Fundamentals of Continuum Mechanics

John W. Rudnicki received his bachelor, master’s and PhD degrees from Brown University in the USA, the last in 1977. He has been on the faculty of Northwestern University since 1981, where he is now Professor of Civil and Environmental Engineering and Mechanical Engineering. He is a Fellow of the American Society of Mechanical Engineers. He has been awarded the Biot Medal from the American Society of Civil Engineers, the Brown Engineering Alumni Medal, the Daniel C. Drucker Medal from the American Society of Mechanical Engineers, and the Engineering Science Medal from the Society of Engineering Science. His research has been primarily in geomechanics, specifically the inelastic behavior and failure of geomaterials. He has been especially interested in deformation instabilities in brittle rocks and granular media, including their interactions with pore fluids, with applications to the mechanics of earthquakes and environment- and resource-related geomechanics

Preface xiii

Nomenclature xv

Introduction 1

Part One Mathematical Preliminaries 3

1 Vectors 5

1.1 Examples 9

1.1.1 9

1.1.2 9

Exercises 9

Reference 11

2 Tensors 13

2.1 Inverse 15

2.2 Orthogonal Tensor 16

2.3 Principal Values 16

2.4 Nth-Order Tensors 18

2.5 Examples 18

2.5.1 18

2.5.2 18

Exercises 19

3 Cartesian Coordinates 21

3.1 Base Vectors 21

3.2 Summation Convention 23

3.3 Tensor Components 24

3.4 Dyads 25

3.5 Tensor and Scalar Products 27

3.6 Examples 29

3.6.1 29

3.6.2 29

3.6.3 29

Exercises 30

Reference 30

4 Vector (Cross) Product 31

4.1 Properties of the Cross Product 32

4.2 Triple Scalar Product 33

4.3 Triple Vector Product 33

4.4 Applications of the Cross Product 34

4.4.1 Velocity due to Rigid Body Rotation 34

4.4.2 Moment of a Force P about O 35

4.5 Non-orthonormal Basis 36

4.6 Example 37

Exercises 37

5 Determinants 41

5.1 Cofactor 42

5.2 Inverse 43

5.3 Example 44

Exercises 44

6 Change of Orthonormal Basis 47

6.1 Change of Vector Components 48

6.2 Definition of a Vector 50

6.3 Change of Tensor Components 50

6.4 Isotropic Tensors 51

6.5 Example 52

Exercises 53

Reference 56

7 Principal Values and Principal Directions 57

7.1 Example 59

Exercises 60

8 Gradient 63

8.1 Example: Cylindrical Coordinates 66

Exercises 67

Part Two Stress 69

9 Traction and Stress Tensor 71

9.1 Types of Forces 71

9.2 Traction on Different Surfaces 73

9.3 Traction on an Arbitrary Plane (Cauchy Tetrahedron) 75

9.4 Symmetry of the Stress Tensor 76

Exercise 77

Reference 77

10 Principal Values of Stress 79

10.1 Deviatoric Stress 80

10.2 Example 81

Exercises 82

11 Stationary Values of Shear Traction 83

11.1 Example: Mohr–Coulomb Failure Condition 86

Exercises 88

12 Mohr’s Circle 89

Exercises 93

Reference 93

Part Three Motion and Deformation 95

13 Current and Reference Configurations 97

13.1 Example 102

Exercises 103

14 Rate of Deformation 105

14.1 Velocity Gradients 105

14.2 Meaning of D 106

14.3 Meaning of W 108

Exercises 109

15 Geometric Measures of Deformation 111

15.1 Deformation Gradient 111

15.2 Change in Length of Lines 112

15.3 Change in Angles 113

15.4 Change in Area 114

15.5 Change in Volume 115

15.6 Polar Decomposition 116

15.7 Example 118

Exercises 118

References 120

16 Strain Tensors 121

16.1 Material Strain Tensors 121

16.2 Spatial Strain Measures 123

16.3 Relations Between D and Rates of EG and U 124

16.3.1 Relation Between Ė and D 124

16.3.2 Relation Between D and U 125

Exercises 126

References 128

17 Linearized Displacement Gradients 129

17.1 Linearized Geometric Measures 130

17.1.1 Stretch in Direction N 130

17.1.2 Angle Change 131

17.1.3 Volume Change 131

17.2 Linearized Polar Decomposition 132

17.3 Small-Strain Compatibility 133

Exercises 135

Reference 135

Part Four Balance of Mass, Momentum, and Energy 137

18 Transformation of Integrals 139

Exercises 142

References 143

19 Conservation of Mass 145

19.1 Reynolds’ Transport Theorem 148

19.2 Derivative of an Integral over a Time-Dependent Region 149

19.3 Example: Mass Conservation for a Mixture 150

Exercises 151

20 Conservation of Momentum 153

20.1 Momentum Balance in the Current State 153

20.1.1 Linear Momentum 153

20.1.2 Angular Momentum 154

20.2 Momentum Balance in the Reference State 155

20.2.1 Linear Momentum 156

20.2.2 Angular Momentum 157

20.3 Momentum Balance for a Mixture 158

Exercises 159

21 Conservation of Energy 161

21.1 Work-Conjugate Stresses 163

Exercises 165

Part Five Ideal Constitutive Relations 167

22 Fluids 169

22.1 Ideal Frictionless Fluid 169

22.2 Linearly Viscous Fluid 171

22.2.1 Non-steady Flow 173

Exercises 175

Reference 176

23 Elasticity 177

23.1 Nonlinear Elasticity 177

23.1.1 Cauchy Elasticity 177

23.1.2 Green Elasticity 178

23.1.3 Elasticity of Pre-stressed Bodies 179

23.2 Linearized Elasticity 182

23.2.1 Material Symmetry 183

23.2.2 Linear Isotropic Elastic Constitutive Relation 185

23.2.3 Restrictions on Elastic Constants 186

23.3 More Linearized Elasticity 187

23.3.1 Uniqueness of the Static Problem 188

23.3.2 Pressurized Hollow Sphere 189

Exercises 191

Reference 194

Index 195