Cyclic Plasticity of Engineering Materials Experiments and Models

New contributions to the cyclic plasticity of engineering materials

Written by leading experts in the field, this book provides an authoritative and comprehensive introduction to cyclic plasticity of metals, polymers, composites and shape memory alloys. Each chapter is devoted to fundamentals of cyclic plasticity or to one of the major classes of materials, thereby providing a wide coverage of the field.

The book deals with experimental observations on metals, composites, polymers and shape memory alloys, and the corresponding cyclic plasticity models for metals, polymers, particle reinforced metal matrix composites and shape memory alloys. Also, the thermo-mechanical coupled cyclic plasticity models are discussed for metals and shape memory alloys.

Key features:

• Provides a comprehensive introduction to cyclic plasticity
• Presents Macroscopic and microscopic observations on the ratchetting of different materials
• Establishes cyclic plasticity constitutive models for different materials.
• Analysis of cyclic plasticity in engineering structures.

This book is an important reference for students, practicing engineers and researchers who study cyclic plasticity in the areas of mechanical, civil, nuclear, and aerospace engineering as well as materials science.

Professor Guozheng Kang achieved his Bachelor Degree from Tsinghua University, China in 1992, and then he obtained his Master and PhD degrees from Southwest Jiaotong University, China in 1994 and 1997, respectively. Kang joined Southwest Jiaotong University, China as a lecturer in 1997 and was promoted to associate professor and professor in 2003 and 2005, respectively. He has received the "Alexander von Humboldt Fellowship", "Outstanding Young Investigator Award of NSFC", "Cheung Kong Chair Professor of MOE, China", and "Program for Ten Thousands Talent, China". His research interests focus on the cyclic constitutive models of advanced materials, fatigue and fracture, and meso-mechanics analysis of composites. Kang has published 5 books, 4 book chapters and 130 international journal papers. Currently, he is a member of the editorial board for five international peer-reviewed journals, including the International Journal of Plasticity, ZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik, Acta Mechanica Sinica and the Journal of the Mechanical Behavior of Materials.

Dr. Qianhua Kan obtained his Bachelor Degree in Civil Engineering with first class honors from Zhengzhou University in 2002. He obtained his Master Degree in Solid Mechanics from Southwest Jiaotong University in 2005 and his PhD degree from the same University in 2009. Following this, Dr. Kan joined Southwest Jiaotong University as a lecturer in 2009 and was promoted to associate professor in 2012. Dr. Kan visited Monash University (Australia) as an award holder of the Endeavour Research Fellowship for six months in 2011. His research interests include fatigue failures of smart materials, wheel-rail contact, biomechanics and finite element analysis. Dr. Kan has been awarded 5 research grants from NSFC and National Key Laboratories since 2009. Currently, he is supervising and co-supervising 13 postgraduate research students. Dr. Kan has published 5 books, 2 book chapters and 45 international journal papers.

Introduction 1

I.1 Monotonic Elastoplastic Deformation 1

I.2 Cyclic Elastoplastic Deformation 3

I.2.1 Cyclic Softening/Hardening Features 3

I.2.2 Mean Stress Relaxation 6

I.2.3 Ratchetting 7

I.3 Contents of This Book 9

References 10

1 Fundamentals of Inelastic Constitutive Models 13

1.1 Fundamentals of Continuum Mechanics 13

1.1.1 Kinematics 13

1.1.2 Definitions of Stress Tensors 15

1.1.3 Frame‐Indifference and Objective Rates 16

1.1.4 Thermodynamics 17

1.1.4.1 The First Thermodynamic Principle 17

1.1.4.2 The Second Thermodynamic Principle 17

1.1.5 Constitutive Theory of Solid Continua 18

1.1.5.1 Constitutive Theory of Elastic Solids 18

1.1.5.2 Constitutive Theory of Elastoplastic Solids 19

1.2 Classical Inelastic Constitutive Models 22

1.2.1 J₂ Plasticity Model 23

1.2.2 Unified Visco‐plasticity Model 24

1.3 Fundamentals of Crystal Plasticity 25

1.3.1 Single Crystal Version 25

1.3.2 Polycrystalline Version 27

1.4 Fundamentals of Meso‐mechanics for Composite Materials 28

1.4.1 Eshelby’s Inclusion Theory 29

1.4.2 Mori–Tanaka’s Homogenization Approach 30

References 32

2 Cyclic Plasticity of Metals: I. Macroscopic and Microscopic Observations and Analysis of Micro-mechanism 35

2.1 Macroscopic Experimental Observations 35

2.1.1 Cyclic Softening/Hardening Features in More Details 35

2.1.1.1 Uniaxial Cases 35

2.1.1.2 Multiaxial Cases 43

2.1.2 Ratchetting Behaviors 47

2.1.2.1 Uniaxial Cases 48

2.1.2.2 Multiaxial Cases 62

2.1.3 Thermal Ratchetting 75

2.2 Microscopic Observations of Dislocation Patterns and Their Evolutions 77

2.2.1 FCC Metals 80

2.2.1.1 Uniaxial Case 80

2.2.1.2 Multiaxial Case 86

2.2.2 BCC Metals 95

2.2.2.1 Uniaxial Case 95

2.2.2.2 Multiaxial Case 103

2.3 Micro‐mechanism of Ratchetting 111

2.3.1 FCC Metals 111

2.3.1.1 Uniaxial Ratchetting 111

2.3.1.2 Multiaxial Ratchetting 114

2.3.2 BCC Metals 115

2.3.2.1 Uniaxial Ratchetting 115

2.3.2.2 Multiaxial Ratchetting 117

2.4 Summary 118

References 119

3 Cyclic Plasticity of Metals: II. Constitutive Models 123

3.1 Macroscopic Phenomenological Constitutive Models 124

3.1.1 Framework of Cyclic Plasticity Models 124

3.1.1.1 Governing Equations 124

3.1.1.2 Brief Review on Kinematic Hardening Rules 126

3.1.1.3 Combined Kinematic and Isotropic Hardening Rules 131

3.1.2 Viscoplastic Constitutive Model for Ratchetting at Elevated Temperatures 136

3.1.2.1 Nonlinear Kinematic Hardening Rules 136

3.1.2.2 Nonlinear Isotropic Hardening Rule 137

3.1.2.3 Verification and Discussion 138

3.1.3 Constitutive Models for Time‐Dependent Ratchetting 144

3.1.3.1 Separated Version 146

3.1.3.2 Unified Version 152

3.1.4 Evaluation of Thermal Ratchetting 161

3.2 Physical Nature‐Based Constitutive Models 163

3.2.1 Crystal Plasticity‐Based Constitutive Models 163

3.2.1.1 Single Crystal Version 163

3.2.1.2 Application to Polycrystalline Metals 167

3.2.2 Dislocation‐Based Crystal Plasticity Model 175

3.2.2.1 Single Crystal Version 175

3.2.2.2 Verification and Discussion 177

3.2.3 Multi‐mechanism Constitutive Model 183

3.2.3.1 2M1C Model 187

3.2.3.2 2M2C Model 188

3.3 Two Applications of Cyclic Plasticity Models 189

3.3.1 Rolling Contact Fatigue Analysis of Rail Head 189

3.3.1.1 Experimental and Theoretical Evaluation to the Ratchetting of Rail Steels 190

3.3.1.2 Finite Element Simulations 194

3.3.2 Bending Fretting Fatigue Analysis of Axles in Railway Vehicles 197

3.3.2.1 Equivalent Two‐Dimensional Finite Element Model 199

3.3.2.2 Finite Element Simulation to Bending Fretting Process 201

3.3.2.3 Predictions to Crack Initiation Location and Fretting Fatigue Life 203

3.4 Summary 209

References 211

4 Thermomechanically Coupled Cyclic Plasticity of Metallic Materials at Finite Strain 219

4.1 Cyclic Plasticity Model at Finite Strain 221

4.1.1 Framework of Finite Elastoplastic Constitutive Model 221

4.1.1.1 Equations of Kinematics 221

4.1.1.2 Constitutive Equations 221

4.1.1.3 Kinematic and Isotropic Hardening Rules 222

4.1.1.4 Logarithmic Stress Rate 223

4.1.2 Finite Element Implementation of the Proposed Model 224

4.1.2.1 Discretization Equations of the Proposed Model 224

4.1.2.2 Implicit Stress Integration Algorithm 227

4.1.2.3 Consistent Tangent Modulus 228

4.1.3 Verification of the Proposed Model 230

4.1.3.1 Determination of Material Parameters 230

4.1.3.2 Simulation of Monotonic Simple Shear Deformation 230

4.1.3.3 Simulation of Cyclic Free‐End Torsion and Tension–Torsion Deformations 231

4.1.3.4 Simulation of Uniaxial Ratchetting at Finite Strain 235

4.2 Thermomechanically Coupled Cyclic Plasticity Model at Finite Strain 239

4.2.1 Framework of Thermodynamics 239

4.2.1.1 Kinematics and Logarithmic Stress Rate 239

4.2.1.2 Thermodynamic Laws 239

4.2.1.3 Generalized Constitutive Equations 241

4.2.1.4 Restrictions on Specific Heat and Stress Response Function 243

4.2.2 Specific Constitutive Model 244

4.2.2.1 Nonlinear Kinematic Hardening Rule 246

4.2.2.2 Nonlinear Isotropic Hardening Rule 247

4.2.3 Simulations and Discussions 249

4.3 Summary 261

References 262

5 Cyclic Viscoelasticity–Viscoplasticity of Polymers 267

5.1 Experimental Observations 268

5.1.1 Cyclic Softening/Hardening Features 268

5.1.1.1 Uniaxial Strain‐Controlled Cyclic Tests 269

5.1.1.2 Multiaxial Strain‐Controlled Cyclic Tests 273

5.1.2 Ratchetting Behaviors 275

5.1.2.1 Uniaxial Ratchetting 275

5.1.2.2 Multiaxial Ratchetting 288

5.2 Cyclic Viscoelastic Constitutive Model 299

5.2.1 Original Schapery’s Model 302

5.2.1.1 Main Equations of Schapery’s Viscoelastic Model 302

5.2.1.2 Determination of Material Parameters 303

5.2.1.3 Simulations and Discussion 303

5.2.2 Extended Schapery’s Model 304

5.2.2.1 Main Modification 304

5.2.2.2 Simulations and Discussion 307

5.3 Cyclic Viscoelastic–Viscoplastic Constitutive Model 310

5.3.1 Main Equations 310

5.3.1.1 Viscoelasticity 313

5.3.1.2 Viscoplasticity 314

5.3.2 Verification and Discussion 315

5.3.2.1 Determination of Material Parameters 315

5.3.2.2 Simulations and Discussion 316

5.4 Summary 327

References 327

6 Cyclic Plasticity of Particle‐Reinforced Metal Matrix Composites 331

6.1 Experimental Observations 332

6.1.1 Cyclic Softening/Hardening Features 332

6.1.2 Ratchetting Behaviors 335

6.1.2.1 Uniaxial Ratchetting at Room Temperature 335

6.1.2.2 Uniaxial Ratchetting at 573 K 338

6.2 Finite Element Simulations 341

6.2.1 Time‐Independent Cyclic Plasticity 342

6.2.1.1 Main Equations of the Time‐Independent Cyclic Plasticity Model 343

6.2.1.2 Basic Finite Element Model and Simulations 346

6.2.1.3 Effect of Interfacial Bonding 351

6.2.1.4 Results with 3D Multiparticle Finite Element Model 362

6.2.2 Time‐Dependent Cyclic Plasticity 367

6.2.2.1 Finite Element Model 368

6.2.2.2 Simulations and Discussion 368

6.3 Meso‐mechanical Time‐Independent Plasticity Model 373

6.3.1 Framework of the Model 373

6.3.1.1 Time‐Independent Cyclic Plasticity Model for the Matrix 374

6.3.1.2 Extension of the Mori–Tanaka Homogenization Approach 374

6.3.2 Numerical Implementation of the Model 376

6.3.2.1 Under the Strain‐Controlled Loading Condition 376

6.3.2.2 Under the Stress‐Controlled Loading Condition 378

6.3.2.3 Continuum and Algorithmic Consistent Tangent Operators 379

6.3.3 Verification and Discussion 380

6.3.3.1 Determination of Material Parameters 380

6.3.3.2 Simulations and Discussion 380

6.4 Meso‐mechanical Time‐Dependent Plasticity Model 387

6.4.1 Framework of the Model 388

6.4.1.1 Time‐Dependent Cyclic Plasticity Model for the Matrix 389

6.4.1.2 Mori–Tanaka Homogenization Approach 390

6.4.2 Numerical Implementation of the Model 390

6.4.2.1 Generalized Incrementally Affine Linearization Formulation 390

6.4.2.2 Extension of Mori–Tanaka’s Model 391

6.4.2.3 Algorithmic Consistent Tangent Operator and Its Regularization 393

6.4.2.4 Numerical Integration of the Viscoplasticity Model 394

6.4.3 Verification and Discussion 395

6.4.3.1 Under Monotonic Tension 395

6.4.3.2 Under Strain‐Controlled Cyclic Loading Conditions 395

6.4.3.3 Time‐Dependent Uniaxial Ratchetting 395

6.5 Summary 398

References 401

7 Thermomechanical Cyclic Deformation of Shape‐Memory Alloys 405

7.1 Experimental Observations 407

7.1.1 Degeneration of Super‐Elasticity and Transformation Ratchetting 407

7.1.1.1 Thermomechanical Cyclic Deformation Under Strain‐Controlled Loading Conditions 407

7.1.1.2 Thermomechanical Cyclic Deformation Under Uniaxial Stress‐Controlled Loading Conditions 411

7.1.1.3 Thermomechanical Cyclic Deformation Under Multiaxial Stress‐Controlled Loading Conditions 419

7.1.2 Rate‐Dependent Cyclic Deformation of Super‐Elastic NiTi SMAs 426

7.1.2.1 Thermomechanical Cyclic Deformation Under Strain‐Controlled Loading Conditions 428

7.1.2.2 Thermomechanical Cyclic Deformation Under Stress‐Controlled Loading Conditions 434

7.1.3 Thermomechanical Cyclic Deformation of Shape‐Memory NiTi SMAs 441

7.1.3.1 Pure Mechanical Cyclic Deformation under Stress‐Controlled Loading Conditions 441

7.1.3.2 Thermomechanical Cyclic Deformation with Thermal Cycling and Axial Stress 451

7.2 Phenomenological Constitutive Models 452

7.2.1 Pure Mechanical Version 452

7.2.1.1 Thermodynamic Equations and Internal Variables 452

7.2.1.2 Main Equations of Constitutive Model 453

7.2.1.3 Predictions and Discussions 457

7.2.2 Thermomechanical Version 464

7.2.2.1 Strain Definitions 464

7.2.2.2 Evolution Rules of Transformation and Transformation‐Induced Plastic Strains 469

7.2.2.3 Simplified Temperature Field 473

7.2.2.4 Predictions and Discussions 477

7.3 Crystal Plasticity‐Based Constitutive Models 489

7.3.1 Pure Mechanical Version 489

7.3.1.1 Strain Definitions 489

7.3.1.2 Evolution Rules of Internal Variables 492

7.3.1.3 Explicit Scale Transition Rule 494

7.3.1.4 Verifications and Discussions 495

7.3.2 Thermomechanical Version 500

7.3.2.1 Strain Definitions 502

7.3.2.2 Evolution Rules of Internal Variables 503

7.3.2.3 Thermomechanical Coupled Analysis for Temperature Field 505

7.3.2.4 Verifications and Discussions 507

7.4 Summary 524

References 525

Index 531

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