Elements of Continuum Mechanics and Thermodynamics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1 Introduction......Page 13
1.2 Rectangular Cartesian Coordinate Systems......Page 15
1.3 Suffix and Symbolic Notation......Page 16
1.4 Orthogonal Transformations......Page 18
1.5 Concept of a Second-Order Tensor......Page 23
1.6 Tensor Algebra......Page 31
1.7 Invariants of Second-Order Tensors......Page 32
1.8 Cayley-Hamilton Theorem......Page 36
1.9 Higher-Order Tensors......Page 37
1.10 Determinants......Page 40
1.11 Polar Decomposition Theorem......Page 44
1.12 Tensor Fields......Page 50
1.13 Integral Theorems......Page 56
EXERCISES......Page 58
SUGGESTED READING......Page 60
2.1 Description of Motion......Page 62
2.2 Spatial and Referential Descriptions......Page 65
2.3 Material Surface......Page 66
2.4 Jacobian of Transformation and Deformation Gradient F......Page 68
2.5 Reynolds Transport Theorem......Page 71
2.6 Continuity Equation......Page 75
2.7 Velocity Gradient......Page 76
2.8 Rate of Deformation and Spin Tensors......Page 77
2.9 Polar Decomposition of F......Page 82
2.10 Further Decomposition of F......Page 86
2.11 Strain......Page 87
2.12 Infinitesimal Strain......Page 90
EXERCISES......Page 93
3.1 Contact Forces and Body Forces......Page 96
3.2 Cauchy or True Stress......Page 99
3.3 Spatial Form of Equations of Motion......Page 102
3.4 Principal Stresses and Maximum Shearing Stress......Page 103
3.5 Pure Shear Stress and Decomposition of the Stress Tensor......Page 106
3.6 Octahedral Shearing Stress......Page 107
3.7 Nominal Stress Tensor......Page 110
3.8 Second Piola-Kirchhoff Stress Tensor......Page 112
3.9 Other Stress Tensors......Page 113
EXERCISES......Page 114
4.1 Stress Power......Page 117
4.2 Principle of Virtual Work......Page 122
4.3 Energy Equation and Entropy Inequality......Page 126
EXERCISES......Page 131
REFERENCES......Page 132
5.2 Rigid Bodies......Page 133
5.3 Ideal Inviscid Fluid......Page 135
5.4 Incompressible Inviscid Fluid......Page 136
5.5 Newtonian Viscous Fluid......Page 137
5.6 Classical Elasticity......Page 138
5.7 Linear Thermoelasticity......Page 142
5.8 Determinism, Local Action, and Material Frame Indifference......Page 150
EXERCISES......Page 159
REFERENCES......Page 161
6.2 Cauchy Elasticity......Page 162
6.3 Hyperelasticity......Page 167
6.4 Incompressible Hyperelastic Solid......Page 172
6.5 Alternative Formulation......Page 174
EXERCISES......Page 178
REFERENCES......Page 179
7.1 Introduction......Page 180
7.2 Strain Energy Functions and Stress-Strain Relations......Page 181
7.3 Simple Shear of a Rectangular Block......Page 182
7.4 Simple Tension......Page 184
7.5 Extension and Torsion of an Incompressible Cylindrical Bar......Page 187
7.6 Spherically Symmetric Expansion of a Thick-Walled Shell......Page 192
7.7 Eversion of a Cylindrical Tube......Page 197
7.8 Pure Bending of an Hyperelastic Plate......Page 201
7.9 Combined Telescopic and Torsional Shear......Page 207
EXERCISES......Page 212
REFERENCES......Page 213
8.2 Basic Relations for Finite Deformation Thermoelasticity......Page 214
8.3 Ideal Rubber-Like Materials......Page 221
8.4 Isentropic Simple Tension of Ideal Rubber......Page 226
8.5 Isentropic Simple Tension of Compressible Rubber......Page 227
EXERCISES......Page 230
REFERENCES......Page 231
9.1 Newtonian Viscous Fluids......Page 232
9.2 A Non-Newtonian Viscous Fluid......Page 235
9.3 Linear Viscoelastic Medium......Page 236
9.4 Viscoelastic Dissipation......Page 249
9.5 Some Thermodynamic Considerations in Linear Thermoelasticity......Page 252
9.6 Simple Shear Problem......Page 256
EXERCISES......Page 258
REFERENCES......Page 260
A1.2 Curvilinear Coordinates......Page 261
A1.3 Orthogonality......Page 264
A1.4 Cylindrical and Spherical Polar Coordinate Systems......Page 265
A1.5 The = Operator......Page 268
A1.6 The = Operator......Page 273
Appendix 2 Physical Components of the Deformation Gradient Tensor......Page 275
Appendix 3 Legendre Transformation......Page 278
Appendix 4 Linear Vector Spaces......Page 281
Index......Page 284