 245

12,000 Figure 9.30 The roll separating high reduction, low speed: empty symbols force as a function of the roll high reduction, high speed: full symbols surface roughness; low-carbon steel strips are rolled. Source: Dick and Lenard (2005). 8000

4000 Roll force (N/mm) Roll force dry 10% Kutwell 10% Walzoel 10% FSG

0 0.0 0.4 0.8 1.2 1.6 2.0 R μ Roll surface roughness a ( m) at 40C. Oil FSG is a high-viscosity, high-VI oil with natural ester lubricity agents and zinc- and phosphorus-containing antiwear agents. Its viscosity is 185 mm2/s at 40C and 16.75 mm2/s at 100C. The supplier estimates the droplets to be between 5 and 10 μminsize. The results indicate that the roll separating forces depend on the roughness of the work roll in a very significant manner, as shown in Figure 9.30. Two sets of data are given in Figure 9.30, both for high reduction. The empty symbols indicate the forces at low rolling speeds, while the full symbols indicate the same at higher rolling velocities. The forces increase almost in a linear fashion as the roll roughness is increasing. The speed effect is also observable from Figure 9.30 and as above, under most conditions the forces drop as the speed increases. In a recent manuscript Lenard (Lenard, 2004), dealing with cold rolling of 6061-T6 aluminium alloy strips, using a low-viscosity mineral seal oil and progressively rougher work rolls, the slopes of the roll forceroll roughness plots indicated sudden increases of the slope at approximately 1 μm Ra. These changes revealed the relative contributions of the adhesive and ploughing forces to friction and indicated that the effect of ploughing overwhelms that of adhesion. While it is possible that increasing the roll roughness beyond 1.76 μm Ra would lead to similar behaviour, no comparable observations can be made in the present study and within the range of the parameters, as no sudden changes of the slopes are demonstrated. The different observations result from the significant differences of the viscosities of the lubricants used. In the study of Lenard (2004) the mineral seal oil’s viscosity was 4.4 mm2/s while in the present study the lightest oil’s viscosity is twice that. The low viscosity created a very low film thickness and the sharp asperities of the sand-blasted work roll must have pierced through the film as soon as contact was established at the entry. In the present work, the sharp asperities must also have pierced the oil particles but because of the higher viscosities, these likely have 246 Primer on Flat Rolling occurred later and to a lesser extent. Trijssenaar (2002) discusses the mechanisms that may cause continuous oil films: the coalescence and the break-up of the oil droplets and concludes that break-up is not likely to occur because of the low Weber numbers present during cold rolling. In the presence of the sharp edges of the asperities, created by the sand blasting process, the surface tension of the droplets may be overcome by the piercing action of the edges, leading to the creation of thin oil films.

9.5.4 The Dependence of the Roll Separating Force on the Lubricant’s Viscosity In the study mentioned above (Dick and Lenard, 2005) the effect of the viscosity of the emulsion was also examined17. The details of the emulsions are given above. The dependence of the roll force per unit width and by its association on the coefficient of friction, as well on the viscosity of the oils in the emulsions is shown in Figure 9.31. The origin at a viscosity of zero indicates dry conditions. The roll forces under two process conditions are shown: high reduction at high speeds (corresponding to a nominal reduction of 50% and roll surface velocity of 0.5 m/s) and low reduction at low speeds (corresponding to a nominal reduction of 12% and a roll surface velocity of 0.2 m/s). As expected, and as predicted by the Stribeck curve, increasing viscosity should lead to lower loads, at least in the boundary and in the mixed lubrication regimes. The figure leads to a surprising observation: the viscosity appears not to affect the loads on the mill at low speed

12,500 Figure 9.31 The roll separating Roll roughness 0.32 (ground) force as a function of the 0.35 lubricant viscosity; low-carbon 10,000 0.91 steel strips are rolled. 1.31 Source: Dick and Lenard (2005). 1.76 7500 50% at 0.5 m/s

5000

12% at 2500 0.2 m/s Roll separating force (N/mm) force Roll separating

0 0 40 80 120 160 200 240 280 Viscosity (mm2/s)

17Considering here that the viscosity of the oil is the same as in the emulsion is legitimate since the dynamic concentration theory is expected to hold, implying that the strip/roll contact is lubricated by almost-neat oils. Tribology 247 and at lower reduction. There is a minor drop of the forces as any emulsion is introduced but no meaningful change is observed. At the higher loads and speeds the effect of viscosity is clear, but it is not as pronounced as expected. The behaviour of the forces appears to depend on both the viscosity and the roll roughness. The drop on the forces from dry conditions to any lubricant is evident once again. Further, as long as the roughness is under 1 μm, increasing viscosity leads to lower forces. When the roughness of the work roll was increased to 1.31 and 1.76 μm, increasing viscosity created increasing forces. This condition implies the presence of a hydrodynamic lubrication regime but measurements of the roughness of the rolled strips contradict this possibility as the roughness of the rolled strips was always lower than it was before the pass. The implication is that the lubrication regime was close to hydrodynamic but was still in the mixed region. There must have been a large number of lubricating pockets and only some metal-to-metal contact. It is recalled that similar results were obtained by Shirizly and Lenard (2000) while rolling low-carbon steel strips.

9.5.5 The Dependence of the Coefficient of Friction on the Temperature While hot rolling low-carbon steels, the coefficient of friction has been shown to decrease with increasing temperature (Munther and Lenard, 1999), caused partly by the decreasing strength of the adhesive bonds between the strip and the rolls which are easier to break at the higher temperatures. The coefficient decreased with increasing velocity and the attendant lower time available for the formation of bonds between the strip and the roll surfaces. Friction increased with increasing reduction, where the increasing size of the deformation zone and the longer contact time caused lower surface temperatures, higher strength and more adhesive bonds. Jin et al. (2002) hot rolled 430 ferritic stainless steel strips their results are shown in Figure 9.14.While there is a large scatter in all three figures, one may conclude that the coefficient drops with increasing speeds, temperatures and reduction.

9.5.5.1 The Layer of Scale (Munther and Lenard, 1999) The rollmetal interface in hot rolling of steels always includes a layer of scale. The secondary scale formed during and after roughing is removed before the bar enters the finishing train. The 515 s that separate the scale breakers and the first stand are sufficient to form a new tertiary scale layer, about 10 μm thick, immediately on the hot steel surface, the behaviour of which may be ductile or brittle, depending on its temperature and thickness. This interface affects the frictional conditions, resulting in changes in the required roll forces, torques and power consumption, as well as the overall roll wear and surface quality. Understanding the formation and behaviour of the scale interface is important when examining the tribological phenomena that take place. Three types of iron oxide phases make up the scale on the steel surface. These are, with increasing oxygen content, wu¨stite, FeO, magnetite, Fe3O4 and haematite, Fe2O3. Usually a scale with all three types of oxide phases is present on the steel 248 Primer on Flat Rolling surface, with wu¨stite being closest to the steel matrix, followed by the intermediate magnetite layer and the outermost haematite layer. In order for an oxide to grow, it must overcome kinetic barriers. These energy barriers are consequences of the temperature and the pressure as well as the environment, available space and the steel’s chemical composition. As oxidation starts, the oxide layer is discontinuous and begins by lateral extension of discrete nuclei. The mass transportation of ions occurs in a direction normal to the surface when the nuclei are interwoven. The iron diffuses as cations and electrons through the oxide film. At the gasoxide interface on the surface, oxygen is reduced to oxygen ions. The zone of the oxide formation is at the gasoxide interface. The diffusion of oxygen in oxides is slower than that of iron in oxides (Samsonov, 1973; So´rensen, 1981). Scale growth on steels therefore requires the Fe ions from the matrix to diffuse outward through the scale layers. Since there is an abundance of oxygen at the surface, the diffusivity of iron in the oxides determines the growth rate. This means that the temperature and the time control the distance the iron atoms can travel; a high temperature in combination with long time will allow the iron atoms to travel far and the concentration gradient will decrease with time. This will aid the growth of the oxide layer, since more iron atoms are made available to react with oxygen to form oxides. The kinetics of the scale growth is commonly seen as parabolic with regard to time and exponential with regard to temperature (Roberts, 1983). Shaesby et al. (1984) investigated the role and morphology of the scale formed on the surface of a low-carbon steel at 1200 C. They reported FeO:Fe3O4:Fe2O3 ratios of 95:4:1 which is consistent with the information reported by others. Birks and Meier (1983) attributed this fact to the greater mobility of defects in wu¨stite. Matsuno (1980) studied blistering and hydraulic removal of relatively thin scale films on AISI 1008 steel. Samples were heated in vacuum and were allowed to oxidize for times no longer than 3 min. It was reported that the scale consisted of wu¨stite to the greatest extent followed by magnetite and haematite. This appeared to be independent of temperature. It was also observed that the growth kinetics followed the parabolic rate law with respect to time. According to Birks and Meier (1983), the growth kinetics can be evaluated by either continuous or discontinuous methods. A continuous method involves uninter- rupted monitoring of the sample, but requires sophisticated equipment. There are two types of continuous methods: those that monitor gas consumption and those that monitor mass gain. A discontinuous method involves either mass gain, in which case the mass of oxygen taken in the scale is measured, or the loss of mass if the oxide layer is stripped from the sample. Another method is to measure the scale thickness. However, this may result in inaccuracies if it is carried out at room temperature, since the scale layer is likely to crack and spall upon cooling. The disadvantages of the discontinuous methods are that many specimens are needed and that the progress is not observed between the data points. When oxides are present in hot metal working operations, the stresses in the oxides are generally so high that the deformation of the scale is dominated by dislocation glide. According to the HuberMises criterion, plastic deformation Tribology 249 requires dislocation glide on five active and independent slip systems. Because of the ionic bonding in the oxides this is only likely to happen at temperatures above 0.5Tm, indicating that the oxide may behave in a viscoplastic manner and deform with the bulk. The most important attributes of the scale, at least when metal working is considered, are its hardness and yield strength, since these indicate whether the oxides are abrasive. Luong and Heijkoop (1981) have reported room temperature hardness values of 460 HV for FeO, 540 HV for Fe3O4 and 1050 HV for Fe2O3. Funke et al. (1978) analysed data obtained by and Stevens et al. (1971) and concluded that the hardness of the oxides is temperature dependent. They found the hardness of magnetite exceeding the hardness of cementite (which was the only carbide present in the roll material investigated) at all temperatures. Lundberg and Gustaffson (1993) have reported hardness values at 900 C for FeO, Fe3O4 and Fe2O3. These were 105, 366 and 516 HV, respectively. Blazevic (1983a, 1983b, 1985) and Ginzburg (1989) discussed scaling on steels during re-heating, roughing, finishing and coiling. The occurrence of surface defects caused by the scale and its location on the strip surface, including their causes and remedies, has also been described. The role of the tertiary scale was considered by Blazevic (1996). According to Blazevic, the scale layer that enters the finishing train may be considered either thin and hot or thick and cold. In the first case of thin, hot scale, the scale fractures along fine lines as it is being compressed and elongated during deformation in the first stand. Hot metal is then extruded partially through the fine fractures as the deformation proceeds. At the same time, the hot metal deforms in the rolling direction, resulting in a simultaneous roughening and smoothing of the steel surface. In the second case of cold, thick scale, the scale is less plastic and therefore fractures severely upon elongation. The fractured scale is depressed into the steel surface, while hot metal extrudes outwards, causing a rough surface that is present even after pickling. The reason is that the metal that extruded upwards in the early stands will be over-pickled and will leave a mirror image of the prior roughness. This image remains although the cold rolling process creates an elongated and reduced image on the final product. The presence of a red scale on finished products is another serious surface defect that has been addressed by Fukugawa et al. (1994). This defect results in two phenomena. First is an aesthetic defect as stripes of red and black scales are found in the rolling direction. The second affects the roughness and shape of the strip since the scale is pushed into the surface of the steel. While Fukagawa et al.’s emphasis was on Si-added steels, the red oxide scale is a potential problem with other steels as well, especially when the steel is not descaled properly. This may occur when a scale layer consists of the three phases with the majority as wu¨stite. The area of wu¨stite particles exposed to the air increases considerably as the scale is fractured by hot rolling. Fewer iron atoms are available to these particles, which become completely separated from the wu¨stite matrix, where there is a ready supply of oxygen. This results in the accelerated reaction of FeO ! Fe3O4 ! Fe2O3.The final product is the red haematite. 250 Primer on Flat Rolling

Scale formation has a very significant effect on friction and hence on the quality of the rolled surfaces as well as on the commercial value of the product. El-Kalay and Sparling (1968) were among the first to investigate the effect of scale on frictional conditions in hot rolling of low-carbon steel. Different conditions were studied in a laboratory: light, medium and heavy scaling with both smooth and rough rolls at various velocities. Load and torque functions, according to Sims’ equations, were calculated for these conditions. It was hypothesized that the scale acts as a poor lubricant and that its effect on the frictional conditions varies along the arc of contact as it fractures. It was found that the presence of scale could reduce the roll loads by as much as 25%. A thick scale reduced the loads more than a thin scale since the thick scale breaks up into islands that transmit the load from the rolls to the strip. The islands become separated as the strip is elongated. Hot metal then extrudes between the islands and sticks to the rolls while the sliding islands move further apart and promote tensions applied to the sticking portion, thereby reducing the load. It was also found that thin scale promotes sliding friction with smooth rolls, but sticking friction with rough rolls. The load functions increased with temperature in rolling with rough rolls, but decreased with temperature for smooth rolls. Roberts (1983) used the data of El-Kalay and Sparling (1968) to empirically model the coefficient of friction in terms of scale thickness, roll roughness and temperature. The model predicts an increase in the coefficient of friction with increasing roll surface roughness, decrease in scale thickness or increased temperature. Li and Sellars (1996) found that sticking friction takes place in hot forging of scaled low-carbon steel, but a certain degree of forward slipping, indicating partly or completely sliding friction, occurs in the rolling of the same material. Comments, similar to Blazevic’s (1996), were made on the break-up of the scale. They found a limited number of cracks on specimens with thin scale. A scale layer can follow a similar reduction and elongation as the steel only if its hot strength is equal to or lower than that of the hot steel. Schunke et al. (1988) presented a hypothesis on the effect of partial oxygen pressure on friction coefficients at room temperature, although additional information for temperatures below 600C was presented for various Fe alloys. While analysing data obtained by other researchers they found that the coefficient of friction during sliding was dependent on the partial pressure of oxygen as well as the sliding length. Generally, the coefficient of friction decreased with increased oxygen pressure and temperature, as these cause an oxide layer to grow more rapidlyonthesurface.Thedropinfrictionwasexplainedasfollows:theoxide particles are fragmented when deformed and become further oxidized and com- pacted onto the metal surfaces where they form islands in the next cycle. When these islands grow in area a large portion of the shearing is at these islands, causing the total contact area to be reduced. Friction is then lowered because of the brittle nature of the oxide particles that are being sheared. Shaw et al. (1995) determined fracture energies of oxidemetal and oxidesilicide interfaces. It was concluded that the fracture energy depends primarily on interfacial bond strength, although roughness of the interface, microstructure of the compounds and porosity also have some effect. Tribology 251

An up-to-date exposition of the role of the layer of oxide in the hot rolling process was given by Krzyzanowski and Beynon (2002). The effect of the layer of scale was investigated using an AISI 1018 steel, containing 0.18% C and 0.71% Mn, as well as an HSLA steel, containing 0.067% C and 0.0764% Nb. The samples, 12.65 mm in thickness and 50.8 mm in width, were heated under closely controlled conditions; hence the scale growth was also well controlled. The scale index, γ, is defined in terms of the ratio of the growth rate of the scale and the time

pﬃﬃ γ 5 kp t ð9:68Þ where the growth rate is given in terms of the activation energy for scale formation, Qscale, the universal gas constant and the absolute temperature: 2Q k 5 k exp scale ð9:69Þ p e RT

where ke is a constant. A finite-element code, Elroll, shown to yield results in excellent agreement with both laboratory experiments and industrial data is used in the present study. The output parameters are temperature profiles in the sample and the work roll, roll pressure, roll separating force and torque, as well as the forward slip, defined as the relative difference in roll surface/strip exit velocity. Two parameters the coefficients of heat transfer and friction may be chosen at will. The heat transfer coefficient has been determined in previous experiments and a value of 1030 kW/m2 K has been established, depending on pressure, contact time and scale thickness. The coefficient of friction is then chosen such that the calculated and measured values of the roll force, the roll torque and the forward slip agree as closely as possible. Following Wankhede and Samarasekera (1997) and Chen et al. (1993), the heat transfer coefficient, α, is modelled as solely pressure dependent. However, previous investigations have shown that their predictions may overestimate the coefficient of heat transfer for these laboratory conditions, especially in rolling of highly scaled steel. The coefficient of heat transfer is therefore described empirically as:

p α 5 2 40 ð9:70Þ 3 where p is the average roll pressure in MPa, taken as the ratio of the roll separating force and the projected contact area. This results in a coefficient of heat transfer that ranges between 10 and 30 kW/m2, values that apply strictly to the laboratory mill used in the present study. 252 Primer on Flat Rolling

9.5.5.2 The Effect of the Scale Thickness on Friction The effect of scale thickness on the frictional conditions can be seen in Figure 9.32, in which the coefficient of friction is plotted against the temperature for a range of thickness. The reduction and the velocity were kept constant at 25% and 170 mm/s, respectively. It is evident that the coefficient of friction has its highest value for the thinnest scale layer of 0.015 mm, ranging between 0.35 at 825C and the lower value, 0.30 at 1050C. The scale thickness is then increased, first to 0.29 and then to 1.01, followed by 1.59 mm. This results in a reduction of the coefficient of friction at all temperatures, to values ranging between 0.22 and 0.30 for a scale thickness of 0.29 mm. A scale thickness of 1.01 mm results in a variation in the coefficient of friction from 0.19 to 0.24. The lowest values are seen for the thickest scale, yielding a coefficient of friction between 0.195 and 0.219. It is realized, of course, that the thickness of the scale under industrial conditions is much below these values. The scale thickness appears to have a significant effect on the frictional conditions. In analysing the experimental data, gain in thickness was taken into account along with changes in the heat transfer coefficient due to the insulating effect the scale provides.

9.6 Heat Transfer

The transfer of thermal energy at the contacting surfaces is affected by the same independent variables and parameters that affect frictional resistance. These have been discussed above in some detail and their interconnections were shown in Figure 9.1. In an approach similar to that followed when discussing the coefficient of friction or the friction factor, it is necessary to be realistic and to limit attention to those parameters that may be measured, identified, determined or at the very

0.5 Figure 9.32 The dependence of the coefficient of friction on Average temperature (°C) the temperature and the 830 0.4 870 thickness of the layer of scale. 960 Source: Munther and Lenard (1999). 0.3

0.2 Coefficient of friction

0.1 Low-carbon steel

0.0 0.0 0.4 0.8 1.2 1.6 2.0 Scale thickness (mm) Tribology 253 least, may be assumed with some reasonable degree of assurance. In the present context, these are the areas in contact, the normal and shear forces on the contacting surfaces, their relative velocities and their bulk and surface temperatures. While the thickness of oxide layers and the use of lubricants must have a significant effect on the transfer of heat, in most instances only their presence or the absence is considered. Surface roughness, a significant parameter affecting friction, is not considered here, largely because its effect on the heat transfer has not been established in detail in the technical literature18. The relationship of the temperatures of the surfaces in contact is governed by the heat transfer coefficient, α, defined as the ratio of the heat flux the amount of heat transferred per unit area and unit time and the difference of the temperatures of the hot and the cold surfaces. The usual manner of mathematical representation is to give the heat flux, q_, in terms of α and the average temperatures of the hot and the cold surfaces:

q_ 5 αðThot 2 TcoldÞð9:71Þ where α is the heat transfer coefficient. The deficiencies of the formulation are apparent immediately. It is usually assumed in Eq. (9.71) that the surface temperatures are uniform across the contacting surfaces, which, of course, is not correct. The heat transfer coefficient is also taken as a constant and in all likelihood, it varies with time as well as location and surface parameters. Nevertheless, Eq. (9.71) works, in the sense of “for all practical purposes”, and is used almost universally. The thermal boundary conditions at the interface are usually formulated in terms of the heat transfer coefficient. Relatively little work has been reported regarding procedures that lead to an estimate of the interface heat transfer coefficient in bulk forming processes. There are essentially two approaches by which the heat transfer coefficient may be estimated. One of these is the “inverse technique” in which one chooses α such that calculated and measured temperature distributions or, at least, average surface temperatures will agree closely. The other is to use the experimentally established timetemperature profiles to estimate the temperatures of the two contacting surfaces and use the definition of the heat transfer coefficient as the ratio of the heat flux and the temperature difference of the surfaces. Naturally, both of the methods have limitations. In the former, success depends on the quality, accuracy and rigour of both the measurements that are to match the predictions of a model and those of the model itself. The latter is also dependent on the measurements in addition to the technique of determining the surface temperatures and hence, their difference. A detailed discussion of previous studies on the heat transfer coefficient was presented by Lenard et al. (1999). While the information given is important, only a summary will be reproduced here, in the form of Table 9.1.

18At a recent conference in Vienna, Austria (Second World Tribology Congress, 2001), the present author heard the statement, made by a prominent researcher, “I am now working on the development of an asperity-based heat transfer model”. Risking the display of some ignorance or inadequate literature search, I have not yet seen the publication of the study. 254 Primer on Flat Rolling

Table 9.1 Heat Transfer Coefficients

Reference Material α (kW/Km2) Comments

Chen et al. (1993) Aluminium 1054 α varies along the arc of contact Stevens et al. (1971) Steel 38.7 Water-cooled rolls Preisendanz et al. (1967) Steel 1.2 and 2.5 MJ/m2 s Obtained on a at 700 and 1100C production mill Silvonen et al. (1987) Steel 70 Obtained on a production mill Bryant and Chiu (1982) Steel 7000 Obtained on a production mill Harding (1976) Steel 2.055 at 700C and Research mill 9.1 at 1100C

Table 9.2 Heat Transfer Coefficients, Obtained on a Laboratory Mill

2 % red. hentry (mm) rpm α (kW/m K)

7 15 4 12.78 7 15 10 19.35 6 19.4 3 19.27 10 19.5 4 10.85 21 19 4 12.78 21 19 10 13.99 19 19 4 9.8 20 19 4 12.31 11 30.9 4 13.06 20 38 4 19.93 19 38 4 11.79 20 38 10 20.76 18 20.1 12 13.74 24 18.3 12 9.6

Source: Karagiozis (1986) and Pietrzyk (1994).

Chen et al. (1993) present a relationship of the heat transfer coefficient and the interfacial pressure in the form:

α 5 0:695p 2 34:4 ð9:72Þ where p is the pressure in MPa and α is the coefficient of heat transfer in kW/m2C. Karagiozis (1986) and Pietrzyk and Lenard (1988, 1991) hot rolled carbon steel slabs, instrumented with several embedded thermocouples. The values of the coefficient at the interface are given in the last column of Table 9.2. Tribology 255

900 Figure 9.33 The temperaturetime Centre temperature profile during hot rolling of a Surface temperature low-carbon steel slab. Entry 850

Tbulk = 30°C Tsurf =800°C 800

bulk, ave. T entry

Temperature (°C) Temperature Exit 750 bulk, ave. T T surf, ave. exit

700 0.00 0.20 0.40 0.60 0.80 1.00 Time (s)

A comparison of the predictions of Chen et al. (1993) and the numbers in Table 9.2 indicates some difficulties. The average roll pressure in the experiment using the 19-mm-thick strip, reduced by 21% at 4 rpm, is estimated to be 162 MPa. Equation (9.69) predicts a heat transfer coefficient of 78 kW/m2 K, while the calculations, based on the experimental data, give 12.78 kW/m2 K. It is apparent that Eq. (9.69) doesn’t include all of the significant variables to be useful in a general case; the relative speed of the work roll or the time of contact, the temperature and the thickness of the layer of scale should also be accounted for.

9.6.1 Estimating the Heat Transfer Coefficient on a Laboratory Rolling Mill It is possible to estimate the coefficient of heat transfer with reasonable accuracy, as long as temperature data concerning the rolled strip are available. Such data are shown in Figure 9.33. The dependence of the temperature at the centre of the strip and near its surface as a function of the elapsed time is shown in Figure 9.33. A fairly thick strip of 38 mm entry thickness was reduced by 20% at a roll speed of 52 mm/s (4 rpm) in a laboratory rolling mill. Two thermocouples were embedded in the tail end of the strip to a depth of 25 mm. One of the thermocouples was located at the centre of the strip while the other was 2.5 mm from the surface19. The suggested approach to estimate the coefficient of heat transfer is as follows. The heat flux has been defined above as the work done (W) on the rolled strip

19Attempts to place a thermocouple closer to the surface were not successful as the stress concentration caused by the hole and the hard sheathing of the thermocouple caused cracking. 256 Primer on Flat Rolling per unit surface area (A) and time (Δt). A simple model may then be written, giving q_ as:

W q_ 5 ð9:73Þ AΔt which, realizing that ΔTbulk 5 W=ðVρcÞ, may be re-written in terms of the temperature drop of the strip (ΔTbulk)as:

ρcΔT h q_ 5 bulk ave ð9:74Þ 2Δt where ρ is the density in kg/m3, c is the specific heat in J/K kg and have 5 ðhentry 1 hexitÞ=2 is the average thickness of the strip in the pass. The volume of the material in the deformation zone is V 5 Lwhave and the contact area is A 5 2Lw, where w is the width of the rolled strip. All of the information, necessary to calculate the heat flux, is available from Figure 9.33. The density of steel may be taken as 7850 kg/m3; the specific heat is 625 J/K kg. The average thickness is 34.2 mm and the temperature loss of the strip is estimated as 30 K. The time elapsed from the entry to exit is read off the figure as 0.5 s. The heat flux is then obtained as 9.034 J/m2 s. Now using Eq. (9.71), the original definition of the heat transfer coefficient, the average surface temperature of the strip (800C) and the average roll surface temperature as 100C, the heat transfer coefficient is obtained as 7191 W/m2 K. The limitations of the approach just described need to be clearly understood. There are essentially three difficulties. The first is the use of average temperatures, based on the data obtained from only two thermocouples. The other is the estimate of the time elapsed from entry to exit, based on the reaction of the thermocouple near the surface. The third is the use of the data from the thermocouple, located 2.5 mm from the surface, as the temperature of the surface, ignoring the changes to the surface. Nevertheless, the magnitude calculated is quite realistic.

9.6.2 Measuring the Surface Temperature of the Roll Tiley and Lenard (2003) conducted hot rolling tests, using low-carbon steel strips of 9.08 mm thickness. The rolling mill was instrumented with two optical pyrometers which allowed the monitoring of the entry and the exit temperatures of the strip. Force and torque transducers measured the roll separating forces and the roll torques. A shaft encoder measured the rolling speed. Eight thermocouples were embedded in the work roll, such that their tips were 0.5 mm from the roll surface, abutting 0.5 mm copper washers. The data collected were used to determine the interface heat transfer coefficient under three separate surface conditions:

G The strips were rolled with the scales on and no lubricants were used. G The strips were descaled before the pass and no lubricant was used. G The strips were not descaled and a mineral seal oil was used. Tribology 257

16 Figure 9.34 The heat transfer coefficient as a function of the interfacial pressure, on a ) 2 laboratory-size rolling mill. 12 Source: Tiley and Lenard (2003).

Surface conditions With scales; no oil Descaled; no oil 4 With scales; with oil Heat transfer coefficient (kW/Km Heat transfer

0 100 200 300 400 Roll pressure (MPa)

The results are shown in Figure 9.34, plotting the experimentally determined interfacial heat transfer coefficient against the interfacial pressure for the three conditions listed above. All three surface conditions cause the heat transfer coefficient to increase with the increasing pressure, albeit at very different rates. The magnitudes of the coefficients are of the same order of magnitude as those shown in Tables 1 and 2, but they are considerably lower than the predictions of Eq. (9.72).

9.6.3 Hot Rolling in Industry the Heat Transfer Coefficient on Production Mills Even though direct measurements of the heat transfer coefficient under industrial conditions are rare, there is a consensus among the researchers and users that it appears to be significantly larger than values obtained in the laboratory. The difficulties in conducting trials using a full-scale strip mill are probably impossible to overcome and this necessitates the use of inverse calculations, in which a variable is measured and calculations are used to match that variable by treating the heat transfer coefficient as a free parameter. Recall that this approach is also used often to determine the coefficient of friction. Calculations were performed using data obtained from several hot strip mills. In the first instance, the heat transfer coefficient that best matched the temperature of the surface of the transfer bar before entry to the finishing train and after exit from the last stand was 50,000 W/m2 K. In the second instance, when strip surface temperatures at the entry to each stand were available, the heat transfer coefficient varied from a low of 75,000 W/m2 K at the first stand to 88,000 W/m2 Katthelast. 258 Primer on Flat Rolling

It is emphasized here that these numbers depend, in a very significant manner, on the data available from mill logs. Traditionally, these include the surface temperature of the strip after the rougher and before coiling but they don’t provide stand-to-stand temperature data.

9.7 Roll Wear

Czichos’ (1993) comments, made quite some time ago, are valid at this time as well. He estimates that nearly 20% of the energy generated in the industrialized world is consumed by friction and that the losses form a significant portion of the gross national product. While he estimates 12% of the GNP is lost because of friction and wear, Rabinowicz (1982) gave the much higher figure of 6%. The recent review “Tribology in Materials Processing” by Batchelor and Stachowiak (1995) underlines these concerns and suggests that the costs begin when the ore is extracted from the ground. They define wear and friction, and hence tribology, as chaotic processes in which predictions are not possible. They state categorically that an analytical approach to wear is impossible. There appears to be agreement with this view in the technical literature. For example, a few years earlier, Barber (1991), considering a tribological system, wrote that accurate point- wise simulation of such a system is inconceivable at the present time. The author continued to describe a caricature of a tribological research paper, commenting on the complexities of the physical system and the need for assumptions in the mathematical model. The opinions of Barber should be taken very seriously. He is absolutely correct in writing that predictions of models, without adequate experimental data supporting those predictions, are of little value. One possible addition to that sentence may be to request experimenters to compare their measurements to the predictions of models. In this way, the accuracy of the models’ assumptions may be determined. In spite of these comments, the relation, given by Roberts (1983), to estimate the change in the radius of a work roll, is found to be useful. The ratio of the change in the roll radius, ΔR, and the rolled length, λ, is given by: μL KμL2rσexp ΔR h ð2 2 rÞ 5 entry ð : Þ 2 9 75 λ D σroll where K is the wear constant, L is the contact length, r is the reduction in decimals and σ and σroll are the flow strength of the strip and roll, respectively. While the wear constant is not easy to determine exactly (Roberts, 1983, gives some further data on K), the formula gives realistic numbers for the loss of roll radius. Letting K 5 8 3 1025, the coefficient of friction equal to 0.4, the roll diameter equal to 400 mm, 40% reduction, the flow strength of the strip as 250 MPa, and that of the Tribology 259 roll to be 600 MPa, the loss of the roll radius, after 100 strips of 1000 m length each, is estimated to be 7 mm, a reasonable number. Batchelor and Stachowiak (1995) also discuss the mechanisms of friction and wear. Mechanisms of wear, shown in Figure 9.35, include abrasive, fatigue, erosive, cavitational and adhesive wear. Abrasive wear is caused by the ploughing action between the contacting asperities. Erosive wear is the result of impact of solid or liquid particles. Repeated contact causes fatigue wear and liquid droplet erosion causes cavitational wear.

(A) Abrasive wear

Direction of abrasive grit Direction of abrasive grit Cracks Grit

Grain pullout

Direction of abrasive grit Direction of abrasive grit Fatigue Repeated deformations by subsequent Grain about grits to detach

(B) Erosive wear High angle of impingement 3 Low angle of impingement 2 1

Abrasion Fatigue

Collapsing bubble

Impact of solid and liquid

Deformation or fracture of solids resulting in wear

Figure 9.35 Mechanisms of wear. Source: Batchelor and Stachowiak (1995). 260 Primer on Flat Rolling

Fitzpatrick and Lenard (2001) define the three phenomena that occur between two contacting surfaces that control the wear process, regardless of what mechanisms are causing the wear. These are as follows:

G the chemical and physical interactions of the surface with lubricants and other constituents of the environment; G the transmission of forces at the interface through asperities and loose wear particles; and G the response of a given pair of solid materials to the forces at the surface. These phenomena are not independent and any changes to these aspects have a dramatic effect on wear and wear rates. Adhesive wear results when the contacting surfaces form bonds between the asperities. Fatigue wear is caused by repeated application of the loads. Abrasive wear is observed when hard particles come in contact with the surface under load. Tribochemical reactions at the surfaces cause chemical wear. Hard, solid particles, causing impact, result in erosion. Similar to this is impact wear, occurring when the two surfaces come in contact under impact conditions. Finally, fretting wear is found when the contact surfaces experiences oscillation with small displacements in the tangential direction. Czichos (1993) presented a well-thought out overview of wear mechanisms. Previous studies show that roll wear rates are highest at temperatures of 850950C, precisely the temperatures used in the finishing stands of hot strip mills. Roll wear is also a function of the specific load, sliding length and abrasive and corrosive particles in the cooling water. The low speeds of the roughing stands cause most of the wear and slippage as a result of too low friction also causes excessive roll wear. The parameters are many and the complex problem can be caused by either excessive or diminishing friction. The fairly recent introduction of tool steel rolls20 gave a significant impetus to research on wear during both hot and cold rolling of steels. The papers presented at the 37th Mechanical Working and Steel Processing Conference agree that the change resulted in a very significant drop of the rates of roll wear (Arnaud, 1995; Webber, 1995; Barzan, 1995; Hashimoto et al., 1995a,b; Hill and Kerr, 1995 and Auzas et al., 1995). More recent work emphasized the improvements (Medovar et al., 2000; Gaspard et al., 2000 and Saltavets et al., 2001). Tool steels rolls, once implemented correctly, do provide benefits that offset their higher costs. The impact of lubricant interactions with these new roll chemistries have not been fully explored (Nelson, 2006).

9.8 Nanotribology

A click in Google for “Nanotribology” yielded over 90,000 hits (August, 2013). While the topic is not new by far, attention of researchers increased with the advent

20See also Chapter 3 for a discussion of roll wear. Tribology 261 of miniaturization of components, in electronic devices, computers and the like. A nanometre equals 1029 metre. In traditional manufacturing, engineers deal with asperity measures and lubricant films in the micrometer scale so it is clear that nanotribology requires original thinking and new approaches. The components one deals with have uniquely large surface to volume ratios. The field involves experimental and theoretical studies of adhesion, wear, friction and lubrication at the atomic and molecular level. A quotation, given below, from the home page of the Fifth Vienna International Conference on Nanotechnology underscores the need for the original approaches:

Mechanical aspects of Nanotechnology up to now are rather poorly understood and developed. Nevertheless there are many future-orientated efforts all over the world in order to increase the knowledge. This has not only a fundamental importance but moreover offers an immense range of practical applications. For this reason, Nanotechnology is now a rather rapidly growing field. Nanomaterials show significantly different mechanical properties. Material strength, scratch hardness as well as friction and wear are strongly influenced by the nanostructure. In addition, in the presence of gaseous or liquid media (e.g. lubricants) the material properties vary over a wide range. Subsequently, due to mechanical activation the chemistry of the near-surface volume changes the engineering in this field has to consider these sometimes sophisticated facts. These aspects will influence many practical applications, ranging from precision systems to precision surface engineering, tribological systems, microsystems and actuators, and others.

A random selection of a few titles of presentations at the Joint ICTP-FANAS Conference on Trends in Nanotribology, held in Trieste, Italy, in 2011, further indicates the breadth of the field. Dynamical behaviour of small organic molecules on metallic and graphitic substrate; Atomic scale energy dissipation mechanisms in NC-AFM; Exploring the atomic underpinnings of macroscopic friction. It is clear that there is a significant need to explore, study, research and understand Nanotribology.

9.9 Conclusions

The independent variables that affect surface interactions friction, lubrication, heat transfer and roll wear in the process of flat rolling have been identified and are classified below, according to the parameters of the process and the three components of the metal rolling system: the rolling mill, the rolled metal and their interface. In the present context, surface interactions refer to the transfer of mechanical and thermal energies at the contact and are characterized numerically by the coefficients of friction and heat transfer. 262 Primer on Flat Rolling

μ changes with increasing reduction

The interfacial pressure increases

Strain hardening The asperities Surface Viscosity The bite leads to fewer flatten leading temperature increases angle grows bonds to more bonds increases μ falls More oil is delivered μ falls Viscosity μ grows at entry decreases It is harder The roll flattens More oil is squeezed to break the μ grows and the contact μ falls out of the cavities asperities area grows - more bonds may form μ grows μ falls μ grows

Figure 9.36 The effect of increasing reduction on the coefficient of friction.

The rolling mill the roll material and its diameter; surface roughness and its direction; surface hardness. The rolled metal the resistance to deformation; surface roughness and its direction; surface hardness. The interface lubricant/emulsion viscosity; flow rate; pressure and temperature sensitivity; density. The process the rolling speed; reduction; temperature. It is recognized, of course, that the above list differs in a most significant manner from the one given in Figure 9.1 as it is much reduced and much less comprehensive, driven by the need to retain only the most important parameters. The interactions of these parameters with the coefficients of friction are presented in Figures 9.36 and 9.37. The first versions of these figures were published by Lenard (2000). They are updated here, making use of recently accumulated experience. The first step in constructing Figures 9.36 and 9.37 is the decision concerning the most important independent variables. Once the metal to be rolled, its chemical composition, dimensions and surface roughness are selected, the lubricant or the emulsion and their flow rates are prescribed, the rolling mill is chosen (bringing Tribology 263

μ changes with increasing relative speed

More oil may be drawn in the contact zone

Surface temperature The surface The surface It is harder increases due roughness is roughness is to flatten to strain rate random in the direction the asperities of rolling hardening The oil is μ falls distributed The oil is not Viscosity evenly distributed decreases The strength is well lowered due to μ falls the rise of the μ grows μ falls μ grows temperature

The film thickness Shearing the oil It is easier will grow is harder to form the bonds μ falls μ grows μ grows

Figure 9.37 The effect of increasing relative velocity on the coefficient of friction. with it the roll dimensions and geometry, its surface hardness and roughness), the remaining decisions concern the reduction per pass and the speed of rolling. These two are then considered in Figures 9.36 and 9.37, respectively. It has been shown above that when the reduction is increased, in most cases the coefficient of friction drops. Increasing the reduction brings with it several changes, the first of which is the correspondingly increasing roll pressure which, in turn, increases the stresses within the rolled strip. The metal may then experience strain hardening and, compared to a softer or less strain hardening material, it may be harder to flatten its asperities. Hence, keeping all other parameters identical, the harder metal will cause the coefficient of friction to drop. A contradictory mechanism may also be observed here: when the metal’s strength grows, the strength of its asperities also grows. The relative movement of the roll and the strip may cause the asperities to break which then requires more effort, resulting in an increase of the coefficient of friction. The increasing normal pressure will also affect the lubricant trapped in the valleys between the asperities. The oil is likely to be squeezed out, wetting the nearby surfaces, causing a drop in the coefficient. The lubricant’s viscosity will be affected by two contradictory events. The lubricant’s temperature will increase leading to a drop of the viscosity and an increase of the coefficient of friction, while at the same time the increasing pressure will increase the viscosity, leading to a drop of the coefficient. Changes of the geometry of the pass will also cause changes. The contact area will grow because the roll will flatten and more adhesive bonds may develop. As well, the bite angle will increase, delivering more oil to the contact zone. The 264 Primer on Flat Rolling mechanisms that cause a drop of the coefficient of friction appear to overwhelm the others in most instances. Similar arguments may be made when the effects of the increasing relative velocity on the coefficient of friction are examined, shown in Figure 9.37. In preparing Figure 9.37 it was assumed that no starvation is present. It is generally agreed that when the relative motion between the roll and the rolled metal increases, more oil is available to be drawn in to the contact zone. The nature of the roll’s surface roughness will determine if the lubricant is spread evenly on the contacting surfaces. If yes, lower coefficient of friction will result. Further, more oil will result in thicker lubricant films and lower coefficients. A limit may be reached here when the contact zone is saturated and the coefficient will not fall any more. The increasing speed may also contribute to increasing coefficient of friction, as long as the hydrodynamic condition is reached. The temperature may rise due to increasing strain rate hardening; the increasing speeds will increase the shear stress needed to shear the lubricant; the metal may experience strain rate hardening and this, in turn, may make it harder to flatten the asperities. There is a contradictory phenomenon here, as well: the increasing temperature will lower the lubricant’s viscosity and the coefficient of friction will fall. The increasing temperature may also cause some softening of the rolled metal; however, this is not expected to be a very significant contributor. The time rates at which these changes occur, the interaction of which will determine the behaviour of the coefficient, are not known at this stage. The heat transfer coefficient, the ratio of the amount of heat transferred by unit area and unit time to the difference of the temperatures of the hot and the cold surfaces, is also affected by the reduction and the relative velocity between the roll and the rolled strip. Increasing the reduction appears to have caused the heat transfer coefficient to increase, implying either a higher heat flux or a lower temperature difference. The true area of contact as well as the time of contact would increase with the reduction, lowering the coefficient. The temperature difference would also decrease and that is expected to overwhelm the other effects, resulting in an increase of the heat transfer coefficient. Increasing relative velocities would decrease the time of contact and the temperature difference would increase as there would be less time for the heat to be transferred. The coefficient would therefore rise. It is possible to make a few recommendations regarding the coefficients of friction and heat transfer, to be used in predictive modelling of the flat rolling process. While the numbers given below cannot replace values arrived at by independent experimentation, they nevertheless should aid in improving the quality of predictions when used in mathematical models of the flat rolling process.

9.9.1 Heat Transfer Coefficient When hot rolling steel in the laboratory, using relatively small rolling mills, values of 420 kW/m2 K appear to be the correct magnitudes. If modelling hot rolling of steel under industrial conditions, values of 50120 kW/m2 K are more Tribology 265 useful. In both cases, the layer of scale is an important parameter. Scale is an insulator so its presence slows the cooling of the surface of the rolled metals. Since it is difficult to determine the heat transfer coefficient in experiments, use of the inverse method is recommended. The heat transfer coefficient during cold rolling of steel varies from a low of about 20 kW/m2 Kupto40kW/m2 K. Somewhat higher magnitudes, by about 15%, are appropriate during cold rolling of aluminium strips.

9.9.2 The Coefficient of Friction 9.9.2.1 Cold Rolling When analysing the flat rolling process of cold rolling steel strips without lubrication, the magnitude of the coefficient of friction is likely to be in the range of 0.150.4. If efficient lubricants or emulsions are used, the coefficient drops to 0.050.15; use the lower values when the thickness of the oil film is high and the higher values when the roll’s surface roughness is high. While rolling soft aluminium strips, the coefficient appears to be approximately 20% higher; when harder alloys are rolled, the coefficient is about 10% higher than that for steels. In general terms, increasing viscosity, speed and reduction cause a drop of the coefficient.

9.9.2.2 Hot Rolling While the suggestion by early researchers, indicating that sticking friction exists during the hot rolling process, has often been shown to be incorrect, the coefficient of friction is found to be significantly higher in the process. The constant existence of a layer of scale is one of the most important contributors that affect the magnitude of the coefficient of friction. The adhesion of the oxide layer to the work roll may also have an effect on the tribological conditions at the contact. Values of the coefficient range from about 0.2 to 0.45 in lubricated hot rolling of steels. In contradiction to that experience found during the cold rolling process, increasing reductions appear to cause larger coefficients, due to the softer scale on the steel’s surface. Higher velocities and thicker layers of scale cause a reduction of the coefficient. While hot rolling aluminium, values about 10% higher are recommended, caused by the accumulating coating on the surface of the work rolls.

9.9.3 Roll Wear Roll wear was discussed briefly. Roberts’ formula, predicting the loss of roll material, was shown to yield realistic numbers. The mechanisms of wear were discussed. There appears to be a general agreement that the use of tool steel rolls reduces the rate of wear of the rolls. 266 Primer on Flat Rolling

9.9.4 What Is Still Missing While several attempts at obtaining functional relationships for various surface interactions as functions of some of the independent variables have been reviewed above, a general equation of the form:

Surface interaction5f ðload; speed; temperature; strength; roughness; viscosity... has not yet been presented. Its availability would ease modelling in a most significant manner. 10 Applications and Sensitivity Studies

10.1 The Sensitivity of the Predictions of the Flat Rolling Models

The importance of the ability to predict the rolling variables prior to designing the draft schedule has been emphasized before. These predictions, which are usually performed using off-line models of the process, are dependent on several known and not well-known parameters. This dependence is examined in the rest of this chapter.

10.1.1 The Sensitivity of the Roll Separating Force and the Roll Torque to the Coefficient of Friction and the Reduction One of the less well-known but arguably one of the most important parameters is the coefficient of friction. The dependence of the roll separating force and the roll torque on the coefficient is examined here. In the calculations, the predictions of two models the empirical model and the refined 1D model are compared at various reductions as functions of the coefficient of friction; recall that both models are capable of reliable predictions. Cold rolling of low-carbon steel strips is consid- : ered. The true stresstrue strain relation of the steel is σ 5 150ð11234εÞ0 251 MPa; the entry thickness is 1 mm and the roll radius is 125 mm. Neither model accounts for the speed of rolling, not a significant problem since the metal’s resistance to deformation is independent of the rate of strain. In a situation where the strain rate affects the metal’s strength, a constitutive relation reflecting that dependence needs to be used. The rate dependence of the roll separating force will then be demonstrated through the stressstrain relation. The dependence of the roll separating force on the reduction and the coefficient of friction is demonstrated in Figure 10.1. The coefficient of friction is plotted on the abscissa and the force on the ordinate. As expected, the forces increase as the reduction and the coefficient of friction increase. The unexpected observation concerns the extremely steep rise of the roll separating force, as predicted by the 1D model which, it is to be recalled, uses the friction hill idea to obtain the roll pressure distribution, the integration of which over the contact yielding the force. At a reduction of 3040% and a coefficient of friction of 0.2 and 0.25, reasonable process parameters if rolling with no lubricants

40,000 Figure 10.1 The dependence of Reduction F F 1D Schey the roll separating force on the 0.1 0.2 reduction and the coefficient of 0.3 friction; the predictions of the 30,000 0.4 empirical model of Schey and that of the refined 1D model are shown.

20,000 σ = 150 (1 + 234ε)0.251 h = 1 mm R = 125 mm Roll force (N/mm) 10,000

0 0.00 0.10 0.20 0.30 0.40 0.50 Coefficient of friction

150 Figure 10.2 The dependence of the σ = 150 (1 + 234ε)0.251 roll torque on the reduction and h = 1 mm the coefficient of friction; the R = 125 mm predictions of the empirical model Reduction F F 1D Schey of Schey and that of the refined 1D 100 0.1 0.2 model are shown. 0.3 0.4

50 Roll torque (Nm/mm)

0 0.00 0.10 0.20 0.30 0.40 0.50 Coefficient of friction and a fairly rough roll surface are considered, the 1D model’s predictions are approximately three times that of the empirical model. This rise, which is deemed quite unreasonable, is directly attributable to the use of the friction hill. Similar deductions may be made by examining Figure 10.2 showing the dependence of the roll torque (for both rolls) on the same two parameters as in Figure 10.1. The rise of the torque, as indicated by the 1D model, is still too steep. It is unclear at this stage which of these two models is to be recommended for off-line use without more calibration. Based on Figure 5.6, however, in which the predictive capabilities of three models were compared, the 1D model is expected to be the most reliable. As well, it is appropriate to recall what should be expected of a model’s predictive capabilities, mentioned already in Chapter 3. The two requirements are the accuracy and the consistency of the predictions. Of these two, Applications and Sensitivity Studies 269

10,000 Figure 10.3 The dependence of the F F Reduction 1D Schey roll separating force on the strain 0.15 hardening coefficient. 8000 0.40

6000 σ = 150 (1 + 234ε)n h = 1 mm 4000 R = 125 mm Roll force (N/mm) 2000

0 0.00 0.10 0.20 0.30 0.40 0.50 Strain hardening coefficient – n

60 Figure 10.4 The dependence of the F F Reduction 1D Schey roll torque on the strain hardening 0.15 coefficient. 0.40

σ = 150 (1 + 234ε)n h = 1 mm R = 125 mm 20 Roll torque (Nm/mm)

0 0.00 0.10 0.20 0.30 0.40 0.50 Strain hardening coefficient – n consistency, shown by a low standard deviation of the difference between the measurements and the calculations, is more important since the data may always be adjusted to be accurate. A model whose accuracy is good only sometimes is essentially useless. These concepts were demonstrated by Murthy and Lenard (1982) by comparing the mean and the standard deviation of the differences of the predictions and the experimental data of several models of flat rolling. In general, increased rigour resulted in decreasing standard deviations.

10.1.2 The Sensitivity of the Roll Separating Force and the Roll Torque to the Strain Hardening Coefficient Figures 10.3 and 10.4 show the dependence of the forces and the torques on the strain hardening coefficient and the reduction, respectively. It is understood that 270 Primer on Flat Rolling

30,000 Figure 10.5 The dependence of the roll separating force on the entry σ = 150 (1 + 234ε)0.251 R = 125 mm thickness at high and low μ =0.1 reductions. Reduction 20,000

50%

10,000 1D model Roll force (N/mm) Schey’s model

10%

0 0 5 10 15 Entry thickness (mm)

500 Figure 10.6 The dependence of the Reduction σ = 150 (1 + 234ε)0.251 roll torque on the entry thickness at R = 125 mm high and low reductions. 400 μ = 0.1 50%

300

200 1D model Schey’s model Roll torque (Nm/mm) 100

10%

0 0 5 10 15 Entry thickness (mm) while changing the strain hardening coefficient leaves the yield strength unchanged, it raises the ultimate strength and therefore it raises the average flow strength of the rolled metal in the pass. This is clearly demonstrated in Figures 10.3 and 10.4. As expected, both the forces and the torques rise with increasing hardening.

10.1.3 The Dependence of the Roll Separating Force and the Roll Torque on the Entry Thickness The changing thickness in subsequent rolling passes affects the roll separating forces and the torques. These effects are illustrated in Figures 10.5 and 10.6 Applications and Sensitivity Studies 271 showing the roll forces and the roll torques, respectively. As above, the calculations by the 1D model and the empirical model are contrasted. The same low-carbon steel and rolling mill are used and the coefficient of friction is taken to be 0.1. Figure 10.5 shows how the roll separating forces are affected by the increasing thickness at entry, at low and high reductions. Figure 10.6 shows the effect of the entry thickness on the roll torques at low and high reductions.

10.2 A Comparison of the Power Predictions Required for Plastic Deformation of the Strip

Three independent models were presented in Chapter 5, each developed to estimate the power required to drive the rolling mill. This power includes two parts: the power to produce permanent deformation of the rolled strip and the power to overcome friction losses in the drive system. The power needed to reduce the strip, as calculated by each of the three models, is compared to the measurements in Figure 10.7 for two nominal reductions of 15% and 50%1. The data used have been developed by McConnell and Lenard (2000) and have been referred to in Figure 10.7. Briefly, low-carbon steel strips were cold rolled in rolls of 250 mm diameter using various lubricants in the process. In the figure, Figure 10.7 the experimental data on the power were obtained by using the measured torque and the rolling speed. The coefficients of friction are needed in both the 1D model and the one presented by Roberts (1978); these were obtained by inverse calculations using the 1D model in the process. The friction factor is needed in the calculations using the upper bound method, which was obtained from Section 5.7.

40,000 Figure 10.7 A comparison of the Experimental data Reduction powers needed to cause permanent 1D model plastic deformation measured and Roberts predicted by various approaches. 30,000 Upper bound

~50% 20,000

10,000

~15% Power for plastic deformation (watts) 0 0 1000 2000 3000 Roll surface speed (mm/s)

1Since friction losses in the bearings would be the same, they are not considered in Figure 10.7. 272 Primer on Flat Rolling

0 Figure 10.8 A comparison of the Data roll pressure distributions as R = 112.5 mm calculated by the 1D model using a h –40 1 = 20 mm h μ constant coefficient of friction of 2 = 16 mm = 0.34 σ = 124 MPa m = m(φ) 0.34 and by the model using m 5 mðφÞ. –80

–120 Roll pressure (MPa) –160

–200 0 5 10 15 20 25 Distance from the exit (mm)

Both the 1D and Roberts models are remarkably close to the experimental data in their predictions. As expected, the upper bound prediction is conservative, yielding numbers much higher than the others.

10.3 The Roll Pressure Distribution

While rarely used in simple models, the variation of the coefficient of friction or the friction factor from the entry to the exit in the roll gap is well acknowledged. The predicted roll pressure distributions using a constant coefficient of friction and a variable friction factor are compared in Figure 10.8. The process parameters used to obtain experimentally the roll pressure distribution by Lu et al. (2002) are employed in the calculations. These measurements were referred in Section 5.7.2.2; briefly, the tests involved measuring the interfacial normal and shear stresses using pins and transducers embedded in the work roll. The shapes of the distribution curves are quite different. The saddle point, resulting from the friction hill and the use of the 1D model with μ 5 0:34, is not realistic, while the rounded top, resulting from the smooth variation of the friction factor, is expected to be close to reality2. The most significant shortcoming of the models that use the friction hill approach is their inability to yield realistic predictions when modelling passes in which large reductions of thin, hard strips are considered and the roll diameter to strip thickness ratios are much larger than unity. The reasons for this are clearly demonstrated in Figure 10.9. Cold reduction of 0.1 mm low-carbon steel strips is considered with work rolls of 250 mm diameter and rolling to progressively larger reductions is modelled. In both models rigid rolls are used. In the 1D model the

2See also Figure 5.11 where the measurements of Lu et al. (2002) are compared to the predictions of the model using m 5 mðφÞ. Applications and Sensitivity Studies 273

0 Figure 10.9 The pressure distribution as a function of the reduction. The distribution using the friction hill method becomes –2000 μ = 0.05 unrealistic at high reductions. The m = m(φ) model, using the variable friction 40% Reduction factor, predicts distributions which –4000 appear to be closer to expected 50% Reduction experimental data. 60% Reduction Roll pressure (MPa) –6000 Cold rolling 0.1 mm steel strips rigid rolls

–8000 0123 Distance from the exit (mm) coefficient of friction is taken to be 0.05, a realistic number when a light lubricant is employed in the pass. At the lower reduction of 40%, both models lead to comparable distributions of the roll pressure. As the reduction is increased, the failure of the friction hill model becomes evident. The top of the saddle point rises to unrealistic magnitudes.

10.4 The Statically Recrystallized Grain Size

Several empirical equations, relating the statically recrystallized grain size in a rolling pass, were presented in Chapter 53. Their predictions are compared in Figures 10.10 and 10.11 as functions of the strain (Figure 10.10) and as functions of the temperature (Figure 10.11). In the computations, the models of Sellars (1990), Roberts (1983), Laasraoui and Jonas (1991a,b), Choquet et al. (1990) and Hodgson and Gibbs (1992) are used. In Figure 10.10, the initial austenite grain diameter is assumed to be 50 µm, the temperature is 1173 K and the rate of strain in the pass is 1 s21. As expected, the predictions indicate that increasing strains cause the grain sizes to drop. The predictions are bunched into two, one giving numbers approximately twice that of the other. The predictions of Sellars (1990) and Laasraoui and Jonas (1991a,b) are close to one another and are much smaller than those of the other three researchers. The dependence of the statically recrystallized grain sizes on the temperature of deformation is illustrated in Figure 10.11. The relations from the same studies as above are used here; the strain is taken as 0.4 and the initial austenite grain size is assumed to be 100 µm. The expectations are that the grain diameters grow with the temperatures

3See Section 5.11.2. 274 Primer on Flat Rolling

40 Figure 10.10 The statically D = 50 µm Sellars recrystallized grain sizes as a m) T µ = 1173 K Roberts function of the strain predicted by ε =1s–1 Laasraoui and Jonas Sellars (1990), Roberts (1983), 30 Choquet Hodgson and Gibbs Laasraoui and Jonas (1991a,b), Choquet et al. (1990) and Hodgson and Gibbs (1992). 20

10 Statically recrystallized grain size ( 0 0.2 0.4 0.6 0.8 1.0 Strain

60 Figure 10.11 The statically Sellars recrystallized grain sizes as a m) Roberts µ Laasraoui and Jonas function of the temperature predicted Choquet by Sellars (1990), Roberts (1983), Hodgson and Gibbs Laasraoui and Jonas (1991a,b), 40 Choquet et al. (1990) and Hodgson and Gibbs (1992).

D = 100 µm ε = 0.4 ε =1s–1 Statically recrystallized grain size ( 0 1000 1100 1200 1300 1400 Temperature (K)

and the predictions of Roberts (1983), Choquet et al. (1990) and Hodgson and Gibbs (1992) indicate this dependence. The two other equations (Sellars, 1990; Laasraoui and Jonas, 1991a,b) do not include the temperature as an independent variable.

10.5 The Critical Strain

The shapes of high temperature stressstrain curves were discussed in Chapter 4 and the strain corresponding to the plateau of the curve was identified as the peak strain. At that strain the rate of hardening equals the rate of softening and just Applications and Sensitivity Studies 275 before that, the process of dynamic recrystallization has begun. The strain at which this restoration phenomenon becomes active has been identified as the critical strain. Equation (3.78) may be used to predict the magnitude of the critical strain as a function of the ZenerHollomon parameter (which is defined in terms of the strain rate, the temperature and the activation energy) and the austenite grain size. The dependence of the critical strain on the temperature, the rate of strain and the austenite grain size is demonstrated in Figure 10.12. Two strain rates (0.1 and 50 s21), three grain sizes (25, 50 and 100 µm) and the constants and exponents of Sellars (1990) are used in the calculations. The process of dynamic recrystallization begins when a sufficient amount of mechanical and thermal energy has been given to the deforming material. This is indicated clearly in the figure. Increasing temperatures, decreasing strain rates and grain sizes reduce the magnitude of the critical strain.

10.6 The Hot Strength of Steels Shida’s Equations

The shape of the stressstrain curves at high temperature was mentioned in Section 8.7.2 and the metallurgical events occurring as the deformation is proceeding were discussed. The peak stress, the strain corresponding to the peak stress, the competing rates of the hardening and restoration mechanisms and the attendant microstructures were described. In the conclusions of Chapter 8, the recommendation was made: if no independent testing programme to determine the material’s resistance to deformation at high temperatures is possible, use Shida’s equations in modelling the hot, flat rolling process. In what follows, Shida’s equations will be examined. In the first instance, the shape of the stressstrain curve will be compared to the expected configuration. This will be followed by the estimated rise of

2.00 Figure 10.12 The critical strain required for the initiation of dynamic 1.75 ε (s–1) D (µm) recrystallization. 1.50 0.1 25 0.1 50 1.25 0.1 100 50 25 50 50 1.00 50 100

0.75 Critical strain

0.50

0.25

0.00 600 800 1000 1200 1400 Temperature (°C) 276 Primer on Flat Rolling

300 Figure 10.13 The shape of Shida’s stressstrain curve.

ε (s–1) 200 50

Stress (MPa) 100 0.1

0.3% C Steel 900°C Furnace temperature

0 0.0 0.4 0.8 1.2 1.6 Strain the temperature as a hot compression process is proceeding. Finally, the ability of the relations to model the stresses in the two-phase ferriteaustenite region will be considered. Axially symmetrical samples of low-carbon steel, containing 0.3% carbon, are being compressed at an initial temperature of 900C.

10.6.1 The Shape of the StressStrain Curve as Predicted by Shida Figure 10.13 shows the shapes of the curves at various rates of strain. These should be examined in comparison to Figures 8.13 and 8.14, which show high temperature stressstrain curves for a NbV steel. The initial shapes of the curves are as expected, as the stress rises with strain and the slopes begin to drop slowly, indicating the competing rates of hardening and dynamic recovery. The peak stresses are also reached, but not at the expected strains. The strains, corresponding to the peak stresses, are expected to increase as the strain rates increase and the times for the metallurgical phenomena drop. While the magnitudes of the strains are close to those predicted by Sellars (1999) see Figure 10.12 the growth is not reflected in Shida’s curves. Further, while the grain size is not an independent variable in Shida’s relations, it is included in Sellars’ equation to some extent, the comparison is again one of apples and oranges. The rise of the temperature of the sample during the compression process has been mentioned already and the need to correct for it was emphasized. This temperature rise is plotted against the strain in Figure 10.14 for the steel dealt with above. The initial temperature, rates of strain and the carbon content are also the same. The importance of conducting isothermal tests, or correcting for the temperature rise during the application of the loads, is emphasized while examining the details of Figure 10.14. At a strain rate of 50 s21 and a strain of 1.2, the rise is Applications and Sensitivity Studies 277

60 Figure 10.14 The temperature rise ε –1 0.3% C Steel (s ) during compression as predicted by 900°C Furnace temperature 50 Shida.

) 10 40 1

0.1

20 Temperature rise (°C

0 0.0 0.4 0.8 1.2 1.6 Strain

300 Figure 10.15 The flow strength, Carbon equivalent calculated by Shida’s formula, for 0.1 various carbon contents showing the 0.2 steel’s behaviour in the two-phase 0.3 region. 200 0.4 Source 0.5 : Reproduced with permission from Lenard et al. (1999).

100 Flow strength (MPa)

0 700 800 900 1000 1100 1200 Temperature (°C)

calculated to be approximately 45C. The steel’s flow strength at 900C, at that strain, is 184 MPa. At the higher temperature of 945C, the strength is estimated to be 159 MPa, a nearly 14% difference. Using the wrong magnitude for the flow strength in a set-up model of the hot strip mill would lead to incorrect settings. Shida’s equations correctly predict the steel’s behaviour in the two-phase austeniteferrite region as well. When the temperature is decreased, the deformation resistance of the steel is expected to increase. When the temperature indicating the appearance of the first ferrite grains is reached, at Ar3, the strength is expected to fall with a further temperature drop, since the strength of the ferrite is lower than 278 Primer on Flat Rolling that of the austenite. This phenomenon continues while all of the austenite transforms to ferrite, and beyond the temperature indicating the end of the transformation, at Ar1, the strength increases again. The dependence of the strength on the temperature is predicted properly in Figure 10.15. As observed, the influence of the carbon is present only at lower temperatures in the two-phase region. The strength of the austenite appears to be independent of the carbon content. 11 Hot Rolling of Aluminium1

11.1 Introduction

Flat rolled aluminium products have for many years made up the largest volume of aluminium products manufactured around the world (Menzie et al., 2010). Figure 11.1 shows the typical product mix of aluminium products produced in 2009. As shown, flat rolled products made up 33% of the product mix for all aluminium products. This includes products such as aluminium foil, flat rolled sheet for automotive and beverage can applications and fin stock. In 2011, the amount of flat rolled aluminium sheet produced globally was B19 Mt. Around the globe, demand for flat rolled aluminium is expected to grow 34% or B6 million tonnes to 25 Mt by 2016 with most of the growth expected to occur in Asia. Aluminium sheet is a remarkably versatile material in part due to the wide variety of surface finishes it can be given from painting to anodizing, textur- ing and polishing. Flat rolling aluminium and its alloys is one of the most important ways of fabri- cating cast aluminium rolling ingots into a useable form. Typically, the starting material for most rolled aluminium products is the direct chill (DC) ingot. These ingots can be produced using metal produced directly from the aluminium smelter or from metal that has been recycled and remelted. Commercial purity aluminium obtained from the smelter typically contains 9999.7% aluminium with the remainder made up of other elements such as iron and silicon. Such metal when rolled is relatively soft and ductile but small quantities of other alloy additions are usually added to produce the final desired properties and microstructure required for the application. Figure 11.2 shows the typical alloy additions and alloy designation for wrought aluminium alloys. The size of the DC-cast ingot depends on the size of the DC unit available, the hot rolling mill capacity and to some extent the alloys being cast. Ingots with dimensions of 500600 mm thickness, 2000 mm width and up to 10 m in length are typically manufactured. During hot rolling of these ingots, it is possible to reduce their thickness from a starting value of 500600 mm down to reroll or slab gauge material with thicknesses of 25020 mm. Afterwards, hot rolling of these slabs can be continued to as low as a final hot rolled sheet thicknesses of 2 mm

1Contributed by Professor M.A. Wells, Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, Canada.

Figure 11.1 Graph showing product mix from 2009 (Menzie et al., 2010). 23% 11%

4% 29%

33%

Forgings and other Wire and cable Extrusions Castings Flat rolled products

Figure 11.2 Alloy additions made to 1xxx 2xxx wrought aluminium and the resulting alloy Al Al–Cu designations.

3xxx 2xxx Al–Mn Al–Cu–Mg

5xxx 6xxx Al–Mg Al–Mg–Si–Mn

5xxx 7xxx Al–Mn–Mg Al–Mg–Zn

Work hardening alloys 7xxx Al–Mg–Zn–Cu

8xxx Al–Li–Cu–Mg

Precipitation hardening alloys which can then be further reduced by cold rolling. Cold rolling can produce sheets to thicknesses as low as 0.006 mm for foil applications. Other routes to produce coiled sheet for cold rolling are near net shape casting operations, including twin roll casters in which the solidification and hot deformation are combined in one unit and belt casters where the aluminium is solidified as a slab on the belt and then subsequently warm or hot rolled. For these near net shape casting operations, the alloy compositions have to be modified to suit the Hot Rolling of Aluminium 281 casting process and hence the type of alloys that can be cast using these techniques is limited. Historically, only aluminium alloys used for foil, fin stock and building products are produced using these techniques (Grydin et al., 2011).

11.2 Hot Rolling Process

The hot rolling process must produce a slab, plate or sheet with not only accurate dimensions through the thickness across the width and along the length but also with certain surface characteristics, flatness and edge quality. In addition, the final as-hot rolled microstructure plays an important role in terms of how the material will respond to the cold rolling process and the final mechanical properties the sheet obtains. The hot rolling operations not only have to convert economically the ingot/slab to the required dimensions but they also have to achieve specified properties. A typical schematic of a hot mill for coiled aluminium sheet is shown in Figure 11.3. To prepare the ingots for hot rolling, after DC casting, the ends of the ingots are cut and a scalper removes the surface layer of the entire ingot to eliminate any surface oxides and surface perturbations which could have an impact on the final sheet quality. Depending on the aluminium alloy system being cast, the as-cast surface can vary dramatically in terms of the surface perturbations (Figure 11.4). Usually, commercial processing for plate/sheet aluminium alloys consists of casting the alloy in a DC casting machine into large ingots. After casting, the ingots undergo a preheat treatment prior to hot rolling; frequently this preheat is extended in time and called a homogenization heat treatment. The homogenization heat treatment is often varied and can take place over many hours at temperatures higher than those actually used for hot deformation. This homogenization heat treatment not only provides the high temperatures required for subsequent hot rolling but it also allows metallurgical reactions to occur within the cast ingot to ensure that the chemical composition becomes more uniform. During this process, constituent intermetallic particles formed during casting can change their shape and composition and small precipitates called dispersoids can come out of solid solution into

Homogenization Hot tandem rolling

DC casting Breakdown rolling

Figure 11.3 Typical aluminium hot rolling mill schematic. 282 Primer on Flat Rolling

Figure 11.4 Typical as-cast surface for different aluminium alloys (Li, 1999). the matrix. The volume fraction, size and chemical composition of these dispersoids depend not only on the thermal history experienced by the ingot during the homogenization treatment but also on the alloy’s initial chemical composition. The initial homogenized microstructure will play an important role in terms of how the material’s microstructure will respond to subsequent hot deformation in terms of the recovery, recrystallization of the grains during and after deformation and the texture that forms in the sheet. Aluminium hot mills fall into two categories: single-stand reversing mills or a tandem mill. The single-stand reversing mills are typically either 2 or 4 high with the product being coiled after the last pass. Such a line must have either small starting ingots, thick reroll gauge slabs or very long run-out tables which would also include slab cooling. To mitigate some of these limitations, larger hot rolling mills typically have a reversing mill with a tandem mill in line. The tandem mills are usually designed to have three to six stands which use a wateroil emulsion for lubrication. Figure 11.5 shows a schematic of one stand for a hot rolling mill and Table 11.1 indicates representative hot deformation conditions for an aluminium hot breakdown and tandem mill. For most rolled products, the actual gauge of thickness is less important than the spatial variation of the gauge across the width and along the length of the rolled product. Small gauge errors are relatively easy to correct in cold rolling, whereas gauge variation is much more difficult to fix. For plate products or situations where the metal is sold directly from the hot mill, accuracy of the gauge to customer specifications is obviously important. Surface control and the quality of the surface finish are also important in hot rolling. In hot rolling of aluminium, the defect that occurs to a greater or lesser extent on the aluminium surface is called “pick-up”. During rolling, the peaks on the freshly ground roll surface become worn down while lubricant, debris and fines are pushed into the valleys between the grooves. The aluminium or its oxide, Hot Rolling of Aluminium 283

Figure 11.5 Schematic of a hot rolling stand. Roll V roll V entry h h V entry exit exit

V roll Roll

V V strip roll Velocity Neutral point

Table 11.1 Typical Deformation Conditions During Hot Rolling

Parameter Roughing/Breakdown Tandem hentry (mm) 500600 25 hexit (mm) 25 2.5 Tentry ( C) 480580 400450 Texit ( C) 400450 280350 Average strain rate (ε_)(s21)110 10100 Strain/pass 0.6 0.6 alumina, is torn from the slab surface and because of the relative slip between the roll and the rolled surface can become smeared onto the roll.

11.3 Heat Transfer

Temperature control of the material being rolled is most important to ensure that the final product properties are met. Temperature will also have the largest effects in terms of the stored energy and microstructure evolution during and after hot rolling. Accurate knowledge of the heat transfer coefficient (h) in the roll gap is essential to achieving these goals, because h determines the temperature distributions in the rolls and the rolled strip which, in turn, influences the microstructural evolution and ultimately the mechanical properties of the strip, as well as the shape of the rolls and rolled strip. Roll temperatures are important in hot mills where good performance of hot rolling emulsions is vital to control surface finish. This includes avoiding pick-up and “plate out” of the oil from the emulsion on to rolls, both of which are temperature dependent. The roll temperatures overall and locally are also important in hot mills for control of thickness profile and in cold mills for control of flatness. 284 Primer on Flat Rolling

Heat transfer in the roll bite between the roll and material plays an important part in controlling roll temperature and depends upon the thermal conductivities, k, of the roll and the aluminium and even the lubricant itself, where there are significantly large oil film thicknesses. Another important factor is the detailed interactions between the strip and the roll during deformation, including the roll/sheet roughness and interface pressure. The physical mechanism that determines the heat transfer between the strip and the roll is the real contact area with the majority of the heat flow between the roll and the strip occurring at points of direct metalmetal contact at discrete locations where the asperities of each surface meet (Figure 11.6). Hlady et al. (1995) have developed an equation to predict the heat transfer coefficient (h) during hot rolling of metals, including aluminium: hC P m 5 r ð11:1Þ k σðTs; ε_Þ where Pr is the mean rolling pressure, σ is the flow stress of the work piece surface, Ts is the surface temperature of the strip and ε_ is the mean strain rate. C is a general roughness term, m is an experimentally determined parameter based on heat transfer measurements with a value close to unity and k is the combined thermal conductivity of the strip being rolled and the roll itself, given by

k k k 5 strip roll ð11:2Þ kstrip 1 kroll

Figure 11.6 Two microrough surfaces in contact representing the roll and the strip Top roll during hot rolling. Source: Adapted from Essadiqi et al. (2012).

Aluminium strip

Roll surface

Al strip

Asperity contacts Hot Rolling of Aluminium 285

Rolling direction

9.6 mm 5.7 mm

2 mm Before rolling After rolling

Figure 11.7 Deformation of the strip during hot rolling at 448C to a strain of 0.52 showing elongation and reduction in thickness for the elements (Wells et al., 2003).

11.4 Deformation

During hot rolling the strip undergoes significant plastic deformation. The strain experienced while reducing the strip from 9.6 to 5.7 mm is shown in Figure 11.7. The distributions of the strain and the temperature at the centre of the strip and on its surface are shown in Figure 11.8. As demonstrated, the thermomechanical history experienced at the centre of the strip is quite different from that experienced at the surface. In particular, the temperature histories vary dramatically with the surface experiencing a rapid decrease in temperature as the strip comes into contact with the roll, whereas the centre experiences a temperature increase due to plastic work. Quickly after exit, the temperature between the surface and centre of the roll returns to equilibrium.

11.5 Microstructure Changes During Hot Rolling

The microstructure changes that occur during hot rolling influence the final properties of the sheet. Dynamic changes occur in the roll bite and can include both recovery and recrystallization. Since aluminium has a relatively high stacking fault energy (SFE) as compared to other metals and hence the dislocations can climb and cross slip easily, dynamic recovery tends to be the main dynamic softening mechanism in aluminium and its alloys. In some cases solute additions, such as Mg in the AA5xxx alloys, will reduce the rate of dynamic recovery. Recovery com- prises all processes that lead to a reduction of the dislocation density and the arrangement of the remaining dislocations in sub-boundaries. Figure 11.9 shows the characteristic form of the stressstrain curves typical of aluminium alloys. As shown, the softening due to dynamic recovery offsets the increase in flow stress due to work hardening so that a steady state flow stress is obtained at strains greater than εm. At that strain, the steady state stress regime is 286 Primer on Flat Rolling

(A) 0.5 480 460 0.4 440 0.3 420 400 0.2 380 Strain Strain 0.1 Temperature (°C) 360 Temperature (°C) Temperature 340 0.0 320 300 0.50 0.55 0.60 0.65 0.70 Time (s)

(B) 0.5 480 460 0.4 440 0.3 420 400 0.2

Strain 380 Strain 0.1 Temperature (°C) 360 Temperature (°C) Temperature 340 0.0 320 300 0.50 0.55 0.60 0.65 0.70 Time (s)

Figure 11.8 Model predicted strain history in the roll bite at the centre and surface of the strip: (A) centre of the strip and (B) surface of the strip (Wells et al., 2003).

Figure 11.9 Typical flow stress Grains elongate and become pancaked curve for an aluminium alloy during hot deformation under constant strain rate and temperature. Source: Adapted from Sellars (1990). Subgrains Subgrain size develop constant Stress

0.0 0.1 0.2 0.3 0.4 0.5 Strain Hot Rolling of Aluminium 287 attained, the rates of dislocation annihilation and generation are equal, and the subgrains remain constant in size, misorientation and dislocation wall density. The stable or steady state dynamically recovered subgrain size, d, depends on the equilibrium dislocation density which is established by the balance between the generation rate and the annihilation rate of the dislocations. The subgrain size has been shown by many researchers (Chen et al., 1992; Zaidi and Sheppard, 1982; McQueen et al., 1967) to be directly related to the ZenerHollomon parameter (Z)

d21 5 a 1 b ln Z ð11:3Þ where Z, the ZenerHollomon parameter is, in fact, a temperature compensated strain rate Q Z 5 ε_ exp def ð11:4Þ RTdef

The activation energy of deformation, Qdef, has been found to equal 156165 KJ/mol by several researchers (McQueen and Conrod, 1985; Hlady et al., 1990). As the strain increases, the grains become progressively elongated and the subgrains within the grains become more equiaxed. When the flow stress does not increase (i.e. steady state is reached) the subgrain size and misorientation of the subgrains remain constant. It is generally accepted that in some instances dynamic recrystallization can occur in aluminium provided that the alloy content is high (i.e. Al6% Mg) (McQueen and Conrod, 1985; McQueen et al., 1985; Lintermanns and Kuhn, 1986). This has been attributed to the suppression of dynamic recovery and/or the presence of large ( . 0.6 µm) second phase particles in the aluminium alloy during hot rolling. The static recovery mechanisms slowly change dislocation tangles within the subgrains into neat arrays and the subgrains grow larger as some of the sub- boundaries disappear. Often the dislocations are attracted into arrays of similar dislocations, thereby increasing the misorientation caused by the array. If this process continues, some sub-boundaries obtain a misorientation of more than 10, becoming grain boundaries capable of migrating, and the region surrounded by such a boundary can become a nucleus for recrystallization (McQueen and Jonas, 1975). Referring to Figure 11.10, static recrystallization occurs by nucleation and growth of new grains in the statically recovered microstructures, in much the same way that classical recrystallization proceeds on annealing after cold working. The differences arise as a result of the much lower dislocation densities resulting from the dynamic and/or static recovery processes preceding recrystallization during hot working, and hence the lower driving forces that are available for recrystallization. In aluminium, nucleation from subgrains within the original grains appears to occur more frequently than in steels. This method of nucleation is favoured in materials 288 Primer on Flat Rolling

Roll bite Interstand Dynamic processes Static processes • Recovery • Recovery • Recrystallization • Recrystallization

Pancaked grains

Equiaxed grains Recrystallized grains

Figure 11.10 Schematic representation of the microstructure changes that occur during hot rolling of aluminium. of large grain sizes in which the deformation encourages the formation of deformation bands that act as effective nucleation sites for subgrains having a relatively large misorientation between them (B10). In contrast to recovery, dislocations are removed as a result of recrystallization by migrating grain boundaries. Nucleation of a recrystallized grain encompasses all processes that lead to the generation of a mobile grain boundary. For nucleation to occur, three criteria must be satisfied:

G Thermodynamic instability (critical nucleus size) a nucleus will be able to grow only when a reduction in free energy results from its expansion G Mechanical instability (imbalance of driving forces) for stimulation of a nucleus, a particular direction of growth must be favoured (i.e. initiation of subgrain expansion through subgrain coalescence) G Kinetic instability (grain boundary mobility) a viable nucleus needs a large-angle grain boundary, at least for part of its surface.

11.6 Summary

Hot rolling is one of the most common and cost-effective techniques to convert aluminium cast slab to flat sheet down to 2 mm thickness, which becomes the starting stock for cold rolling. Hot rolling is one of the most important manufacturing processes for flat rolled aluminium alloys as it allows the economical reduction of the DC-cast slab to thin slab and sheet products. From an energy efficiency perspective, it is more economical to develop near net shape casting processes such as twin roll and twin belt casting, which can cast starting strips of 63 mm and 1220 mm thickness, respectively. The suitability of this process is restricted to a limited number of aluminium alloys which can tol- erate the faster cooling rates during solidification without segregation. Hot Rolling of Aluminium 289

Historically, microstructure control during hot rolling of aluminium sheet was somewhat limited and was controlled by process parameters such as roll speed, temperature and reduction per pass. With the latest developments in microstructure modelling techniques, advanced sensors and the arrival of powerful real-time computing systems, direct control of microstructure development during hot rolling and knowledge of its influence during subsequent cold rolling are becoming possible. 12 Temper Rolling

12.1 The Temper Rolling Process

Temper rolling is a particular form of flat rolling. Its primary purpose is to suppress the yield point extension which, if present, would create Lu¨der’s lines, a form of surface defect, shown in Figure 12.1. The presence of this defect in subsequent sheet metal operations such as deep drawing, stretch forming and their combinations would have very deleterious effects on the resulting products. The temper rolling process subjects the flat product to a very low reduction of thickness, typically 0.55%. Other possible reasons for a temper pass include

G production of the required metallurgical properties; G production of the required surface finish; G production of the required flatness; G creation of magnetic properties; and G correct surface flaws and shape defects. The difficulties with temper rolling include the creation of non-uniform residual stresses in addition to possibly pre-existing non-uniform flatness of the starting product. Both of these may cause further processing difficulties. Essentially, mathematical modelling of the temper rolling process needs to account for the same phenomena that were included in the traditional models of flat rolling. Pawelski (2000) presents a list of the differences, which require additional attention, however. These are

G the nearly equal elastic and plastic regions of the deforming strip, caused by the very small reductions per pass; G the pronounced flattening of the work roll, which, if neglected or not accounted for carefully, would introduce large errors in the predictions of the roll separating forces; and G the order of magnitude of the thickness reduction, which is comparable to that of the surface roughness.

12.2 The Mechanism of Plastic Yielding

Specialized textbooks dealing with plastic deformation of metals discuss the transformation of the recoverable elastic behaviour to that of permanent plastic equilibrium and flow. The two most often used yield and flow criteria, the maximum shear stress and the maximum distortion energy theorems (developed by Tresca

Figure 12.1 Lu¨der’s lines (Wiklund and Sandberg, 2002).

and by HuberMises, respectively), form the mathematical bases of this change of the response of the metal1. Very briefly, the metal will enter the plastic deformation mode when and where the elastic stresses first satisfy either one of the criteria2. The elastic and the plastic regions will be separated by the elasticplastic boundary, the location and the shape of which become extra unknowns to determine. The mechanism of plastic yielding in a cold rolling pass has been discussed extensively by Johnson and Bentall (1969). They consider the rolling of thin strips and the longitudinal stresses acting on it. These include the tensile stresses as a result of the compression by the rolls and the elongation caused by them, in addition to the compressive stresses imparted by the interfacial shear stresses which always act towards the centre of the roll gap. When thinner and thinner strips are rolled, the compressive longitudinal and normal stresses subject them to hydrostatic compression in the deformation zone. This then implies that attempting further reduction of the metal would instead flatten the rolls more and would not result in any change of the strip’s thickness; a limiting reduction is reached. This phenomenon is, of course, critically dependent on the frictional conditions at the roll/strip contact. They identify a neutral region note that others identify a neutral point, not a region near the centre of the arc of contact where there is no relative motion between the roll and the strip. In the analysis, Johnson and Bentall (1969) present, two aspects are not considered: that of the bending of the rolls and the possibility that the rolls touch outside the width of the strip3. In the numerical analysis that follows, the authors show that there are two regions of slip, one near the entry and another near the exit. Elsewhere, there is no relative motion between the rolls and the strip. Further analysis indicates that yielding will occur on a plane perpendicular to the direction of rolling and will be restricted to the entry and to the exit regions. There will be a state of plastic equilibrium in the neutral region with no change in thickness.

1Refer to the last section of Chapter 1 where further reading material on plastic deformation is listed. 2Neither criterion is a law of nature. When one of them is assumed to govern the metal’s behaviour, one may write of a “Tresca material” or a “HuberMises material”. 3Johnson and Bentall (1969) acknowledge that the two phenomena just mentioned may affect the results of a more advanced analysis. Most other researchers simply ignore these two possibilities. Temper Rolling 293

12.3 The Effects of Temper Rolling

12.3.1 Yield Strength Variation Cold working the steel results in increased strength and decreased ductility. Roberts (1988) presents data on the effect of the elongation during a temper pass on the yield strength, showing that the yield strength decreases for reductions of 1 less than /2% but when the reduction is increased beyond that, the metal’s strength grows very significantly. Fang et al. (2002) studied the effect of temper rolling on several mechanical attributes of two CMn steels. They found that the lower yield strength increases with the equivalent strain, according to the relations:

σy 525:8εeq 1 328:9 ð12:1Þ for the steel with 0.135% C and

σy 529:4εeq 1 309:2 ð12:2Þ for the steel containing 0.019% C. In both equations above the stress is in MPa and the strains are in percentage. Fang et al. (2002) also found some drop of the yield strength at low strains, as did Roberts (1978). The tensile strength of both steels increased monotonically with the strain. The uniform elongation, the yield point elongation and the strain hardening exponent dropped for both steels as the equivalent strain increased. If the temper pass is performed at above ambient temperature, for the same elongation the strip hardness increases with the temperature. Considering tinplate rolling, increasing the strip’s entry temperature from 70F to 300F, at 1% elongation, causes an increase in hardness of two points on the Rockwell scale. For 2.5% elongation, the same temperature rise will cause an increase of four points. However, increasing the temperature of the strip results in a decrease of the ductility. The same temperature rise will cause a drop in the ductility of about 17%.

12.4 Mathematical Models of the Temper Rolling Process

12.4.1 The Fleck and Johnson Models (1987, 1992) Fleck and Johnson (1987) analyse cold rolling of thin foils and take careful account of the deformation of the work roll and the frictional conditions in the zone of contact. They also make use of the “planes remain planes” assumption and consider rolling of an isotropic, elastic-perfectly plastic thin strip. The elastic and contained plastic deformation of the strip in a direction parallel to a line connecting the roll centres is ignored. Their analysis leads to similar results to those of Johnson and Bentall (1969). They also find that there is a neutral region in the contact zone 294 Primer on Flat Rolling where contained plastic flow occurs with no change in thickness. Plastic deformation is predicted to occur only at the entry and the exit regions. The authors include some caution in the application of their model in their conclusions and it is illumi- nating to quote their comments exactly:

It is thought that the new model provides a physical picture of the foil rolling process which is qualitatively correct. We express caution with regard to the quantitative results, as the location of plastic deformation in the roll bite and the rolling loads and torques are sensitive to the model chosen for deformation of the rolls. A more realistic treatment of the rolls is required in order to determine the accuracy of the present results.

Fleck et al. (1992) refined the work of Fleck and Johnson (1987) by introducing an advanced treatment of roll flattening in the analysis, treating the rolls as elastic half-spaces. They retained the previous assumptions of having an elastic-perfectly plastic strip, homogeneous compression in the roll bite and a constant coefficient of friction. The general conclusions have not changed in that plastic deformation occurred near the entry and the exit regions, separated by a neutral region. Using the coefficient of friction, obtained by equating predicted and measured roll forces, to calculate the roll torque resulted in a 23% difference, attributed to the use of a constant coefficient of friction. Another quotation from their conclusions appears to be appropriate here:

There seems to be no point in refining the rolling model until more is known about the nature of friction in the roll bite.

Roberts (1988) quotes Hundy (1955), who wrote that in the temper rolling process homogeneous compression of the rolled strips defined as uniform deformation across the thickness of the work piece doesn’t occur. Only some portions of the metal deform sufficiently to enter the plastic range. It is known that the traditional mathematical models of the flat rolling process fail when applied at reductions experienced by the temper rolled strip. The causes of the inability to provide reasonable predictions of the roll separating forces are

G the small reductions; G the unexpected high values of the coefficient of friction; G the need to include roll flattening in the model; and G the lack of published experimental data to which a model’s predictions may be compared. Fleck and Johnson (1987) state that the conventional models analysing the cold rolling process fail when the thickness is less than 100 μm.

12.4.2 Roberts’ Model An approximate model has been given by Roberts (1978) and is reproduced below. The model gives the magnitude of the roll separating force (P) in terms of the Temper Rolling 295

2 minimum pressure required to deform the strip (σp, MPa or lb/in ), the thickness of the strip at the entry (hentry, mm or in), the reduction (r), the coefficient of friction (μ) and the length of the arc of contact (L):

½μ = ð 2 Þ 5 σ ð 2 Þ exp L hentry 1 r ð : Þ P p hentry 1 r μ 12 3 with the length of the arc of contact given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 L 5 0:5 Drμ=2 1 ðDrμ=2Þ 1 2 Dhentryr ð12:4Þ and the minimum pressure required to deform the strip is prescribed in terms of the metal’s yield strength ðσÞ in the form:

σ 5 : ½σ 1 ð ε_Þ 2 : ðσ 1 σ Þð: Þ p 1 155 alog10 1000 0 5 entry exit 12 5 where the effect of the stresses at the entry and at the exit is taken into account. As recommended by Roberts, the material constant a is to be taken as 7500 lb/in2 or 52 MPa. The roll separating force may be estimated by hentry σpð1 2 rÞ μL Pr 5 exp 2 1 ð12:6Þ μ hentryð1 2 rÞ

Assuming that the stresses at the entry and at the exit are equal, the torque to cause the deformation of the strip is given by

M 5 0:5 Dhentryrðσc 2 σaÞð1 1 μL=hentryÞð12:7Þ where σc is the constrained dynamic yield strength of the metal and σa is the average tensile strength in the strip. Roberts (1978) shows that the predictions of Eq. (12.6) compare quite well to measurements taken when a low-carbon steel strip was temper rolled. Temperature rise in a temper rolled strip or sheet may be estimated by assuming that all of the work done on the rolled metal is converted into heat. As before, the rise of the temperature is then obtained from

ΔT 5 power=ðmass flow 3 specific heatÞð12:8Þ

The power is obtained from the roll torque, for both rolls, in terms of the roll velocity

Power 5 Mvr=R ð12:9Þ 296 Primer on Flat Rolling

where the roll surface velocity, vr, is given by vr 5 2π rpm/60, the mass flow is obtained by assuming mass conservation:

Mass flow 5 entry velocity 3 entry thickness 3 width 3 density ð12:10Þ and the specific heat (the heat necessary to increase the mass of the body by one degree) is given for steels as 500650 J/kg K. The temperature rise of the rolled strip, as calculated by the above formula, is an average value. It is important to realize that the temperature is not uniform across the strip. The strains experienced by the strip are the highest near the contact surface (Lenard, 2003) and that indicates that the rise of temperature there, due to plastic work, is the largest. Counteracting this rise of the temperature near the surface is the cooling effect of the work roll and the lubricant, if wet temper rolling is performed.

12.4.3 The Model of Fuchshumer and Schlacher (2000) Fuchshumer and Schlacher (2000) considered temper rolling as the last possibility to exert an influence on the strip by rolling. They developed a mathematical model for an industrial temper mill, the schematic diagram of which is given in Figure 12.2. The authors acknowledge that the conventional models don’t apply here. They list the important parameters of the process: the forward and backward tensions, entry and the exit thickness, material parameters, roll velocity, slip conditions at the roll/strip interface, the roll force and mill dynamics. The deformation of the rolls is accounted for by Jortner et al.’s (1960) analysis. The rolling regimes characterized by a central region of contained plastic flow, considered by Fleck and

Hydraulic adjustment system Upper back-up roll Upper Snubber ωr work roll Unwinder φ roll L M L φ Snubber eco 2 r 3 xco (pay-off reel) roll Rewinder (exit R tension reel) sr R β β sr xco LA eco L Pass- b φ F line xco eco Lower L R 4 eco work roll φ L R eco 1 xco Lower M M back-up roll xco fr,xco M M ω ω eco fr,xco xco Mill housing xco

Figure 12.2 A temper rolling mill (Fuchshumer and Schlacher, 2000). Temper Rolling 297

Johnson (1987), are not included here. The coefficient of friction is taken to remain constant in the roll gap. The assumptions in the derivation of the model are that

G plane strain flow is present; G planes remain planes; and G the transition between elastic and plastic zones occurs abruptly. The result is a multi-input, multi-output system which is then used to control the process. The authors include the results of a typical example, showing the shape of the contact arc and the corresponding roll pressure distribution. The roll shape appears to include an indented portion but the flat portion, predicted by the Fleck and Johnson model, is not observed. The roll pressure distribution indicates the traditional friction hill model, with a sharp point at the pressure peak.

12.4.4 The Gratacos and Onno Model (1994) Gratacos and Onno (1994) agree that the classical models cannot predict the rolling parameters in the temper rolling process. They attribute the difficulties to convergence problems and/or unrealistic asymptotic behaviour which occurs when the strip entry thickness to roll diameter ratio is much less than unity. Gratacos et al. (1992) review the roll deformation models. These are Hitchcock’s formula; influence functions, FE modelling, Grimble’s approach and a contact mechanics technique. They comment that when modelling the rolling of thin strips, foils and temper rolling, the manner of coupling the deformation of the work roll and that of the strip is the most important step. They apply their model to temper rolling and find their predictions of the roll separating force acceptable. Gratacos and Onno (1994) employed two different two-dimensional models to analyse the flat rolling process. They considered both thick and thin strips and applied their model to temper rolling, as well. One of the models was a full finite- element formulation and the other was a slab/finite-element approach. In both models, the elastic deformation of the work roll was calculated by the finite-element approach. The rolled metal’s behaviour was described by the PrandtlReuss elastic plastic relations and Tresca or Coulomb friction was employed. The difficulties listed by the authors include the lack of precise evaluation of the sliding velocity between the non-circular deformed roll and the strip and the very high computational times4. Considering 32% reduction of a 0.05 mm-thick sheet, the central portion of the roll profile was found to be flat, as predicted by the Fleck and Johnson models. The roll pressure distribution was rounded on top. Both models yielded similar results. The temper rolling process was modelled next. The entry thickness was 0.4 mm, the reduction was 1.5% and the friction factor was assumed to be 0.12. Calculations of the equivalent strains in the deformation zone indicated that the

4The work of Gratacos and Onno was completed in 1994. It is very likely that computational times, using the high-speed computers available in 2013, would greatly reduce these times. 298 Primer on Flat Rolling deformation was not homogeneous and that the “slab hypothesis seems also insuffi- cient locally under the flat part of the roll”. The shape of the deformed roll was strongly dependent on the entry thickness, which was taken to be 1.4 mm at first. No flatness was found. As the thickness was decreased, flatness began to be noted at an entry thickness of 0.6 mm. The roll pressure curves were rounded at the top.

12.4.5 The Model of Domanti et al. (1994) The mathematical model of Domanti et al. (1994) builds on the work of Fleck et al. (1992) by keeping the general ideas and by adding several improvements. The existence of a region within the roll gap where the roll contour is essentially flat and the strip is in a state of contained plastic flow is retained. The additional work includes a material model that accounts for the effects of the strain rate and the temperature on the rolled metals resistance to deformation. A rare comparison of the predictions to experimental data was also presented. The roll force and the torque, as predicted by the model using a roll which remained circular under load, differed from the measurements in a most significant manner. When the non-circular roll profile was used, however, the predictions and the measurements were quite close.

12.4.6 The Chandra and Dixit Model (2004) A rigid-plastic finite-element model was used to study the temper rolling process. Roll deformation was analysed by assuming the roll to be an elastic half-space. The results indicated that the deformation in the roll gap is not homogeneous. The roll is found to flatten in the central zone where a rigid actually elastic region is found. Further, the authors report that under some conditions “the roll shape takes concave shape locally”. Comparison to experimental data is also included in the study. However, one of the references quoted (Shida and Awazuhara, 1973) gave roll force and torque data for cold rolling, not temper rolling. Since elastic behaviour appears to be a significant contributor to the roll force and torque in temper rolling, conclusions, using a rigid-plastic material, may not reflect the accuracy of the predictions.

12.4.7 The Models of Wiklund (1996a,b, 1999, 2002) Wiklund and Sandberg (2002) reviewed and summarized their studies of the temper rolling process in a state-of-the-art review. They described the application of several models and discussed their advantages and their abilities to predict the roll force and the contact length. They also considered the applicability of these models for online use. The first model is based on a finite-element approach, using an implicit Lagrangian code and four-node elements. Both the strip and the work roll were modelled. Both strain and strain rate were included in the material model. While mostly rounded roll pressure distributions were obtained, in one instance, a double peak was observed, similar to the pressure peak at entry in the Fleck and Temper Rolling 299

350 Figure 12.3 A comparison of the calculated and the measured roll 300 separating forces (Wiklund and

250 Sandberg, 2002).

200

150

Predicted roll force 100 Data from SSAB, 50 Sweden

0 0 50 100 150 200 250 300 350 Measured roll force

Johnson model. When the strip thickness was decreased, the roll profile demonstrated the flat portion, again as postulated by Fleck and Johnson (1987) and Fleck et al. (1992). In these computations, surprisingly low coefficients of friction were used, in the range of 0.10.2. Homogeneous compression of the rolled strips are shown in Figure 15.9A and B of Wiklund and Sandberg (2002). The next model combined the FE method and neural networks and a rare comparison of measurements and predictions. The approach is shown to be very successful; see Figure 12.3, where the calculated and the measured roll separating forces are compared. An interesting concept is introduced by Wiklund and Sandberg (2002): the “flattening risk factor”, α, defined in terms of the contact length and the strip thickness at entry: rffiffiffiffiffiffiffi L R0 α 5 5 ε ð12:10Þ h Δh pffiffiffiffiffiffiffiffiffiffiffi where L 5 R0Δh, is the contact length, R0 is the roll radius as calculated by Hitchcock’s formula and h is the entry thickness. The authors predict that severe roll flattening is expected when the risk factor exceeds 10. In conclusion, Wiklund and Sandberg (2002) write that good prediction abilities were obtained by the use of neural and hybrid modelling. The use of cylindrical roll deformation models was found to be valid when the steel strips are thicker than 0.4 mm. Non-circular roll deformation models are necessary under that thickness, leading to flat contact regions in the roll gap.

12.4.8 The Model of Liu and Lee (2001) Reasonable predictive abilities were demonstrated by the model, in which the preliminary displacement principle of Kragelsky et al. (1982) was applied. The authors present an argument for the need for a physically based model of the temper rolling 300 Primer on Flat Rolling process in light of the significant differences between it and the conventional cold rolling process. Further, the authors question the existence of the flat contact region in the roll gap, stating that “...the force of normal temper rolling is usually not large enough to build a central flat region because the soft annealed material is rolled and the reduction is slight”. According to the model, the strip in the roll gap has both plastic and elastic regions and the main contact part of the strip is elastic. Hence, the friction in the contact region is mostly governed by the contact between two elastic bodies. Kragelsky et al. (1982) give two expressions for the friction stress in the elastic contact region, depending on the preliminary displacement, δ, and the limiting preliminary displacement, ½δ, given by the following expressions:

ðθ δ 5 hn 2 ðθÞ θ ð : Þ ðθÞ 1 a d 12 11 φn h where hn is the strip thickness at the neutral point and 2 2 v ½δ 5 μ R ε ð12:12Þ 2ð1 2 vÞ max

In Eq. (12.12), v is the surface roughness coefficient, μ is the coefficient of friction and Rmax is the maximum height of the work roll asperities. The elastic deformation of the work roll is given by aðθÞ, evaluated using influence functions:

ðφ aðθÞ 5 Uðθ 2 tÞpðtÞdt 1 R ð12:13Þ 0 where Uðθ 2 tÞ is the influence function (Grimble et al., 1978), p(t) is the roll pressure and R is the undeformed roll radius. The symbol ε indicates the relative approach of the contacting bodies, to be taken as unity. With these definitions, the interfacial shear stress is given by () δ ð2v11Þ=2 τ 5 μ δ ½δ τ 5 μ 2 2 p for larger than and p 1 1 ½δ otherwise ð12:14Þ

The calculations yield the roll profile, the roll pressure and shear stress distributions, all of which are compared to Grimble’s predictions. A rounded pressure distribution is found but no flat portions of the deformed roll are located.

12.4.9 The Experiments of Sutcliffe and Rayner (1998) Experimental verification of the predictions of a neutral region in which no reduction occurs (Fleck and Johnson, 1987; Fleck et al., 1992) has been given by Temper Rolling 301

Sutcliffe and Rayner (1998). Plasticine strips were rolled in elastomer rolls with steel cores. The roll diameters were 24 and 50 mm. Chalk was used as the lubricant and the ring compression method was used to determine the coefficient of friction5. Low rolling speeds were used (0.008 m/s). The rolls were stopped and were separated fast after a certain part of the plasticine was rolled and the partially rolled strip profile was carefully measured. The authors conclude that when thin strips are rolled, a clearly noticeable reduction near the inlet is observed. As well, a central region exists which is relatively flat, confirming two of the Fleck and Johnson predictions. There are two more predictions of the theory, however, which are not confirmed: the measured profiles don’t show plastic reduction at the exit and the measured roll loads are almost an order of magnitude lower than predicted.

12.4.10 The Model of Pawelski (2000) In order to read Pawelski’s (2000) statement in context, that the magnitude of the thickness change in a temper pass is comparable to that of the surface roughness, the examination of some numbers is helpful. Consider the entry thickness to be 0.25 mm, as in Pawelski’s Figure 9, and the reduction to be 5%. The corresponding thickness change is 0.0125 mm or 12.5 μm. The surface roughness of a steel strip, ready for temper rolling, may be 12 μm and the combined roll/strip roughness may be somewhat larger. If, however, a reduction of 1% in the pass is considered, the two numbers are very much closer, making Pawelski’s idea very interesting and appropriate. In his model of temper rolling he accounts for the roll deformation, considering it to be an elastic, semi-infinite space. This deformation, UðsÞ, supported at a distance R below the surface and loaded by a unit line load, is written as 1 1 v s 2 UðsÞ 52 1 1 ð1 2 vÞln ð12:15Þ πE R where E is the elastic modulus of the roll, v is Poisson’s ratio and s is the distance between the load and the deforming roll. The vertical displacement of the roll, due to a pressure, p(x), then can be calculated by

ðN uðxÞ 5 pðξÞUðx 2 ξÞ dξ ð12:16Þ 2N

Pawelski then connects the pressure with the fractional area of contact of the surface asperities, using a slip-line field approach. Further, he takes the characteristics of the roll pressure and strip thickness distribution as was done by Fleck and

5The ring compression test was discussed in Chapter 9, Tribology; see Section 9.3.1.3. 302 Primer on Flat Rolling

Johnson (1987). Rounded roll pressure distributions are obtained. Coefficient of friction values from 0.09 to 0.25 are determined.

12.5 Summary

The temper rolling process was described and the differences between it and the conventional cold rolling process were presented. The objectives of the process were listed; these include the effect of the process on mechanical and metallurgical attributes. Several mathematical models were presented, including empirical models, one-dimensional models and finite-element models, as well as the use of artificial intelligence in predicting the rolling variables. Only in very rare instances were the predictions of the models compared to experimental data. It appears that the efforts to model the mechanics of the temper rolling process are not quite finished and in the opinion of the present writer, the assumptions used in the models are to blame. These include

G The “planes remain planes” assumption. This step assures that the stresses across the thickness will remain constants and that ordinary differential equations result when forces are balanced on a slab of the deforming material. Integration of these equations results in the usual friction hill which has been shown not to represent actual conditions. G The assumption that the coefficient of friction is usually constant along the arc of contact. Experimental evidence exists that this is not the case. G The assumption that the roll deformation models are realistic. However, they need to be coupled to the interfacial friction stresses, which, when non-constant frictional coefficients are used, may yield different flattened roll profiles than have been demonstrated so far. 13 Severe Plastic Deformation Accumulative Roll Bonding1

13.1 Introduction

The interest in bulk nanostructured materials, processed by methods of severe plastic deformation, is justified by the unique physical and mechanical properties of the resulting products. The advantage of these materials over other processes is concerned with overcoming the difficulties connected with residual porosity in com- pacted samples, impurities from ball milling, processing of large-scale billets and the practical application of the resulting materials. Methods of severe plastic deformation create ultra-fine-grained structures with prevailing high-angle grain boundaries. They should also be able to create uniform nanostructures within the whole volume of a sample to provide stable properties of the processed materials and they should not suffer mechanical damage when exposed to large plastic deformations.

13.2 Manufacturing Methods of Severe Plastic Deformation

There are several different SPD methods: high-pressure torsion (HPT), equal channel angular pressing (ECAP), accumulated roll bonding (ARB), multiple forging (MF), repetitive corrugations and straightening (RCS). These methods can be divided into four main groups: methods based on torsion and compression, methods using the extrusion, methods based on rolling and methods based on forging. In what follows, a review of these methods and possible industrial applications are given.

13.2.1 High-Pressure Torsion The Bridgman anvil type device (Bridgman, 1952) in which an ingot is held between two anvils and strained in torsion under the applied pressure of several GPa magnitude was the first equipment utilizing this method. The approach has since been used by several researchers. The lower holder rotates and the surface

1Most of the information in this chapter was provided by Dr. Krallics of the Budapest University of Technology and Economics. The experimental portion is based on Krallics and Lenard (2004).

Primer on Flat Rolling. DOI: http://dx.doi.org/10.1016/B978-0-08-099418-5.00013-5 © 2014 Elsevier Ltd. All rights reserved. 304 Primer on Flat Rolling friction forces deform the ingot by shear. During the process the sample experiences quasi-hydrostatic compression. As a result, in spite of the very large strains, the deformed sample is not destroyed. The samples, processed by severe torsion straining, are usually of a disc shape, from 1020 mm in diameter and 0.20.5 mm in thickness. While significant changes of the microstructure are observable after 1/2 rotation, several rotations are required to produce a homogeneous nanostructure, however. In spite of the very large deformation in which true strains of more than 100 may be reached without defects, the process is used mostly in laboratory experiments and to the best of the authors’ knowledge, no industrial applications are available. Recent investigations showed that severe torsional straining can be used successfully not only for the refinement of the microstructure but also for the consolidation of powders (Valiev, 1996; Alexandrov et al., 1998).

13.2.2 Equal Channel Angular Pressing Segal and co-workers developed the method of ECAP in the beginning of the 1980s, creating the deformation of massive billets via pure shear (Segal et al., 1981, 1984). The goal of the method was to introduce intense plastic strain into materials without changing their cross-sectional area. In the early 1990s, the method was further developed and applied as an SPD method for the processing of structures with submicron and nanometre grain sizes (Valiev et al., 1991; Furukawa et al., 2001). Since the cross-sectional dimensions of the sample remain unchanged with a single passage through the die, the sample may be pressed repeatedly through the die in order to achieve very high total strains. The overall shearing characteristics within the crystalline sample may be changed by rotation between the individual pressings. It is possible to define three distinct processing routes: route A in which the sample is not rotated between repeated pressings, route B in which the sample is rotated by 90 between each pressing and route C in which the sample is rotated by 180 between each pressing. A further possibility may be introduced when it is noted that route B may be undertaken either by rotating the sample by 90 in alternate directions between each individual pressing, termed route BA, or by rotating the sample by 90 in the same direction between each individual pressing, termed route BC. To obtain a desired microsructure using ECAP, about 810 passes are typically required. For each pass, the pressed billet must be removed from the die and re-inserted for the next pass, often after reheating in a separate furnace. This makes the process inefficient and difficult to control. A new ECAP method using a rotary die was developed to make ECAP more industrially viable (Nishida et al., 2001). Most of the experiments so far used work pieces with diameters of 1520 mm, but the results of Horita et al. (2001) demonstrate the feasibility of scaling ECAP to large sizes (40 mm) for use in industrial applications. An important variable is the length/diameter ratio of the specimen. In current practice that ratio is 67. Current implementations of the ECAP in industry require labour-intensive handling of the work pieces between process steps. Severe Plastic Deformation Accumulative Roll Bonding 305

13.2.3 Cyclic ExtrusionCompression Very large deformations are imposed by the cyclic extrusioncompression method, which, as the name indicates, combines the extrusion and compression processes. The sample is placed in a two-piece sectional die consisting of an upper and lower chamber of equal diameters. The chambers are connected by a constriction whose diameter is smaller than that of the dies. The deformation proceeds by the cyclic flow of metal from one chamber to the other. Compression occurs simultaneously with the extrusion so that the sample is restored to its initial diameter. It has been found (Reichert et al., 2001) that strain localization in the long-range shear bands crossing the whole volume of the samples is the main deformation mechanism. As a result of mutually crossing shear bands and micro-bands, nearly equiaxial subgrains are formed, creating a homogeneous structure.

13.2.4 Multiple Forging Another method for the formation of nanostructures in bulk billets is MF (Valiahmetov et al., 1990; Imayev et al., 1992). The process of MF is usually associated with dynamic recrystallization. The principle of MF assumes multiple repeats of free forging operations: setting and drawing while changing the axis of the applied load. The homogeneity of strain provided by MF is lower than in the case of ECAP pressing and torsional straining. However, the method allows one to obtain a nanostructured state in rather brittle materials because processing starts at elevated temperatures. As well, the specific loads on the tooling are low. The choice of appropriate temperaturestrain rate regimes of deformation leads to a minimal grain size.

13.2.5 Continuous Confined Strip Shearing The continuous confined strip shearing process is based on ECAP. The process is designed to apply simple shear to the metal strip in a continuous mode. A specially designed feeding roll with grooves on its surface is used, delivering the power required to feed the metal strip through the ECAP channel at a given speed. The experimental results indicate that ECAP can be used as a means not only for enhancing the tensile strength but also for controlling the texture of the strips suitable for subsequent sheet forming applications.

13.2.6 Repetitive Corrugation and Straightening In this process a work piece is repeatedly bent and straightened without significantly changing its cross-section. Large plastic strains are imparted to the material, which lead to the refinement of the microstructure. The RCS process can be easily adapted to large-scale industrial production. 306 Primer on Flat Rolling

13.2.7 Accumulative Roll Bonding Flat rolling is acknowledged as the most applicable deformation process for continuous production of bulk sheets (Saito et al., 1998). It is often stated that up to 90% of all metals are rolled at some point in the manufacturing process. The rolling process has serious limitations2, however. One of these concerns the possible total reduction in thickness, i.e. the total strain achieved per pass, which is limited because of the resulting tensile straining and the attendant cracking at the edges. Accumulative roll bonding, developed by Saito et al. (1998), is one of the techniques capable of creating the metallurgical and mechanical attributes demanded of metals with very small grains. While the major objective of the accumulative roll- bonding process is to produce very small grains within the rolled metal, another, interrelated objective is to achieve this without damage. This requires the minimization of the development of the tensile cracking of the edges. The process is simple. The roll surface is cleaned carefully and a strip of the metal is rolled to 50% reduction, usually without lubricants. After rolling, it is cut into two parts, cleaned very carefully and stacked, one on top of the other part, resulting in a strip whose dimensions are practically identical to the starting work piece. The stacked sheets are rolled again to 50% reduction and the two sheets are cold-bonded during the rolling pass while creating bulk material. Hence ARB is not only a deformation process but also a roll-bonding process. After the second pass the process is repeated and continued until edge cracking is severe such that the resulting product may not be usable any further. To achieve good, strong bonding, surface treatments such as degreasing with a strong, non-greasy detergent and wire brushing, preferably using stainless steel brushes, of the sheet surface are done before stacking. Rolling at elevated temperature is advantageous for joining ability and workability, though too-high temperatures may cause recrystallization and cancel the accumulated strain. Therefore, the rolling (roll bonding) in the accumulative roll-bonding process is preferably carried out at “warm” temperatures. Research indicates that the process may be repeated numerous times, and while rolling strips of several layers the occurrence of edge cracking, if not eliminated completely, is reduced in a significant manner. Saito et al. (1998) rolled strips of fully annealed commercially pure aluminium of 1 mm thickness using the ARB process, with no lubrication. The strips were held in the furnace at 473 K for 300 s. The pre-rolled, pre-heated grain diameters were measured to be 37 µm. The reduction in each pass was 50% at a mean strain rate of 12 s21. No cracks were observed even after eight cycles. While the process created ultra-fine grains of 670 nm mean grain diameter after the eight cycles, the grain diameter was under 1 µm after the third rolling pass. After six cycles the grain distribution was uniform. The tensile strength of the metal increased from about 90 MPa to nearly 300 MPa and the elongation decreased from about 40% to under 10% after eight cycles. Tsuji et al. (1999a,b) used the accumulative roll- bonding process to reduce the grain size of 5083 aluminium alloy from 18 µmto

2The limitations of the flat rolling process have been discussed in Chapter 4. Severe Plastic Deformation Accumulative Roll Bonding 307

280 nm, in five cycles of rolling. Testing at higher temperatures after the roll- bonding process indicated that the metal had become superplastic, elongating to nominal strains of 200400%. Saito et al. (1999) rolled Ti added interstitial free steel strips at 773 K, employing the accumulative roll-bonding process. The pre- rolled average grain size was measured to be 27 µm. After five cycles the grains decreased to less than 500 nm. The changes in tensile strength and elongation of the IF steel were given by Tsuji et al. (1999a,b), indicating that the strength increased from about 280 MPa to over 800 MPa after seven cycles of roll bonding and the elongation dropped from just under 60% to under 5%. The accumulative roll-bonding process and other techniques that create ultra-fine grains have been reviewed recently by Tsuji et al. (2002). Park et al. (2001) used the ARB process to create grains of 0.4 µm in 6061 aluminium alloys after five passes, starting with a grain size of approximately 40 µm. The rolling passes were performed at 523, 573 and 623 K, at strain rates of 18 s21. The authors showed that no delamination of the rolled sheets was observed. Lee et al. (2002) examined the effect of the shear strain experienced by the samples in accumulative roll bonding. The authors write that during the rolling pass the effect of the interfacial conditions between the rolls and the rolled metal on the characteristics of the deformation process is most significant. They subjected commercially pure aluminium sheets to eight cycles of the ARB process, without any lubrication. A pin was inserted into the samples and the distortion of the pin was used to infer the amount of shearing in the passes. The distribution of the shear strain across the thickness of the sheets was found to correspond well to the grain size distribution. Lee et al. (2002) used 6061 aluminium alloy sheets and found that after eight cycles of ARB, the tensile strength increased from 120 to 350 MPa, while the elongation dropped from about 30% to a low of 5%. Xing et al. (2002), rolling AA3003 aluminium alloys, reduced the grains from a starting magnitude of 10.2 to 700800 nm, in six cycles of the ARB process. The results and the conclusions were similar to those of Lee et al. (2002). The review indicates that the first concerns of the researchers are the changes in the metallurgical attributes of the multi-layered strips, demonstrating the very pronounced decrease of the grain diameters accompanying the accumulative roll- bonding process. The increasing tensile strength and the loss of ductility were also indicated for a number of materials, including an ultra-low-carbon steel, containing 0.0031% C, and several aluminium alloys. While the surface hardness, the strength of the bonds and the bending strength of the multi-layered strips following the process would contribute to the success of potential industrial applications, these have been treated less intensively in the technical literature. Further, the parameters of the successive rolling passes have not been given. Experiments were conducted to study these phenomena and in what follows, a detailed account of the work will be presented. These form the topics of the next section, as does a discussion of the potential industrial use of the multi-layered strips. In what follows, the effect of the progressively increasing number of layers on the mechanical attributes of the multi-layered strips after rolling and cooling is 308 Primer on Flat Rolling examined3. An ultra-low-carbon steel, containing 0.002% C, somewhat less than contained in the steel of Tsuji et al. (1999a,b), is used. The parameters of the warm rolling process are documented. The changes of the hardness, the yield and tensile strengths, the corresponding loss of ductility and the behaviour of the multi-layered strips in three-point bending are followed as the number of layers is increased. The strength of the bond is determined. The number of layers that may be bonded without producing edge cracking is indicated and the causes of the cracking of the edges are discussed. A suggestion for a potential industrial use of the multi-layered strips is presented.

13.3 A Set of Experiments

13.3.1 Material An ultra-low-carbon steel was used in the tests. The chemical composition of the steel is given in Table 13.1. The steel is comparable to that of Tsuji et al. (1999a,b) except for the lower carbon content. The grain structure of the as-received steel, obtained using a scanning electron microscope, is shown in Figure 13.1, indicating grain sizes of 2535 µm. The true stresstrue strain curve of the metal, determined in a uniaxial tension test at 22C, is

: σ 5 183:2ð1151:7εÞ0 317 MPa

Table 13.1 The Chemical Composition of the Steel (wt%)

C Mn P S Cu Nb Ti Al N Si Fe

0.002 0.133 0.01 0.0099 0.02 0.005 0.053 0.048 0.0067 0.009 Rest

Figure 13.1 The grain structure of the as-received steel.

3Most of the work, reported here, was performed during Dr. Krallics’ tenure as a Visiting Professor in the Department of Mechanical Engineering, University of Waterloo. Dr. Krallics is Associate Professor in the Budapest University of Technology and Economics. Severe Plastic Deformation Accumulative Roll Bonding 309

13.3.2 Preparation and Procedure The ultra-low-carbon steel strips, nominally 2.5 mm thick, were cut into samples of 25 mm width and 300 mm length. The surfaces of the strips were roughened by using a wire brush, removing as much of the layer of scale as possible and creating a somewhat random surface of roughness of 1.51.8 µm Ra. After brushing, the surfaces were cleaned using acetone. The strips were then joined on the roughened surfaces and while holding them in a vice to ascertain that they lay flat against one another, the leading and the trailing edges were spot welded4. The leading edge was tapered to ease entry to the roll gap. The strips were then placed in a furnace, pre-heated to 515C and held there for 10 min, in air, before rolling. After the soaking period the strips were rolled, without any lubrication, to a nominally 50% reduction, at a velocity of 0.39 m/s (50 rpm), creating strain rates of approximately 20 s21. Ten pairs of strips were prepared. Ten two-layered strips were rolled in the first pass, four layers in the second, eight in the third and so on. The rolled strips were visually inspected for the appearance of edge cracking and for successful bonding. One of the strips was removed for mechanical testing. The remaining nine strips were cut into two samples of equal length and the procedure was repeated, rolling the four-layered strips, at the same temperature and the same rolling speed to the same nominal reduction. The experiments were stopped when the cracking of the edges became pronounced.

13.3.3 Equipment All experiments are carried out on a 15 kW, two-high, STANAT laboratory mill with a four-speed transmission and tool steel work rolls of 150 mm diameter, hardened to Rc 5 55 and having a surface roughness Ra 5 1.7 µm, obtained by sand blasting. The surface roughness is expected to be random and is also expected to be helpful in drawing the strips into the roll bite. The mill is instrumented with two load cells positioned over the bearing blocks of the top roll. The data are collected and are stored in a personal computer. The top speed of the mill is 1 m/s and the maximum roll force is 800 kN. The furnace is located beside the mill so transfer of the strips for rolling caused minimal loss of heat and the entry temperature may be safely assumed to be very close to the furnace temperature.

13.4 Results and Discussion

13.4.1 Process Parameters A typical experimental matrix and the observations for three sets of tests are given in Table 13.2. The entry thickness and the width, the exit thickness and the width,

4In hindsight, welding the edges was not the best approach to keep the strips from sliding over one another during the rolling pass. A soft wire, wrapped around the strips, would have been much better and is used in current practice. 310 Primer on Flat Rolling

Table 13.2 The Experimental Matrix and Observation While Roll Bonding the Ultra-Low-Carbon Steel hin win hout wout Reduction Pr Layers Comments (mm) (mm) (mm) (mm) (%) (N/mm)

First set of experiments

5.6 29.50 2.40 33.20 57.1 12,024 2 Good bond 4.8 33.20 2.38 34.55 50.4 10,614 4 Good bond 4.76 34.55 2.47 35.70 413.1 10,901 8 Good bond 4.94 35.70 2.80 37.65 43.3 10,716 16 Small cracks 5.6 37.65 2.55 40.00 54.4 12,870 32 Large cracks, bonding Second set of experiments

5.6 29.50 2.42 33.20 56.8 12,799 2 Good bond 4.84 33.20 2.45 34.65 49.4 10,853 4 Good bond 4.90 34.65 2.43 36.90 50.4 11,391 8 Good bond 4.86 36.90 2.80 37.65 42.4 10,437 16 Small cracks 5.60 37.65 3.50 40.00 37.5 11,703 32 No bonding, few cracks Third set of experiments

5.61 29.50 2.40 N/A 57.2 12,024 2 Good bond 4.80 33.20 2.38 N/A 50.4 10,614 4 Good bond 4.76 34.55 2.47 N/A 413.1 10,901 8 Good bond 4.94 35.70 2.8 N/A 43.3 10,716 16 Good bond 5.60 37.65 2.6 N/A 53.6 12,870 32 Edge cracking, bonding

the reduction per pass and the measured roll separating forces per unit width are given in the table. The number of layers and qualitative observations concerning the bonds and the appearance of cracking of the edges are also indicated. In the first few passes the bonds are generally well formed, as long as the reduction the strips experience is above a certain limit, estimated to be approximately 50%. When the reduction is much below that level, bonding appears unsuccessful, as in the test where the reduction reached only 37.5%. Minor cracking of the edges appears after 16 layers have bonded well. When the 32-layered strip is rolled to a reduction, beyond 50%, bonding is acceptable but cracking of the edges is pronounced. When the reduction is less, only 37.5%, no bonding, as expected, and much less edge cracking are observed. The similar magnitudes of the roll separating forces per pass are to be pointed out. As the following sections will indicate, the room-temperature strengths of the rolled strips depend on the number of layers and the amount of cold or, more Severe Plastic Deformation Accumulative Roll Bonding 311

Figure 13.2 The hardness on the edges of the rolled samples, 300 measured in the transverse direction. The averages are given.

200 Vickers hardness Vickers 100 Ultra-low carbon steel Vickers hardness 200 g force

0 0 10203040 Layers precisely, warm rolling process they are subjected to. The cumulative effect of the repeated warm working and the accumulation of residual stresses after cooling are observed to be causing the increasing resistance to deformation. The similar magnitudes of the measured roll forces/pass in the warm rolling process are expected to be caused by the nearly ideally plastic behaviour of the ultra-low- carbon steel, which at the rolling temperature of 500C experiences dynamic recovery only. Using the simple, empirical model of the flat rolling process5, inverse calculations give the effective flow strength of the strip in each pass and, in general, a slowly increasing trend is noted, indicating some accumulation of strain.

13.4.2 Mechanical Attributes at Room Temperature 13.4.2.1 Hardness The hardness on the edges of the rolled samples was measured, in the transverse direction, and the averages of the measurements are shown in Figure 13.2.The average Vickers hardness, obtained using a force of 200 g, is given on the ordinate and the number of layers rolled is shown on the abscissa. As expected, the hardness increases as the number of passes is increased. The hardness of the as-received steel, prior to rolling, is 111 HV. At the end of the fifth pass, after rolling the 32- layered strip, the hardness became 293 HV, indicating significant hardening. It is observed that the largest increase in the hardness, nearly 100%, was created when the two-layered strip was rolled. The hardness increases in the subsequent passes but at a progressively lower rate. It is probable that if the cracking of the edges did not limit the process, a limiting hardness would have been reached.

5See Section 5.2. 312 Primer on Flat Rolling

1000 100 Figure 13.3 The yield strength, the tensile strength and the elongation. 800 80

600 60

Ultra-low carbon steel 400 40 Yield strength Elongation (%) Tensile strength Elongation 200 20 Yield and tensile strength (MPa)

0 0 010203040 Layers 13.4.2.2 Yield, Tensile Strength and Ductility The yield strength, the tensile strength and the elongation have been determined in standard tensile tests. The results are shown in Figure 13.3, plotted against the number of layers contained in the rolled strips. The strengths change as a result of the cumulative effect of warm working, in a manner similar to the hardness, increasing by approximately the same percentage. The major increase is again observed to occur in the first pass. The yield strength increases from a low of 183 MPa before the rolling process to a high of 695 MPa, after five cycles of rolling. The tensile strength increases from 300 to 822 MPa. At the same time, however, the ductility decreases from a high of nearly 75% to 4%, indicating a very pronounced loss of formability.

13.4.2.3 The Bending Strength Three-point bending tests were performed in order to observe the behaviour of the multi-layered strips in a potential sheet metal forming operation. These tests subject the sample to significant tensile and compressive stresses in their plane in addition to shear stresses which vary from maximum at the neutral axis to zero at the outermost surfaces. The results of the tests are given in Figure 13.4, plotting the force/ sample width versus the vertical displacement. Up to the displacement shown in the figure the tensile strains were not excessive and no fractures occurred. Some, but not excessive, delamination of the bonded layers was observed. As the number of layers increased, fewer instances of delamination were noted, indicating that the bond strength increased after repeated rolling passes.

13.4.2.4 The Cross-Section of the Roll-Bonded Strips The cross-sections of the rolled strips are indicated in Figure 13.5A, showing two layers and in Figure 13.5B, showing the 32 layers. The interfaces are visible only slightly, indicating the possibility that the roll-bonding process was successful. Severe Plastic Deformation Accumulative Roll Bonding 313

300 Figure 13.4 The results of the three- 2 layers v = 5 mm/m point bending tests. 4 layers d = 12 mm 250 8 layers 16 layers 32 layers 40 200

150 Load (N/mm) 100

0 04812 Vertical displacement (mm)

(A) (B) 1.41 mm 151 µm (2 layers) 2.42 mm 1.41 mm

Figure 13.5 The cross-sections of the rolled strips: (A) two layers and (B) 32 layers.

40 Figure 13.6 The test for the bond strength. 10 0.5 Bond strength to be tested 1.21

2.42 1.21

0.5 Force of the grips NOT TO SCALE

13.4.2.5 The Strength of the Bond The bond strength was tested, following a test procedure, shown schematically in Figure 13.6. The figure indicates a four-layered strip and the strength of the bond in the middle, just formed, is to be tested. As shown, two narrow slots are milled at about 10 mm from each end of the sample, to carefully controlled depths. Tension 314 Primer on Flat Rolling tests, conducted at a speed of 1 mm/m, are then performed on an Instron tensile tester. Two tests, performed to test the bond in the middle of a four-layered and an eight-layered strip, indicated that the shear stress necessary to separate the bond is in the order of 5253 MPa, somewhat less than expected but still indicating that reasonably successful bonding was achieved. The third test, performed on a 32- layered strip, was designed to test the strength of the bond on the second layer from the surface. The sample broke as a result of the tension test, at a tensile stress of 730 MPa, while the shear stress at the designated layer reached nearly 100 MPa; however, it did not separate.

13.5 The Phenomena Affecting the Bonds

Several phenomena, mechanical and metallurgical in nature, are involved in the accumulative roll-bonding process. The drastic decrease of the grain size and the attendant changes of the mechanical properties are among the major features. The strength of the bonds, created when several layers are rolled, also contributes to the success or otherwise of the process. Further, the ability of the multi-layered strips to resist edge cracking is of interest. The cumulative hardening and the loss of ductility during cold or warm rolling are well understood. As predicted by the HallPetch equation, the decreasing grain sizes and the increasing strength are clearly related. The loss of ductility associated with these changes has also been discussed in the technical literature. As mentioned above, the focus in this study is on the post-rolling, room-temperature mechanical attributes, the strength of the adhesive bonding between the layers and the occurrence of cracking of the edges. The roll-bonding process is a form of cold welding. In the process, two sheets, usually but not exclusively metals, are rolled and hence, bonded together. The strength of the bond depends on providing the appropriate conditions for adhesion of the materials: cleanliness, closeness and pressure. When contact is made, the phenomena there are best explained in terms of the adhesion hypothesis (Bowden and Tabor, 1950), which examines the origins of the resistance to relative motion in terms of adhesive bonds formed between the two contacting surfaces that are absolutely clean and are an interatomic distance apart. Bowden and Tabor (1973) credit the French scientist Desaguliers, living and working in the eighteenth century, with this idea and reproduce his account of an experiment with two lead balls which, when pressed and twisted together by hand, created what must have been adhesive bonds. The top ball held the bottom ball, a load of nearly 7.3 kg. The parameters that influence the adhesion of metals are discussed in detail by Gilbreath (1967). He lists the material properties, the interfacial pressure, the duration of the contact, the temperature and the environment as those that affect the adhesion coefficients, defined as the ratio of the strength of the bond to the strength of the parent metal. The study, conducted in high vacuum, indicates that while adhesion is inversely proportional to hardness, it increases with increasing loads, Severe Plastic Deformation Accumulative Roll Bonding 315 the time of contact and the temperature. Further, even small amounts of oxygen or air decrease adhesion. In the present set of tests, these parameters were kept constant. Another parameter of importance is the roughness of the surfaces to be joined, also kept constant here. The roughness of the surfaces, created manually by wire brushing, was measured to be in the order of Ra 5 1.51.8 µm. These would create large true areas of contact that would be expected to aid adhesion. Since the normal pressures are several times the metal’s resistance to deformation, the major change of the true area of contact is expected to occur in the first pass. Subsequent passes, during which the rolled metal experiences pressures of similar magnitude, would likely not increase the true area of contact by any significant measure. As several rolling passes have indeed increased the bond strength, this is likely due to the increasing chemical affinity which would result in stronger interfacial adhesive bonds.

13.5.1 Cracking of the Edges The occurrence of edge cracking is indicated in Table 13.2 and it is observed that in most instances 16 layers of the steel were rolled successfully while the edges did not crack much. Only in the last pass, when rolling the 32 layers, was cracking pronounced. The process was ended at that point. It is recalled that Saito et al. (1999) trimmed the cracked edges and continued to roll the multi-layered strips. In the accumulative roll-bonding process, as followed in this work, the true strain experienced by the strips in each pass is near 0.13. The total true strain, i.e. the sum of the strains per pass of the strips, is approximately 3.54. This corre- sponds to a reduction of over 97%, much larger than what can be achieved in one conventional pass, without edge cracking. The maximum reduction obtainable in one pass of the cold rolling process is limited by the metal’s ductility, the through-thickness and the transverse non- homogeneity and when lubricants are used, by the directionality of the roll’s surface roughness. Since the rolling passes were performed dry, only the non- homogeneity of the deformation needs to be considered as the probable limiting mechanism. Non-homogeneity through the thickness may cause alligatoring. Since the ratio of the roll diameter and the strip thickness was quite large and the shape factor was significantly larger than unity, alligatoring was not expected nor was it observed. Transverse non-homogeneity may cause splitting of the rolled samples in the direction of rolling. This limit of workability was not observed when the ultra- low-carbon steel strips were rolled. It is worthwhile in this context to refer to a few unsuccessful accumulative roll-bonding tests using medium carbon steel strips. Following the procedure of the ultra-low-carbon steel, the medium carbon steel strips split in the direction of rolling, no doubt due to metallurgical reasons in addition to transverse non-homogeneity. In the present set of tests the process was limited by the appearance of significant cracking of the edges. In a study concerning the workability of aluminium alloys in the hot rolling process (Duly et al., 1998), the occurrence and the direction of edge cracking were identified to have occurred as a result of the state of 316 Primer on Flat Rolling

Figure 13.7 Cracking at the edges.

2.42 mm

Individual layers, on average 76.5 µm thick stress at that location. The same approach indicates that at the centre of the sample, the tensile stresses in the direction of rolling and the compressive stresses transverse to that direction cause the maximum shear stresses to occur in a direction of 45. The majority of the cracks of the 32-layer strip, as shown in Figure 13.7, are in general, in that orientation. Also noted are several cracks in other directions, at various angles, not 45. The reasons for the orientation of the cracks lie in the complex stress distribution at the edge during the rolling passes and are considered beyond the scope of this study. The reasons for the ability of the strips to resist edge cracking, however, may be explained by considering the ultra-low-carbon steel’s resistance to deformation at the rolling temperature of 500C. At that temperature the true stresstrue strain curves exhibit dynamic recovery and almost perfect ideally plastic behaviour. The dynamic recovery process, in which some of the stored internal energy is relieved by dislocation motion without affecting the size of the grains, allows the samples to recover some, but not all, of their original softness. Some of the strain is then retained and as the passes are repeated, these strains accumulate. When the accumulation is sufficient to reach the limit of workability, cracking occurs at the most highly stressed location near the edges. The origin of the cracks in Figure 13.7 cannot, of course, be identified at this time.

13.6 A Potential Industrial Application: Tailored Blanks

Tailor welded blanks are made up of two sheets of unequal thickness which are welded to form a blank for subsequent sheet metal operations involving bending in one or two directions, such as in the deep drawing or the stretch forming processes. While welding techniques are well advanced and the interruption of material continuity can be accounted for in the design of the forming processes, the strength of the welds is often less than that of the parent metal (Worswick, 2002). The accumulative roll-bonding process may lead to blanks of uniform thickness but Severe Plastic Deformation Accumulative Roll Bonding 317 significantly different strength and formability from one portion of the blank to another. A limited number of tests have been performed and the refinement of the technique is continuing but in essence, the procedure is, as follows. The surface of a strip is roughened by a wire brush and cleaned with acetone, as above. Two strips are then placed on one another such that over half of the length of the strip is made up of two layers. The strip is then warm rolled, dry, to a reduction of 50%. The end result is a strip made up of two bonded layers over about half the length of the sample, the remaining part being a single layer. The bonded portion has smaller grains, increased strength and reduced ductility. The unrolled portion’s mechanical attributes do not change. Tests performed so far allow some cautious optimism that the removal of the welding process and the attendant discontinuity may result in improved formability.

13.7 A Combination of ECAP and ARB6

An aluminium alloy, used in the automotive industry, is processed by ECAP and by repeated rolling. First, the metallurgical attributes caused by one pass of the ECAP process are examined. The alloy is then rolled in several passes and the changes of its attributes are monitored. The objective is to determine whether repeated applications of the rolling process are able to create grains of magnitude smaller than those that were produced by several passes of the ECAP process. 6082-T3 aluminium alloys were used, possessing an initial yield strength of 125 MPa and a tensile strength of 180 MPa. The ductility of the metal is 55%. The samples were processed by ECAP and the effect of different processing routes on the development of the tensile strengths and that of the grain structure were investigated. While rotating the sample around its longitudinal axis after each of a total of eight passes by 1130, the tensile strength increased from 180 to 260 MPa (Krallics et al., 2004). The pre-ECAP grain size of 2.5 µm was reduced to 300500 nm after one pass and did not decrease any further after seven subsequent passes (Krallics et al., 2002). In what follows, the potential advantages of combining the rolling process with the ECAP are examined. Using the aluminium alloy, two processing steps are employed. First, the alloys are subjected to one pass through the ECAP dies and their grain structures are examined. These are followed by repeated, unlubricated rolling passes and the influence of the combination of the processes on the resulting metallurgical attributes is monitored. All rolling experiments are conducted at a nominal roll speed of 170 mm/s. The ECAP tests were performed using the press, described by Krallics et al. (2002). Before the experiments, all samples were annealed at 420C for one hour and allowed to cool with the furnace at a rate of approximately 1C/s.

6This study was conducted by R. Boga´r, doctoral candidate, in the Department of Mechanical Engineering, University of Waterloo. 318 Primer on Flat Rolling

Figure 13.8 (A) The microstructure after heat treatment at 420C for 1 h and cooling in the furnace. The transverse section is shown. (B) The microstructure after heat treatment at 420C for 1 h and cooling in the furnace. The longitudinal section is shown.

Figure 13.9 (A) The microstructure after heat treatment at 420C for 1 h and cooling in the furnace. The transverse section is shown. (B) The microstructure after heat treatment at 420C for 1 h and cooling in the furnace. The longitudinal section is shown.

13.7.1 The ECAP Process The grain size of the aluminium sample, after the heat treatment but before the ECAP process, was 2.5 µm for the 6082 alloy; see Figure 13.8A and B.The figures show the transverse (1a) and the longitudinal sections (1b). The grain boundaries are clearly visible in the transverse sections and the effects of the prior extrusion process are also observable in the longitudinal directions. After one ECAP pass the yield strength increased to 190 MPa, the tensile strength to 230 MPa and the ductility dropped to 45%. The grain structure after one pass of the ECAP process is shown in Figure 13.9A and B, taken in the transverse and the longitudinal directions, respectively. The diffraction patterns are also indicated in the lower left corners of the figures. Severe Plastic Deformation Accumulative Roll Bonding 319

13.7.2 The Rolling Process The samples were subjected to essentially flat rolling passes, even though the cross-sections were circular at the start. The circular shape changed to an almost completely flat cross-section by the end of three passes of 50% reduction each. During the passes the roll separating forces per unit width were measured and they are reported in Figure 13.10 as a function of the effective strain. The actual width of the contact was measured before and after each pass and the average was used in the calculation of the specific roll force. In the figures the specific force is plotted on the ordinate and the effective strain is given on the abscissa. Figure 13.10 allows a comparison of the behaviour of the alloy as a result of various processing routes. Note that the deformation of the samples was far from homogeneous and as a result, the distribution of the strains was also highly non-homogeneous. The roll pressure was applied on a fairly small area, in contact with the work rolls, in the first pass. The contact area increased in each pass until the work pieces became practically completely flat. It is interesting to note that in spite of the increasing contact area and the attendant increasing resistance to frictional forces, the roll force, after an initial increase in the first pass, decreases as the strains accumulate. When the nominal reduction per pass is increased to 50%, the picture changes in a significant manner. As observed in Figure 13.10, the roll forces increase with the effective strain, indicating the effect of the strain hardening of the metal. It is noted that the roll separating forces in four 50% reduction passes with no ECAP (indicated by the open squares) and three 50% reduction passes following one press through the ECAP die (indicated by the diamonds) are practically identical. In the rolling passes the transverse edges, especially near the centreline of the samples, experienced uniaxial tensile stresses and strains. The presence or the lack

12,000 Figure 13.10 The roll No ECAP; 50% Reduction/pass separating force as a function of One-pass ECAP; 50% Reduction/pass the effective strain. Rolling only One-pass ECAP, 20% Reduction/pass and rolling after ECAP are One-pass ECAP, 10% Reduction/pass shown. 8000

6082 alloy; all passes 4000 at room temperature Roll separating force (N/mm) force Roll separating

0 012345 Effective strain 320 Primer on Flat Rolling of tensile cracking there is indicative of the ductility of the sample. No cracking was observed in the first 50% reduction and some minor cracks were created after the second and the third passes.

13.7.3 The Microstructure After ECAP and the Rolling Passes The transmission electron microscope photographs shown in Figures 13.1113.14 indicate the development of the microstructure as a result of the ECAP and the rolling processes. In Figures 13.11 and 13.12, the effect of one pass at 50% reduction

Figure 13.11 The microstructure after heat treatment at 420C for 1 h, cooling in the furnace and subjected to one pass of the ECAP process and a rolling pass of 50% reduction. The longitudinal section is shown.

Figure 13.12 The diffraction pattern after heat treatment at 420C for 1 h, cooling in the furnace and subjected to one pass of the ECAP process and a rolling pass of 50% reduction. Severe Plastic Deformation Accumulative Roll Bonding 321

Figure 13.13 The microstructure after heat treatment at 420C for 1 h, cooling in the furnace and subjected to one pass of the ECAP process and three rolling passes of 50% reduction each. The longitudinal section is shown.

Figure 13.14 The diffraction pattern after heat treatment at 420C for 1 h, cooling in the furnace and subjected to one pass of the ECAP process and three rolling passes of 50% reduction each.

is demonstrated while in Figures 13.13 and 13.14, the effects of three passes are given. The microstructure and the diffraction patterns are given. It is found that the most intense reduction of the grain size occurs in the first pass through the ECAP die. Subsequent rolling passes caused no significant further reduction.

13.8 Conclusions

Ultra-low-carbon steel strips were warm rolled, following the accumulative roll- bonding process. Strips made up of 32 layers were rolled and bonded successfully, 322 Primer on Flat Rolling as long as the reductions/passes were above 50%. The process was limited by the occurrence of cracking of the edges, caused by the state of stress at that location. The effect of cumulative warm working was monitored, and the hardness, the yield and the tensile strengths increased significantly as the process continued. The ductility decreased to very low levels, indicating that post-rolling sheet metal forming processes may have to be planned with care. The most pronounced changes of the mechanical attributes were observed to occur in the first rolling pass. The bond strength was also investigated in selected instances. The shear stress necessary to separate the centre bond was found to be about half of the metal’s original yield strength in shear. The strength of the adhesive bonds near the edge appeared to be higher, affected by the number of rolling passes. Edge cracking was most likely initiated when the strains, retained after dynamic recovery, reached the limit of workability of the metal. A possible industrial application is discussed: that of the creation of tailored blanks of uniform thickness in which part of the blank is stronger and less ductile while the remainder’s attributes are unchanged. Combining the ECAP and the ARB processes resulted in sharply reduced grain sizes. Most of the reduction, however, occurred in the first ECAP pass, and subsequent rolling passes contributed little to the formation of small grains. 14 Roll Bonding1

14.1 Introduction

Since components produced by cold pressure welding include automotive parts, bimetal products and household items, understanding the mechanisms and the details of the process is of significant industrial importance. The solid-state joining technique can be used on a large number of materials, which may be the same, possessing identical attributes, or may be different, possessing widely varying mechanical and metallurgical properties. As mentioned by Bay (1986), materials that cannot be welded by traditional fusion often respond well to cold welding. The cold welding process causes bonding by adhesion and as described by Bowden and Tabor (1950) this requires the surfaces to be clean and to be an interatomic distance apart. Considering the comment of Batchelor and Stachowiak (1995) that “surfaces are always contaminated” and Figure 3.1B of Schey (1983), reproduced as Figure 5.2 in Chapter 5, showing the layers of oxides and adsorbed films on the surface of a metal, cleanliness of a surface is difficult to achieve without a controlled atmosphere and significant plastic deformation, large enough to break up the contaminants. Under industrial or laboratory conditions without the provision of protective environments, complete cleanliness is simply not achievable. The normal pressures are expected to be sufficiently large to satisfy the second criterion of the adhesion hypothesis that the surfaces be close to one another and that at least some new metal surface be created. Wu et al. (1998) write that diffusion bonding and mechanical bonding are two types of solid-state bonding. They define the first as bonding that occurs in a considerable amount of time and it involves the application of temperature and pressure. Mechanical bonding, on the other hand, occurs practically instantaneously or over a very short time and depends, among other things, on the forces of attraction between the atoms. In their experiments on several metals they find that the bond strength depends on the exponential of the temperature, implying that diffusion plays a role. It is the shear strength of the bond that determines the usefulness of the two- layered component in subsequent metal forming processes in which bending in two directions takes place, such as in deep drawing, stretch forming or a combination of the two. The process parameters affecting the bond strength involve the surface expansion and normal pressure, the surface roughness, the storage time between

1Based on Yan and Lenard (2004).

Primer on Flat Rolling. DOI: http://dx.doi.org/10.1016/B978-0-08-099418-5.00014-7 © 2014 Elsevier Ltd. All rights reserved. 324 Primer on Flat Rolling surface preparation and the welding process, in addition to the time during which the normal pressure is applied (Bay, 1986). Gilbreath (1967) also includes the temperature and vibratory loading as two further parameters that may affect the strength of the bond. Kolmogorov and Zalazinsky (1998) add the strain at the interface to the list of parameters. As shown by Bay (1986), the bond strength of a cold welded AlAl combination may approach the strength of the parent metals at high levels of surface expansion, defined as the increase in total bonding area, as compared to either the initial or the final area. As mentioned above, the large magnitude of surface expansion is required to cause the oxide layer to break up and to allow the fresh metal in between the cracks to make contact and thus adhere. The pre-bonding preparation of the surface is also shown to affect the strength of the bond (Clemenson et al., 1986). Scratch brushing, using a brush with medium stiffness at high speeds, was found to create the strongest bond while the normal pressure during brushing was found to have no effect. A model of the cold welding process was presented by Zhang and Bay (1997), making use of the observation that the strength of the weld between absolutely clean surfaces is approximately equal to the applied normal stress (Zhang and Bay, 1997; Bay, 1979). Kolmogorov and Zalazinsky (1998) base their model for the bond strength on the kinetic energy of micro-damage accumulation, resulting in the rupture of oxide films. They applied their model to the production of steelaluminium wires. Manesh and Taheri (2005) used the upper bound theorem to examine the rolling of bimetal strips. Their model correctly predicted the measured peel strength as a function of the composite reduction. Cold welding by rolling, i.e. roll bonding, is well suited to the creation of two- layered strips or plates. The rolling process is capable of producing the high interfacial pressures required to cause strong adhesion of the components. The process was studied by Hwang et al. (2000), presenting a mathematical model of rolling sandwich sheets. The model is an extension of an earlier study and is based on the upper bound method, using stream functions to define the velocity field. Mean contact pressures under 10 MPa magnitudes were considered. Experimental data, obtained while rolling aluminium and copper sheets, were correctly predicted by the model. A mathematical model of roll bonding, hot and cold, was presented by Tzou et al. (2002), deriving the stress field during the process of roll bonding. They concluded that the important parameters needed to create strong bonds include the reduction, the friction factor at the interface and the tension, enlarging the bonding length during the bonding process. Zhang and Bay (1997) identified the threshold surface expansion, caused by plastic deformation, necessary to initiate cold welding. Weld efficiency, defined as the ratio of the strength of the weld and the strength of the base metal, was examined by Madaah-Hosseini and Kokabi (2002) during cold roll bonding of an aluminium alloy. The strain hardening of the metal was included in their model, which predicted the weld efficiency with good accuracy. The expansion of the surfaces in contact is the result of the application of the normal pressures, or, in other words, the work done in the rolling process. In this chapter, the energy needed to cause successful and strong bonds in the roll-bonding Roll Bonding 325 process is considered. It is hypothesized that the shear strength of the bond will approach that of the parent metal when a sufficient amount of energy the activation energy to initiate the bonding process has been given to the two components to be joined. This energy may be provided by heating and/or by mechanical means. A discussion and examination of the process parameters that create bonds by rolling between two layers of an aluminium alloy, whose strengths are comparable to the strength of the original metal, is given below. The independent parameters are the reduction/pass, the entry temperature and the rolling speed. The work done during the passes, necessary to create the bonds, is identified and a correlation between it and the activation energy of bond formation is demonstrated.

14.2 Material, Equipment, Sample Preparation and Parameters

14.2.1 Material Strips of cold-rolled aluminium alloy Al 6111, commonly used in the automotive industry, are experimented within the tests. The chemical composition of the alloy, by weight%, is given in Table 14.1. The stressstrain curve of the 6111 alloy, obtained in a uniaxial tension test, is : closely approximated by the relation σ 5 150ð11157εÞ0 245 MPa. The pre-rolling surface roughness of the Al 6111 strips was Ra 5 0.5 μm.

14.2.2 Equipment The two-high experimental mill has 249.8 mm diameter 3 150-mm-long D2 tool steel rolls hardened to Rc 5 63. The surface roughness is Ra 5 0.4 μm and the roughness direction is random, created by manually finishing the surface using 100 grit carbide paper. The mill is driven by a 42 kW, constant torque, DC motor. Two force transducers are located under the bearing blocks of the lower work roll, measuring the roll separating force. Two torque transducers are placed in the drive spindles leading to the roll torque. A shaft encoder monitors the rotational velocity of the roll. A digital data acquisition system is used to collect and process the results. The screwdown is governed by a hydraulic feedback control system.

14.2.3 Sample Preparation The samples for two-layered bonding are prepared as follows. The strips of 300 mm length, 15 mm width and 0.97 mm in thickness are cut from a cold-rolled sheet,

Table 14.1 The Chemical Compositions of the Alloy (wt%)

Cu Mn Si Zn Mg Al

6111 0.82 0.21 0.21 0.02 ,0.01 Rest 326 Primer on Flat Rolling parallel to the original rolling direction. All edges are deburred. The leading edges of the strips are tapered to ease entry to the roll bite. The surfaces are cleaned using acetone. The two pieces of the strip are stacked together and bound, near the leading edge, using a soft aluminium wire. No special surface preparation is done prior to performing the warm roll-bonding tests. When cold roll bonding is performed, the surfaces to be bonded are roughened manually, using a stainless steel brush. The average surface roughness after brushing is in the range of 45 μm. The surface roughness is random. After brushing, the surfaces are cleaned using acetone.

14.2.4 Parameters The effects of the temperature, reduction, speed and annealing temperature on the shear strength of the bond were considered in the experiments. In one set of tests, the entry temperature was raised to the warm rolling level while in the next set, room temperatures were used. Prior to the tests at 22C, the strips were annealed at various temperatures for 2 h and cooled in air. The ranges of the parameters used in the tests were

G the reduction, varied from 40% to 70%; G the entry temperature, varied from 22 C to 280 C; G the rolling speed, varied from 50 to 380 mm/s; and G annealing temperatures: 350 C to 450 C.

14.2.5 Testing of the Shear Strength of the Bond The shear strength was measured, following the approach of Bay (1979), by carefully cutting narrow slots approximately 1 mm wide on both sides of the roll- bonded samples so when pulling, the bonded areas should experience shearing only. The tests were conducted using an Instron electrohydraulic testing system. Crosshead speed was 5 mm/s and temperature was 22C. The bonding area after fracture was measured using an optical travelling microscope. It is noted that in several previous studies, the peel test was used to determine the strength of the bond; in that test, a normal stress is applied to the bonded surfaces. Since the bonded work pieces would eventually be formed further in sheet metal forming processes, such as deep drawing or stretch forming, in which the stresses at the bonds would be created mostly by a shearing action, the shear test is deemed more relevant to evaluate the success or otherwise of the bonding process.

14.3 Results and Discussion

14.3.1 The Roll Force and the Torque The specific roll separating force and the roll torque, measured during warm roll bonding, are shown in Figure 14.1A and B, respectively, as a function of the reduction and the entry temperatures. Roll Bonding 327

(A) (B) 30,000 200

Entry temperature Entry temperature 25,000 250°C 250°C 280°C 150 280°C 20,000

15,000 100

10,000

Roll torque (Nm/mm) 50

Roll separating force (N/mm) 5000 Speed of rolling: 52 mm/s Speed of rolling: 52 mm/s

0 0 30 40 50 60 70 30 40 50 60 70 Reduction (%) Reduction (%)

Figure 14.1 (A) The roll force as a function of the reduction and the entry temperature. (B) The roll torque as a function of the reduction and the entry temperature.

As expected, the roll forces increase as the reductions increase. The rollp torquesffiffiffiffiffiffiffiffiffiffi reach a maximum at a reduction where the contact length, defined as L 5 RΔh, and hence, the moment arm, begin to drop. The entry temperatures and the speed of rolling are also shown. The 4 rpm roll speed or the 52 mm/s linear speed indicate that the contact time is approximately 220 ms. The effect of the slightly increased entry temperature and the attendant drop of the metal’s resistance to deformation are also observable.

14.3.2 The Shear Strength of the Bond 14.3.2.1 The Effect of the Speed of Rolling The shear strength of the bond, obtained at an entry temperature of 280C, is presented in Figure 14.2 as a function of the rolling speed. The data are given for two nominal reductions, 55% and 65%. The shear strength is observed to decrease sharply as the speed is increased, leading to shorter times of contact, confirming the conclusions of Gilbreath (1967). Also observed is the effect of the reduction; at the higher reductions the bond strengths are higher. At the higher reduction and at a low speed of 52 mm/s the time of contact is about 220 ms, sufficient to create a bond whose shear strength is near 230 MPa, approximately equivalent to the shear strength of the original material. Reducing the time of contact leads to sharply lower bond strength; at the reduction of 55% and a speed of 350 mm/s the contact time is about 30 ms and the shear strength of the bond is reduced to 120 MPa, nearly half that at the lower speed. Since the bonding mechanism is expected to be adhesion and probably some limited amount of diffusion, the effect of the time of contact is expected and understandable. 328 Primer on Flat Rolling

300 Figure 14.2 The shear strength of the bond as a function of the rolling Nominal reduction speed and the reduction. 250 55% 65%

200

150

100 Shear strength (MPa)

50 Entry temperature = 280°C

0 0 100 200 300 400 500 Speed (mm/s)

300 Figure 14.3 The shear strength of the bond as a function of the roll pressure. 250

200

150

100 Shear strength (MPa) Speed of rolling ≅ 52 mm/s Entry temperature = 280°C 50

0 800 1200 1600 2000 Roll pressure (MPa)

14.3.2.2 The Effect of the Normal Pressure The results, shown in Figure 14.3, were produced at an entry temperature of 280C and at a rolling speed of 52 mm/s, giving ample time for the bonds to form. The roll pressure, estimated by dividing the roll force/unit width by the projected contact length, varied from a low of 950 MPa to a high of 2000 MPa, corresponding to reductions of 4068%. At the roll pressure of 950 MPa, a very small drop of the rolling speed resulted in a steep rise of the bond strength, emphasizing again the importance of time in creating the adhesive bonds. The bond strength increased with the increasing normal pressures and the attendant growth of pure metal-to-metal contact, as expected. Roll Bonding 329

A maximum of the shear strength of the bond was reached at a pressure of approximately 1100 MPa and no further increase in the strength was obtained. When one surface approaches the other the forces of attraction between the atoms increase until the attraction equals the repellent forces. Beyond that the adhesive forces and hence, the bond strength, remain stable. Note that the normal pressures in the present set of tests are orders of magnitudes higher than those considered by Hwang et al. (2000).

14.3.2.3 The Effect of the Entry Temperature Warm Bonding The relationship between shear strength of the bond and the entry temperature of the strips is shown in Figure 14.4, at a reduction of 66% and a nominal rolling speed of 88 mm/s. It is observed that with increasing temperatures the shear strength of the bond also increases at a relatively fast rate at the lower magnitudes. The rate decreased and a maximum strength was reached at 280C with no significant change beyond that temperature. Bonding experiments were also conducted at 200C entry temperature at a reduction of 66%. No successful bonds were created. The temperature affected the bonding properties greatly by influencing and accelerating the adhesion process and the break-up of the oxide layer. At the higher temperatures the strengths of the oxide layer, of the parent metal and of the bond between the oxide and the strip all drop. As well, the rate sensitivity of all three increases. These phenomena enhance the possibility of the two surfaces conforming to one another leading to the higher bond strength. While Peng et al. (1999) bonded aluminium and copper sheets at various temperatures, they also indicated that an optimum temperature to reach the maximum bond strength exists. Further, they identified the formation of intermetallic phases at the bonded surfaces, a step not followed in the present tests.

300 Figure 14.4 The shear strength of the bond as a function of the entry temperature. 250

200

150

100 Shear strength (MPa) Speed of rolling = 88 mm/s Nominal reduction = 66% 50

0 240 260 280 300 320 Entry temperature (°C) 330 Primer on Flat Rolling

60 Figure 14.5 The shear strength of the bond as a function of the Annealing temperature 350°C annealing temperature and the 400°C reduction. 450°C 40

20 Shear strength (MPa)

Speed of rolling ≅ 88 mm/s

0 60 64 68 72 Reduction (%)

14.3.2.4 The Effect of the Entry Temperature Cold Bonding Roll-bonding experiments of the 6111 alloy at room temperatures were not successful at first. The two strips of aluminium adhered together but it was easy to pry them apart manually. Annealing the strips at various temperatures for 2 h and air cooling before bonding, however, resulted in relatively strong bonds. The results are given in Figure 14.5 showing the bond strength in terms of the reduction and the annealing temperature. Annealing at the highest temperature of 450C created the strongest bonds. When the reduction was high, nearly 72%, the bond strength reached 50 MPa, about 30% of the strength produced by warm bonding. Decreasing annealing temperatures caused sharply lower strength. The annealing process affected the microstructure of the aluminium strips. At the lower temperatures, static recovery did not change the grain sizes but caused the dislocations to move in such a way that the effect of some or all of the prior cold working was removed and the material regained some lost softness. At the higher temperatures, static recrystallization created strain free, new grains and hence, the softening. Both processes enhanced the deformation of the asperities and allowed the formation of larger true areas of contact, leading to higher bond strength.

14.4 Examination of the Interface

Using scanning electron microscopy, the appearance of the bonded surfaces, after separation, has been examined. In evaluating these surfaces, it is to be kept in mind that some damage during separation by the shear test must have affected their appearance. Results obtained during warm bonding are given first followed by the Roll Bonding 331 observations in cold bonding. Following these, the side views of the warm and cold bonds are shown.

14.4.1 Warm Bonding Three bonds of various strengths have been chosen for examination. In Figure 14.6 the surface of a strong bond is indicated. In Figure 14.7 a bond of medium strength is given while in Figure 14.8 a poor bond is presented. In each figure, the process parameters are also shown and while they are not identical, the differences are not expected to affect the surface appearance. The differences as a function of the bond strength are clearly observable. Figure 14.6, showing a strong bond, indicates significant permanent deformation of the asperity tops, giving rise to relatively large true areas of contact, resulting in a large number of adhesive bonds. Figure 14.7, demonstrating the average bond strength, indicates similar behaviour but the contact surfaces are somewhat smaller than those in Figure 14.6. The surfaces of the poorly bonded samples are shown in Figure 14.8 demonstrating much lower roughness and a noticeably smaller true area of contact. The appearance of the fracture

Flattened asperity tips Figure 14.6 The bonded surface: entry temperature, 280C; rolling speed, 52 mm/s; reduction, 67.5%; bond strength, 244 MPa.

44 μm

Flattened asperity tips Figure 14.7 The bonded surface: entry temperature, 280C; rolling speed, 78.5 mm/s; reduction, 66.5%; bond strength, 195 MPa.

44 μm 332 Primer on Flat Rolling

Flattened asperity tips Figure 14.8 The bonded surface: entry temperature, 280C; rolling speed, 73.9 mm/s; reduction, 48.5%; bond strength, 140 MPa.

50 μm

Flattened asperity tips Figure 14.9 The surface of the bond: annealing temperature, 450C, 2 h; rolling speed, 90 mm/s; reduction, 70.7%; bond strength, 50.8 MPa.

50 μm

surface suggests little permanent deformation of the asperity tops and that bonding occurred only at the top of the asperities. Figures 14.614.8 may be compared with the SEM photograph of Dick and Lenard (2005), obtained after the surfaces were separated by the peel test. The figures are similar in a general sense. The method of separation shear versus the peel test, applying shear or normal stresses, respectively, to cause the separation contributed to the appearance of the surfaces and it is expected that the peel test interfered less with the asperities.

14.4.2 Cold Bonding The surface of contact in cold bonding, after separation by the shear test, is shown in Figure 14.9. The bond strength was approximately 50 MPa, obtained at a reduction of 70.7%. The surface structure appears to be significantly different from that of the warm-bonded specimens. It is indicated that the bonds occurred only at the asperity tops. Also indicated is a possible relative movement of the strips during bonding, evidenced by asperity deformation in the direction of rolling, similar to the surfaces seen after rolling, using sand-blasted rolls (Dick and Lenard, 2005). Roll Bonding 333

14.4.3 Side View of the Bond The side view of the bond of a two-layer warm-rolled blank is shown in Figure 14.10. The process parameters were: 280C entry temperature, 52 mm/s rolling speed and 66.5% reduction. The shear strength of the bond was 239 MPa. The large bonded areas are clearly evident as is the conformance of the surfaces to one another. The cross-section of the bond of a cold-rolled specimen is shown in Figure 14.11, produced at 90 mm/s speed, a reduction of 70.7%, after annealing at 450C for 2 h. The bond strength was nearly 50 MPa. The difference between the warm and cold bonds is immediately noticeable. A much smaller contact area as well as much less conformance of the surfaces are noted when the process was conducted at room temperature. There are several clearly visible gaps between the two layers. It is indicated again that bonding happened only at the asperity tips of two surfaces.

Bonded surface Figure 14.10 The cross-section of a warm-bonded specimen: entry temperature, 280C; rolling speed, 52 mm/s; reduction, 66.5%.

100 μm

Figure 14.11 The cross-section of Separation Bonded surface a cold-bonded specimen: annealing temperature, 450C, 2 h; rolling speed, 90 mm/s; reduction, 70.7%.

100 μm 334 Primer on Flat Rolling

14.5 The Phenomenon of Bonding

Metallic bonding, the mechanism that is found in aluminium, occurs when the valence electrons, which are not bound, form a sea of electrons (Callister, 2000), holding the metal together. The bonding energy of aluminium is given by Callister (2000) as 324 kJ/mol. The figure of Schey, referred in Schey (1983), needs to be considered when the events, occurring during the roll-bonding process, are studied. Both surfaces are covered with contaminants in addition to a layer of aluminium oxide of approximately 12 μm thickness. The oxide is very hard; Rabinowicz gives its hardness as 2150 kg/mm2 (Rabinowicz, 1965) but it is brittle and it breaks at tensile strains of approximately 0.00060.0008 (Callister, 2000). As the two strips placed on one another enter the roll gap, the oxide layers are subjected to increasing normal pressures and when the strips deform in the direction of rolling, the oxides cannot follow. The layers then break up into islands, exposing the new, clean, virgin metal surfaces, a prerequisite for the formation of adhesive bonds. This phenomenon is similar to the break-up of the iron oxides during hot rolling of steels, studied by Li and Sellars (1998) and Krzyzanowski and Beynon (2002), who concluded that the hot, new material, extruding in between the cracks in the oxides and contacting the colder roll surface, increases the coefficients of heat transfer and friction. The asperities of the freshly created aluminium surfaces in contact are now subjected to pressures in the range of 9002000 MPa (Figure 14.3). Calculations, using Elroll, a commercially available finite element program designed to predict the variables during flat rolling of metals, indicate that the interfacial temperatures may reach as much as 180C, softening the materials and enhancing the creation of the adhesive bonds. (In the calculation of the temperature at the centre of the two-strip work piece, a heat transfer coefficient of 20,000 W/m2 K, entry temperature of 22C and a coefficient of friction of 0.25 were used, subjecting the strips to a reduction of 70% at a speed of 4 rpm.) In order to identify the mechanisms that may be involved in the creation of the bonds, a brief summary of the results follows:

G no successful bonds were created when rolling the as-received metal at room temperatures; G bonding was successful when the strips were rolled at room temperatures only after they were annealed for 2 h prior to rolling. The bond strength, which was a fraction of the strength of the parent metal, increased with increasing annealing temperatures; G higher entry temperatures resulted in higher bond strength; and G the time of contact in one pass was in the order of 30220 ms. Diffusion bonding needs time, pressure and temperature to proceed. It is hypothesized that in the present set of experiments, mechanical bonding, caused by the force of attraction of the atoms, overwhelmed diffusion bonding. This is based mainly on the time available for diffusion, and when that is at most 220 ms, not much of mass transfer may have occurred, in spite of the large interfacial pressures. Annealing softened the metals and softened the adhesion in between the oxide layer Roll Bonding 335 and the parent metal, making it easier to fracture the aluminium oxide and creating larger true areas of contact. The softness allowed the metals to conform to each other well. The strength of the softened aluminium alloy was somewhat rate sensitive, partially accounting for the observation of the effect of the time of contact. As also indicated by Zhang and Bay (1997), when the clean, new metals make contact, mechanical bonding occurs. The activation energy required for bond formation, a material property, may be defined in terms of the shear strength of the bond, and noting the strong temperature dependence, an Arrhenius-type relation may be formulated as Q τ 5 τ 2 b ð : Þ bond 0 exp RT 14 1 where the shear strength of the bond is designated by τbond, the shear strength of the original metal is τ0, Qb is the activation energy required for bond formation, R is the universal gas constant and T is the entry temperature in K. The activation energy needed to form bonds is plotted versus the energy input during the rolling pass (Figure 14.12), where Qb is given on the ordinate and the energy input for rolling calculated using the measured torque, the speed of rolling and the time taken to travel through the roll gap on the abscissa. Three distinct sets of points are shown for room temperature and for 250C and 280C entry temperatures. Bond formation appears to require the least amount of activation energy when the entry temperature is the highest, at 280C, near the amount given by Callister (2000) as 324 kJ/mol. While the figure indicates that as the energy input is increased that is the reduction is increased less energy is required to cause successful bonds, this change is not very pronounced and one may infer that the activation energy is not strongly dependent on the energy input. It is, however, strongly dependent on

8000.0 Figure 14.12 The activation energy τ ave = 20 MPa Entry temperature of bonding as a function of the b 22°C energy input and the temperature. 250°C 6000.0 280°C

50–90 mm/s speed 4000.0 50–70% reduction τ ave = 167 MPa b

2000.0 τ ave Activation energy (kJ/mol) = 244 MPa b

0.0 0 200 400 600 Energy (Nm) 336 Primer on Flat Rolling

1.2 Figure 14.13 The ratio of the bond

50–90 mm/s speed strength to the strength of the 50–70% reduction original material as a function of the energy input.

0.8 τ b τ 0

0.4 Entry temperature 22°C 250°C 280°C

0.0 0 200 400 600 Energy (Nm) the temperature-induced structural changes, which are more pronounced at higher rolling temperatures. A limitation of the physical meaning of Eq. (14.1) needs to be mentioned. The equation indicates that the bond strength reaches the shear strength of the metal when the temperature reaches infinity. The only conclusion that should be arrived at is that the ratio of the shear strength of the metal to the bond strength will approach unity as the temperature is increasing. The energy input is related to the ratio of the shear strength of the bond to the shear strength of the original metal in Figure 14.13. The process parameters are as given in Figure 14.12. The figure confirms earlier observations that the creation of stronger bonds requires higher effort. As before, the strongest bonds have been produced at the highest temperature and the weakest bonds at the lowest temperature.

14.6 Conclusions

G Strong bonds whose shear strength is comparable to that of the parent metal can be created with the Al 6111 alloy by warm roll bonding. G Entry and annealing temperatures, the reduction and the rolling speed are the four important parameters of the roll-bonding process. The most important factor is the temperature. Warm roll bonding creates higher bond shear strength than cold roll bonding. In cold bonding, it was possible to approach but not to reach the same strength as the parent material. G Mechanical bonding was the dominant mechanism. 15 Flexible Rolling

15.1 Introduction

One of the most important current objectives of the automotive industry is to develop the technology to produce lightweight components. New materials are being introduced, among them the interstitial free steels, transformation-induced plasticity steels, the high-strength low-alloy steels, bake-hardenable steels, dual phase steels, martensitic and manganeseboron steels, having yield strengths that vary from a low of 200 up to 1250 MPa. The elongation of these steels decreases as the strength increases, from a high of nearly 40% to a low of 45%, affecting their subsequent applications (Flaxa and Shaw, 2003). A recent review indicates, however, that light construction steels with induced plasticity possess tensile strength in the order of 1000 MPa and total elongation of 6070% (Ehrhardt et al., 2004). New processes are also being examined in order to satisfy the objectives of reduced weight and increased strength. One of the methods combines advanced computing techniques to determine the exact load-carrying needs of a component with the application of tailor-welded blanks. Tailor-welded blanks are made up of two sheets of unequal thickness which are welded to form a blank for subsequent sheet metal operations involving bending in one or two directions, such as in the deep-drawing or the stretch-forming processes. Their importance in the sheet metal forming industry is indicated by a statement extracted from the American Iron and Steel Instie website, AISI.com, in 2005;

Demand is expected to jump from 20 million tailor-welded blanks in 1999 to more than 90 million in 2005.

While welding techniques are well advanced and the interruption of the geometry may be accounted for in the design of the forming processes, the strength and the ductility of the welds are often different from that of the parent metal. As Worswick (2002) writes, the weld metal strength in welded aluminium alloy blanks is less than that of the parent metal which contrasts the behaviour of steel welds that tend to be overmatched. Chan et al. (2003) developed forming limit diagrams of tailor-welded blanks, made of cold rolled steel sheets, butt-welded by Nd: YAG lasers. They concluded that while the strengths of the tailor-welded blanks and the parent metals were similar, the formability of the blanks was lower and depended significantly on the thickness ratio. Higher thickness ratios resulted in lower formability. Min and Wang (2000), testing two steels with 0.12% and 0.35%

C contents, agree. As well, there is agreement that higher gauge ratios tend to push failure away from the weld into the more ductile thinner gauge material. Figure 15.1, reproduced from Worswick (2002), shows a typical weld cross-section of an aluminium alloy. It is observed that the weld is not ideal: it exhibits an undercut at the top surface and shows some weld metal porosity. The weld seams cause discontinuities and thus affect formability as well as the fatigue resistance of the subsequent components, produced using the tailor-welded blanks. Kampusˇ and Balicˇ (2003) compare the deep-drawing capabilities of tailor-welded blanks using laser and MIG welding and four steels of similar yield strengths. They comment that the use of laser welding of thin sheet metals is questionable because of the high power requirements and hence, lasers are used less often. Using MIG welding, the microhardness in the weld of two of their metals increased to 400 HV and fractures always appeared in the weld. When using laser welding, the hardness increased to 700 HV. They found that drawing the laser-welded cups was not successful, even after annealing. Venkat et al. (1997) evaluated the effect of autogenous CO2 laser butt welding using 6111-T4 aluminium alloys on their elongation under uniaxial tension. The uniform elongation of the welds varied from a low of 8.3% to 18.3%, compared to 24% of the base metal. Ahmetoglu et al. (1995) conducted deep-drawing experiments using aluminium killed deep-drawing quality steel sheets. They found that failure occurred at the flat bottom, parallel to the weld line. They attributed this to the non-uniform distribution of the deformation and concluded that the difficulties occasioned by the use of tailor-welded blanks in sheet metal forming operations make process planning more difficult and that new design guidelines need to be developed. It may be concluded that while tailor-welded blanks represent a major advance in the quest for weight reduction of automotive components, the discontinuity caused by the weld seam often causes difficulties. Merklein and Geiger (2002) write about the need to develop new materials and new processes to produce innovative lightweight construction, of special importance in the automotive industry. Kopp et al. (2002) and Kopp (1996) describe a fairly new technique to produce tailored blanks that of flexible rolling. In that process, the roll gap is changed during the pass and, depending on the desired

Figure 15.1 The cross-section of the weld in a tailor-welded blank (Worswick, 2002). Flexible Rolling 339 final product, it may be increased or decreased, producing a work piece whose attributes geometrical, mechanical, metallurgical vary along the length. The authors also conducted limiting dome height tests and found that when the thickness transition is appropriate the heights were near that of the parent metal. They further comment that “this is a great advantage compared to Tailor-welded Blanks, which are known for their decreased formability ... which can be attributed to the presence of the welding seam”. Further, while rarely reported, it is expected that the fatigue strength of the continuous material, rolled but not interrupted by welding, will be improved. In the more-reduced portion the grain sizes are decreased, strength is increased and the ductility is decreased when compared to the less-reduced portion. The weld and the attendant discontinuity are eliminated. The manner and the rate of change of the roll gap will affect the stress concentration at the changeover, however. The advantages and disadvantages of strip profile rolling and flexible rolling when compared to tailor-welded blanks are discussed by Kopp et al. (2005), in addition to a description of the behaviour of the rolled products in subsequent sheet metal forming processes. They conclude that both processes may offer cost savings over the welded blanks. It is worthwhile to quote directly from the website Mubea.com in 2004 describing the first commercial application of flexible rolling.

Flexible Rolling is a new rolling process with considerable lightweight construction potential for the automotive industry. This technology is characterised by the fact that the roller gap is selectively altered during the rolling process by means of a control system, enabling the metal thickness to be adapted everywhere along the length of the sheet steel to suit the respective loads to which the component is to be subjected. Furthermore, Flexible Rolling shortens the process of manufacturing the sheet metal and enables it to be formed by further processing stages, such as deep drawing or hydroforming. The reduction will directly affect the steels’ strength and ductility and as such, is expected to be one of the most important process parameters.

A more recent check of Mubea.com shows how the product line on tailor-rolled sheet metals has evolved. Recently, Danieli of Italy commissioned a flexible rolling mill for Mubea. Hirt et al. (2005) write that up to 50% thickness changes are now possible using Strip Profile Rolling. They also considered wrinkling during deep drawing of the tailor-rolled blanks and describe the die modifications that minimize the problems use of an elastic blank holder was most beneficial. In the present chapter the relation of the process and material parameters to the resulting formability of the tailor-rolled blanks is examined. Three materials are experimented with: an aluminium alloy, often used in the automotive industry, and two low-carbon steels. The roll gap is changed as a step function shortly after entry into the roll gap; however, the inertia of the screw-down system leads to a gradual change of the thickness. The reduction in the two parts of the sample is varied in a systematic manner, keeping the total reduction to 5556%. The results of the two- stage passes are compared to those obtained in single-stage passes. The roll 340 Primer on Flat Rolling separating forces, the changes in the roll gap and the resulting changes to the rolled metals’ resistance to deformation and strain at fracture are examined.

15.2 Material, Equipment, Procedure and Sample Preparation

15.2.1 Material Two low-carbon steel strips and an aluminium alloy were experimented with. The first set of steel samples was made of AISI 1030, cold drawn steel alloy and the other was of AISI 1008, deep-drawing quality steel. The engineering stressstrain relation for the cold drawn AISI 1030 steel, obtained in a conventional, uniaxial tension test is σ 5 400ð11169εÞ0:083 MPa and for the 1008, deep-drawing quality : (DQSK) it is σ 5 243ð11139:5εÞ0 095 MPa. The strain at fracture of the 1030 steel, before rolling, is 2527% and that of the 1008 steel is, surprisingly, only slightly over 18%. The strength of the aluminium alloy, Al 6111-T4, an automo- : tive grade, is well described by the relation σ 5 150ð11157εÞ0 245 MPa. The aluminium alloy is identified as among those that provide the best forming properties along with very good weldability, demonstrating a fracture strain of nearly 45%, though in the present tests only 22% strains were obtained. The carbon steels are comparable to the materialsusedbyMinandKang(2000)and Chan et al. (2003) (Table 15.1).

15.2.2 Equipment The two-high experimental mill has been described in Section 14.2.2.

15.2.3 Procedure The rolls and the strip to be rolled are cleaned using acetone. The roll gaps for the single or the two stages of rolling and the roll speed are set and the roll pass is started. The digital data acquisition system begins the data collection after the roll force reaches a pre-determined level and stops automatically when the rolled strip exits. All passes are performed without lubrication.

Table 15.1 The Chemical Compositions of the Alloys, Weight %

Cu Mn Si Zn Mg (max) C (max) P (max) S (max) Fe Al

6111 0.82 0.21 0.21 0.02 0.01 Rest 1030 0.60.9 0.280.34 0.4 0.05 Rest 1008 0.30.5 0.1 0.04 0.05 Rest Flexible Rolling 341

15.2.4 Sample Preparation The steel samples are 1.32 (AISI 1030) and 1.12 mm (AISI 1008) thick, 25 mm wide and 600 mm long. The aluminium samples are 1.02 mm thick, 25 mm wide and 600 mm long. The edges are deburred and the strips are cleaned using acetone. The long dimension is always in the direction of previous rolling passes.

15.3 Results and Discussion

15.3.1 Roll Separating Forces and the Roll Gap 15.3.1.1 AISI 1030 Steel, Cold Drawn

The variations of the roll force during single- and two-stage rolling passes are shown in Figure 15.2. The specific roll force is plotted on the ordinate while the time elapsed from the start of the pass is given on the abscissa. All tests are conducted at a roll surface speed of 0.26 m/s. The roll gap is changed at approximately the middle of the pass, 0.5 s after entry. No lubricants are used. The data from a single-stage and four two-stage experiments are indicated in the figure and the reductions/passes in each test are also presented. In all four tests the aim was to roll to progressively larger reduction in the first stage and to keep the total reduction to as near 55% as possible so the reduction in the second stage was progressively smaller. The objective in the choice of these reductions was to test the metals’ reaction to cold working when subjected to various sequences. The roll force obtained in a single-stage pass of 55% reduction is shown for comparison. The roll force in the single-stage pass is 11,370 N/mm, steady for most of the pass, dropping slightly near the exit. In the second pass of the two-stage reductions all of

14,000 Figure 15.2 The roll forces in single- and two-stage rolling; 12,000 AISI 1030 alloy.

10,000 Reduction (%) 8000 15/56 21/60 35/55 6000 44/54 55 4000 Single and two-stage rolling

Roll separating force (N/mm) force Roll separating AISI 1030 steel 2000 0.26 m/s No lubrication 0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) 342 Primer on Flat Rolling which are approximately of similar total magnitudes all of the forces rise beyond that value, with one exception. While the differences are not very large, of the order of approximately 1000 N/mm, about 10%, they are still noticeable and are not expected to be caused by experimental scatter. It is hypothesized that tribological factors are at play. Since no lubricants are used, the interfacial tribological mechanisms include adhesion and ploughing. The rolls were ground and finished manually, creating shallow-angle asperities, making the contribution of adhesive forces overwhelm those of ploughing. Hence, as implied by the adhesion hypothesis, the creation and the rate of growth of adhesive bonds would control the transfer of energy and the frictional conditions at the contact surface. The reductions in the first stages of the four two-stage tests are smaller than in the 55% single-pass experiment so the flattening of the asperities up to the change in the roll gap is not as pronounced as in the second stage. It is the sudden change of the roll gap that is responsible for the change of tribological conditions. Interface events depend, among other mechanisms, on the size and the relative movement of the contacting surfaces. The roll gap change, completed in about 100 ms in a direction perpendicular to the direction of rolling, adds to the usual relative movement along the roll surface, increases the contact areas and adds to the creation of adhesive bonds; hence the approximately 10% growth of the roll separating forces. The changes of the roll gap during the two-stage passes, while rolling the 1030 steel alloy, are shown in Figure 15.3. The roll gap is plotted on the ordinate and as before, the time is given along the abscissa. In addition to the changes of the roll gap during the two stages of rolling, the dynamic response of the mill is also observable in the figure. As soon as the strip enters the roll gap, the upper and the lower rolls separate and in about 100 ms the

2.0 Figure 15.3 The roll gap Two-stage rolling changes during two-stage Reduction (%) AISI 1030 steel rolling; AISI 1030 alloy. 0.26 mm/s 15/56 No lubrication 21/60 35/55 1.6 44/54

Roll gap (mm) 1.2

0.8 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Flexible Rolling 343 requested roll gap is reached. The command to change the distance between the rolls is according to a step function and approximately another 100 ms are needed to reach the steady-state roll gap distance of the second pass. The 100 ms delays are contributed to by two events: the response of the servovalves with a frequency response of 100 Hz and the inertia of the massive top roll and its connections to the hydraulic apparatus.

15.3.1.2 AISI 1008 Steel Cold Drawn

The roll separating forces obtained in a single-stage pass of 55% reduction and two two-stage passes of 14/39% and 20/33.6% reduction are given in Figure 15.4, measured when rolling the softer AISI 1008, deep-drawing quality steel. Figures 15.2 and 15.4 are quite similar. The dynamic response of the mill is also similar to the previously observed behaviour. As expected, the roll separating forces increase with the reduction and remain fairly steady as the process continues. At the time the roll gap is changed, the roll forces increase to approximately the same magnitude, as the total reduction in the two passes are nearly identical.

15.3.1.3 Al 6111 Aluminium Alloy

Figures 15.5 and 15.6 demonstrate the behaviour of the 6111 aluminium alloy when subjected to single- and two-stage rolling passes. The roll forces are shown in Figure 15.5 and the roll gap changes are given in Figure 15.6. The reductions in the single- and double-stage passes are approximately the same; see Figure 15.5. The roll forces in the two-stage passes don’t rise beyond the single- stage data, opposing the observations made when the harder steel alloy was rolled. This is caused again by the flattening of the asperities. When the softer aluminium alloy enters the roll gap, its asperities flatten rapidly and even at the smaller reductions

14,000 Figure 15.4 The roll forces in AISI 1008 steel single- and two-stage rolling; 12,000 AISI 1008 alloy.

10,000

8000

6000

Reduction (%) 4000 55 14/39 Roll separating force (N/mm) force Roll separating 2000 20/33.6

0 0.0 0.2 0.4 0.6 0.8 Time (s) 344 Primer on Flat Rolling

12,000 Figure 15.5 The roll forces in Two-stage rolling single- and two-stage rolling; Al 6111 aluminium 6111-T4 alloy. 10,000 0.26 m/s No lubrication

8000

6000 Reduction (%) 4000 16/53 24/53 35/55 Roll separating force (N/mm) force Roll separating 2000 45/54 56

0 0.0 0.5 1.0 1.5 Time (s)

1.6 Figure 15.6 The roll gap changes during two-stage rolling; 6111-T4 alloy. 1.4 Reduction (%) 16/53 24/53 1.2 35/55 45/54

1.0 Roll gap (mm) 0.8 Two-stage rolling AI 6111 aluminium 0.6 0.26 m/s No lubrication 0.4 0.0 0.5 1.0 1.5 Time (s) they reach their maximum deformation; hence, no changes in the tribological behaviour result. The dynamic response, see Figure 15.6, is as observed before.

15.4 Predictions of a Simple Model

The measured roll separating forces are realistic and consistent with earlier results. Using a simple one-dimensional model of the flat rolling process, in which the Flexible Rolling 345

Table 15.2 Comparison of the Measurements with the Predictions of a Simple Model; Rolling of the 1030 Steel Strips

Reduction Roll Force (N/mm) Roll Force (N/mm) Coefficient of (%) Measured Calculated Friction

15 (Single stage) 4580 4610 0.095 21 (Single stage) 5400 5510 0.085 35 (Single stage) 7400 7403 0.078 44 (Single stage) 8800 8803 0.078 55 (Single stage) 11,370 11,316 0.082 55 (Second pass) 12,000 11,892 0.086

elastic deformation of the roll is analysed using the theory of elasticity (Roychoudhury and Lenard, 1984) and the stressstrain relations, the forces are calculated and found to be quite close to those measured. The predictions and the measurements are compared in Table 15.2 for the 1030 steel. The coefficient of friction values, obtained by matching measured and calculated roll forces, are also indicated. In the first five rows the first passes are modelled, including the single pass 55% reduction. As mentioned above, the 55% reduction in the second passes caused the roll forces to increase due to increased frictional resistance and the predictions of the model indicate that this is, in fact, the case. The force measured in the second stage of the 35/55% two-pass rolling is approximately 12,000 N/mm. Calculations demonstrate that the model predicts a force of 11,892 N/mm, when using a coefficient of friction of 0.086, a 5% increase over the value needed to predict the force on the single-stage pass. Admittedly this is not much of a change, but it is not taken to be insignificant. When designing pass schedules for flexible rolling, the changes of tribological conditions, however small, need to be taken into account. Table 15.3 gives similar results for the 1008 steel. The magnitudes of the coefficient of friction are again realistic and are as published elsewhere (Lenard, 2001). As expected, they decrease as the reduction is increased. As well, their magnitudes are higher than those found in the case of the 1030 steel, showing the applicability of the adhesion hypothesis and the enhanced ease of the formation of adhesive bonds with the softer steel. The predictions of the model for the Al 6111 aluminium alloy are given in Table 15.4 and the conclusions are as for the two steel alloys. The aluminium alloy is the softest of the three metals and the formation of adhesive bonds occurs most easily, hence the increase of the coefficient of friction with increasing loads.

15.5 Strain at Fracture

The longitudinal strains at fracture in a conventional tension test indicate the metals’ formability. These were determined using the 6111 aluminium and the 346 Primer on Flat Rolling

Table 15.3 Comparison of the Measurements with the Predictions of a Simple Model; Rolling of the 1008 Steel Strips

Reduction Roll Force (N/mm) Roll Force (N/mm) Coefficient of (%) Measured Calculated Friction

35.6 6788 6897 0.15 56.8 8935 8962 0.105

Table 15.4 Comparison of the Measurements with the Predictions of a Simple Model; Flexible Rolling of the Aluminium Strips

Reduction Roll Force (N/mm) Roll Force (N/mm) Coefficient of (%) Measured Calculated Friction

16 2000 2023 0.04 24 3150 3110 0.068 35 4500 4557 0.075 45 5100 5141 0.06 56 7380 7388 0.07

AISI 1008 steels, as a function of the processing parameters, and are shown in Figures 15.715.9. The loss of ductility as a result of cold working is very pronounced in the aluminium alloy; see Figure 15.7. The fracture strain, approximately 22% in the as-received metal, dropped to about 1% after two-stage rolling to a total of 59% reduction, indicating the impos- sibility of further plastic forming. The AISI 1008 steel shows similar behaviour, as demonstrated in Figure 15.8. The strain at fracture before rolling is almost 19%. When rolled to 58.5% in a single pass or to 52.5% in two passes, the strains reduce to about 1%, as with the aluminium. Some, but not all, of this loss is regained after annealing, indicated in Figure 15.9, confirming the conclusions of Dyment (2004), whose experiments in tube bending and annealing showed significant recapture of ductility. It is evident that careful annealing before subsequent processing is needed. This would increase the formability, albeit at the expense of the increased strength. The ductility of the 6111-T4 alloy, obtained after rolling, may be compared to that obtained after laser welding. Venkat et al. (1997) used a 3 kW laser beam to weld two pieces of the alloy of the same thickness. The ductility of the weld was measured in traditional tension tests. The longitudinal elongation varied, depending on the velocity of the laser beam, from a low of 8.6% to as high as 18.7%, much higher than in the present study, emphasizing again the importance of annealing after flexible rolling. The results of tension tests on the welds of tailor-welded aluminium samples of the 5000 designation were given by Davies et al. (2001). The Flexible Rolling 347

600 Figure 15.7 The fracture strain as a function of the process parameters; Al 6111 aluminium 6111-T4 alloy.

As received 400 32% and 59% reduction

200 Tensile stress (MPa)

0 0.00 0.05 0.10 0.15 0.20 0.25 Strain

800 Figure 15.8 The fracture strain as a function of the process parameters; AISI 1008 steel AISI 1008 steel.

600 As received 58.5% reduction, single-stage 25.4% and 52.5% reduction 400

Tensile stress (MPa) 200

0 0.00 0.04 0.08 0.12 0.16 0.20 Strain strain hardening exponents of the alloys, taken at the onset of instability, varied from a low of 0.22 to 0.257. Some, but not major, losses of ductility were reported, with the fracture strains of the welds approaching that of the parent metal. Min and Kang (2000) tested the effect of flash welding on an S35C steel of 2 mm in thickness. The S35C steel is similar to the AISI 1030 in both chemical composition and tensile strength; it is somewhat more ductile in the as-received state, demonstrating an elongation of 53%. After welding the elongation reduced to 15%. In a sense, the comparison of the formability of the tailor-welded blanks and that of the tailor-rolled blanks may well be misleading. The welded blanks were not subjected to cold working so the loss of formability is the direct function of the welding process. The rolled samples lose ductility through the rolling passes; it can, however, be restored, at least to some extent, by well-designed annealing. 348 Primer on Flat Rolling

400 Figure 15.9 The fracture strain as a function of the process parameters; AISI 1008 steel, annealed.

300

200 AISI 1008 steel, annealed

As received

Tensile stress (MPa) 100 58.5% reduction, single-stage 18.6% and 60.2% reduction

0 0.00 0.04 0.08 0.12 0.16 0.20 Strain

Two further aspects became clear as the present project was drawing to a close. These concern the stress concentration at the thickness change and its effect on the fatigue strength of the rolled blanks, especially on samples that will eventually be used to produce load-carrying components. These should then be compared to that of tailor-welded blanks. The results will allow a clear choice concerning the superi- ority or otherwise of tailor-rolled blanks over tailor-welded blanks.

15.6 Conclusions

Flexible rolling, a recently developed technology to create sheet metal with varying thickness, was discussed. Details of a set of experiments, examining the effect of the process on the rolling variables, were presented. The effect of the reduction on the roll separating forces, resistance to deformation and subsequent strain to fracture during the production of tailor-rolled blanks was considered. An aluminium alloy, used in the automotive industry, and two low-carbon steels were subjected to single- and two-stage, continuous rolling passes. The reduction in the first stage was progressively increased while the total reduction was kept to 55%. The metals’ strength increased as a result of cold working and their ductility decreased drastically. Some of the ductility was regained after annealing. It became apparent that in order for the tailor-rolled blanks to be competitive with tailor-welded blanks, post-rolling annealing is unavoidable. 16 Problems and Solutions

Part 1 Problems

Chapter 1 Introduction Problem 1 Figure 16.1 shows the undeformed and the deformed shapes of a plate. The strain distribution in the deformed plate is given by

2 3 εx 5 0:005 xy 1 0:005 y and εx 5 0:006 yx 1 0:007ðy 1 xÞ where x and y are in metres. Determine the displacement components u and v of point D. Determine the shear strain at point C. Assuming a state of plane stress determine the stress component at point C. Let E 5 207 3 103 MPa and ν 5 0.3.

Problem 2 Considering logarithmic strains, show that

εI 1 εII 1 εIII 5 0 indicates incompressibility. Note that the Roman numeral subscripts indicate the principal strains.

Problem 3 The radial and circumferential stresses, respectively, in a rotating, hollow disk are given by the expressions 0 1 3 1 v σ 5 @ Aρω2 a2 1 b2 2 ðab=rÞ2 2 r2 r 8 0 1 2 3 1 1 @3 vA 24 2 2 2 1 3v 25 σϑ 5 ρω a 1 b 2 ðab=rÞ 2 r 8 3 1 v

Figure 16.1 The undeformed and A B the deformed shapes of a plate. Undeformed 25 mm u

Deformed C D

40 mm v

where the internal radius is a 5 25 mm, the external radius is b 5 125 mm, ν 5 0.3, 3 the yield strength is σy 5 300 MPa and the density is 7800 kg/m . Determine the rotational speed and the location at which yielding will begin, according to the maximum shear stress criterion.

Problem 4 Consider the problem of tube eversion in which a thin-walled tube is compressed longitudinally by two dies and in the process the ends of the tube are everted shown schematically below. Show that if the material behaves according to the Tresca criterion of plastic flow, the wall thickness is not expected to change (Figure 16.2).

Problem 5 In open die forging of a right, circular cylinder, the work piece is compressed by two rigid, rough platens. Define mathematically the boundary conditions for the stresses and velocities.

Problem 6 Write down the strain tensor for a state of plane stress.

Problem 7 Give the definitions for the three modes of heat transfer: conduction, convection and radiation.

Problem 8 Does the volume of a piece of metal, deforming elastically, remain constant? If not, how does it change?

Problem 9 Estimate the dimensional error of the rolled strip’s thickness across the 2000 mm width when the roll axes make 0.5 with one another while the rolls don’t flatten or bend.

Problem 10 Check the World Bank’s Yearbook for the amount of steel produced in the last few years. Problems and Solutions 351

P Figure 16.2 A schematic diagram of the tube eversion process.

σθ dθ Dies

Tube

σθ

a P t β o

D α σα + dσα t + dt dα ρ R α d μp f

p σα dθ σθ sin 2 r′ r dr

Problem 11 Look for statistics that indicate the number of people employed in steel production over the years. Has the number increased or decreased during the last few decades?

Problem 12 Contact used equipment sales companies to find out the cost of various used rolling mills.

Problem 13 The work rolls are connected to the driving motor through flexible joints. It is essential that the angular velocity of the rolls be constant within each turn. What type of flexible coupling, with its flexible joints, will allow this to happen?

Chapter 4 Flat Rolling Problem 1 Sketch the free-body diagram1 of one of the driven back-up rolls of a four-high rolling mill.

1The free-body diagram shows all forces acting on a body. 352 Primer on Flat Rolling

μ 5 φ Problem 2 Show that the minimum coefficient of friction is given by min tan 1 where ϕ1 is the bite angle.

Problem 3 Consider a heavy reduction of a fairly thick plate, such that the “planes remain planes” assumption is not valid any longer. How would an originally straight cross-sectional plane change by the time it reaches the exit?

Problem 4 The wide strip to be rolled is initially of a uniform thickness across its width. It is rolled using a non-cambered, small diameter work roll, in a two-high mill. How would the strip’s velocity at the exit vary across the width?

Problem 5 Estimate the radial strain experienced by the steel work roll during temper rolling. Assume that the original roll diameter is 400 mm. The entry thickness of the strip is 1 mm, and it is reduced by 2%, to a final thickness of 0.98 mm. Also assume that the roll becomes completely flat in the contact zone. Is this strain large enough to cause plastic deformation of the high-carbon steel roll?

Problem 6 Consider a two-high mill. The work roll is 200 mm in diameter and 1500 mm long. Estimate the magnitude of the force in the direction of rolling that would cause the work roll to deflect by 2 mm in that direction. Let E 5 200,000 MPa.

Problem 7 The entry speed of the transfer bar into the first stand of the finishing mill is 1000 mm/s. The entry thickness is 20 mm while the exit thickness is 2 mm. Estimate the exit velocity of the strip.

Problem 8 Estimate the time of contact during the pass in the last stand of the finishing mill. Take the entry thickness as 2.7 mm and the exit thickness as 2 mm. The work roll diameter is 800 mm and its rotational speed is 200 rpm.

Problem 9 Show that in flat rolling, assuming a state of plane-strain plastic flow of a HuberMises material, the stress in the direction of vanishing strain is given by

σ 1 σ σ 5 1 2 3 2

Problem 10 Sketch how the actual entry of a strip into the roll gap may look like, including the thickening of the leading edge of the strip and the slight compression of the roll.

Chapter 5 Mathematical Modelling pffiffiffiffiffiffiffiffiffiffiffi Problem 1 Show that the projected contact length is given by L 5 R0Δh, where 0 R is the radius of the flattened roll and Δh 5 hentry 2 hexit. Problems and Solutions 353

Problem 2 Compare the predicted power requirements of a rolling mill, using Eqs. (5.8), (5.9) and (5.62). Under what conditions do they compare well? Comment critically on the predictions.

Problem 3 Estimate the magnitude of the mill stretch in a small laboratory rolling mill. Compare it to the stretch of a full-size industrial mill. Make realistic assumptions for the dimensions of both mills. Consider rolling a thin, low-carbon steel strip.

Problem 4 The entry speed of the transfer bar into the first stand of the finishing mill is 1000 mm/s. There are seven mill stands and the reduction at each is 25%. The entry thickness is 20 mm. Estimate the exit velocity of the strip.

Problem 5 Derivations Show that the following equations can be derived from first principles: Derive Eq. (5.18), using a balance of forces on a slab of the deforming material in the roll gap. Derive Eq. (5.19), using the HuberMises yield criterion and the condition of plane- strain plastic flow. Derive Eq. (5.20), using geometry of the deforming strip. Derive Eq. (5.32), considering the independent variable to be the distance in the direction of rolling. Derive Eq. (5.48) for the stress due to the force of inertia.

Problem 6 A strip of 6061-T6 aluminium is to be produced. The dimensions of the rolled strip are to be as follows: 0.25 mm thickness, 25 mm width and 1000 mm length. The starting material is available in strips of 10 mm thickness, 25 mm width and 2000 mm length. The plates are to be hot rolled, at 500C, to a thickness of 2 mm. The thickness of 0.25 mm is to be obtained subsequently by cold rolling, in which a light mineral oil base is used as the lubricant, with oleic acid as the boundary additive. Design the flat rolling process by prescribing the reductions to be done in both the hot and the cold rolling processes. The details of the two rolling mills are: Hot mill: Roll radius: 150 mm Roll speed: 50 rpm Sticking friction Motor power: 45 kW Cold mill: Roll radius: 125 mm Roll speed: 100 rpm Coefficient of friction: 0.1 Motor power: 100 kW 354 Primer on Flat Rolling

The true stresstrue strain relations, obtained in uniaxial tests, are as follows: For hot deformation: σ 5 37ε_0:17 MPa For cold deformation: σ 5 450ε0:03 MPa What are the reductions to be in each of the hot and the cold rolling processes? How would the introduction of a light mineral oil in the roll gap affect the roll force and the roll torque? How would increasing the speed of rolling affect the roll force and the roll torque, when rolling with a lubricant? Why? Should a lubricant be applied in the hot rolling process? What would you expect it to achieve?

Problem 7 Derive the equation of motion for the metal in the roll gap. Under what conditions is the inertia term significant? (Give your answer in numerical terms of the roll speed and the reduction; assume a set of parameters for your calculations.)

Problem 8 The roll pressure distribution, obtained experimentally during rolling of an 1100 H14 aluminium strip, is shown below. Determine the changes of the coefficient of friction along the roll gap. Let σfm 5 50 MPa; R 5 125 mm (Figure 16.3).

Problem 9 A two-high rolling mill, for both hot and cold rolling of steel strips, is to be designed. The diameter of the work roll is 250 mm. Specifically, the dimensions for three components are needed:

G Diameter of one of the drive spindles (recall that the shear stress in a spindle, subjected to a torque of T, is given by τ 5 Tc=J; J 5 πd4=32; c is the radius and J is the polar moment of inertia of the cross-section). The maximum allowable shear stress in the spindle is 100 MPa.

Figure 16.3 The roll pressure 504°C, central temp. distribution. 120 6.32 mm entry thickness 80 rpm roll speed 30.4% reduction 2% oil – water emulsion 80

40 Roll Pressure

Interfacial stresses (MPa) 0

–40 0.00 0.04 0.08 0.12 0.16 Angle (radians) Problems and Solutions 355

G Load carrying area (A) of the frame such that the mill stretch is limited to 1 mm (recall that the elongation of a member is given by Δ 5 PL=AE; the effective height, L, of the frame is 1 m; its modulus of elasticity is 200,000 MPa and P is the total load carried). G Power of the driving motor. The requirements are as follows: Hot rolling (sticking friction): Low-carbon steel strips (5 mm thick, 50 mm wide, 500 mm long) are to be rolled at 1000C, at 0.5 m/s entry velocity, to a reduction of 60%; the true stresstrue strain rate curve of the metal, obtained in uniaxial tension, is given by σ 5 100 ε_0:1 MPa. The rolls are turning at 45 rpm. Cold rolling (μ 5 0.05): Low-carbon steel strips (5 mm thick, 50 mm wide, 500 mm long) are to be rolled, at 22C, at 0.5 m/s entry velocity, to a reduction of 60%; the true stresstrue strain curve of the metal, obtained in uniaxial tension, is given by σ 5 600 ε0:25 MPa. The rolls are turning at 45 rpm (Figure 16.4).

Problem 10 Show that, as long as the roll remains rigid, the strain rate in a flat rolling pass is given by

φφ_ ε_ 5 2R sin h2 1 2Rð1 2 cos φÞ

Problem 11 The roll separating forces and the roll torques, measured during lubricated cold rolling of a low-carbon steel strip, are given in Figure 16.5. The details of the rolling mill: The roll diameter is 250 mm and its length is 150 mm. The centre-to-centre distance of the bearings on either side of the rolls is 250 mm. The rolls are made of D2 tool steel, hardened to Rc 5 63. The surface roughness of the roll is Ra 5 0.2 μm. The mill is driven by a 42 kW, constant torque, DC motor. The rolled metal: Cold rolled strips containing 0.05% C steel are used, having a : true stresstrue strain curve σ 5 150ð11 234 εÞ0 251 MPa, obtained in uniaxial tension.

Figure 16.4 The schematic diagram of Frame Spur gears a rolling mill.

Driving motor

Spindle 356 Primer on Flat Rolling

10,000 70 Nominal reduction Nominal reduction 9000 15% 60 15% 8000 27% 27% 35% 35% 7000 50% 50 50%

6000 40 5000 30 4000

3000 20 Roll force (N/mm) 2000 Roll torque (Nm/mm) 10 1000 0 0 0 1000 2000 3000 0 1000 2000 3000 Roll surface speed (mm/s) Roll surface speed (mm/s)

Figure 16.5 The roll force and the torque.

The metal’s strength is essentially independent of the rate of strain. The initial surface roughness of the samples is approximately Ra 5 0.8 μm, in both the rolling and transverse directions. Note that ð ða1bxÞn11 ða1bxÞn dx 5 ðn 1 1Þb

The sample dimensions: The to-be-rolled samples are 1 mm thick, 25 mm wide and 300 mm long. Respond to the following: A pair of force and torque data is indicated on the figures, by small circles. Using an assumed value of the coefficient friction of 0.1, calculate the forces and the torques. Are the measurements and the calculation different? Why? (Account for at least three phenomena.) Using the measured force, infer what the coefficient of friction should be. The roll separating forces decrease as the rolling speed increases. Why? The strip, rolled to a reduction of 50% in the first pass, is rolled once more. In the second pass it is reduced by another 30%. Calculate the roll separating force in the second pass. Let the coefficient of friction be 0.1.

Problem 12 Steel strips of 0.5 mm thickness are to be produced by cold rolling in three passes. The strip is supplied in 5 mm thickness, 50 mm width and 400 mm length. The true stresstrue strain curve of the steel is given by σ 5 950 ε0:12 MPa, obtained in uniaxial tension. The steel is reduced to 3 mm in the first pass at a roll speed of 50 rpm; to 1 mm in the second pass and to 0.5 mm in the third pass, both at 100 rpm. A light lubricant is used in all passes and the coefficient of friction is to be taken as 0.1 in all three passes (Figure 16.6). Problems and Solutions 357

Figure 16.6 Three-pass rolling Stand 1 Stand 2 schedule. Stand 3 h h h h 0 1 2 3

The rolls are 250 mm in diameter. The mill power is 100 kW. Is the motor power sufficient for all three passes? If not, what do you suggest to remedy the problem? (list at least two possibilities; no calculations are needed.) How long will the strip be after the third pass?

Problem 13 6061-T6 aluminium strips are to be rolled in two passes to produce the 0.2 mm final thickness. The available strips are of 2 mm thickness, 25 mm width and 300 mm length. In the first pass the strips are to be rolled to 1 mm, using roll speed of 100 rpm and an entry velocity of 1250 mm/s. In the second pass the final thickness of 0.2 mm is produced. The roll speed is 200 rpm and the entry velocity of the strip is 2500 mm/s. The rolling mill is of a two-high design, having work rolls of 125 mm radius. The surface roughness of the work roll is 0.2 μm and the surface hardness is Rc 5 64. During the two passes a light mineral oil with oleic acid as the boundary additive is used as the lubricant. The true stresstrue strain curve of the aluminium, obtained in a uniaxial tension test, is given by σ 5 450 ε0:03 MPa. Determine the required power in the two passes. Take the coefficient of friction in the first pass to be 0.1 and in the second, 0.05. Ignoring the heat loss in between the two passes, what will be the temperature of the strip after the second pass? Let the specific heat of the aluminium be 900 J/kg K and its density be 2800 kg/m3. The initial temperature of the strip is 22C. Do you expect the width of the rolled strip to be different from the width at entry to the first pass? If there is any change, would the final width be larger or smaller than the initial width? In another pass, the light mineral oil is exchanged for one with a higher viscosity. Do you expect the roll separating force and the roll torque to change? How?

Problem 14 1 mm thick and 25 mm wide AISI 1008 low-carbon steel strips are to be produced by cold rolling. The metal is available in 8 mm thick, 25 mm wide bars so several passes will be necessary to reduce it to the final thickness. Its true stresstrue strain curve, obtained in uniaxial tension is represented by σ 5 600 ε0:25 MPa. The same two-high rolling mill, having 250 mm diameter rolls, will be used in all passes. 358 Primer on Flat Rolling

When rolling the strips, the time in between passes is to be as short as possible, so there would be no heat losses. A low viscosity lubricant is used so the coefficient of friction in each pass is 0.1. The lubricant breaks down at a temperature of 200C. The power of the driving motor is 150 kW. Assume that in all three passes the thickness of the strip at the neutral point equals the average of the entry and exit thickness of that pass. Let the specific heat of the steel be 500 J/kg K and its density be 7830 kg/m3. Three consecutive passes are planned to produce the 1 mm thick strips. In the first, the strips are reduced to 4 mm at a roll speed of 100 rpm; in the second to 2 mm, at 180 rpm, and in the third, to the final thickness of 1 mm at 240 rpm. Is the equipment capable of carrying out the task? Will the lubricant provide the needed lubricating action in all of the passes?

Problem 15 The experimentally determined roll torques per unit width (for both rolls), during two-pass cold rolling of 3 mm thick carbon steel strips, are given below:

Pass # rpm Torque (Nm/mm) hin (mm) hout (mm) 1 100 60 3 1.8 2 100 100 1.8 0.6

The roll diameter is 200 mm. The strips are 50 mm wide. The true stresstrue strain curve of the metal, obtained in uniaxial tension is σ 5 600 ε0:25 MPa. The torque is measured by torque transducers, located in the drive spindles, as shown in the schematic diagram of the rolling mill. The schematic diagram of the two-high rolling mill is given in Figure 16.7. Determine The coefficient of friction in each of the two passes. Can the second pass be performed without front pull? Why or why not? The temperature rise of the sample in the second pass. The density is 7800 kg/m3 and the specific heat is 500 J/kg K. The necessary motor power.

Bearings Work rolls Torque transducers Driving motor

Figure 16.7 The rolling mill. Problems and Solutions 359

Note: If plastic deformation occurs at the entry, followed by the no-slip region, the strip must speed up very fast to reach the roll’s surface speed. This would cause large inertia forces; would these be significant? Other data show that increasing pressures and speeds lower the coefficient of friction; how does that go with the high friction, which is supposed to occur in temper rolling? If there is a further reduction at the exit, the strip would have to speed up there, to exit at a velocity higher than the roll’s.

Chapter 8 Materials Problem 1 True stresstrue strain curves, obtained during constant strain rate hot compression of a carbon steel show the characteristic behaviour of steels during hot deformation, indicating that the peak stresses are reached at progressively higher strains as the strain rates are increased. Why? What metallurgical phenomena are active in the steady-state region?

Problem 2 True stresstrue strain curves, obtained in compression at 900C for a NbV, are shown below. Determine the strain rate-hardening coefficient at a true strain of 0.4. At high temperatures the true stressstrain curve is given by σ 5 Cε_m (Figure 16.8).

Problem 3 Washers are to be made of commercially pure aluminium by open die forging. Cylinders of 10 mm in diameter and 15 mm in length are available. Two possible approaches are possible: fast forging and slow forging. In both cases the force of compression is the same. Which of the two processes will result in larger reduction?

280 Figure 16.8 True stresstrue strain ε (1/s) curves of a NbV steel. 240 5

200 0.5

0.05 160 σ

(MPa) 120 0.005

40 Nb–V steel temperature, 900°C

0 0.00.4 0.8 1.2 1.6 ε 360 Primer on Flat Rolling

The true stressstrainstrain rate curve is given by σ 5 90 ε0:2 ε_0:1 MPa; it may be assumed that during the deformation the strain rate remains constant. In the first forging process the average rate of strain is 1 s21 and in the next it is 100 s21. The force reaches a maximum of 20,000 N during the deformation process and it increases linearly, from zero to its maximum value. The density of the aluminium is 2500 kg/m3 and its specific heat is 900 J/kg K.

Problem 4 Which of the two empirical relations represents the stressstrain curve of a steel better:

σ 5 K εn or σ 5 Yð11BεÞn

Why? Would the same argument hold for soft aluminium?

Problem 5 A cylindrical sample is subjected to torsion at high temperatures. Would dynamic recrystallization occur at the same time across the cross-section? Why or why not? Would static recrystallization occur at the same time across the cross-section as the cylinder is being twisted?

Problem 6 The true stresstrue strain curves of a NbV steel, obtained in uniaxial, isothermal compression, are shown below. Determine the activation energy for dynamic recrystallization (Figure 16.9).

Problem 7 A drop hammer for open die forging of brass cylinders of originally 25 mm height and 28 mm diameter is to be designed. Specifically, the height from which the 1000 N hammer is to be dropped is to be determined. The true

280 240 ε (1/s) ε (1/s) 5 240 200 5 200 0.5 160 0.5 160 0.05 σ σ 120 0.05 (MPa) 120 0.005 (MPa) 0.005 80 80

Nb–V steel 40 40 Nb–V steel temperature, 900°C temperature, 950°C 0 0 0.0 0.4 0.8 1.2 1.6 0 0.4 0.8 1.2 1.6 ε ε

Figure 16.9 True stresstrue strain curves of a NbV steel. Problems and Solutions 361 stresstrue strain relation of the brass is given by σ 5 800 ε0:33 MPa. Its ductility, expressed in terms of total elongation to fracture ½ef 5 ðlf 2 l0Þ=l0 is 0.5. Determine the height from which the hammer is to be dropped, such that the maximum permissible reduction of the brass cylinder is achieved. What will be the temperature rise of the cylinder after the drop? Let the density of the brass be 8530 kg/m3 and its specific heat be 385 J/kg K. Will the distribution of the temperature be uniform? If not, how will it vary?

Problem 8 A cylindrical part (10 mm high, 158 mm diameter) is to be produced by open die forging using the drop tower, which has a weight of 20,000 N to be dropped on the sample from a height of 10 m. The stock from which it is to be made is 100 mm high and 50 mm in diameter. The metal is low-carbon steel and its true stresstrue strain curve in cold forming is given by σ 5 600 ε0:25 MPa. Can the part be made in one drop? If a second drop is necessary, anneal the part first. From what height should the weight be dropped to produce the 10 mm final height? What should be the ductility of the metal to avoid tensile cracking? How is the velocity to change during the compression process for the strain rate to remain constant? (Don’t calculate.)

Problem 9 An applicant is being interviewed for the position of master forger in a company and is given a test. The assignment is to hot forge a railcar wheel, the schematic diagram of which is shown below. The block from which the wheel is to be made is also shown in Figure 16.10. The wheel is made of a medium-carbon steel whose true stresstrue strain curve is given by σ 5 120 ε0:10 MPa, while formed at 1000C. (Note that since the true stresstrue strain curve is given, there is no need to consider the effects of the rate of strain.) The ductility of the steel, at 1000C is 0.6, based on the tensile strain. The equipment available includes a high-temperature furnace, and a drop forge which can drop a weight of 10,000 N from a height, adjustable from a minimum of 0.5 m to a maximum of 10 m. Several more high-powered forges are also available. The density of the steel is 7800 kg/m3 and the specific heat is 500 J/kg K. The applicant decides to use the drop forge and to forge the wheel following the steps given below: Pre-heat the furnace to 1000C and place the block in the furnace when 1000Cis reached. The wheel is then held there for 4 h, in air, during which it reaches a uniform

D Figure 16.10 The railcar 1400 mm wheel.

H 45 mm

Block Railcar wheel 362 Primer on Flat Rolling

temperature of 1000C. The wheel is then taken to the drop forge; the transportation takes 2 min and the rate of cooling is 1C/s. The weight is raised to a height of 10 m and is dropped onto the wheel. What should the height and the diameter of the block be to produce the railcar wheel? Is the 10 m height sufficient to forge the wheel in one drop? The first drop took 15 s. The cooling rate of the wheel, in contact with the cold dies, is 2C/s. If a second drop is necessary, should the wheel be placed back in the furnace for reheating? Would you recommend that the air in the furnace be replaced by argon or helium (both inert gases)? Why? Would you hire the applicant?

Problem 10 A drop hammer is available to produce, by cold forging, flat, low- carbon steel washers of circular shape. The dimensions of the washers are to be 2 mm thickness, 50 mm in diameter. The maximum height from which the 1000 N weight of the drop hammer may be dropped is 4 m. The true stresstrue strain rate curve of the steel, obtained in uniaxial tests, is σ 5 600ε0:25 MPa. The specific heat of the steel is 500 J/kg K and its density is 7800 kg/m3. Determine the initial size of the sample; The initial temperature of the work piece is 22C. What will be the temperature at the end of the compression process? Would it be possible to drop the weight from a height such that the sample would melt? (Tmelt D 1600 C) What would that height be? The original sample was machined from a cold rolled sheet of steel to a right, circular cylinder. Would you expect that the flattened, deformed sample is circular? Why or why not? Sketch the true stresstrue strain curve you would expect to see for the low-carbon steel, obtained in uniaxial tension. How would the curve change if the steel is subjected to 40% cold working before the test?

Problem 11 An extensometer, attached to the sample, is used to monitor its extension in the tension test (see Figure 8.7). A linear variable differential transformer (LVDT), attached to the two compression platens, is used to monitor the deformation of a sample, subjected to compression (see Figure 8.8). How and why would the deformation of either sample be affected by the deformation of the tension tester or the compression tester?

Problem 12 Open die forging of a low-carbon steel cylindrical sample (50 mm diameter, 75 mm high) is being performed on a hydraulic press, at 900C. The compression platens are not heated. The metal’s true stresstrue strain curve, at 900C, is given by σ 5 100 ε0:1 MPa; its density is 7800 kg/m3 and its specific heat is 500 J/kg K. Its ductility at 900C, as measured by the fracture strain, is 45%. The procedure is as follows: Preheat the furnace to 1000C. Hold the sample in the furnace, in air, for 1 h, until its temperature distribution is uniform. Problems and Solutions 363

(A) (B) Figure 16.11 The distortion of an axially symmetrical, cylindrical sample, subjected to compression.

Compression force Figure 16.12 The open die forging process. 20 mm Upper compression platen (moving at a velocity V)

Cylinder to be forged 50 mm

Lower compression platen (stationary) Compression force

Move the sample to the lower compression platen, position it in the centre and wait until it reaches 900C. Reduce its height by 75%, using a compression velocity of 1000 mm/s. Determine The work done during the compression process The temperature rise during the compression process The strain rate at the end of the compression process Whether tensile cracking will limit the 75% compression process. The cylindrical sample is expected to deform as shown in Figure 16.11A. Instead, the sample deformed non-symmetrically, as shown in Figure 16.11B. What possible phenomena may have caused the distortion?

Problem 13 An open-die upsetting operation is shown in Figure 16.12. The cylinder is made of AISI 1015 steel. The upsetting operation is performed at an initial temperature of 800C and at a constant rate of strain of 10 s21. A glassalcohol suspension is used as a lubricant in between the compression platens and the cylinder and it may be assumed that perfect lubrication exists. During the compression process the compression force and the change in height of the sample were monitored. The true stresses and strains were calculated and they are given in the table below. Determine the force of compression and the current height of the sample at a true strain of 0.25. 364 Primer on Flat Rolling

Determine the temperature of the cylinder at a true strain of 0.6. Let the density be 7800 kg/m3 and the specific heat be 500 J/kg K. The ductility of the metal at the temperature of 800C is 0.6. Will the cylinder, compressed to a true strain of 0.8, experience tensile cracking on its outside circumference? Under what conditions will the cylinder remain cylindrical during the compression process?

True Strain True Stress (MPa)

00 0.05 75 0.1 150 0.15 180 0.2 190 0.25 200 0.3 198 0.35 195 0.4 191 0.45 190 0.5 188.84 0.6 188.84 0.7 188.84 0.8 188.84

Problem 15 The force-deformation data, obtained during compression of a super- purity aluminium cylinder, are given below. The original dimensions are 10 mm diameter and 15 mm high (Figure 16.13). Determine The engineering stressengineering strain and the true stresstrue strain curve of the metal; no more than two points in each are required. The constants in the relation σ 5 Kεn where σ is the true stress and ε is the true strain.

20,000 Figure 16.13 The force-deformation data.

16,000

12,000

8000 Force (N)

4000

0 0 2 4 6 8 10 Deformation (mm) Problems and Solutions 365

The temperature rise of the sample at a true strain of 0.6; let the density be 2600 kg/m3 and the specific heat be 900 W/kg K. As the compression is proceeding, what phenomenon would limit the deformation process?

Problem 16 The circumferential engineering strain (defined as the difference of the current and original circumference divided by the original circumference) during hot compression (at 1000C) of a low-carbon steel cylinder has been measured and is given in Figure 16.14 as a function of the force needed for the compression. The initial dimensions of the cylinder are also given in the figure. The compression velocity is constant, at 10 mm/s. The specific heat of the metal is 500 J/kg K and the density is 7830 kg/m3. Determine the true stresstrue strain curve, in the direction of the compression, of the alloy; no more than four pairs of stresses and strains need to be calculated note that care is to be exercised in the choice of these points! What is the height of the sample at the end of the compression process? What is the rate of strain at a circumferential engineering strain of 0.4? A thermocouple, located at mid-height of the sample, indicated a temperature rise of 125 K. What is the temperature rise based on the conversion of the total work/volume? Are the measured and the predicted temperatures different? Why?

Problem 17 Let the stressstrain relation for a metal be σ 5 Yð11BεÞn; determine the average flow strength: ð εmax σ 5 1 σðεÞ ε fm ε d max 0

90,000

80,000

70,000

60,000

50,000

40,000 15 Force (N) 30,000

20,000

10,000 10

0 0.00 0.20 0.40 0.60 0.80 Dimensions are Circumferential engineering strain in mm

Figure 16.14 The force-deformation data and the sample dimensions. 366 Primer on Flat Rolling

Chapter 9 Tribology Problem 1 Two pieces of a flat, strain-hardening metal are in non-lubricated contact under increasing normal pressure, high enough to cause plastic deformation. As well, there is relative, constant velocity motion between the two. What are the mechanisms, parameters and variables that affect the coefficient of friction at the contact surface? How do they affect the coefficient?

Problem 2 Define the friction factor and the coefficient of friction. Under what conditions does the coefficient of friction lose meaning? Which one would you recommend for use in bulk metal forming? Why?

Problem 3 Figure 16.15 shows the true stresstrue strain curves of a NbV steel, obtained at 950C and at a constant true rate of strain of 0.05 s21. Curve #1 was obtained using a cylindrical sample with flat ends and with no lubrication. Curve #3 was obtained after corrections for friction were introduced. Determine the friction factor (m) necessary to get the corrected curve.

Problem 4 Figure 16.16 shows experimental results, obtained while cold rolling a : low-carbon steel, whose true stresstrue strain curve is σ 5 150ð11234 εÞ0 251 MPa. The letters AF refer to different lubricants used during the rolling passes. The roll diameter was 250 mm; the strips’ dimensions were 1 mm thick, 25 mm wide, 300 mm long. The roll separating forces are given as a function of the surface speed of the roll, the lubricant and the reduction. Choose one of the lubricants and one of the reductions. Use Hill’s relation to determineÐ the coefficient of friction for each of σ 5 ð =εÞ ε σ ε 0 5 5 the five speeds. Note that fm 1 0 d and R R 125 mm.

200 Figure 16.15 The stressstrain curves. 190 d = 10 mm 180 0 h 1 170 0 =15mm 160 150 2 140 130 120 3 110

σ 100

(MPa) 90 80 70 1. Specimen with flat ends 60 2. Specimen with recessed ends 3. Correction of curve-1 for friction 50 40 Nb–V steel, 850°C, ε = 0.05 1/s 30 20 10 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ε Problems and Solutions 367

10,000 Figure 16.16 The roll force as a Lubricant function of the speed, the reduction A and the lubricant. 8000 B C D E 6000 Nominal reduction: 50% F

4000 Nominal reduction: 15% Roll force (N/mm)

2000

0 0 500 1000 1500 2000 2500 3000 3500 Roll surface speed (mm/s)

Plot the variation of the coefficient of friction as a function of the speed for the two reductions.

Problem 5 A drop hammer in which a large weight is dropped on a sample is used to compress cylindrical steel samples. Two tests are performed. In both, the weight is dropped from the same height, first on a completely dry sample and then on a sample, covered with highly viscous lubricant. Assume that in the second test friction was eliminated completely. Which of the samples will be compressed more? Why?

Problem 6 The viscositypressure relationships indicate that as the pressure increases, the viscosity increases. In the boundary and in the mixed lubrication regimes, the Stribeck diagram indicates that the coefficient of friction decreases as the viscosity increases. When rolling soft aluminium without lubrication the coefficient of friction was found to increase with reduction. The opposite was found during rolling of hard steel. How would you explain the apparent contradiction?

Problem 7 In a lubricated contact the coefficient of friction is expected to decrease as the load is increased. What is the most important mechanism that contributes to this phenomenon?

Problem 8 Give the definition of the true area of contact.

Problem 9 How do the lubrication regimes affect the resulting surface roughness of the product?

Problem 10 Lubricating oils lower the coefficient of friction. Would water do the same? 368 Primer on Flat Rolling

900 Reduction = 20.0% Roll speed = 4.0rpm Strain rate = 0.38/s

800 C) ° 50.8 700 2.5 thermocouple #2 thermocouple #4 Thermocouple #1 19.0 Thermocouple #3 2.8 5.1 Thermocouple #2 thermocouple #1 600 Thermocouple #4 thermocouple #3 Temperature (

500 0 1 2 3 4 5 6 7 8 Time (s)

Figure 16.17 Temperaturetime data.

Problem 11 Why does increasing viscosity lead to lower friction in the mixed lubrication regime but higher friction in the hydrodynamic regime?

Problem 12 A steel strip is rolled, successively in two passes. The reduction in both passes is 50%. All process parameters are kept the same. Would you expect the coefficient of friction to remain the same in the two passes? If not, how would it change and why?

Problem 13 A layer of scale is an insulator. Would that increase or decrease the coefficient of heat transfer?

Problem 14 Figure 16.17 shows the temperature distribution obtained during hot rolling of a low-carbon steel slab. The locations of the measuring thermocouples are also shown. Assuming that the roll temperature is 40C, determine the coefficient of heat transfer at the roll-slab contact surface.

Chapter 10 Applications and Sensitivity Studies Reproduce the figures in the chapter.

Chapter 12 Temper Rolling Class discussion: What are the differences between temper rolling and conventional flat, cold rolling? Why is the coefficient of friction high in temper rolling? Problems and Solutions 369

Chapter 13 Severe Plastic Deformation Accumulative Roll Bonding Class discussion: What are the advantages and disadvantages of the accumulative roll-bonding process?

Chapter 14 Roll Bonding Class discussion: Discuss the paper of Gilbreath (1967).

Chapter 15 Flexible Rolling Class discussion: Would tailor-rolled blanks replace tailor-welded blanks in the automotive industry?

Part 2 Solutions

Chapter 1 Introduction 1. Integrate the expressions for the strains, realizing that they are the partial derivatives of the displacements therefore integration functions will result after the integration. Using the boundary conditions that u 5 0atx 5 0 and v 5 0 at all y, the integration functions are shown to vanish. 27 26 At point D, therefore, uD 5 2.25 3 10 m and vD 5 9.187 3 10 m. The shear strain at point C is obtained by using its original definition in terms of the sum of partial derivatives of the displacements; the shear strain is then obtained as 1.75 3 1024 radians. The stress components at point C are obtained using the two-dimensional version of Hooke’s law as σx 5 12:65 MPa, σy 5 40 MPa and τxy 5 13:93 MPa. 2. Consider a cube. For no volume change the ratio of the original and the current volumes equals unity. The ratio can also be expressed as ratio of the product of the three sides of the undeformed cube to the product of the three sides of the deformed cube which will, of course, also equal unity. Take natural logarithm of both sides; on the left side one gets zero while on the right side the sum of the three logarithmic strains appears. Hence, the vanishing of the trace of the strain tensor (i.e. the sum of the three logarithmic strains) implies constancy of volume. 3. According to the maximum shear stress criterion the largest principal stress will be in the circumferential direction and the smallest in the axial direction (perpendicular to the plane of the disk). So yielding will begin at r 5 a (the inside radius) at a rotational speed when the stress in the circumferential direction equals the yield strength, at a speed of 1720 r/s. 4. In the tube eversion process the largest principal stress will be in the circumferential direction and the smallest in the longitudinal direction. The intermediate principal direction therefore is perpendicular to the wall of the tube. According to the flow rule, associated with the maximum shear stress criterion of Tresca, the strain rate in the intermediate principal direction vanishes; the strain in that direction will therefore be constant and the wall thickness will remain unchanged. (Note: this answer may be arrived at by considering that while the tensile stress in the circumferential direction causes thinning of the wall, the compressive stress in the longitudinal direction causes thickening. The two processes therefore may cancel one another.) 370 Primer on Flat Rolling Ð 5 σ 5 τ 5 π R σ 5 56 = _ 5 52 = 5. At r R: r 0; rz 0 at all y;2 0 zr dr Faty L 2 and w 0aty L 2 and w_ 5 v at y 5 L=2. 6. εxz 5 εyz 5 0; all others components are non-zero. 7. Conduction is heat transfer by means of molecular agitation within a material without any motion of the material as a whole. Convection is the flow of heat through a bulk, macroscopic movement of matter from a hot region to a cool region. Radiation is heat transfer by the emission of electromagnetic waves which carry energy away from the emitting object. 8. No; the volume changes under elastic loading, controlled by the bulk modulus. 9. 17.4 mm 10. Check the website of the World Bank for the necessary information (worldbank.org). 11. Find the information from Statistics Canada or from the website of the American Iron and Steel Institute. 12. Use Google or any search engine to find addresses for used companies that sell used rolling mills. 13. A three-component flexible coupling with constant input speed will ensure constant output speed.

Chapter 4 Flat Rolling 1.

Roll separating force

Roll torque

Shear stress of the work roll

Pressure of the work roll

μ 5 = φ 5 φ μ 5 φ = φ 5 φ 2. F P; for entry: F cos 1 P sin 1;so sin 1 cos 1 tan 1;

φ P sin 1 φ 1 P F φ F cos 1

3. See Figure 4.4, Chapter 4. Problems and Solutions 371

4. The not-very-stiff work roll would bend under the forces it experiences during the pass, creating a strip which is thicker in the centre than at the sides; hence, the edges would move faster than the centre. For complete flattening, the radius would decrease by 0.01 mm. The radial strain is then estimated to be 0.01/200 5 0.00005, much less than necessary to cause plastic flow. Consider the work roll to be a simply supported beam, loaded by a uniformly distributed force along the 1500 mm length. The force is obtained as 714,887 N, distributed uniformly over the 1500 mm. Using mass conservation, the exit velocity would be 10,000 m/s. The roll’s surface speed is 8377.58 mm/s. The projected contact length is 16.73 mm so the time of contact is 0.001998 s. (The result is reached using the assumptions of rigid rolls and that the strip’s speed equals the roll’s surface speed.) The plane-strain flow condition indicates that the strain in the direction perpendicular to the direction of rolling, ε3, (the width direction) vanishes. According to the flow rule associated with the HuberMises flow criterion, then

σ 1 σ ε 5 0 5 σ 2 1 2 3 3 2

While the work roll’s radius of curvature would increase, an indentation is not likely to occur. The leading edge of the strip would thicken somewhat.

Chapter 5 Mathematical Modelling pffiffiffiffiffiffi 5 02 0 2 1 Δ 2 1. The projected length of contact, from the geometry of the pass, is L R R 2 h as long as Δh2=4 is much less than unity, true in strip rolling; the relationship is valid. 2. See Section 5.10 for a discussion. 3. Consider the frames of both the laboratory and the industrial mills to consist of four columns, subjected to a force along their lengths. For the laboratory mill, take the load- carrying area of one of the columns to be 100 mm 3 100 mm, 800 mm long and the force to be 400,000 N. The stretch would then be 0.04 mm. For the industrial mill, the load-carrying area is to be 1 m2; the length to be 4 m. Let the roll force be 60 3 106 N. The extension is then 0.3 mm. 4. Using mass conservation, the exit velocity from the seventh stand will be 7490.6 mm/s. 5. Derivations Equation 5.18: Refer to Figure 5.2. After some simplifications, the balance of the forces on the slab leads to

0 0 ðdσxhÞ 7 2τ R cos φ dφ 1 2pR sin φ dφ 5 0

Note that from geometry cos φ 5 dx=R0 dφ and sin φ 5 dh=2R0 dφ; Use τ 5 μp to obtain the required relation. Equation 5.19: The HuberMises flow criterion is written in terms of the principal 2 2 2 2 stresses as ðσ1 2σ2Þ 1 ðσ2 2σ3Þ 1 ðσ1 2σ3Þ 5 6k ;letσ1 5 σx; σ2 2 p; σ3 5 ðσx 2 pÞ=2; substitute; simplify to get the required equation. Equation 5.20: Refer to Figure 5.2. The required equation is obtained immediately. The simplified version is valid as long as the angles are much smaller than unity. 372 Primer on Flat Rolling

Equation 5.32: Take Eq. (5.18) as the starting point. Examine the conditions of equilibrium in the direction perpendicular to the rolled metal, at the roll/strip interface to get σy 5 p 7 τ tan φ; use the yield criterion to get σy 2 σx 5 2k; hence σx 5 p 2 2k 7 τ tan φ; substitute this into Eq. (3.18); recognize that tan φ 5 dy=dx to get the required equation. Equation 5.48: See Section 5.5.2 for the detailed derivation. 6. Hot rolling: Use Schey’s approach. Check if there is sufficient motor power to reduce the strip from 10 to 2 mm in one pass. The flow strength is obtained as 68.2 MPa, at a strain rate of 36.49 s21. The contact length is 34.64 mm; the pressure intensification factor is 2.4; so the roll force is 163,000 N and the required power is 29.6 kW; one hot rolling pass is sufficient, but just barely. Cold rolling: Repeat the above steps; the flow strength is now 446.6 MPa; the contact length is 14.79 mm; the factor is 2 and the power needed is 58.8 kW so again, one pass is sufficient. The reductions are then 80% in the hot rolling pass; 87.5% in the cold rolling pass. (Note: tensile cracking may be present at such high reductions; the ductility of each metal should be checked to see if the reductions need to be reduced.) The lubricant would reduce the power requirements. Increasing the speed would increase the strain rate and the metal’s resistance to deformation; at the same time more oil would be drawn into the contact; the two contradictory events may cancel one another; Yes; lower power consumption would result. 7. Refer to Figure 5.2. Add the force of inertia, mass 3 acceleration, to the free-body diagram of the slab. Balance of the forces in the direction of rolling will result in Eq. (5.45). It is now necessary to express the mass and the acceleration in terms of the independent variables so the result can then be integrated. Mass conservation requires that dðvhÞ 5 0 so the acceleration is then a 5 dv=dt 52ðv=hÞdh=dt. The time rate of change of the strip’s thickness is obtained from the simplified version of Eq. (5.20), as dh=dt 5 2xv=R0 where the velocity of the strip is v 5 vrðhnp=hÞ. The mass of the slab, per unit width is m 5 ρ dxh. Use the HuberMises criterion to write: σx 5 2k 2 p; substitute into Eq. (5.45); the required Eq. (5.46) will result. See Section 5.5.2 for the numerical work. 8. Invert the equation of equilibrium so the coefficient of friction is on one side and all other terms are on the other side. All of these terms can be obtained numerically from the figure.

Angle Pressure h (mm) dp/dhk/h μ

0.012567 45.65 4.979741 315.8733 5.020341 0.42614 0.016756 50.5 4.995095 2 103.844 5.004909 2 0.1804 0.020945 48.45 5.014837 2 78.7463 4.985207 2 0.18152 0.025134 46.55 5.038965 2 61.3711 4.961337 2 0.18047 0.029323 44.8 5.06748 2 48.6293 4.933419 2 0.17766 0.033512 43.2 5.100382 0 4.901594 2 0.01939 0.037701 43.2 5.137671 2 2.39947 4.866018 2 0.03258 0.04189 43.1 5.179347 20.62403 4.826864 0.079522 0.046079 44.05 5.225409 58.47409 4.784314 0.293474 0.050268 47 5.275859 60.17875 4.738565 0.312832 (Continued) Problems and Solutions 373

(Continued) Angle Pressure h (mm) dp/dhk/h μ 0.054457 50.3 5.330696 57.40959 4.689819 0.304259 0.058646 53.7 5.389919 43.23186 4.638288 0.227175 0.062835 56.45 5.45353 16.91241 4.584187 0.074837 0.067024 57.6 5.521527 37.99164 4.527733 0.215003 0.071213 60.35 5.593911 13.67699 4.469145 0.060779 0.075402 61.4 5.670683 14.78593 4.40864 0.072266 0.079591 62.6 5.751841 18.11909 4.346435 0.10072 0.08378 64.15 5.837386 23.90693 4.282739 0.149608 0.087969 66.3 5.927318 25.97568 4.217759 0.171116 0.092158 68.75 6.021637 12.66388 4.151695 0.068709 0.096347 70 6.120343 2 9.21499 4.084738 2 0.11204 0.100536 69.05 6.223436 16.09516 4.017074 0.064136 0.078932

9. In hot rolling, for a reduction of 60% at a speed of 45 rpm, the flow strength is 139.5 MPa; the roll force is 373,576 N and the torque is 7247 Nm. The motor power to reduce the strip is 34.15 kW. In cold rolling, the required power is 54 kW; since only one mill is being designed, use the larger power. The diameter of one of the spindles is then 68 mm; the load-carrying area is 0.0031 m2; and the motor power is 57 kW. Note that no allowance was made either for friction losses or for a generous factor of safety. 10. Start with Eq. (5.20). The strain is ε 5 lnðhentry=hÞ; differentiate with respect to time to get the required relation. 11. Read the force and the torque data off the figure; Pr 6600 N=mmand M 41 Nm=mm. Using the given data, the force is calculated to be 7080 N/mm and the torque is found as 56 Nm/mm. The differences are most likely due to a too high value of the coefficient of friction; Using the measures data and inverse approach, the coefficient of friction is estimated to be 0.08. Note that Figure 5.1 allows only approximations, not exact values; The roll force will drop with the speed as more oil is drawn into the contact zone; In the second pass the flow strength becomes 570 MPa and the roll force is 5110 N/mm. 12. Pass #1: The roll force is 853,512 N; the power is 70.7 kW , 200 kW; OK. Pass #2: The flow strength is now 952 MPa; the roll force is 1.38 3 106 N; the required power is 229 kW; the available motor is not powerful enough. Remedy: Only the installation of a more powerful motor will solve the problem; extra lubricant would help but it would not be sufficient. 13. Pass #1: The roll force is 208,300 N so the power needed is 24.4 kW; Pass #2: The power needed now is 27.4 kW Temperature rise: Use Eq. (5.12) to determine the temperature rise. In the first pass the rise is obtained as 154.9C and in the second, 173.9C. The strip temperature will then be 350.8C.The width will change somewhat but not in any significant manner; the plane-strain flow assumption is quite close to reality. Higher-viscosity lubricant will likely lower the loads on the mill. 14. Design the schedule for three passes. Pass #1: Reduce the strip from 8 to 4 mm. The flow strength is then 438 MPa; the roll force is 337,800 N and the power is 79 kW , 150 kW; OK. 374 Primer on Flat Rolling

Pass #2: Another 50% reduction; account for the effect of cumulative strain hardening so the flow strength is now 603.8 MPa; the force is 356,700 N and the power is 106.3 kW; Pass #3: 50% reduction; the strength is 687.5 MPa; the power is 86.9 kW. All three passes are possible. In order to check the lubricant for possible breakdown, the temperature rise in the three passes needs to be calculated. As suggested, assume that there is no heat loss in between the passes. Use Eq. (5.12) to get ΔTpass1 5 102:9 C; ΔTpass2 5 153:6 C so the lubricant will break down in the second pass. 15. Inverse calculations are needed. Pass #1: The torque is given so the roll force can be obtained as 5477 N/mm (use Eq. (5.6)). The only unknown in Eq. (5.1) is now the pressure intensification factor; it is obtained as 1.42; so the coefficient of friction is then 0.15. Pass #2: Similarly, using the idea of cumulative strain hardening, the coefficient of friction is obtained as 0.04; note that this is below the minimum necessary coefficient to initiate bite, so in the second pass the strip needs some front pull to assist entry. The rise of temperature in the second pass is calculated as 214C. The motor power is 104 kW, twice what was used to calculate the temperature rise of the strip, needed to overcome friction losses in the bearings.

Chapter 8 Material Attributes 1. The metallurgical phenomena are time dependent. To have sufficient time available for the hardening and softening rates to reach equilibrium as the rate of strain is increased, a larger strain hence, a little more time is needed to get to the peak stress, corresponding to the peak strain. 2. Read the stress values at a strain of 0.4, at various strain rates. Prepare a lnln plot of the stresses at that strain versus the strain rates. The slope of the line is the strain rate hardening exponent, obtained as 0.16. ε 5 ð = Þ ε 5 ð = Þ 3. The strain in slow forging is s ÀÁln 15 hs and in fast forging f ln 15 hf . The stress σ 5 = 5 ð = Þ 0:2 5 ðπ= Þ 2 3 = in slow forging is s F As 90 ln 15 hs ÀÁwhere As 4 10 15 hs 0:2 0:1 2 The stress in fast forging is σf 5 F=Af 5 90 lnð15=hfÞ 100 where Af 5 ðπ=4Þ10 3 15=hf The force is the same in both cases, at 20,000 N. Hence, in slow forging hs 5 5.334 mm and in fast forging the final height is 7.8 mm. The slow forging creates higher reduction. 4. The relation σ 5 Yð11BεÞn represents the behaviour of the metal better, since it indicates that at a strain of zero, the metal possesses some strength. The same argument holds for either steel or aluminium. 5. No. Dynamic recrystallization would occur sooner on the outer fibres since the stresses and strains there are the highest. No; static recrystallization would not occur since the sample is loaded dynamically (this is intended as a trick question). 6. The procedure is outlined below:

G perform a number of stressstrain tests at several temperatures and rates of strain; G obtain the peak stresses and prepare a loglog plot of the peak stresses versus the temperatures; G at an arbitrary stress level obtain from the plot two temperatures and the corresponding rates of strain; and G determine the activation energy from the slope Q Δðln ε_Þ=Δð21=RTÞ Problems and Solutions 375

Non-linear regression analysis is used to obtain the relationships of the strain rate in terms of the peak stress, using the lines from Figure 5.16. For 950C

ε_ 5 expbðln σp 2 5:244Þ=0:1178c and at 900C

ε_ 5 expbðln σp 2 5:4006Þ=0:10411c The average activation energy for dynamic recrystallization is then obtained as 483 kJ/mole. 7. The limit of deformation is decided by the circumferential strain which cannot exceed 0.5. The diameter after the drop is then obtained from εθ 5 ðπdf 2 28πÞ=28π 5 0:5 so the maximum diameter will be 42 mm. The sample’s height after the drop is obtained from mass conservation as 11.11 mm. The work done per volume is the integral of the stressstrain curve over the strain ðε 5 lnð25=11:11Þ 5 0:811Þ is 455.26 MPa. The potential energy/volume at the start of the drop must match that magnitude; so the height is 7 m (excluding friction losses). The rise of the temperature is 138C. Since the strain distribution will not be uniform within the sample, the temperature distribution will also not be uniform. The centre is expected to be warmer than the surfaces. 8. The available energy is 1018.6 MPa; the required work done is 1359.6 MPa so the part cannot be produced in one drop. The height for the second drop: first determine the height from the first drop from 1018:6 3 106 5 ðε1:25=1:25Þ so the height is 16.1 mm. In the second drop this is to be reduced to 10 mm. Since the sample was annealed, the original stressstrain curve may be used. Equate the potential energy to the required work/volume:

: 20; 000 H 600 3 106 16:1 1 25 5 ln 0:000196 1:25 10 so the height is H 5 1.86 m. 9. The diameter and the height of the block: There are two criteria: the volume remains constant; tensile cracking is to be avoided. Constant volume indicates that 0:06927 m3 5 ðπ=4ÞD2H the diameter is obtained from the limiting circumferential strain: 0.6 5 (1.4 2 D)/D; D 5 0.875 m; the height is then 0.115 m. Check if the 10 m drop height is sufficient: The available potential energy/volume is 10 3 10,000/0.06927 5 1.4436 3 106 MPa; the required work/unit volume is the integral of the stressstrain curve over the strain of 0.938; it is obtained as 101.7 3 106 MPa; the wheel can be forged in one drop. The temperature drop during transportation is 120C; in the forge, it is 30C. The wheel must be reheated for the second drop. No need for protective atmosphere as the layer of scale will crack during the forging process. 10. The available energy is 4000 Nm. The final volume of the sample is 3.927 3 106 m3. Equate the available energy/volume to the required energy in terms of the strain; from the strain the original height is obtained as 12.41 mm. 376 Primer on Flat Rolling

The temperature rise will be 261C; yes, in theory it is possible to drop the height so the sample will melt; in practice this may not be achievable, though. Taking the melting temperature to be 1600C, dropping the weight from a height of 23.2 m may cause melting. 11. The extensometer in the tension test is attached directly to the sample so the deformation of the machine will have no effect on its readings. The LVDT in the compression test is attached to the compression platens so the deformation of the machine will have a significant effect on its readings.

Chapter 9 Tribology 1. The parameters that will affect the coefficient of friction are Pressure Surface roughness Surface hardness Relative velocity 2. The coefficient of friction is the ratio of the tangential and the normal forces. The friction factor is the ratio of the friction stress and the yield strength of the interface. The coefficient of friction loses meaning when the interfacial pressures are several times the strength of the material. The friction factor is recommended for bulk forming since its meaning doesn’t depend on the pressure. 3. Read the data off the curves:

Strain p#1 p#3 5 σf hdm

0.2 128 125 12.28096 16.05406 0.138549 0.4 132 128 10.0548 12.21662 0.133645 0.6 128 120 8.232175 13.50145 0.211215 0.8 126 116 6.739934 14.92141 0.202334 1 131 115 5.518192 16.49071 0.241914 1.2 145 115 4.517913 18.22506 0.336027

mave 5 0.210614

4. Hill’s equation is given in the text; see Eq. (9.26). The results below are obtained for 50% reduction and lubricant F. The flow strength is 432 MPa. The calculations are given in the table:

0 hentry hexit Speed Pr Strain Sigma Average R 5 Coefficient (mm) (mm) (mm/s) (N/mm) (MPa) 125 mm

1 0.5 260 7300 0.693147181 432.6655243 125 0.1563177 1 0.5 500 7000 0.693147181 432.6655243 125 0.1475527 1 0.5 1300 6700 0.693147181 432.6655243 125 0.1387877 1 0.5 1850 6500 0.693147181 432.6655243 125 0.1329444 1 0.5 2300 6050 0.693147181 432.6655243 125 0.1197969 Problems and Solutions 377

5. The sample will be compressed more in the test with the lubricant as less of the available energy is spent on overcoming friction losses. 6. While rolling soft aluminium, the strip’s asperities flattened more easily than those of the steel, creating more sites for adhesive bonds to form, thus increasing the effort needed to cause relative motion. 7. Increasing loads cause the viscosity to increase. In the boundary and the mixed lubrication regimes this will lower the coefficient of friction. 8. The true area of contact is the sum of all areas in contact at the asperity tips. 9. In the boundary and the mixed regimes the resulting surfaces will be smoother and will reflect the surface roughness of the rolls. In the hydrodynamic regime the strip will roughen as a result of the free deformation at the surface. 10. Yes; even though its viscosity is very low. In the hydrodynamic regime there is complete separation between the strip and the roll. Relative motion in this regime means that the oil film is being sheared. The force needed to shear a more viscous lubricant is higher than for one of lower viscosity. In the second pass the steel has hardened due to strain hardening. Its asperities will be somewhat harder to flatten so the creation of adhesive bonds will be harder; the coefficient of friction will be lower than in the first pass. 11. The coefficient of heat transfer would decrease. Appendix

List 1. Early USA Hot Strip Mills (Throughputs in Short Tons)

1 American Rolling Mill Co., Ashland, KY, 1924, 58”, 484,000 TPY 2 American Rolling Mill Co., Butler, PA, 1926, 48”, 504,000 TPY 3 Republic Steel, Warren, OH, 1927, 42”, 435,000 TPY 4 Weirton Steel, Weirton, WV, 1927, 66”, 830 PIW, 720,000 TPY 5 Carnegie Illinois Steel, Gary, IN, 1927, 42”, 493,000 TPY 6 American Rolling Mill Co., Middletown, OH, 1928, 80”, 500 PIW, 678,000 TPY 7 Wheeling Steel, Steubenville, OH, 1929, 66”, 941,000 TPY 8 Great Lakes Steel, Ecorse, MI, 1930, 38”, 599,000 TPY 9 Carnegie Illinois Steel, South Chicago, IL, 1931, 96”, 806,000 TPY 10 Otis Steel, Cleveland, OH, 1932, 77”, 420,000 TPY 11 Inland Steel, Indiana Harbor, IN, 1932, 76”, 560 PIW, 672,000 TPY 12 Allegheny Ludlum, Brackenridge, PA, 1932, 36”, 550 PIW, 308,000 TPY 13 Youngstown Sheet & Tube, Campbell, OH, 1935, 79”, 650 PIW, 840,000 TPY 14 Carnegie Illinois Steel, Gary, IN, 1935, 38”, 326,000 TPY 15 Ford Motor Co., Dearborn MI, 1935, 66”, 325 PIW, 393,000 TPY 16 Carnegie Illinois Steel, McDonald, OH, 1935, 43”, 470,000 TPY 17 Bethlehem Steel, Lackawanna, NY, 1935, 79”, 400 PIW, 806,000 TPY 18 Carnegie Illinois Steel, Gary, IN, 1936, 80”, 450 PIW, 717,000 TPY 19 Great Lakes Steel, Ecorse, MI, 1936, 96”, 941,000 TPY 20 Granite City Steel, Granite City, IL, 1936, 90”, 440 PIW, 470,000 TPY 21 Carnegie Illinois Steel, Homestead, PA, 1936, 100”, 806,000 TPY 22 Jones and Laughlin Steel, Pittsburgh, PA, 1937, 96”, 500 PIW, 806,000 TPY 23 Republic Steel, Cleveland, OH, 1937, 98”, 901,000 TPY 24 Bethlehem Steel, Sparrows Point, MD, 1937, 56”, 570 PIW, 672,000 TPY 25 Tennessee Coal & Iron, Fairfield, AL, 1937, 48”, 504,000 TPY 26 Inland Steel, Indiana Harbor, IN, 1937, 44”, 350 PIW, 560,000 TPY 27 Carnegie Illinois Steel, Dravosburg, PA, 1938, 80”, 672,000 TPY 28 Youngstown Sheet & Tube, Indiana Harbor, IN, 1939, 54”, 400 PIW, 672,000 TPY

Total Annual Capacity: 17,616,000 TPY

List 2. Some of the Early Steckel Mills 1 Dominion Foundry & Steel, Hamilton, ON, Canada, 1935, 36”, 4 high 2 McLouth Steel, Detroit, MI, 1936, 20”, 2 high 380 Primer on Flat Rolling

3 Crucible Steel, Midland, PA, 1949, 66”, 4 high 4 Newport Steel, Newport, KY, 1949, 66”, 4 high 5 McLouth Steel, Trenton, MI, 1949, 42”, 4 high 6 AM Byers, Ambridge, PA, 1949, 38”, 2 high 7 Arbed, Dudelange, Luxembourg, 1950, 66”, 4 high 8 Com Sid Beigo Maniera, Brazil, 1951, 40”, 2 high 9 Fagersta Bruks A.G., Fagersta, Sweden, 1951, 29”, 4 high 10 Altos Hornos Mexico, Monclova, Mexico, 1952, 48”, 4 high 11 Lone Star Steel, Dangerfield, TX, 1953, 72”, 4 high 12 Hojalata Y. Lamina, Monterrey, Mexico, 1953, 48”, 4 high 13 Dominion Foundry and Steel, Hamilton, ON, 1954, 60”, 4 high 14 Eregli Iron & Steel, Eregli, Turkey, 1962, 68”, 4 high 15 Iligan Steel, Iligan City, Philippines, 1964, 68”, 4 high

List 3. USA Generation I Hot Strip Mills (Throughputs in Short Tons)

1 Weirton Steel, Weirton, WV, 1927 (1955), 54”, 830 PIW, 3,100,000 TPY 2 Allegheny Ludlum, Brackenridge, PA, 1932 (1952), 56”, 550 PIW, 600,000 TPY 3 Inland Steel, East Chicago, IN, 1932 (1958), 76”, 560 PIW, 1,700,000 TPY 4 Bethlehem Steel, Lackawanna, NY, 1935 (1963), 79”, 400 PIW, 2,520,000 TPY 5 Ford Motor, Dearborn, MI, 1935 (1961), 66”, 325 PIW, 1,985,000 TPY 6 United States Steel, McDonald, OH, 1935 (1966), 43”, 1,326,000 TPY 7 Youngstown S & T, Campbell, OH, 1935 (1961), 79”, 650 PIW, 2,477,000 TPY 8 Great Lakes Steel, Ecorse, MI, 1936 (1967), 96”, 2,400,000 TPY 9 United States Steel, Gary, IN, 1936 (1948), 80”, 450 PIW, 3,084,000 TPY 10 Bethlehem Steel, Sparrows Point, MD, 1937 (1960), 56”, 570 PIW, 2,700,000 TPY 11 Inland Steel, East Chicago, IN, 1937 (1951), 44”, 350 PIW, 2,000,000 TPY 12 Jones & Laughlin, Pittsburgh, PA, 1937 (1964), 98”, 500 PIW, 1,500,000 TPY 13 Republic Steel, Cleveland, OH, 1937 (1957), 98”, 2,184,000 TPY 14 United States Steel, Fairfield, AL, 1937 (1965), 68”, 1,622,000 TPY 15 United States Steel, Dravosburg, PA, 1938 (1963), 80”, 3,076,000 TPY 16 Youngstown S & T, East Chicago, IN, 1939 (1962), 54”, 400 PIW, 2,028,000 TPY 17 United States Steel Geneva, Utah, 1944 (1947), 132”, 450 PIW, 2,062,000 TPY 18 Bethlehem Steel, Sparrows Point, MD, 1948 (1965), 68”, 550 PIW, 3,120,000 TPY 19 Kaiser Steel, Fontana, CA, 1950 (1958), 86”, 525 PIW, 1,670,000 TPY 20 Detroit Steel, Portsmouth, OH, 1952 (1965), 56”, 1,200,000 TPY 21 Armco Steel, Ashland, KY, 1953 (1965), 80”, 400 PIW, 1,700,000 TPY 22 Pittsburgh Steel, Allenport, PA, 1953 (1964), 66”, 500 PIW, 900,000 TPY 23 United States Steel, Fairless Hills, PA, 1953 (1964), 80”, 600 PIW, 2,800,000 TPY 24 McLouth Steel, Trenton, MI, 1954 (1967), 60”, 670 PIW, 1,800,000 TPY 25 Armco Steel, Butler, PA, 1957, 58”, 500 PIW, 915,000 TPY 26 Cyclops Steel, Mansfield, OH, 1957 (1960), 56”, 400,000 TPY 27 Jones & Laughlin, Aliquippa, PA, 1937 (1957), 44”, 630 PIW, 1,600,000 TPY 28 Republic Steel, Gadsden, AL, 1957 (1965), 54”, 1,235,000 TPY 29 Crucible Steel Midland, PA, 1961, 56”, 400 PIW, 425,000 TPY 30 Sharon Steel, Farrell, PA, (1965), 60”, 1,680,000 TPY Appendix 381

Total Annual Capacity: 55,809,000 TPY

List 4. USA Generation II Hot Strip Mills (Throughputs in Short Tons) 1 Republic Steel, Warren, OH, 1961, 58”, 940 PIW, 2,000,000 TPY 2 Great Lakes Steel, Ecorse, MI, 1961, 80”, 1000 PIW, 3,600,000 TPY 3 J & L Steel, Cleveland, OH, 1964, 80”, 1000 PIW, 2,400,000 TPY 4 Wheeling Pgh Steel, Mingo Junction, OH, 1965, 80”, 1000 PIW, 2,400,000 TPY 5 Inland Steel, East Chicago, IN, 1965, 80”, 1100 PIW, 2,800,000 TPY 6 Bethlehem Steel, Burns Harbor, IN, 1966, 80”, 1200 PIW, 3,300,000 TPY 7 Granite City Steel, Granite City, IL, 1967, 80”, 1100 PIW, 2,400,000 TPY 8 US Steel, Gary, IN, 1967, 84”, 1220 PIW, 3,500,000 TPY 9 Armco Steel, Middletown, OH, 1968, 86”, 1100 PIW, 3,500,000 TPY 10 Youngstown S & T, East Chicago, IN, 1968, 84”, 1250 PIW, 3,800,000 TPY 11 Republic Steel, Cleveland, OH, 1970, 84”, 1250 PIW, 2,600,000 TPY 12 Ford Motor Co., Dearborn, MI, 1974, 68”, 1040 PIW, 2,400,000 TPY

Total Annual Capacity: 34,700,000 TPY

List 5. Other Generation I and II Hot Strip Mills 1 Sumitomo, Japan, 1961, 2032 mm (80”), 900 PIW, Generation II 2 Tokai, Japan, 1961, 1727 mm (68”), 900 PIW, Generation II 3 Usinas Gustave Boel SA, Belgium, 1961, 1829 mm (72”), 1000 PIW, Generation I (II) 4 Kawasaki, Japan, 1962, 2032 mm (80”), 900 PIW, Generation II 5 Yawata, Japan, 1962, 1422 mm (56”), 900 PIW, Generation II 6 RTB, Great Britain, 1962, 1727 mm (68”), 1000 PIW, Generation II 7 Usinor Dunkirk, France, 1962, 2032 mm (80”), 1000 PIW, Generation II 8 Algoma, Sault Ste. Marie, ON, Canada, 1963, 2692 mm (106”), 1000 PIW, Generation II 9 Altos Hornos, Spain, 1963, 1676 mm (66”), 670 PIW, Generation I 10 Cosider, Italy, 1963, 1727 mm (68”), 1000 PIW, Generation II 11 Salzgitter, Germany, 1963, 2083 mm (82”), 1000 PIW, Generation II 12 August Thyssen Hutte Germany, 1964, 2235 mm (88”), 1000 PIW, Generation II 13 Nissan, Kure, Japan, 1965, 1524 mm (60”), 660 PIW, Generation I 14 Hoogovens, Holland, 1966, 2235 mm (88”), 1180 PIW, Generation II 15 NKK Fukuyama, Japan, 1966, 2032 mm (80”), 1000 PIW, Generation II 16 British Steel, Great Britain, 1967, 2032 mm (80”), Generation II 17 Sidmar, Belgium, 1967, 2032 mm (80”), 980 PIW, Generation II 18 Sumitomo Kashima, Japan, 1967, 1778 mm (70”), 1300 PIW, Generation II 19 Kawasaki Mizushima, Japan, 1968, 2286 mm (90”), 1600 PIW, Generation II Super 20 Skopje, Yugoslavia, 1968, 1727 mm (68”), Generation I 21 Yawata Kimizu, Kimizu, Japan, 1969, 2286 mm (90”), (2000) PIW, Gen II Super 22 Russia, 1969, 2032 mm (80”), 1150 PIW, Generation II 23 Kloeckner, Germany, 1970, 2286 mm (90”), 1280 PIW, Generation II 24 Altos Hornos, Mexico, 1971, 1727 mm (68”), 722 PIW, Generation I 25 Fuji Oita, Japan, 1971, 2248 mm (88.5”), 1600 PIW, Generation II Super 26 Kobe Steel Shinko Japan, 1971, 2184 mm (86”), 1300 PIW, Generation II 382 Primer on Flat Rolling

27 NKK Fukuyama, Japan, 1971, 1778 mm (70”), 1200 PIW, Generation II 28 Rautarukki, Finland, 1971, 2032 mm (80”), 985 PIW, Generation II 29 CAP, Chile, 1972, 1727 mm (68”), 320 PIW, Generation I 30 Italsider, Italy, 1972, 2286 mm (90”), 1350 PIW, Generation II 31 Posco, South Korea, 1972, 1422 mm (56”), 690 PIW, Generation I 32 Solmer, France, 1973, 2286 mm (90”), 1506 PIW, Generation II Super 33 Iscor, South Africa, 1974, 2032 mm (80”), 1100 PIW, Generation II 34 Russia, 1975, 2000 mm (79”), 1090 PIW, Generation II 35 Smederevo, Yugoslavia, 1976, 2235 mm (88”), 525 PIW, Generation I (II) 36 Erdemir, Turkey, 1976, 1676 mm (66”), 1000 PIW, Generation II 37 Benxi, China, 1977, 1702 mm (67”), Generation II 38 Nippon Kokan Ohgishima, Japan, 1979, 2400 mm (94”), 975 PIW, Generation II 39 Companhia Siderurgica, Brazil, 1979, 1730 mm (68”), 1430 PIW, Generation II

List 6. Worldwide Coil Box Hot Strip Mills

1 1 John Lysaght, Western Port, Australia, 1978, 2032 mm (80”), new mill 2 Stelco (US Steel), Hamilton, ON, Canada, 1980, 1422 mm (56”), existing mill 3 Svenkst Stal SSAB, Sweden, 1981, 1900 mm (75”) 4 Algoma (Essar Steel), Sault Ste. Marie, ON, Canada, 1982, existing mill 5 Thyssen Krupp Bochun, Germany, 1982 6 Boel (Duferco La Louviere SA), Belgium, 1983 7 Stelco Lake Erie (US Steel), Nanticoke, ON, Canada, 1983, 2032 mm (80”), new mill 8 BSC (Tata Corus), Port Talbot, Wales, UK, 1985, 2032 mm (80”), existing mill 9 BHP (Bluescope), Port Kembla Australia, 1987, existing mill 10 New Zealand Steel (Bluescope), New Zealand, 1988 11 Dunaferr Steelworks Co. Ltd., Hungary, 1989 12 An Feng, Kaoshiung, Taiwan, 1990, 1676 mm (66”), new mill, 2,000,000 MTPY 13 Bethlehem Steel, Sparrows Point, MD, 1991, 1727 mm (68”), existing mill 14 Tokyo Steel, Japan, 1991 15 Sharon Steel (Duferco), Farrel, PA, 1992, existing mill 16 Pan Zhi Hua Iron & Steel, China, 1993 17 TISCO, Jamshedpur, India, 1993, 1550 mm (61”), new mill, 2,000,000 MTPY 18 Wheeling Pittsburgh Steel, Mingo Junction, OH, 1993, existing mill 19 APM SA de CV Monterrey, Mexico, 1994, existing mill 20 Chendge, China, 1994 21 Lingyuan, China, 1994 22 Suhaviriya Steel Industries, Thailand, 1994 23 Erdemir Eregli, Turkey, 1995, 1727 mm (68”), existing mill 24 Essar Steel, Hazira, India, 1995, 2000 mm (79”), new mill, 2,000,000 MTPY 25 Kawasaki Steel Chiba No 3, Japan 1995, 2032 mm (80”) 26 (Yieh Loong) Chun Hung Taiwan, 1997, 1727 mm (68”) 27 Arcelor Eko Stahl, Germany, 1997, 1640 mm (65”), 1,500,000 MTPY 28 Jindal Vijayanagar Steel Ltd., Bellares, India, 1997, 1422 mm (56”), new mill 29 Nippon Steel Oita, Japan, 1997, 2388 mm (94”) 30 Trico Nucor Steel, Decatur, AL, 1997, 1778 mm (70”), new mill Appendix 383

31 Rourkela Steel, Sail Rourkela, India, 1997, 1422 mm (56”), existing mill 32 Severstal JSC Cherepovets, Russia, 1997 33 Hadeed, Al-Jubail, S.A., 1999, 1650 mm (65”), new mill, 2,000,000 MTPY 34 Siam Steel Pipe Group, Thailand, 1999 35 Beta Steel (NLMK Indiana Corp), Portage, IN, 2000 36 Rautaruukki Oy, Finland, 2000 37 Anshan, China, 2001 38 Arcelor Brazil CST Vitoria, Brazil, 2002, 1880 mm (74”), 2,000,000 MTPY 39 Baosteel, China, 2003 40 Laiwu Iron & Steel, Laiwu, China, 2005, 1500 mm (59”) 41 Taiyuan Iron & Steel, Shanxi, China, 2006, 2250 mm (89”), 4,000,000 MTPY 42 Shougang Iron & Steel Qian’an, China, 2006, 2160 mm (85”), 4,000,000 MTPY 43 Tangshan Iron & Steel Tangshan, China, 2006, 1700 mm (67”) 44 Hyundai Steel, Dangjin, South Korea, 2006, 2080 mm (82”) 45 Posco, South Korea, 2006 46 Bhushan Steel, Orissa, India, 2007, 1850 mm (73”), 2,500,000 MTPY 47 Han Zhong Steel, China, 2007 48 Zaporozhstal JSC Zaporozhye Ukraine, 2007, 1680 mm (66”) 49 Chengde Iron & Steel Chengde, China, 2008, 1780 mm (70”) 50 Colakoglu, Gebze, Turkey, 2009, 1850 mm (73”), new mill 3,500,000 MTPY 51 Rizhao Iron & Steel Rizhao, China, 2009, 2150 mm (85”) 52 Ansteel, Yingkou, China, 2009, 1580 mm (62”) 53 Lingyuan Iron & Steel, Loudi, China, 2009, 2250 mm (86”) 54 Anling, Chaoyang, China 2011, 1700 mm (67”) 55 Magnitogorsk Iron & Steel, Magnitogorsk, Russia, 2012, 2500 mm (98”)

List 7. SMS Siemag Thin Slab Hot Strip Mills

1 Nucor Crawfordsville, IN, 1989, 1350 mm (53”), 1,800,000 MTPY 2 Nucor Hickman, Arkansas, 1992, 1560 mm (61.4”), 2,000,000 MTPY 3 Thyssenkrupp Aciaia Special Terni, Italy, 1992, 1560 mm (61.4”) 4 Hylsa, Mexico, 1995, 1370 mm (54”), 1,450,000 MTPY 5 INI Steel, South Korea, 1995, 1560 mm (61.4”), 2,000,000 MTPY 6 Gallatin, Ghent, KY, 1995, 1560 mm (61.4”), 1,000,000 MTPY 7 Steel Dynamics, Butler, IN, 1995, 1560 mm (61.4”), 2,600,000 MTPY 8 Arcelor ACB, Spain, 1996, 1560 mm (61.4”), 1,850,000 MTPY 9 Mittal Steel (Acme), Riverdale, IL, 1996, 1560 mm (61.4”), 1,000,000 MTPY 10 Nucor Berkeley, Mt. Pleasant, SC, 1996, 1680 mm (66.1”), 2,400,000 MTPY 11 NSM Chonburi, Bor Win, Thailand, 1997, 1600 mm (63”), 1,200,000 MTPY 12 Ispat Industries, Dolvi, India, 1998, 1560 mm (61.4”), 2,400,000 MTPY 13 Zhujiang Steel, Guangzhou, China, 1999, 1380 mm (54.3”), 1,700,000 MTPY 14 Megasteel Selangor, Malaysia, 1998, 1575 mm (62”), 3,200,000 MTPY 15 Handan Iron & Steel, Handan, China, 1999, 1680 mm (66.1”), 2,475,000 MTPY 16 Baotou Iron & Steel, Baotou, China, 2001, 1560 mm (61.4”), 2,000,000 MTPY 17 Thyssenkrupp Steel, Duisburg, Germany, 1999, 1600 mm (63”), 2,400,000 MTPY 18 ANSDK Alexandria, Egypt, 1999, 1600 mm (63”), 1,000,000 MTPY 384 Primer on Flat Rolling

19 Thyssenkrupp Acciai Special Terni, Italy, 2001, 1560 mm (61.4”), 900,000 MTPY 20 Maanshan Iron & Steel, Anhui, China, 2003, 1600 mm (63”), 2,000,000 MTPY 21 Lianyuan Iron & Steel, Hunan, China, 2004, 1600 mm (63”), 2,350,000 MTPY 22 Jiuquan Iron & Steel, Gansu, China, 2005, 1680 mm (66.1”), 2,000,000 MTPY 23 Bhushan Ltd., Kolkata, India, 2006, 1300 mm (51.2”), 800,000 MTPY 24 Severcorr, Columbus, Ms, 2007, 1880 mm (74”), 1,350,000 MTPY 25 Wuhan Iron & Steel, China, 2008, 1600 mm (63”), 2,500,000 MTPY 26 Essar Steel, Hazira, Guajarat, India, 2009, 1680 mm (66.1”), 2,500,000 MTPY 27 Tata Iron & Steel, Jamshedpur, India, 2010, 1680 mm (66.1”), 2,400,000 MTPY

List 8. Mannesmann Demag Thin Slab Hot Strip Mills

1 Arvedi, Cremona, Italy, 1992, 1330 mm (52”), 700,000 MTPY 2 Posco, Kwangyang, Korea, 1996, 1350 mm (53”), 1,800,000 MTPY 3 Saldanha Steel, South Africa, 1998, 1560 mm (61”), 1,400,000 MTPY 4 Posco, Kwangyang, Korea, 1998, 1350 mm (53”), 2,000,000 MTPY

List 9. Some of the Danieli Thin Slab Hot Strip Mills

1 Algoma, Sault Ste. Marie, ON, Canada, 1997, 1600 mm (63”), 2,000,000 MTPY 2 North Star BlueScope Steel, Delta, OH, 1997, 1549 mm (61”), 2,700,000 MTPY 3 Ezz Steel, Suez, Egypt 4 Tangshan Iron & Steel, China, 2005, 3,250,000 MTPY 5 OMK, Russia, 2009 6 Posco CEM, Korea, 2010 7 Baosteel Group Shanghai, Meishan Co. Ltd., Shanghai, China, 2010, 2,500,000 MTPY 8 Lucchini Severstal Group Piombino, Italy, 2010, 1600 mm (63”), 1,700.000 MTPY 9 MMK Metalurji Iskenderun, Turkey, 2011, 1570 mm (62”), 2,300,000 MTPY

List 10. Misubishi Hitachi Thin Slab Hot Strip Mills

1 Posco, Kwangyang, Korea, 1996, 1350 mm (53”), 1,800,000 MTPY 2 Nucor Steel (Trico Steel), Decatur, AL, 1996, 1650 mm (65”), 2,300,000 MTPY 3 Posco, Kwangyang, Korea, 1998, 1350 mm (53”), 2,000,000 MTPY 4 G Steel (Siam Strip Mill) Rayong, Thailand, 1998, 1549 mm (61”) 5 Corus Hoogovens Ijmuiden, Netherlands, 2000, 1560 mm (62”) 6 Thangshen Iron & Steel Thangshen, China, 2003, 1680 mm (66”), 1,500,000 MTPY 7 Benxi Iron & Steel, China, 2005, 1750 mm (69”), 1,500,000 MTPY 8 Tonghua Iron & Steel, China, 2005, 1560 mm (61”), 1,400,000 MTPY 9 Dongbu Steel, Asan, Korea, 2009, 1650 mm (65”), 2,500,000 MTPY Appendix 385

List 11. Arvedi Cremoni/Siemens VAI ESP Thin Slab Hot Strip Mill

1 Arvedi, Cremoni, Italy, 2009, 2,000,000 MTPY

List 12. Some of the Newer Generation II Hot Strip Mills

1 Krakatau Steel, Cilegon, Indonesia, 1980, 2080 mm (82”) 2 Ilva Bagnoli, Bagnoli, Italy, 1980, 1320 mm (52”) 3 Posco, Pohang, Korea, 1980, 2050 mm (81”) 4 Nisshin Steel Kure, Japan, 1980, 1810 mm (71”) 5 Dofasco, Hamilton, ON, Canada, 1982, 1730 mm (68”), 4,200,000 MTPY 6 China Steel No 1 Kaoshiung, Taiwan, 1982, 1730 mm (68”), 3,400,000 MTPY 7 Nippon Steel, Yawata, Japan, 1982, 1700 mm (67”) 8 Nippon Steel Hirohata, Japan, 1984, 1830 mm (72”) 9 Posco, Kwangyang, Korea, 1986, 1780 mm (70”) 10 Baoshan Iron & Steel, China, 1989, 1900 mm, (75”), 4,500,000 MTPY 11 Posco, Kwangyang, Korea, 1990, 1700 mm (67”) 12 Posco, Kwangyang, Korea, 1992, 2135 mm (84”) 13 Nisco Mobarakeh Isfaha´n, Iran, 1992, 1880 mm (74”), 3,100,000 MTPY 14 Sahaviriya, Bang Saphan, Thailand, 1994, 1550 mm (61”), 2,400,000 MTPY 15 Kawasakik, Chiba, 1995, 2050 mm (81”) 16 China Steel, Kaohsiung, Taiwan, 1996, 1880 mm (74”), 2,700,000 MTPY 17 Baosteel Shanghai, China, 1996, 1580 mm (62”) 18 Hanbo Steel, Asan, Korea, 1997, 2080 mm (82”) 19 Anshan Anshan, China, 2000, 1780 mm (70”), 3,500,000 MTPY 20 Wuhan Iron & Steel, China, 2003, 2130 mm (84”), 4,500,000 MTPY 21 Maanshan Iron & Steel, Anhui, China, 2007, 2130 mm (84”), 5,500,000 MTPY 22 Baosteel Shanghai, China, 2007, 1880 mm (74”), 3,700,000 MTPY 23 Shougang Jingtang Caofeidian, Hebei, China, 2008, 2130 mm (84”), 5,500,000 MTPY 24 Handan Iron & Steel, Hebei, China, 2008, 2130 mm (84”), 4,500,000 MTPY 25 Isdemir, Iskenderun, Turkey, 2008, 2200 mm (87”), 3,500,000 MTPY 26 Benxi Iron & Steel, China, 2009, 2150 mm (85”), 5,150,000 MTPY 27 JSW Steel, Toranagallu, India, 2009, 2300 mm (91”), 3,500,000 MTPY 28 Dragon Steel, Taichung, Taiwan, 2010, 2030 mm (80”), 3,040,000 MTPY 29 Hyundai Steel, Dangjin, Korea, 2010, 2150 mm (85”), 3,500,000 MTPY 30 Thyssenkrupp Steel, Calvert, AL, 2010, 1890 mm (74”), 5,300,000 MTPY 31 Cosipa Cubatao, Brazil, 2011, 2200 mm (87”), 2,300,000 MTPY

List 13. Some of the Modern Steckel Mills

1 Southern Cross, Middleburg, South Africa, 1982, 1980 mm (78”), 150,000 MTPY 2 Haynes International, Kokomo, IN, 1983, 1320 mm (52”) 3 Highveld Steel & Vanadium, South Africa, 1984, 2540 mm (100”), 400,000 MTPY 386 Primer on Flat Rolling

4 Tuscaloosa Steel (Nucor), AL, 1985, 2591 mm (102”), 750,000 MTPY 5 Outokumpu Stainless, Tornio, Finland, 1988, 1625 mm (64”), 1,600,000 MTPY 6 Outokumpu Stainless, Avesta, Sweden, 1991, 2100 mm (83”), 1,000,000 MTPY 7 Fabrique de Fer(Arcelor), Belgium, 1992, 3000 mm (118”), 500,000 MTPY 8 Yieh United, Kaohsiung, 1994, 1600 mm (63”), 600,000 MTPY 9 Salem Steel (Sail), India, 1995, 1275 mm (50”), 200,000 MTPY 10 LPN Plate Mill, Samutprakarn, Thailand, 1996, 3086 mm (122”), 600,000 MTPY 11 Gunung Raja Paksi, Indonesia, 1996, 1600 mm (63”), 500,000 MTPY 12 Oregon Steel Mills, Portland, OR, 1997, 3050 mm (120”), 910,000 MTPY 13 Ipsco, Montpellier, Muscatine, IA, 1997, 2438 mm (96”), 1,180,000 MTPY 14 Stelco, Hamilton, ON, Canada, 1998, 3099 mm (122”), 900,000 MTPY 15 Nova Hut (Mittal), Czech Republic, 1999, 1575 mm (62”), 1,000,000 MTPY 16 Kunming Iron & Steel, China, 2002, 1575 mm (62”), 1,200,000 MTPY 17 Maghreb Steel, Casablanca, Morocco, 2008, 1575 mm (62”), 1,000,000 MTPY 18 Baosteel Special Steel Bran, Shanghai, China, 2009, 1300 mm (51”), 282,200 MTPY References

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