A First Course in Continuum Mechanics

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ISBN: 9780521714242 | 0521714249
Cover: Paperback
Copyright: 2/18/2008

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The modeling and simulation of fluids, solids and other materials with significant coupling and thermal effects is becoming an increasingly important area of study in applied mathematics and engineering. Necessary for such studies is a fundamental understanding of the basic principles of continuum mechanics and thermodynamics. This book is a clear introduction to these principles. It is designed for a one- or two-quarter course for advanced undergraduate and beginning graduate students in the mathematical and engineering sciences, and is based on over nine years of teaching experience. It is also sufficiently self-contained for use outside a classroom environment. Prerequisites include a basic knowledge of linear algebra, multivariable calculus, differential equations and physics.

Preface	p. xv
Tensor Algebra	p. 1
Vectors	p. 1
Vector Algebra	p. 2
Scalar and Vector Products	p. 3
Projections, Bases and Coordinate Frames	p. 4
Index Notation	p. 6
Summation Convention	p. 6
Kronecker Delta and Permutation Symbols	p. 8
Frame Identities	p. 8
Vector Operations in Components	p. 9
Epsilon-Delta Identities	p. 10
Second-Order Tensors	p. 11
Definition	p. 11
Second-Order Tensor Algebra	p. 13
Representation in a Coordinate Frame	p. 13
Second-Order Dyadic Products, Bases	p. 14
Second-Order Tensor Algebra in Components	p. 15
Special Classes of Tensors	p. 16
Change of Basis	p. 19
Traces, Determinants and Exponentials	p. 21
Eigenvalues, Eigenvectors and Principal Invariants	p. 23
Special Decompositions	p. 25
Scalar Product for Second-Order Tensors	p. 26
Fourth-Order Tensors	p. 27
Definition	p. 27
Fourth-Order Tensor Algebra	p. 28
Representation in a Coordinate Frame	p. 28
Fourth-Order Dyadic Products, Bases	p. 29
Symmetry Properties	p. 30
Isotropic Tensor Functions	p. 31
Bibliographic Notes	p. 32
Exercises	p. 33
Answers to Selected Exercises	p. 38
Tensor Calculus	p. 44
Preliminaries	p. 44
Points, Tensors and Representations	p. 44
Standard Norms, Order Symbols	p. 45
Differentiation of Tensor Fields	p. 46
Derivatives, Gradients	p. 46
Divergence	p. 50
Curl	p. 52
Laplacian	p. 53
Integral Theorems	p. 54
Divergence Theorem	p. 54
Stokes' Theorem	p. 57
Localization Theorem	p. 58
Functions of Second-Order Tensors	p. 59
Scalar-Valued Functions	p. 60
Tensor-Valued Functions	p. 63
Bibliographic Notes	p. 65
Exercises	p. 65
Answers to Selected Exercises	p. 70
Continuum Mass and Force Concepts	p. 75
Continuum Bodies	p. 75
Mass	p. 76
Mass Density	p. 76
Center of Mass	p. 77
Force	p. 77
Body Forces	p. 77
Surface Forces	p. 78
The Stress Tensor	p. 82
Equilibrium	p. 85
Preliminaries	p. 85
Necessary Conditions	p. 86
Local Equations	p. 87
Basic Stress Concepts	p. 90
Simple States of Stress	p. 90
Principal, Normal and Shear Stresses	p. 92
Maximum Normal and Shear Stresses	p. 93
Spherical and Deviatoric Stress Tensors	p. 94
Bibliographic Notes	p. 95
Exercises	p. 96
Answers to Selected Exercises	p. 102
Kinematics	p. 112
Configurations and Deformations	p. 112
The Deformation Map	p. 113
Measures of Strain	p. 114
The Deformation Gradient F	p. 114
Interpretation of F, Homogeneous Deformations	p. 115
The Cauchy-Green Strain Tensor C	p. 120
Interpretation of C	p. 121
Rigid Deformations	p. 124
The Infinitesimal Strain Tensor E	p. 125
Interpretation of E	p. 126
Infinitesimally Rigid Deformations	p. 128
Motions	p. 128
Material and Spatial Fields	p. 129
Coordinate Derivatives	p. 130
Time Derivatives	p. 130
Velocity and Acceleration Fields	p. 131
Rate of Strain, Spin	p. 134
Rate of Strain Tensor L and Spin Tensor W	p. 134
Interpretation of L and W	p. 135
Vorticity	p. 135
Rigid Motions	p. 136
Change of Variables	p. 136
Transformation of Volume Integrals	p. 137
Derivatives of Time-Dependent Integrals	p. 139
Transformation of Surface Integrals	p. 141
Volume-Preserving Motions	p. 144
Bibliographic Notes	p. 144
Exercises	p. 145
Answers to Selected Exercises	p. 154
Balance Laws	p. 167
Motivation	p. 167
Balance Laws in Integral Form	p. 170
Conservation of Mass and Laws of Inertia	p. 170
First and Second Laws of Thermodynamics	p. 172
Integral Versus Local Balance Laws	p. 179
Localized Eulerian Form of Balance Laws	p. 179
Conservation of Mass	p. 179
Balance of Linear Momentum	p. 182
Balance of Angular Momentum	p. 183
Characterization of Net Working	p. 185
First Law of Thermodynamics	p. 186
Second Law of Thermodynamics	p. 187
Summary	p. 189
Localized Lagrangian Form of Balance Laws	p. 190
Conservation of Mass	p. 190
Balance of Linear Momentum	p. 191
Balance of Angular Momentum	p. 193
Characterization of Net Working	p. 193
First Law of Thermodynamics	p. 195
Second Law of Thermodynamics	p. 196
Summary	p. 197
Frame-Indifference	p. 199
Superposed Rigid Motions	p. 199
Axiom of Frame-Indifference	p. 200
Material Constraints	p. 202
Isothermal Considerations	p. 204
Bibliographic Notes	p. 207
Exercises	p. 208
Answers to Selected Exercises	p. 213
Isothermal Fluid Mechanics	p. 221
Ideal Fluids	p. 222
Definition	p. 222
Euler Equations	p. 223
Frame-Indifference Considerations	p. 225
Mechanical Energy Considerations	p. 226
Initial-Boundary Value Problems	p. 226
Motion Map, Other Boundary Conditions	p. 228
Irrotational Motion, Bernoulli's Theorem	p. 229
Elastic Fluids	p. 234
Definition	p. 235
Elastic Fluid Equations	p. 235
Frame-Indifference Considerations	p. 236
Mechanical Energy Considerations	p. 237
Initial-Boundary Value Problems	p. 238
Irrotational Motion, Generalized Bernoulli's Theorem	p. 239
Linearization	p. 243
Newtonian Fluids	p. 245
Definition	p. 245
Navier-Stokes Equations	p. 246
Frame-Indifference Considerations	p. 248
Mechanical Energy Considerations	p. 250
Initial-Boundary Value Problems	p. 250
Kinetic Energy of Fluid Motion	p. 252
Bibliographic Notes	p. 256
Exercises	p. 256
Answers to Selected Exercises	p. 262
Isothermal Solid Mechanics	p. 271
Elastic Solids	p. 272
Definition	p. 272
Elasticity Equations	p. 274
Frame-Indifference Considerations	p. 275
Initial-Boundary Value Problems	p. 277
Isotropy, Simplified Response Functions	p. 279
Hyperelastic Solids	p. 282
Definition	p. 282
Frame-Indifference Considerations	p. 283
Mechanical Energy Considerations	p. 285
Common Models	p. 287
Linearization of Elasticity Equations	p. 288
Linearized Equations, Elasticity Tensors	p. 289
Initial-Boundary Value Problems	p. 290
Properties of Elasticity Tensors	p. 291
Equation of Linearized, Isotropic Elasticity	p. 295
Linear Elastic Solids	p. 296
Definition	p. 296
General Properties	p. 298
Bibliographic Notes	p. 301
Exercises	p. 302
Answers to Selected Exercises	p. 312
Thermal Fluid Mechanics	p. 324
Perfect Gases	p. 325
Definition	p. 325
Gas Dynamics Equations	p. 328
Frame-Indifference Considerations	p. 330
Thermodynamical Considerations	p. 331
Entropy Formulation, Isentropic Equations	p. 334
Compressible Newtonian Fluids	p. 336
Definition	p. 337
Compressible Navier-Stokes Equations	p. 339
Frame-Indifference Considerations	p. 341
Thermodynamical Considerations	p. 343
Initial-Boundary Value Problems	p. 345
Bibliographic Notes	p. 347
Exercises	p. 348
Answers to Selected Exercises	p. 350
Thermal Solid Mechanics	p. 355
Thermoelastic Solids	p. 356
Definition	p. 356
Thermoelasticity Equations	p. 359
Thermodynamical Considerations	p. 361
Frame-Indifference Considerations	p. 364
Initial-Boundary Value Problems	p. 367
Linearization of Thermoelasticity Equations	p. 370
Preliminaries	p. 370
Linearized Equations, Thermoelasticity Tensors	p. 371
Initial-Boundary Value Problems	p. 374
Bibliographic Notes	p. 376
Exercises	p. 377
Answers to Selected Exercises	p. 381
Bibliography	p. 386
Index	p. 390
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A First Course in Continuum Mechanics

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