The mechanical and thermodynamical theory of plasticity

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C M Y CM MY CY CMY K

The Mechanical and

Thermodynamical Theory

of Plasticity

Mehrdad Negahban

The Mechanical and

Thermodynamical Theory of Plasticity

Negahban

7230

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The Mechanical and Thermodynamical Theory of Plasticity

Born out of fifteen years of courses and lectures on continuum mechanics, nonlinear mechanics,

continuum thermodynamics, viscoelasticity, plasticity, crystal plasticity and thermodynamic plasticity,

The Mechanical and Thermodynamical Theory of Plasticity represents one of the most

extensive and in-depth treatises on the mechanical and thermodynamical aspects of plastic and

visicoplastic flow. Suitable for student readers and experts alike, it offers a clear and comprehensive

presentation of multi-dimensional continuum thermodynamics to both aid in initial understanding

and introduce and explore advanced topics.

Features:

•Includes more than 200 figures and an extensive number of exercises and

computer simulation problems

•Reviews background mathematics, continuum thermodynamics, and the extension of bars,

to provide a broad perspective for student readers

•Offers a look at special advanced topics, including infinitesimal theory and solutions;

the finite deformation theories; and the common perspective that connects them

•Explores the common perspectives and similarities of the mechanical and thermodynamical

theories, with extensive use of analogs to help connect the ideas

•Connects the first gradient theory to elasticity, plasticity, visicoelasticity, visicoplasticity,

and crystal plasticity

•Demonstrates extensive use of the representation theory to provide tools for

constructing complex models

Covering a wide range of foundational subjects and presenting unique insights into the unification

of disparate theories and practices, this book offers an extensive number of problems, figures,

and examples to help the reader grasp the subject from many levels. Starting from one-dimensional

axial motion in bars, the book builds a clear understanding of mechanics and continuum

thermodynamics during plastic flow. This approach makes it accessible and applicable for a varied

audience, including students and experts from Engineering Mechanics, Mechanical Engineering,

Civil Engineering, and Materials Science.

CIVIL AND MECHANICAL ENGINEERING

The Mechanical and

Thermodynamical Theory

of Plasticity

7230_FM.indd 1 3/27/12 2:18 PM

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The Mechanical and

Thermodynamical Theory

of Plasticity

Mehrdad Negahban

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

Boca Raton London New York

7230_FM.indd 3 3/27/12 2:18 PM

CRC Press

Taylor & Francis Group

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Boca Raton, FL 33487-2742

© 2012 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works

Version Date: 20120419

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With Love

To my parents

Meriam and Ezat

for letting me become what I wanted to be and for helping me in this process

To my wife

Setareh

for reminding me that some will never understand

To my children

Arman and Shahdi

for the strength you give me

and

with the hope that

you each find what you want to become

you reach to achieve it

and

you enjoy the process

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Acknowledgments

The author would like to acknowledge the help of many people who provided insight,

sometimes unknowingly, and people who helped in the preparation of this book. These

include colleagues and students. Many of the students taking my classes pointed out cor￾rections and provided suggestions, which was very much appreciated. Particular thanks go

to Ashwani Kumar Goel, Lili Zhang, Kyle Strabala, Benjamin Polly, and Saeed Eghtedar

Doust, who each read part or all of the text and provided corrections and comments, and

Yenan Wang for doing the steel and aluminum tests. I owe a particular debt of gratitude

to Ashwani Kumar Goel, who did most of the simulations, and Lili Zhang, who care￾fully read and checked all the equations and representations; they were true partners in this

endeavor.

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ix

Contents

Preface xix

1 Plasticity in the 1-D bar 1

1.1 Introduction to plastic response . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The bar and the continuum assumption . . . . . . . . . . . . . . . . . . . . 2

1.3 Motion and temperature of points on a bar . . . . . . . . . . . . . . . . . . 4

1.4 Stretch ratio, strain, velocity gradient, temperature gradient . . . . . . . . . 9

1.5 Superposition of deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Elastic, plastic, and thermal strains . . . . . . . . . . . . . . . . . . . . . . . 14

1.7 Examples of constitutive models . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7.1 Elastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7.2 Thermoelastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7.3 Viscous fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7.4 Elastic-plastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7.5 Fourier’s law for heat conduction . . . . . . . . . . . . . . . . . . . . 20

1.8 Mechanical theory of rate-independent plasticity . . . . . . . . . . . . . . . 21

1.9 Mechanical models for plasticity . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.9.1 Elastic perfectly plastic . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.9.2 Elastic-plastic with isotropic hardening: strain hardening . . . . . . . 26

1.9.3 Elastic-plastic with isotropic hardening: work hardening . . . . . . . 28

1.9.4 Elastic-plastic with kinematic hardening . . . . . . . . . . . . . . . . 29

1.9.5 Elastic-plastic with arbitrary hardening . . . . . . . . . . . . . . . . 31

1.9.6 Elastic-plastic with combined isotropic and kinematic hardening . . 34

1.9.7 Isotropic hardening with changing elastic modulus . . . . . . . . . . 36

1.10 Temperature-dependent plasticity . . . . . . . . . . . . . . . . . . . . . . . . 40

1.11 An infinitesimal theory of thermoplasticity . . . . . . . . . . . . . . . . . . . 44

1.12 Rate-dependent models for plasticity . . . . . . . . . . . . . . . . . . . . . . 49

1.13 Load control as opposed to strain control . . . . . . . . . . . . . . . . . . . . 54

1.14 Numerical integration of constitutive equations . . . . . . . . . . . . . . . . 56

1.14.1 Rate-independent and temperature-independent plasticity . . . . . . 56

1.14.2 Rate-independent and temperature-dependent plasticity . . . . . . . 63

1.14.3 Load control as opposed to strain control . . . . . . . . . . . . . . . 65

1.14.4 Rate-dependent plasticity . . . . . . . . . . . . . . . . . . . . . . . . 66

1.15 The balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

1.15.1 Calculating physical properties from their distributions . . . . . . . . 68

1.15.2 The material time derivative of physical properties . . . . . . . . . . 69

1.15.3 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

1.15.4 Balance of linear momentum . . . . . . . . . . . . . . . . . . . . . . 73

1.15.5 Balance of work and energy . . . . . . . . . . . . . . . . . . . . . . . 74

1.15.6 The entropy production inequality . . . . . . . . . . . . . . . . . . . 75

1.16 Thermodynamic restrictions on constitutive equations . . . . . . . . . . . . 77

1.16.1 Thermoelasticity using (, θ, g) . . . . . . . . . . . . . . . . . . . . . . 78

1.16.2 Thermoelasticity using (

e

, θ

, θ, g) . . . . . . . . . . . . . . . . . . . . 79

1.16.3 Thermoplasticity using (, p

, θ

, ξ, θ, g) . . . . . . . . . . . . . . . . . 81

1.16.4 Thermoplasticity using (

e

, p

, θ

, ξ, θ, g) . . . . . . . . . . . . . . . . . 83

1.16.5 Rate-dependent thermoplasticity . . . . . . . . . . . . . . . . . . . . 86

1.17 Heat generation and flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

1.18 Equilibrium and quasi-equilibrium problems . . . . . . . . . . . . . . . . . . 90

1.18.1 Elastic bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

1.18.2 Elastic-plastic bar with isotropic hardening . . . . . . . . . . . . . . 93

1.18.3 Elastic-viscoplastic bar . . . . . . . . . . . . . . . . . . . . . . . . . . 94

1.19 Dynamic loading problems: Numerical solution . . . . . . . . . . . . . . . . 95

1.20 Dealing with discontinuities: Jump conditions . . . . . . . . . . . . . . . . . 104

1.21 Plastic drawing of bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

1.22 Elastic and plastic (shock) waves in a bar . . . . . . . . . . . . . . . . . . . 109

1.23 General comment on selection of moduli . . . . . . . . . . . . . . . . . . . . . 112

1.24 Notation and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2 Vectors and tensors 117

2.1 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.1.1 Indicial notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.1.2 Basic matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . 119

2.1.3 The eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . 124

2.1.4 Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . . . . . . . 128

2.1.5 Polar decomposition theorem . . . . . . . . . . . . . . . . . . . . . . 129

2.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

2.2.1 Frames and coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 130

2.2.2 Vectors and basic vector operations . . . . . . . . . . . . . . . . . . . 131

2.2.3 Base vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

2.2.4 Change of base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

2.2.5 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 138

2.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

2.3.1 Extracting components of a tensor . . . . . . . . . . . . . . . . . . . . 147

2.3.2 The eigenvalue problem for tensors . . . . . . . . . . . . . . . . . . . 149

2.3.3 Length of tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

2.4 Tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

2.4.1 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

2.4.2 Divergence and Laplacian . . . . . . . . . . . . . . . . . . . . . . . . 154

2.4.3 Curl and circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

2.4.4 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

2.4.5 Time derivatives and partial derivatives . . . . . . . . . . . . . . . . . 165

2.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

3 Describing motion, deformation, and temperature 169

3.1 Position, velocity, acceleration, and temperature . . . . . . . . . . . . . . . . 169

3.2 Configurations of material bodies . . . . . . . . . . . . . . . . . . . . . . . . 169

3.3 Streamlines and pathlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

3.4 Deformation gradient and temperature gradient . . . . . . . . . . . . . . . . 175

3.5 Stretch and strain tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

3.6 Velocity gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

3.7 Relative deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

3.8 Triaxial extension, simple shear, bending, and torsion . . . . . . . . . . . . . 188

3.9 Small deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Contents xi

3.9.1 Small stretch and small rigid body rotation . . . . . . . . . . . . . . 196

3.9.2 Small stretch and large rigid body rotation . . . . . . . . . . . . . . 197

3.10 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

4 Elastic, plastic, and thermal deformation 201

4.1 Elastic and plastic deformation gradients . . . . . . . . . . . . . . . . . . . . 201

4.2 Elastic and plastic strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

4.3 Elastic and plastic velocity gradients . . . . . . . . . . . . . . . . . . . . . . 204

4.4 Infinitesimal elastic and plastic deformations . . . . . . . . . . . . . . . . . . 208

4.5 Large rigid body rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

4.5.1 Small elastic and plastic strain . . . . . . . . . . . . . . . . . . . . . 211

4.5.2 Small elastic strain and moderate plastic strain . . . . . . . . . . . . 211

4.6 Thermal deformation and thermal strain . . . . . . . . . . . . . . . . . . . . 212

4.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

5 Traction, stress, and heat flux 217

5.1 The traction vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

5.2 The relation between tractions on different surfaces . . . . . . . . . . . . . . 220

5.3 The stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.4 Isotropic invariants and the deviatoric stress . . . . . . . . . . . . . . . . . . 224

5.5 Examples of elementary states of stress . . . . . . . . . . . . . . . . . . . . . 226

5.6 True stress as opposed to engineering stress . . . . . . . . . . . . . . . . . . 229

5.7 The Piola-Kirchhoff, rotated, and convected stresses . . . . . . . . . . . . . 233

5.8 Heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.9 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6 Balance laws and jump conditions 239

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

6.2 Transport relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

6.3 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

6.4 Balance of linear momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.5 Balance of angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 254

6.6 Balance of work and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

6.7 Entropy and the entropy production inequality . . . . . . . . . . . . . . . . 258

6.8 Heat flow and thermodynamic processes . . . . . . . . . . . . . . . . . . . . . 260

6.9 Infinitesimal deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

6.10 The generalized balance law . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

6.11 Jump conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

6.11.1 Divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

6.11.2 Transport theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

6.11.3 The jump conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6.12 Perturbing a motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

6.13 Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 278

6.14 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7 Infinitesimal plasticity 285

7.1 A mechanical analog for plasticity . . . . . . . . . . . . . . . . . . . . . . . . 285

7.2 Elastic perfectly plastic response . . . . . . . . . . . . . . . . . . . . . . . . . 290

7.3 Common assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

7.3.1 Linear elastic response . . . . . . . . . . . . . . . . . . . . . . . . . . 293

7.3.2 Yield function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

7.3.3 Hardening parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

xii The Mechanical and Thermodynamical Theory of Plasticity

7.3.4 Flow rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

7.3.5 Dissipation for a mechanical model . . . . . . . . . . . . . . . . . . . 303

7.4 Von Mises yield function with combined isotropic and kinematic hardening . 304

7.4.1 Simple shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

7.5 Thermoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

7.5.1 Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

7.5.2 Temperature-dependent mechanical theory . . . . . . . . . . . . . . . 309

7.5.3 Thermodynamical model . . . . . . . . . . . . . . . . . . . . . . . . . 310

7.5.4 Balance of work and energy . . . . . . . . . . . . . . . . . . . . . . . 312

7.5.5 Heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

7.6 Free energy of quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . 314

7.6.1 Thermodynamic constraint . . . . . . . . . . . . . . . . . . . . . . . . 315

7.6.2 Balance of work and energy . . . . . . . . . . . . . . . . . . . . . . . 316

7.6.3 Flow rule, hardening rule, and yield function . . . . . . . . . . . . . . 317

7.7 Scalar stress and hardening functions . . . . . . . . . . . . . . . . . . . . . . 319

7.8 Multiple elements in parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

7.8.1 A model made of perfectly plastic elements . . . . . . . . . . . . . . . 333

7.9 Multiple elements in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

7.9.1 A model made of a finite number of plastic elements . . . . . . . . . . 337

7.10 Rate-dependent plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

7.11 Deformation plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

7.12 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

8 Solutions for infinitesimal plasticity 345

8.0.1 Simulation of response . . . . . . . . . . . . . . . . . . . . . . . . . . 348

8.1 Homogeneous deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

8.1.1 Equal triaxial stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

8.1.2 Pure shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

8.1.3 Uniaxial tension/compression . . . . . . . . . . . . . . . . . . . . . . . 350

8.1.4 Equal biaxial extension . . . . . . . . . . . . . . . . . . . . . . . . . . 353

8.2 Torsion-extension of a thin circular cylindrical tube . . . . . . . . . . . . . . 355

8.3 Compression in plane strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

8.3.1 During elastic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

8.3.2 During plastic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

8.4 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

8.4.1 Elastic bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

8.4.2 Plastic bending under a single monotonic moment . . . . . . . . . . . 366

8.5 Torsion of circular members . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

8.5.1 Elastic torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

8.5.2 Elastic-plastic torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

8.6 Unloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

8.7 Torsion of prismatic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

8.7.1 Elastic torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

8.7.2 Perfect plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

8.8 Nonuniform loading of bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

8.8.1 Equilibrium problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

8.8.2 Assuming small rotational inertia in bending . . . . . . . . . . . . . . 390

8.8.3 Statically determinate problems . . . . . . . . . . . . . . . . . . . . . 390

8.8.4 Rotating beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

8.9 Cylindrical and spherical symmetry . . . . . . . . . . . . . . . . . . . . . . . 394

8.9.1 Antiplane shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

8.9.2 Internal pressure under plane strain . . . . . . . . . . . . . . . . . . . 402

Contents xiii

8.9.3 Other problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

8.10 Two-dimensional problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

8.10.1 Plane-stress problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

8.10.2 Plane-strain problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

8.11 Heat and its generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

9 First-gradient thermomechanical materials 411

9.1 First-gradient theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

9.2 Superposition of pure translations . . . . . . . . . . . . . . . . . . . . . . . . 412

9.3 Superposition of rigid body rotations . . . . . . . . . . . . . . . . . . . . . . 413

9.4 Material symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

9.5 First-gradient state variable models . . . . . . . . . . . . . . . . . . . . . . . 420

9.6 Higher-gradient and nonlocal models . . . . . . . . . . . . . . . . . . . . . . 422

9.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

10 Elastic and thermoelastic solids 425

10.1 The thermoelastic solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

10.2 The influence of pure rigid body translation on the constitutive response . . 428

10.3 The influence of pure rigid body rotation on the constitutive response . . . 429

10.4 Material symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

10.4.1 Isotropic elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . 436

10.4.2 Transversely isotropic elastic materials . . . . . . . . . . . . . . . . . 439

10.4.3 Orthotropic elastic materials . . . . . . . . . . . . . . . . . . . . . . . 441

10.5 Change of reference configuration . . . . . . . . . . . . . . . . . . . . . . . . 442

10.6 A thermodynamically consistent model . . . . . . . . . . . . . . . . . . . . . 444

10.6.1 Isotropic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

10.6.2 Isotropic rubber-elasticity-based model . . . . . . . . . . . . . . . . . 450

10.6.3 Isotropic Ogden type models . . . . . . . . . . . . . . . . . . . . . . . 453

10.7 Models based on F

e and F

θ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

10.7.1 Models depending on (F

e

, θ, g) . . . . . . . . . . . . . . . . . . . . . . 460

10.8 Specific free energy of quadratic form in strain . . . . . . . . . . . . . . . . . 466

10.8.1 Models based on Ee

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

10.8.2 Infinitesimal strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

10.9 Heat generation and heat capacity . . . . . . . . . . . . . . . . . . . . . . . . 473

10.10 Material constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

10.11 Multiple material constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 480

10.12 Superposition of deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 483

10.12.1Wave propagation in preloaded bodies . . . . . . . . . . . . . . . . . 486

10.13 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

11 Finite deformation mechanical theory of plasticity 493

11.1 General mechanical theory of plasticity . . . . . . . . . . . . . . . . . . . . . 494

11.1.1 Consistency condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

11.2 Rigid body motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

11.3 Material symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

11.4 Stress depending only on elastic deformation gradient . . . . . . . . . . . . . 505

11.4.1 Initially isotropic materials . . . . . . . . . . . . . . . . . . . . . . . 506

11.5 Stress depending on both elastic deformation and plastic strain . . . . . . . . 507

11.6 General comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

11.7 Deformation plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

11.8 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

xiv The Mechanical and Thermodynamical Theory of Plasticity

12 Thermoplastic solids 513

12.1 A simple thermomechanical analog . . . . . . . . . . . . . . . . . . . . . . . . 513

12.2 Thermoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

12.3 Thermodynamic constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

12.3.1 Thermal deformation gradient . . . . . . . . . . . . . . . . . . . . . . 518

12.3.2 Thermoelastic range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

12.3.3 Flow rule and hardening parameter . . . . . . . . . . . . . . . . . . . 520

12.4 Isotropic examples with J2 type yield functions . . . . . . . . . . . . . . . . . 522

12.4.1 Perfectly plastic material . . . . . . . . . . . . . . . . . . . . . . . . . 522

12.4.2 Isotropic hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

12.4.3 Kinematic hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

12.4.4 Combined isotropic and kinematic hardening . . . . . . . . . . . . . . 531

12.5 Superposition of rigid body motions . . . . . . . . . . . . . . . . . . . . . . . 535

12.6 Material symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

12.7 An initially isotropic material . . . . . . . . . . . . . . . . . . . . . . . . . . 541

12.8 Models depending on Cp

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

12.8.1 Isotropic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

12.9 Heat generation and heat flow . . . . . . . . . . . . . . . . . . . . . . . . . . 548

12.10 Specific free energy of quadratic form in strain . . . . . . . . . . . . . . . . . 550

12.10.1 Small strain approximation . . . . . . . . . . . . . . . . . . . . . . . . 552

12.11 Plasticity models based on Green strains . . . . . . . . . . . . . . . . . . . . 553

12.11.1Models using elastic and plastic Green strains . . . . . . . . . . . . . 554

12.11.2 Separable free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

12.12 Heat flux vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

12.13 Material constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

12.14 Models based on F = F

eF

θF

p

. . . . . . . . . . . . . . . . . . . . . . . . . . 563

12.15 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

13 Viscoelastic solids 567

13.1 One-dimensional linear viscoelasticity . . . . . . . . . . . . . . . . . . . . . . 567

13.1.1 Standard linear solid . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

13.1.2 General linear viscoelastic material . . . . . . . . . . . . . . . . . . . 575

13.2 One-dimensional nonlinear viscoelasticity . . . . . . . . . . . . . . . . . . . . 580

13.2.1 A standard quasi-nonlinear viscoelastic solid . . . . . . . . . . . . . . 580

13.2.2 A standard nonlinear viscoelastic solid . . . . . . . . . . . . . . . . . 585

13.2.3 Generalization to multiple elements . . . . . . . . . . . . . . . . . . . 588

13.3 Three-dimensional linear viscoelasticity . . . . . . . . . . . . . . . . . . . . . 590

13.3.1 Standard linear solid . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

13.3.2 General linear viscoelastic model . . . . . . . . . . . . . . . . . . . . . 592

13.4 A one-element thermoviscoelastic model . . . . . . . . . . . . . . . . . . . . . 594

13.4.1 One-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . 594

13.4.2 A one-element three-dimensional thermodynamically consistent model 595

13.4.3 Rigid body motions and the symmetry of the Cauchy stress . . . . . 597

13.4.4 Material symmetry constraints . . . . . . . . . . . . . . . . . . . . . . 600

13.4.5 Free energies of quadratic form in Green strains . . . . . . . . . . . . 602

13.4.6 Linear flow rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

13.4.7 Infinitesimal strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

13.4.8 An example model with quadratic free energy . . . . . . . . . . . . . 605

13.4.9 An isotropic coupled elastic-viscoelastic free energy . . . . . . . . . . 609

13.4.10 An example model motivated by rubber elasticity . . . . . . . . . . . 612

13.5 Multielement thermodynamic viscoelastic model . . . . . . . . . . . . . . . . 618

13.5.1 Rigid body motions and the symmetry of the Cauchy stress . . . . . 620