Continuum Mechanics of Solids 0198864728, 9780198864721

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OUP CORRECTED PROOF – FINAL, 15/6/2020, SPi

Continuum Mechanics of Solids

OUP CORRECTED PROOF – FINAL, 15/6/2020, SPi

OUP CORRECTED PROOF – FINAL, 15/6/2020, SPi

Continuum Mechanics of Solids Lallit Anand Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Sanjay Govindjee Department of Civil and Environmental Engineering, University of California, Berkeley, Berkeley, CA 94720, USA

1

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Lallit Anand and Sanjay Govindjee 2020 The moral rights of the authors have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2020938849 ISBN 978–0–19–886472–1 DOI: 10.1093/oso/9780198864721.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Preface Continuum Mechanics is a useful model of materials which views a material as being continuously divisible, making no reference to its discrete structure at microscopic length scales well below those of the application or phenomenon of interest. Solid Mechanics and Fluid Mechanics are subsets of Continuum Mechanics. The focus of this book is Solid Mechanics, which is concerned with the deformation, flow, and fracture of solid materials. A study of Solid Mechanics is of use to anyone who seeks to understand these natural phenomena, and to anyone who seeks to apply such knowledge to improve the living conditions of society. This is the central aim of people working in many fields of engineering, e.g., mechanical, materials, civil, aerospace, nuclear, and bio-engineering. This book, which is aimed at first-year graduate students in engineering, offers a unified presentation of the major concepts in Solid Mechanics. In addressing any problem in Solid Mechanics, we need to bring together the following major considerations: • Kinematics or the geometry of deformation, in particular the expressions of stretch/strain and rotation in terms of the gradient of the deformation field. • Balance of mass. • The system of forces acting on the body, and the concept of stress. • Balance of forces and moments in situations where inertial forces can be neglected, and correspondingly balance of linear momentum and angular momentum when they cannot. • Balance of energy and an entropy imbalance, which represent the first two laws of thermodynamics. • Constitutive equations which relate the stress to the strain, strain-rate, and temperature. • The governing partial differential equations, together with suitable boundary conditions and initial conditions. While the first five considerations are the same for all continuous bodies, no matter what material they are made from, the constitutive relations are characteristic of the material in question, the stress level, the temperature, and the timescale of the problem under consideration. The major topics regarding the mechanical response of solids that are covered in this book include: Elasticity, Viscoelasticity, Plasticity, Fracture, and Fatigue. Because the constitutive theories of Solid Mechanics are complex when presented within the context of finite deformations, in most of the book we will restrict our attention to situations in which the deformations may be adequately presumed to be small—a situation commonly encountered in most structural applications. However, in the last section of the book on Finite Elasticity of elastomeric materials, we will consider the complete large deformation theory, because for such materials the deformations in typical applications are usually quite large.

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PREFACE

In addition to these standard topics in Solid Mechanics, because of the growing need for engineering students to have a knowledge of the coupled multi-physics response of materials in modern technologies related to the environment and energy, the book also includes chapters on Thermoelasticity, Chemoelasticity, Poroelasticity, and Piezoelectricity—all within the (restricted) framework of small deformations. For pedagogical reasons, we have also prepared a companion book with many fully solved example problems, Example Problems for Continuum Mechanics of Solids (Anand and Govindjee, 2020). The present book is an outgrowth of lecture notes by the first author for first-year graduate students in engineering at MIT taking 2.071 Mechanics of Solid Materials, and the material taught at UC Berkeley in CE 231/MSE 211 Introduction to Solid Mechanics by the second author. Although the book has grown out of lecture notes for a one-semester course, it contains enough material for a two-semester sequence of subjects, especially if the material on coupled theories is to be taught. The different topics covered in the book are presented in essentially self-contained “modules”, and instructors may pick-and-choose the topics that they wish to focus on in their own classes. We wish to emphasize that this book is intended as a textbook for classroom teaching or selfstudy for first-year graduate students in engineering. It is not intended to be a comprehensive treatise on all topics of importance in Solid Mechanics. It is a “pragmatic” book designed to provide a general preparation for first-year graduate students who will pursue further study or conduct research in specialized sub-fields of Solid Mechanics, or related disciplines, during the course of their graduate studies.

Attributions and historical issues Our emphasis is on basic concepts and central results in Solid Mechanics, not on the history of the subject. We have attempted to cite the contributions most central to our presentation, and we apologize in advance if we have not done so faultlessly. Lallit Anand Cambridge, MA Sanjay Govindjee Berkeley, CA

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Acknowledgments Lallit Anand is indebted to his teacher Morton Gurtin of CMU, who first taught him continuum mechanics in 1975–76, and once again during the period 2004–2010, when he worked with Gurtin on writing the book, The Mechanics and Thermodynamics of Continua (Gurtin et al., 2010). He is also grateful to his colleague David Parks at MIT—who has used a preliminary version of this book in teaching his classes—for his comments, helpful criticisms, and his contributions to the chapters on Fracture and Fatigue. Discussions with Mary Boyce, now at Columbia University, regarding the subject matter of the book are also gratefully acknowledged. Sanjay Govindjee is indebted to his many instructors and in particular to his professors at the Massachusetts Institute of Technology and at Stanford University, who never shied away from advanced concepts and philosophical discussions. Special gratitude is extended to the late Juan Carlos Simo, Jerome Sackman, and Egor Popov. Additional thanks are given to current colleague Robert L. Taylor and countless students whose penetrating questions have formed his thinking about solid mechanics.

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Contents

I Vectors and tensors

1

1 Vectors and tensors: Algebra

3

1.1 Cartesian coordinate frames. Kronecker delta. Alternating symbol 1.1.1

Summation convention

1.2 Tensors 1.2.1

3 4 6

What is a tensor?

6

1.2.2 Zero and identity tensors

7

1.2.3 Tensor product

7

1.2.4 Components of a tensor

7

1.2.5 Transpose of a tensor

8

1.2.6 Symmetric and skew tensors

9

1.2.7 Axial vector of a skew tensor

9

1.2.8 Product of tensors

10

1.2.9 Trace of a tensor. Deviatoric tensors

11

1.2.10 Positive definite tensors

12

1.2.11 Inner product of tensors. Magnitude of a tensor

12

1.2.12 Matrix representation of tensors and vectors

13

1.2.13 Determinant of a tensor

14

1.2.14 Invertible tensors

15

1.2.15 Cofactor of a tensor

15

1.2.16 Orthogonal tensors

15

1.2.17 Transformation relations for components of a vector and a tensor under a change in basis

15

Transformation relation for vectors

16

Transformation relation for tensors

17

1.2.18 Eigenvalues and eigenvectors of a tensor. Spectral theorem

17

Eigenvalues of symmetric tensors

19

Other invariants and eigenvalues

20

1.2.19 Fourth-order tensors Transformation rules for fourth-order tensors

20 22

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CONTENTS

2 Vectors and tensors: Analysis

23

2.1 Directional derivatives and gradients

23

2.2 Component expressions for differential operators

24

2.2.1 Divergence, curl, and Laplacian

25

2.3 Generalized derivatives

27

2.4 Integral theorems

29

2.4.1 Localization theorem

29

2.4.2 Divergence theorem

30

2.4.3 Stokes theorem

31

II Kinematics

33

3 Kinematics

35

3.1 Motion: Displacement, velocity, and acceleration 3.1.1

35

Example motions

37

Simple shear

37

Elementary elongation

37

Rigid motion

37

Bending deformation

38

3.2 Deformation and displacement gradients

39

3.2.1 Transformation of material line elements

40

3.2.2 Transformation of material volume elements

41

3.2.3 Transformation of material area elements

42

3.3 Stretch and rotation

43

3.3.1 Polar decomposition theorem

44

3.3.2 Properties of the tensors U and C

45

Fiber stretch and strain Angle change and engineering shear strain 3.4 Strain

46 47 48

3.4.1 Biot strain

48

3.4.2 Green finite strain tensor

48

3.4.3 Hencky’s logarithmic strain tensors 3.5 Infinitesimal deformation

49 49

3.5.1 Infinitesimal strain tensor

49

3.5.2 Infinitesimal rotation tensor

51

3.6 Example infinitesimal homogeneous strain states

52

3.6.1 Uniaxial compression

52

3.6.2 Simple shear

53

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3.6.3 Pure shear

54

3.6.4 Uniform compaction (dilatation)

55

3.6.5 Infinitesimal rigid displacement

56

3.7 Volumetric deviatoric split 3.7.1

Volume changes

3.7.2 Strain deviator

56 56 57

3.8 Summary of major kinematical concepts related to infinitesimal strains

58

APPENDICES

59

3.A Linearization

59

3.B Compatibility conditions

60

3.B.1 Proof of the compatibility theorem

63

III Balance laws

67

4 Balance laws for mass, forces, and moments

69

4.1 Balance of mass

69

4.2 Balance of forces and moments. Stress tensor. Equation of motion

71

4.3 Local balance of forces and moments

73

4.3.1 Pillbox construction

73

4.3.2 Cauchy’s tetrahedron construction

75

4.3.3 Cauchy’s local balance of forces and moments

76

4.3.4 Cauchy’s theorem

77

4.4 Physical interpretation of the components of stress

79

4.5 Some simple states of stress

80

4.5.1 Pure tension or compression

80

4.5.2 Pure shear stress

81

4.5.3 Hydrostatic stress state

82

4.5.4 Stress deviator

82

4.6 Summary of major concepts related to stress

5 Balance of energy and entropy imbalance 5.1 Balance of energy. First law of thermodynamics 5.1.1

Local form of balance of energy

5.2 Entropy imbalance. Second law of thermodynamics 5.2.1 Local form of entropy imbalance 5.3 Free-energy imbalance. Dissipation 5.3.1 Free-energy imbalance and dissipation inequality for mechanical theories

83

85 85 87 88 89 89 90

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CONTENTS

6 Balance laws for small deformations

91

6.1 Basic assumptions

91

6.2 Local balance and imbalance laws for small deformations

92

IV Linear elasticity

95

7 Constitutive equations for linear elasticity

97

7.1 Free-energy imbalance (second law): Elasticity

97

7.1.1

Voigt single index notation

100

7.2 Material symmetry

102

7.3 Forms of the elasticity and compliance tensors for some anisotropic linear elastic materials

103

7.3.1

Monoclinic symmetry

104

7.3.2 Orthorhombic or orthotropic symmetry

105

7.3.3 Tetragonal symmetry

106

7.3.4 Transversely isotropic symmetry. Hexagonal symmetry

108

7.3.5 Cubic symmetry

110

Inter-relations between stiffnesses and compliances: Cubic materials 7.4 Directional elastic modulus 7.4.1

Directional elastic modulus: Cubic materials Anisotropy ratio for a cubic single crystal

7.5 Constitutive equations for isotropic linear elastic materials 7.5.1

111 114 114 116 116

Restrictions on the moduli μ and λ

118

7.5.2 Inverted form of the stress-strain relation

119

7.5.3 Physical interpretation of the elastic constants in terms of local strain and stress states

120

Limiting value of Poisson’s ratio for incompressible materials

122

7.5.4 Stress-strain relations in terms of E and ν

122

7.5.5 Relations between various elastic moduli

123

7.6 Isotropic linear thermoelastic constitutive relations

124

8 Linear elastostatics

128

8.1 Basic equations of linear elasticity

128

8.2 Boundary conditions

129

8.3 Mixed, displacement, and traction problems of elastostatics

130

8.4 Uniqueness

130

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8.5 Superposition

131

8.6 Saint-Venant’s principle

132

8.7 Displacement formulation. The Navier equations

133

8.8 Stress formulation. The Beltrami—Michell equations

134

8.9 Two-dimensional problems

137

8.9.1 Plane strain

137

Navier equation in plane strain

139

Equation of compatibility in plane strain

139

8.9.2 Plane stress

140

Navier equation in plane stress

142

Compatibility equation in terms of stress

142

8.9.3 Similarity of plane strain and plane stress equations

143

8.9.4 Succinct form of the governing equations for plane strain and plane stress

144

9 Solutions to some classical problems in linear elastostatics

145

9.1 Spherical pressure vessel

145

9.2 Cylindrical pressure vessel

149

9.2.1 Axial strain 0 for different end conditions

153

9.2.2 Stress concentration in a thick-walled cylinder under internal pressure

154

9.3 Bending and torsion 9.3.1 Bending of a bar

155 155

9.3.2 Torsion of a circular cylinder

159

9.3.3 Torsion of a cylinder of arbitrary cross-section

162

Displacement-based formulation

164

Stress-based formulation. Prandtl stress function

165

9.4 Airy stress function and plane traction problems

173

Displacements in terms of the Airy stress function

174

9.4.1 Airy stress function and plane traction problems in polar coordinates

175

Displacements in terms of the Airy stress function in polar coordinates

176

9.4.2 Some biharmonic functions in polar coordinates

177

9.4.3 Infinite medium with a hole under uniform far-field loading in tension

177

9.4.4 Half-space under concentrated surface force

182

9.4.5 A detailed example of Saint-Venant’s principle

194

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CONTENTS

9.5 Asymptotic crack-tip stress fields. Stress intensity factors

196

9.5.1 Asymptotic stress fields in Mode I and Mode II

197

Asymptotic Mode I field

199

Asymptotic Mode II field

202

9.5.2 Asymptotic stress field in Mode III Solution for uz (r, θ) in the vicinity of the crack

205 207

9.6 Cartesian component expressions at crack-tip

209

9.6.1 Crack-opening displacements

211

V Variational formulations 10 Variational formulation of boundary-value problems 10.1 Principle of virtual power

213

215 215

10.1.1 Virtual power

216

10.1.2 Consequences of the principle of virtual power

217

10.1.3 Application of the principle of virtual power to boundary-value problems

219

10.2 Strong and weak forms of the displacement problem of linear elastostatics

11 Introduction to the finite element method for linear elastostatics

221

223

12 Principles of minimum potential energy

and complementary energy

228

12.1 A brief digression on calculus of variations

228

12.1.1 Application of the necessary condition δI = 0

230

Euler–Lagrange equation

230

Essential and natural boundary conditions

230

12.1.2 Treating δ as a mathematical operator. The process of taking “variations” The process of taking “variations”

232 233

12.2 Principle of minimum potential energy

235

12.3 Principle of minimum potential energy and the weak form of the displacement problem of elastostatics

238

12.4 Principle of minimum complementary energy

239

12.5 Application of the principles of minimum potential energy and complementary energy to structural problems

244

12.5.1 Castigliano’s first theorem

245

12.5.2 Castigliano’s second theorem

245

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CONTENTS

VI Elastodynamics. Sinusoidal progressive waves 13 Elastodynamics. Sinusoidal progressive waves

249

251

13.1 Mixed problem of elastodynamics

251

13.2 Sinusoidal progressive waves

252

VII Coupled theories 14 Linear thermoelasticity

255

257

14.1 Introduction

257

14.2 Basic equations

258

14.3 Constitutive equations

258

14.3.1 Maxwell and Gibbs relations

260

14.3.2 Specific heat

261

14.3.3 Elasticity tensor. Stress-temperature modulus. Evolution equation for temperature

261

14.3.4 Entropy as independent variable

263

14.3.5 Isentropic and isothermal inter-relations

264

14.4 Linear thermoelasticity

266

14.5 Basic field equations of linear thermoelasticity

268

14.6 Isotropic linear thermoelasticity

269

14.6.1 Homogeneous and isotropic linear thermoelasticity

270

14.6.2 Uncoupled theory of isotropic linear elasticity

271

14.7 Boundary and initial conditions

272

14.8 Example problems

274

15 Small deformation theory of species diffusion coupled

to elasticity

286

15.1 Introduction

286

15.2 Kinematics and force and moment balances

286

15.3 Balance law for the diffusing species

287

15.4 Free-energy imbalance

288

15.5 Constitutive equations

288

15.5.1 Thermodynamic restrictions

289

15.5.2 Consequences of the thermodynamic restrictions

289

15.5.3 Chemical potential as independent variable

291

15.5.4 Drained and undrained inter-relations

293

15.6 Linear chemoelasticity

294

xv

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CONTENTS

15.6.1 Isotropic linear chemoelasticity

295

15.6.2 Governing partial differential equations

297

Boundary and initial conditions

298

16 Linear poroelasticity

299

16.1 Introduction

299

16.2 Biot’s theory

300

16.2.1 Fluid flux

302

16.2.2 Material properties for poroelastic materials

303

16.3 Summary of the theory of linear isotropic poroelasticity

304

16.3.1 Constitutive equations

304

16.3.2 Governing partial differential equations

305

Alternate forms of the governing partial differential equations 16.3.3 Boundary conditions

305 306

16.4 Microstructural considerations

311

16.4.1 Constitutive equations of poroelasticity in terms of the change in porosity u

16.4.2 Estimates for the material parameters α, K , and M 16.5 Linear poroelasticity of polymer gels: Incompressible materials

313 314 315

16.5.1 Governing partial differential equations

317

Boundary and initial conditions

317

16.5.2 Alternate forms for the governing partial differential equations

318

17 A small deformation chemoelasticity theory for energy

storage materials

320

17.1 Introduction

320

17.2 Basic equations

320

17.3 Specialization of the constitutive equations

321

17.3.1 Strain decomposition

321

17.3.2 Free energy

322

17.3.3 Stress. Chemical potential

323

17.4 Isotropic idealization

324

17.4.1 Chemical strain

324

17.4.2 Elasticity tensor

324

17.4.3 Species flux

324

17.4.4 Stress. Chemical potential. Species flux

325

17.5 Governing partial differential equations for the isotropic theory

325

Partial differential equations

325

Boundary and initial conditions

326

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CONTENTS

18 Linear piezoelectricity

327

18.1

Introduction

327

18.2

Kinematics and force and moment balances

328

18.3

Basic equations of electrostatics

329

18.4

External and internal expenditures of power. Power balance

330

18.5

Dielectric materials

331

18.6

Free-energy imbalance for dielectric materials

332

18.7

Constitutive theory

332

18.7.1 Electric field as independent variable

333

18.8

Piezoelectricity tensor, permittivity tensor, elasticity tensor

334

18.9

Linear piezoelectricity

336

18.10 Governing partial differential equations Boundary conditions

337 337

18.11 Alternate form of the constitutive equations for linear piezoelectricity with σ and e as independent variables

338

18.12 Piezocrystals and poled piezoceramics

339

18.12.1 Voigt notation

339

18.12.2 Properties of piezoelectric crystals

340

18.12.3 Properties of poled piezoceramics

340 350

APPENDICES

18.A Electromagnetics

18.B

350

18.A.1 Charge conservation

350

18.A.2 Quasi-static limit

351

18.A.3 Energy transport in the quasi-static limit

351

Third-order tensors

352

18.B.1 Components of a third-order tensor

352

18.B.2 Action of a third-order tensor on a second-order tensor

353

18.B.3 Transpose of a third-order tensor

354

18.B.4 Summary of a third-order tensor and its action on vectors and tensors

354

VIII Limits to elastic response. Yielding and plasticity 19 Limits to elastic response. Yielding and failure

355

357

19.1

Introduction

357

19.2

Failure criterion for brittle materials in tension

358

19.3

Yield criterion for ductile isotropic materials

359

19.3.1

361

Mises yield condition

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CONTENTS

19.3.2 Tresca yield condition

362

19.3.3 Coulomb–Mohr yield criterion for cohesive granular materials in compression

364

19.3.4 The Drucker–Prager yield criterion

365

20 One-dimensional plasticity 20.1

369

Some phenomenological aspects of the elastic-plastic stress-strain response of polycrystalline metals

369

20.1.1

371

Isotropic strain-hardening

20.1.2 Kinematic strain-hardening

372

20.1.3 Strain-rate and temperature dependence of plastic flow

373

20.2 One-dimensional theory of rate-independent plasticity with isotropic hardening

374

20.2.1 Kinematics

375

20.2.2 Power

375

20.2.3 Free-energy imbalance

376

20.2.4 Constitutive equation for elastic response

376

20.2.5 Constitutive equation for the plastic response

377

Plastic flow direction

377

Flow strength

378

Yield condition. No-flow conditions. Consistency condition

378

Conditions describing loading and unloading. The plastic strain-rate ˙p in terms of ˙ and σ

380

20.2.6 Summary of the rate-independent theory with isotropic hardening

381

20.2.7 Material parameters in the rate-independent theory

382

Some useful phenomenological forms for the strain-hardening function 20.3 One-dimensional rate-dependent plasticity with isotropic hardening

383 385

20.3.1 Power-law rate dependence

386

20.3.2 Power-law creep

388

20.3.3 A rate-dependent theory with a yield threshold

388

20.3.4 Summary of the rate-dependent theory with isotropic hardening

390

20.4 One-dimensional rate-independent theory with combined isotropic and kinematic hardening 20.4.1 Constitutive equations

391 391

Defect energy

392

Plastic flow direction

393

Flow resistance

394

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Yield function. No-flow conditions. Consistency condition

394

The flow rule

395

20.4.2 Summary of the rate-independent theory with combined isotropic and kinematic hardening 20.5 One-dimensional rate-dependent power-law plasticity with combined isotropic and kinematic hardening

21 Three-dimensional plasticity with isotropic hardening

396 404

406

21.1

Introduction

406

21.2

Basic equations

406

21.3

Kinematical assumptions

407

21.4

Separability hypothesis

407

21.5

Constitutive characterization of elastic response

408

21.6

Constitutive equations for plastic response

408

21.6.1

409

Flow strength

21.6.2 Mises yield condition. No-flow conditions. Consistency condition

410

21.6.3 The flow rule

412

21.7

Summary

413

21.8

Three-dimensional rate-dependent plasticity with isotropic hardening

416

21.9

Summary

418

21.9.1

419

21.8.1

Power-law rate dependence Power-law creep form for high homologous temperatures

417

22 Three-dimensional plasticity with kinematic and isotropic

hardening

421

22.1

Introduction

421

22.2

Rate-independent constitutive theory

421

22.2.1 Yield function. Elastic range. Yield set

423

22.2.2 The flow rule

424

22.3

Summary of the rate-independent theory with combined isotropic and kinematic hardening

425

22.4

Summary of a rate-dependent theory with combined isotropic and kinematic hardening

427

23 Small deformation rate-independent plasticity based on a

postulate of maximum dissipation

429

23.1

Elastic domain and yield set

429

23.2

Postulate of maximum dissipation and its consequences

429

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CONTENTS

23.3

Application to a Mises material

432

APPENDICES

433

23.A Kuhn–Tucker optimality conditions

433

24 Some classical problems in rate-independent plasticity 24.1

Elastic-plastic torsion of a cylindrical bar

434

24.1.1

434

Kinematics

24.1.2 Elastic constitutive equation

435

24.1.3 Equilibrium

435

24.1.4 Resultant torque: Elastic case

436

Shear stress in terms of applied torque and geometry

24.2

24.3

434

437

24.1.5 Total twist of a shaft

437

24.1.6 Elastic-plastic torsion

437

Onset of yield

438

Torque-twist relation

439

Spring-back

442

Residual stress

442

Spherical pressure vessel

444

24.2.1 Elastic analysis

444

24.2.2 Elastic-plastic analysis

447

Onset of yield

447

Partly plastic spherical pressure vessel

448

Fully plastic spherical pressure vessel

451

Residual stresses upon unloading

451

Cylindrical pressure vessel

452

24.3.1 Elastic analysis

452

24.3.2 Elastic-plastic analysis

455

Onset of yield

455

Partly plastic cylindrical pressure vessel

455

Fully plastic cylindrical pressure vessel

459

25 Rigid-perfectly-plastic materials. Two extremal principles

460

25.1

Mixed boundary-value problem for a rigid-perfectly-plastic solid

460

25.2

Two extremal principles for rigid-perfectly-plastic materials

461

25.2.1 Extremal principle for power expended by tractions and body forces

461

25.2.2 Extremal principle for velocity

463

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CONTENTS

25.3

Limit analysis

465

25.4

Lower bound theorem

467

25.5

Upper bound theorem

468

25.5.1 Upper bound estimates obtained by using block-sliding velocity fields

469

Example problems

471

25.6.1 Bounds to the limit load for a notched plate

471

25.6

Lower bound

471

Upper bound

472

25.6.2 Hodograph

474

25.6.3 Plane strain frictionless extrusion

475

25.6.4 Plane strain indentation of a semi-infinite solid with a flat punch

478

IX Fracture and fatigue 26 Linear elastic fracture mechanics

483

485

26.1

Introduction

485

26.2

Asymptotic crack-tip stress fields. Stress intensity factors

486

26.3

Configuration correction factors

489

26.4

Stress intensity factors for combined loading by superposition

490

26.5

Limits to applicability of KI -solutions

491

26.5.1 Limit to applicability of KI -solutions because of the asymptotic nature of the KI -stress fields

491

26.5.2 Limit to applicability of KI -solutions because of local inelastic deformation in the vicinity of the crack-tip

493

26.5.3 Small-scale yielding (ssy)

495

26.6 Criterion for initiation of crack extension

496

26.7

497

Fracture toughness testing

26.8 Plane strain fracture toughness data

503

27 Energy-based approach to fracture

506

27.1

Introduction

506

27.2

Energy release rate

507

27.3

27.2.1

Preliminaries

507

27.2.2

Definition of the energy release rate

508

Griffith’s fracture criterion

509

27.3.1

510

An example

xxi

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CONTENTS

27.4 27.5

Energy release rate and Eshelby’s tensor

514

27.4.1

515

A conservation integral I

J -integral

27.5.1

J -integral and the energy release rate G

516 518

27.6

Use of the J -integral to calculate the energy release rate G

521

27.7

Relationship between G , J , and KI , KII , and KIII

522

27.8

Use of J as a fracture parameter for elastic-plastic fracture mechanics

523 525

APPENDICES

27.A

Equivalence of the two main relations for G(a)

525

28 Fatigue

529

28.1

Introduction

529

28.2

Defect-free approach

532

28.3

28.2.1 S-N curves

532

28.2.2 Strain-life approach to design against fatigue failure

534

High-cycle fatigue. Basquin’s relation

535

Low-cycle fatigue. Coffin–Manson relation

536

Strain-life equation for both high-cycle and low-cycle fatigue

537

Mean stress effects on fatigue

538

Cumulative fatigue damage. Miner’s rule

538

Defect-tolerant approach

539

28.3.1 Fatigue crack growth

539

28.3.2 Engineering approximation of a fatigue crack growth curve

542

28.3.3 Integration of crack growth equation

543

X Linear viscoelasticity

547

29 Linear viscoelasticity

549

29.1

Introduction

549

29.2

Stress-relaxation and creep

550

29.3

29.2.1 Stress-relaxation

551

29.2.2 Creep

553

29.2.3 Linear viscoelasticity

554

29.2.4 Superposition. Creep-integral and stress-relaxation-integral forms of stress-strain relations

554

A simple rheological model for linear viscoelastic response

556

29.3.1 Stress-relaxation

557

29.3.2 Creep

559

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CONTENTS

29.4

Power-law relaxation functions

561

29.5

Correspondence principle

562

29.5.1 Correspondence principle in one dimension

563

29.5.2 Connection between Er (t) and Jc (t) in Laplace transform space

564

29.6

29.7

29.8 29.9

Correspondence principles for structural applications

565

29.6.1 Bending of beams made from linear viscoelastic materials

566

First correspondence principle for bending of beams made from a linear viscoelastic material

566

Second correspondence principle for bending of beams made from a linear viscoelastic material

568

29.6.2 Torsion of shafts made from linear viscoelastic materials

570

Linear viscoelastic response under oscillatory strain and stress

572

29.7.1

Oscillatory loads

572

29.7.2

Storage compliance. Loss compliance. Complex compliance

573

29.7.3

Storage modulus. Loss modulus. Complex modulus

574

Formulation for oscillatory response using complex numbers

575

29.8.1 Energy dissipation under oscillatory conditions

576

More on complex variable representation of linear viscoelasticity

580

29.9.1





E (ω), E (ω), and tan δ(ω) for the standard linear solid

581

29.10 Time-integration procedure for one-dimensional constitutive equations for linear viscoelasticity based on the generalized Maxwell model

588

29.11 Three-dimensional constitutive equation for linear viscoelasticity

592

29.11.1 Boundary-value problem for isotropic linear viscoelasticity

594

29.11.2 Correspondence principle in three dimensions

595

29.12 Temperature dependence. Dynamic mechanical analysis (DMA) 29.12.1 Representative DMA results for amorphous polymers

598

29.12.2 DMA plots for the semi-crystalline polymers

600

29.13 Effect of temperature on Er (t) and Jc (t). Time-temperature equivalence 29.13.1 Shift factor. Williams–Landel–Ferry (WLF) equation

XI Finite elasticity 30 Finite elasticity 30.1

597

Brief review of kinematical relations

601 607

609

611 611

30.2 Balance of linear and angular momentum expressed spatially

613

30.3 First Piola stress. Force and moment balances expressed referentially

614

30.4 Change of observer. Change of frame

617

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CONTENTS

30.4.1 Transformation rules for kinematic fields

617

30.4.2 Transformation rules for the Cauchy stress and the Piola stress

618

30.4.3 Form invariance of constitutive equations. Material-frameindifference

619

30.5 Free-energy imbalance under isothermal conditions 30.5.1 The second Piola stress 30.6 Constitutive theory 30.7

619 620 621

30.6.1 Thermodynamic restrictions

621

Summary of basic equations. Initial/boundary-value problems

622

30.7.1

622

Basic field equations

30.7.2 A typical initial/boundary-value problem

623

30.8 Material symmetry

623

30.9 Isotropy

625

30.10 Isotropic scalar functions

626

30.11 Free energy for isotropic materials expressed in terms of invariants

627

30.12 Free energy for isotropic materials expressed in terms of principal stretches

628

30.13 Incompressibility

629

30.14 Incompressible elastic materials

630

30.14.1 Free energy for incompressible isotropic elastic bodies

631

30.14.2 Simple shear of a homogeneous, isotropic, incompressible elastic body

633

APPENDICES

635

30.A Derivatives of the invariants of a tensor

635

31 Finite elasticity of elastomeric materials

637

31.1

Introduction

637

31.2

Neo-Hookean free-energy function

639

31.3

Mooney–Rivlin free-energy function

640

31.3.1

643

The Mooney–Rivlin free energy expressed in an alternate fashion

31.4

Ogden free-energy function

643

31.5

Arruda–Boyce free-energy function

644

31.6

Gent free-energy function 31.6.1

31.7

I2 -dependent extension of the Gent free-energy function

647 650

Slightly compressible isotropic elastomeric materials

651

31.7.1

Models for the volumetric energy

652

31.7.2

Slightly compressible form of the Arruda–Boyce free energy

654

31.7.3

Slightly compressible form of the Gent free energy

654

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CONTENTS

XII Appendices A

B

655

Cylindrical and spherical coordinate systems

657

A.1

Introduction

657

A.2

Cylindrical coordinates

657

A.3

Spherical coordinates

664

Stress intensity factors for some crack configurations

Bibliography Index

671 679 689

xxv

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PA R T I

Vectors and tensors

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1

Vectors and tensors: Algebra

The theories of Solid Mechanics are mathematical in nature. Thus in this chapter and the next, we begin with a discussion of the essential mathematical definitions and operations regarding vectors and tensors needed to discuss the major topics covered in this book.1 The approach taken is a pragmatic one, without an overly formal emphasis upon mathematical proofs. The end-goal is to develop, in a reasonably systematic way, the essential mathematics of vectors and tensors needed to discuss the major aspects of mechanics of solids. The physical quantities that we will encounter in this text will be of three basic types: scalars, vectors, and tensors. Scalars are quantities that can be described simply by numbers, for example things like temperature and density. Vectors are quantities like velocity and displacement—quantities that have both magnitude and direction. Tensors will be used to describe more complex quantities like stress and strain, as well as material properties. In this chapter we deal with the algebra of vectors and tensors. In the next chapter we deal with their calculus.

1.1 Cartesian coordinate frames. Kronecker delta. Alternating symbol Throughout this text lower case Latin subscripts range over the integers {1, 2, 3}.

A Cartesian coordinate frame for the Euclidean point space E consists of a reference point o called the origin together with a positively oriented orthonormal basis {e1 , e2 , e3 } for the associated vector space V . Being positively oriented and orthonormal, the basis vectors obey ei · ej = δij ,

ei · (ej × ek ) = eijk .

Here δij , the Kronecker delta, is defined by ⎧ ⎨1, δij = ⎩0,

if i = j, if i = j,

1

(1.1.1)

(1.1.2)

The mathematical preliminaries in this chapter and the next are largely based on the clear expositions by Gurtin in Gurtin (1981) and Gurtin et al. (2010).

Continuum Mechanics of Solids. Lallit Anand and Sanjay Govindjee, Oxford University Press (2020). © Lallit Anand and Sanjay Govindjee, 2020. DOI: 10.1093/oso/9780198864721.001.0001

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VECTORS AND TENSORS: ALGEBRA

while eijk , the alternating symbol,2 is defined by ⎧ ⎪ ⎪ 1, if {i, j, k} = {1, 2, 3}, {2, 3, 1}, or {3, 1, 2}, ⎪ ⎨ eijk = −1, if {i, j, k} = {2, 1, 3}, {1, 3, 2}, or {3, 2, 1}, ⎪ ⎪ ⎪ ⎩ 0, if an index is repeated,

(1.1.3)

and hence has the value +1, −1, or 0 when {i, j, k} is an even permutation, an odd permutation, or not a permutation of {1, 2, 3}, respectively. Where convenient, we will use the compact notation def {ei } = {e1 , e2 , e3 }, to denote our positively oriented orthonormal basis.

1.1.1 Summation convention For convenience, we employ the Einstein summation convention according to which summation over the range 1, 2, 3 is implied for any index that is repeated twice in any term, so that, for instance, ui v i = u1 v 1 + u2 v 2 + u3 v 3 , Sij uj = Si1 u1 + Si2 u2 + Si3 u3 , Sik Tkj = Si1 T1j + Si2 T2j + Si3 T3j .

In the expression Sij uj the subscript i is free, because it is not summed over, while j is a dummy subscript, since Sij uj = Sik uk = Sim um . As a rule when using the Einstein summation convention, one can not repeat an index more than twice. Expressions with such indices indicate that a computational error has been made. In the rare case where three (or more) repeated indices must appear in a single term, an explicit 3 summation symbol, for example i=1 , must be used. Note that the Kronecker delta δij modifies (or contracts) the subscripts in the coefficients of an expression in which it appears: ai δij = aj ,

ai bj δij = ai bi = aj bj ,

δij δik = δjk ,

δik Skj = Sij .

This property of the Kronecker delta is sometimes termed the substitution property. Further, when an expression in which an index is repeated twice but summation is not to be performed we state so explicitly. For example, ui v i

(no sum)

signifies that the subscript i is not to be summed over.