Principles of Continuum Mechanics: A Basic Course for Physicists

Preface......Page 8
Contents......Page 9
 Scalars......Page 14
 Vectors......Page 15
 Tensors......Page 16
 Operators......Page 17
 Other Symbols......Page 18
 1.1 Bodies, Configurations and Motion......Page 19
 1.3 Referential and Spatial Coordinates......Page 23
 1.4 Lagrangian and Eulerian Variables......Page 26
 1.5 Deformation Gradients......Page 28
 1.6 Polar Decomposition of the Deformation Gradient......Page 33
 1.7 Measures of Deformation......Page 36
 1.8 Length and Angle Changes......Page 38
 1.9 Area and Volume Changes......Page 41
 1.10 Strain Invariants and Principal Strains......Page 42
 1.11 The Displacement Vector......Page 45
  1.12.1 Linearised Analysis of Deformation......Page 47
  1.12.2 Length and Angle Changes......Page 49
  1.12.3 Area and Volume Changes......Page 50
 2.1 Material and Spatial Time Derivatives......Page 52
 2.2 Time Derivatives of Some Geometric Quantities......Page 54
 2.3 Reynolds Transport Theorem......Page 59
 2.4 Modified Reynolds Transport Theorem......Page 61
 3.1 Mass and Density......Page 65
 3.2 Body and Surface Forces......Page 66
 3.3 Cauchy Traction Principle......Page 67
 3.4 Cauchy Stress Formula......Page 68
 3.5 Other Measures of Stress......Page 72
 4.1 Global Conservation Principles......Page 75
  4.2.1 Continuity Equation......Page 79
  4.2.3 Symmetry of the Cauchy Stress Tensor......Page 80
  4.2.4 Energy Equation......Page 82
  4.2.6 Overview of Local Conservation Principles in the Eulerian Form......Page 84
  4.2.7 Jump Conditions in Special Cases......Page 85
  4.3.1 Continuity Equation......Page 87
  4.3.2 Equation of Motion......Page 88
  4.3.4 Energy Equation......Page 89
  4.3.5 Entropy Inequality......Page 91
 5.1 Observer Transformation......Page 92
 5.2 Frame-Indifference......Page 96
 5.3 Frame-Indifference of Some Kinematic Quantities......Page 97
 5.4 Observer Transformation of the Deformation Gradient......Page 100
 5.5 Frame-Indifferent Time Derivatives......Page 102
 5.6 The Postulate of Frame-Indifference......Page 105
 5.7 Observer Transformation of Basic Field Equations......Page 107
 6.1 The Need for Constitutive Equations......Page 109
 6.2 Formulation of Thermomechanical Constitutive Equations......Page 110
 6.3 Simple Materials......Page 112
 6.4 The Principle of Material Frame-Indifference......Page 114
 6.5 Reduction by Polar Decomposition......Page 116
 6.6 Kinematic Constraints......Page 118
 6.7 Material Symmetry......Page 121
 6.8 Material Symmetry of Reduced-Form Constitutive Functionals......Page 126
 6.9 Noll's Rule......Page 127
  6.10.2 Fluids......Page 129
  6.10.3 Solids......Page 130
 6.11 Constitutive Equations for Isotropic Materials......Page 131
 6.12 The Present Configuration as the Reference......Page 132
 6.13 Isotropic Constitutive Functionals in the Relative Representation......Page 133
 6.14 The General Constitutive Equation for Fluids......Page 136
 6.15 The Principle of Bounded Memory......Page 137
 6.16 Representation Theorems for Isotropic Functions......Page 139
  6.16.1 Representation Theorem for a Scalar-Valued Isotropic Function......Page 140
  6.16.2 Representation Theorem for a Vector-Valued Isotropic Function......Page 141
  6.16.3 Representation Theorem for a Symmetric Tensor-Valued Isotropic Function......Page 142
  6.17.1 Elastic Solids......Page 144
  6.17.2 Thermoelastic Solids......Page 146
  6.17.4 Elastic Fluids......Page 147
  6.17.5 Thermoelastic Fluids......Page 148
  6.17.7 Incompressible Viscous Fluids......Page 149
  6.17.8 Viscous Heat-Conducting Fluids......Page 150
 7.1 The Clausius–Duhem Inequality......Page 152
 7.2 Application of the Clausius–Duhem Inequality to a Thermo-Elastic Solid......Page 154
 7.3 Application of the Clausius–Duhem Inequality to a Viscous Fluid......Page 158
 7.4 Application of the Clausius–Duhem Inequality to an Incompressible Viscous Fluid......Page 163
 7.5 The Müller–Liu Entropy Principle......Page 166
 7.6 Application of the Müller–Liu Entropy Principle to a Thermoelastic Solid......Page 168
 7.7 Application of the Müller–Liu Entropy Principle to a Viscous Heat-Conducting Fluid......Page 173
 8.1 Linear Elastic Solids......Page 181
 8.3 Isotropic Linear Elastic Solids......Page 185
 8.4 Constraints on Elastic Coefficients......Page 191
  8.5.1 Isotropic Linear Elastic Solids......Page 193
  8.5.2 Incompressible, Isotropic Linear Elastic Solids......Page 194
  8.5.3 Isotropic Linear Thermoelastic Solids......Page 195
 9.1 Equations for the Initial Configuration of a Body......Page 197
 9.3 Lagrangian and Eulerian Increments......Page 199
 9.4 Linearised Continuity Equations......Page 200
 9.5 Increments in Stress......Page 202
 9.6 Linearised Equations of Motion......Page 203
  9.7.2 Dynamic Jump Conditions......Page 206
 9.8 Linearised Elastic Constitutive Equations......Page 212
 9.9 The Gravitational Force......Page 213
  9.9.2 Increments in Gravitation......Page 214
  9.9.3 Linearised Poisson's Equations......Page 216
  9.9.4 Linearised Jump Conditions for Potential Increments......Page 217
  9.9.5 Linearised Jump Conditions for Increments in Gravitation......Page 218
 9.10 Equations of Motion for a Self-gravitating Body......Page 221
A Vector and Tensor Differential Identities......Page 223
 B.1 Local Base Vectors......Page 226
 B.2 Tangent Vectors and Tensors......Page 227
 B.4 Differential Identities......Page 228
 B.6 Divergence of a Vector......Page 229
 B.7 Gradient of a Vector......Page 230
 B.8 Spatial and Surface Invariants......Page 231
 C.1 Coordinate Transformations......Page 232
 C.2 Base Vectors......Page 234
 C.3 Derivatives of Unit Base Vectors......Page 237
 C.4 Derivatives of Vectors and Tensors......Page 240
  C.5.1 Gradient of a Scalar......Page 241
  C.5.2 Divergence of a Vector......Page 243
  C.5.3 Curl of a Vector......Page 244
  C.5.4 Gradient of a Vector......Page 245
  C.5.5 Divergence of a Tensor......Page 247
  C.5.6 Laplacian of a Scalar and a Vector......Page 249
D Selected References for General Reading......Page 250
 References......Page 251
Index......Page 253