' Methodology for Design and Control Of

Home , Process function

' METHODOLOGY FOR DESIGN AND CONTROL

OF THERMOMECHANICAL PROCESSES

A Dissertation Presented to

The Faculty of the College of Engineering and Technology

Ohio University

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

by James C. Malas, Ill

November, 1991 @ 1991

James C. Malas, Ill

MALAS, JAMES C., Ill. Ph.D. November, 1991 Engineering

odoloqv for Oes~gnand Contra of Thermomechanlcal Prwsses. (202~~4 Director of Dissertation: Jay S. Gunasekera The overall goal of this research is to develop an interdisciplinary approach for the design and control of material behavior throughout a sequence of thermomechanical processes involving hot-working and heat- treatment operations. The specific objectives of the research are as follows:

1) to establish a scientific methodology for determining the processing conditions under which material behavior is stable and predictable; 2) to establish an axiomatic approach for determining the initial condition of the workpiece material, the number and sequence of thermomechanical operations required, and the processing conditions corresponding to each operation; and 3) to establish a scientific methodology for determining the optimal microstructural evolution path and the corresponding optimal control inputs for a given metalforming process. Material-behavior modeling consisted of 1) evaluation of the flow-stress data for a given material over a wide range of temperatures and strain rates, 2) determination of constitutive relations, 3) stability analysis of flow-stress data using the dynamic-material-modeling approach, and 4) microstructural characterization of the deformed structures. Design of thermomechanical processes was accomplished using the axiomatic design approach developed by Suh. Design matrices which relate functional requirements and design parameters were derived such that the Independence Axiom was satisfied. From these various "decoupled" design solutions, the best one was selected on the basis of the Information Axiom. An open-loop control system was designed for a hot- working operation employing optimai control techniques to yield the desired final microstructures. The present scientific investigation has resulted in significant advancements in the areas of material-behavior modeling and in the development of process-design methods and process-control strategies for thermomechanical processes. In material-behavior modeling, the apparent activation energy for metallurgical processes has been discovered to be a fundamental link between atomistic and continuum phenomena. Together with stability analyses, the magnitude of activation energy and its variation provide the means of identifying the processing regimes within which the material behavior is essentially deterministic. Using this approach, a more precise definition of the "processing window" for a particular material and its boundaries was established. The roles of activation energy and stability in material-behavior modeling have been demonstrated to be valid in different systems of varying complexity. A new method for designing thermomechanical processes based upon scientific principles from design theory, materials science, and process mechanics has been established. Existing design axioms and related concepts are applied to this problem for the first time. As with other applications of these axioms, the exploitation of functional independence and minimal information content has been found to be the key to successful design solutions. The fundamental roles of material characteristics and geometrical shape in the creation and elimination of process-design solutions have been demonstrated. In fact, a measure of the information content for a material-processing system has been found to be the temperature and strain-rate domain which overlaps the material processing window and the capacity range of the forming equipment. It is TABLE OF CONTENTS

I. INTRODUCTION ...... 1 Current Approaches to Design of Metalforming Processes ...... 1 Current Approaches to Process Control ...... 5 Scope and Objectives of Research ...... 7 II. APPROACH ...... 11 Systems Approach to Metals Processing ...... 11 Global Design of Thermomechanical Processes ...... 15 Detailed Design of Thermomechanical Operations ...... 17 Ill. MATERIAL-BEHAVIOR MODELING OF METAL ALLOYS UNDER PROCESSING CONDITIONS ...... 19 Constitutive Relationships ...... 20 Basic Concepts ...... 21 Required Experimental Data ...... 23 Flow-Stress Behavior of Ductile and Brittle Materials ...... 23 Constitutive Equations for Hot-Deformation Behavior ...... 28 Hot Workability ...... 33 Ashby-Frost Deformation Maps ...... 35 Damage Nucleation Maps ...... 36 Dynamic Material Model ...... 38

Microstructural Evolution During Hot Working ...... 46

Dynamic-Recovery Relationships ...... 49 Dynamic-RecrystallizationRelationships ...... 51 Static-Recrystallization Relationships ...... 53 Microstructural Evolution of Plain C-Mn Steel ...... 54 TABLE OF CONTENTS (CONTINUED)

IV . HYBRID APPROACH TO MATERIAL-BEHAVIOR MODELING ..... 59 Assumptions ...... 60 Isotropic and Homogeneous Material Behavior ...... 60 Dynamic Material Behavior ...... 60 Nature of Thermodynamic System ...... 61 Material Deformation ...... 61

Metallurgical Interpretation of the Dynamic Material Model ...... 62

Values of Strain-Rate-Sensitivity Parameter ...... 63 Variation of m with Strain Rate ...... 64 Values of Excess-Entropy-Production Rate ...... 65 Variation of s with Strain Rate ...... 66 Relationship Between Constitutive Equations and Dynamic Material Modeling ...... 68

Case I: a = K Zm ...... ; ...... 69 Case II: Z = A[sinh(ow)]n ...... 70 Evaluation of Case II Using Material-Data-Analysis Software ... 74 Relationship Between Dynamic Material Modeling and Microstructure Development ...... 77

Role of Activation Energy ...... 78 Role of Stability ...... 81

V . MODEL VALIDATION: CASE STUDIES ...... 84

Thermomechanical Processing of a Titanium-Aluminide Alloy ..... 84 Effect of Thermomechanical Processing Upon Microstructure ...... 85 Dynamic Recrystallization of a Gamma-TiAl Alloy ...... 89 TABLE OF CONTENTS (CONTINUED)

Hot-Working Behavior of 2024 Aluminum Materials ...... Information Content of Material Processing Systems ...... Ingot 2024 Aluminum ...... P/M 2024 Aluminum ...... P/M 2024 Aluminum with Silicon-Carbide Whisker ...... Hot Working Behavior of a Nickel-Base Superalloy ...... P/M Rene 95 ...... Effects of Thermomechanical History Upon Material Behavior ...... VI . AXIOMATIC APPROACH TO PROCESS DESIGN ...... Review of Key Concepts of Axiomatic Design ...... Methodology for Systematic Evaluation of Design Alternatives .... Example: Axiomatic Design of High-Performance Aircraft-Engine Component ...... Selection of Thermomechanical Processes ...... VII . PROCESS-DESIGN METHODOLOGY FOR PRODUCING REQUIRED MICROSTRUCTURE ...... Design Matrices for Recrystallization Processes ...... Dynamic Recrystallization of Plain C-Mn Steel ...... Static Recrystallization of Plain C-Mn Steel ...... Design Matrix for Hot-Plane-RollingProcesses ...... Example: Design of a Hot-Rolling Process for Producing Steel Sheets with a Fine-Grained Microstructure ......

Case I: Processing Variables for Producing d(1) = 50 ym ..... Case II: Processing Variables for Producing d(1) = 30 ym ..... TABLE OF CONTENTS (CONTINUED)

VIII. OPTIMAL CONTROL STRATEGY FOR DYNAMIC- RECRYSTALLIZATION PROCESSES ...... 153 Control-Problem Formulation ...... 154

Semi-Linearization of Yada's Equations for Dynamic Recrystallization ...... 154 Optimality Criterion for Dynamic Recrystallization ...... 160 IX. SUMMARY AND CONCLUSIONS ...... 164 Material-Behavior Modeling ...... 165 Design Methodology ...... 166 Process-Control Methodology ...... 167 Futurework ...... 168 REFERENCES ...... 169 APPENDIX: CONTROL-PROBLEM SOLUTION ...... 177

ABSTRACT

vii LIST OF FIGURES

1 Framework of Functions Involved in Computer- lntegrated Manufacturing System (From Ref. 1) ...... 2 2 Schematic Representation of Two Basic Thermo- mechanical Processes as Systems Composed of Material, Equipment, and Control Devices. (a) Hot- Deformation Processing System, (b) Thermal- Processing System ...... 12 3 Overall Approach Used in Development of Method- ology for Design and Control of Thermomechanical Processes...... 14 4 Four Domains of Suh's Design World (From Ref. 21) ...... 16 5 Block Diagram of Open-Loop Control of Metalwork ing Processes ...... 18 6 Typical Flow Curves for Ductile Material (Solid) and for Brittle Material (Dashed) ...... 24 7 Typical Stress-Strain Rate Relationships for Ductile and Brittle Materials ...... 25

8 Constant-Strain-Rate, Isothermal-Compression Test Results of Ti-49.5AI-2.5Nb-1.1 Mn Alloy. (a) Relation of Flow Stress to Strain Rate, (b) Relation of Peak Stress to Strain Rate ...... 27

9 Typical Stress-Temperature Relationship for Metallic Systems ...... 28 10 Flow Stress Versus Reciprocal Temperature Relation- ship for Ti-49.5AI-2.5Nb-1.1 Mn Alloy ...... 29

11 Flow Stress Versus Zener-Holloman Parameter Rela- tionship Using (a) Power Law and (b) Hyperbolic Sine Function ...... 32 LIST OF FIGURES (CONTINUED)

12 Sample of Round Products Extruded at Temperature of 500°F and Speeds of (a) 1.5 in./min, (b) 0.9 in./min ...... 34 13 Ashby-Frost Deformation Map for Pure Nickel (From Ref. 47) ... 36

14 Processing Map by Raj Based Upon Various "Damage" Mechanisms for Aluminum (From Ref. 48) ...... 37

15 Schematic Diagram Showing Constitutive Relation of Material System as Energy Dissipator During Forming ......

16 Processing Map for Ti-6242 P Microstructure with Stable Regions Identified (From Ref. 19) ...... 47

17 Typical Dynamic and Static Metallurgical Processes in Operation During (a) Hot Rolling and (b) Hot Extru- sion (From Ref. 55). Microstructure Development Generally Dependent Upon Stacking-Fault Energy of Metallic System as well as Temperature, Strain, and Strain Rate of Process ...... 50

18 Prediction of Critical Strain for Dynamic Recrystalliza- tion of Plain C-Mn Steel Using Yada Model...... 56

19 Prediction of Grain Size During Dynamic Recrystalliza- tion of Plain C-Mn Steel Using Yada Model ...... 57

20 Prediction of Kinetics of Dynamic Recrystallization of Plain C-Mn Steel Using Yada Model...... 58

21 Primary Material-Behavior Models Used in Hot- Deformation Studies. (a) Disjointed Approach, (b) Hybrid Approach ...... 60

22 Schematic Representation of Variation of Log o with Log E for Two Cases: (a) a(log ~)/a(logE) > 0, (b) a(log o)/a(log ti) < o ...... 64 LIST OF FIGURES (CONTINUED)

23 Schematic Representation of Variation of Log a with 1/T for Two Cases. For Dynamic Recrystallization: a(ln ~)/a(lIT) > T; For Dynamic Recovery: OE, ...... 67

25 Plot of Bracketed Hyperbolic Function Given in Eq. (50) forDomainofOcaoc1.2 ...... 73 26 Flow-Stress Values (MPa) for Plain C-Mn Steel Com- puted from Eq. (55) for Strain-Rate Range 1o-~ - 1O2 s-' and Temperature Range 800 - 1200°C ...... 75 27 DMM Processing Maps for Plain C-Mn Steel. Contour Map of Each Stability Criterion Shown for Strain = 1.O. Shaded Regions Identify Where Particular Stability Criterion is Satisfied: (a) m-Values, (b) am/a(log E), (c) s-Values, (d) as/a(log E) ...... 76

28 Change in Free Energy of Atom as It Takes Part in Tran- sition. "Reaction Coordinate" is Any Variable Defining Progress Along Reaction Path (From Ref. 66) ...... 79

29 Schematic Representation of Various Dissipative Paths Which Material System Can Take During Transition to Lower Its Free Energy ...... 82

30 Examples of Various Types of Stability from Field of Systems Engineering (From Ref. 68) ...... 83

31 Typical Thermomechanical Processing Sequence for Gamma-TiAl Alloys...... 86

32 Effect of Thermomechanical Processing upon Micro- structure of Ti-48AI-2Nb-2Cr Alloy. (a) Cast + HIP, (b) Cast + HIP + lsoforge + Homogenize, (c) Cast + HIP + lsoforge + Homogenize + lsoforge + Heat Treat ...... 88 LIST OF FIGURES (CONTINUED)

DMM Processing Map for Ti-49.5AI-2.5Nb-1.1 Mn Alloy in Cast + HIP Condition. Contours of Apparent Activation Energy (kcal/mole) with DMM Stability Infor- mation for Strain = 0.5. Shaded Regions Satisfy All Four DMM Stability Criteria...... 90

Microstnrcture Development Associated with Dynamic Recrystallization of Ti-49.5AI-2.5Nb-1.1Mn Alloy in Cast + HIP Condition. Microstructure Shown for Differ- entStrainLevels: (a)~=0.3,(b)~=0.6,(~)~=1.0...... 92 Observed Relationship Between Volume Fraction Recrystallized and Strain for Ti-49.5AI-2.5Nb-1.1 Mn Alloy in Cast + HIP Condition (From Ref. 72) ...... 93 Logarithmic Representation of Relationship Between Volume Fraction Recrystallized and Strain for Ti-49.5AI-2.5Nb-1.1Mn Alloy in Cast + HIP Condi- tion (From Ref. 72) ...... 94

Observed Relationship Among Recrystallized Grain Size, Temperature, and Strain Rate for Ti-49.5AI-2.5Nb-1.1Mn Alloy in Cast + HIP Condi- tion (From Ref. 72)...... 95

Logarithmic Representation of Relationship Between Recrystallized Grain Size and Temperature- Compensated Strain Rate for Ti-49.5AI-2.5Nb-1.1Mn Alloy in Cast + HIP Condition (From Ref. 72) ...... 96 Influence of Solute Content upon Melting and Solution Temperatures and, Therefore, upon Forgeability (From Ref. 38) ...... 99

Schematic Diagram of System and Design Ranges for Hot-Working Processes. Contours of Activation Energy Shown with Desirable One Identified as Q*. Shaded Region Satisfies All Four DMM Stability Criteria ...... 100 LIST OF FIGURES (CONTINUED)

41 Flow-Stress Curves for Ingot 2024 A1 ...... 101

42 DMM Processing Map for Ingot 2024 A1 . For E = 0.3, Contours of Apparent Activation Energy (kcal/mole) with DMM Stability lnformation Are shown. Shaded Regions Satisfy All Four DMM Stability Criteria ...... 102 43 DMM Processing Map for P/M 2024 Al in Vacuum Hot Pressed + Extruded Condition. For E = 0.3, Contours of Apparent Activation Energy (kcal/mole) with DMM Stability lnformation Are Shown. Shaded Regions Satisfy All Four DMM Stability Criteria...... 104

44 DMM Processing Map for P/M 2024 Al with Sic Whis- kers in Vacuum Hot Pressed + Extruded Condition. For E = 0.3, Contours of Apparent Activation Energy (kcal/mole) with DMM Stability lnformation Are Shown. Shaded Regions Satisfy All Four DMM Stability Criteria ...... 106

45 DMM Processing Map for -150 Mesh P/M Rene 95 in Compacted Condition. For E = 0.3, Contours of Apparent Activation Energy (kcal/mole) with DMM Stability lnformation Are Shown. Shaded Regions Satisfy All Four DMM Stability Criteria ...... 109

46 DMM Processing Map for -270 Mesh P/M Rene 95 in Compacted Condition. For E = 0.3, Contours of Apparent Activation Energy (kcal/mole) with DMM Stability lnformation Are Shown. Shaded Regions Satisfy All Four DMM Stability Criteria ...... 1 10

47 Hierarchial Tree Structures of Functional Requirements and Design Parameters (From Ref. 84) ...... 1 15

48 Turbofan Aircraft Engine. (a) United Technologies Pratt and Whitney F100-PW-200 Design, (b) Revolutionary Design of the Future ...... 119 LIST OF FIGURES (CONTINUED)

49 General View of Design Domains and Constraints Considered in Example Design Problem of High- Performance Aircraft-Engine Component ...... 120

50 Consumer and Functional Domains Considered in Example Design Problem of High-Performance Aircraft-Engine Component ...... 121

51 Functional and Physical Domains Considered in Example Design Problem of High-performance Aircraft-Engine Component ...... 123

52 Physical and Process Domains Considered in Example Design Problem of High-Performance Aircraft-Engine Component ...... 126

53 lsogram of Process Variables Mapped Onto Physical Domain. PV Axes are Hot Deformation and Thermal Processes; DP Axes are Geometrical Shape and Microstructure; xo is Initial State; Xj is Intermediate State; xf is Final State ...... 129

54 Schematic Diagram of Rolling Process. Tempera- ture, Strain-Rate, and Plastic-Strain Profiles Given for Typical Thermomechanical Cycle ...... 132

55 Schematic Diagram of Extrusion Process. Tempera- ture, Strain-Rate, and Plastic-Strain Profiles Given for Typical Thermomechanical Cycle ...... 133

56 Schematic Diagram of Forging Sequence for Producing Integral-Blade and Rotor Component. Temperature, Strain-Rate, and Plastic-Strain Profiles Given for Typical Thermomechanical Cycle ...... 135

57 System Ranges of Rolling, Extrusion, and Forging Proc- esses Compared to Design Range for Producing Hypo- thetical Disk Component ...... 137 LIST OF FIGURES (CONTINUED)

58 Three-Dimensional Plot Showing Dependence of Recrystallized Grain Size of Plain C-Mn Steel Upon StrainRateandTemperature(Strain=l.O) ...... 156

59 Three-Dimensional Plot Showing Dependence of Volume Fraction Recrystallized of Plain C-Mn Steel Upon Strain and Temperature (Strain Rate = 1.0 s-1) ...... 157 LIST OF TABLES

Table Paae

1 Typical Efficiency Ranges for Hot-Deformation Mechanisms in Aluminum (From Ref . 51) ...... 42

2 Summary of Reported Recrystallization and Grain Growth Relationships (From Ref . 58) ...... 55 3 Results of Stability Analyses ...... 74

4 Typical Hot-Working Information for Some Common Metallic Systems (From Ref . 67) ...... 80 5 Characteristic Design Vectors for Material Systems ...... 139 FOREWORD

In the design of manufacturing processes, a significant gap exists in mapping from design of the product to implementation of process control on the shop floor. This situation presents major challenges to process designers who are faced with ever increasing constraints on quality, cost, and delivery. Such high-technology enterprises as the aerospace and automotive industries are increasingly requiring manufacturers to produce net-shape components having controlled microstructures and properties. In fact, the need for improvements in process design and control exists throughout the manufacturing community. The fundamental problem is to determine an effective thermomechanical path for achieving product requirements--a path which is controlled by the flow of energy, materials, and signals. A systems approach is required for integrating the interdisciplinary aspects of the design of manufacturing processes. Only through a comprehensive understanding of materials and processes can an effective solution to the process design and control problem be found. The objective of the research described herein was to bridge the gap between product design and process control by combining the fundamental principles of materials science, mechanics, control theory, and design theory. A scientific methodology for designing and controlling thermomechanical processes of metals and alloys is presented. In this research emphasis was placed on the development of process-control strategies based upon a fundamental understanding of the behavior of material during processing. In this dissertation it is assumed that the reader is familiar with the following related subjects: Information Theory, Stability Theory, Mechanics, Irre- versible Thermodynamics, Finite Element Analysis, Constitutive Equations, and

xvi Optimal Control Theory. Relevant principles are outlined where necessary, but details on these topics are not provided. Chapter I contains an introduction to the subject, including background on the nature of the problem, a statement of the specific research objectives, and a summary of results. In Chapter II the overall approach to the research is presented. A systems approach is used to connect the material, equipment, and control systems. The knowledge of material behavior of metallic alloys which are described as stochastic, dynamical systems is the basis for the design and control of therrnomechanical processes. The intrinsic workability, defined in terms of temperature and strain-rate space, provides the design range for processing a given metal alloy. Microstructural-evolution models are used to describe the kinetics of a given metallurgical process. Through the use of design axioms, a methodology for designing thermomechanical processes is formulated for in- depth evaluation of materials and processes. Furthermore, the construction of control-system designs is based upon control targets and system dynamics which are directly extracted from the material behavior. Hence, the material behavior is assumed to be the predominant factor in the design and control of t hermomechanical processes. In Chapter Ill the primary material characteristics relevant to hot- deformation processes are described. Previous efforts in the development of constitutive relations, in hot workability, and in microstructure development are reviewed. Chapter IV presents new perspectives in material-behavior modeling. Complementary relationships which exist among constitutive equation, dynamic material modeling, and microstructural evolution are established. In Chapter V case studies are examined for validation of the material- modeling approach as well as elucidation of the role of material behavior in process design. The material behavior of three different metallic systems, namely, titanium aluminide, 2024 aluminum alloy, and a nickel-base superalloy, are examined. Other aspects of material behavior such as information content and prior thermomechanical history are discussed. In Chapter VI the material and geometry aspects of product and process design are addressed through the use of Suh's axiomatic approach to design. The fundamental concept of functional, physical, and process domains clarifies the material and geometrical aspects of the design problem. The design of a new aircraft-engine component is used to demonstrate some important concepts of axiomatic design which pertain to thermomechanical processes. Cer- tain of these processes, e.g., forging, extrusion, rolling, and heat treatment, are characterized in terms of their typical range of capability for producing components of a certain shape, size, and "fineness" of microstructure. Chapter VII presents a scientific methodology for establishing a control strategy for thermomechanical processes. A systematic approach based on Suh's axioms for design is outlined for evaluating the relationship among property, microstructure, and processing. The required thermomechanical path is determined for producing a final desired microstructure, subject to material and equipment constraints. It is shown that a process-control strategy can be derived from the required therrnomechanical path. This strategy includes the selection of process design and control parameters such as the number and sequence of thermomechanical operations, the type of deformation process, the processing variables for each operation, and the starting condition of the workpiece material. xviii In Chapter Vlll the control-system design of a hot-metal deformation process is evaluated using optimal control theory. Formulation of the state-space model and its constraints and performance index is presented. Optimal conditions for producing the desired microstructural-evolution path and corresponding control inputs are determined for a plain carbon-steel alloy. Chapter IX contains the summary and conclusions. ACKNOWLEDGMENTS

I wish to express my deep gratitude and sincere appreciation to my advi- sor, Prof. Jay S. Gunasekera, for his exceptional guidance, valuable advice, and personal commitment throughout these investigations. I am extremely grateful to Dr. Venkat Seetharaman for his invaluable comments and suggestions, especially those pertaining to relevant aspects of materials science. I wish to thank Prof. Nam Suh and colleagues of the MIT Laboratory for Manufacturing and Productivity for discussions on the key aspects of axiomatic design. I am also grateful to Prof. Dennis Irwin for collabo- rative efforts in the area of optimal control techniques. I wish to acknowledge Dr. Steven LeClair, Dr. Lee Semiatin, Prof. Ramana Grandhi, and Mr. James Morgan for their encouragement and useful comments. Thanks are due to Messrs. Srivats Gopinath, Douglas Lang, Tim Voegeli, and Robert Goetz for their assistance in computation and graphics. I am also extremely grateful to Mrs. Marian Whitaker for editorial assistance and preparation of the dissertation, to Mr. Dale Fox, Mrs. Cynthia Sanders, and Mrs. Bethann Thompson for graphics assistance, and to Messrs. Jeffrey Pitsinger and Clarence Randall for photographic assistance. I wish to thank Dr. Harris Burte, Dr. Norman Tallan, Mr. Allan Gunderson, Mr. Robert Rapson, and Dr. Walter Griffith of the Materials Directorate of Wright Laboratory at Wright-Patterson Air Force Base for giving me the opportunity to conduct this research in a stimulating and invigorating environment. I also wish to acknowledge the encouragement and assistance which I received during the course of this research from Dr. Gordon Schanzenbach of Graduate Student ,Services and Dean T. Richard Robe, Prof. Nicholas Dinos, Prof. Jerrel Mitchell, and other members of the faculty of the College of Engineering and Technology at Ohio University. I am grateful to Drs. Harold Gegel and Sokka Doraivelu for introducing me to the new and exciting field of Processing Science. I express my sincere gratitude to my parents for years of loving support and encouragement. I also wish to acknowledge all other members of my family and the Moriarty Clan for their love and encouragement. Finally, I express my utmost love and gratitude to my wife, Eileen, for her endless patience, optimism, and devotion.

xxi NOMENCLATURE

Product design matrix Process design matrix Set of desired attributes Set of design parameters Set of field quantities Set of functional requirements Set of processing variables Dynamic recrystallized grain size, pm Final grain size, pm Desired final grain size, pm Initial grain size, pm Subgrain size, pm Static recrystallized grain size, pm

Power-dissipator content (rate of kinetic energy dissipated), J/(s m3) Instantaneous height, m Final height, m Initial height, m Information content Power-dissipator co-content (rate of potential energy dissipated), J/(s m3) Strain-rate-sensitivity parameter Applied power to material system, J/(s m3) Activation energy, kJ/mol Desired activation energy, kJ/mol Universal gas constant, 8.31 4 J/(g mol K) Roll radius, m Reduction in height, % Internal material variable Entropy applied to material system, J/(mol K) Entropy production by material system, J/(mol K) Temperature-sensitivity parameter Absolute temperature, K Initial workpiece temperature, K Desired initial workpiece temperature, K Melting temperature, K Hot-working temperature, K Desired hot-working temperature, K Time, s Dwell time, s

Ram velocity, mls Desired ram velocity, m/s Roll velocity, m/s Liapunov function for mechanical stability Liapunov function for material stability Volume fraction dynamically recrystallized Volume fraction not recrystallized Volume fraction statically recrystallized

State of material system (e.g., microstructure and geometry) Initial state of material system Final state of material system Zener-Holloman parameter, s-1 Shear strain rate, s-1 True, effective strain Desired effective strain Critical plastic strain for dynamic recrystallization Critical plastic strain for dynamic recovery Peak strain Steady-state strain Fracture strain Plastic strain for 50-volume-percent recrystallization Effective strain rate, s-1 Desired effective strain rate, s-1 Efficiency of power dissipation with respect

to metallurgical processes, % Principle stresses, MPa True effective stress, MPa Fracture stress, MPa Peak flow stress, MPa Steady-state flow stress, MPa Shear stress, MPa I. INTRODUCTION

! For ,centuries, skillful artisans have employed thermomechanical operations such as rolling, extrusion, forging, and heat treatment in the production of metallic goods. These manufacturing processes usually involve a series of thermomechanical operations which transform the workpiece material into a useful finished product characterized by specific shape, size, and properties. A typical sequence of thermomechanical operations consists of a hot-deformation process followed by a heat-treatment process. The shape of the product is achieved through the deformation-processing step(s). The properties of products are, in general, dependent upon the entire thermomechanical-processing history of the product. New materials are being developed for use in cases where more precise, special processing is often required to achieve the desired properties. Hence, knowledge of material behavior under processing conditions is important in the design and control of thermomechanical processes-- especially in the case of alloy systems developed for high-technology applications.

CURRENT APPROACHES TO DESIGN OF METALFORMING PROCESSES

Techniques for the design of manufacturing processes do not give adequate consideration to the role of materials. The typical framework for the designing, planning, and manufacturing activities in a computer-integrated manufacturing system' is shown in Fig. 1. Techniques have been developed for primary support of management and marketing functions. Indeed, the criteria used for optimizing the design and control of manufacturing processes are /- \ 'EXTERNAL \ ( INFORMATlIIN \ -7'

1 MANAGEMENT PLANNING I I I I

7--

I -----1 / - ," RAW '\ I - MANUFACTURING . ( MATE"AL ,kj/ ACOUISITION \ SUPPLIER / CAM) '--1 I I 1 I - + I I QUALITY PAODUCTlON I I CONTPOL CONTROL 1 I CAT I (LEGEND) I I = = a Maletla1 flow I 1 - I - - - - t Technolcg~almlormaton lblr I-____-______I [CAM] I - - - Management inlormalw fbw

Figure 1. Framework of Functions Involved in Computer-Integrated Manufacturing System (From Ref. 1). as follows: 1) maximize the production rate, 2) minimize production costs, and 3) maximize the profit rate. These approaches are quite suitable for conventional materials such as steels, copper, and aluminum-based alloys which can be processed successfully under a wide range of processing conditions. In other words, these materials can be said to possess wide "processing windows" in terms of workpiece temperature and strain rate. Such approaches to process 3 design and production planning are mainly concerned with the ease of obtaining the finished shape and size of the final product, and specific information concerning the material behavior is lacking. For example, total production time may be minimized with little regard for the ability of the material to respond favorably at the imposed production rate. In contrast, materials such as superalloys, intermetallics, ordered alloys, and metal-matrix composites have restricted or narrow "processing windows." During processing of these materials, variables such as temperature and strain rate must be monitored and controlled continuously, lest the material experience different types of instabilities. In the extreme case, these materials may undergo dynamic fracture if processed at high deformation rates; this problem can generally be overcome by selection of higher processing temperatures. Thus, knowledge of material behavior must be explicitly taken into account in the design and control of manufacturing processes, especially for difficult-to-process materials. In the case of advanced materials, quality assurance is increasingly being integrated into the process design. Since t hermomechanical processing usually involves a sequence of operations, it is necessary to control processing variables during each operation in order to ensure that product quality is achieved along with desired microstructures and mechanical properties. Among the important processing variables are 1) the initial condition of the workpiece material, 2) the number and sequence of thermomechanical operations, and 3) the processing conditions which correspond to each operation. Processing sequences have traditionally been designed through the use of various empirical rules which have evolved as a result of many years of experience in shape making. Build-and-test methods have commonly been used for die design and for the selection of processing variables such as working temperature and ram velocity. These methods result in high tooling and set-up costs and in long lead times before production which, in turn, adversely affect the manufacturing enterprise.2

I In build-and-test methods, success is strongly dependent upon the avail- ability of appropriate experience. These methods are simply not feasible for processing of newly developed alloys such as 2024 Al + 20 vol.% Sic whisker- reinforced composite material where control of processing conditions is unusually strict.3 Processing difficulties prevent many emerging materials from reaching their potential in a wide range of applications. Consequently, conventional materials continue to be used as substitutes, leading to performance trade-offs: For example, processing problems with titanium-aluminide alloys have forced designers to consider a nickel-base superalloy as an alternative material for the next generation of high-performance turbine engine^.^ Recently many limitations of build-and-test methods have been overcome through the use of process models based upon computer-simulation techniques. In several industries state-of-the-art process modeling has reached a high level of sophistication and acceptance as a process-analysis to01.~~~Cur- rent process models are capable of analyzing very complex material-flow operations such as three-dimensional, non-isothermal deformation processes with a very high degree of, accuracy. For example, in the forging industry detailed numerical analyses of the phenomenon of the workpiece filling the forging die, the resulting die stresses, and the post-deformation heat treatment of the workpiece are being applied increasingly for verification of forging and heat-treatment process designs. l~heuse of process-modeling techniques contributes to reduced overall production costs, improved quality, and reduced lead times? However, significant improvements in the capabilities of process 5 models must be made before sequences of thermomechanical processes can be analyzed adequately. Furthermore, the current approach to computer-aided design coupled with process modeling relies on iterative methods for the selection of the processing conditions for a given step in the thermomechanical process. Despite these limitations, the use of process models has greatly facilitated the processing of advanced materials. Computer-based process models have made it possible to design by iteration. This approach is quite attractive because frequent changes in the process design can be made without fabrication of tooling. However, even this approach to design remains ad hoc. It is important to develop a systematic methodology for design based upon fundamental scientific principles. Such an approach would minimize the number of iterations required to achieve an acceptable solution and, at the same time, lead to alternative and unconven- tional solutions which are beyond the scope of ad hoc approaches. Recently, several innovative attempts have been made to develop such a scientific approach. These include the design methodology proposed by ~uh,7 Doraivelu, et al.,8 Ham and Kumara,g ~ixon,loand Yoshikawa.11 The general goal of these approaches is to eliminate iterations in the design process. How- ever, some iterations may be required due to incomplete knowledge of the relationships which exist between processing variables and product quality.

CURRENTAPPROACHESTOPROCESSCONTROL

Monitoring and control of thermomechanical processes plays a major role in achievement of the quality goals of the final product. Present control instrumentation on metalforming equipment essentially consists of open-loop 6 systems with either manual or microprocessor-operated throttles and valves. Closed-loop control systems have been implemented in only a few metalforming processes such as multistand rolling operations where these systems are employed primarily for gauge or dimension control. A specific example of inadequate process control involves the aluminum precision-forging industry where rejection rates are as high as 35x.12 The production of high-quality products on a reproducible basis can be achieved only through the use of automated control systems. Computer-based control and data-acquisition systems for metalforming processes are being incorporated increasingly on production equipment. Commands are preprogrammed in the control system for 1) executing specified ram-velocity profiles, 2) achieving final ram displacements, and 3) maintaining desired temperature distributions. Modern metalforming equipment such as servo-hydraulic forge presses and section rolling mills possess the necessary dynamic-response characteristics for implementing the process-control algorithms mentioned above. In a recent study conducted by the National Research Council,l3 several reasons for the lag in advances in the control of metal processing were identified, in addition to the most obvious one involving re-equipping factories. Approaches to control system design and execution of metalforming processes must be improved if the total quality benefits of automation are to be realized. Methodologies are needed for establishing control strategies and determining performance specifications of actuators and sensors. Classical control methods and sophisticated artificial-intelligence methods are at opposite ends of the spectrum in terms of complexity in modeling, sensing, and controlling processes. Approaches with intermediate degrees of complexity such as modern control theory have not been widely exploited. Classical control methods 7 employ a "single-inputlsingle-output" approach which cannot effectively treat the nonlinear, time-varying multivariable nature of metal-deformation processes. Sophisticated artificial-intelligence methodologies in the context of expert systems are reported to have only qualified applicability with respect to specific alloys and processes.13-15 Neither of the extreme approaches offers generic utility in the control-system design and execution of metalforming processes. Hence, control techniques can be improved by explicit incorporation of fundamental knowledge of the material behavior and the physics of the process.

SCOPE AND OBJECTIVES OF RESEARCH

Research and development efforts in the area of thermomechanical processing of metals and alloys have experienced steady growth over the past several decades. Multidisciplinary approaches have been adopted in attempts to understand the fundamental aspects of the problem. The scientific disci- plines involved in this research area include Metallurgy, Mechanics, Thermody- namics, and Control Theory. In previous eff0rts,16~17significant progress was made in material-behavior and process modeling and studies on tribological effects and forming-equipment characteristics. The results of these efforts have been effectively utilized in understanding and analyzing materials and processes. Recent advances in material-behavior modeling and design theory have aided in the design and control of material processes. The development of "dynamic material modeling" has led to a better understanding of the behavior of complex engineering alloys under hot-forming conditions. Dynamic material modelingl8-20 is based on the combined approach of irreversible 8 thermodynamics and continuum mechanics for describing the manner in which a material dissipates power during deformation processing. Design theory has also experienced a major scientific breakthrough in the establishment of fundamental axioms and theorems which can be universally applied to any design domain.7121 The new science of design provides a systematic approach to "direct design" which reduces the need for iterative analysis while assuring that the design will fully satisfy the requirement. Recent advances in material- behavior modeling and design theory have provided a more fundamental understanding of property/microstructure/'processing relationships for metals and alloys. Thus, the overall goal of the present research was to develop a methodology for design and control of microstructure throughout a sequence of thermomechanical processes. The specific objectives of the research described herein are as follows: 1) To establish a scientific methodology for determining the processing conditions under which material behavior is stable and predictable. 2) To establish an axiomatic approach for determining the initial condition of the workpiece material, the number and sequence of thermomechanical operations required, and the processing conditions corresponding to each operation. 3) To establish a scientific methodology for determining the optimal microstructural evolution path and the corresponding optimal control inputs for a given metalforming process. The present scientific investigation has resulted in significant advancements in the areas of material-behavior modeling, process-design methods, and process-control strategies for thermomechanical processes. In material- behavior modeling, the apparent activation energy for metallurgical processes 9 has been discovered to be a fundamental link between atomistic and continuum phenomena. Together with stability analyses, the magnitude of the activation energy and its variation provide the means of identifying the processing regimes within which the material behavior is essentially deterministic. Through the use of this approach, a more precise definition of the "processing window" for a particular material and its boundaries has been established. The roles of activation energy and stability in material-behavior modeling have been demonstrated to be valid in different systems of varying complexity. Further- more, these new perspectives are shown to elucidate the effects of thermomechanical history and the degree of structural intricacy upon material behavior. These advancements in material-behavior modeling have enabled the explicit incorporation of material phenomena into process design and control methodologies. A new method for designing thermomechanical processes has been established; this method is based upon scientific principles from design theory, materials science, and process mechanics. Existing design axioms and related concepts are applied to this problem for the first time. As in other applications of these axioms, exploitation of functional independence and minimal information content has been found to be the key to successful design solutions. An unforeseen result of this research has been the revelation of the fundamental role of material characteristics and geometrical shape in the creation and elimination of process-design solutions. In fact, a measure of the information content for a material-processing system has been found to be the temperature and strain-rate domain which overlaps the material-processing window and the capacity range of the forming equipment. Through the use of example problems, it has been demonstrated that the axiomatic approach is realistic and 10 technically feasible for both global and detailed design of thermomechanical processes. A new strategy for controlling thermomechanical processes has been established; this strategy is based upon scientific principles from optimal control theory and materials science. Novel contributions of this part of the research are: 1) the first direct application of optimal control techniques to manufacturing processes, and 2) the first control method which is explicitly based upon analytical models of the governing material-behavior phenomena. The formulation for controlling dynamic-recrystallization processes has been derived from existing models of microstructure development. 11. APPROACH

SYSTEMS APPROACH TO METALS PROCESSING

The basic approach adopted in this research effort was to consider a sequence of thermomechanical operations and the total system associated with each operation. The basis for design and control is the behavior of the workpiece material under processing conditions. For a given processing operation, the total system consists of several subsystems, including equipment, material, and control subsystems. It is important to note that while the total system is considered to be isolated (i.e., neither matter nor energy is exchanged across its boundaries), the individual subsystems are regarded as closed (i.e., only energy is exchanged across their boundaries). A sequence of thermomechanical operations is considered to be a series of events which transforms the state of the workpiece material in terms of its shape, size, microstructure, and properties. Hot-deformation processes and thermal treatments are the two basic types of thermomechanical operations considered in this research for the production of metallic components having controlled microstructures and properties. In the case of hot-deformation operations (forging, rolling, extrusion), the tooling, lubrication, and workpiece material are considered to be a material subsystem; the metalforming press and furnace which deliver the thermomechanical energy to the material subsystem for carrying out the process are considered to be an equipment subsystem; and the devices which regulate the energy delivered are considered to be a control subsystem. A schematic representation of the total system associated with hot-deformation processes is shown in Fig. 2(a). The subsystems are closely linked through identifiable input CONSlTUTlVE EQUATION FOR PLASTIC DEFORMATION AND HEAT TRANSFER 1

METALWORKING MATERIAL SYSTEM EQUIPMENT WORKPIECE SYSTEM DIES 0LUBRICATION

CONTROL DYNAMIC MATERIAL MODEL ALGORITHM +MICROSTRUCTURE- EVOLUTION MODELS

CONTROL SYSTEM

CONSTITUTIVE EQUATION / FOR HEAT TRANSFER

MATERIAL SYSTEM 0WORKPIECEf

TIME-TEMPERATURE- SET-POINT TRANSFORMATION DIAGRAMS CONTROL + MICROSTRUCTURE- EVOLUTION MODELS

CONTROL SYSTEM

Figure 2. Schematic Representation of Two Basic Thermomechanical Processes as Systems Composed of Material, Equipment, and Control Devices. (a) Hot-Deformation Processing System, (b) Thermal-Processing System. and output variables which govern the flow of energy, materials, and signals. The equipment subsystem is connected to the material subsystem through constitutive equations; these mathematical expressions govern plastic deformation and heat flow during processing. The material subsystem is, in turn, interfaced with the control subsystem through hot-workability models and microstructure-evolution models. Hot-workability models are used to determine the feasible design range for control in strain-rate and temperature state space. The dynamic-material-modelingl8-20 approach to hot workability was used in this research. Within feasible design ranges, models of the microstructure development were used to find a desirable metallurgical path. The control and equipment subsystems are interfaced by control algorithms for regulating process parameters such as tooling and furnace temperature and press ram movement and force. Similarly, thermal processes such as heat treatment, diffusion bonding, and surface hardening require systems composed of material, equipment, and control devices. A schematic representation of the total system used for thermal processes is shown in Fig. 2(b). In this case, the material and the equipment subsystems are simply the workpiece and furnace, respectively. The material subsystem is related to the control system through time-temperature- transformation diagrams and microstructure-evolution models. These diagrams are employed for determining the feasible design range for thermal processing, while the models are used for finding particular processing conditions within the feasible range which produce a desired microstructure. The control subsystem is related to the equipment by a simple algorithm for regulating the temperature of the workpiece material. The systems approach to metals processing, with emphasis on material behavior, is incorporated into the existing analysis and optimization techniques for developing an improved design and control methodology. The mathematical models which describe the relationships among the material, equipment, and control systems are used in the conceptual through detail stages of design. Figure 3 is a schematic diagram of the overall approach used in the methodology development. The dashed box and arrows identify those aspects of the problem which are beyond the scope of this dissertation (and, therefore, are not described in detail) but have been included in the figure to

I I PRODUCT DESIGN I n I STRUCTURAL - - -1 SPECIFY ALLOY AND MICROSTRUCTURE ANALYSES I SPECIFY GEOMETRY AND TOLERANCES

PROCESS DESIGN SIMPLIFIED ANALYSES P I I SPECIFY INITIAL MICROSTRUCTURE OF THE PROCESS SPECIFY INITIAL BILLET SIZE -i I SPECIFY NUMBER AND SEQUENCE OF THERMOMECHANICAL PROCESSES DESIGN AXIOMS SPECIFY NOMINAL PROCESSING CONDITIONS FOR EACH OPERATION s

OPERATION DESIGN

CONTROL-SYSTEM DESIGN OPTIMAL CONTROL TECHNIQUES - SPECIFY TEMPERATURE CONTROLS - SPECIFY DIE-MOVEMENT CONTROLS TOOLING-SYSTEM DESIGN - SPECIFY DIE GEOMETRY FOR EACH OPERATION ANALYSES OF THE - SPECIFY DIEIWORKPIECE PROCESS LUBRICATION AND COATING SYSTEM

Figure 3. Overall Approach Used in Development of Methodology for Design and Control of Thermomechanical Processes. 15 provide a comprehensive view of the product and process development. The nature of the material behavior under processing conditions is explicitly taken into account in all design activities. The conceptual design of the product and the manufacturing process should be accomplished simultaneously. However, in this research the material and geometry information of the product design is assumed to be known, along with constraints related to cost and delivery schedules. The basic framework for establishing the manufacturing process, in general terms, is provided in order to satisfy effectively the product-design requirements in the application of design axioms. Simplified techniques for analyzing hot deformation and thermal processes are used in the conceptual stages of process design for evaluation of manufacturing alternatives. For a particular thermomechanical operation, optimal control techniques and finite- element methods are employed for control- and tooling-system design, respectively.

GLOBAL DESIGN OF THERMOMECHANICAL PROCESSES

Development of a methodology for the design of thermomechanical processes was based upon the application of the design axioms and concepts according to Suh21-22 whose design world is divided into four domains, as shown in Fig. 4. First is the consumer domain where the needs of customers are expressed as the desired attributes of the product. In the second or functional domain, the attributes desired by the customer have been transformed into a set of functional requirements {FRs). The chosen {FRs) are then mapped to the physical domain as a set of design parameters {DPs) through the mani- festation of a product. Once the product design has been completed through CONSUMER FUNCTIONAL PHYSICAL (PRODUCT) PROCESS DOMAIN DOMAIN DOMAIN DOMAIN

Figure 4. Four Domains of Suh's Design World (From Ref. 21). the establishment of {DPs}, a manufacturing process can be designed in the process domain, which is characterized by a set of process variables {PVs). The two fundamental axioms established by Suh for systematic design are called the Independence Axiom and the Information Axiom. Other important concepts in axiomatic design include 1) the existence of hierarchies, 2) decomposition of characteristic vectors, and 3) the zig-zagging requirement for decomposition. The methodology of Su h provides a systematic approach for incrementally deriving design solutions at various levels of detail. Of course, design solutions are nonunique because a given design depends upon the {FRs) specified by the designer, the {DPs) selected by the designer, and the constraints imposed by the designer. However, Su h's methodology provides techniques for evaluating the 'relative merit of a given design. In this research, the specific domains investigated were mechanical properties, microstructures, and material processes. In the context of Suh's design world, mechanical properties are assumed to be a subset of {FRs} in the functional domain. Similarly, microstructures are assumed to be a subset of {DPs) in the physical domain and material processes to be a subset of {PVs) in the processing domain. Relationships between these domains are evaluated in 17 terms of the two design axioms for finding a nominal solution for effective production of the required product. Simple techniques such as slab analysis of deformation processes and lumped heat-transfer analysis of thermal processes provide useful estimates of processing variables associated with a particular operation.

DETAILED DESIGN OF THERMOMECHANICAL OPERATIONS

Details of the selected thermomechanical operations are determined after the global process design has been established. In this research control- and tooling-system design are considered to be detailed design activities. The control systems are designed to provide the dynamic conditions which ultimately affect the state of the material in a desirable way. Details of the tooling systems are designed to control the spatial distribution of microstructure development throughout the workpiece material. Through the use of optimal control techniques,23-z4 an open-loop control system can be designed for a given operation which will provide the necessary control inputs for producing the desired final microstructure. The open-loop control approach for hot-deformation processes is shown in the block diagram in Fig. 5. Models of microstructural development are used to determine the desired field variables of the process such as strain (E*), strain rate (E'), and deformation temperature (T;) for transforming the initial microstructure (do) into the desired final microstructure (d;). Subsequently, process models are used to find the corresponding desired equipment control parameters such as ram velocity (V'), and billet temperature (T;). Treatment of the differences between the desired states and the actual states which occur in practice is not within the scope of this investigation. 18 MATERIAL EQUIPMENT CONSTRAINTS CONSTRAINTS

A t r 1 do MICRO- E* '-'*(I) STRUCTURE - ' t - & * DEVELOPMENT -PROCESSMODEL MODEL T$ Tg* (f)

b b-

L P V*(t) V(t) t METAL- WORKING MATERIAL d f SYSTEM - Ti(1) EQUIPMENT ' Ts(f) - SYSTEM C

_I *

Figure 5. Block Diagram of Open-Loop Control of Metalworking Processes.

In the production of complex components, a common problem is the development of nonuniform temperature and/or plastic-deformation distribution(~)throughout the volume of the workpiece material. Finite-element methods can be used to predict the magnitude and variation of field quantities during a thermomechanical process. This method of analysis assumes that any continuously varying quantity such as temperature or velocity, which is defined over a region of space, can be approximated by a set of interrelated elements of the problem. A thermal visco-plastic finite-element program ALP ID^^ was used for the analysis of hot-deformation processes. The optimal solution for open-loop control is given as an input to the finite-element model of the process. From ALPlD predictions, details of the tooling system are adjusted for controlling the location and distribution of microstructure development in the deforming workpiece. Ill. MATERIAL-BEHAVIOR MODELING OF METAL ALLOYS UNDER PROCESSING CONDITIONS

The behavior of metallic systems during thermomechanical processing must be studied both macroscopically and microscopically. Flow and fracture phenomena as well as the microstructural mechanisms which are operative during hot deformation and/or during post-deformation heat treatment are important processing characteristics of the workpiece material. These characteristics collectively govern the size, shape, and service properties of finished parts which are producible by thermomechanical processes. Plastic flow and fracture behavior can be regarded as macroscopic phenomena which generally restrict the complexity and dimensions of part geometries producible without gross instabilities and defects. The mechanical properties of metallic components are dependent upon the specific features of the microstructure such as volume fraction, size, shape, distribution, and orientation of the existent solid- state phases. The desired microstructure is usually not obtained in the original casting or ingot but is produced by carefully controlled processes involving hot working and heat treatment. The primary material characteristics relevant to hot-deformation processing are inelastic-flow properties, hot workability, and operative metallurgical mechanisms. The particular flow behavior of a metallic system is modeled through the use of constitutive equations which relate the flow stress to field quantities such as effective strain rate and temperature. Only the constitutive relations applicable for a rigid plastic deformation are considered in the present research, i.e., elastic behavior is neglected. Several workability models have been developed for determining the relative ease with which a metal can be shaped through plastic deformation. A continuum and thermodynamic

19 20 approach to hot workability, called dynamic material modeling (DMM), was used in the present investigation because of its applicability to complex metallic systems with a minimum of assumptions. The metallurgical mechanisms which are operative during hot-deformation processing (i.e., dynamic mechanisms) can be identified by detailed metallographic studies of test samples. Changes in microstructure are quantified, and the kinetics of the metallurgical processes involved are mathematically modeled. An understanding of the primary material characteristics of the workpiece is necessary in the design and control of efficient thermomechanical processes. In this chapter basic concepts and previous work relative to constitutive relationships, hot workability, and modeling of microstructure development are reviewed.

CONSTITUTIVE RELATIONSHIPS

Constitutive equations describe the nonlinear relationship which exists among such processing quantities as effective stress, effective strain rate, and temperature at different deformation levels. These equations which are unique and specific for each material under each processing condition are developed through the use of data obtained under simplified experimental conditions which can be extended to complex situations by means of well-known hypothe- ses. For the case of plastic deformation, the von Mises yield criterion is commonly used for extending data from uniaxial tension tests (or compression tests) to multi-axial deformation conditions. Both types of equations are used in con- junction with processing models such as slab analysis and finite-element analysis to predict deformation loads and the processing behavior of the 21 material. The mathematical form of the constitutive equations affect the numerical convergence and accuracy of the processing models.

In general, the constitutive equations for material modeling are expressed as follows:

o = f($ T, S), where ;is the effective strain rate, T is the absolute temperature, and S is an internal variable whose magnitude is dependent upon the deformation history of the material and is related to the current microstructure. These variables are discussed in detail in this section. According to the von Mises yield criterion, the flow stress of homogeneous and isotropic materials can be expressed by

where o is the effective stress and GI, 02, and 03 are the principal stresses.

In the case of a uniaxial state of stress (02 = 03 = O), the von Mises yield criterion leads to 5 = 01. Therefore, during uniaxial loading, the stress o applied externally onto the cross-sectional area of the specimen is equal to the flow stress and is given by 22 where F is the instantaneous force and A the instantaneous cross-sectional area. Hereafter, 3 = o = flow stress. The true strain for compression, z, is defined as

where h, ho, and hf are the instantaneous, initial, and final heights of the specimen, respectively. Hereafter, i = e = true strain. The effective strain rate is defined as

- '-dE=dh=Idh=VE - dt hdt h dt h '

where V is the instantaneous velocity of the cross head. Hereafter, = E = effective strain rate. The instantaneous temperature, T, of the specimen during deformation is referred to as the adiabatic temperature and is defined as T = To + AT, where To the initial temperature and AT the temperature increase during deformation. The current state of the material is defined by an internal variable S which is related to its instantaneous microstructure. Some examples of history- dependent attributes of the microstructure are size, morphology, volume fraction and composition of constituent phases of the material. Experimental techniques such as quantitative metallography can be used to determine the functional relationship between S and the other state variables given in Eq. (1). However, it is generally sufficient to express S implicitly in the constitutive equation by referring to the thermomechanical history of the starting material. In other 23 words, the relationship among o,&, and T is expressed with respect to the initial condition of the material S. Furthermore, the effect of S upon the flow behavior of the material is minimal in hot-deformation processes involving large strains (i.e., typically E > 0.6).

Values of flow stress for different temperatures and strain rates are required for the development of constitutive equations. Such values are generally obtained from flow curves which have been generated using standard compression tests for metalforming applications. Compression tests are pre- ferred over other testing methods such as tensile and torsion tests because most metalforming operations involve predominantly compressive states of stress. Other advantages of compression testing include ease of preparing test specimens and the ability to obtain large strain and strain-rate values. Tooling setup required for these tests and data acquisition are treated in a number of paper~.~6-28Also, the same flow-stress values are used in the development of DMM processing maps, as discussed in a later section.

Flow-Stressand Brittle Materials

The effect of deformation parameters upon plastic flow and fracture can be generally observed in the flow-stress behavior. The relationship between flow stress and true strain is frequently referred to as the flow curve because it determines the stress required to cause the material to flow plastically under any given strain. A typical flow curve for a ductile material having low stacking 24 fault energy is shown schematically in Fig. 6. At the onset of deformation, the material undergoes a hardening stage where the flow stress increases rapidly with strain. As the deformation proceeds, the rate of hardening decreases until the flow stress reaches a peak value. The strain value, ep, corresponding to the peak stress, 9, represents the critical amount of deformation required for a balance of the hardening and softening contributions to the flow stress of the material. Subsequent deformation involves a softening stage in which the flow stress gradually decreases until it reaches a steady-state value, as,at the corresponding strain value, ES. For typical ductile materials, the fracture strain, &fr, is relatively large, i.e., (ep + E,) < efr. The significance attached to the steady-state stress, o,, is transferred to the peak stress, op, in the case of brittle materials. As shown in Fig. 6, the flow stress appears to continue to decrease and does not reach a steady-state value for materials undergoing fracture.

Figure 6. Typical Plow Curves for Ductile Material (Solid) and for Brittle Material (Dashed). The relationship between flow stress and strain rate is studied using the power-law or logarithmic expression. A typical plot of log o as a function of log

E is shown schematically in Fig. 7. In general, log o increases with log E. The rate of change of stress with strain rate at constant levels of strain and temperature is known as the strain-rate-sensitivity parameter, m, which is defined as follows:

m = [a(log o)/a(log E)], ,T. (6)

As log o increases, the m-value typically decreases. For typical brittle materials, the fracture stress, of,., is exceeded at relatively low stress values, as shown in Fig. 7.

The gamma-TiAl-based alloy containing 49.5% Al, 2.5% Nb, and 1 .l% Mn is an example of a material which exhibits characteristics similar to brittle

DUCTILE LOG ofr ------BRllTLE ---

LOG o

J__

LOG i FigureI 7. Typical Stress-Strain Rate Relationships for Ductile and Brittle Materials. 26 deformation at high temperatures. Seetharaman and Lombard29 investigated the hot-deformation behavior of this alloy using the constant-strain-rate, isothermal-compression test technique. Their results are used to illustrate the flow-stress behavior described above. For six different temperatures, the log op vs. log ti behavior of this TiAl alloy is shown in Fig. 8(a). For the six curves shown, m varies from 0.4 to 0.0 in a similar manner. At strain-rate values of - 1

s-1 (i.e., log E = 0), m = 0 and severe cracking is observed. Figure 8(b) shows

the corresponding log op vs. log E behavior. In this case, a much more linear

relationship is observed, with m being constant up to 1 s-1 (i.e., log E = 0). The flow-stress dependence upon the reciprocal temperature can be related to metallurgical mechanisms which are operative during the process. A schematic diagram of a typical log o vs. 1TT relationship is shown in Fig. 9. In general, log o decreases with increasing temperature. The temperature sensitivity of the flow stress is analyzed in terms of a parameter, s which is defined as follows:

As shown in Fig. 9, the s-values associated with dynamic recrystallization and dynamic phase transformations are generally greater than those associated with other metallurgical restoration processes. The temperature dependence of the Ti-49.5AI-2.5Mb-1.1 Mn alloy can be considered typical of the high-temperature flow behavior of metallic systems. Compression test results indicating the temperature dependence of flow stress for this material are shown in Fig. 10.29 The relationship between log o and 1000TT is similar for five different decades of strain rates. The equivalent slopes of the five curves imply that the same mechanism is dominant throughout this LOG (PEAK STRESS, MPa) LOG (FLOW STRESS, MPa) DYN. RECRYS. + DYN. RECOV. DYN. RECOV. . E =

DYN. GRAIN

PHASE TRANS- FORMATION / / / < - 1IT

Figure 9. Typical Stress-Temperature Relationship for Metallic Systems. range of strain rates. Dynamic recrystallization is the corresponding dominant metallurgical mechanism in this TiAl alloy under these test conditions. Further- more, the slope of all five curves rapidly decreases at a reciprocal temperature corresponding to - 6.3 x 10-4 K-1 (T = 1300~C).This sudden drop in flow stress with respect to temperature results from a concurrent phase transformation (y -+ y + a) of the Ti-49.5AI-2.5Nb-1.1Mn alloy which occurs above 1300°C.

. . ConsUtlve FqWions for Hot-Deformation Behavlor

Constitutive laws for describing the inelastic deformation behavior of metals have been under development for the past four decades. ~homas30 summarized the results of these studies in a comprehensive overview article on flow-based analysis of metal-deformation processes. Recently, several researchers31-33 have proposed phenomenological approaches to the construction of constitutive equations based upon micromechanics of plastic flow. Figure 10. Flow Stress Versus Reciprocal Temperature Relationship for Ti-49.5AI-2.5Nb-1 .I Mn Alloy.

An objective of these efforts is to establish a concise relationship between deformation behavior and microstructure. The practical feasibility of these sophisticated constitutive models with respect to accuracy and computational efficiency has not yet been established. Hence, empirical relationships based upon the power law or logarithmic expressions are generally used to describe the high-temperature deformation behavior of materials.34 30 Common relationships used to describe the effects of temperature and strain rate upon the flow stress are summarized below in Eqs. (8) - (11 ):

and

In Eqs. (8) and (9), the relationships between flow stress and strain rate are given for constant-temperature and large-strain (i.e., E > eS) conditions. The constant K in these equations is a temperature-dependent strength parameter. The m-value used in Eqs. (9) and (10) is the strain-rate-sensitivity parameter which was defined in Eq. (6). For complex alloys at a given deformation, the m- value varies as a function of strain rate and temperature. In Eqs. (10) -and (11 ) the flow stress is related to the quantity Z which is called the Zener-Holloman pararneter;ss Z is also referred to as a temperature-compensated strain rate

where Q is the activation energy and R is the universal gas constant. Equation (10) is a phenomenological relationship developed for creep deformation which also adequately describes the flow-stress behavior during hot working. Equa- tion (1 1) is a particularly useful relation proposed by Sellars and ~egart36~3~ 31 because it generally describes the flow-stress behavior over a wider range of temperatures and strain rates than does Eq. (10). As demonstrated by Sellars and ~egart,36~37Eq. (1 1) approximates Eq. (10) at low (a o < 0.8) stresses. In Eq. (1 1) A, a, and n are experimentally determined constants. The advantages of the hyperbolic sine function given in Eq. (1 1) are exhibited in the constitutive behavior of the Ti-49.5AI-2.5Nb-1.1 Mn alloy. The apparent activation energy, Q, for this alloy has been reported by Seetharaman and Lombard29 to be 327 10 kJ/mole in the strain-rate regime 10-3 - 10 s-l and the temperature regime 1000 - 12500C. They also showed that this value of activation energy for plastic flow correlates with dynamic recrystallization and intragranular deformation processes such as dislocation climb and cross slip. The Zener-Holloman parameter, Z, is computed using Q = 327 kJ/mole for the Ti-AI-Nb-Mn alloy. The flow-stress value corresponding to each temperature- compensated strain rate, Z, is shown in Fig. 11. In Fig. 11(a), the logarithm of o versus the logarithm of Z corresponds to the power law given in Eq. (10). The slope of the curve-fit line shown in Fig. 11(a) is the m-value given in Eq. (10). In the range of 8 < log Z < 15, the m-value is shown to be positive and to decrease with increasing log Z. The constitutive equation given in Eq. (1 1) has been verified, and the results are shown in Fig. 11 (b). A straight-line relationship between log[sinh(a a)] and !og Z is shown with a slope of 3.9. The following constitutive equation29 represents peak stresses for this alloy:

If the power-law approach given by Eq. (10) were used instead of the sine- hyperbolic-functionapproach (i.e., unified constitutive equation) given by Eq. (1 I),then the resultant linear relationship between log Z and log a would break LOG (Z,s-l) (a)

LOG (Z,s-1) (b)

Figure 1 1. Flow Stress Versus Zener-Holloman Parameter Relationship Using (a) Power Law and (b) Hyper- bolic Sine Function. 33 down at - log Z = 11-5. Thisis consistent with the observation that the power- law approach is valid for low stresses only.36137

HOT WORKABILITY

The Metals Handbook38 loosely describes workability as the relative ease with which a metal can be shaped through plastic deformation. Previous developments in workability theory and testing methods have been reviewed by Dieter,38-40 Semiatin and Jonas,41 Kubzn,42 and Gegel.20 In 1967 Hart43 formalized a method for studying flow locaiization through normalized flow- softening rate parameters and strain-rate-sensitivity parameters. The most widely used fracture criterion was proposed in 1969 by Cockcroft and L~itham~~ which simply recognizes the joint roles of tensile stress and plastic strain in producing fracture. In these studies and others,45-46 workability is defined in a number of ways. For example, ~homas30refers to workability as the maximum amount of strain (defined locally) which a workpiece can experience without undergoing failure. In some cases this can be related to measured ductility parameters such as percent reduction of area. According to Dieter38 workability is not considered to be a unique property of a given material since it is dependent upon process variables such as strain, strain rate, temperature, friction conditions, and the stress system imposed by the process. In brief, workability is considered to be equal to a scalar product of two functions--a material function, f,, and a process function, f2. However, these functions have not been defined adequately, and their independence has not been established. In the present study workability is defined as a material property which should be independent of changes in geometry, die design, press 34 characteristics, and lubrication condition. For example, center-burst phenomena observed during extrusion, as shown in Fig. 12(a) for one set of conditions, strongly suggest poor workability for the workpiece material. However, for the same material and die geometry, center bursting is not observed, as shown in Fig. 12(b), for a different set of processing conditions. Favorable workability

Figure 12. Sample of Round Products Extruded at Temperature of 500°F and Speeds of (a) 1.5 in./min, (b) 0.9 in./rnin. 35 in this case can be attributed to the ability of the material to dissipate power and reduce tensile stresses by a favorable mechanism, which avoids fracture and plastic instability. More recently, attempts have been made by Frost and Ashby4' and by Raj48 to describe the deformation and fracture processes which occur during deformation processing. Both approaches are deterministic with shear strain rate equations, valid for the steady state, being formulated by assuming that the contribution from a number of basic atomistic processes such as dislocation motion, diffusion, grain-boundary sliding, twinning, and phase transformation are independent and, therefore, additive. A brief description of these approaches is given below.

In principle, deformation maps showing the area of dominance of each flow mechanism can be constructed in normalized stress vs. absolute temperature space for any polycrystalline material. Each flow mechanism must have an equation relating shear strain rate (i),shear stress (r), temperature (T), and structure. The term "structure" includes all parameters describing the atomic structure such as: bonding, crystal structure, defect structure, grain size, dislocation density and arrangement, solute concentration, and volume fraction of second-phase particles. Maps of this type are limited to pure materials, simple alloys, and steady-state conditions. A typical example of an Ashby-Frost deformation map is given in Fig. 13. TEMPERATURE, OC

I IDEAL ~ I 0.1 -STRENGTH, I : - 104 I 1 GLIDE ~ ! I I , I I ,103

I I 1 (NABARRO- ,0.01 I I 1 HERRING BOUNDARY DIFFUSION I (COBLE CREEP) I ' CREEP) 10-8 I I I 1 i I 1 103 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 HOMOLOGOUS TEMPERATURE, TIT,

Figure 13. Ashby-Frost Deformation Map for Pure Nickel (From Ref. 4

Raj48 extended the deformation-map concept to a processing map representing the nucleation of damage as a function of temperature T and effective strain rate &. A Raj processing map, such as that shown in Fig. 14, is a composite map based upon various "damage" mechanisms for aluminum. A processing map is very useful in that it shows those regions where it is safe to process the workpiece material and avoid defect nucleation. The boundaries are the loci of bifurcation points, where the dynamic behavior of this system follows a HOMOLOGOUS TEMPERATURE TITm

0.4 0.6 0.8

300 500 700 900 TEMPERATURE (K)

Figure 14. Processing Map by Raj Based Upon Various "Damage" Mechanisms for Aluminum (From Ref. 48). 38 new path (corresponding to a different mechanism of power dissipation) or where the flow process changes from stable to unstable. The Raj processing map is highly idealized in much the same way as the Ashby-Frost deformation maps. Neither approach is suitable for evaluating the workability of complex engineering alloys because these models consider only the deterministic aspect of predicting the behavior of a workpiece material under processing conditions. Experience in designing bulk-forming processes clearly indicates that predicting the future behavior of a material involves not only deterministic aspects but also statistical and stochastic ones. Furthermore, rate equations describing the various synergistic mechanisms contributing to deformation, fracture, and microstructure development in complex engineering alloys cannot be explicitly formulated without the benefit of a large body of prior experimental knowledge.

Pvnamic Material Model

Recently, Gege149 and Prasad, et al.,l8 developed a methodology called "dynamic material modeling" (DMM). This approach makes use of the constitutive equations for plastic flow determined from hot-compression tests performed at different temperatures and strain rates. Parameters such as the strain-rate sensitivity are related to the manner in which the workpiece dissipates power instantaneously during hot deformation. According to this model, power P (rate of work done) is dissipated coherently by partitioning it in part into J, whose energy primitive is potential energy; it is also dissipated incoherently as heat by partitioning some of P into G. Kinetic energy is the primitive for G. The partitioning is given by where

G = \oi odi and

Power dissipated by plastic work is denoted by the area under the curve, G which is illustrated in Fig. 15 and designated the dissipator content. J is the area above the curve and is designated the dissipator co-content. The fundamental assumption by Gegel49 and Prasad, et a1.,18 is that J is related to structural changes and G is related to continuum effects. Plastic instability and fracture processes are associated with G, and microstructural evolution is associated with J. The irreversible driving force for the creation of structure is the negative gradient of the potential energy. J and G are complementary functions and are related by a Legendre transformation.50 It follows from Eq. (14) that the partitioning of power between J and G is given by

This ratio is generally known as the strain-rate-sensitivity parameter, m, of the workpiece material. Making use of this parameter, the following dynamic constitutive equation can be developed: -a STRAIN RATE -+ &

Figure 15. Schematic Diagram Showing Constitutive Relation of Material System as Energy Dissipator During Forming.

This equation is called a dynamic constitutive equation because it results from integrating along the dynamic (actual) path which an element of material takes according to the principle of least action, when it is instantaneously deformed with a particular strain rate. The strain-rate sensitivity is assumed to be constant along this trajectory up to that strain rate and at the temperature and effective strain under consideration. The strain-rate-sensitivity parameter, calculated from continuous flow-stress values measured at different constant strain rates, generally has been observed to be independent of temperature and strain rate for pure metals; however, in engineering alloys it has been shown to vary with temperature and strain rate. 4 1 J and G can be expressed in terms of m. Substitution of Eq. (16) into Eqs. (14b) and (14c) and integration will yield the following expressions for J and G:

and

Because flow localization in the deformation zone is avoided, the material undergoes enhanced elongation. This type of behavior is observed generally when superplastic materials, which have m-values near unity, are tested in tension. Also, this condition gives rise to a dissipative rate function which is geometrically equivalent to an ideal ellipsoid. Hence, m = 1 is assumed to be the upper limit for stability in metals and alloys in the present study. This upper limit of m = 1 fixes the maximum value of J at 0.5 P. By normalizing instantaneous J with this maximum value, an efficiency factor, q, can be defined as

The q-factor describes the amount of total power applied to the material system which is partitioned to metallurgical processes. Processing maps representing the variation of efficiency of power dissipation with temperature and strain rate show regimes where processes such as dynamic recovery, dynamic recrystallization, cavitation, wedge cracking, and phase transformation are dominant. Table 1 gives the typical efficiency ranges for various hot- deformation mechanisms in aluminum. Favorable metallurgical processes as well as fracture processes are observed to be very efficient. Therefore, Table 1

Typical Efficiency Ranges for Hot-Deformation Mechanisms in Aluminum (From Ref. 51 )

1. Strain Hardening < 5 Most power dissipated as viscoplastic heat and in atomic-defect generation. 2. Dynamic Recovery 10-20 Dislocation annihilation by climb and cross slip. 3. Dynamic Recrystallization > 30 Efficiency generally increases with increasing strain rate until instability OCCU rs.

4. Void Formation at Hard Particles (Ductile Fracture) > 35 Higher temperature, lower strain-rate process; efficiency decreases with temperature.

5. Wedge Cracking > 35 Higher temperature, lower strain-rate process; efficiency increases with increasing temperature and decreasing st rain rate until grain growth occurs.

6. Superplasticity near 100 Some wedge cracking occurs with superplasticity; hence, the two cannot be totally separated. 7. Dynamic Strain Aging 0 or negative Material acts as a power store. 43 additional information is required for a priori studies of the dominant metallurgical processes and for determining the optimal processing conditions. Subsequently, Malaslg introduced stability theory into the dynamic- material-modeling (DMM) approach established by Gegel and Prasad. The temporal behavior of q was observed to possess the fundamental properties of a Liapunov function. The Liapunov function is a system quantity associated with Liapunov stability criteria. The Liapunov technique for stability analyses is regarded as a general and accepted method in engineering design.=* This function is an arbitrary positive definite quantity which is related to changes in the total energy of a given system. The Liapunov criteria for stability require the system to lower its total energy continuously. It was observed19 that q is a dimensionless parameter which reaches a maximum value at the lowest energy state under stable conditions. Therefore, the Liapunov function V1 was formulated as shown below:

and the condition for stability in the sense of Liapunov is given by

This condition ensures that the system is approaching steady state in which it experiences a minimum energy level and maximum value of efficiency without fracture. Similarly, another dimensionless parameter was derived19 for evaluating the influence of temperature upon material behavior and workability. The 44

Principle of Maximum Rate of Entropy Production is an extremum principle which gives rise to the following relations:

and

where Sapp is the rate of entropy applied and ssysthe rate of entropy produced by the system. Furthermore,

where

Hence, a new coefficient, s, is defined as follows:

The minus sign in Eqs. (23) - (25) can be neglected, and the s-value can be treated as a positive because flow stress decreases with increasing temperature. Note that the s-factor is the same material-behavior parameter which was defined earlier in Eq. (7). The rate of entropy production by the system is given by

where P/T can be thought of as the applied rate of entropy input, Sapp. Accord- ing to the second law of thermodynamics, s should be greater than unity for stable material flow. This implies that the workpiece should store entropy at least as fast as the entropy production rate of working heat for stable flow. The s-factor is also observed19 to possess the time-varying properties of a Liapunov function. Therefore, the Liapunov function V2 is formulated as shown below:

v2 = S (log &) > 0, and the condition for stability in the sense of Liapunov is given by

v2 = asla(10g E) < 0.

Using flow-stress values calculated at different temperatures and effective strain rates, the parameters q and s are determined as functions of temperature and effective strain rate. From these, &/13(log &)and ds/a(log &) values are calculated, and all values less than zero are grouped to determine the stable region(s). The stable region(s) and the contours of q are presented as a processing map with respect to effective strain rate and temperature. The processing map provides the conditions necessary for working a material under stable processing conditions. It represents a conservative estimate of the temperature and strain-rate conditions. A typical example of a DMM processing 46 map is shown in Fig. 16. For a detailed description of the Ti-6242 processing map, refer to Malaslg and Gegel.20 The DMM methodology realistically describes the intrinsic workability of complex engineering alloys. Dynamic Material Modeling explicitly considers the deterministic as well as the statistical and stochastic aspects of predicting the behavior of the workpiece material under processing conditions. The DMM maps define processing conditions which give rise to stable material flow. Thus, the regions where different stable conditions of material flow processes occur in temperature and strain-rate processing space are identified from the processing maps before the metallurgical mechanisms are characterized through microstructure studies.

MICROSTRUCTURAL EVOLUTION DURING HOT WORKING

Microstructural changes which occur during hot working are conse- quences of complex metallurgical phenomena such as recovery, recrystallization, grain growth, phase transformations, and precipitation/dissolution reactions. These phenomena may occur dynamically during hot-deformation processing or statically during post-deformation cooling or heat treatment. The mechanisms and kinetics of these phenomena as well as the associated changes in size, morphology, distribution, volume fraction, and composition of the constituent phases are strongly dictated by the macroscopic heat-flow and material-flow processes. While the temperature distribution in the workpiece is controlled primarily by the interface heat transfer between the workpiece and the dieslrolls or platens, frictional-heating and deformation-heating effects also contribute significantly. At the same time, the distributions of strain, strain rate,

48 effective stress, and hydrostatic stress within the deforming body are influenced by the material-flow behavior as well as the thermal history. Thus, the microstructural evolution during hot working is strongly dependent upon the specific details of both heat flow and material flow. The modeling of microstructural development is a vital component in the comprehensive modeling of thermomechanical processing operations. The microstructural state of a material strongly influences the mechanical properties both during processing and in the final product. Furthermore, laboratory testing methods such as uniaxial compression, plane-strain compression, and torsion are generally incapable of emulating the thermomechanical conditions associated with industrial multi-stage hot-working processes. Material testing techniques also have specific limitations in the range of attainable strain and strain rates as well as applicable stress states and velocity fields. Thus, the need exists for reliable modeling of microstructural evolution during hot working, especially for multi-stage thermomechanical processes. The phenomena of dynamic recovery and recrystallization during hot working and creep deformation have been studied extensively and reviewed comprehensively by Jonas, et a1.,53 Sellars,54 McQueen and Jonas,55 and by Roberts.56-57 These two metallurgical processes have significant influence upon the shape of flow curves obtained under hot-working conditions. If dynamic recovery alone is in operation, the flow stress increases progressively with strain up to a steady-state value, os,which is determined by a balance between the accumulation and elimination of dislocations by climb or cross-slip. Examples of materials exhibiting this behavior include aluminum and its dilute alloys and ferritic steels (e.g., Fe-Cr, Fe-Si). Metals and alloys having a face- centered-cubic (FCC) crystal structure (e.g., copper, nickel, austenitic steels) are 49 generally characterized by low or intermediate values of stacking-fault energy. In this case, dynamic recovery proceeds very slowly until the resultant dislocation density reaches a critical value where dynamic recrystallization occurs. The resulting flow curves have characteristic maxima followed by steady-state behavior. Hot-working processes are normally characterized by high values of temperature-compensated strain rate, Z, which implies that the critical strain, Q, for the onset of dynamic recrystallization (DRX) is quite high. Thus, DRX is expected to manifest itself only in operations involving large increments of plastic deformation such as hot extrusion and forging processes. As shown in Fig. 17, DRX is commonly observed in extrusion processes which involve very large strains (typically E > 1.0). In contrast, hot-rolling operations invotve small increments of strain (typically E < 0.5) and are accomplished by means of dynamic recovery (DR). Dynamic metallurgical processes are immediately followed by post-deformation restoration processes such as static recovery (SR) and static recrystallization (SRX). Furthermore, if the grain size is refined progressively by static recrystallization (SRX) between rolling passes, then it is likely that DRX will manifest itself to a significant extent. This change in dynamic softening mechanism follows from a net reduction in the original grain size, do, which lowers and also accelerates the kinetics of DRX.

IC- Recoverv Reldonsh~~~

Dynamic recovery leads to a well-developed steady-state subgrain structure at deformations greater than the critical plastic strain, em, which increases HOT ROLLING 50% REDUCTION

DYNAMIC HOT ROLLING

HUT EXTRUSION 99% REDUCTION

RECRYSTALLIZATIO

HOT EXTRUSION 99% REDUCTION

RECRYSTALLIZATION

Figure 17. Typical Dynamic and Static Metallurgical Processes in Operation During (a) Hot Rolling and (b) Hot Extrusion (From Ref. 55). Microstructure Development Generally Dependent Upon Stacking-Fault Energy of Metallic System as well as Temperature, Strain, and Strain Rate of Process. 51 with an increase in imposed & and/or a decrease in T. The combined influence of T and & upon Em can be represented as

where p = constant = 2 x 10-3 and

Z = ti exp(QlRT), with Q being activation energy for dynamic recovery and R the universal gas constant. The subgrain size, d,, can be expressed as follows:

where o, and K are constants. As shown in Eqs. (10) and (1 I),the steady-state flow stress, as, can be generally expressed as a power law or hyperbolic sine function of the temperature-compensated strain rate, Z. In this case, the ds given in Eq. (30) can be also expressed in terms of Z.

Materials characterized by low-stacking-fault energy have a high propen- sity for undergoing restoration via dynamic recrystallization and recovery. Under practical hot-working conditions, these materials exhibit flow curves containing single maxima. When the plastic strain exceeds a critical value, cc, dynamically recrystallized grains nucleate principally through a mechanism involving strain-induced grain-boundary migration. Upon continuation of deformation, additional nuclei form until all grain-boundarylincoherent-twin-boundary sites are exhausted; thereafter, the reaction proceeds via nucleation at the boundary between recrystallized and unrecrystallized regions until the recrystallized grains impinge upon one another at the center of the original grains. Although nucleation of dynamic recrystallization is generally associated with pre-existing grain boundaries in polycrystalline materials, course-grained materials and, of course, single crystals exhibit intragranular nucleation near twin boundaries or inclusions associated with large misfit strains. Experimental studies on the kinetics of dynamic recrystallization have yielded a standard Avrami expression for the recrystallized fraction, Xdyn:

where 6 and v are constants. The exponent v is generally considered to be

independent of E, TI and do; the kinetic dependence on these parameters resides in 5. The 6 term increases with a decrease in do or in Z. At hot-working strain rates, decreases steadily with Z, reaching a minimum value at very low

Since dynamic recrystallization involves repeated nucleation but very limited growth of the new grains, the recrystallized grain size, ddyn, varies only slightly as Xdyn increases and is virtually independent of do. A phenomenological relationship between the steady-state stress, os, and the steady-state recrystallized grain size, ddyn, has been reported:57

where the exponent, q, ranges in value from 0.5 to 0.8 and is independent of temperature and strain rate. If a constitutive relation between o and Z can be 53 established, then the dynamically recrystallized microstructure can be related to the imposed test variables T and E.

During static recrystallization, a finite incubation period exists for the initiation of recrystallization (provided E c Q during prior deformation). The volume fraction of statically recrystallized grains as a function of time after termination of deformation yields an Avrami relationship:

where k, and n, are constants and k, contains the dependence upon prior strain, temperature, and grain size. The reference time, toss,required for 50 volume percent of the material to undergo static recrystallization has been shown to exhibit the following empirical dependence:

where a, b and c are constants. Among these parameters, the variation of to.5 with E was found to be negligible. One can write Xsta in terms of 10.5:

The grain size obtained at the completion of SRX is strongly dependent upon the amount of prior strain, E, and the original grain size, do: The exponents a' and b' are in the range 0.5-1 .O, whereas c' is very small (c' < 0.1 ). If the strain accumulated during prior deformation is higher than ec , then the recrystallization events occurring after the cessation of deformation are collectively known as metadynamic recrystallization. Since nuclei of recrystallized regions already exist during deformation, their growth is not subject to an incubation period after deformation. The most notable difference between metadynamic- and static-recrystallization phenomena is related to the temperature dependence of the recrystallized grain size. Metadynamic recrystallization exhibits a very strong temperature dependence compared to static recrystallization.

JVli~~~~alFvolution of Plain C-Mn Steel

Of all the metallic alloys, steel has received the most study. Recently,

Devadas, et al.,58 reviewed previous work related to the development of models to predict microstructural evolution during thermomechanical processing of plain C-Mn steel. The compilation of the equations for both static and dynamic conditions which have been determined by well-known research groups is given in Table 2. The microstructural-evolution models of Sellars, Yada, Saito, and Perdrix are shown for dynamic recrystallization (DRX), static recrystallization (SRX), and grain growth. It is interesting to note that the expressions employed by different researchers are similar in mathematical form. Furthermore, models associated with the static metallurgical processes are Table 2 Summary of Reported Recrystallization and Grain Growth Relationships (From Ref. 58)

University of Sheffield Nippon Steel Kawasaki Steel IRSID Laboratories

Dynamic Recrystallization

&, = 4.9.10-*d~~~0'J E, = 3.68. 10-',?"9d~' not incorporated e, = a&, d, = 2.82. 104Z-0U Q Q Z = E exp - d, = 22,6002-On Z= iexp- RT Q RT Q = 312 kJ/mol 2 = & exp - RT Q = 312 kJ/mol (rnetadynamic) Q = 267.1 kl/mol 1-ex-069()) X-=l-exp 6420 E, = 1.144. 10-'d~xEOwexp (T)

Static Recrystallization

x = 1 - p (-0.693(2) X = 1 - exp (-0,693(k)') X = 1 - exp (-0.693(;)') X = 1 - exp (-0.693(-!-in]

E < 0.8~~ to,, = 2.2. tor = 3.67. 10-"eP'i-Ok' 30,000 to, = 2.5. 10-'9~-'d~exp &-. exp (?) d:" exp (32

Grain Growth

dlo = dAO + At exp (;$)- dl = d: + At exp T> 1100°C Qu T> 1100°C cr = 0.195 A = 3.87.10" A = 1.44.10" -R = 32,100 A = 3.87.10" (C-Mn-A1 steels) Q,, = 400 kJ/mol Q,, = 400 kJ/rnol a = 0.098 T< 110O0C T< ll0OoC (Nb grades) A = 1.31.1@' A = 1.31.10" Q,. = 914 kJ/mol Q,, = 914 kJ/rnol generally more available and more highly developed than those for dynamic processes. The Yada equations59 for modeling the DRX of plain C-Mn steel were used extensively in this research. This DRX model provides a comprehensive description of the relationship between the microstructure and the field quantities. Hence, the equations are considered to be universal in their application to modeling DRX of other metallic systems. The Yada DRX model is shown 56 graphically in Figs. 18-20 to illustrate the typical nature of this metallurgical phenomenon. Over the temperature range 1073-1473 K, the Yada corresponding prediction of the critical strain for DRX ranges from 0.8 to 0.1, as shown in Fig. 18. The relationship of grain size to temperature and strain-rate conditions is given in Fig. 19. Yada employs an Avrami expression to represent the kinetics of DRX in terms of volume percent recrystallized as a function of plastic strain. Figure 20 illustrates the strong influence of temperature upon the

TEMPERATURE, K

Figure 18. Prediction of Critical Strain for Dynamic Recrystallization of Plain C-Mn Steel Using Yada Model. TEMPERATURE, K

Figure 19. Prediction of Grain Size During Dynamic Recrystallization of Plain C-Mn Steel Using Yada Model. kinetics, the critical strain for initiating DRX, and the required strain for completing DRX. The sigmoidal shape of the curves exhibited in Fig. 20 is characteristic of the Avrami relationship. STRAIN, &

Figure 20. Prediction of Kinetics of Dynamic Recrystallization of Plain C-Mn Steel Using Yada Model. IV. HY BRlD APPROACH TO MATERIAL-BEHAVIOR MODELING

A comprehensive understanding of the phenomena associated with a particular workpiece material under processing conditions is required for the effective design and control of thermomechanical processes. Fundamental material-processing characteristics such as plastic stability, material stability, and the kinetics of microstructure development strongly influence product quali- ties such as shape, size and mechanical properties. In the case of deformation processes, knowledge of constitutive behavior, hot workability, and microstructure development is needed for process design and control to achieve high- quality, complex-shaped components. Similarly, in heat-treatment processes, knowledge of time-temperature-transformation behavior and the kinetics of microstructure development is required for controlling the microstructure and properties of the product. Although distinct aspects of the thermomechanical behavior of materials are rather well understood, their interrelationship is not-- especially in the case of nonconventional or special-purpose metallic systems. In the present research a novel approach has been taken in identifying the interrelationships of material phenomena, and a more comprehensive methodology for modeling material behavior has been developed for hot- deformation processing of metallic systems. Phenomenological models of the distinct aspects of material behavior have been combined through the use of dynamic material modeling (DMM). This hybrid approach to material behavior modeling is shown schematically in Fig. 21 and is explained in detail in the remainder of this section. CONSTITUTIVE HOT WORKABILITY MICROSTRUCTURE RELATIONSHIPS 1DEVELOPMENT

CONSTITUTIVE MICROSTRUCTURE RELATIONSHIPS DEVELOPMENT

Figure 21. Primary Material-Behavior Models Used in Hot-Deformation Studies. (a) Disjointed Approach, (b) Hybrid Approach.

ASSUMPTIONS

Jsotropic and Homoaeneous Material Behavior

Materials under hot-working conditions are considered to be isotropic and homogeneous; therefore simplified yield criteria, flow rules, and extremum principles are applicable.

IC Material Behavior

A material system under processing conditions is considered to be dynamic because it satisfies the following conditions:

Time is an independent variable and enters the analysis through integration of strain rate imposed on the workpiece material: time always marches for- ward. 61

8 Initial conditions cannot be precisely specified when a strain rate is imposed on the system, i.e., the dynamic (actual) path taken by the material to attain the final condition cannot be specified a priori.

8 The behavior of the workpiece material is also stochastic, since it is impos- sible to predict the occurrence of bifurcations such as fracture, cavitation, and shear localization which lead to workpiece failure. A bifurcation is said to exist if more than one deformation mode (or energy-release mechanism) is present which gives rise to a maximum in the energy input rate and satisfies the same set of boundary conditions and stress equilibrium throughout the body.41

Therefore, the concepts of general dynamic system behaviorso are applicable to a material system. The state of a dynamic system at any particular instant may be loosely defined as the collection of information which is both necessary and sufficient to determine its future behavior.

The metal-processing system is considered to be open with respect to energy exchanges with the surroundings but closed with respect to exchanging matter.

Material Deforw

Material deformation can be divided into two categories--elastic deformation and plastic deformation. Elastic deformation is recoverable and is neglected here since the main concern of this research is the irreversible part of 62 the plastically deforming system. Plastic deformation of the material is considered to be irreversible, no matter how small the plastic strain. When mechanical power is input to a material system, its current state is disturbed and it attempts to return to steady-state conditions. A steady state is considered to be an attractor for irreversibility. In this case the Second Law of Thermodynamics can be extended using the approach described by Prigogineel for establishing the stability of the system. His work shows that nonequilibrium may become a source of order and that irreversible processes may lead to new type of dynamic states of matter called "dissipative structures" which represent dissipative energy states. During unstable flow conditions the dissipative structures may lead to fracture or plastic instability which can be identified with the aid of entropy calculations.

METALLURGICAL INTERPRETATION OF THE DYNAMIC MATERIAL MODEL

The Dynamic Material Model (DMM) plays a key role in the hybrid approach to material-behavior modeling developed in the present research. DMM was developed in response to the need for a unified and macroscopic description of flow, fracture, and workability of complex engineering materials under hot-working conditions. The DMM stability criteria are examined using metallurgical principles for elucidating their relationships with constitutive equations and microstructure development. The DMM stability criteria are given below: and

Values of Strain-Rate-Sensitivitv Parameter

The first criterion, Eq. (37), is derived on the basis of the maximum rate of power dissipation by the material system. Metals and alloys satisfy this criterion under hot-working conditions. Negative values of the strain-rate-sensitivity parameter, m, are obtained only under conditions which promote dynamic strain aging (a product of the interaction of mobile dislocations and solute atoms). However, the imposed strain rates for hot-working operations are, in general, sufficiently high that such interactions are avoided. Another mechanism which can lead to negative m-values is the dynamic propagation of pre-existing or newly formed microcracks in the workpiece which eventually lead to fracture of the workpiece. As the value of m increases, the tendency for localized deformation decreases, which enables extensive elongation of specimens under tensile loading without necking. Similarly, the occurrence of shear-band formation can be inhibited very effectively at high m-values. It has been shown62 that the ease of shear-band formation can be represented by the parameter a which is equivalent to -[a(log o)/a&]/mat constant temperature and strain rate. For a > 5, intense shear bands form during isothermal deformation. Even under conditions of flow softening, high values of m render the shear-localization process ineffective. However, for metals and alloys, m = 1 represents ideal superplastic behavior resulting from the Newtonian flow which is typical of a glass material. The second criterion relating to the variation of m with log E stems from the application of stability analysis to the Liapunov function, m(log k). Figure 22 shows schematically the variation of log o with log k for two cases: (A) a(log o)/ (log E) > 0 and (El) a(log o)/(log E) c 0. If fracture stress is assumed to be independent of strain rate, then Curve A will probably lead to catastrophic failure at

LOG

LOG 2

Figure 22. Schematic Representation of Variation of Log o with Log k for Two Cases: (a) a(log o)/a(log E) > 0,(b) a(10g o)/a(log k) < 0. 65 high strain rates. In contrast, the behavior described by Curve B has a lower probability of inducing fracture in the workpiece. Moreover, if a certain distribu-

tion of local strain rates is assumed to range from E, to & with a nominal value

of E,, then it is apparent that Type-B behavior will lead to more uniform stress fields across the workpiece. The strain-localization tendency for the material exhibiting Type-A behavior increases with decreasing strain rate--a trend which is clearly undesirable. In contrast, for Curve B the shear localization becomes significant only above a critical strain rate, &,, where the value of m is very low.

The use of the extremum principle which states that the net entropy-production rate associated with an irreversible process must be positive leads to the third criterion, i.e., s 2 1. This criterion can be understood qualitatively with reference to Fig. 23. Here, log o is plotted as a function of 1IT at a constant effective strain rate and effective strain levels. It is a well-established fact that when dynamic recovery alone operates as a softening mechanism, the temperature dependence of flow stress is relatively weak. In contrast, when both dynamic recrystallization and dynamic recovery are in operation, the flow stress varies very markedly with temperature. Typical log o versus (In) plots for these two types of processes are shown in Fig. 23. Thus, low values of s in the neigh- borhood of 1 are indicative of the operation of the DR mechanism, while high values of s are usually associated with the DRX mechanism. Finally, it is important to note that the criterion 0 < s < 1 for dynamic recovery actually corresponds to a range of three orders of magnitude for the slope of log o versus (1m) plots. Figure 23. Schematic Representation of Variation of Log o with 1/T for Two Cases. For Dynamic Recrystallization: a(ln o)/a(l/T) > T; For Dynamic Recovery: 0 < a(ln o)/a(l IT) < T.

Variation of s with Strain Rate

The fourth criterion, i.e., as/a(log &) I 0, is derived from the stability argu- ments relative to the Liapunov function, s(log E). Figure 24 schematically illustrates the implication of this criterion. For E2 > El, the slope of log o versus (In) plots at the same temperature must increase in moving from E2 to The following argument provides an aid for understanding this criterion. As the strain rate increases, the extent of adiabatic heating in local regions increases significantly. Hence, in any particular region of the workpiece, if the local strain rate increases above the nominal value, then it is accompanied by a reduction in the local flow stress. However, if the slope of the log o versus (1m) plot increases Figure 24. Schematic Representation of Variation of Log o with 1/T for Two Strain Rates. E2 > El.

with &, then, a very significant thermal softening will be encountered at & which will produce severe strain localization and adiabatic shear bands. This auto- catalytic process and is likely to lead to severe cracking of the workpiece if the material has poor resistance to nucleation and growth of cracks. On the other hand, as/a(log E) s 0 has a mitigating influence upon this tendency for flow localization. Thus, the four stability criteria of the DMM approach have a sound metallurgical basis. Since the material behavior is both dynamic and stochastic, these criteria should be regarded as probabilistic indicators of the behavioral 68 trends exhibited by a material during hot working. The effects of these instabilities upon the workability of a material can be alleviated substantially by proper control of the die design. For instance, die designs which result in a high ratio of mean-to-effective stress can in part offset the adverse effect of am/a(log &)-- an observation exemplified by the remarkable successes associated with processes such as hydrostatic extrusion, pack rolling, and closed-die forging.

RELATIONSHIP BETWEEN CONSTITUTIVE EQUATIONS AND DYNAMIC MATERIAL MODELING

Flow stress, a, is the fundamental material property which links the constitutive behavior to hot workability via dynamic material modeling. In the case of constitutive relations, flow-stress values describe the inherent resistance of the material to plastic deformation as well as the hardening and softening behavior of the material. In dynamic material modeling, the flow-stress sensitivity to changes in temperature and strain rate describes the intrinsic workability in terms of mechanical stability, material stability, and power partitioning. At large plastic strains, the flow-stress value has a negligible dependence upon strain. This phenomenon coincides with a basic premise of dynamic material modeling--that steady state is the attractor for irreversibility and material stability by means of entropy production. Hence, flow stress and the accuracy with which it is measured are important to the understanding of the interrelationships of material behavior. The mathematical forms of commonly used constitutive equations are examined below in terms of their implications with regard to stability as defined by dynamic material modeling. The mathematical models given in Eqs. (10) 69 and (1 1) are evaluated according to the DMM stability criteria given by Eqs. (37) - (40).

Case I: o= KZm

The terms of this constitutive relation were defined earlier in the discus- sion of Eq. (10). Substituting the strain rate and temperature expression for Z yields

Taking the logarithm of Eq. (41),

Qm logo= log K+m log&+-. RT

Inspection of Eq. (42) allows the stability criteria given in Eqs. (37) and (38) to be readily evaluated. The m-value must be positive definite and less than unity for stability according to Eq. (37). Also, m must be a decreasing function of strain rate for stability according to Eq. (38). To evaluate the criteria given by Eqs. (39) and (40), a relationship for the s-value is determined. The values of m, K, and Q are assumed to be positive, constant values for narrow hot-working ranges of temperature and strain rate. Substituting Eq. (41) into Eq. (25),which is the definition of s, results in 70 Hence, for stability according to Eq. (39), the numerator (Qm) must always be greater than the denominator (RT). Furthermore, the dynamic nature of m influences the stability conditions given by Eq. (40) in the same way as it does in the evaluation of Eq. (38).

Case 11: Z = A[sinh(ao)ln

This constitutive equation was discussed earlier [Eq. (1 I)]. The parameters A and a, in general, vary only with strain in this relation. The term n is a constant which generally ranges in value from 3 to 5. The strain rate and temperature expression for Z is substituted, and the terms in Eq. (1 1) are rearranged as follows:

Taking the logarithm of Eq. (44),

Q log E = log A + n log[sinh(ao)] - -log e. RT

Under hot-working conditions, A, a, and n are generally defined as constants and Q is assumed to be constant only for narrow ranges of temperature and strain rate. Under these assumptions, Eq. (45) is differentiated with respect to log & and ultimately yields the following relation for m: 7 1 Since n and (ao) are always positive definite during plastic deformation, then this relation for m is also positive definite. Using the L' Hospital Rule, the limits of Eq. (46) are found to be

lim m(m)=l/n ao + 0 and

lim m(ao) =O. ao+ *

The constitutive relation given by Eq. (1 1) implies that the value of m is less than unity because (ao) cc = and n > 1. The normal range for the product of a and o is from 0.8 to 1.2 and much less than infinity. Therefore, the criterion given in Eq. (37) is satisfied because m is always positive definite and less than unity. For evaluation of the stability conditions given by Eq. (38), Eq. (46) is dif-

ferentiated with respect to log E as follows:

If the quotient rule is used, the derivative of Eq. (49) simplifies to

tan h(ao) (50) (a@

To determine whether Eq. (50) is negative definite, some extreme limits are examined using the L' Hospital Rule as follows: and

The two limits given in Eqs. (51) and (52) imply that the derivative of m, defined by Eq. (50) is zero. However, Eq. (50) is actually negative and nonzero for reasonable values of all the terms in the equation. For example, the derivative of m as defined by Eq. (50) is negative semidefinite for the range of 0 aos 1.2, as shown in Fig. 25. Hence, for all practical purposes the stability condition given by Eq. (38) is always satisfied. The following expression for s is obtained by taking the derivative of Eq. (45) with respect to reciprocal temperature (In):

s=- Q tanh(ao) RTn (ao) '

If Eq. (46) is combined with Eq. (53), the s expression reduces to Eq. (43) which was derived for the constitutive relation analyzed in Case I. Hence, the product of Q and m must always be greater than the product of T and R for stability according to Eq. (39). Furthermore, the s expression given by Eq. (53) is similar to the m expression given by Eq. (46), and the two equations differ only by a factor of Q/(RT). The differential of s and m with respect to log E is also similar since o is assumed to be the only strain-rate-dependent term in this analysis. It follows that

as - mQ sech2(ao) - tan h(ao) a(10g i) RTn log e (ao) Figure 25. Plot of Bracketed Hyperbolic Function Given in Eq. (50) for Domain of 0 c ao < 1.2.

These two case studies clearly show that constitutive relationships have stability information embedded in their mathematical form. The results of the stability analyses with respect to the DMM criteria are shown in Table 3. These results indicate that the more complicated equations may implicitly contain more stability information than the simple ones. Table 3 Results of Stability Analyses

o c m 21 amla log E c o SZ~ as/a log E c o

Case I ------(~m)> (RT) amla log E c o o=KZ"'

Case II Always Always (Qm) > (RT) Always Z = A[sinh(ao)]n Stable Stable Stable

on of Case II Us~naMaterial-Data-Analvs~s Softw

Gopinathe3 developed a computer program for automating the analysis of material-flow-stress data in terms of the DMM stability criteria. This analysis package was further refined and commercialized by Universal Energy Systems Inc. of Dayton, Ohio. The software program which is known as MME (Material Modeling Environment) was used in the present study. The following constitutive equation for plain C-Mn steel64 was used to calculate the flow-stress values for E = 1.0 and for a wide range of strain rates and hot-working temperatures: 75 where Q is assumed to be a constant (258.3 kJ/mol). The MME program was used to plot the computed flow-stress values (Fig. 26) and analyze them with respect to each stability criterion (Fig. 27). The contour plot in Fig. 27(a) shows the m values to be positive and to range from near zero at low T and high E to

> 0.35 at high T and low &. As indicated by the cross-hatching, the stability criterion of 0 < m I1is satisfied throughout the given range of strain rates and

Figure 26. Flow-Stress Values (MPa) for Plain C-Mn Steel Computed from Eq. (55) for Strain-Rate Range 10-2 - 102 S-1 and Temperature Range 800 - 1200°C...... C'. .

m 857 914 mi ion, iwm 1 1198 TEMPERATURE (.IO TEMPERATURE (.C)

21100- 21100- - A - 111 B - 19 ...... C-39 0-4O E -w F -M ...... 0.70 H-M I -0a

......

Ba) 857 914 Wl 1GZE 1085 1 1198 800 857 914 Wl 1028 1085 1142 1198

TEMPERATURE (.C) TEMPERATURE (.C)

Figure 27. DMM Processing Maps for Plain C-Mn Steel. Contour Map of Each Stability Criterion Shown for Strain = 1.O. Shaded Regions Identify Where Particular Stability Criterion is Satisfied: (a) m-Values, (b) am/a(log E), (c) s-Values, (d) as/a(log E). temperatures. In Fig. 27(c) the stability criterion of s r 1 is violated at low T and high &. The boundary for stability corresponds to the limiting condition of

(Qm) = (RT) which occurs at m = 0.03 since Q is assumed to be constant. Regions of instability are shown in Figs 27(b) and 27(d) which correspond to the 77 criteria given by Eqs. (38) and (40), respectively. These instabilities are not consistent with the previous analytical evaluation of the hyperbolic sine function [Eq. (1I)] and probably occur as a result of numerical error arising from differentiation. However, the flow-stress values computed from Eq. (55) generally yield stable material behavior, which is expected for plain C-Mn steel.

RELATIONSHIP BETWEEN DYNAMIC MATERIAL MODELING AND MICROSTRUCTURE DEVELOPMENT

Workability and microstructure development under hot-working conditions are strongly influenced by the deformation mechanism(s) which govern the material system. Typically, several metallurgical mechanisms are operative during processing, especially in the case of complex engineering materials. This situation contributes to the stochastic nature of the material behavior which leads to nonuniform microstructures and inhomogeneous plastic deformation. For most metallic systems, these complicated situations are mitigated under certain processing conditions where a particular softening mechanism domi- nates. Hence, determining the location of these desirable regions is important for producing controlled microstructures and enhanced plastic flow. The dynamic material model (DMM) provides useful macroscopic information for identifying processing regimes where desirable softening mechanisms are in operation and the material behavior is essentially deterministic. From the DMM the activation energy (Q) associated with a prevailing deformation mechanism is computed and mapped over a given range of processing conditions. The DMM analyses provide a description of the desirable processing conditions under which the operative deformation mechanisms are stable and thereby controllable. Using the DMM predictions of Q and stability, 78 the optimal processing conditions for a particular softening mechanism can be identified and the corresponding microstructure development modeled.

Deformation mechanisms can be identified by the amount of potential free energy required for their activation. Microstructural transformations require atomic mobility; and for activation of this mobility, an increase in energy must be provided to transport the atoms from one site to another.66 The change in free energy of an atom during a metallurgical transition is shown schematically in Fig. 28, where the reaction coordinate is any variable which defines progress along the reaction path. FI and FF represent the mean-free energy of an atom in the initial configuration and that after transformation, respectively. AF (= FF - FI) is negative and is the driving force for the transformation. The movement of the atom from the initial state to the final state is opposed by an energy barrier; therefore, until the atom can temporarily acquire the extra energy necessary to carry it over the barrier, it must remain in the initial state. The smallest incre- ment of energy which will allow the atom to go over the barrier is the activation free energy of the metallurgical reaction. An apparent activation energy, Q, is often determined by identifying strain rate and temperature pairs which satisfy a given stress value and by plot- ting In(&)versus 1/T for different s-values.65 From these Arrhenius-type plots, the Q of the controlling deformation mechanism can be determined using the following expression: TRANSITION STATE ,, ------

AFA = ACTIVATION FREE ENERGY I I

I FINAL STATE REACTION CO-ORDINATE

Figure 28. Change in Free Energy of Atom as It Takes Part in Transition. "Reaction Coordinate" is Any Variable Defining Progress Along Reaction Path (From Ref. 66).

Straight isostress lines imply that only one mechanism is dominant throughout the stress and temperature range examined. If multiple mechanisms were act- ing in a concurrent and dependent manner, the In(&) versus 1/T plots would be nonlinear. Some softening mechanisms and the associated Q-values are given67 in Table 4 for common metallic systems under hot-working conditions. Table 4 Typical Hot-Working Information for Some Common Metallic Systems (From Ref. 67)

Metal Softening Mechanism Activation Energy (kJ1rnol)

Al DRV to a high degree; DRV: QHW = 140 DRX only as a result of DRX: QHW = 165 particle enhancement.

N i DRV to a medium degree; QHW = 234 DRX is found in a closed QHW rises rapidly region on a deformation map. as alloy additions increase.

y- Fe DRV to a medium degree is enhanced by carbon content; DRX is unaffected by carbon content.

p - Ti DRV to a high degree reduced little by solute; Strain aging due to Si.

To establish a relationship with DMM, Eq. (56) is evaluated as follows:

and 8 1

If the expressions for m and s given by Eqs. (6) and (7),respectively, are substituted into Eq. (58), the result is

Q=%. (59)

Hence, Eq. (56) and Eq. (43) are shown to be equivalent. Furthermore, Q can be determined from computations of m and s, which are the fundamental material parameters of DMM.

Stability is an important characteristic of the transient behavior of a material system. A stable material system is one which remains under control. Hence, a stable material system will respond in some reasonable manner to an applied thermomechanical condition. The DMM stability criteria discussed earlier provide the necessary--but not sufficient--conditions for instabilities such as shear bands and fractures. Violations of the DMM stability criteria cause ampli- fication of stochastic behavior which results in nonuniform microstructures and deformation. In general, a material system is not usable if it is unstable or cannot be stabilized by an external force. During a thermomechanical process, a material system changes its state in accordance with the imposed path for dissipating energy. In order for a system to achieve a more stable, lower energy state, it must first pass through an intermediate, less stable, higher energy state which acts as a barrier to the transformation unless the necessary activation can be provided. However, in the case of instability, no such barrier exists. As shown schematically in Fig. 29, a material system can lower its free energy by transitioning through a series of 0 FINAL STATE

I I 1 V FINAL STATE A 0 FINAL STATE 0

J REACTION COORDINATE

Figure 29. Schematic Representation of Various Dissipative Paths Which Material System Can Take During Transition to Lower Its Free Energy. dissipative energy states and microstructures. Typically, for a given free-energy state of the material system, the potential exists for a number of dissipative paths. In this situation, the material response must be controlled to such an extent that the microstructure will transform in a stable and efficient way. The degree of stability of the material system can also influence the pre- dictability of microstructural development. Examples of various types of sta- bility68 are shown in Fig. 30. Global asymptotic stability [Fig. 30(a)] represents GLOBAL ASYMPTOTIC GLOBAL STABILITY UNSTABLE UNSTABLE ASYMPTOTIC STABILITY STAB1 LlTY i.s.L. BUT BUT AND STABILITY BUT UNBOUNDED i.s.L. UNBOUNDED BOUNDED UNBOUNDED (a) (4 (c) (d) (8) (0

Figure 30. Examples of Various Types of Stability from Field of Systems Engineering (From Ref. 68). the lowest free-energy state of the material. The stability condition of the intermediate equilibrium states (known as metastable states) could also be described as asymptotically stable but unbounded [(Fig. 30(b)]. The case of global stability in the sense of Liapunov (i.s.L.) given in Fig. 30(c) is generally not observed in metallic systems. A marginally stable situation which is typical of brittle or difficult-to-process materials could be described as stable i.s.L. but unbounded [Fig. 30(d)]. Unstable but bounded behavior [Fig. 30(e)] represents a bifurcation mode of the material where several metallurgical mechanisms operate concurrently. Fracture and flow-localization phenomena can be depicted as unstable and unbounded, as shown in Fig. 30(f). Hence, the shape of the free-energy surface characterizes the degree of stability of a particular material system and varies with time. Modeling of microstructure development seems to be more plausible for the stability situations described by Fig. 30(a) and 30(b) than for the stable i.s.L. but unbounded condition given by Fig. 30(d). V. MODEL VALIDATION: CASE STUDIES

In this chapter selected case studies are presented and analyzed for the specific purpose of validating the material modeling approach as well as, elucidating the role of material behavior in process design. Model predictions for three metallic systems are compared with data from metallography studies, results of hot-deformation processing experiments, and documented industrial experiences. In the first case, the effects of thermomechanical processing upon microstructure development are investigated for a titanium-aluminide alloy. Secondly, different forms of an aluminum alloy are examined for the purpose of demonstrating the influence of microstructural complexity upon hot workability and microstructure development. The third case study illustrates the effect of thermomechanical history upon the mechanical and structural behavior of a powder-metallurgy (P/M) nickel-base superalloy. Some of these case studies provide new results, while others offer improved interpretation and analysis of pre-existing data. These selected case studies, which represent a wide range of advanced metallic systems, verify and demonstrate the wide-ranging applications of the material-modeling approach for designing thermomechanical processes.

THERMOMECHANICAL PROCESSING OF A TITANIUM-ALUMINIDE ALLOY

Gamma-titanium aluminide (TiAI) and its alloys offer great promise as potential materials for applications at temperatures exceeding 800°C, due to their high specific modulus, specific tensile and creep strength, and oxidation resistance.69 The disadvantages of these materials include low ductility at 85 ambient temperature, poor fracture toughness at service temperatures, and difficulties associated with processing. Preparation of these materials is usually accomplished by powder-metallurgy techniques, e.g., plasma rotating electrode process (PREP), or by conventional ingot-metallurgy techniques. While no major problems have been encountered with compaction and extrusion of TiAl powders, ingot breakdown processes do pose problems with respect to mechanical integrity as well as microstructural control of the product. For example, macro- and micro-segregation patterns produced by the ingot casting process cause complex microstructural evolution during thermomechanical processing. Segregation problems associated with ingot metallurgy are usually resolved during the initial stages of thermomechanical processing [primary processing step(s)]. Primary processing typically consists of ingot-breakdown operation(s) followed by homogenization heat treatments. The purpose of the primary processing step(s) is to homogenize and refine the microstructure as well as provide a useful preform shape for subsequent secondary processing operations. In order to achieve a more uniform microstructure, metallurgical mechanisms such as recrystallization and phase transformations are enabled during the deformation and heat-treatment cycle(s) associated with primary processing .

ct of Thermomechanical Processina wnMicrostr-

The microstructure development of nearly single-phase TiAl alloys was investigated. A typical processing sequence for gamma-TiAl is shown in Fig. 31. The microstructures of a gamma-TiAl alloy having a nominal composition of PRIMARY PROCESSING I (INGOT BREAKDOWN)

EXTRUSION I

+/+EAT I EAT kmLLuT

Figure 31. Typical Thermomechanical Processing Sequence for Gamma-TiAl Alloys. 87

Ti-48 AI-2 Nb-2 Cr which corresponds to different steps of a thermomechanical process are shown in Fig. 32. The as-cast microstructure [Fig. 32(a)] indicates a high degree of microsegregation. The equiaxed, pure-gamma grains formed in the interdendritic regions correspond to relatively high levels of Al, and the den- dritic cores correspond to relatively low levels of Al. After one cycle of hot working and heat treatment in the alpha phase field, the microsegregation patterns were eliminated to a great extent, as shown in Fig. 32(b). However, the microstructure contains coarse prior-alpha grains with a mean grain size of 2 mm which have transformed to a lamellar mixture of gamma and alpha-two phases. After a second cycle of hot working and heat treatment, the microstructure was completely altered to yield a mixture of fine equiaxed grains of gamma and alpha-two phases, as shown in Fig. 32(c). The alpha-two grains are fine and distributed uniformly in a matrix of gamma grains. Since the volume fraction of the gamma phase is in excess of 80% at temperatures below 1300°C, the essential microstructural characteristics involved in the thermomechanical processing of nearly single-phase TiAl alloys are characterized. Hot-working operations are especially important in the homogenization of intermetallic systems such as gamma-TiAl. Thermal processing alone is very time consuming for these materials because of the low diffusion rates associated with ordered lattices. In fact, homogenization by thermal processing is only 10-20% effective in mitigating segregation problems. Hence, the kinetics of metallurgical mechanisms such as dynamic recrystallization play a key role in the homogenization of these materials. 'igure 32. Effect of Thermomechanical Processing upon Micro- . structure of Ti-48AI-2Nb-2Cr Alloy. (a) Cast + HIP, (b) Cast + HIP + lsoforge + Homogenize, (c) Cast + HIP + lsoforge + Homogenize + lsoforge + Heat Treat. 89 . . namic Rec-on of a Gamma-TIAI liUpy

The kinetics and associated grain-size refinement of dynamic recrystallization in a gamma-TiAl-based alloy containing 49.5% Al, 2.5% Nb, and 1.l% Mn were investigated using the constant strain rate, isothermal compression test technique. The alloy was supplied by Duriron Company of Dayton, Ohio in the form of induction skull-melted and cast cylindrical ingots, 70 mm in diameter. These ingots were hot isostatically pressed (HIP) at 1175°C and 105 MPa for 4 hr followed by slow cooling. The microstructure after the HlPing process consisted mainly of equiaxed gamma grains (mean grain size of 125 mm) and 10- 15% of grains containing alternate lamellae of gamma and alpha-two phases. Axisymmetric compression tests of the gamma-TiAl alloy in the cast + HIP condition were conducted in the strain-rate range 0.001 to 10.0 s-1 and at temperatures ranging from 1000 to 1400°C.29 Using flow-tress measurements from the hot-compression tests, DMM processing maps were constructed for the gamma-TiAl alloy in the cast + HIP condition. Stable regimes and contours of computed values of apparent activation energy, Q, are graphically shown in Fig. 33 for the range of conditions tested. Unstable behavior in the form of brittle and ductile fractures was readily observed for samples tested at 1-10.0 s-1 and at temperatures ranging from 1000 to 1400°C. The rapidly changing and undesirable behavior associated with the temperature range 1250-1300°C corresponds to the y 4 (y + a) phase transformation. Similarly, the (y + a ) a phase transformation at - 1380°C yields an instability associated with excessive grain growth at higher temperatures. Therefore, the DMM predictions of unstable behavior can be verified by macro- and micro-scopic examination of the compressed samples. TEMPERATURE (OC) Figure 33. DMM Processing Map for Ti-49.5AI-2.5Nb-1.1 Mn Alloy in Cast + HIP Condition. Contours of Apparent Activation Energy (kcal/mole) with DMM Stability Information for Strain = 0.5. Shaded Regions Satisfy All Four DMM Stability Criteria.

The primary interest of this study is the metallurgical phenomena associated with stable material behavior. A stable regime for hot working the gamma-TiAl alloy in the cast + HIP condition is predicted for the strain-rate range 0.001- 1.0 s-1 and in the temperatures range 1000-1200°C. In this stable region, the computed Q values increase steadily from 251 kJ/mole (60 kcal/mole) to 376 kJ/mole (90 kcal/mole) with temperature and are relatively insensitive to strain rate. Using the same flow-stress data, Seetharaman and Lombard29 determined Q to be 327 + 10 kJ/mole from plots of log & versus 1/T at constant values 91 of flow stress, and they compared their Q values with those measured by others.70171 In general, the Q associated with plastic deformation involving dynamic recrystallization has been identified with that for grain-boundary mobility. According to Seetharaman, et a/., Q =327 kJ/mole is a relatively high value for grain-boundary diffusion but can be rationalized by the relatively high strain rates imposed and the cast + HIP condition of the alloy which is characterized by segregation and nonequilibrium phase distribution. Therefore, the activation energy for plastic deformation in this stable processing regime is governed by dynamic recrystallization and other intragranular deformation mechanisms such as dislocation climb and cross-slip. Under stable processing conditions, the kinetics of dynamic recrystallization are investigated for the Ti-49.5AI-2.5Nb-1.I Mn alloy in the cast + HIP condition. The microstructure development for compression-test conditions of 1100°C and 0.1 s-I is shown in Fig. 34. After a true plastic strain of 0.3, the microstructure remains undisturbed, as shown in Fig. 34(a). The gamma grains remain equiaxed with a mean grain size of 125 pm which is the same as that of the starting microstructure. Figure 34(b) shows an early stage of the dynamic- recrystallization process which occurred at a plastic strain of 0.6 under these test conditions. At this stage the recrystallization process has just commenced at the grain boundaries where a fine equiaxed structure is formed. After a true plastic strain of 1.O, the recrystallization progressed significantly [Fig. 34(c)]. In this final stage of deformation the volume fraction recrystallized is - 80%, and the recrystallized grain size is 12-15 pm.

Seetharamad* represented the kinetics of the dynamic recrystallization process in a quantitative manner. Under compression-test conditions of 1100% and 0.1 s-l, a sigmoidal relationship between volume fraction recrystallized, 3 Pe+ & = 10-1 s-1 _I TRUE PLASTIC STRAIN, E

Figure 34. Microstructure Development Associated with Dynamic Recrystalliz on of Ti-49.5AI-2.5Nb-1.1 Mn Alloy in Cast + HIP Condition. Microsdructure Shown for Different Strain Le Is: (a) e = 0.3, (b) E = 0.6, (c) E = 1 .O.

Figure 37. Observed Relationship Among Recrystallized Grain Size, Temperature, and Strain Rate for Ti-49.5AI-2.5Nb-1.1 Mn Alloy in Cast + HIP Condition (From Ref. 72). derived the following power-law relation for predicting the dynamically recrystallized grain size, ddyn:

where ddyn is in micrometers (pm). In conclusion, this case study provided supporting evidence for validation of the hybrid approach to material modeling--especially the relationship between DMM and microstructure development. The complicated hot-working behavior associated with gamma-TiAl alloys is elucidated by the macroscopic information provided by the DMM processing maps. The stability and activation

98 processes. DMM processing maps provide a useful framework for representing a the information content of a material which significantly influences hot workability and microstructure development.

The complexities of a material tend to reduce the size of its processing window which is defined as the range of processing conditions for acceptable material behavior. Generally, pure metals and single-phase alloys exhibit the best workability. Alloys which contain low-melting-point phases tend to be difficult to deform and have a limited range of working temperatures. In general, as the solute content of the alloy increases, the possibility of forming low-melting- point phases increases and the temperature for precipitation of second phases increases. The net result is a decreased region for good workability as shown in Fig. 39. Material-processing systems are evaluated using Suh's definition21 of information content

system range I = log range1 -

In the context of material processing, the system range is dictated by the capabilities of the available thermomechanical equipment. The design range is assumed to be the "processing window" of the particular material system. The common range is defined as the intersecting domain of the system and the design ranges. The relationship between the processing window of a material and the capabilities of a manufacturing system is illustrated in Fig 40. This plot shows schematically the temperatures and strain rates associated with system POOR FORGEABILITY DUE TO PRESENCE OF LOW-MELTING-POINT PHASES

...... GOOD FORGEABILITY ......

:.;.;.>:>...... POOR FORGEABILITY DUE TO FORMATION ...... OF PRECIPITATION-HARDENING PHASES

INCREASING PERCENTAGE OF ALLOYING ADDITIONS THAT FORM SECOND PHASES

Figure 39. Influence of Solute Content upon Melting and Solution Temperatures and, Therefore, upon Forgeability (From Ref. 38). and design ranges for hot-deformation processes. A typical metalforming system is capable of accommodating a much wider range of temperatures and strain rates than a particular material system. The common range is equivalent to the design range in the case shown in Fig 40. In the present study, a reasonable design range corresponds to the processing conditions where a desirable and fairly constant value of Q is operative and the process is stable in that regime. It should be pointed out that this concept represents a new approach to process design. Furthermore, for a given equipment-system range, the

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Figure 42. DMM Processing Map for Ingot 2024 A1 . For e = 0.3, Contours of Apparent Activation Energy (kcallmole) with DMM Stability Information Are shown. Shaded Regions Satisfy All Four DMM Stability Criteria. ranges from 108 to 159 kJ1mole (26 to 38 kcallmole) which corresponds to a known Q value for dynamic-recovery processes of 140 kJ1mole (33.5 kcal/mole). Furthermore, the large stable region indicated on the DMM processing map is comparable to Raj's safe processing region for commercial aluminum48 shown in Fig. 14. The unstable conditions associated with high temperatures and low strain rates correspond to the wedge-cracking phenomenon which has been obselved by others.48 In summary, the rather small variability of Q and large

TEMPERATURE (OC)

Figure 43. DMM Processing Map for P/M 2024 Al in Vacuum Hot Pressed + Extruded Condition. For E = 0.3, Contours of Apparent Activation Energy (kcal/mole) with DMM Stability Information Are Shown. Shaded Regions Satisfy All Four DMM Stability Criteria. regimes where Q is fairly constant and stable are smaller for the PIM alloy than for the ingot alloy. Hence, the P/M material is more difficult to process than its ingot form because the design range is more restrictive.

. . 4 Aluminum with Silicon-Carbide Whiskers

The material behavior of PIM 2024 Al is further complicated with a dispersion of silicon-carbide whiskers. PIM 2024 Al with 20 volume percent Sic 105 whisker material was used in the present study. The material was supplied by ARC0 Metals in the form of vacuum-hot pressed and extruded billets. The details of the billet conditioning are reported elsewhere.76 Cylindrical specimens, 10 mm in diameter and 15 mm in height, were tested in compression over the temperature range 300-550°C and at four strain rates of 1o-~, and 10-I s-l. Testing was conducted on an lnstron testing machine which has provisions for maintaining constant strain rates and controlling temperature to within + 2°C. A MoS2 lubricant was used to ensure homogeneous deformation during compression testing. Load-displacement curves were recorded on a strip-chart recorder from which flow-stress curves were obtained. These flow- stress data were used to generate the DMM processing map shown in Fig. 44. The processing maps for P/M 2024 Al with and without Sic whiskers shown, respectively, in Figs. 44 and 43 indicate that the Sic dispersion significantly reduces the processing window. The Q values computed for the whisker- reinforced material generally range from 377 to 1047 kJImole (90 to 250 kcal/mole) and vary precipitously throughout the temperatures and strain rates tested. In the large stable region, at least two metallurgical processes, namely, dynamic recrystallization and dissolution of second-phase particles, have been observed in the matrix material.3175 Similarly, for the unstable regions, cavitation was observed near conditions of 10-I s-l and 300°C; matrix kinking (plastic instability) was observed in the region of 10-I and 550°C; and dynamic grain growth was reported at 10' 4 s' 1 and 550°C. The optimum processing range for the composite occurs over a very narrow range of temperature (around 470°C) and strain rate (10-2 s- 1) in which dynamic recrystallization in the matrix is the dominant mechanism. The tensile properties of a billet of this composite material which was extruded using the predicted parameters and a TEMPERATURE (OC)

Figure 44. DMM Processing Map for P/M 2024 Al with SIC Whiskers in Vacuum Hot Pressed + Extruded Condition. For E = 0.3, Contours of Apparent Activation Energy (kcallmole) with DMM Stability Information Are Shown. Shaded Regions Satisfy All Four DMM Stability Criteria. streamlined die showed favorable correlation with theoretically calculated val-

In summary, data from microstructure studies and experiences in processing 2024 Al material systems were cited which validated the material- behavior predictions. The DMM processing map was shown to provide a useful representation for identifying the design range of material systems. Structural 107 complexity was shown to have deleterious effects upon the processing window of 2024 Al; this is probably true for other metallic systems also.

HOT-WORKING BEHAVIOR OF A NICKEL-BASE SUPERALLOY

Nickel-base superalloys are generally used at temperatures above 540°C in high-performance applications such as gas-turbine-engine components. The high-temperature strength properties associated with Ni-base superalloys are superior to those in many other metallic systems. However, these superalloys are difficult to deform and are subject to cracking during hot working. According to Couts and ~owson,77Ni-base superalloys are history- sensitive materials, and their processing must be controlled in almost every step of the manufacturing process. The processes that must be reproducibly controlled include primary and secondary melting, homogenization, ingot conver- sion, hot and cold (if any) deformation, and heat treatment.

In this case study, the hot-working behavior of a nickel-base superalloy, namely Rene 95, is examined. Its nominal composition in terms of weight percent is 14 Cr, 8 Co, 3.5 Mo, 3.5 W, 3.5 Al, 3.5 Cb, and 2.5 Ti, with the balance being Ni. Since it is a difficult-to-process material, Rene 95 is commonly hot worked in a powder-consolidated form. In the present research, material produced by inert gas atomization was supplied by Cameron Iron Works, Inc. of Brighton, Michigan, in the form of 76.2-mm- (3-in.-) diameter extrusion cans filled with a uniform size distribution of powder. Two different powder sizes of

Figure 45. DMM Processing Map for -1 50 Mesh PIM Rene 95 in Compacted Condition. For E = 0.3, Contours of Apparent Activation Energy (kcallmole) with DMM Stability Information Are Shown. Shaded Regions Satisfy All Four DMM Stability Criteria.

Effects of Thermomechanical History upon Material Behavior

The thermomechanical history of a material system is generally known to influence its future behavior. The DMM processing maps for PIM Rene 95 indicate that seemingly small differences in material history such as prior powder size can affect workability and microstructure development. The effects of thermomechanical history upon material behavior can be considered analogous to 1070 1100

TEMPERATURE (OC)

Figure 46. DMM Processing Map for -270 Mesh P/M Rene 95 in Compacted Condition. or E = 0.3, Contours of Apparent Activation Energy (kcallmole) with DMM Stability Information Are Shown. Shaded Regions Satisfy All Four DMM Stability Criteria. a nonlinear differential equation whose solution depends upon the given initial conditions. Hence, the starting condition of the material system is an important factor in the design and control of thermomechanical processes. In the processing regime associated with isothermal forging of P/M Rene 95, the information content is apparently greater for the finer prior powder size. As shown in Fig. 46, the design range for the -270 mesh material is reduced because of unstable behavior in the processing regime of interest. Unstable 111 behavior is not predicted under the slow-strain-rate conditions for the -150 mesh material, as shown in Fig. 45. The coarser powder mesh size is apparently less difficult to hot work than the finer one. Therefore, other differences in the characteristics of the powder such as shape, contamination, and size distribution could significantly influence the design range of the material system. In summary, the material behavior of Rene 95 under hot-working conditions was corroborated by processing practices documented in the Metals Handbook.'g The material-behavior model predicted that the prior powder size would influence workability. Other seemingly small differences in the starting condition of the material could significantly affect the size of its processing window. VI. AXIOMATIC APPROACH TO PROCESS DESIGN

The design of thermomechanical processes is a challenging task which requires evaluation of several alternative operations and processing variables in the search for an effective manufacturing solution. One of the important goals of any manufacturing organization is to determine the optimum means for producing defect-free parts on a repeatable basis. Global and local optimization of manufacturing processes is subject to constraints such as cost, product quality, and delivery schedule. The process-design tasks are numerous and range from selecting thermomechanical process(es) to determining equipment control commands. In general, the process-design activity is difficult because of the diversity of knowledge involved and the complex relationships which exist within this collection of knowledge. Futhermore, an in-depth understanding of the material behavior under processing conditions and the advances in manufacturing technology is needed for improving processes of mature and emerging material systems. A number of design theories have been proposed which could simplify the process-design activity by providing a methodology for systematic evaluation of alternatives. Dinsdaleao classified these design philosophies as axiomatic or algorithmic methods. The axiomatic method proposed by Suh21 is based upon fundamental truths which are always valid, and no exception exists. The algorithmic methods proposed by others81182are derived from abstractions of particular experience in a field of technology and generally have limited application to other fields. Quality function deployment (QFD) is an algorithmic design approach which has been successfully employed by various Japanese manufactures.83 The QFD approach provides a method for defining the design 113 problem in terms of a conceptual map which supports interfunctional planning and communications. The design problem is subsequently solved by a comprehensive prioritization scheme. In a more fundamental way, Suh's axiomatic design approach incorporates the basic elements of QFD and other algorithmic approaches. Furthermore, the axiomatic approach provides an effective conceptual mapping scheme for describing complex functional relationships and a scientific method for optimizing design solutions. Suh's axiomatic approach to design is a general design theory which has achieved success in extremely diverse applications such as software design,e4 composite process design? and design of organizations.21 In the present research the axioms and concepts proposed by Suh were assumed to be applicable to all engineering design problems including the design of thermomechanical processes.

REVIEW OF KEY CONCEPTS OF AXIOMATIC DESIGN

Although Suh's axiomatic approach to design is described elsewhere,2l184 the relevant and important aspects are reviewed here to provide a better understanding of its application to the current problem. Suh has described the design process in terms of five basic steps: 1) establishment of design goals, 2) conceptualization of design solutions, 3) analyses of proposed solutions, 4) selection of the best design from those proposed, and 5) implementation. These activities occur between and in different design domains. Suh's design world is divided into four domains, as shown earlier in Fig. 4 of Chapter 11. Design problems are described in terms of consumer, functional, physical, and process domains. The consumer domain is where 114 customer needs reside. Customer needs must be mapped into the functional domain where they are translated into a set of functional requirements (FRs), which constitute a characteristic vector. FRs are defined as a minimum set of independent requirements which completely characterize the functional needs of the product design in the functional domain. These FRs are then mapped into the physical domain where the design parameters (DPs) are chosen to satisfy the FRs. DPs are the key variables which characterize the physical entity created by the design process to fulfill the FRs. The DPs in the physical domain are mapped into the process domain in terms of the process variables (PVs). In the case of thermomechanical process design, the process domains are in the form of initial conditions of the workpiece material, various manufacturing operations, and nominal processing conditions which correspond to each operation. Furthermore, the relationship between the domains is that the domain on the left is "what we want" and the domain on the right is "how we will satisfy what we want." Hence, moving from one domain to another is called mapping which is the synthesis phase of the design process. Another important concept in axiomatic design is the existence of hierarchy in each domain. For example, in the functional domain, the FRs can be decomposed and formed into a hierarchy. However, the decomposition in each domain cannot be made independent of the hierarchies in other domains. In order to decompose a given FR at a given level of the FR hierarchy, a design solution which is characterized by a set of DPs must be conceived in the physical domain; then one moves back to the functional domain and decompose the FR. As shown in Fig. 47, the concepts of decomposition and hierarchy are closely related to the concept of zig-zagging between domains to proceed with the decomposition from one level to the next lower level. represents a leaf FR ---F: zig-zag direction of decomposition

Figure 47. Hierarchial Tree Structures of Functional Requirements and Design Parameters (From Ref. 84).

The most important concept in axiomatic design is the existence of the design axioms, which must be satisfied during the mapping process to arrive at acceptable design solutions. According to Suh,21 good designs are governed by two design axioms. Axiom 1 deals with the relationship between functions and physical variables and Axiom 2, with the complexity of design. The axioms may be stated as follows:

Axiom 1 The independence Axiom Maintain the independence of functional requirements.

Axiom 2 The Information Axiom Minimize the information content of the design.

The first axiom provides the criterion for acceptable design during the mapping process. The "product" design, which is the mapping process between the functional domain and the physical domain, can be represented by a design equation as {FRs) = [A] {DPs), where {FRs) is a vector which describes the functional requirements of the product in terms of its independent components FRi, {DPs} is a vector describing the parameters which define the product in terms of its effect upon {FRs), and [A] is a product-design matrix. The elements of the product design matrix, Aij, are given by

which is a constant in linear design. In this case, the design window is indefi- nitely large. Axiom 1 states that during a design process, the relationship between the FRs in the functional domain and the DPs in the physical domain must be such that a perturbation in a particular DP affects only its referent FR. Hence, [A] must be a diagonal or triangular matrix in order to satisfy the Inde- pendence Axiom. The design which has a diagonal matrix is called an uncou- pled design. When triangular, it is a decoupled design which also satisfies the Independence Axiom, provided DPs are changed in a specific sequence. All other designs are coupled designs. A similar equation for mapping between the physical and the process domains can be written as

(DPs} = [B] {PVS}, (65) where [B] is a process-design matrix and {PVs} is the vector which describes the process variables in the process domain. The elements of the [B] matrix are 117 partial derivatives of DPs with respect to PVs. Again, the [B] matrix must be either diagonal or triangular to satisfy the Independence Axiom. Equation (65) can be substituted in Eq. (63) to relate {FRs) to {PVs) directly. Axiom 2 states that,among all the designs which satisfy Axiom 1, the one having the minimum information content is the best design. The term "best" is used in a relative context, since potentially an infinite number of designs can satisfy a given set of FRs.

METHODOLOGY FOR SYSTEMATIC EVALUATION OF DESIGN ALTERNATIVES

A scientific methodology for the design of thermomechanical processes has been developed in the present research. The principles employed are derived from the two design axioms of Suh and fundamental understanding of material behavior and shapemaking processes. The physical and process domains of Suh's design world define, respectively, the physical requirements are for a given part and the process involved in making that part. The physical domain can be described by a heirarchy of material and geometrical attributes predetermined from the functional domain. A corresponding hierarchy of process variables can be established in the process domain from a level-by-level corroboration with the design axioms. The design range for producing a part can be defined as the common domain of the material and shapemaking design ranges. The material design range defined in Chapter V corresponds to the processing conditions under which a desirable and fairly constant Q is operative and where the process is stable. The design range for thermomechanical processes such as forging, extrusion, rolling, and heat treatment can be defined in terms of their capabilities for producing a certain shape, size, and "fineness" 118 of microstructure. Hence, creative process-design solutions can be derived from the knowledge of alternative shape-making operations and corresponding material-behavior phenomena. Exploring the synergism between material and geometrical shape is extremely important for revolutionary advancements in product and process design. For example, a national program known as Integrated High Perfor- mance Turbine Engine Technology (IHPTET) Initiative has as its goal to double turbine-engine propulsion capability by the turn of the century. The IHPTET plan is to combine advanced-material developments with innovative structural designs. In Fig. 48, state-of-the-art and advanced turbofan engine designs are compared. The four domains of Suh's design world clarify the material and geometrical shape aspects of the design problem. The decomposition of domains can be facilitated by tradeoffs between the material and the geometry. In the functional domain, material properties can be exchanged for geometrical properties. Similarly, in the physical domain, material structures can be traded for geometrical structures. Finally, in the process domain, material processing variables can be exchanged for geometrical shapemaking alternatives. The following example is presented to illustrate the important concepts of the process-design methodology.

le: Axiomatic Oesian of a Hiah-Performance- Aircraft-Fnaine Com~m

An example design problem is presented to clarify the fundamental characteristics of the four domains and their inter-relationship. The axiomatic approach is applied to a hypothetical problem associated with the design and manufacture of a new high-performance aircraft system. Figure 49 gives a Ire 48. Turbofan Aircraft Engine. (a) United Technologies Pratt and Whitney F100-PW-200 Design, (b) Revolutionary Design of the Future. 120

-MANUFACTURING

CONSUMER FUNCTIONAL PHYSICAL PROCESS DOMAIN DOMAIN DOMAIN DOMAIN

Figure 49. General View of Design Domains and Constraints Considered in Example Design Problem of High-Performance Aircraft-Engine Component. general view of the four design domains and the constraints which are assumed in this problem. The aerospace industry is considered to be the end user, which constitutes the consumer domain. In this example an aerospace customer has indicated the need for the design of an aircraft with improved cruise speed, flight range, and envelop of stability. Of course, the customer description includes goals for improved performance, known as desired attributes {DAs}. Subse- quently, the {DAs) are mapped to the functional domain, as shown in Fig. 50. The functional requirements {FRs) are determined after consideration of the {DAs} and the imposed constraints of time, cost and quality. The {FRs) selected for satisfying the {DAs} are greater thrust capacity, lower system weight, and greater system rigidity. The qualitative relationship between the {DAs) and the {FRs) is shown in the following equation:

] = [; ;;] rreater~hrust {F:;!~ Lower Weight , (66) Greater Stability OOX Greater Rigidity

122 where X represents a nonzero element and 0 represents a zero element. The design matrix which is described by X and 0 terms is triangular and, thus, represents a decoupled design. In this case, the sequence of change of {FRs) in Eq. (66) is very important; if it is altered, then Axiom 1 may be violated. Hence, the stability attribute should be determine first, followed by the distance attribute and then the speed attribute. Subsequently, the {FRs} are mapped to the physical domain, as shown in Fig. 51. The design parameters {DPs) are determined after consideration of the {FRs), the technology advancements, and the imposed constraints of time, cost and quality. The {DPs) selected for satisfying the {FRs) are engine configuration, structural configuration, and materials. The qualitative relationship between the {FRs) and the {DPs) is shown in the following equation:

Engine Configuration

This design matrix is unacceptable because it represents a coupled design. One remedy is to reduce aircraft system weight through the use of advanced materials--not innovative structures. This modified-design solution produces a decoupled relationship between the {FRs) and {DPs) given in Eq. (67). A hierarchy is developed in each of the functional and physical domains by the zig-zagging method of decomposition. After returning to the functional domain, each FR at Level 1 is separated into its elements which are known as Level-2 {FRs). In this example, only the Level-1 FR decomposition considered is the requirement for greater thrust, as shown in Fig. 51. Greater thrust FUNCTIONAL DOMAIN PHYSICAL DOMAIN

NEW SYSTEM NEW REOUIREMENTS AIRCRAFT 1 1 I I I I NEW NEW GREATER LOWER GREATER NEW ENGINE STRUCTURE MATERIALS LEVEL 1 THRUST WEIGHT RIGIDITY CUNFIGURAIION CONFIGURATION L L .L I I I 1 t t 1 I 1 -- -- i 1 t I1 IMPROVED PROPULSION INTEGRAL ALUlY STRUCTURES LEVEL 2 I!Ec'p SYSTEM PRoPERTY EFFICIENCY I I I EX BLISK I I 1 i I

MECHANICAL ENVIRONMENTAL HEAT FLUID M~C~RUCTU~PROTECTIVE FOR STREAMLINE LEVEL 3 PROPERTY PROPERTY FLOW FLOW SCALE HEAT TRANSFER L-1- L-1- I-

Figure 51. Functional and Physical Domains Considered in Example Design Problem of High-Performance Aircraft-Engine Component. 124

requirement translates into Level-2 {FRs) such as improved service properties and improved propulsion-system efficiencies. Subsequently, the Level-2 {FRs) are mapped to the same level of the physical domain, as shown in Fig. 51. After consideration of the Level-2 {FRs), the corresponding DPs selected are a specific alloy system and integral structures. An example of an integral structure is a blisk which is a hybrid structure of a turbine disk and blades. The qualitative Level-2 relationship between the {FRs) and {DPs) is shown below:

Service Properties = [:3 [IIOY SYS~~~S) > (68) Propulsion-System Efficiencies Integral Structures

Equation (68) represents a decoupled design which implies that the service- property requirements should be satisfied first and then the propulsion-system efficiency requirements. Hence, the alloy systems should be selected before the integral structures are chosen. Similarly, the Level-3 {FRs) and subsequently the Level-3 {DPs) are determined by zig-zag decomposition of the functional and physical domains, respectively. The level-3 {FRs), the Level-3 {DPs), and the decoupled relationship between them are shown in the following equation: Hence, the required mechanical and environmental properties will be satisfied, respectively, by the microstructure and the protective scale which forms on the alloy selected. Equation (69) indicates that the heat-flow requirement should be addressed first and then the fluid-flow one. These functional and physical domains can be further decomposed. For the purpose of this example, however, the three levels of hierarchial structure given for both domains are sufficient. After completion of the decomposition of functional and physical domains, a heirarchy can be developed for the process domain. The process variables {PVs) are determined after consideration of the {DPs), the manufacturing-technology advancements, and the imposed constraints of time, cost and quality. As shown in Fig. 52, a heirarchy is developed in the process domain by zig-zagging back and forth between the physical and process domains. The Level-1 (DPs), the Level-1 (PVs), and their qualitative relationship are given'in the following equation:

Processes Finishing Operations Structure . (70) Material Removal Processes

Since the number of PVs is greater than the number of DPs, this design solution is not unique. The partitioned design matrix implies that the starting workpiece is considered to be a fixed PV. This assumption is sufficient for at least two reasons. First, all three DPs are affected by the initial conditions of the material, and this coupling should be confined. Secondly, advanced materials are generally supplied in very limited forms. Hence, under this assumption,

Eq. (70) becomes a decoupled design. This design solution implies that the thermomechanical processes for producing the required material state should be determined before the finishing operations or material-removal processes are selected. The next decomposition in the process domain is Level 2. The corresponding relationship between the {DPs) and {PVs) is given in the following equation: phermal Processes I Deformation Processes 1I Systems 1= [X X 0 01 " ITMP History of Billet (71) Integral Structures OXIXXX Nonconventional Machining @eating Process J

This design matrix is decoupled by restricting the thermomechanical processing history of the billet, the nonconventional machining process, and the coating process. The design solution of Eq. (71) indicates that the deformation processes for producing the integral structures should be selected before the heat-treatment processes are chosen. The Level-3 relationship between the {DPs) and {PVs) is given in the following equation:

Net-Shape Forging (streamlined ~eometrj) [X 0 0 0:0 ProcessesI X X 0 0;X X Electrochem. Machining *structure /= , . (72) Heat-Trans. Geometry 1X 0 X 0 lo 01 1Chemical Vapor Dep. Protective Scale 0 0 0 X10 0 Starting Microstructure Starting Billet Size 1 128

If the starting microstructure and billet size are held constant, Eq. (72) represents a decoupled design. The Level-3 design solution for selecting the {PVs) indicates that the net-shape forging process for producing the required streamlined geometrical shape should be addressed before the other processes are considered.

In general, a given part can be manufactured in several different ways. The previous example demonstrated this fact since the multiple combinations of {PVs) in the process domain were shown to satisfy the {DPs) in the physical domain. This situation is advantageous because it fosters the generation of creative design solutions. However, the selection of the best process design from among those proposed is a complicated task. Therefore, at all levels of process design, particular knowledge of {PV) characteristics is required for systematic evaluation of different process design solutions. The relationship between the physical and process domains can be represented graphically to provide further insight into the process design problem. Suh21 has shown how isograms of {DPs} and {PVs} can be superimposed as an aid to visualizing the functional coupling between them. Consider the following decoupled relationship:

(73) Thermal Processes which is subject to material and equipment constraints. Equation (73) represents a fundamental structure-processing relationship where the geometrical 129 shape and microstructure are the design parameters and the hot-deformation and thermal processes are the process variables. The decoupled relationship given by Eq. (73) is graphically shown in Fig. 53. In this figure, the two PV axes are not orthogonal, and the axis associated with the hot-deformation process is not parallel to the axis associated with the geometrical shape. The state xo in the functional domain is represented by the initial microstructure (e.g., grain size) and geometrical shape. Because of the nonorthogonal, nonparallel relationship between the axes, x, in the process domain cannot reach the desired final state, xf, in one processing step. The decoupled relationship of these DPs and PVs requires an intermediate state, xi, as shown in the figure. Hence, the axiomatic approach implies that the process design solution to the given problem is thermomechanical cycles of hot-deformation and post-deformation heat treatment. The number of steps required in the thermomechanical sequence of

GEOMETRY

Figure 53. lsogram of Process Variables Mapped Onto Physical Domain. PV Axes are Hot Deformation and Thermal Processes; DP Axes are Geometrical Shape and Microstructure; xo is Initial State; xj is Intermediate State; xf is Final State. 130 operations is dependent upon the constraints imposed by the particular capabilities of the material, the shapemaking process, and the forming equipment and tools. Several analytical methods are available for identifying the characteristic phenomena associated with particular shape-making processes. Generally, these methods fall into the following categories: (1) the slab method which restricts the change of stress to one direction, (2) the uniform-deformation energy method which neglects redundant work involved in internal shearing due to non-uniform deformation, (3) the slip-line field solution which is limited to rigid-plastic materials under plane-strain conditions, (4) the bounding methods which can provide fairly accurate estimates of the upper and lower limits of the deformation force but cannot provide details of the local stress and strain distributions, and (5) advanced numerical techniques such as finite-element or finite- difference methods which are applicable to complex geometries and take into account realistic friction and thermal boundary conditions as well as material properties as functions of temperature, strain, and strain rate. These methods of analysis are given in increasing order of complexity and ability to predict fine detail.85~86~8~Hence, the simplified methods such as slab analysis are suitable in the global design of thermomechanical processes, and the sophisticated process models such as finite-element analyses are required for detailed process design. Every step of the thermomechanical process (or cycle) is a process variable which should be evaluated with respect to the corresponding material and geometrical shape requirements of the physical domain. Basic shapemaking processes can be characterized by time-varying profiles of strain, strain rate, and temperature which directly affect the state of the material and the 131 producibility of certain geometrical shapes. A post-deformation heat treatment is usually part of the thermomechanical cycle for homogenization, solution annealing, or precipitation hardening. Typical t hermomechanical cycles associated with rolling, extrusion, and forging are examined in the following para- graphs to illustrate some of their distinguishing characteristics. Sheet rolling is a hot or cold forming process for reducing the cross- sectional area of the workpiece material through the use of rotating rolls as shown in Fig. 54. In general, the rolled material becomes elongated and spreads simultaneously, while the cross-sectional area is reduced.16 This process is used to produce strips, sheets,and plates typically ranging from 0.1 to 20 mm in thickness and several centimeters in width and length. During the hot sheet rolling, the mean temperature of the workpiece may increase but generally is dependent upon the balance between deformation heating and the heat losses resulting from conduction to the rolls. The mean-strain-rate profile shown in Fig. 54 is based upon the analytical expression derived by Ford and Alexander.88 In hot-rolling processes strain rates are high and maximum strain levels are relatively low, as compared to those of other metalforming processes. In the process of direct hot extrusion, the heated billet is forced through a die, with the use of some form of lubrication, to obtain a product of desired cross section as shown in Fig 55. This process is used to produce long, straight, semi-finished metal products of constant cross section such as bars, solid and hollow sections, tubes, wires, and strips. The extrusion process is typically capable of producing shapes of 0.5 to > 2 cm in thickness, 2 to 80 cm in diameter, and up to several meters in length. During the hot-extrusion process, the deforming billet can experience a significant increase in temperature due to the large, high rate of deformation typically associated with this process. The PLASTIC STRAIN STRAIN RATE TEMPERATURE TYPICAL THERMOMECHANICAL CYCLE

- 0 I .I HEAT TREATMENT EXTRUDE TIME

u TIME

Figure 55. Schematic Diagram of Extrusion Process. Temperature, Strain-Rate, and Plastic-Strain Profiles Given for Typical Thermomechanical Cycle. 134 mean-strain-rate profile shown in Fig. 55 is based upon a conical extrusion die design and can be modified by using other die geometries. For example, a streamlined die would produce gradual changes in strain rates near the entrance and exit of the die, whereas a sharp variation in strain rate is encountered in the central region of the die.89 The strain rates and maximum strain levels associated with hot extrusion are relatively high in comparison to those of other metalforming processes. Forging is generally defined as the process of deforming a rough workpiece in open or closed dies by means of compressive forces. Open-die forging involves the use of flat surface dies to upset the workpiece. Closed-die forging involves forming to the required shape and size by machined impressions in specially prepared dies which exert three-dimensional control over the workpiece. The forces can be administered by a slow squeeze (press forging) or by impact (drop or hammer forging). The forging process is typically capable of producing axisymmetrical shapes of 0.5 to > 50.0 cm in thickness and 10 to < 100 cm in diameter. An example of the various steps involved in the forging of an integral blade and rotor component is shown in Fig. 56. In this case the billet is upset to the preform shape at a nearly constant temperature and a moderate strain rate and strain level. The shape of the strain-rate profiles shown in Fig. 56 is typical of press-forging operations and can be modified through the use of other types of forging equipment. In subsequent forging steps, the strain rate and strain levels continue to decrease until they are relatively low when compared to those of other metalforming processes. Implications of the information axiom can be used in selecting the best shapemaking process from those considered. The selected process design corresponds to the one which minimizes the information content of the material TYPICAL THERMOMECHANICAL CYCLE

Wa 3 at aW 1 I- TREATMENT

U TlME

W at z sz 0 TlME

3 d z 2 I- V) a4 I I I I * I I I I I I 0 TlME

Figure 56. Schematic Diagram of Forging Sequence for Producing Integral- Blade and Rotor Component. Temperature, Strain-Rate, and Plastic-Strain Profiles Given for Typical Thermomechanical Cycle. 136 processing system, as previously defined in Eq. (62). As shown in Fig. 57, the system ranges associated with alternative thermomechanical processes can be compared to the design range for producing a particular geometrical shape and microstructure. In this figure the system ranges of rolling, extrusion, and forging processes are indicated by double-headed arrows. The design ranges are represented by (shaded) probability density distributions for quantities such as dimensions or grain size for producing a hypothetical disk component. The example product is nominally specified in terms of height (1.5 cm), diameter (10 cm), and grain size (25 mm). The system ranges associated with producing certain grain sizes are derived from the known microstructure response to characteristic profiles of strain, strain rate, and temperature. In this example problem, extrusion is the process of choice because it has a greater overlap of design and system ranges. This axiomatic approach to the selection of thermomechanical processes is applicable to all levels of process design. I. SHAPE SPECIFICATIONS

@ FINAL THICKNESS, Z,

EXTRUDE

0 0.8 1 .O 1.5 2.O z, (cm)

@) FINAL DIAMETER, 0,

II. MATERIAL SPECIFICATIONS

FINAL GRAIN SIZE, d,

4 ROLL- *

Figure 57. System Ranges of Rolling, Extrusion, and Forging Processes Compared to Design Range for Producing Hypothetical Disk Component. VII. PROCESS-DESIGN METHODOLOGY FOR PRODUCING REQUIRED MICROSTRUCTURE

In this chapter material-behavior phenomena are examined in more detail using design axioms and related concepts. As Suh22 pointed out, the property, structure, and processing relationships associated with materials science are no different from those relative to any other design problem. These material-behavior relationships can be expressed in terms of the functional, physical, and process domains.90 The characteristic design vectors considered in the present study are summarized in Table 5. The functional domain is assumed to consist of a set of mechanical properties. The related physical domain is considered to be microstructural features. For convenience, the corresponding process domain is partitioned into two characteristic vectors: the field quantities, {FQs), which describe the thermomechanical path and the process variables, {PVs), which are the equipment parameters. These two vectors associated with the process domain can be combined into one characteristic vector. However, the {FQs) provides a common basis for relating the microstructural development and various types of processing equipment and associated control parameters. The elements of the design vectors given in Table 5 are defined in this chapter. In general, the desired microstructure of a given material system can depend upon a variety of required material properties such as mechanical, chemical, and thermophysical properties. Standard mechanical properties such as yield strength (oYs),ultimate tensile strength (outs), ductility (E~,),and fracture toughness (Klc) are determined substantially by the state of the microstructure. The development of a design matrix for relating the mechanical Table 5 Characteristic Design Vectors for Material Systems

{FRs) : Mechanical Properties, {oysloutsl cC1 KIc. ...)

{DPs) : Microstructure, {dl X,,, Xp, rpl A, ...)

{FQs) : Thermomechanical Path, {el i,Twl t,, iw,... )

{PVs) : Equipment Parameters, {r, VR, TB, ...)

properties to the microstructural features is beyond the scope of this dissertation and is not attempted. Hence, material properties are merely mentioned here for the purpose of providing a more complete view of microstructural relationships with respect to design domains. Microstructural states can be characterized by features such as grain size (d), volume fraction recrystallized (X,,,), volume fraction of the second phase (Xp), size of the second-phase particles (rp), and aspect ratio of the matrix- phase grains (A). The state of a given microstructure is derived from its prior thermomechanical history. Hence, the microstructure is controlled by the conditions imposed by thermomechanical processes and by its initial condition (i.e., previous state). The relationship between the (DPs) of microstructure and the {FQs) of thermomechanical path is examined for both thermal and hot- deformation processes. 140 Relevant field quantities associated with thermal processes are workpiece temperature (T,), dwell time (t,), cooling rate (T~),and prior microstructural states. The qualitative relationship between a given state of microstructure and the {FQs) for thermal processing is expressed in the following equation:

OXXXXXXO

where n and n-1 represent the nth state and the previous state, respectively. In the design matrix, X represents a nonzero element and 0, a zero element. Equation (74) can be decoupled in more than one way. The X-terms in the design matrix can be replaced with analytical expressions from known microstructural-development relations or, alternatively, with empirical coefficients from experimental correlation studies. Similarly, a design matrix can be constructed for relating microstructure to field quantities associated with hot deformation. The relevant {FQs) are plastic strain (E), strain rate (E), workpiece temperature (T,), and prior microstructural states. The corresponding relationship is given in qualitative terms in the following equation: 0XXX0000 ;I:::::::OXXOOXXO 0XX0000X

where j and j-1 represent the jth state and the previous state, respectively.

DESIGN MATRICES FOR RECRYSTALLIZATION PROCESSES

Control of both dynamic and static recrystallization events is the essence of thermomechanical processing. Recrystallization during hot working usually-- but not always--leads to grain refinement which generally corresponds to favorable mechanical properties. For materials with inferior hot ductility, the formation of intergranular cracks can be inhibited through the onset of dynamic recrystallization. Both recrystallization events can enhance the hot workability of a material.

. . namlc R-n of Plain C-Mn Steel

The kinetics of dynamic recrystallization in plain C-Mn steels have been studied by several researchers, as shown earlier in Table 2 of Chapter Ill. The activation energy for this metallurgical phenomenon has been reported to range 142 from 267 to 312 kJ/mol. Microstructural-evolution models have been developed to predict the critical strain for initiation of dynamic recrystallization (E,), grain size [d(j)], and volume fraction of recrystallization (X,,,) as a function of temperature (T,), plastic strain (E), strain rate (E), and initial grain size [d(j-l)]. Yada's equations59 were used in the present study to derive a design matrix. They were chosen because of their completeness and generality in describing the dynamic recrystallization process. Yada's equations follow:

cC = 4.76 x 1 exp (Y,,

d(j) = 22,600 E exp - -0.2' [ (:w)!

X,,, = 1 - exp [ -0.693 w.1.-

where Q = 267.1 kJ/mol, R = 8.314 J/(g . mol . K), and ~,.5 is the plastic strain for 50 volume percent recrystallization. The units of T, ti, and d are Kelvin, per second, and micrometers, respectively. In the evaluation of these equations, it is convenient to linearize terms through a logarithmic operation.91 For describing the grain size during dynamic recrystallization, Eq. (77) can be written as 143

For evaluation of changes in grain size with strain rate at a fixed temperature, Eq. (80) is differentiated as follows:

To evaluate how the grain size changes with temperature at a fixed strain rate, Eq. (80) is differentiated as follows:

In the evaluation of Xrxn, it is convenient to consider the volume fraction dynamically recrystallized, Xnot, where

If Eqs. (78) and (83) are combined,

XnOt= exp [ -0.693 r:o:)j. -

Taking the logarithm of Eq. (84) yields

Again, taking the logarithm of Eq. (85) linearizes the relationship as follows: The logarithm of Eq. (79) yields the following relationship:

For evaluation of the changes in volume fraction of recrystallization with strain rate at a fixed temperature, strain, and initial grain size, Eq. (86) [with the Eq. (87) substitution] is differentiated as follows:

For evaluation of the sensitivity of Xnot with temperature, Eq. (86) [with Eq. (87) substitution] is differentiated as follows:

Similarly, differentiation of Eq. (86) with respect to In(&- E~)yields

The sensitivity of Xnot with respect to the initial grain size is given by the differentiation of Eq. (86) [with Eq. (87) substitution] as follows: The microstructural attributes of grain size and volume fraction of a dynamically recrystallized grains are defined as the set of design parameters {DPs). The corresponding field quantities {FQs) of the process are defined as temperature, strain, strain rate, and previous grain size. Hence, a system of equations can be written through the use of Eqs. (81) - (82) and (88) - (91). That is,

If the temperature and previous grain size are fixed, Eq. (92) reduces to the following decoupled design:

where a = ln(22.600) - 0.27 Q/(RTw) and P = ln(0.693) - 2 ln(1.144 x - 12,840/Tw - 0.56 In[d(j-1)].

Static Recrvstallization of Plain C-Mn Steel

A design matrix for static recrystallization of plain C-Mn steel is derived based upon the following equations of Saito, et a/. :92 X = 1 - exp 10.693 I&)*],

and to., = 2.5 10-lg r4dg exp ($1 ,

where Qs = 300 kJ/mol, R = 8.31 4 Jl(g mol K), and to.5 is the time intetval for 50 volume percent recrystallization. It should be noted that the units of T, t,, and d are Kelvin, seconds, and micrometers, respectively. A system of equations can be derived for static recrystallization through the use of the mathematical approach employed previously for dynamic recrystallization. Hence, the microstructural attributes of grain size and volume fraction of dynamically recrystallized grains is defined as the set of design parameters {DPs). The corresponding field quantities {FQs) of the process are defined as previous grain size, plastic stain, temperature, and dwell time. The processing relationship associated with static recrystallization is as follows:

Therefore, Eq. (97) can be readily reduced to a decoupled design by fixing the temperature and previous grain size. DESIGN MATRIX FOR HOT-PLANE-ROLLING PROCESSES

In this section, a design matrix is derived for relating the {FQs} associated with hot-deformation processes to the (PVs} associated with hot-plane-rolling processes. The equipment parameters considered in this investigation are percent reduction in height (r), velocity of the rolls (VR), and initial workpiece temperature (TB). These three processing variables are related to the plastic strain

(E), strain-rate (E), and working temperature (Tw) using the following simple mathematical models:

E = In (1 - r),

and

Equation (98) can be derived from the definition of true strain. The h, and R in Eq. (99) are defined as initial height of the workpiece and roll radius, respectively. This well-known expression for strain rate was determined by Ford and Alexander.88 Equation (100) is an assumed empirical relation for working temperature which is known to have some dependence upon r, VR, and TB.

Natural-logarithm operations tend to be a convenient means of decou- pling relationships, as shown in previous design problems. However, this is not always the case, which can be shown by the expression in Eq. (98). Further- more, in the axiomatic approach, eloquent expressions of design relationships are not required, if reasonably simplified ones are justified. For example, the 148 relationship between e and r can be simplified for relatively low strain processes. That is, for E I 0.2; then,

Fortunately, the plastic strain levels encountered in plane-rolling processes are low, and this simplifying assumption is considered to be valid. The FQs and PVs are related by

(FQs} = [B] (PVS} + {C} , (102) where Bij = aFQ@PVi and Ci is a vector of constants. If one takes the logarithm of Eq. (101), the following equation results:

In (E)= In (r) . (1 03)

Evaluation of the B1, elements yields

and

If the logarithm of Eq. (99) is taken, the result is In & = In VR + 0.5 In r - 0.5 In (R h,).

Evaluation of the corresponding design-matrix coefficients yields the following:

and

If the logarithm of Eq. (100) is taken, the result is

InT,,,,= Ink +InTB+aInr+bInvR, (111)

Evaluation of the corresponding design-matrix coefficients yields the following:

and 150 If the coefficients are collected, the corresponding design matrix can be expressed as

EXAMPLE: DESIGN OF A HOT-ROLLING PROCESS FOR PRODUCING STEEL SHEETS WITH A FINE-GRAINED MICROSTRUCTURE

The purpose of this example is to demonstrate how the design matrices given in Eqs. (93) and (1 15) can be employed to solve a given problem. In this example, the following conditions are assumed:

Initial grain size, d(O), = 100 pm, Isothermal process with Tw = TB = 1173 K, Initial sheet thickness, ho, = 12.5 mm, and

Roll radius, R, = 300 mm.

The design solutions for producing two different microstructures are examined.

In both cases, the required volume fraction recrystallized is X,,, = 0.1 0, but the required grain sizes are d(1) = 50 pm in the first case and d(1) = 30 pm in the second case. It should be noted that dynamic-recrystallization processes achieve relatively low levels of completion during hot-rolling operations. Hence, numerous thermomechanical operations involving combinations of static and dynamic processes are employed for full recrystallization of the workpiece material. 151 The given assumptions can be used to determine the constant terms of Eqs. (93) and (1 IS), where a = 2.630851, P = -0.345463, and C2 = -2.8723. Hence, Eq. (93) can be rearranged and expressed as

Similarly, Eq. (1 15) can be rearranged and expressed as

By definition [Eq. (83)], it can be shown that xnOt= 0.90 for x,, = 0.10.

Case I: Processina Variables for Prsducina d(1) = 50 urn

If d(l) = 50 pm and Xnot = 0.90 are substituted into Eq. (1 16) and the system of equations solved, the result is In(&- E,) = -1.1 897 and In(&) = -4.7451.

From Eq. (76), the critical strain E, at 1173 K can be determined to be 0.44.

Hence, the corresponding total plastic strain, E, is 0.74 and the strain rate, E, is 8.7 x 10-3 s-1. The selected {FQs) are substituted into Eq. (1 17) to determine the characteristic vector of PVs which consists of percent reduction, r, and roll velocity, VR. The equipment parameters are found to be In(r) = -0.301 1 and In(VR)= -1.7222, which corresponds to r = 0.74 and VR = 0.179 mm s-1. Case II: Processina Variables for Producina dfl! = 30 um

If d(l) = 30 pm and XnOt = 0.90 are substituted into Eq (1 16) and the system of equations solved, the result is In(&- E,) = -1.0951 and In(&) = -2.8531.

Hence, the corresponding total plastic strain, E, is 0.77 and strain rate, &, is 5.77 x 10-2 s-1. The selected {FQs) are substituted into Eq, (117) to determine the characteristic vector of PVs which consists of percent reduction, r, and roll velocity, VR. The equipment parameters are found to be In(r) = -0.2614 and

In(VR)= 0.1 498 which corresponds to r = 0.77 and VR = 1.1 62 mm s-1. In summary, design matrices were determined for relating the required microstructural parameters to the processing variables. Decoupled design matrices were derived for dynamic- and static-recrystallization processes, as well as for hot-plane-rolling processes. A hypothethical example problem was employed to demonstrate the relationship between the derived design matrices. Hence, it appears feasible to begin with the given {FRs), to proceed through the intermediate relationships, and finally to select the corresponding {PVs). VIII. OPTIMAL CONTROL STRATEGY FOR DYNAMIC- RECRYSTALLIZATION PROCESSES

The goals of process control can be met through several different control- system design strategies ranging from simple open-loop to sophisticated closed-loop control systems. The direction of mainstream research is to employ various advanced computer technologies for sensing, analyzing, making deci- sions, and implementing corrective actions if needed. Hence, the resultant control strategies are primarily determined from heuristic knowledge of the manufacturing process and the workpiece material. In the present research, a new control-system design strategy was investigated--a strategy explicitly based upon the state of the material under hot-deformation conditions. The design of metalforming processes is strongly influenced by the spatial and temporal nature of the thermomechanical field quantities associated with the deforming material. Field quantities such as plastic strain, strain-rate, and temperature actually control the material behavior. The spatial distribution of these field quantities is determined primarily by the die and workpiece geometries and the heating system for the process. This aspect of the control problem has been addressed in part by other researchers93 and is not pursued here. The focus of this investigation was the use of kinetic models of microstructural processes to determine the admissible trajectories in plastic strain, strain- rate, and temperature space. The dynamic control inputs for a metalforming process could be determined from the kinetic requirements of a desirable metallurgical mechanism. Through the use of variational techniques, an optimal control method has been formulated by Malas and Irwin94 for the dynamic-recrystallization process. 154 In the present study Yada's m0de158~59for the dynamic recrystallization of plain C-Mn steel which were defined earlier in Eqs. (76) - (79) was used. A major problem was encountered in applying the variational approach due to the

highly nonlinear nature of the equations relating strain (E), strain rate (E), and temperature (T) to grain size (d) and volume percent recrystallized (X). Hence, Yada's set of equations were semi-linearized through a combination of nonlinear transformation of variables, perturbation analysis, and linearization. From the semi-linearized set of equations, an optimal control problem could then be formulated. Using standard variational techniques, the necessary conditions for optimality were derived.

CONTROL-PROBLEM FORMULATION

...... eml-l 1near17a1onof Yada's Fam~onsfor Dvnam~cRecrys-

This section includes the derivation of the semi-linearized equations used in the optimization procedure. The initial nonlinear transformations, the perturbation analysis, and the linearization procedure are presented. Yada's equations for dynamic recrystallization are given here in expanded form:

and

X= 1 - [ '-K3 1 , K4d2iK9 exp 155 where K1 = 22,600, K2 = -0.693, Kg = 4.76 x 10-4, Kq = 1.I 44 x 10-3, Kg =

8000, Kg = 6420, K7 = -0.27, Kg = 0.28, and Kg = 0.05. The equations can be written more compactly by defining the critical strain (E,) as

Q = K3 exp (y)

and the strain for 50 percent recrystallization (E~.~)as

~0.5= K4 d!$ tiK9 exp (9- .

Yada's equations are then d=K1 eexp-(EDK7 and X = 1 - exp [K2 (&io:T]& -

From Eq. (122) the grain size can be plotted as a function of T and &, as shown in Fig. 58. Similarly, the fraction recrystallized can be plotted as a function of T and & from Eq. (123), as shown in Fig. 59. Clearly, Yada's equations are highly nonlinear functions of strain, strain-rate, and temperature. This situation was alleviated in part by taking the natural logarithm of Eqs. (1 22) and (123) as follows:

In(d) = In(K1) + K7 In(&)+ K7Q 1 and .C- 2 (3- a* Q) r N I1 -a c C -- ac6s a," a 2 * 3 Oaa,.- 2 8. a, € -0 a, %I- ams a9c a 'f cr 0 C as .-g ,a P c I ao, -32 - .o % a + ca Q1c .-€2 ah 6, s-2 .z I-L

158

In Eq. (124) the quantities In(&) and 1IT appear as linear functions of ln(d). Unfortunately, a linear relationship was not established in Eq. (125). Some further simplification of Eq. (125) was obtained by performing another logarithmic operation and expanding the terms as follows:

In[-ln(1 - X)] = In(-K2) + 21n (1 26)

The term In(&-&,) remains in Eq. (128) as a nonlinear function of strain and temperature. The nonlinear term was then expanded for linearization as follows:

where and

The required partial derivatives are given by and

Equations (132) and (133) can be substituted into Eq. (129), and the result can be used in Eq. (128). Equations (130) and (131) can be substituted into Eqs. (124) and (128) and the resulting equations written as follows:

and

where

-~K~K,exp j+\ and

The right-hand side of Eqs. (134) and (135) is linear in the variables Ah(&), In(&), and A(1lT). This situation has made it possible to define an optimality criterion which is both realistic and mathematically tractable.

Optimalitv Criterion for Dvnamic Recrystallization

Several goals can be considered in choosing a strain, strain-rate, and temperature profile for a metalforming process. In this case the assumed goals are: 1) to maintain a small grain size, 2) to ensure that the recrystallization process proceeds at a proportionate rate, and 3) to regulate temperature. A corresponding optimality criterion which is to be subsequently minimized can be given as

+ I:[., {lnjd) - In[-ln(1 - X)] l2+ a2[ln(d)]' + a3 [XIdt9 where PI, P2, all a2 , and a3 are design parameters which can be varied and xf and df are the desired final percent recrystallization and desired final grain size, respectively. The integrand of Eq. (144) can be represented as f(xl, x2, x3), where Xi = Ah(&), and

Hence,

Before the necessary conditions for optimality can be stated, the following constraint must be enforced:

or, equivalently,

which has the advantage of being expressed in terms of xl, xp, and x3. This constraint can be enforced via a lagrange multiplier p to yield the following augmented integrand:

From standard calculus of variations,23 the necessary conditions for optimality are given as 3 .a[&]=o, ax, dt ax,

and

The necessary conditions for optimality can be evaluated by substituting Eqs. (1 48) and (151). The resulting equations are given below:

and where and

This control problem was subsequently solved by 1rwin.95 His solution of the necessary conditions for optimality is given in the Appendix. An important result of the solution was that the strain rate should be maintained constant. An optimal strategy for choosing strain, strain-rate, and temperature trajectories has been developed in this chapter. Yada's equations for dynamic recrystallization have been transformed and partially linearized. The resulting equations have been used to form a practical criterion for choosing the required trajectory, and the optimal (or minimizing) trajectory has been derived. For the criteria considered here, the optimal strategy for dynamic recrystallization was shown to require a constant strain rate throughout the hot-deformation process. IX. SUMMARY AND CONCLUSIONS

A methodology for the design and control of thermomechanical processes was developed based upon a comprehensive understanding of the material behavior under manufacturing conditions. A scientific methodology was established for determining the processing conditions under which material behavior is stable and predictable. An axiomatic approach was applied to the design of thermomechanical processes for determining such variables as the initial conditions of the workpiece, the number and sequence of operations required, and the processing conditions corresponding to each operation. Finally, a new method for establishing control strategies was formulated in order to determine the optimal microstructural evolution path and the corresponding control inputs. The fundamental approach adopted in this research was to unravel the material aspects of the design and control problem at all levels. Perhaps the overall design methodology developed in this study will enable metallurgical synthesis in the way Burte and Gegelg6 originally envisioned it over a decade ago. In fulfilling the goals of metallurgical synthesis, this research provides a means for (1) achieving the desired part geometry (size, shape, and tolerance) with adequate defect control, (2) developing a controlled microstructure to yield the desired properties and in-service performance, and (3) optimizing the cost of production. Conclusions relative to various aspects of the research are summarized in the following sections. MATERIAL-BEHAVIOR MODELING

Relationships among constitutive equations, hot workability, and microstructure development were established to provide a more comprehensive understanding of material behavior. The dynamic-material-modeling (DMM) approach to hot workability provided advantageous relationships relative to constitutive behavior and microstructure development through stability analysis and prediction of apparent activation energy, respectively. The level of difficulty involved in hot deformation of a particular metallic system was evaluated by determining the information content of that system. The information content was defined in terms of the temperature and strain-rate ranges associated with both the processing equipment and the processing window of the particular material. The processing window of a given metallic system was defined as the conditions under which material behavior is stable and a desirable and relatively constant value of apparent activation energy exists. Three different metallic systems--namely, TiAI, All and Ni-base alloys-- were employed to validate the material-modeling approach and clarify the role of material behavior in process design. For the case of a gamma-titanium-aluminide alloy, DMM predictions of stability and apparent activation energy were useful in identifying the processing regimes where desirable softening mechanisms are in operation and the material behavior is essentially deterministic. Structural complexity was shown to have deleterious effects upon the processing window of 2024 aluminum. The material behavior of P/M Rene 95 was shown to be significantly affected by seemingly small differences in thermomechanical history. Successful prediction of processing windows in these case studies which represent very different types of materials implies that the 166 material-modeling approach has wide applicability to hot-deformation processing of a variety of materials.

DESIGN METHODOLOGY

The axiomatic approach was effectively applied to the design of thermomechanical processes for different levels of detail. The four design domains were shown to clarify the material and geometrical shape aspects of product and process design problems. In the functional domain, requirements were identified in terms of material and geometrical-shape properties. In the physical domain, design parameters were identified in terms of material and geometrical structures. In the process domain, variables were identified in terms of the initial conditions of the workpiece material and thermal and shape-making processes. Processing variables were selected by comparing the system ranges of alternative processing variables with the design range of a component in terms of geometrical shape and material specifications. Hence, the implications of the independence and information axioms provide a fundamental basis for formu- lating and solving process design problems. The property, microstructure, and processing relationships which are associated with material science were shown to be the same as those for any design problem. Material-behavior relationships were expressed in terms of the functional, physical, and process domains. Decoupled design matrices were derived for dynamic- and static-recrystallization processes of plain C-Mn steel and for a hot-lane-rolling operation. The nonlinear relationships which are prevalent in the modeling of microstructural evolution and thermomechanical operations were either simplified or mathematically transformed for construction 167 of process design relationships. Hence, the results of this research suggest that it is feasible to begin with a target microstructure, proceed through intermediate relationships, and finally select the corresponding workpiece and equipment parameters.

PROCESS-CONTROL METHODOLOGY

An optimal control methodology based upon phenomenological models of material behavior was developed for open-loop control of hot-deformation processes. Control systems can be designed to maintain stable material- processing conditions and comply with the kinetics of desired microstructural changes. This approach to process control leads to robustness of the material- processing system with respect to internal perturbations such as stochastic material responses and process transients. Robustness with respect to external disturbances must be realized through feedback control methods; such methods were not developed in this research. A control strategy was developed for determining an optimal strain, strain rate, and temperature trajectory for dynamic recrystallization of plain C-Mn steel. The corresponding equations which describe the microstructure development during hot-deformation processes were transformed and partially linearized. These equations were used to formulate an optimality criterion for control-system design which was based upon achieving the desired dynamic material response and considering the practicality of real-time temperature control. From standard calculus of variational techniques, the necessary conditions for optimality were derived. An optimal strain, strain rate, and temperature trajectory for dynamic recrystallization of plain C-Mn steel was shown to exist. The 168 formulation and solution of the optimal control strategy developed in this study are sufficiently generic to permit application to other material systems and the study of other diffusion-controlled transformation processes.

FUTURE WORK

As a continuation of the research reported in this dissertation, the following future investigations are recommended. Additional studies in material- behavior modeling should include a statistical treatment of flow stress and processing maps, detailed metallurgical evaluations of bifurcation phenomena at the boundaries of stable and unstable processing regimes, and detailed characterization of micromechanisms other than dynamic recrystallization. For improvement of the process-design methodology, efforts should be focused on expansion of design matrices to incorporate additional functional requirements, incorporation of quantitative measures for functional independence such as reangularity and semangularity defined by Suh,21 and application of the design approach to thermomechanical processes other than hot rolling. Future work in process-control strategies should include simulation of the derived optimal trajectories to determine the acceptable range of the design parameters, analytical and experimental verification of the optimal trajectories, derivation of optimal trajectories based upon the complete non-linearized models of the material behavior, and development of methods for optimizing spatial variation of state variables (i.e., field quantities) during the process. REFERENCES

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CONTROL-PROBLEM SOLUTION

The following solution of the necessary conditions for optimality was derived by Irwin.95 It can be easily verified that the following choices for XI, x2, x3, and p satisfy the nonlinear differential relations of Eqs. (1 55) - (158) for some choice of constants:

x3 = c4 + cg In (cl t), (A-3) and

Hence, six undetermined constants remain and tf which must be determined from the boundary conditions. Three of the seven required equations for solution can be given by the initial conditions xl (to), In[d(t,)], and In[-ln(1-x(to))]. The other four required equations can be derived from the following optimality considerations. Let Then the required boundary conditions are

and

Hence, seven equations can be derived to determine the six constants and tf. However, the most important result can be realized by substituting Eq. (143) into Eq. (169), which results in an optimal control strategy of maintaining a constant strain rate.