Boundary element analysis of cracks in shear deformable plates and shells

Illustrated throughout, this book presents a new set of boundary element formulations for the solution of bending problems in plates and shells. The book is part of the Topics in Engineering series.

Preface	p. ix
List of Notations	p. xv
Chapter 1 Introduction	p. 1
1.1 General	p. 1
1.2 Numerical Models for Fracture Mechanics Analysis of Plates and Shells	p. 5
1.3 Overview of the Present Work	p. 7
Chapter 2 Basic Concepts	p. 9
2.1 Introduction	p. 9
2.2 Basic Definitions of Shallow Shells	p. 9
2.3 Governing Equations of Shallow Shells	p. 12
2.3.1 Strain-displacement relationships	p. 12
2.3.2 Stress resultants and stress couples	p. 14
2.3.3 Equilibrium equations	p. 17
2.3.4 Stress resultant-strain relationships	p. 19
2.3.5 Stress resultant-displacement relationships	p. 20
2.3.6 Equilibrium equations in term of displacements	p. 20
2.4 Governing Equations of Flat Plates	p. 21
2.4.1 Stress resultants and stress couples	p. 21
2.4.2 Strain-displacement relationships	p. 22
2.4.3 Equilibrium equations	p. 22
2.4.4 Stress resultant-strain relationships	p. 23
2.4.5 Equilibrium equations in terms of displacements	p. 24
2.5 Basic Concepts of Fracture Mechanics	p. 24
2.5.1 Crack tip elastic fields	p. 25
2.5.2 Stress resultant intensity factors evaluation	p. 27
2.5.3 J-integral	p. 29
2.5.4 Fatigue crack growth	p. 30
2.5.5 Crack growth direction	p. 31
2.6 Note on Classical Plate and Shell Theories	p. 31
2.7 Summary	p. 33
Chapter 3 The Boundary Element Method	p. 35
3.1 Introduction	p. 35
3.2 The Integral Representations	p. 37
3.2.1 Rotations and out-of-plane displacement integral representations	p. 39
3.2.2 Fundamental solutions	p. 43
3.2.3 In-plane displacements integral representations	p. 44
3.2.4 Fundamental solutions	p. 47
3.3 Boundary Integral Equations	p. 47
3.4 Numerical Implementation	p. 50
3.4.1 Discretisation	p. 50
3.4.2 Treatment of singularities	p. 54
3.4.3 Boundary conditions	p. 55
3.5 Transformation of Domain Integrals	p. 56
3.6 Internal Stress Resultants	p. 59
3.7 Evaluation of Boundary Stress Resultants	p. 61
3.8 Boundary Integrals of Shear Deformable Plate Bending and Two-Dimensional Plane Stress	p. 63
3.9 Numerical Examples	p. 64
3.9.1 Circular shallow spherical shells: uniformly loaded	p. 64
3.9.2 Circular shallow spherical shells with a hole in the centre	p. 68
3.9.3 A simply supported square shallow spherical shell: uniformly loaded	p. 70
3.9.4 A simply supported square shallow cylindrical shell: uniformly loaded	p. 76
3.9.5 A rectangular plate with a rectangular opening	p. 78
3.10 Summary	p. 81
Chapter 4 Hypersingular Integral Equations	p. 83
4.1 Introduction	p. 83
4.2 Hypersingular Integral Equations for Shear Deformable Shallow Shells	p. 83
4.3 The Traction Integral Equations	p. 86
4.4 Evaluation of Domain Integrals Using the Dual Reciprocity Technique	p. 87
4.5 Numerical Implementation	p. 89
4.5.1 Treatment of singularities	p. 91
4.6 Traction Integral Equations of Shear Deformable Plate Bending and Two-Dimensional Plane Stress	p. 93
4.7 Summary	p. 94
Chapter 5 The Dual Boundary Element Method	p. 97
5.1 Introduction	p. 97
5.2 The Dual Boundary Integral Equations	p. 97
5.2.1 Shallow shell problems	p. 98
5.2.2 Plate bending and tension problems	p. 105
5.3 Numerical Implementation	p. 106
5.3.1 Crack modelling strategy	p. 106
5.3.2 Special crack-tip elements	p. 108
5.3.3 Discretisation strategy	p. 108
5.3.4 Modelling consideration of the dual reciprocity technique	p. 109
5.3.5 Treatment of the singularities	p. 110
5.4 Stress Resultant Intensity Factors Evaluation	p. 110
5.4.1 The crack surface displacements extrapolation technique (CSDE)	p. 111
5.4.2 The J-integral technique	p. 112
5.5 Numerical Examples	p. 115
5.5.1 A rectangular plate with a centre crack loaded by bending and tension	p. 115
5.5.2 A simply supported square plate with a centre crack: uniform pressure	p. 118
5.5.3 A plate with symmetric double edge cracks	p. 120
5.5.4 A plate with an edge crack	p. 123
5.5.5 A single crack emanating from a hole in a finite width plate	p. 123
5.5.6 Two symmetric cracks emanating from a hole in a finite width plate	p. 125
5.5.7 An infinite plate with a slant centre crack loaded by bending and bi-axial tension	p. 130
5.5.8 Clamped and simply supported square spherical shells with a centre crack: uniformly loaded	p. 133
5.5.9 Clamped and simply supported square cylindrical shells with a centre crack: uniformly loaded	p. 140
5.5.10 Symmetric cracks emanating from a hole in a square cylindrical shell	p. 148
5.6 Summary	p. 148
Chapter 6 Crack Growth Simulation	p. 151
6.1 Introduction	p. 151
6.2 Crack Growth Simulation	p. 152
6.2.1 Crack propagation direction	p. 153
6.2.2 Fatigue life prediction	p. 155
6.2.3 Multiple cracks growth simulation	p. 156
6.2.4 Computational procedure	p. 157
6.3 Numerical examples	p. 158
6.3.1 A rectangular plate with a centre crack loaded by bending and tension	p. 158
6.3.2 An infinite plate with a slant centre crack loaded by bending and bi-axial tension	p. 160
6.3.3 A rectangular plate with symmetric edge cracks loaded by torsion and tension	p. 162
6.3.4 Multiple site damage problem of a plate with three holes	p. 164
6.3.5 Crack propagation in a cylindrical shell	p. 169
6.3.6 Crack propagation in a spherical shell	p. 172
6.4 Summary	p. 177
Chapter 7 A Multi-Domain BEM Formulation for Assembled Plate-Structures	p. 179
7.1 Introduction	p. 179
7.2 Multi-Region Formulation of Plate Structure	p. 179
7.3 Numerical Examples	p. 182
7.3.1 A cantilever plate with variable thickness and modulus of elasticity	p. 183
7.3.2 An L-shape plate structure	p. 184
7.3.3 A cantilever box subjected to bending and torsion	p. 186
7.3.4 A quarter of a long cylinder, clamped and loaded by internal pressure	p. 189
7.3.5 A cantilever box with a centre crack on the upper skin, subjected to bending	p. 190
7.4 Summary	p. 193
Chapter 8 Conclusions and Future Work	p. 195
8.1 Conclusions	p. 195
8.2 Future Research	p. 197
References	p. 198
Appendix A Evaluation of Modified Bessel Functions	p. 211
A.1 Polynomial Approximations of I[subscript 0] (x)	p. 211
A.2 Polynomial Approximations of I[subscript 1] (x)	p. 212
A.3 Polynomial Approximations of K[subscript 0] (x)	p. 213
A.4 Polynomial Approximations of K[subscript 1] (x)	p. 213
Appendix B The Limits and Jump Terms of the Integral Equations for Shallow Shells	p. 215
B.1 The Displacement Integral Equations	p. 215
B.2 The Bending Stress Resultant Integral Equations	p. 217
B.3 The Shear Stress Resultant Integral Equation	p. 220
B.4 The Membrane Stress Resultant Integral Equations	p. 222
Appendix C Treatment of Singularities	p. 227
C.1 Bi-cubic Nonlinear Coordinate Transformation	p. 227
C.2 Singularity SubstractionMethod	p. 228
C.2.1 Strongly singular integrals	p. 228
C.2.2 Hypersingular integrals	p. 232
C.3 Analytical Integration of Membrane Fundamental Solutions	p. 237
C.4 Triangle to Square Transformation	p. 238
Appendix D Particular solutions	p. 241
D.1 Particular solutions for two-dimensional plane stress	p. 241
D.2 Particular solutions for plate bending	p. 243
Appendix E Decompositions for the J- Integral Technique	p. 247

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