Numerical Fracture Mechanics

The purpose of this book is to present, describe and demonstrate the use of numerical methods in solving crack problems in fracture mechanics. The text concentrates, to a large extent, on the application of the Boundary Element Method (BEM) to fracture mechanics, although an up-to-date account of recent advances in other numerical methods such as the Finite Element Method is also presented. The book is an integrated presentation of modem numerical fracture mechanics, it contains a compilation of the work of many researchers as well as accounting for some of authors' most recent work on the subject. It is hoped that this book will bridge the gap that exists between specialist books on theoretical fracture mechanics on one hand, and texts on numerical methods on the other. Although most of the methods presented are the latest developments in the field of numerical fracture mechanics, the authors have also included some simple techniques which are essential for understanding the physical principles that govern crack problems in general. Different numerical techniques are described in detail and where possible simple examples are included, as well as test results for more complicated problems. The book consists of six chapters. The first chapter initially describes the historical development of theoretical fracture mechanics, before proceeding to present the basic concepts such as energy balance, stress intensity factors, residual strength and fatigue crack growth as well as briefly describing the importance of stress intensity factors in corrosion and residual stress cracking.

1. Basic Fracture Mechanics.- 1.1 Introduction.- 1.2 Energy balance.- 1.3 Stress intensity factors.- 1.4 Residual strength.- 1.5 Fatigue crack growth.- 1.6 Stress corrosion cracking.- 1.7 Limitations (plasticity).- 1.8 Residual stress effects.- 1.9 Concluding remarks.- 1.10 Bibliography.- 1.11 References.- 2. Development of Fracture Mechanics Analysis.- 2.1 Introduction.- 2.2 Basic equations of elasticity.- 2.3 Airy stress functions.- 2.4 Muskhelishvili’s complex functions.- 2.5 Complex stress function analysis.- 2.6 Westergaard’s stress function.- 2.7 William’s eigenfunction series expansion.- 2.8 Papkovich-Neuber potentials.- 2.9 The energy principle.- 2.10 Elastic-plastic fracture.- 2.11 Three-dimensional stress field.- 2.12 Mixed mode fracture.- 2.13 Concluding remarks.- 2.14 References.- 3. Numerical Methods in Linear Elastic Fracture Mechanics.- 3.1 Introduction.- 3.2 Superposition.- 3.3 Stress concentrations.- 3.4 Local stress distributions.- 3.5 Green’s functions/Weight functions.- 3.6 Compounding method.- 3.7 Boundary collocation methods.- 3.8 Integral transforms/continuous dislocations.- 3.9 Body force method.- 3.10 Method of lines.- 3.11 Edge function method.- 3.12 Finite element method.- 3.13 Alternating technique.- 3.14 Concluding remarks.- 3.15 References.- 4. The Boundary Element Method.- 4.1 Introduction.- 4.2 The boundary element formulation in elasticity.- 4.3 Fundamental solutions.- 4.4 Numerical discretization.- 4.5 Assembly of system of equations.- 4.6 Stress and displacement at interior points.- 4.7 Numerical evaluation of coefficient matrices.- 4.8 Evaluation of boundary stresses.- 4.9 Multi-domain formulation.- 4.10 Body force.- 4.11 Concluding remarks.- Appendix A Betti’s reciprocal theorem.- Appendix B Evaluation of the free term Cijfor smooth boundaries.- Appendix C Evaluation of local boundary stresses.- 4.12 References.- 5. Application of Boundary Element Methods to Fracture Mechanics.- 5.1 Introduction.- 5.2 Difficulties in crack modelling.- 5.3 Crack-tip elements.- 5.4 Crack Green’s function.- 5.5 Displacement discontinuity method.- 5.6 Energy method.- 5.7 J-integral.- 5.8 Subtraction of singularity.- 5.9 Concluding remarks.- 5.10 References.- 6. Weight Function Techniques.- 6.1 Introduction.- 6.2 Basic principles of Green’s functions.- 6.3 Stress intensity factors as Green’s functions.- 6.4 Systematic use of Green’s functions.- 6.5 Available Green’s functions.- 6.6 Weight functions.- 6.7 Numerical weight functions.- 6.8 Approximate weight functions.- 6.9 Application of the BEM to weight functions.- 6.10 Weight functions for strip yield cracks.- 6.11 Weight functions for residual stress fields.- 6.12 Concluding remarks.- Appendix A Closed form weight functions.- Appendix B Three-dimensional stress and displacement fields for a semi-infinite crack.- Appendix C Near-tip displacement and stress fields.- 6.13 References.