Continuum mechanics and linear elasticity

Preface......Page 7
Contents......Page 10
Part I Elements of Continuum Mechanics......Page 15
 1.1 Matrices......Page 16
 1.2 Vector Spaces......Page 21
 1.3 Euclidean Vector Spaces......Page 24
 1.4 Euclidean Point Spaces......Page 31
 1.5 Second-Order Tensors: Fundamentals......Page 33
 1.6 Examples of Tensors: Elementary Projections......Page 38
 1.7 Basic Properties of Tensors......Page 39
 1.8 Linear Mappings as Geometric Transformations......Page 45
 1.9 Transformation Rules for a Change of Basis......Page 48
 1.10 Higher Order Tensors......Page 51
 1.11 A Special Property: Isotropy......Page 53
 1.12 Pseudo-scalars/vectors/tensors......Page 54
 1.13 Other Uses of Vector and Tensor Products......Page 56
 1.14 Fourth-Order Tensors......Page 63
 1.15 Finite Rotations......Page 69
 1.16 On the Relationship Between Skw and mathscrV......Page 72
 1.17 More General Bases......Page 74
  1.17.1 Cylindrical Polar Coordinates......Page 76
  1.17.3 Physical Components for Vectors and Tensors......Page 78
 1.18 Invariants......Page 80
 1.19 Eigenvalues......Page 84
 1.20 Some Important Theorems......Page 89
 1.21 Projections in Lin......Page 94
 1.22 Vector and Tensor Calculus......Page 95
  1.22.1 Differential Calculus in Finite-Dimensional Euclidean Vector Spaces......Page 98
  1.22.2 Differential Operators: DIV, GRAD, CURL......Page 104
  1.22.3 Vector and Tensor Identities......Page 109
  1.22.4 Non-Cartesian Coordinates......Page 112
  1.22.5 Some Important Integral Theorems......Page 115
 1.23 Differentiation of Tensor Functions......Page 118
 1.24 Exercises......Page 125
 Bibliography......Page 133
 2.1 Deformable Bodies: Definition and Generalities......Page 134
 2.2 Examples of Deformations/Motions......Page 136
 2.3 Velocity and Acceleration Fields. Material Time Derivatives......Page 140
 2.4 The Deformation Gradient......Page 145
 2.5 Changes in Area and Volume......Page 149
 2.6 Strain Tensors......Page 152
 2.7 Examples of Particular Strain Tensors......Page 156
 2.9 The Spatial Gradient of Velocity......Page 160
 2.10 Transport Formulae......Page 164
 2.11 Exercises......Page 167
 Bibliography......Page 172
3 Balance Laws......Page 174
 3.1 The Principle of Mass Conservation......Page 175
 3.2 Body and Surface Forces......Page 179
 3.3 Global Form......Page 183
 3.4 Local Form......Page 184
 3.5 Some Technical Proofs......Page 186
 3.6 Further Properties of the Cauchy Stress Tensor......Page 191
 3.7 Particular States of Stress......Page 197
 3.8 The Piola–Kirchhoff Stress Tensors......Page 199
 3.9 Exercises......Page 202
 Bibliography......Page 209
4 Constitutive Relationships......Page 210
 4.1 The Principle of Material Frame-Indifference......Page 211
 4.2 Other Important Constitutive Principles......Page 219
 4.3 Cauchy-Elastic Materials......Page 222
 4.4 Material Symmetry......Page 223
 4.5 Hyperelastic Solids......Page 230
 4.6 Constitutive Representations for Isotropic Hyperelastic Solids......Page 236
 4.7 Internal Constraints......Page 240
 4.8 Particular Forms of the Strain-Energy Function......Page 244
 4.9 Simple Fluids......Page 245
 4.10 Exercises......Page 250
 Bibliography......Page 253
Part II Topics in Linear Elasticity......Page 254
 5.1 Introduction......Page 255
 5.2 Linearised Kinematics......Page 256
 5.3 Distortional and Spherical Strain......Page 257
 5.4 Linearised Constitutive Behaviour......Page 258
 5.5 Linearised Field Equations......Page 263
 5.6 Restrictions on the Elastic Constants......Page 267
 5.7 The Navier–Lamé Equations......Page 270
 5.8 Principle of Superposition......Page 272
 5.9 Saint-Venant's Principle......Page 274
 5.10 Worked Examples......Page 275
 5.11 Standard Simplifications......Page 283
  5.11.1 Plane Strain......Page 284
  5.11.2 Plane Stress......Page 285
  5.11.3 Antiplane Strain/Stress......Page 286
 5.12 Exercises......Page 287
 Bibliography......Page 292
 6.1 Introduction......Page 293
 6.2 Simply and Multiply Connected Domains......Page 295
 6.3 The `Incompatibility' Operator......Page 297
 6.4 Conservative Fields......Page 300
 6.5 Cesàro–Volterra Formula......Page 301
 6.6 Alternative Forms of the Compatibility Equation......Page 308
 6.7 Beltrami–Michell Equations......Page 311
 6.8 Explicit Calculations and Examples......Page 313
 6.9 Weingarten-Volterra Dislocations......Page 319
 6.10 Exercises......Page 323
 Bibliography......Page 330
 7.1 Introduction......Page 331
 7.2 Some Auxiliary Notation......Page 333
 7.3 Governing Equations......Page 335
 7.4 Circular Cylinder......Page 336
 7.5 Non-circular Cylinder......Page 339
 7.6 A Closer Look at the Torsional Rigidity......Page 346
 7.7 Prandtl Stress Function......Page 351
 7.8 Modified Stress Function......Page 354
 7.9 Multiply Connected Domains......Page 357
 7.10 The Shear Stress......Page 359
 7.11 Complex Variables Formulation......Page 362
 7.12 Worked Examples......Page 366
 7.13 Exercises......Page 377
 Bibliography......Page 381
 8.1 Introduction......Page 383
 8.2 Plane Strain......Page 385
 8.3 Plane Stress......Page 387
 8.4 Generalised Plane Stress......Page 388
  8.5.2 The Governing Equation for Φ......Page 391
  8.5.3 Physical Interpretation of the Boundary Conditions......Page 393
  8.5.4 The Displacement Field......Page 396
 8.6 Worked Examples......Page 397
 8.7 Volterra Distortions......Page 416
 8.8 Exercises......Page 420
 Bibliography......Page 430
 9.1 The Bi-harmonic Equation via Fourier Transforms......Page 431
 9.3 The Direct Approach......Page 441
 9.4 A Modification of the Method......Page 446
 9.5 The Elastic Quarter-Plane......Page 449
 9.6 Displacement Boundary-Value Problems......Page 454
  9.6.1 The Papkovitch–Neuber Representation......Page 455
  9.6.2 The Stress Tensor in Terms of ψ and B......Page 458
  9.6.3 Particular Case: Plane Elasticity......Page 459
 9.7 Exercises......Page 464
 Bibliography......Page 470
Appendix A Vector and Tensor Identities......Page 471
B.1  Cylindrical Coordinates......Page 473
B.2  Spherical Coordinates......Page 476
Appendix C Geometry of Areas......Page 479
D.1  Definitions and General Properties......Page 486
D.2  Fourier Transforms for Multivariate Functions......Page 488
D.3  The Dirac Delta Function......Page 490
D.4  The Use of Complex Variables......Page 494
D.5  Some Calculations for Chap. 9......Page 498
D.6  Useful Sine and Cosine Fourier Transforms......Page 499
E.1  Cartesian Coordinates......Page 501
E.2  Complex Variables......Page 505
E.3  Polar Coordinates......Page 514
Appendix  References......Page 522
Index......Page 524