Offers data, examples, and applications supporting the use of the mechanical threshold stress (MTS) model Written by Paul S. Follansbee, an international authority in the field, this book explores the underlying theory, mechanistic basis, and implementation of the mechanical threshold stress (MTS) model. Readers are introduced to such key topics as mechanical testing, crystal structure, thermodynamics, dislocation motion, dislocation obstacle interactions, hardening through dislocation accumulation, and deformation kinetics. The models described in this book support the emerging theme of Integrated Computational Materials Engineering (ICME) by offering a foundation for the bridge between length scales characterizing the mesoscale (mechanistic) and the macroscopic. Fundamentals of Strength begins with a chapter that introduces various approaches to measuring the strength of metals. Next, it covers: Structure and bonding Contributions to strength Dislocation obstacle interactions Constitutive law for metal deformation Further MTS model developments Data analysis: deriving MTS model parameters The next group of chapters examines the application of the MTS model to copper and nickel, BCC metals and alloys, HCP metals and alloys, austenitic stainless steels, and heavily deformed metals. The final chapter offers suggestions for the continued development and application of the MTS model. To help readers fully understand the application of the MTS model, the author presents two fictional materials along with extensive data sets. In addition, end-of-chapter exercises give readers the opportunity to apply the models themselves using a variety of data sets. Appropriate for both students and materials researchers, Fundamentals of Strength goes beyond theory, offering readers a model that is fully supported with examples and applications.

PAUL S. FOLLANSBEE, PhD, is a materials scientist and engineer with thirty-five years of experience at Los Alamos National Laboratory, Howmet Castings, General Electric Corporate Research and Development, and Pratt and Whitney Aircraft. He joined Saint Vincent College in 2008 as the James F. Will Professor of Engineering Sciences. His research centers on deformation modeling and constitutive behavior at low temperatures and high strain rates and the application of these models to materials processing and performance. Dr. Follansbee proposed and developed an internal state variable constitutive model, the mechanical threshold stress model, and has applied it to Cu, Ni, Ti-6Al-4V, and several other metals.

FOREWORD xi

PREFACE xiii

ACKNOWLEDGMENTS xv

HOW TO USE THIS BOOK xvii

LIST OF SYMBOLS xxi

1 MEASURING THE STRENGTH OF METALS 1

1.1 How Is Strength Measured? 1

1.2 The Tensile Test 3

1.3 Stress in a Test Specimen 6

1.4 Strain in a Test Specimen 6

1.5 The Elastic Stress versus Strain Curve 7

1.6 The Elastic Modulus 8

1.7 Lateral Strains and Poisson's Ratio 9

1.8 Defining Strength 11

1.9 StressStrain Curve 12

1.10 The True StressTrue Strain Conversion 16

1.11 Example Tension Tests 18

1.12 Accounting for Strain Measurement Errors 22

1.13 Formation of a Neck in a Tensile Specimen 25

1.14 Strain Rate 27

1.15 Measuring Strength: Summary 29

Exercises 29

References 35

2 STRUCTURE AND BONDING 36

2.1 Forces and Resultant Energies Associated with an Ionic Bond 36

2.2 Elastic Straining and the Force versus Separation Diagram 39

2.3 Crystal Structure 40

2.4 Plastic Deformation 42

2.5 Dislocations 46

2.6 Summary: Structure and Bonding 51

Exercises 52

References 53

3 CONTRIBUTIONS TO STRENGTH 54

3.1 Strength of a Single Crystal 54

3.2 The Peierls Stress 59

3.3 The Importance of Available Slip Systems and Geometry of HCP Metals 61

3.4 Contributions from Grain Boundaries 63

3.5 Contributions from Impurity Atoms 66

3.6 Contributions from Stored Dislocations 68

3.7 Contributions from Precipitates 71

3.8 Introduction to Strengthening: Summary 71

Exercises 72

References 75

4 DISLOCATIONOBSTACLE INTERACTIONS 76

4.1 A Simple DislocationObstacle Profile 76

4.2 Thermal Energy: Boltzmann's Equation 77

4.3 The Implication of 0 K 78

4.4 Addition of a Second Obstacle to a Slip Plane 79

4.5 Kinetics 80

4.6 Analysis of Experimental Data 83

4.7 Multiple Obstacles 87

4.8 Kinetics of Hardening 88

4.9 Summary 89

Exercises 90

References 92

5 A CONSTITUTIVE LAW FOR METAL DEFORMATION 94

5.1 Constitutive Laws in Engineering Design and Materials Processing 94

5.2 Simple Hardening Models 98

5.3 State Variables 102

5.4 Defining a State Variable in Metal Deformation 103

5.5 The Mechanical Threshold Stress Model 104

5.6 Common Deviations from Model Behavior 109

5.7 Summary: Introduction to Constitutive Modeling 112

Exercises 113

References 115

6 Further MTS Model Developments 117

6.1 Removing the Temperature Dependence of the Shear Modulus 117

6.2 Introducing a More Descriptive Obstacle Profile 119

6.3 Dealing with Multiple Obstacles 122

6.4 Defining the Activation Volume in the Presence of Multiple Obstacle Populations 131

6.5 The Evolution Equation 132

6.6 Adiabatic Deformation 133

6.7 Summary: Further MTS Model Developments 135

Exercises 137

References 141

7 DATA ANALYSIS: DERIVING MTS MODEL PARAMETERS 142

7.1 A Hypothetical Alloy 142

7.2 Pure Fosium 143

7.3 Hardening in Pure Fosium 145

7.4 Yield Stress Kinetics in Unstrained FoLLyalloy 146

7.5 Hardening in FoLLyalloy 150

7.6 Evaluating the Stored DislocationObstacle Population 151

7.7 Deriving the Evolution Equation 160

7.8 The Constitutive Law for FoLLyalloy 163

7.9 Data Analysis: Summary 164

Exercises 165

8 APPLICATION TO COPPER AND NICKEL 167

8.1 Pure Copper 168

8.2 Follansbee and Kocks Experiments 169

8.3 Temperature-Dependent StressStrain Curves 177

8.4 Eleiche and Campbell Measurements in Torsion 181

8.5 Analysis of Deformation in Nickel 187

8.6 Predicted StressStrain Curves in Nickel and Comparison with Experiment 192

8.7 Application to Shock-Deformed Nickel 195

8.8 Deformation in Nickel plus Carbon Alloys 198

8.9 Monel 400: Analysis of Grain-Size Dependence 200

8.10 CopperAluminum Alloys 205

8.11 Summary 211

Exercises 213

References 214

9 APPLICATION TO BCC METALS AND ALLOYS 216

9.1 Pure BCC Metals 217

9.2 Comparison with Campbell and Ferguson Measurements 225

9.3 Trends in the Activation Volume for Pure BCC Metals 228

9.4 Structure Evolution in BCC Pure Metals and Alloys 231

9.5 Analysis of the Constitutive Behavior of a Fictitious BCC Alloy: UfKonel 232

9.6 Analysis of the Constitutive Behavior of AISI 1018 Steel 237

9.7 Analysis of the Constitutive Behavior of Polycrystalline Vanadium 248

9.8 Deformation Twinning in Vanadium 256

9.9 A Model for Dynamic Strain Aging in Vanadium 258

9.10 Analysis of Deformation Behavior of ...