Introduction to Mathematical Elasticity

Contents......Page 14
Foreword......Page 6
Preface......Page 8
Some Notation......Page 12
1.1 Orientation......Page 18
1.2 Some Words on the Fundamentals of Our Subject......Page 19
1.3 Metric Spaces and Spaces of Particles......Page 21
1.4 Vectors and Vector Spaces......Page 25
1.5 Normed Spaces and Inner Product Spaces......Page 28
1.6 Forces......Page 33
1.7 Equilibrium and Motion of a Rigid Body......Page 38
1.8 D\'Alembert\'s Principle......Page 40
1.9 The Motion of a System of Particles......Page 42
1.10 The Rigid Body......Page 48
1.11 Motion of a System of Particles; Comparison of Trajectories; Notion of Operator......Page 50
1.12 Matrix Operators and Matrix Equations......Page 57
1.13 Complete Spaces......Page 61
1.14 Completion Theorem......Page 65
1.15 Lebesgue Integration and the Lp Spaces......Page 71
1.16 Orthogonal Decomposition of Hilbert Space......Page 77
1.17 Work and Energy......Page 80
1.18 Virtual Work Principle......Page 84
1.19 Lagrange\'s Equations of the Second Kind......Page 87
1.20 Problem of Minimum of a Functional......Page 91
1.21 Hamilton\'s Principle......Page 100
1.22 Energy Conservation Revisited......Page 102
2.2 Two Main Principles of Equilibrium and Motion for Bodies in Continuum Mechanics......Page 106
2.3 Equilibrium of a Spring......Page 108
2.4 Equilibrium of a String......Page 112
2.5 Equilibrium Boundary Value Problems for a String......Page 117
2.6 Generalized Formulation of the Equilibrium Problem for a String......Page 122
2.7 Virtual Work Principle for a String......Page 125
2.8 Riesz Representation Theorem......Page 129
2.9 Generalized Setup of the Dirichlet Problem for a String......Page 132
2.10 First Theorems of Imbedding......Page 133
2.11 Generalized Setup of the Dirichlet Problem for a String, Continued......Page 137
2.12 Neumann Problem for the String......Page 139
2.13 The Generalized Solution of Linear Mechanical Problems and the Principle of Minimum Total Energy......Page 143
2.14 Nonlinear Model of a Membrane......Page 145
2.15 Linear Membrane Theory: Poisson\'s Equation......Page 148
2.16 Generalized Setup of the Dirichlet Problem for a Linear Membrane......Page 149
2.17 Other Membrane Equilibrium Problems......Page 162
2.18 Banach\'s Contraction Mapping Principle......Page 168
3.1 Introduction......Page 174
3.2 An Elastic Bar Under Stretching......Page 175
3.3 Bending of a beam......Page 185
3.4 Generalized Solutions to the Equilibrium Problem for a Beam......Page 192
3.5 Generalized Setup: Rough Qualitative Discussion......Page 196
3.6 Pressure and Stresses......Page 198
3.7 Vectors and Tensors......Page 205
3.8 The Cauchy Stress Tensor, Continued......Page 213
3.9 Basic Tensor Calculus in Curvilinear Coordinates......Page 219
3.10 Euler and Lagrange Descriptions of Continua......Page 224
3.11 Strain Tensors......Page 225
3.12 The Virtual Work Principle......Page 231
3.13 Hooke\'s Law in Three Dimensions......Page 235
3.14 The Equilibrium Equations of Linear Elasticity in Displacements......Page 238
3.15 Virtual Work Principle in Linear Elasticity......Page 241
3.16 Generalized Setup of Elasticity Problems......Page 244
3.17 Existence Theorem for an Elastic Body......Page 248
3.18 Equilibrium of a Free Elastic Body......Page 249
3.19 Variational Methods for Equilibrium Problems......Page 252
3.21 Countable Sets and Separable Spaces......Page 260
3.22 Fourier Series......Page 262
3.23 Problem of Vibration for Elastic Structures......Page 266
3.24 Self-Adjointness of A and Its Consequences......Page 269
3.25 Compactness of A......Page 272
3.26 Riesz–Fredholm Theory for a Linear, Self-Adjoint, Compact Operator in a Hilbert Space......Page 279
3.27 Weak Convergence in Hilbert Space......Page 284
3.28 Completeness of the System of Eigenvectors of a Self- Adjoint, Compact, Strictly Positive Linear Operator......Page 289
3.29 Other Standard Models of Elasticity......Page 294
Appendix A Hints for Selected Exercises......Page 298
Bibliography......Page 310
Index......Page 312