Introduction to Continuum Mechanics Gonzales - Stuart

Introduction to Continuum Mechanics

by

O. Gonzalez and A.M. Stuart

c Copyright 2005

All Rights Reserved

Preface

This book is designed for a one- or two-quarter course in continuum me-

chanics for first-year graduate students and advanced undergraduates

in the mathematical and engineering sciences. It is based on courses

taught over the past nine years at Stanford University, USA and at the

University of Warwick, UK. This book started to take shape in 1995

when we taught the graduate Continuum Mechanics course at Stanford

ME238. The notes have been developed and improved over four years

of teaching at Stanford, followed by a further four years of teaching the

course to advanced Mathematics undergraduates at Warwick MA3G2.

The resulting text is, we believe, suitable for use by both applied math-

ematicians and engineers.

Our perception is that there is a need for a book which simultane-

ously captures the essence of the clean, coordinate free, mathematical

presentations of the subject (such as the book by Gurtin (1981)) and

engineering presentations in which the basic thermo-mechanical balance

laws are stated and derived in a particular coordinate system, using in-

dices (such as the book by Mase (1970)). Our book might be viewed as

an attempt to synthesize the strengths of these two approaches: aim-

ing to state all important results in an absract coordinate free fashion,

whilst developing a clear understanding of the derivation of all the basic

results by means of working in coordinates, with index notation.

The first five chapters provide a general introduction to tensor analy-

sis in three-dimensional Euclidean space and introduce various concepts

that form the foundation of continuum mechanics: continuum mass and

force distributions, deformation and strain, the balance laws of mass,

momentum, energy and entropy, and the principle of material frame-

indifference. Both index and direct notation are used throughout. Clas-

sical theories of fluids and solids are covered in the three final chapters.

ii

Preface iii

We first consider various isothermal theories for ideal, compressible and

viscous fluids, and for linear and nonlinear elastic solids. We then gener-

alize these material models to the full thermo-mechanical case where the

temperature field may vary in space and time. We emphasize the formu-

lation of typical initial-boundary value problems for the various material

models, study important qualitative properties and, in several cases, il-

lustrate how the technique of linearization can be used to simplify the

problems under appropriate assumptions.

A short bibliography appears at the end of each chapter, pointing to

material which underpins, or expands upon, the presentation of the sub-

ject that we give here. As well as the general text books on Continuum

Mechanics by Gurtin (1981) and Mase (1970), we also recommend the

books by Chadwick (1976) and Malvern (1969). In the area of fluid me-

chanics, the books by Chorin and Marsden (1990) and by Temam (1984)

build on the material as presented here; in the area of solid mechanics

a similar role is played by the texts of Ogden (1984), Ciarlet (1983),

Marsden and Hughes (1983) and Antman (1995). The encyclopedia ar-

ticles by Truesdell and Toupin (1960), Truesdell and Noll (1965), Serrin

(1959), and Gurtin (1972) contain a wealth of information on both the

subject of, and history of, the various classic field theories of continuum

mechanics.

We are endebted to many of our colleagues at Stanford and at War-

wick, especially to Huajian Gao who gave a version of the course which

we sat through in the 1993-1994 and 1994-1995 academic years, to Tom

Hughes who gave us considerable encouragement to develop the notes

into a book, as well as guidance on the choice of material, and to Robert

Mackay who read and commented upon an early draft of the book. It

is also a pleasure to thank the many students at Stanford and War-

wick who helped to make this book possible. Special thanks go to Nuno

Catarino, Doug Enright, Gonzalo Feijoo, Liam Jones, Paul Lim, Teresa

Langlands and Matthew Lilley.

Contents

iv

Contents v

1 Tensor Algebra page 1

1.1 Vectors 1

1.1.1 Vector Algebra 2

1.1.2 Dot and Cross Product 3

1.1.3 Projections, Bases, and Coordinate Frames 4

1.2 Index Notation 6

1.2.1 Summation convention 6

1.2.2 Kronecker delta and permutation symbols 8

1.2.3 Frame identities 8

1.2.4 Vector operations in components 9

1.2.5 Epsilon-delta identities 10

1.3 Second-Order Tensors 11

1.3.1 Definition and Examples 11

1.3.2 Second-Order Tensor Algebra 13

1.3.3 Representation in a Coordinate Frame 13

1.3.4 Dyadic Products 14

1.3.5 Special Classes of Tensors 15

1.3.6 Change of Basis 18

1.3.7 Traces, Determinants and Exponentials 20

1.3.8 Eigenvalues, Eigenvectors and Principal Invariants 22

1.3.9 Special Decompositions 24

1.3.10 Scalar Product for Second-Order Tensors 25

1.4 Fourth-Order Tensors 26

1.4.1 Definition and Example 26

1.4.2 Fourth-Order Tensor Algebra 27

1.4.3 Representation in a Coordinate Frame 27

1.4.4 Symmetry Properties 28

1.5 Isotropic Tensor Functions 29

vi Contents

2 Tensor Calculus 43

2.1 Points, Tensors and Representations 43

2.2 Differentiation of Tensor Fields 44

2.2.1 Scalar Fields 44

2.2.2 Vector Fields 45

2.2.3 Divergence, Curl and Laplacian 47

2.3 Integral Theorems 52

2.3.1 Divergence Theorem 52

2.3.2 Stokes Theorem 54

2.4 Functions of Second-Order Tensors 55

2.4.1 Scalar-Valued Functions 56

2.4.2 Tensor-Valued Functions 59

Contents vii

3 Continuum Mass and Force Concepts 70

3.1 Continuum Bodies 70

3.2 Mass 71

3.2.1 Mass Density 71

3.2.2 Center of Mass 72

3.2.3 Conservation of Mass 72

3.3 Force 73

3.3.1 Body Forces 73

3.3.2 Surface Forces 74

3.4 The Stress Tensor 76

3.5 Equilibrium Conditions 77

3.5.1 Preliminaries 77

3.5.2 Integral Form 78

3.5.3 Localization 79

3.5.4 Local Form 80

3.6 Basic Stress Concepts 83

3.6.1 Simple States of Stress 83

3.6.2 Principal, Normal and Shear Stresses 85

3.6.3 Maximum Normal and Shear Stresses 86

3.6.4 Spherical and Deviatoric Stress Tensors 87

viii Contents

4 Kinematics 106

4.1 Configurations and Deformations 106

4.2 The Deformation Map 107

4.3 Measures of Strain 108

4.3.1 The Deformation Gradient F 108

4.3.2 Interpretation of F , Homogeneous Deformations 109

4.3.3 The Cauchy-Green Strain Tensor C 114

4.3.4 Interpretation of C 115

4.3.5 Rigid Deformations 118

4.3.6 The Infinitesimal Strain Tensor E 119

4.3.7 Interpretation of E 120

4.3.8 Infinitesimally Rigid Deformations 122

4.4 Motions 122

4.4.1 Material and Spatial Fields 123

4.4.2 Coordinate Derivatives 124

4.4.3 Time Derivatives 124

4.4.4 Velocity and Acceleration Fields 125

4.4.5 The Concept of Rate of Strain 128

4.4.6 Rate of Strain Tensor L and Spin Tensor W 128

4.4.7 Interpretation of L andW 129

4.4.8 Rigid Motions 129

4.5 Change of Variables 130

4.5.1 Transformation of Volume Integrals 130

4.5.2 Derivatives of Time-Dependent Integrals 133

4.5.3 Transformation of Surface Integrals 135

4.6 Volume-preserving motions 137

Contents ix

5 Balance Laws 160

5.1 Balance Laws in Integral Form 160

5.1.1 Motivation From Particle Mechanics 161

5.1.2 Conservation of Mass and Laws of Inertia 163

5.1.3 First and Second Laws of Thermodynamics 164

5.1.4 Integral Versus Local Balance Laws 168

5.2 Localized Eulerian Balance Laws 168


5.2.2 Balance of Linear Momentum 171

5.2.3 Balance of Angular Momentum 173

5.2.4 Mechanical Energy Identity 174

5.2.5 First Law of Thermodynamics 175

5.2.6 Second Law of Thermodynamics 176

5.2.7 Summary 178

5.3 Localized Lagrangian Balance Laws 179


5.3.2 Balance of Linear Momentum 180

5.3.3 Balance of Angular Momentum 182

5.3.4 First Law of Thermodynamics 182

5.3.5 Mechanical Energy Identity 185

5.3.6 Second Law of Thermodynamics 185

5.3.7 Summary 187

5.4 Material Constraints 188

5.5 Frame-indifference 190

5.5.1 Frame-indifferent Fields 190

5.5.2 Axioms of Frame-indifference 191

5.5.3 Transformation Rules 191

5.5.4 Examples 194

x Contents

6 Eulerian Formulations 204

6.1 Ideal Fluids 205

6.1.1 Definition 205

6.1.2 Euler Equations 207

6.1.3 Frame-Indifference Considerations 208

6.1.4 Initial-Boundary Value Problems 209

6.1.5 Irrotational Motion and Bernoullis Theorem 211

6.1.6 Steady, Irrotational Motions 214

6.2 Elastic Fluids 215


6.2.2 Elastic Fluid Equations 217



6.2.5 Irrotational Motion, Generalized Bernoullis Theorem 219

6.2.6 Steady, Irrotational Motions 220

6.2.7 Linearization: Acoustic Equations 221

6.3 Newtonian Fluids 223


6.3.2 Navier-Stokes Equations 225



6.4 Kinetic Energy of Fluid Motion 228

6.4.1 Preliminaries 229

6.4.2 Energy Bounds 230

Contents xi

7 Lagrangian Formulations 244

7.1 Elastic Solids 245


7.1.2 Elasticity Equation 247



7.2 Isotropic Elastic Solids 250


7.2.2 Simplified Response Functions 252

7.3 Mechanical Energy Inequality 253

7.3.1 Integral Form 253

7.3.2 Local Form 254

7.4 Hyperelastic Solids 255



7.4.3 Mechanical Energy Considerations 258

7.5 Linearized Elasticity 259

7.5.1 Linearization 260


7.5.3 Properties of the Elasticity Tensors 263

7.5.4 Equation of Linearized, Isotropic Elasticity 267

7.6 Linear Elastic Solids 267


7.6.2 Hyperelastic Structure 268

xii Contents

8 Thermal Effects 291

8.1 Thermoelastic Solids 291


8.1.2 Thermodynamical Considerations 294




8.2 Compressible Ideal Gases 304


8.2.2 Compressible Ideal Gas Equations 307



8.3 Compressible Newtonian Fluids 310


8.3.2 Compressible Newtonian Fluid Equations 312


Contents xiii

Bibliography 320

Documents

Introduction to Continuum Mechanics Gonzales - Stuart