Table Of ContentSOLUTION OF CRACK PROBLEMS
SOLID MECHANICS AND ITS APPLICATIONS
Volume 44
Series Editor: G.M.L. GLADWELL
Solid Mechanics Division, Faculty ofE ngineering
University ofW aterloo
Waterloo, Ontario, CanadaN2L3Gl
Aims and Scope of the Series
The fundamental questions arising in mechanics are: Why?, How?, and How much?
The aim of this series is to provide lucid accounts written by authoritative research
ers giving vision and insight in answering these questions on the subject of
mechanics as it relates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it
includes the foundation of mechanics; variational formulations; computational
mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations
of solids and structures; dynamical systems and chaos; the theories of elasticity,
plasticity and viscoelasticity; composite materials; rods, beams, shells and
membranes; structural control and stability; soils, rocks and geomechanics;
fracture; tribology; experimental mechanics; biomechanics and machine design.
The median level of presentation is the first year graduate student. Some texts are
monographs defining the current state of the field; others are accessible to final
year undergraduates; but essentially the emphasis is on readability and clarity.
For a list ofr elated mechanics titles, see final pages.
Solution of
Crack Problems
The Distributed Dislocation Technique
by
D.A.HILLS
Department ofE ngineering Science,
University of Oxford,
Oxford, U.K.
P. A. KELLY
The Oxford Orthopaedic Engineering Centre,
Nuffield Orthopaedic Centre,
Oxford, U.K.
D.N.DAI
Department of Engineering Science,
University of Oxford,
Oxford, U.K.
and
A. M. KORSUNSKY
Department ofM aterials Science and Metallurgy,
University of Cambridge,
Cambridge, U.K.
Springer-Science+Business Media, B.V.
Library of Congress Cataloging-in-Publication Data
Solution of crack problems; the distributed disc location technique I
by D. A. Hi I Is. .. [et a 1. l.
p. cm. -- (Solid lIIechanics and its appl ications ; v. 44)
Includes bibliographical references and index.
1. Fracture mechanics. 2. Stresses and strains--Mathematical
models. I. Hills, D. A. <David Anthony), 1955- II. Series.
TA409.S65 1996
620.1' 126--dc20 95-47426
ISBN 978-90-481-4651-2 ISBN 978-94-015-8648-1 (eBook)
DOI 10.1007/978-94-015-8648-1
Printed on acid-free paper
All Rights Reserved
© 1996 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1996.
Softcover reprint of the hardcover 1s t edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
to Edwin Hills and Alphonsus Kelly
Contents
Preface Xl
1 Introduction to Fracture Mechanics 1
1.1 Designing Against Failure ..... . . . . . . . . 1
1.2 Review of Linear Elastic Fracture Mechanics . . . 3
1.2.1 Asymptotics of Plane Elasticity Problems 5
1.2.2 Westergaard Solution ............ 8
1.2.3 Crack Tip Mechanics . . . . . . . . . . . . 13
1.2.4 Three-dimensional Aspects of Crack Mechanics 15
1.2.5 Elements of the Mechanics of Interfaces . 18
1.2.6 Limited Crack Tip Plasticity. 19
1.3 Determining Stress Intensity Factors 20
1.3.1 Analytical Methods . . . . . . 21
1.3.2 Numerical Methods . . . . . . 24
1.4 Singular Integral Equation Formulations 26
2 Distributed Dislocation Fundamentals 29
2.1 Dislocations .......... 31
2.2 Introduction to the Technique 34
2.2.1 Exact Inversion .... 37
2.2.2 Mode II Loading ... 39
2.3 Numerical Solution: Cauchy Kernel 40
2.3.1 Gauss-Chebyshev Quadrature 40
2.3.2 Further Quadrature Schemes. 44
2.4 Cracks Near a Straight Free Boundary 45
2.4.1 Buried Crack, Normal to the Free Surface 47
2.4.2 Buried Crack, Inclined to the Free Surface 49
2.5 Buried Cracks in a General Geometry . . . . . . . 52
2.6 Surface-Breaking Cracks .............. 56
2.6.1 Crack Normal to the Free Surface of a Half-Plane 56
2.6.2 Surface-Breaking Slant Crack .......... 62
2.6.3 A Further Example of a Surface-Breaking Crack 64
Vll
Vlll CONTENTS
3 Further Topics in Plane Crack Problems 69
3.1 Closed Cracks .................... . 69
3.1.1 Principles of Formulation .......... . 71
3.1.2 Proportional Loading: Uncoupled Problems 74
3.1.3 Proportional Loading: Coupled Problems. 76
3.1.4 Incremental Loading ........ . 80
3.1.5 Advancing Stick ........... . 84
3.1.6 Closed Cracks: Concluding Remarks 86
3.2 Multiple and Kinked Cracks ....... . 87
3.2.1 Two Cracks with Finite Separation 88
3.2.2 The Kinked Crack .. . 91
3.3 Curved Cracks ........ . 93
3.4 Arbitrarily Shaped Boundaries . 97
3.4.1 Dislocation Formulation 99
3.4.2 Dislocation and Boundary Element Formulation 101
3.5 Crack Tip Plasticity .................. . 103
4 Interface Cracks 107
4.1 The Interface Crack. 107
4.2 Interfacial Fracture Mechanics 108
4.2.1 The Open Model .. 108
4.2.2 The Contact Model .. 113
4.3 Modelling Interface Cracks . . 117
4.4 Solution of Interface Crack Problems 118
4.4.1 The Open Model ...... . 119
4.4.2 The Contact Model . . . . . . 123
4.4.3 Other Standard Interface Cracks 126
4.5 Singular Points in Plane Elasticity. . . . 127
4.5.1 Evaluating Stress Singularities .. 127
4.5.2 Straight Cracks Terminating with Interfaces 128
4.5.3 Kinked Cracks Interacting with Interfaces . 131
4.6 Cracks Terminating, or Kinking, at an Interface . . 133
4.6.1 A Perpendicular Crack Touching an Interface 133
4.6.2 Further Examples .. 135
4.7 The Kinked Interface Crack ........ . 135
5 Solution ofAxi-Symmetric Crack Problems 137
5.1 Introduction ....... . 137
5.2 Dislocation Loops ............ . 138
5.3 Plane Dislocation Dipoles ....... . 140
5.3.1 Dipoles in Plane Crack Problems 145
5.4 Ring Dipoles ............... . 151
5.4.1 Stress Fields due to Ring Dipoles 153
CONTENTS IX
5.4.2 Separation of the Ring Dipole Stress Singularity 156
5.4.3 Ring Dipoles and Plane Dipoles 158
5.5 Axi-symmetric Crack Problems 159
5.5.1 The Annular Crack .. 159
5.5.2 The Cylindrical Crack 161
5.5.3 Cone-Shaped Cracks .. 162
5.6 Axi-symmetric Cracks in Bonded Half-Spaces 164
5.6.1 Half-Space Crack Problems ..... . 167
6 Three-Dimensional Cracks: An Introduction 171
6.1 Introduction ......... . 171
6.2 Infinitesimal Dislocation Loop 172
6.3 Singular Integral Equations 174
6.4 A Closed-Form Solution 176
6.5 Hyper-Singular Integrals .. 179
6.6 Numerical Treatment .... 184
6.6.1 Piecewise Constant Approximation 184
6.6.2 Piecewise Linear Approximation. 192
6.7 Numerical Examples ........... . 202
7 Three-Dimensional Cracks: Further Concepts 211
7.1 Introduction ......... . 211
7.2 Infinitesimal Dislocation Loop 211
7.3 Singular Integral Equations 214
7.4 Hyper-Singular Integrals .. . 216
7.5 Numerical Treatment .... . 219
7.5.1 Discretization of the System 219
7.5.2 Calculation of Matrix Elements Ki] 220
7.5.3 Determination of Stress Intensity Factors 222
7.6 Examples and Advanced Applications . . . . . . 223
8 Concluding Remarks 231
A Dislocation Influence Functions 235
A.l Notation .................. . 235
A.2 A Dislocation Outside a Circular Inclusion 236
A.2.1 A Dislocation Near a Circular Hole 239
A.3 A Dislocation Near a Straight Interface 239
A.3.1 A Dislocation in a Half-Plane 241
AA The Interfacial Dislocation ...... . 241
AA.1 Interfacial Stresses ...... . 242
A.5 A Dislocation on the Boundary of a Circular Inclusion 244
A.6 Displacements Due to a Dislocation .......... . 245
x CONTENTS
A.7 Transformation Rules for the Dislocation 248
A.S Dundurs' Parameters .......... . 250
B Numerical Solution of SIEs with Cauchy Kernel 253
B.1 The Standard Gaussian Quadrature Formulae .. 253
B.2 Gaussian Quadrature for SIEs with Cauchy Kernel 255
B.3 SIEs Arising in Surface-Breaking Crack Problems 257
B.3.1 The Method of Boiko and Karpenko 257
B.3.2 Expansion Method ............ . 258
B.3.3 Series Expansion ............. . 259
BA Numerical Quadrature: Generalised Cauchy Kernel 259
C Plane and Ring Dipole Influence Functions 261
C.l Plane Influence Functions ........ . 261
C.2 Plane Dipole Transformation Rules .... . 262
C.3 The Modified Lipschitz-Hankel Integrals J 264
mnp
C.3.l Asymptotic Behaviour of the Integrals 266
C.3.2 Numerical Considerations 268
CA Finite Part Integrals ......... . 269
D Contour Integral and Kernel Function 271
D.1 Closed-Form of Certain Contour Integrals 271
Ki
D.2 Explicit Expression for Kernel Function 272
j
References 277
Index 293
