Applied solid mechanics

452 Pages·2009·7.751 MB·English

by  Peter Howell, Gregory Kozyreff, John Ockendon

Applied Solid Mechanics
Cambridge Texts in Applied Mathematics
Editorial Board
Mark Ablowitz, University of Colorado, Boulder
S. Davis, Northwestern University
E.J. Hinch, University of Cambridge
Arieh Iserles, University of Cambridge
John Ockendon, University of Oxford
Peter Olver, University of Minnesota
Applied Solid Mechanics
Peter Howell
University of Oxford
Gregory Kozyreff
Universit´e Libre de Bruxelles
John Ockendon
University of Oxford
published by the press syndicate of the university of cambridge
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Contents
List of Illustrations page xi
Prologue xvii
1 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
1.11.1 Cartesian coordinates 19
1.11.2 Cylindrical polar coordinates 20
1.11.3 Spherical polar coordinates 22
Exercises 24
2 Linear Elastostatics 27
2.1 Introduction 27
2.2 Linear displacements 28
2.2.1 Isotropic expansion 28
2.2.2 Simple shear 30
2.2.3 Uniaxial stretching 30
2.2.4 Biaxial strain 33
2.2.5 General linear displacement 34
2.3 Antiplane strain 36
v
vi Contents
2.4 Torsion 38
2.5 Multiply-connected domains 42
2.6 Plane strain 46
2.6.1 Definition 46
2.6.2 The Airy stress function 46
2.6.3 Boundary conditions 48
2.6.4 Plane strain in a disc 50
2.6.5 Plane strain in an annulus 52
2.6.6 Plane strain in a rectangle 55
2.6.7 Plane strain in a semi-infinite strip 57
2.6.8 Plane strain in a half-space 61
2.6.9 Plane strain with a body force 64
2.7 Compatibility 66
2.8 Generalised stress functions 69
2.8.1 General observations 69
2.8.2 Plane strain revisited 70
2.8.3 Plane stress 72
2.8.4 Axisymmetric geometry 74
2.8.5 The Galerkin representation 76
2.8.6 Papkovich–Neuber potentials 77
2.8.7 Maxwell and Morera potentials 79
2.9 Singular solutions in elastostatics 81
2.9.1 The delta function 81
2.9.2 Point and line forces 83
2.9.3 The Green’s tensor 85
2.9.4 Point incompatibility 89
Exercises 91
3 Linear Elastodynamics 100
3.1 Introduction 100
3.2 Normal modes and plane waves 101
3.2.1 Normal modes 101
3.2.2 Waves in the frequency domain 107
3.2.3 Scattering 109
3.2.4 P-waves and S-waves 110
3.2.5 Mode conversion in plane strain 112
3.2.6 Love waves 114
3.2.7 Rayleigh waves 118
3.3 Dynamic stress functions 119
3.4 Waves in cylinders and spheres 121
Contents vii
3.4.1 Waves in a circular cylinder 121
3.4.2 Waves in a sphere 125
3.5 Initial-value problems 129
3.5.1 Solutions in the time domain 129
3.5.2 Fundamental solutions 130
3.5.3 Characteristics 134
3.6 Moving singularities 135
3.7 Concluding remarks 140
Exercises 141
4 Approximate Theories 147
4.1 Introduction 147
4.2 Longitudinal displacement of a bar 148
4.3 Transverse displacements of a string 149
4.4 Transverse displacements of a beam 151
4.4.1 Derivation of the beam equation 151
4.4.2 Boundary conditions 153
4.4.3 Compression of a beam 154
4.4.4 Waves on a beam 155
4.5 Linear rod theory 156
4.6 Linear plate theory 160
4.6.1 Derivation of the plate equation 160
4.6.2 Boundary conditions 162
4.6.3 Simple solutions of the plate equation 165
4.6.4 An inverse plate problem 166
4.6.5 More general in-plane stresses 167
4.7 Von K´arm´an plate theory 169
4.7.1 Assumptions underlying the theory 169
4.7.2 The strain components 169
4.7.3 The Von K´arm´an equations 172
4.8 Weakly curved shell theory 174
4.8.1 Strain in a weakly curved shell 174
4.8.2 Linearised equations for a weakly curved shell 176
4.8.3 Solutions for a thin shell 177
4.9 Nonlinear beam theory 183
4.9.1 Derivation of the model 183
4.9.2 Example: deflection of a diving board 186
4.9.3 Weakly nonlinear theory and buckling 188
4.10 Nonlinear rod theory 192
4.11 Geometrically nonlinear wave propagation 194
viii Contents
4.11.1 Nonlinearity and solitons 194
4.11.2 Gravity-torsional waves 195
4.11.3 Travelling waves on a beam 196
4.11.4 Weakly nonlinear waves on a beam 198
4.12 Concluding remarks 201
Exercises 202
5 Nonlinear Elasticity 213
5.1 Introduction 213
5.2 Stress and strain revisited 214
5.2.1 Deformation and strain 214
5.2.2 The Piola–Kirchhoff stress tensors 216
5.2.3 The momentum equation 217
5.2.4 Example: one-dimensional nonlinear elasticity 218
5.3 The constitutive relation 219
5.3.1 Polar decomposition 219
5.3.2 Strain invariants 220
5.3.3 Frame indifference and isotropy 222
5.3.4 The energy equation 224
5.3.5 Hyperelasticity 226
5.3.6 Linear elasticity 228
5.3.7 Incompressibility 229
5.3.8 Examples of constitutive relations 230
5.4 Examples 231
5.4.1 Principal stresses and strains 231
5.4.2 Biaxial loading of a square membrane 232
5.4.3 Blowing up a balloon 233
5.4.4 Cavitation 235
5.5 Conclusion 237
Exercises 237
6 Asymptotic Analysis 243
6.1 Introduction 243
6.2 The linear plate equation 246
6.2.1 Nondimensionalisation and scaling 246
6.2.2 Dimensionless equations 247
6.2.3 Leading-order equations 249
6.3 Boundary conditions and St Venant’s principle 251
6.3.1 Boundary layer scalings 252
6.3.2 Equations and boundary conditions 253
6.3.3 Solvability conditions 255
Contents ix
6.3.4 Asymptotic expansions 256
6.4 The von K´arm´an plate equations and weakly curved shells 260
6.4.1 Background 260
6.4.2 Scalings 261
6.4.3 Leading-order equations 262
6.4.4 Equations for a weakly curved shell 265
6.5 The Euler–Bernoulli plate equations 267
6.5.1 Dimensionless equations 267
6.5.2 Asymptotic structure of the solution 269
6.5.3 Leading-order equations 269
6.5.4 Longitudinal stretching of a plate 271
6.6 The linear rod equations 273
6.7 Linear shell theory 274
6.7.1 Geometry of the shell 274
6.7.2 Dimensionless equations 275
6.7.3 Leading-order equations 276
6.8 Conclusions 279
Exercises 279
7 Fracture and Contact 283
7.1 Introduction 283
7.2 Static Brittle Fracture 284
7.2.1 Physical background 284
7.2.2 Mode III cracks 285
7.2.3 Mathematical methodologies for crack problems 290
7.2.4 Mode II cracks 292
7.2.5 Mode I cracks 299
7.2.6 301
7.2.7 Dynamic fracture 304
7.3 Contact 306
7.3.1 Contact of elastic strings 306
7.3.2 Other thin solids 310
7.3.3 Smooth contact in plane strain 313
7.4 Conclusions 316
Exercises 317
8 Plasticity 323
8.1 Introduction 323
8.2 Models for granular material 325
8.2.1 Static behaviour 325
8.2.2 Granular flow 326
x
8.3 Dislocation theory 332
8.4 Perfect plasticity theory for metals 338
8.4.1 Torsion problems 338
8.4.2 Plane strain 343
8.4.3 Three-dimensional yield conditions 346
8.5 Plastic flow 351
8.6 Simultaneous Elasticity and Plasticity 357
9 More General Theories 367
9.1 Introduction 367
9.2 Viscoelasticity 368
9.2.1 Introduction 368
9.2.2 Springs and dashpots 369
9.2.3 Three-dimensional linear viscoelasticity 373
9.2.4 Large-strain viscoelasticity 375
9.3 Thermoelasticity 378
9.4 Composite Materials and Homogenisation 381
9.4.1 One-dimensional homogenisation 381
9.4.2 Two-dimensional homogenisation 385
9.4.3 Three-dimensional homogenisation 391
9.5 Poroelasticity 393
9.6 Anisotropy 398
9.7 Epilogue 402
Epilogue 410
Appendix 412
Appendix 412
Appendix 2 Elementary differential geometry 415
Appendix 3 Orthogonal curvilinear coordinates 419
BibliographyOrthogonal curvilinear coordinatesA3.8 Time-dependent co-
ordinatesBIBLIOGRAPHYBIBLIOGRAPHY 433