Cambridge Introduction To Continuum Mechanics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
1.1 Concept of a Continuum......Page 17
1.2 Sequence of Topics......Page 18
SUGGESTED READING......Page 19
2.1 Index Notation and Summation Convention......Page 20
2.2 Kronecker Delta and Permutation Symbol......Page 22
2.2.1 Example: Skew Symmetry......Page 23
2.3 Coordinate System......Page 24
2.4 Coordinate Transformations......Page 25
2.5 Vectors......Page 28
2.6.1 Examples of Tensors......Page 30
2.6.2 Quotient Rule......Page 33
2.7 Quadratic Forms and Eigenvalue Problems......Page 34
2.7.1 Example: Eigenvalue Problem......Page 36
2.7.2 Diagonalization and Polar Decomposition......Page 37
2.7.3 Example: Polar Decomposition......Page 39
SUGGESTED READING......Page 40
3 General Tensors......Page 43
3.1 Vectors and Tensors......Page 47
3.3 Tensor Calculus......Page 49
3.4 Curvature Tensors......Page 51
3.5.1 Example: Incompressible Flow......Page 52
3.5.2 Example: Equilibrium of Stresses......Page 53
SUGGESTED READING......Page 54
4.1 Gauss Theorem......Page 58
4.2 Stokes Theorem......Page 60
SUGGESTED READING......Page 63
5.1 Lagrangian and Eulerian Descriptions......Page 65
5.2 Deformation Gradients......Page 67
5.2.1 Deformation Gradient Vectors......Page 68
5.2.2 Curvilinear Systems......Page 70
5.3 Strain Tensors......Page 71
5.3.1 Decomposition of Displacement Gradients......Page 72
5.3.2 Stretch......Page 73
5.3.4 Infinitesimal Strains and Rotations......Page 74
5.3.5 Deformation Ellipsoids......Page 76
5.3.6 Polar Decomposition of the Deformation Gradient......Page 79
5.3.7 Stretch and Rotation......Page 80
5.3.8 Example: Polar Decomposition......Page 81
5.3.9 Example: Square Root of a Matrix......Page 83
5.5 Change of Volume......Page 84
5.6 Change of Area......Page 85
5.7 Compatibility Equations......Page 86
5.8 Spatial Rotation and Two-Point Tensors......Page 87
5.9 Curvilinear Coordinates......Page 88
SUGGESTED READING......Page 89
6.1 Material Derivative......Page 92
6.1.1 Some Terminology......Page 93
6.1.2 Example: Path Line, Stream Line, and Streak Line......Page 95
6.2 Length, Volume, and Area Elements......Page 96
6.2.2 Volume......Page 97
6.3.1 Line Integrals......Page 98
6.4 Deformation Rate, Spin, and Vorticity......Page 99
6.5 Strain Rate......Page 102
SUGGESTED READING......Page 103
7.1 Mass......Page 106
7.2.2 Balance of Linear Momentum......Page 107
7.3 Axiom of Material Frame Indifference......Page 108
7.4 Objective Measures of Rotation......Page 110
7.5 Integrity Basis......Page 111
SUGGESTED READING......Page 112
8.2 Internal Forces and Moments......Page 114
8.3 Cauchy Stress and Couple Stress Tensors......Page 116
8.3.2 Principal Stresses......Page 117
8.3.3 Shear Stress......Page 118
8.3.4 Hydrostatic Pressure and Deviatoric Stresses......Page 119
8.3.5 Objective Stress Rates......Page 120
8.4.1 Conservation of Mass......Page 122
8.4.3 Balance of Moment of Momentum (Angular Momentum)......Page 123
8.5 Material Description of the Equations of Motion......Page 124
8.5.1 First Piola–Kirchhoff Stress Tensor......Page 125
8.5.2 Second Piola–Kirchhoff Stress Tensor......Page 126
SUGGESTED READING......Page 127
9.1 Classical Thermodynamics......Page 130
9.2 Balance of Energy......Page 131
9.3 Clausius–Duhem Inequality......Page 132
9.3.1 Fourier’s Law of Heat Conduction......Page 134
9.3.4 Strain Energy Density......Page 135
9.3.5 Ideal Gas......Page 136
9.4.1 Legendre or Contact Transformation......Page 137
9.5 Method of Jacobians in Thermodynamics......Page 139
SUGGESTED READING......Page 142
10 Constitutive Relations......Page 145
10.1.1 Principles of Exclusion......Page 146
10.1.3 Principle of Spatial Invariance......Page 147
10.2 Simple Materials......Page 148
10.3.1 Elastic Materials of Cauchy......Page 150
10.3.2 Elastic Materials of Green......Page 151
10.4 Stokes Fluids......Page 153
10.5 Invariant Surface Integrals......Page 154
SUGGESTED READING......Page 156
11.1 Finite Elasticity......Page 158
11.1.1 Homogeneous Deformation......Page 159
11.1.2 Simple Extension......Page 160
11.1.4 Simple Shear......Page 161
11.1.5 Torsion of a Circular Cylinder......Page 162
11.2 Approximate Strain Energy Functions......Page 164
11.2.2 Small-Strain Approximation......Page 165
11.3 Integrated Elasticity......Page 166
11.3.1 Example: Incremental Loading......Page 167
11.4 A Variational Principle for Static Elasticity......Page 168
11.5 Isotropic Thermoelasticity......Page 170
11.5.1 Specific Heats and Latent Heats......Page 171
11.5.3 Adiabatic and Isothermal Elastic Modulus......Page 172
11.5.4 Example: Rubber Elasticity......Page 173
11.6 Linear Anisotropic Materials......Page 175
11.7 Invariant Integrals......Page 176
SUGGESTED READING......Page 177
12.1 Basic Equations......Page 180
12.3 Newtonian Fluids......Page 181
12.5 Shearing Flow......Page 183
12.6 Pipe Flow......Page 185
12.7 Rotating Flow......Page 188
12.9 Incompressible Flow......Page 191
12.11 Inviscid Flow......Page 192
12.11.1 Speed of Sound......Page 193
12.11.2 Method of Characteristics......Page 194
12.12 Bernoulli Equation......Page 195
SUGGESTED READING......Page 196
13.1 Kelvin–Voigt Solid......Page 200
13.2 Maxwell Fluid......Page 201
13.3 Standard Linear Solid......Page 203
13.4 Superposition Principle......Page 205
13.6 Three-Dimensional Linear Constitutive Relations......Page 206
13.7 Anisotropy......Page 207
13.8 Biot’s Theory......Page 208
13.9 Creep in Metals......Page 210
13.10 Nonlinear Theories of Viscoelasticity......Page 211
13.11 K-BKZ Model for Viscoelastic Fluids......Page 212
SUGGESTED READING......Page 213
14.1.1 Rigid Perfectly Plastic Material......Page 216
14.1.3 Elastic Linearly Hardening Material......Page 217
14.2 Three-Dimensional Theories......Page 219
14.3.1 Levy–Mises Flow Rule......Page 222
14.3.2 Prandtl–Reuss Flow Rule......Page 223
14.4 General Yield Condition and Plastic Work......Page 224
14.4.2 Rigid Plasticity and Slip-Line Field......Page 225
14.4.3 Example: Symmetric External Cracks......Page 228
14.5 Drucker’s Definition of Stability......Page 229
14.6 Il´ıushin’s Postulate......Page 231
14.7.2 Isotropic Hardening......Page 232
14.7.3 Kinematic Hardening......Page 233
14.8 Endochronic Theory of Valanis......Page 235
14.9 Plasticity and Damage......Page 237
14.10 Minimum Dissipation Rate Principle......Page 240
SUGGESTED READING......Page 241
Author Index......Page 245
Subject Index......Page 247