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Fundamentals of Strength: Principles, Experiment, and Applications of an Internal State Variable Constitutive Formulation

ISBN: 978-1-118-80854-2
518 pages
December 2013
Fundamentals of Strength: Principles, Experiment, and Applications of an Internal State Variable Constitutive Formulation (1118808541) cover image

Description

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.

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Table of Contents

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 Stress–Strain Curve 12

1.10 The True Stress–True 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 DISLOCATION–OBSTACLE INTERACTIONS 76

4.1 A Simple Dislocation–Obstacle 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 Dislocation–Obstacle 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 Stress–Strain Curves 177

8.4 Eleiche and Campbell Measurements in Torsion 181

8.5 Analysis of Deformation in Nickel 187

8.6 Predicted Stress–Strain 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 Copper–Aluminum 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 Polycrystalline Niobium 263

9.11 Summary 272

Exercises 275

References 280

10 APPLICATION TO HCP METALS AND ALLOYS 282

10.1 Pure Zinc 283

10.2 Kinetics of Yield in Pure Cadmium 288

10.3 Structure Evolution in Pure Cadmium 292

10.4 Pure Magnesium 296

10.5 Magnesium Alloy AZ31 300

10.6 Pure Zirconium 311

10.7 Structure Evolution in Zirconium 317

10.8 Analysis of Deformation in Irradiated Zircaloy-2 325

10.9 Analysis of Deformation Behavior of Polycrystalline Titanium 333

10.10 Analysis of Deformation Behavior of Titanium Alloy Ti–6Al–4V 350

10.11 Summary 356

Exercises 360

References 364

11 APPLICATION TO AUSTENITIC STAINLESS STEELS 367

11.1 Variation of Yield Stress with Temperature and Strain Rate in Annealed Materials 367

11.2 Nitrogen in Austenitic Stainless Steels 372

11.3 The Hammond and Sikka Study of 316 381

11.4 Modeling the Stress–Strain Curve 382

11.5 Dynamic Strain Aging in Austenitic Stainless Steels 386

11.6 Application of the Model to Irradiation-Damaged Material 391

11.7 Summary 394

Exercises 396

References 400

12 APPLICATION TO THE STRENGTH OF HEAVILY DEFORMED METALS 403

12.1 Complications Introduced at Large Deformations 404

12.2 Stress Dependence of the Normalized Activation Energy goε 404

12.3 Addition of Stage IV Hardening to the Evolution Law 408

12.4 Grain Refinement 411

12.5 Application to Large-Strain ECAP Processing of Copper 417

12.6 An Alternative Method to Assess ECAP-Induced Strengthening 423

12.7 A Large-Strain Constitutive Description of Nickel 427

12.8 Application to Large-Strain ECAP Processing of Nickel 431

12.9 Application to Large-Strain ECAP Processing of Austenitic Stainless Steel 435

12.10 Analysis of Fine-Grain Processed Tungsten 444

12.11 Summary 447

Exercises 449

References 452

13 SUMMARY AND STATUS OF MODEL DEVELOPMENT 455

13.1 Analyzing the Temperature-Dependent Yield Stress 456

13.2 Stress Dependence of the Normalized Activation Energy goε 459

13.3 Evolution 460

13.4 Temperature and Strain-Rate Dependence of Evolution 461

13.5 The Effects of Deformation Twinning 466

13.6 The Signature of Dynamic Strain Aging 468

13.7 Adding Insight to Complex Processing Routes 473

13.8 Temperature Limits 479

13.9 Summary 483

References 485

INDEX 488

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Author Information

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.

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