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Elements of Fracture Mechanics by Prasant Kumar

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Elements of Fracture Mechanics by Prasant Kumar Elements of Fracture Mechanics Prashant Kumar Former Professor Department of Mechanical Engineering IITKanpur LNVM (NEEMRANA) 7084 11111111111111111 IIIII IIII IIII Library McGraw Hill Education (India) Private Limited NEW• DELHI McGraw Hi/11:ducation Offices New Delhi New York St Louis San Francisco Auci<land Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal San Juan Santiago Singapore Sydney Tokyo Toronto McGraw Hill Education (India) Private Limited Published by McGraw Hill Education (India) Private Limited, P-24, Green Park Extension, New Delhi 110 016. Copyright © 2009, by McGraw Hill Education (India) Private Limited Seventh reprint 2014 RLZXCRCURQZAR No part of this publication may be reproduced or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior written permission of the publishers. The program listings (if any) may be entered, stored and executed in a computer system, but they may not be reproduced for publication. This edition can be exported from India only by the publishers, McGraw-Hill Education (India) Private Limited. ISBN (13): 978-0-07-065696-3 ISBN (10): 0-07-065696-7 Managing Director: Kaushik Be!lani Asst. Manager-Production: Sohan Gaur Manager-Sales & Marketing: S Girish Product Manager-Science, Technology and Computing: Rekha Dhyani General Manager-Production: Rajender P Ghansela Information contained in this work has been obtained by McGraw-Hill, from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render professional services. If such services are required, the assistance of an appropriate professional should be sought. Typeset at Script Makers, 19, Al-B, DDA Market, Pasehim Vihar, New Delhi 110 063 and printed at AP Offset Pvt. Ltd., Del.hi 110 095 Cover Printer: AP Offset Pvt. Ltd. Cover Designer: Kapil Gupta To Many excellent teachers who motivated me from my school days to PhD program. Some of them are: K. Kumar, Lucknow Montessori School, Lucknow (1956-57) T. Joshi, Lucknow Christian College, Lucknow (1960-62) N. C. Dahl, IIT Kanpur, Visiting from MIT, USA (1963-64) A. J. Erickson, IIT Kanpur, Visiting from MIT, USA (1963-65) P W. Fay, IIT Kanpur, Visiting from Cal. Tech., USA (1965-66) C. W. Radcliff, University of California, Berkeley, USA (1967-68) Y. Takahashi,J.Jniversity of California, Berkeley, USA (1967-68) R.f. Clifton, Brown University, USA (1970-77) L.B. Freund, Brown University, USA (1971-74) A.C. Pipkin, Brown University, USA (1972-73) P.C. Paris, Brown University, Visiting from Lehigh University, USA (1973-74) Barton Roessler, Browh University, USA (1973-74) J.R. Rice, Brown University, USA (1972-73) and My uncle Dr. Ram Krishan and aunt Mrs. Sheela Krishan who raised me and played the most pivotal role in shaping my thought process. Preface There are many components of machines, process plants and household goods that fail through fatigue and fracture, which can be avoided by applyingfracture mechanics. Although the modern fracture mechanics was born in 1948-49, its acceptability was quite slow for several decades because of its complex mathematics and involved concepts. Initially, the fracture mechanics was applied to high-risk products like nuclear plants, airplanes, space vehicles, submarines, etc. Now this field is becoming popular at the grassroot levels too. Therefore, it is important to explain this concept in simple, well-disposed and easy-to-understand manner without compromising on the rigor-this book is intended to do the same. The book has gone through several phases of revisions to improve its readability further. Several analogies and anecdotes placed appropriately in this book have also helped in accomplishing this goal. Two most important mechanisms-fatigue failure and environment-assisted fracture that result in the growth of subcritical cracks to catastrophic failure of structural components-have been discussed in Chapter 9. Along with that, an additional chapter has talked about various non­ destructive test methods for identifying cracks in the structural components. The detailed illustrations developed with the help of latest graphic software have added more clarity in understanding the concept. This book has adopted the approach of different courses on fracture mechanics at the advanced undergraduate and postgraduate levels. The readers novice in this field can explore this complex but beautiful concept effortlessly through this book. The responses of students have directed me in choosing the right pace and the appropriate levels while presenting the subject. Readers with prior knowledge of fracture mechanics may find it rudimentary at few places. But with experience I have learned to be on elementary side rather than leaving a reader confused. Unlike in most books on fracture mechanics where basic concepts are discussed shortly usually leaving the reader confused, this book gradually discusses the fundamental theory with the results and their applications. With my industrial experience and interest in product design, I view the theories of fracture mechanics as useful tools to be applied to practical problems. Once the fundamentals of fracture mechanics are mastered properly, the reader can easily move to advanced books or research journals. Therefore, special tricks involving highly complex mathematics to solve problems of minor interest are not included here. I hope this book will be appreciated by the readers for its salient features like lucid content, articulation, anecdotes, and presentation of the subject in small and effective steps, and will prove to be beneficial in introducing the subject in general. PRASHANT KUMAR Acknowledgements • IIT Kanpur for encouragement and financial support to prepare the manuscript • ARDB, Sena Bhavan, New Delhi for supporting various research grants in the field of fracture mechanics • Prof. N.N. Kishore (IIT Kanpur) for contributing a chapter in the book • Prof. Raju Sethuraman (IIT Madras) for contributing a chapter in the book • Prof. K. Ramesh (IIT Madras) for contributing a figure on photoelastic fringes • Profs. Bishakh Bhattacharya, J. Ramkumar and K. K. Kar for constructive criticism of the manuscript • Raj Mulani, Rahul Ranjan Pandey, Ramesh Chandra and Yatendra for preparing the illustrations using latest graphic softwares • Vinay Pahlajani and Rajamanohar for their help in graphics through latest softwares • Mr. Divakar to coordinate various activities in preparing the manuscript • Mr. Anurag Goel to meet various needs of the manuscript preparation • Mrs. Sandhya Agnihotri and S.L. Yadav for typing the manuscript • Mr. Kalyan Kumar Singh for constructive suggestions and help in proof reading •- My son Saurabh Vishal on persistent encouragement to have the book published • McGraw-Hill Education (India) Ltd. for publishing the book Contents Preface vii Acknowledgements ix 1. Background 1 1.1 KindsofFailure 1 1.2 Historical Aspects 3 1.3 Brittle and Ductile Fracture 4 1.4 Modes of Fracture Failure 5 1.5 How Potent is a Crack? 6 1.6 Point of View 7 1.7 Damage Tolerance 7 References 7 2. Energy Release Rate 9 2.1 Introduction 9 2.2 Griffith's Dilemma 9 2.3 Surface Energy 10 2.4 Griffith's Realization 10 2.5 Griffith's Analysis 11 2.6 Energy Release Rate 14 2.6.1 Definition 14 2.6.2 Mathematical Formulation 15 2.6.3 Change in Compliance Approach 16 2.6.4 Change in the Strain Energy Approach 20 2.7 Energy Release Rate of DCB Specimen 21 2.8 Anelastic Deformation at Crack-tip 24 2. 9 Crack Resistance 25 2.10 Stable and Unstable Crack Growth 26 2.11 R-curve for Brittle Cracks 27 2.12 Thin Plate vs Thick Plate 28 2.13 Critical Energy Release Rate 29 2.14 Closure 31 Questions 31 Problems 32 References 34 xii Contents 3. Stress Intensity Factor 35 3.1 Introduction 35 3.1.1 Why Should Investigations be Closer to the Crack Tip? 35 3.1.2 Linear Elastic Fracture Mechanics (LEFM) 35 3.2 Stress and Displacement Fields in Isotropic Elastic Materials 36 3.3 Stress Intensity Factor 38 3.4 Background for Mathematical Analysis 41 3.4.1 Field Equations 42 3.4.2 Elementary Properties of Complex Variables 45 3.5 Westergaard's Approach 47 3.5.1 Model (Opening Mode) 47 3.5.2 Mode II (Sliding Mode) 56 3.5.3 Mode III (Tearing Mode) 58 3.6 Concluding Remarks 59 Questions 60 Problems 60 References 61 4. SIF of More Complex Cases 62 4.1 Other Applications of Westergaard Approach 62 4.1.1 Wedge Loads on Cracked Surfaces 62 4.1.2 Collinear Cracks in an Infinitely Long Strip 64 4.2 Application of the Principle of Superposition 66 4.2.1 Internal Pressure on Cracked Faces 67 4.2.2 Wedge Load at the Surface of a Crack Face 68 4.3 Crack in a Plate of Finite Dimensions 69 4.4 Edge Cracks 71 4.5 Embedded Cracks 72 4.5.1 Elliptical Crack 73 4.5.2 Semi-elliptical Cracks 74 4.5.3 Quarter or Comer Cracks 75 4.6 The Relation between G1 and K1 75 4.7 Critical Stress Intensity Factor 78 4.8 Bending and Twisting of Cracked Plates 80 4.8.1 Terminology of the Plate Theory 80 4.8.2 Through-the-Thickness Crack in a Plate 81 4.8.3 Bending Moment on a Centre-Cracked Plate 82 4.9 Closure 84 APPENDIX 4A General Approach to Determine Stress and Displacement Fields 84 Contents xiii APPENDIX 4B SIF of Some Important Cases 89 Questions 95 Problems 96 References 98 5.
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