| Date: 14 April 2011, 07:18
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The world around us, natural or man-made, is built and held together by solid materials. Understanding their behaviour is the task of solid mechanics, which is in turn applied to many areas, from earthquake mechanics to industry, construction to biomechanics. The variety of materials (metals, rocks, glasses, sand, flesh and bone) and their properties (porosity, viscosity, elasticity, plasticity) is reflected by the concepts and techniques needed to understand them: a rich mixture of mathematics, physics and experiment. These are all combined in this unique book, based on years of experience in research and teaching. Starting from the simplest situations, models of increasing sophistication are derived and applied. The emphasis is on problem-solving and building intuition, rather than a technical presentation of theory. The text is complemented by over 100 carefully-chosen exercises, making this an ideal companion for students taking advanced courses, or those undertaking research in this or related disciplines.
Contents
List of illustrations page viii
Prologue xiii
Modelling solids 1
1.1 Introduction 1
1.2 Hooke’s law 2
1.3 Lagrangian and Eulerian coordinates 3
1.4 Strain 4
1.5 Stress 7
1.6 Conservation of momentum 10
1.7 Linear elasticity 11
1.8 The incompressibility approximation 13
1.9 Energy 14
1.10 Boundary conditions and well-posedness 16
1.11 Coordinate systems 19
Exercises 24
Linear elastostatics 28
2.1 Introduction 28
2.2 Linear displacements 29
2.3 Antiplane strain 37
2.4 Torsion 39
2.5 Multiply-connected domains 42
2.6 Plane strain 47
2.7 Compatibility 68
2.8 Generalised stress functions 70
2.9 Singular solutions in elastostatics 82
2.10 Concluding remark 93
Exercises 93
Linear elastodynamics 103
3.1 Introduction 103
3.2 Normal modes and plane waves 104
3.3 Dynamic stress functions 121
3.4 Waves in cylinders and spheres 124
3.5 Initial-value problems 132
3.6 Moving singularities 138
3.7 Concluding remarks 143
Exercises 143
Approximate theories 150
4.1 Introduction 150
4.2 Longitudinal displacement of a bar 151
4.3 Transverse displacements of a string 152
4.4 Transverse displacements of a beam 153
4.5 Linear rod theory 158
4.6 Linear plate theory 162
4.7 Von K?arm?an plate theory 172
4.8 Weakly curved shell theory 177
4.9 Nonlinear beam theory 187
4.10 Nonlinear rod theory 195
4.11 Geometrically nonlinear wave propagation 198
4.12 Concluding remarks 204
Exercises 205
Nonlinear elasticity 215
5.1 Introduction 215
5.2 Stress and strain revisited 216
5.3 The constitutive relation 221
5.4 Examples 233
5.5 Concluding remarks 239
Exercises 239
Asymptotic analysis 245
6.1 Introduction 245
6.2 Antiplane strain in a thin plate 246
6.3 The linear plate equation 248
6.4 Boundary conditions and Saint-Venant’s principle 253
6.5 The von K?arm?an plate equations 261
6.6 The Euler–Bernoulli plate equations 267
6.7 The linear rod equations 273
6.8 Linear shell theory 278
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