Continuum mechanics: Elasticity, plasticity, viscoelasticity

CONTINUUM MECHANICS

Elasticity, Plasticity, Viscoelasticity

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CONTINUUM MECHANICS

Ellis H. Dill

Elasticity, Plasticity, Viscoelasticity

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CRC Press

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Library of Congress Cataloging-in-Publication Data

Dill, Ellis Harold, 1932-

Continuum mechanics : elasticity, plasticity, viscoelasticity / by Ellis Harold

Dill.

p. cm.

Includes bibliographical references and index.

ISBN 0-8493-9779-0 (alk. paper)

1. Continuum mechanics. 2. Elasticity. 3. Plasticity. 4. Viscoelasticity. I.

Title.

QA808.2.D535 2006

531--dc22 2006048958

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Preface

This book is intended as a reference book for professional engineers and a text for

first-year graduate students in engineering. Books on continuum mechanics are

usually limited to elasticity and fluid mechanics. I have included a thorough treatment

of the constitutive relations for the theory of plasticity and viscoelasticity, and an

introduction to the continuum model for fatigue and fracture mechanics. Because

the complete equations for such materials can rarely be solved in closed form, I have

also included the formulation of numerical solution by the finite element method.

The last chapter provides the mathematical tools and results that are used in the

remainder of the book. I have reversed the usual order of the chapters so that readers

who have the necessary mathematical background will not be subjected to a review

of mathematics on the first page, and so that instructors may cover only the math￾ematical results that are not familiar to their class. Study of the subject will normally

begin with some part of Chapter 7.

Chapter 1 covers the general and exact formulation of the continuum model for

the thermomechanics of materials. The exact equations of geometry of deformation

and motion, conservation of mass, balance of momentum, and balance of energy are

derived. The mechanics and thermodynamics of Kelvin-Voigt viscoelastic materials

and viscous fluids are fully treated.

Chapter 2 treats the exact nonlinear equations of elasticity and thermoelasticity.

This includes material symmetries, conjugate measures of stress and strain, objective

stress rates, and energy principles. Examples of explicit constitutive models that

have been found to be important in engineering applications are fully developed.

Chapter 3 contains a full treatment of the equations of small deformations of an

elastic body. The equations are derived in both tensor notation and in the matrix

notation that is most commonly used for numerical solution. Energy principles that

underlay the numerical methods are developed for a general assembly of finite

elements. The solution of the equations by the finite element method is presented

with examples.

Chapter 4 begins with the classical theory of plasticity for small deformations.

It includes a detailed treatment of isotropic hardening, kinematic hardening, and

combined hardening models. The general formulation in strain space is developed

along with the work postulates of Drucker and Ilyushin. The method of solution of

the plasticity equations by the finite element method is derived. The chapter con￾cludes with an introduction to the theory of large deformations of plasticity and

thermoplasticity.

Chapter 5 introduces the classical theory of linear viscoelasticity by means of

rheological models and proceeds to the formulation of the general relations as

hereditary integral equations. The dependence on temperature and the thermorheo￾9779_C000.fm Page v Tuesday, October 10, 2006 6:24 AM

logically simple model are included. The exact theory for large deformations and

the thermodynamics of materials with fading memory are fully developed.

Chapter 6 begins with linear elastic fracture mechanics wherein initiation of

crack propagation is determined by the critical value of the stress intensity factor.

The growth of a crack according to the Paris formula leads to Miner’s formula for

fatigue life.

Direct solutions of the basic equations are not discussed beyond the simple

deformations, such as the tensile test, which are needed to illustrate the significance

of the material parameters. It is assumed that the first course in continuum mechanics

is followed by a course on numerical methods (e.g., the finite element method), which

provides the tools needed to solve the equations in practical situations, and by classical

courses on elasticity and plasticity. However, I have included a complete derivation

of the basic equations and solution procedures for the finite element method.

I recognize that the treatment is concise. I have attempted to cover only the

essentials of the subject and to provide the tools necessary for comprehension of

the technical literature and the commercial finite element programs. I apologize in

advance to all of the originators of this material. I have long ago forgotten where I

learned the theory.

First causes are not known to us, but they are subject to simple and constant

laws that can be discovered by observation.

Jean Baptiste Joseph Fourier (1768–1830),

Théorie Analytique de la Chaleur, 1822

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Author

Ellis Harold Dill earned his B.S., M.S., and Ph.D. from the University of Califor￾nia—Berkeley in civil engineering. He taught aeronautical engineering at the Uni￾versity of Washington, in Seattle, from 1956 to 1977. He was dean of engineering

at Rutgers, the State University of New Jersey, from 1977 to 1998. He is currently

university professor at Rutgers, teaching mechanical and aerospace engineering. His

principal research areas include aircraft structures, analysis of plates and shells, solid

mechanics, and the finite element method of analysis.

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Contents

Chapter 1 Fundamentals of Continuum Mechanics............................................. 1

1.1 Material Models ............................................................................................... 1

1.2 Classical Space-Time....................................................................................... 3

1.3 Material Bodies................................................................................................ 4

1.4 Strain ................................................................................................................ 6

1.5 Rate of Strain ................................................................................................. 10

1.6 Curvilinear Coordinate Systems .................................................................... 13

1.7 Conservation of Mass .................................................................................... 16

1.8 Balance of Momentum .................................................................................. 18

1.8.1 Virtual Work....................................................................................... 22

1.8.2 Physical Components......................................................................... 23

1.9 Balance of Energy.......................................................................................... 25

1.10 Constitutive Equations ................................................................................... 26

1.11 Thermodynamic Dissipation.......................................................................... 28

1.12 Objectivity: Invariance for Rigid Motions .................................................... 29

1.13 Coleman-Mizel Model ................................................................................... 32

1.13.1 Nonlinear Kelvin-Voigt Materials...................................................... 34

1.14 Fluid Mechanics............................................................................................. 35

1.14.1 Objectivity.......................................................................................... 36

1.14.2 Dissipation Principle.......................................................................... 37

1.14.3 Thermodynamics of Fluids ................................................................ 40

1.14.4 Couette Flow ...................................................................................... 43

1.15 Problems for Chapter 1.................................................................................. 45

1.16 Bibliography................................................................................................... 48

Chapter 2 Nonlinear Elasticity ........................................................................... 49

2.1 Thermoelasticity............................................................................................. 49

2.2 Material Symmetries...................................................................................... 53

2.3 Isotropic Materials ......................................................................................... 55

2.3.1 Principal Stresses and Principal Extensions...................................... 58

2.3.2 Tensile Test......................................................................................... 61

2.4 Incompressible Materials ............................................................................... 63

2.5 Conjugate Measures of Stress and Strain...................................................... 65

2.6 Some Symmetry Groups................................................................................ 68

2.7 Rate Formulations for Elastic Materials........................................................ 73

2.8 Energy Principles ........................................................................................... 77

2.8.1 Potential Energy................................................................................. 77

2.8.2 Complementary Energy ..................................................................... 79

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2.9 Geometry of Small Deformations ................................................................. 79

2.10 Linear Elasticity ............................................................................................. 81

2.10.1 Anisotropic Materials......................................................................... 85

2.11 Special Constitutive Models for Isotropic Materials .................................... 86

2.11.1 Linear Stress–Strain Model ............................................................... 91

2.11.2 Kirchhoff Model ................................................................................ 93

2.11.3 Blatz-Ko Model.................................................................................. 94

2.11.4 Generalized Mooney-Rivlin Model ................................................... 96

2.11.5 Ogden Foam....................................................................................... 99

2.11.6 Logarithmic Strain ........................................................................... 100

2.11.7 Gent Model ...................................................................................... 101

2.11.8 Yeoh Model...................................................................................... 103

2.12 Mechanical Restrictions on the Constitutive Relations .............................. 105

2.12.1 Tensile Test....................................................................................... 106

2.12.2 Volumetric Strain ............................................................................. 107

2.12.3 The Pressure-Compression (P-C) Inequality................................... 107

2.12.4 The Tension-Extension Inequality ................................................... 107

2.12.5 Extension-Tension (E-T) Inequalities.............................................. 108

2.12.6 Ordered Forces (O-F) Inequalities................................................... 108

2.12.7 General Condition of Monotonicity (GCM) ................................... 108

2.13 Problems for Chapter 2................................................................................ 110

2.14 Bibliography................................................................................................. 112

Chapter 3 Linear Elasticity ............................................................................... 113

3.1 Basic Equations............................................................................................ 113

3.2 Plane Strain .................................................................................................. 117

3.3 Plane Stress .................................................................................................. 118

3.4 Properties of Solutions................................................................................. 119

3.5 Potential Energy........................................................................................... 122

3.5.1 Proof of Minimum Potential Energy............................................... 124

3.6 Special Matrix Notation............................................................................... 126

3.7 The Finite Element Method of Solution ..................................................... 127

3.7.1 Basic Equations in Matrix Notation................................................ 133

3.7.2 Basic Equations Using Virtual Work............................................... 135

3.7.3 Displacements Are Underestimated................................................. 136

3.7.4 Dynamical Equations ....................................................................... 137

3.7.5 Example Problem............................................................................. 139

3.8 General Equations for an Assembly of Elements ....................................... 139

3.8.1 Generalized Variational Principle .................................................... 141

3.8.2 Potential Energy............................................................................... 142

3.8.3 Hybrid Displacement Functional..................................................... 143

3.8.4 Hybrid Stress and Complementary Energy..................................... 143

3.8.5 Mixed Methods of Analysis............................................................. 145

3.8.6 Nearly Incompressible Materials..................................................... 148

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3.9 Finite Element Analysis for Large Deformations ....................................... 149

3.9.1 Example Problem............................................................................. 155

3.10 Problems for Chapter 3................................................................................ 156

3.11 Bibliography................................................................................................. 158

Chapter 4 Plasticity........................................................................................... 159

4.1 Classical Theory of Plasticity...................................................................... 159

4.2 Work Principle ............................................................................................. 163

4.3 von Mises-Type Yield Criterion................................................................... 165

4.4 Hill Yield Criterion for Orthotropic Materials ............................................ 168

4.4.1 Orthotropic Materials....................................................................... 169

4.4.2 Transverse Isotropy.......................................................................... 170

4.4.3 Bauschinger Effect........................................................................... 171

4.5 Isotropic Hardening...................................................................................... 172

4.5.1 Strain Hardening .............................................................................. 172

4.5.2 Work Hardening............................................................................... 176

4.6 Kinematic Hardening ................................................................................... 177

4.6.1 Theory of Prager .............................................................................. 177

4.6.2 Theory of Ziegler............................................................................. 181

4.7 Combined Hardening laws........................................................................... 182

4.7.1 Isotropic Strain Hardening with Prager’s Rule

for Kinematic Hardening ................................................................. 182

4.7.2 Isotropic Work Hardening and Prager’s Rule ................................. 184

4.7.3 Isotropic Strain Hardening with Ziegler’s Rule

for Kinematic Hardening ................................................................. 185

4.7.4 Chaboche’s Model ........................................................................... 186

4.8 General Equations of Plasticity ................................................................... 187

4.9 Strain Formulation of Plasticity................................................................... 190

4.9.1 Work Postulate ................................................................................. 191

4.10 Finite Element Analysis............................................................................... 197

4.10.1 Example Problem............................................................................. 198

4.11 Large Deformations ..................................................................................... 199

4.11.1 Approximation for the Materially Linear Case............................... 203

4.11.2 Work Postulate ................................................................................. 204

4.11.3 Rate Formulations............................................................................ 206

4.12 Thermodynamics of Elastic-Plastic Materials............................................. 207

4.13 Problems for Chapter 4................................................................................ 210

4.14 Bibliography................................................................................................. 211

Chapter 5 Viscoelasticity .................................................................................. 213

5.1 Linear Viscoelasticity................................................................................... 213

5.1.1 General Model ................................................................................. 221

5.1.2 Slow or Rapid Deformations ........................................................... 226

5.1.3 Symmetry of the Relaxation Modulus ............................................ 226

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5.2 Effect of Temperature .................................................................................. 230

5.3 Nonlinear Viscoelasticity ............................................................................. 234

5.4 Thermodynamics of Materials with Fading Memory ................................. 235

5.5 Problems for Chapter 5................................................................................ 238

5.6 Bibliography................................................................................................. 238

Chapter 6 Fracture and Fatigue ........................................................................ 239

6.1 Fracture Criterion......................................................................................... 239

6.2 Plane Crack through a Sheet ....................................................................... 241

6.3 Fracture Modes ............................................................................................ 244

6.4 Calculation of the Stress Intensity Factor ................................................... 245

6.5 Crack Growth............................................................................................... 247

6.6 Problems for Chapter 6................................................................................ 251

6.7 Bibliography................................................................................................. 251

Chapter 7 Mathematical Tools for Continuum Mechanics .............................. 253

7.1 Sets of Real Numbers .................................................................................. 253

7.1.1 Indicial Notation .............................................................................. 253

7.1.2 Summation Convention.................................................................... 253

7.1.3 The Kronecker Delta........................................................................ 254

7.1.4 The Permutation Symbol ................................................................. 256

7.1.5 Symmetry and Skew-Symmetry ...................................................... 257

7.1.6 Integral Transformations.................................................................. 259

7.2 Matrices........................................................................................................ 260

7.2.1 Matrix Notation................................................................................ 260

7.2.2 Matrix Addition and Multiplication................................................. 263

7.2.3 Special Matrices............................................................................... 264

7.2.4 Kronecker Products.......................................................................... 266

7.2.5 Determinants .................................................................................... 269

7.2.6 Inverse Matrix .................................................................................. 271

7.2.7 Linear Algebraic Equations ............................................................. 271

7.3 Vector Analysis ............................................................................................ 272

7.3.1 Vector Algebra ................................................................................. 272

7.3.2 Derivatives of Vectors ...................................................................... 276

7.3.3 Base Vectors ..................................................................................... 276

7.3.4 Curvilinear Coordinates and Covariant Base Vectors ..................... 280

7.3.5 Gradient, Divergence, and Curl of Vectors ..................................... 284

7.3.6 Cylindrical Coordinates ................................................................... 285

7.4 Tensors ......................................................................................................... 287

7.4.1 Tensor Algebra ................................................................................. 287

7.4.2 Tensor Product of Vectors and Components of Tensors ................. 289

7.4.3 Derivatives of Tensors...................................................................... 293

7.4.4 Trace of a Tensor ............................................................................. 294

7.4.5 Transpose of a Tensor and Dot from the Left................................. 295

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7.4.6 Principal Invariants of Tensors ........................................................ 297

7.4.7 Regular, Singular, and Inverse Tensors ........................................... 299

7.4.8 Eigenvalues, Eigenvectors, and Eigenprojections

of Symmetric Tensors ...................................................................... 301

7.4.9 Canonical Representation of Tensors .............................................. 304

7.4.10 Polar Decomposition of Tensors...................................................... 307

7.4.11 Cayley-Hamilton Theorem and Alternate Invariants....................... 308

7.4.12 Higher-Order Tensors....................................................................... 309

7.5 Isotropic Functions....................................................................................... 311

7.5.1 Scalar-Valued Functions of One Tensor.......................................... 311

7.5.2 Scalar-Valued Functions of Two Tensors ........................................ 312

7.5.3 Linear Isotropic Tensor-Valued Functions of a Tensor................... 314

7.5.4 General Isotropic Tensor-Valued Functions of a Tensor................. 315

7.6 Abstract Derivatives..................................................................................... 316

7.6.1 Real-Valued Functions of Vectors ................................................... 316

7.6.2 Vector-Valued Functions of Vectors ................................................ 318

7.6.3 Scalar-Valued Functions of Tensors ................................................ 319

7.6.4 Tensor-Valued Functions of Tensors ............................................... 322

7.6.5 Multiple Arguments and the Chain Rule......................................... 323

7.7 Some Basic Mathematical Definitions and Theorems ................................ 324

7.7.1 Fields ................................................................................................ 324

7.7.2 Vector Spaces ................................................................................... 324

7.7.3 Metric Spaces................................................................................... 325

7.7.4 Normed Spaces, Banach Spaces...................................................... 326

7.7.5 Scalar Product and Hilbert Space.................................................... 327

7.7.6 Fading Memory Space..................................................................... 327

7.7.7 Derivative of a Function .................................................................. 328

7.8 Problems for Chapter 7................................................................................ 329

7.9 Bibliography................................................................................................. 331

Index...................................................................................................................... 333

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