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Chapter 1 Mathematical Prerequisites 1

1.1 Index Notation 1

1.1.1 Range convention 1

1.1.2 Summation convention 2

1.1.3 The Kronecker delta 3

1.1.4 The permutation symbol 4

1.2 Vector Operations and Some Useful Integral Theorems 5

1.2.1 The scalar product of two vectors 6

1.2.2 The vector product of two vectors 6

1.2.3 The scalar triple product 7

1.2.4 The gradient of a scalar function 7

1.2.5 The divergence of a vector function 8

1.2.6 The curl of a vector function 9

1.2.7 Laplacian of a scalar function 9

1.2.8 Divergence theorem（Gauss's theorem） 9

1.2.9 Stokes'theorem 10

1.2.10 Green's theorem 10

1.3 Cartesian Tensors and Transformation Laws 13

Problems 1 17

Chapter 2 Analysis of Stress 20

2.1 Continuum 20

2.2 Forces 21

2.3 Cauchy's Formula 21

2.4 Equations of Equilibrium 24

2.5 Stress as a Second-order Tensor 28

2.6 Principal Stresses 30

2.7 Maximum Shears 33

2.8 Yields Criteria 35

Problems 2 36

Chapter 3 Analysis of Strain 40

3.1 Differential Element Considerations 40

3.2 Linear Deformation and Strain 43

3.3 Strain as a Second-order Tensor 46

3.4 Principal Strains and Strain Measurement 46

3.5 Compatibility Equations 48

3.6 Finite Deformation 52

Problems 3 54

Chapter 4 Linear Elastic Materials,Framework of Problems of Elasticity 56

4.1 Introduction 56

4.2 Uniaxial Stress-Strain Relations of Linear Elastic Materials 57

4.3 Hooke's Law 59

4.3.1 Isotropic materials 59

4.3.2 Orthotropic materials 60

4.3.3 Transversely isotropic materials 61

4.4 Generalized Hooke's Law 62

4.5 Elastic Constants as Components of a Fourth-order Tensor 65

4.6 Elastic Symmetry 65

4.6.1 One plane of elastic symmetry（monoclinic material） 66

4.6.2 Two planes of elastic symmetry 67

4.6.3 Three planes of elastic symmetry（orthotropic material） 68

4.6.4 An axis of elastic symmetric（rotational symmetry） 68

4.6.5 Complete symmetry（spherical symmetry） 69

4.7 Elastic Moduli 70

4.7.1 Simple tension 70

4.7.2 Pure shear 71

4.7.3 Hydrostatics pressure 71

4.8 Formulation of Problems of Elasticity 73

4.9 Principle of Superposition 74

4.10 Uniqueness of Solution 75

4.11 Solution Approach 76

Problems 4 78

Chapter 5 Some Elementary Problems 81

5.1 Extension of Prismatic Bars 81

5.2 A Column under Its Own Weight 83

5.3 Pure Bending of Beams 85

5.4 Torsion of a Shaft of Circular Cross Section 88

Problems 5 89

Chapter 6 Two-dimensional Problems 92

6.1 Plane Strain 92

6.2 Plane Stress 93

6.3 Connection between Plane Strain and Plane Stress 94

6.4 Stress Function Formulation 95

6.5 Plane Problems in Cartesian Coordinates 96

6.5.1 Polynomial solutions 96

6.5.2 Product solutions 99

6.6 Plane Problems in Polar Coordinates 104

6.6.1 Basic equations in polar coordinates 104

6.6.2 Stress function in polar coordinates 105

6.6.3 Problems with axial symmetry 106

6.6.4 Problems without axial symmetry 112

6.7 Wedge Problems 120

6.7.1 A wedge subjected to a couple at the apex 122

6.7.2 A wedge subjected to concentrated loads at the apex 123

6.7.3 A wedge subjected to uniform edge loads 124

6.8 Half-plane Problems 125

6.9 Crack Problems 130

Problems 61 34

Chapter 7 Torsion and Flexure of Prismatic Bars 141

7.1 Saint-Venant's Problem 141

7.2 Torsion of Prismatic Bars 143

7.2.1 Displacement formulation 143

7.2.2 Stress function formulation 146

7.2.3 Illustrative examples 148

7.3 Membrane Analogy 153

7.4 Torsion of Multiply Connected Bars 157

7.5 Torsion of Thin-walled Tubes 160

7.6 Flexure of Beams Subjected to Transverse End Loads 162

7.6.1 Formularion and solution 162

7.6.2 Illustrative examples 166

Problems 7 172

Chapter 8 Complex Variable Methods 175

8.1 Summary of Theory of Complex Variables 175

8.1.1 Complex functions 175

8.1.2 Some results from theory of analytic functions 176

8.1.3 Conformal mapping 178

8.2 Plane Problems of Elasticity 181

8.2.1 Complex formulation of two-dimensional elasticity 181

8.2.2 Illustrative examples 186

8.2.3 Complex representation with conformal mapping 192

8.2.4 Illustrative examples 195

8.3 Problems of Saint-Venant's Torsion 199

8.3.1 Complex formulation with conformal mapping 199

8.3.2 Illustrative examples 203

Problems 82 6

Chapter 9 Three-dimensional Problems 209

9.1 Introduction 209

9.2 Displacement Potential Formulation 210

9.2.1 Galerkin vector 211

9.2.2 Papkovich-Neuber functions 213

9.2.3 Harmonic and biharmonic functions 215

9.3 Some Basic Three-dimensional Problems 216

9.3.1 Kelvin's problem 216

9.3.2 Boussinesq's problem 220

9.3.3 Cerruti's problem 222

9.3.4 Mindlin's problem 224

9.4 Problems in Spherical Coordinates 225

9.4.1 Hollow sphere under internal and external pressures 226

9.4.2 Spherical harmonics 227

9.4.3 Axisymmetric problems of hollow spheres 231

9.4.4 Extension of an infinite body with a spherical cavity 233

Problems 92 35

Chapter 10 Variational Principles of Elasticity and Applications 238

10.1 Introduction 238

10.1.1 The shortest distance problem 238

10.1.2 The body of revolution problem 238

10.1.3 The brachistochrone problem（the shortest time problem） 239

10.2 Variation Operation 240

10.3 Minimization of Variational Functionals 243

10.4 Illustrative Examples 246

10.5 Principle of Virtual Work 251

10.6 Principle of Minimum Potential Energy 253

10.7 Principle of Minimum Complementary Energy 259

10.8 Reciprocal Theorem 261

10.9 Hamilton's Principle of Elastodynamics 263

10.10 Vibration of Beams 265

10.11 Bending and Stretching of Thin Plates 271

10.12 Equivalent Variational Problems 277

10.12.1 Self-adjoint ordinary differential equations 277

10.12.2 Self-adjoint partial differential equations 280

10.13 Direct Methods of Solution 285

10.13.1 The Ritz method 285

10.13.2 The Galerkin method 286

10.14 Illustrative Examples 288

10.15 Closing Remarks 296

Problems 10 296

Chapter 11 State Space Approach 301

11.1 Introduction 301

11.2 Solution of Systems of Linear Differential Equations 302

11.2.1 Solution of homogeneous system 303

11.2.2 Solution of nonhomogeneous system 309

11.3 State Space Formalism of Linear Elasticity 310

11.3.1 State variable representation of basic equations 310

11.3.2 Hamiltonian formulation 312

11.3.3 Explicit state equation and output equation 314

11.4 Analysis of Stress Decay in Laminates 316

11.5 Application to Two-dimensional Problems 319

11.5.1 Infinite-plane Green's function 320

11.5.2 Half-plane Green's functions 321

11.5.3 A half-plane under line load 323

11.5.4 Extension of infinite plate with an elliptical hole 323

11.6 Symplectic Characteristics of Hamiltonian System 325

11.6.1 Simple and semisimple systems 326

11.6.2 Non-semisimple system 327

11.7 Application to Three-dimensional Elasticity 328

Problems 11 330

References 333

Appendix A Basic Equations in Cylindrical and Spherical Coordinates 335

Appendix B Fourier Series 339

Appendix C Product Solution of Biharmonic Equation in Cartesian Coordinates 341

Appendix D Product Solution of Biharmonic Equation in Polar Coordinates 342

Index 345

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