Preface ........................................................ xv
Acknowledgments .............................................. xvii
Chapter 1 Introduction ......................................... 1
1.1 What is Mechanics? ......................................... 1
1.2 Continuum Mechanics ........................................ 1
1.3 An Example of an Ad-Hoc Approach ........................... 2
Chapter 2 Mathematical Preliminaries ........................... 5
2.1 Summation Convention, Dummy Indices ........................ 5
2.2 Free Indices ............................................... 6
2.3 Kronecker Delta ............................................ 7
2.4 Index Notation ............................................. 8
2.5 Permutation Symbol ........................................ 10
2.6 Manipulations with the Indicial Notations ................. 12
2.7 Translation and Rotation of Coordinate Axes ............... 13
2.8 Tensors ................................................... 19
2.9 The Divergence Theorem .................................... 27
2.10 Differentiation of Tensor Fields .......................... 28
References ..................................................... 28
Exercises ...................................................... 28
Chapter 3 Kinematics .......................................... 31
3.1 Description of Motion of a Continuum ...................... 31
3.2 Referential and Spatial Descriptions ...................... 34
3.3 Displacement Vector ....................................... 35
3.4 Restrictions on Continuous Deformation of a Deformable
Body ...................................................... 36
3.5 Material Derivative ....................................... 38
3.6 Finding Acceleration of a Particle from a Given
Velocity Field ............................................ 40
3.7 Deformation Gradient ...................................... 44
3.8 Strain Tensors ............................................ 58
3.9 Principal Strains ......................................... 63
3.10 Deformation of Areas and Volumes .......................... 71
3.11 Mass Density, Equation of Continuity ...................... 72
3.12 Rate of Deformation, Strain-Rate Tensor, Spin ............. 76
3.13 Polar Decomposition ....................................... 84
3.14 Infinitesimal Deformations ................................ 92
3.15 Infinitesimal Deformations Superimposed upon Finite
Deformations ............................................. 102
3.16 Volumetric and Deviatoric Strains ........................ 102
3.17 Transformation of Tensors Under a Change of Bases ........ 104
3.18 Plane Strain Deformation ................................. 105
Appendix A: Solution of a Cubic Equation ...................... 106
References .................................................... 108
Exercises ..................................................... 108
Chapter 4 The Balance Laws, Stress Tensors ................... 115
4.1 Kinetics of a Continuous Media ........................... 115
4.2 Traction Boundary Conditions ............................. 121
4.3 The Nominal Stress Tensor ................................ 123
4.4 Transformation of Stress Tensors Under the Rotation of
Axes ..................................................... 125
4.5 Principal Stresses; Maximum Shear Stress ................. 130
4.6 Relations Among Stress Tensors for Infinitesimal
Deformations ............................................. 133
4.7 Plane Stress ............................................. 134
4.8 Deviatoric Stress, von-Mises Stress ...................... 134
4.9 Balance of Energy ........................................ 135
4.10 Entropy Inequality, The Clausius-Duhem Inequality ....... 138
4.11 Summary of Equations Governing Deformations of a Body ... 140
4.12 Nonuniqueness of Solutions for Static Problems .......... 141
Appendix В: The Transport Theorem ............................. 142
Exercises ..................................................... 143
Chapter 5 Constitutive Relations ............................. 147
5.1 Introductory Remarks ..................................... 147
5.2 Thermoelastic Material ................................... 147
5.3 Principle of Material Objectivity ........................ 151
5.4 Linear Constitutive Relations for Finite Deformations
of a Thermoelastic Body .................................. 153
5.5 Isotropic Thermoelastic Materials ........................ 157
5.6 Comparison of Results from Four Linear Constitutive
Relations in Isotropic Finite Elasticity ................. 161
5.7 Transversely Isotropic Thermoelastic Materials ........... 177
5.8 Orthotropic Thermoelastic Materials ...................... 181
5.9 Coincidence of Principal Axes of Stress and Strain in
Isotropic Elastic Materials .............................. 183
5.10 Coincidence of Principal Axes of Stress and Strain in
Transversely Isotropic Elastic Materials ................. 185
5.11 Incompressible Elastic Materials ......................... 186
5.12 Comparison of Results from Constitutive Relations ........ 188
5.13 Constitutive Relations for Infinitesimal Deformations
of Elastic Materials ..................................... 194
5.14 Constitutive Relations for Special Isotropic Nonlinear
Elastic Materials ........................................ 205
5.15 Infinitesimal Deformations Superimposed upon Finite
Deformations of an Isotropic Elastic Body ................ 208
5.16 Constitutive Relations for Plane Deformations of a
Thermoelastic Body ....................................... 210
5.17 Thermoviscoelastic Materials ............................. 216
5.18 Summary .................................................. 225
References .................................................... 225
Exercises ..................................................... 226
Chapter 6 Torsion of a Circular Cylinder ..................... 229
6.1 Torsion of a Linear Elastic Circular Cylinder ............ 229
6.2 Torsion of a Second-Order Elastic Circular Cylinder ...... 234
6.3 Infinitesimal Twist of a Finitely Stretched Circular
Cylinder ................................................. 239
6.4 Finite Torsion of a Circular Cylinder .................... 241
Appendix C: A Uniqueness Theorem .............................. 246
References .................................................... 248
Exercise ...................................................... 248
Chapter 7 Fluid Flow ......................................... 249
7.1 Steady Flow Between Two Parallel Plates .................. 249
7.2 Steady Isothermal Flow of an Incompressible Fluid Down
an Inclined Plane ........................................ 252
7.3 Steady Flow of an Incompressible Fluid in a Horizontal
Circular Pipe ............................................ 257
Exercise ...................................................... 263
Chapter 8 Bending of Beams ................................... 265
8.1 Bending of a Rectangular Beam ............................ 265
8.2 Bending of a Nonlinear Elastic Rectangular Beam .......... 270
8.3 Airy Stress Function for Bending of a Beam ............... 276
Exercises ..................................................... 279
Chapter 9 Wave Propagation ................................... 281
9.1 Singular Surface ......................................... 281
9.2 Kinematics of a Singular Surface ......................... 284
9.3 Acceleration Waves in Linear Elasticity .................. 286
9.4 Progressive Waves ........................................ 291
9.5 Incompressible Linear Elastic Materials .................. 291
9.6 Acceleration Waves in Nonlinear Elastic Bodies ........... 293
9.7 Infinitesimal Deformations Superimposed upon Finite
Deformations ............................................. 296
Exercises ..................................................... 301
Chapter 10 Spherical and Cylindrical Pressure Vessels ......... 303
10.1 Radial Expansion of a Spherical Pressure Vessel .......... 303
10.2 Radial Expansion of an Incompressible Hookean Sphere
with Shear Modulus a Function of Radius .................. 307
10.3 Radial Expansion of a Cylindrical Pressure Vessel ........ 309
10.4 Radial Expansion of an Inhomogeneous and Incompressible
Hookean Cylinder ......................................... 311
10.5 Finite Radial Expansion of a NeoHookean Cylinder ......... 313
Index ......................................................... 317
Supporting Materials .......................................... 327
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