Preface ...................................................... xiii
1 Introduction ................................................. 1
1.1 Continuum Mechanics ..................................... 1
1.2 A Look Forward .......................................... 4
1.3 Summary ................................................. 5
Problems ..................................................... 6
2 Vectors and Tensors .......................................... 8
2.1 Background and Overview ................................. 8
2.2 Vector Algebra .......................................... 9
2.2.1 Definition of a Vector ........................... 9
2.2.2 Scalar and Vector Products ...................... 11
2.2.3 Plane Area as a Vector .......................... 16
2.2.4 Components of a Vector .......................... 17
2.2.5 Summation Convention ............................ 18
2.2.6 Transformation Law for Different Bases .......... 22
2.3 Theory of Matrices ..................................... 24
2.3.1 Definition ...................................... 24
2.3.2 Matrix Addition and Multiplication of a Matrix
by a Scalar ..................................... 25
2.3.3 Matrix Transpose and Symmetric Matrix ........... 26
2.3.4 Matrix Multiplication ........................... 27
2.3.5 Inverse and Determinant of a Matrix ............. 29
2.4 Vector Calculus ........................................ 32
2.4.1 Derivative of a Scalar Function of a Vector ..... 32
2.4.2 The del Operator ................................ 36
2.4.3 Divergence and Curl of a Vector ................. 36
2.4.4 Cylindrical and Spherical Coordinate Systems .... 39
2.4.5 Gradient, Divergence, and Curl Theorems ......... 40
2.5 Tensors ................................................ 42
2.5.1 Dyads and Polyads ............................... 42
2.5.2 Nonion Form of a Dyadic ......................... 43
2.5.3 Transformation of Components of a Dyadic ........ 45
2.5.4 Tensor Calculus ................................. 45
2.5.5 Eigenvalues and Eigenvectors of Tensors ......... 48
2.6 Summary ................................................ 55
Problems .................................................... 55
3 Kinematics of Continua ...................................... 61
3.1 Introduction ........................................... 61
3.2 Descriptions of Motion ................................. 62
3.2.1 Configurations of a Continuous Medium ........... 62
3.2.2 Material Description ............................ 63
3.2.3 Spatial Description ............................. 64
3.2.4 Displacement Field .............................. 67
3.3 Analysis of Deformation ................................ 68
3.3.1 Deformation Gradient Tensor ..................... 68
3.3.2 Isochoric, Homogeneous, and Inhomogeneous
Deformations .................................... 71
3.3.3 Change of Volume and Surface .................... 73
3.4 Strain Measures ........................................ 77
3.4.1 Cauchy-Green Deformation Tensors ................ 77
3.4.2 Green Strain Tensor ............................. 78
3.4.3 Physical Interpretation of the Strain
Components ...................................... 80
3.4.4 Cauchy and Euler Strain Tensors ................. 81
3.4.5 Principal Strains ............................... 84
3.5 Infinitesimal Strain Tensor and Rotation Tensor ........ 89
3.5.1 Infinitesimal Strain Tensor ..................... 89
3.5.2 Physical Interpretation of Infinitesimal
Strain Tensor Components ........................ 89
3.5.3 Infinitesimal Rotation Tensor ................... 91
3.5.4 Infinitesimal Strains in Cylindrical and
Spherical Coordinate Systems .................... 93
3.6 Rate of Deformation and Vorticity Tensors .............. 96
3.6.1 Definitions ..................................... 96
3.6.2 Relationship between D and Ё .................... 96
3.7 Polar Decomposition Theorem ............................ 97
3.8 Compatibility Equations ............................... 100
3.9 Change of Observer: Material Frame Indifference ....... 105
3.10 Summary ............................................... 107
Problems ................................................... 108
4 Stress Measures ............................................ 115
4.1 Introduction .......................................... 115
4.2 Cauchy Stress Tensor and Cauchy's Formula ............. 115
4.3 Transformation of Stress Components and Principal
Stresses .............................................. 120
4.3.1 Transformation of Stress Components ............ 120
4.3.2 Principal Stresses and Principal Planes ........ 124
4.3.3 Maximum Shear Stress ........................... 126
4.4 Other Stress Measures ................................. 128
4.4.1 Preliminary Comments ........................... 128
4.4.2 First Piola-Kirchhoff Stress Tensor ............ 128
4.4.3 Second Piola-Kirchhoff Stress Tensor ........... 130
4.5 Equations of Equilibrium .............................. 134
4.6 Summary ............................................... 136
Problems ................................................... 137
5 Conservation of Mass, Momenta, and Energy .................. 143
5.1 Introduction .......................................... 143
5.2 Conservation of Mass .................................. 144
5.2.1 Preliminary Discussion ......................... 144
5.2.2 Material Time Derivative ....................... 144
5.2.3 Continuity Equation in Spatial Description ..... 146
5.2.4 Continuity Equation in Material Description .... 152
5.2.5 Reynolds Transport Theorem ..................... 153
5.3 Conservation of Momenta ............................... 154
5.3.1 Principle of Conservation of Linear Momentum ... 154
5.3.2 Equation of Motion in Cylindrical and
Spherical Coordinates .......................... 159
5.3.3 Principle of Conservation of Angular Momentum .. 161
5.4 Thermodynamic Principles .............................. 163
5.4.1 Introduction ................................... 163
5.4.2 The First Law of Thermodynamics: Energy
Equation ....................................... 164
5.4.3 Special Cases of Energy Equation ............... 165
5.4.4 Energy Equation for One-Dimensional Flows ...... 167
5.4.5 The Second Law of Thermodynamics ............... 170
5.5 Summary ............................................... 171
Problems ................................................... 172
6 Constitutive Equations ..................................... 178
6.1 Introduction .......................................... 178
6.2 Elastic Solids ........................................ 179
6.2.1 Introduction ................................... 179
6.2.2 Generalized Hooke's Law ........................ 180
6.2.3 Material Symmetry .............................. 182
6.2.4 Monoclinic Materials ........................... 183
6.2.5 Orthotropic Materials .......................... 184
6.2.6 Isotropic Materials ............................ 187
6.2.7 Transformation of Stress and Strain
Components ..................................... 188
6.2.8 Nonlinear Elastic Constitutive Relations ....... 193
7 Linearized Elasticity Problems ............................. 210
7.1 Introduction .......................................... 210
7.2 Governing Equations ................................... 211
7.3 The Navier Equations .................................. 212
7.4 The Beltrami-Michell Equations ........................ 212
7.5 Types of Boundary Value Problems and Superposition
Principle ............................................. 214
7.6 Clapeyron's Theorem and Reciprocity Relations ......... 216
7.6.1 Clapeyron's Theorem ............................ 216
7.6.2 Betti's Reciprocity Relations .................. 219
7.6.3 Maxwell's Reciprocity Relation ................. 222
7.7 Solution Methods ...................................... 224
7.7.1 Types of Solution Methods ...................... 224
7.7.2 An Example: Rotating Thick-Walled Cylinder ..... 225
7.7.3 Two-Dimensional Problems ....................... 227
7.7.4 Airy Stress Function ........................... 230
7.7.5 End Effects: Saint-Venant's Principle .......... 233
7.7.6 Torsion of Noncircular Cylinders ............... 240
7.8 Principle of Minimum Total Potential Energy ........... 243
7.8.1 Introduction ................................... 243
7.8.2 Total Potential Energy Principle ............... 244
7.8.3 Derivation of Navier's Equations ............... 246
7.8.4 Castigliano's Theorem I ........................ 251
7.9 Hamilton's Principle .................................. 257
7.9.1 Introduction ................................... 257
7.9.2 Hamilton's Principle for a Rigid Body .......... 257
7.9.3 Hamilton's Principle for a Continuum ........... 261
7.10 Summary ............................................... 265
Problems ................................................... 265
8 Fluid Mechanics and Heat Transfer Problems ................. 275
8.1 Governing Equations ................................... 275
8.1.1 Preliminary Comments ........................... 275
8.1.2 Summary of Equations ........................... 276
8.1.3 Viscous Incompressible Fluids .................. 277
8.1.4 Heat Transfer .................................. 280
8.2 Fluid Mechanics Problems .............................. 282
8.2.1 Inviscid Fluid Statics ......................... 282
8.2.2 Parallel Flow (Navier-Stokes Equations) ........ 284
8.2.3 Problems with Negligible Convective Terms ...... 289
8.3 Heat Transfer Problems ................................ 293
8.3.1 Heat Conduction in a Cooling Fin ............... 293
8.3.2 Axisymmetric Heat Conduction in a Circular
Cylinder ....................................... 295
8.3.3 Two-Dimensional Heat Transfer .................. 297
8.3.4 Coupled Fluid Flow and Heat Transfer ........... 299
8.4 Summary ............................................... 300
Problems ................................................... 300
9 Linear Viscoelasticity ..................................... 305
9.1 Introduction .......................................... 305
9.1.1 Preliminary Comments ........................... 305
9.1.2 Initial Value Problem, the Unit Impulse, and
the Unit Step Function ......................... 306
9.1.3 The Laplace Transform Method ................... 307
9.2 Spring and Dashpot Models ............................. 311
9.2.1 Creep Compliance and Relaxation Modulus ........ 311
9.2.2 Maxwell Element ................................ 312
9.2.3 Kelvin-Voigt Element ........................... 315
9.2.4 Three-Element Models ........................... 317
9.2.5 Four-Element Models ............................ 319
9.3 Integral Constitutive Equations ....................... 323
9.3.1 Hereditary Integrals ........................... 323
9.3.2 Hereditary Integrals for Deviatoric
Components ..................................... 326
9.3.3 The Correspondence Principle ................... 327
9.3.4 Elastic and Viscoelastic Analogies ............. 331
9.4 Summary ............................................... 334
Problems ................................................... 334
References .................................................... 339
Answers to Selected Problems .................................. 341
Index ......................................................... 351
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