Boundary element formulations for fracture mechanics problems

$: Boundary element formulations for fracture mechanics problems$
University of Wollongong Thesis Collections

University of Wollongong Thesis Collection

University of Wollongong Year

Boundary element formulations for

fracture mechanics problems

Wei-Liang WuUniversity of Wollongong

Wu, Wei-Liang, Boundary element formulations for fracture mechanics problems, PhDthesis, School of Mathematics and Applied Statistics, University of Wollongong, 2004.http://ro.uow.edu.au/theses/253

This paper is posted at Research Online.

http://ro.uow.edu.au/theses/253

BOUNDARY ELEMENT FORMULATIONS FOR

FRACTURE MECHANICS PROBLEMS

By

Wei-Liang Wu

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

AT

UNIVERSITY OF WOLLONGONG

NORTHFIELDS AVENUE, WOLLONGONG NSW 2522

AUSTRALIA

NOVEMBER 2004


SCHOOL OF

MATHEMATICS AND APPLIED STATICS

This thesis is submitted to the University of Wollongong, and

has not been submitted for a higher degree to any other university or

institution.

Wei-Liang Wu

November 2004

ii


Date: November 2004

Author: Wei-Liang Wu

Title: Boundary Element Formulations for Fracture

Mechanics Problems

School: Mathematics and Applied Statics

Degree: Ph.D.

Permission is herewith granted to University of Wollongong to circulate

and to have copied for non-commercial purposes, at its discretion, the above

title upon the request of individuals or institutions.

Signature of Author

THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, ANDNEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAYBE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’SWRITTEN PERMISSION.

THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINEDFOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THISTHESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPERACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USEIS CLEARLY ACKNOWLEDGED.

iii

To My Parents with Love and Gratitude.

iv

Acknowledgements

I would like to acknowledge the many people who have helped me in my life and who

helped bring this thesis to fruition.

In particular, I want to thank my supervisor, Dr. Xiaoping Lu who had constantly

been a source of knowledge and expertise on mathematical techniques which have been

utilised in this thesis. I thank her for her constant support, encouragement and her

willingness to help at any time despite her hectic schedule.

Above all, I want to thank my teachers – both in Taiwan and Australia. To those

who helped me learn in the classroom. To those who helped me learn in the world

beyond the classroom.

I would also like to thank the colleagues of the University of Wollongong who

devoted time and effort to provide me with critical feedback on earlier drafts, point-

ing inaccuracies, suggesting better examples, and gently noting things that needed

smoothing. I would like to mention especially Dr. Ahmed A. El-Feki, Dr. Anna Maria

Milan, Dr. Bin Liu and Frank Bierbrauer.

In addition, my deepest thanks are owed to my family who have always inspired

me to greater things particularly my parents for their love, sacrifice and support.

v

Table of Contents

Acknowledgements v

List of Tables ix

List of Figures x

1 Introduction 1

1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Modes of Crack Tip Deformation . . . . . . . . . . . . . . . . . . . . 4

1.3 Other Fracture Characterising Parameters . . . . . . . . . . . . . . . 6

1.3.1 The J-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 The Crack Tip Opening Displacement . . . . . . . . . . . . . 7

1.4 Numerical Methods in Linear Elastic Fracture Mechanics . . . . . . . 8

1.4.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . 8

1.4.3 Boundary Element Method . . . . . . . . . . . . . . . . . . . 9

1.5 Advanced Formulations in Boundary Element Method . . . . . . . . . 11

1.5.1 Dual Boundary Element Method . . . . . . . . . . . . . . . . 11

1.5.2 Subregion Boundary Element Method . . . . . . . . . . . . . . 12

1.5.3 Dual Reciprocity Boundary Element Method . . . . . . . . . . 13

2 Dual Boundary Integral Formulation for Two Dimensional Crack

Problems 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 The Dual Boundary Integral Formulation . . . . . . . . . . . . . . . . 18

2.3 Stress Intensity Factor Calculation . . . . . . . . . . . . . . . . . . . 22

2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

vi

3 An Efficient Dual Boundary Element Method for Crack Problems

with Anti-plane Shear Loading 43

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 The Dual Boundary Integral Equation

for Anti-plane Problems . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Calculation of the Mode III Stress Intensity Factor . . . . . . . . . . 49


3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 The Evaluation of Stress Intensity Factors 60

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 The Numerical Evaluation of Stress Intensity Factors . . . . . . . . . 63

4.2.1 J-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.2 The Discontinuous Quarter Point Element Method . . . . . . 66

4.2.3 The Special Crack Tip Element Method . . . . . . . . . . . . 69


4.3.1 A Plate with a Central Slant Crack . . . . . . . . . . . . . . . 71

4.3.2 Infinite Plate with Two Inclined Cracks . . . . . . . . . . . . . 77

4.3.3 Infinite Plate with Two Parallel Cracks . . . . . . . . . . . . . 82

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 A New Subregion Boundary Element Technique 87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 The Multi Region Technique of Boundary Element Method . . . . . . 91

5.3 Comparison of Subregion BEM Techniques . . . . . . . . . . . . . . . 94

5.3.1 The Traditional Method . . . . . . . . . . . . . . . . . . . . . 94

5.3.2 Kita & Kamiya’s Method . . . . . . . . . . . . . . . . . . . . 94

5.3.3 The Proposed Method . . . . . . . . . . . . . . . . . . . . . . 96

5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 A Subregion DRBEM Formulation for the Dynamic Analysis of Two

Dimensional Cracks 109

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2 Dual Reciprocity Boundary Element Method . . . . . . . . . . . . . . 113

6.3 Derivation of Particular Solutions . . . . . . . . . . . . . . . . . . . . 118

6.4 The Dynamic Stress Intensity Factors . . . . . . . . . . . . . . . . . . 119


6.5.1 A Rectangular Plate with a Central Crack . . . . . . . . . . . 121

vii

6.5.2 A Rectangular Plate with a Central Slant Crack . . . . . . . . 123

6.5.3 A Rectangular Plate with an Internal Kinked Crack . . . . . . 126

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Conclusion 132

Bibliography 135

viii

List of Tables

2.1 Mode I SIF for an internal kinked crack in a rectangular plate . . . . 29

2.2 Mode II SIF for an internal kinked crack in a rectangular plate . . . . 29

2.3 Comparison of normalised stress intensity factors . . . . . . . . . . . 35

3.1 Normalised mode III stress intensity factor for a straight central crack 52

ix

List of Figures

1.1 The three modes of loading . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Definition of the coordinate axis ahead of a crack tip . . . . . . . . . 5

1.3 Stress normal to the crack plane in mode I . . . . . . . . . . . . . . . 6

2.1 Rectangular plate with a central slant crack (h/w = 2, θ = 45) . . . . 23

2.2 Normalised mode I SIF for the rectangular plate with a central slant

crack: (a) the present method, (b) Reference [80], and (c) Reference [73] 24

2.3 Normalised mode II SIF for the rectangular plate with a central slant

crack: (a) the present method, (b) Reference [80], and (c) Reference [73] 25

2.4 Normalised mode I SIF for the rectangular plate with a central slant

crack (a/w = 0.1): (a) the present method, and (b) Reference [73] . . 26

2.5 Normalised mode II SIF for the rectangular plate with a central slant

crack (a/w = 0.1): (a) the present method, and (b) Reference [73] . . 27

2.6 Rectangular plate with an internal kinked crack (h/w = 2, a/w = 0.1) 28

2.7 Normalised mode I SIF vs. crack ratio b/a at tip A. The angle of the

kinked crack is (a) 30, (b) 45, (c) 60 . . . . . . . . . . . . . . . . . 30

2.8 Normalised mode II SIF vs. crack ratio b/a at tip A. The angle of the


2.9 Normalised mode I SIF vs. crack ratio b/a at tip B. The angle of the


2.10 Normalised mode II SIF vs. crack ratio b/a at tip B. The angle of the


2.11 Rectangular plate with two inclined cracks (h/w = 2, a/W = 0.25) . . 34

2.12 Normalised mode I SIF vs. inclined angle θ: (a) Finite region (b) In-

finite region at tip A, and (c) Finite region (d) Infinite region at tip

B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

x

2.13 Normalised mode II SIF vs. inclined angle θ: (a) Finite region (b)

Infinite region at tip A, and (c) Finite region (d) Infinite region at tip B 37

2.14 Rectangular plate with two parallel cracks (h/w = 2, a/w = 0.025) . . 38

2.15 Normalised mode I SIF for the rectangular plate with two parallel

cracks: (a) the present method, and (b) Reference [93] . . . . . . . . 39

2.16 Normalised mode I SIF vs. s, with a/w given by (a) 0.025, (b) 0.05,

(c) 0.1, (d) 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.17 Normalised mode II SIF vs. s, with a/w given by (a) 0.025, (b) 0.05,

(c) 0.1, (d) 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Rectangular plate with a central slant crack . . . . . . . . . . . . . . 51

3.2 Normalised mode III stress intensity factor (SIF) for the rectangular

plate with a central slant crack (a) θ = 30, (b) θ = 45, and (c) θ = 60 53

3.3 Normalised mode III SIF for the infinite plate with a central slant crack

(a) the analytical solutions, and (b) the present method . . . . . . . 54

3.4 A finite plate with two collinear cracks . . . . . . . . . . . . . . . . . 55

3.5 Normalised mode III SIF for the rectangular plate with two identical

collinear cracks at tip A: (a) the analytical results, (b) the present

method and at tip B: (c) the analytical solutions, (d) the present

method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 A finite plate with two parallel cracks . . . . . . . . . . . . . . . . . 57

3.7 Normalised mode III SIF for the rectangular plate with two parallel

cracks (a) Reference [93] (b) the present method . . . . . . . . . . . 58

4.1 Crack in an infinite plane . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 General continuous quadratic element . . . . . . . . . . . . . . . . . . 63

4.3 Coordinate reference system and contour path for J-integral . . . . . 64

4.4 Singular quarter-point boundary elements . . . . . . . . . . . . . . . 66

4.5 Modeling of the quarter point boundary element . . . . . . . . . . . . 68

4.6 Relative error of Mode I SIF for the infinite plate with a central slant

crack from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elements

and the SCT with (d) 6 elements, (e) 8 elements, and (f) 10 elements 72

4.7 Relative error of Mode II SIF for the infinite plate with a central slant

crack from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elements

and the SCT with (d) 6 elements, (e) 8 elements, and (f) 10 elements 73

xi

4.8 Relative error of Mode III SIF for the infinite plate with a central

slant crack from the QPE with (a) 6 elements, (b) 8 elements, (c) 10

elements and the SCT with (d) 6 elements, (e) 8 elements, and (f) 10

elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.9 Mode I SIF of the finite plate with a central slant crack from the QPE

with aw

= (a) 23, (b) 1

2, (c) 1

5and the SCT with (d) 2

3, (e) 1

2, (f) 1

5. . . 75

4.10 Mode II SIF of the finite plate with a central slant crack from the

QPE with aw

= (a) 23, (b) 1

2, (c) 1


3, (e) 1

2, (f) 1

576

4.11 Relative error of Mode I SIF for the infinite plate with two inclined

cracks at tip A from the QPE with (a) 6 elements, (b) 8 elements, (c)

10 elements and the SCT with (d) 6 elements, (e) 8 elements, and (f)

10 elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.12 Relative error of Mode I SIF for the infinite plate with two inclined

cracks at tip B from the QPE with (a) 6 elements, (b) 8 elements, (c)


10 elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.13 Relative error of Mode III SIF for the infinite plate with two inclined

cracks at tip A from the QPE with (a) 6 elements, (b) 8 elements, (c)


10 elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.14 Relative error of Mode III SIF for the infinite plate with two inclined

cracks at tip B from the QPE with (a) 6 elements, (b) 8 elements, (c)


10 elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.15 Mode I SIF for the infinite plate with two parallel cracks from (a) the

Reference [93], the QPE with (b) 6 elements, (c) 8 elements, (d) 10

elements and the SCT with (e) 6 elements, (f) 8 elements, and (g) 10

elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.16 Mode II SIF for the infinite plate with two parallel cracks from the

QPE with (a) 6 elements, (b) 8 elements, (c) 10 elements and the SCT

with (d) 6 elements, (e) 8 elements, and (f) 10 elements . . . . . . . 84

4.17 Mode III SIF for the infinite plate with two parallel cracks from (a)

the Reference [93], the QPE with (b) 6 elements, (c) 8 elements, (d)

10 elements and the SCT with (e) 6 elements, (f) 8 elements, and (g)

10 elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xii

5.1 A three subregion medium . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2 A perfectly bonded dissimilar elastic semi-strip . . . . . . . . . . . . . 101

5.3 Normalised normal stress distribution on the interface

(E2/E1, ν1, ν2) = (9.0, 0.5, 0.5): (a) Ref. [17], (b) the present method

(E2/E1, ν1, ν2) = (3.0, 0.5, 0.5): (c) Ref. [17], (d) the present method . 102

5.4 A three-layer plate with a centre crack . . . . . . . . . . . . . . . . . 103

5.5 Normalised mode I stress intensity factor (SIF) on the three-layer plate

with a centre crack: (a) l/h2 = 0.1, (b) l/h2 = 0.2, (c) l/h2 = 0.3, (d)

l/h2 = 0.4, (e) l/h2 = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . 104

5.6 A three-layer plate with two identical co-linear cracks . . . . . . . . . 105

5.7 Normalised mode I SIF at tip A on the three-layer plate with two

co-linear cracks: (a) l/h2 = 0.1, (b) l/h2 = 0.2, (c) l/h2 = 0.3, (d)

l/h2 = 0.4, (e) l/h2 = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . 106

5.8 Normalised mode I SIF at tip B on the three-layer plate with two

co-linear cracks: (a) l/h2 = 0.1, (b) l/h2 = 0.2, (c) l/h2 = 0.3, (d)

l/h2 = 0.4, (e) l/h2 = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . 107

6.1 Boundary and internal nodes . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 Crack on the body for subregion method . . . . . . . . . . . . . . . . 118

6.3 Rectangular plate with a central crack . . . . . . . . . . . . . . . . . 121

6.4 Normalised mode I DSIF for the rectangular plate with a central crack

(a) the present method, (b) the dual reciprocity method [6] and (c) the

Laplace transform method [6] . . . . . . . . . . . . . . . . . . . . . . 122

6.5 Rectangular plate with a central slant crack . . . . . . . . . . . . . . 123

6.6 Normalised mode I DSIF for the rectangular plate with a central slant

crack (a) the present method, (b) the dual reciprocity method [6] and

(c) the Laplace transform method [6] . . . . . . . . . . . . . . . . . . 124

6.7 Normalised mode II DSIF for the rectangular plate with a central slant

crack (a) the present method, (b) the dual reciprocity method [6] and

(c) the Laplace transform method [6] . . . . . . . . . . . . . . . . . . 125

6.8 Rectangular plate with an internal kinked crack . . . . . . . . . . . . 126

6.9 Normalised mode I DSIF for the rectangular plate with an internal

kinked crack at tip A (a) b/a = 0.2, (b) b/a = 0.4, and (c) b/a = 0.6 . 128

6.10 Normalised mode I DSIF for the rectangular plate with an internal

kinked crack at tip B (a) b/a = 0.2, (b) b/a = 0.4, and (c) b/a = 0.6 . 129

xiii

6.11 Normalised mode II DSIF for the rectangular plate with an internal

kinked crack at tip B (a) b/a = 0.2, (b) b/a = 0.4, and (c) b/a = 0.6 . 130

xiv

Chapter 1

Introduction

1.1 General Considerations

The design of modern structures necessitates the consideration of fractures. Iron and

steel saw increasing structural use in the 19th century and fracture was a problem.

In Great Britain, there were two hundred people killed per year in railway accidents

during the period 1860-1870 [22]. Most of the accidents were caused by fractures of

wheels, axles or rails, and were detrimental to the economy. In 1861, a leading article

of “The Engineer” reported: “Effects of percussion and frost upon iron... . We need

hardly say that this is one of the most important subjects that engineers of the present

day are called upon to investigate. The lives of many persons, and the property of

many more, will be saved if the truth of the matter be discovered – lost if it be not”

[15]. More time and effort needs to be spent on maintenance and repair of structures

in order to minimize the potential of component fracture and subsequent structure

failure. An economic study estimated the cost of fracture in the United States in

1978 at approximately US$119 billion, about 4% of the gross national product [35].

Some of these failures occurred due to poor designs, but many were due to material

1

Chapter 1: Introduction 2

deficiencies in the form of pre-existing flaws that initiated cracks and thus caused

fractures.

However, no material is flawless and fabrication flaws cannot be avoided. From

a practical point of view, many cracks may be considered as harmless. A structure,

such as a tanker, contains probably several thousand macroscopic cracks and several

million micro-cracks [21]. Essentially, if the cracks reach a certain size and situated in

a highly strained region, they are regarded as a potential cause of fractures. There-

fore, engineers must understand and characterise cracks and their effects, and try to

predict if and when they may become unsafe during the structures operational service

life. The subject of this thesis is to examine numerical techniques that can be used

to efficiently solve fracture problems. These techniques can then aid engineers in

studying and understanding the characteristics of cracks under particular conditions.

Griffith [49], in 1920, was the first systematic investigator of fracture by experi-

mental studies. In order to explain the measured strengths of the crack propagation,

Griffith introduced a model where the global energy balance controls the growth of

cracks. Fracture initiates when the energy stored in the fracture overcomes the surface

energy of the material. This theory is valid for materials with little or no preceding

plastic deformation near the crack tip and the fracture strength depends on flaw size.

From 1940, a major development of fracture theory was the brittle fracture of large

welded structures such as bridges, ships and oil storage containers. Linear elastic

fracture mechanics (LEFM) succeeded in application to the reliability assessment of

aircrafts. A fatigue crack that grew from an astro-navigation window caused the loss

of two aircrafts in 1953 and 1954. LEFM could provide an explanation for the fracture

[55, 96].


In 1956, Irwin [56] developed a new formulation of energy release rate and its

critical value by extending Griffith’s theory for ideally brittle materials. The energy

release rate is defined as the rate of change in potential energy with crack area for a

linear elastic material, and its critical value is a measure of fracture toughness. Thus,

Irwin postulated that crack growth occurs when the energy release rate reaches its

critical value. Another important contribution by Irwin [57], in 1957, was that the

distribution of the crack tip stress field was recognized and solved by the mathematical

procedures of Westergaard [101]. It is shown that the stresses and displacements near

the crack tip could be described by a single constant that was related to the energy

release rate. This crack tip characterising parameter is known as the stress intensity

factor. The stress intensity factor is a measure of the intensity of the singularity in

the stress field near the crack tip. It plays a fundamental role in linear elastic fracture

mechanics anaysis.

In 1988, a Boeing 737 lost a top section of the fuselage over Hawaii and yet it still

managed to land safely. The cause of this fatigue failure was multi-site fatigue, which

is still poorly understood. In 2001, American Airlines Flight 587, an airbus A300,

crashed in Queens after taking off from Kennedy International Airport, New York.

According to a National Transportation Safety Board statement, the plane’s vertical

stabilizer and rudder separated from the fuselage before impact. It has become the

subject of investigator’s scrutiny due to the pre-crash separation. But the cause of

the flight crash remains unknown. There is no doubt that there will be more and

more applications of fracture mechanics.


1.2 Modes of Crack Tip Deformation

The fundamental postulate of linear elastic fracture mechanics is based on the theory

that the intensity of the stress or strain fields surrounding a crack tip may be uniquely

described in terms of the stress intensity factor. A crack can be loaded in three

different modes, see Figure 1.1. These modes represent the local deformation ahead

Figure 1.1: The three modes of loading

of a crack. Normal stresses give rise to the opening mode or mode I loading. The

displacements of the crack surfaces are perpendicular to the plane of the crack. In-

plane shear results in mode II or sliding mode. The displacement of the crack surfaces

slide over each other along the crack line. The tearing or anti-plane mode, or mode

III, is caused by out-of-plane shear. The displacement of the crack surfaces slide over

each other perpendicular to the crack line. The superposition of these three modes is

sufficient to describe the most general case of crack surface displacement.

Irwin showed that the stress field in any linear elastic cracked body is given by


the following series representation of the stress tensor σij:

σij =1√r

KIfIij(θ) + KIIf

IIij (θ) + KIIIf

IIIij (θ) + higher order terms

(1.1)

where r and θ are as defined in Figure 1.2, KI , KII and KIII denote stress intensity

Figure 1.2: Definition of the coordinate axis ahead of a crack tip

factors corresponding to the three basic modes of crack surface displacement. The

higher order terms depend on geometry, but the solution for any given configuration

contains a leading term that is proportional to 1/√

r. As r → 0, the leading term

approaches infinity, but the other terms remain finite or approach zero. Thus stress

near the crack tip varies with 1/√

r, regardless of the configuration of the cracked

body. It can also be shown that displacement near the crack tip varies with√

r. In

equation (1.1), it is seen that r → 0 leads to a stress singularity, as the stress tensor

σij tends to infinity. This is shown in Figure 1.3.


Figure 1.3: Stress normal to the crack plane in mode I

1.3 Other Fracture Characterising Parameters

Two other fracture mechanics parameters that can be used to calculate the stress

intensity factor are the J-integral, and the crack tip opening displacement.

1.3.1 The J-integral

The J-integral represents the energy extracted through the crack tip singularity. The

original concept of the path independent J-integral was developed by Rice [81].

The path independent J−integral characterises the stress-strain field at the crack

tip, whose path is taken sufficiently far from the crack tip for the cracks to be analysed

elastically, where singularities or non-linear elasto-plastic behaviour are not encoun-

tered.

In this method, the physical interpretation in terms of the potential energy is


available for crack extension, but retains its physical significance as a measure of

the intensity of the characteristic crack tip strain field. This is similar to the stress

intensity factor concept, which measures the intensity of the stress field in the vicinity

of the crack tip.

1.3.2 The Crack Tip Opening Displacement

In 1963, Wells [97] proposed that the fracture behaviour in the vicinity of a sharp

crack could be characterised by the crack tip opening displacement. Furthermore,

Wells showed that the concept of crack tip opening displacement was analogous to

the concept of critical crack extension force. It is postulated that failure occurs when

the crack tip opening displacement value reaches a critical value on the assumption

that this value characterises the strain or stress ahead of the crack tip necessary to

initiate the failure mechanisms. Accordingly, it is the plastic strain in the vicinity

of the crack tip region that controls fracture. A measure of the amount of crack

tip plastic strain is the displacement of the crack tip, especially at or very close

to the tip. Thus, it might be expected that, at the onset of fracture, crack tip

opening displacement has a critical value for a given material and accordingly can

be used as a fracture characterising parameter. Burdekin and Stone [23], in 1966,

used Dugdale’s strip yield model [36] and provided an improved basis for the crack

tip opening displacement concept. Burdekin and Stone have also shown that under

linear elastic fracture mechanics conditions the crack tip opening displacement could

be related to the stress intensity factors. Therefore, this technique is compatible with

the stress intensity factor approach.


1.4 Numerical Methods in Linear Elastic Fracture

Mechanics

The solution of engineering problems can be obtained using analytical methods or

numerical methods. Analytical solutions of fracture problems are limited to idealised

situations wherein the domain is considered to be infinite and the boundary conditions

are relatively simple. To deal with complex geometries and boundary conditions, an

accurate and efficient numerical method is essential. The finite difference method [61],

finite element method [108] and boundary element method [18, 20] are well known

numerical methods for solving fracture mechanics problems.

1.4.1 Finite Difference Method

The finite difference method solves the differential field equations by considering dis-

crete values of the mesh points throughout the domain of interest. This results in

a large system of algebraic equations, which require the use of considerable compu-

tational power for calculating the solution. To obtain accurate results for the finite

difference method, a good understanding of the effects of discretisation is important.

The finite difference method is often used in applications that require very fine meshes

and a large number of repeated operations because of the straightforward procedures

in matrix generation and manipulation.

1.4.2 Finite Element Method

The finite element method involves the approximation of the variables over the do-

main, in terms of polynomial interpolation functions. The domain is divided into


regions, called elements. These elements are interconnected at specified joints that

are called nodes or nodal points. These interpolation functions are defined in terms

of the values of the field variable at the nodal points. When the governing equa-

tions in finite element method are integrated over each finite element, the unknowns

will be the nodal values of the field variable. Continuity over adjacent elements is

applied. Assembling all of these together, a banded matrix is produced by solving

the field equations. The nodal value of the field variable can be obtained through-

out the assemblage of elements. Because each mesh is easily graded and a general

type of boundary conditions is incorporated, the finite element method is much more

powerful than the finite difference method. However, the main disadvantage of the

finite element method is that large quantities of data are still required because the

full domain needs to be discretised.

1.4.3 Boundary Element Method

In boundary value problems, an integral equation within a specified domain can be

transformed into a boundary integral equation over the boundary of the domain. In

order to solve the boundary integral equation numerically, the boundary of the domain

must be sub-divided into segments, known as boundary elements. The boundary

element method is based on Betti’s reciprocal theorem [84]. Assuming that (1) is the

loading of interest and (2) is a reference loading with a known solution, then in the

absence of body forces, Betti’s theorem can be stated as follows [11]:∫

S

T(1)i u

(2)i dS =

∫

S

T(2)i u

(1)i dS (1.2)

where S is the boundary of the domain, Ti and ui are components of the traction and

displacement vectors, respectively, and superscripts denote loadings (1) and (2).


Applying Betti’s reciprocal theorem to the boundary conditions, where a unit

force is applied at an interior point X′, resulting in displacements and tractions at

surface point x, leads to

ui(X′) = −

∫

S

Tij(X′,x)uj(x)dS(x) +

∫

S

Uij(X′,x)tj(x)dS(x) (1.3)

where Tij and Uij are the fundamental solutions, ui(X′) is the displacement vector

at the interior point X′, uj(x) and tj(x) are the reference displacement and traction

vectors at the boundary point x. If we let X′ → x′, where x′ is a boundary point,

equation (1.3) becomes:

1

2ui(x

′) +

∫

S

Tij(x′,x)uj(x)dS(x) =

∫

S

Uij(x′,x)tj(x)dS(x) (1.4)

assuming the boundary is smooth. In order to solve for the unknown boundary data,

the boundaries must be divided into elements, and equation (1.4) approximated by a

system of algebraic equations.

The boundary element method has emerged as a powerful alternative to the finite

element method. The most important features of the boundary element method is

that it reduces the dimensionality of the problem by one, resulting in a smaller system

of equations and a considerable reduction in the data required for the analysis. Hence,

the boundary element method programs are easier to use with existing solid modellers

and mesh generators. The advantage is particularly important for problems which

involve many modifications. Meshes can easily be generated and do not require a

complete re-meshing.


1.5 Advanced Formulations in Boundary Element

Method

The boundary element method is one of the most powerful numerical techniques for

the solution of problems in fracture mechanics. While much research has been done

in this area, more accurate and reliable solutions are still worthy to be explored. This

thesis presents some new boundary element formulations which have applications to

a wide range of problems.

1.5.1 Dual Boundary Element Method

The boundary element formulation in elastostatics is based on the displacement

boundary integral equation. It has been successfully applied to linear elastic prob-

lems in domains containing no degenerated area or volume. However, the solution of

general crack problems cannot be achieved in a single-region analysis with the direct

application of the boundary element method, because the coincidence of the crack

surfaces gives rise to a singular system of algebraic equations. The dual boundary

element method is formulated for the analysis of crack problems. The dual equations

are the displacement and the traction boundary integral equations. When the dual

boundary integral equations are applied, the displacement equation on one of the

crack surfaces and the traction equation on the other, general crack problems can be

solved in a single-region formulation.

In this thesis, the formulations for two dimensional linear elastic crack problems

under in-plane tensile and anti-plane shear loading, respectively, are proposed. Dis-

continuous quarter point element method and special crack tip element method are


used to correctly model the displacement in the vicinity of crack tips. The unknowns

on the crack surfaces are the relative displacement between upper and lower crack

surfaces. Once the relative displacements are solved numerically, physical quantities

of interest, such as crack tip stress intensity factors can be easily obtained.

1.5.2 Subregion Boundary Element Method

The need for more accurate and reliable solutions requires more computationally

intensive processes. The implementation of parallel processing techniques is one of

the best economical alternatives because of the advances of multiprocessor computing

technology.

In the last decade many attempts have been made to implement boundary ele-

ment method on parallel processing computers for the solution of various structural

mechanics problems and design problems. Domain decomposition method is very

useful for parallel processing in boundary element method. The method splits the

original, complex domain into smaller, simpler sub-domains assigned to different pro-

cessors. Thereafter the solution of the original problem can be reconstructed through

the solutions of these sub-problems. There are various techniques of domain decom-

position such as the iterative sub-structuring scheme, which uses domain partitions

splitting the original domain into small sub-domains and reduces the original problem

to an interface problem solved by an iterative method. The execution time in parallel

processing is determined by the processor with the largest execution time.

This thesis is concerned with a new subregion boundary element technique, in

which the system of equations for each subregion is transformed into an appropriate

form independently to utilize parallel computing. This technique is more efficient


than traditional methods because it significantly reduces the size of the resulting

matrix, which in turn reduces the required computation time.

1.5.3 Dual Reciprocity Boundary Element Method

The main advantage of the boundary element method is its ability to provide a com-

plete problem solution in terms of boundary values only, which provides substantial

savings in computer time and data preparation. However, non-homogeneous terms

accounting for efforts such as distributed loads were included in the formulation by

means of domain integrals, thus making the boundary element technique lose the

attraction of its “boundary only” character. Many different approaches have been

developed to overcome this problem, such as analytical integration of the domain

integrals, the use of Fourier expansions, the Galerkin vector technique, the multiple

reciprocity method, and the dual reciprocity boundary element method. The dual

reciprocity boundary element method is the only general technique other than cell

integration. It is an accurate, straightforward, and more powerful method. This

technique transforms the domain integrals to the boundary, with the aid of the re-

ciprocal theorem applied for the second time, the first time being when formulating

problem in integral form. Thus, the dual reciprocity boundary element method could

be employed to preserve the dimensionality reduction advantage of boundary ele-

ment method for non-homogeneous problems. It can be used to solve non-linear and

time-dependent problems as well as problems with internal source distributions.

In this thesis, the dual reciprocity boundary element method employing the step

by step time integration technique is developed to analyse two dimensional dynamic

crack problems. The equation of motion is expressed in a boundary integral form using


elastostatic fundamental solutions. The dual reciprocity boundary element method is

combined with an efficient subregion boundary element method to overcome the dif-

ficulty of a singular system of algebraic equations in crack problems. Dynamic stress

intensity factors are calculated using the discontinuous quarter point elements.

Chapter 2

Dual Boundary Integral

Formulation for Two Dimensional

Crack Problems

An efficient integral equation formulation for two-dimensional linear elastic crack

problems is proposed with the displacement equation being used on the outer bound-

ary and the traction equation being used on one of the crack faces. Discontinuous

quarter point elements are used to correctly model the displacement in the vicinity of

crack tips. Using this formulation a general crack problem can be solved in a single-

region formulation, and only one of the crack faces needs to be discretised. Once

the relative displacements of the cracks are solved numerically, physical quantities of

interest, such as crack tip stress intensity factors can be easily obtained. Numeri-

cal examples are provided to demonstrate the accuracy and efficiency of the present

formulation.

15

Chapter 2: Dual Boundary Integral Formulation 16

2.1 Introduction

Fracture mechanics analysis plays a central role in structural integrity assessments.

Many engineering materials have various kinds of defects such as scratches and micro-

cracks which cannot be precluded. These defects tend to intensify the local stress

field and hence reduce the load bearing capacity of a component. At the same time

increasing demands for energy and material conservation requires that structures

be designed with smaller safety margins. Therefore, the study of events of existing

cracks has become increasingly important. The complicated nature of multi-crack

problems means that any meaningful solution has to be obtained numerically. The

need of developing an accurate and effective numerical technique in fracture mechanics

analysis is obvious.

Various numerical techniques have been used by researchers in the fracture me-

chanics community. Among them, the boundary element method is now widely re-

garded as the most accurate numerical tool for the analysis of crack problems in linear

elastic fracture mechanics (see Brebbia & Dominguez [18]). However, the solution of

a general crack problem cannot be achieved in a single region analysis by a direct

application of the boundary element method; the boundary integral equations for

two geometrically coincident points on both surfaces of a crack are identical, thus

resulting in a singular system of algebraic equations.

Some special techniques have been proposed to overcome this difficulty. Among

these are the crack Green’s function method [90], the displacement discontinuity

method [31], the multi-domain formulation [16] and dual boundary element method

(DBEM) [27, 25, 29, 80]. Comparing these methods, DBEM has certain apparent

advantages over others in that crack problems can be solved with a single region


formulation [10, 28, 99, 100]. However, the DBEM formulation employed by Portela et

al. [80] incorporates two independent boundary integral equations; the displacement

equation to model one of the crack surfaces, and the traction equation to model the

other. Since the former equation is also applied to the outer boundary, this approach

results in a large system of equations, and could therefore be time-consuming in

computation, especially for structures with multi-cracks.

Chen & Chen [25] proposed a different DBEM formulation; the displacement

integral equation is applied only on the outer boundary, and the traction integral

equation on one of the crack surfaces. In their formulation, relative displacement

of crack surfaces is introduced instead of the displacement. This has reduced the

total number of degrees of freedom and computational effort. However, constant

elements are used to discretise the crack surface; a virtual boundary connected to

one of the crack surfaces is needed to construct a closed integral path to evaluate the

hypersingular integral. Although good accuracy is reported for the examples given in

[25], to model cracks of arbitrary geometry, quadratic elements are usually necessary.

Also, more elements are needed to yield accurate results, which could lead to a loss

of the advantage in the DBEM.

In this chapter, we propose an integral equation formulation that combines Chen

& Chen’s approach with the crack modelling strategy of Portela et al. with some

modification of the latter by placing discontinuous quarter-point elements at crack

tips. In this formulation, the displacement equation is used on the outer boundary

only while the traction equation is used on one of the crack surfaces. The unknowns

are the relative displacement of the crack faces and the displacement of the outer


boundary. Discontinuous quadratic elements are used to model the relative displace-

ment. These elements not only satisfy the requirement of continuity of both tractions

and strains, but also lead to an easy implementation of collocation at crack tips, crack

kinks and crack-edge corners [80]. With the use of quadratic elements, the number

of elements needed for the crack face is small, which in turn reduces the size of the

resulting system of algebraic equations. Instead of using the more complicated J-

integral method as in Portela et al. [80], the stress intensity factor (SIF) is calculated

by using the near tip relative displacement. More accurate results are obtained by

placing discontinuous quarter-point elements at crack tips, which correctly model the

behaviour of the crack tip displacement. Above all, the present formulation can be

effectively used to study structures with arbitrary number and distribution of cracks,

especially edge cracks and cracks of arbitrary shapes due to the adoption of discon-

tinuous quadratic elements and the relative displacement on crack faces. Numerical

examples are provided to demonstrate the accuracy and efficiency of the present for-

mulation.

2.2 The Dual Boundary Integral Formulation

A finite two dimensional body enclosed by a boundary Γ is considered. The boundary

integral representation of the displacement ui at an internal point X′ is given by (the

body force term is neglected) [32]

ui(X′) +

∫

Γ

Tij(X′,x)uj(x)dΓ(x) =

∫

Γ

Uij(X′,x)tj(x)dΓ(x) (2.1)

where Uij and Tij (i, j = 1, 2) represent the Kelvin displacement and traction fun-

damental solutions respectively, at the boundary integration point x. The Kelvin


fundamental solutions are given as

Uij(X′,x) =

1

8πµ(1 − ν)

[

(3 − 4ν) ln

(

1

r

)

δij +∂r

∂xi

∂r

∂xj

]

(2.2)

and

Tij(X′,x) = − 1

4π(1 − ν)r

∂r

∂n

[

(1 − 2ν)δij + 2∂r

∂xi

∂r

∂xj

]

−(1 − 2ν)

(

∂r

∂xi

nj −∂r

∂xj

ni

)

(2.3)

where r is the distance between X′ and x; µ, ν and δij represent the shear modulus,

Poisson’s ratio and the Kronecker delta, respectively; n denotes the unit outward

normal vector at the point x on the boundary; ni and nj are the direction cosines of

the normal with respect to xi and xj.

In the absence of body forces, consider a homogeneous, isotropic linear elastic

body, occupying the domain Ω with N cracks, and enclosed by the boundary ΓS.

Equation (2.1) can be written as

ui(X′) +

∫

ΓS

Tij(X′,x)uj(x)dΓ(x) +

N∑

n=1

∫

Γ+n

Tij(X′,x+)uj(x

+)dΓ(x)

−N∑

n=1

∫

−Γ−

n

Tij(X′,x−)uj(x

−)dΓ(x) =

∫

ΓS

Uij(X′,x)tj(x)dΓ(x) (2.4)

+N∑

n=1

∫

Γ+n

Uij(X′,x+)tj(x

+)dΓ(x) −N∑

n=1

∫

−Γ−

n

Uij(X′,x−)tj(x

−)dΓ(x)

where x+ and x− are the boundary points on the upper and lower crack surfaces,

respectively, ΓS represents the outer boundary, Γ+n the nth upper crack boundary,

Γ−n the nth lower crack boundary, and Γ = ΓS +

∑Nn=1(Γ

+n + Γ−

n ). Using the facts

that Uij(X′,x+)|Γ+ = Uij(X

′,x−)|−Γ− and Tij(X′,x+)|Γ+ = Tij(X

′,x−)|−Γ− , equation


(2.4) becomes

ui(X′) +

∫

ΓS

Tij(X′,x)uj(x)dΓ(x) +

N∑

n=1

∫

Γ+n

Tij(X′,x+)∆uj(x)dΓ(x)

=

∫

ΓS

Uij(X′,x)tj(x)dΓ(x) +

N∑

n=1

∫

Γ+n

Uij(X,x+)∆tj(x)dΓ(x) (2.5)

where ∆uj = uj(x+) − uj(x

−) and ∆tj = tj(x+) − tj(x

−). However, ∆tj is always

zero on the crack faces. Utilizing this fact and moving the source point X′ to the

outer boundary, we obtain the displacement equation [25]

cij(x′)uj(x

′) +

∫

ΓS

− Tij(x′,x)uj(x)dΓ(x) +

N∑

n=1

∫

Γ+n

Tij(x′,x+)∆uj(x)dΓ(x)

=

∫

ΓS

Uij(x′,x)tj(x)dΓ(x) (2.6)

where∫

− represents for the Cauchy principal value integral, cij(x′) is given by δij/2

for a smooth boundary at the point x′, and δij is the Kronecker delta.

In the absence of body forces, the stress components are obtained by differentiating

equation (2.5), followed by the application of Hooke’s law. The stress components

σij, at an internal point X′, can be expressed as

σij(X′) +

∫

ΓS

Skij(X′,x)uk(x)dΓ(x) +

N∑

n=1

∫

Γ+n

Skij(X′,x+)∆uk(x)dΓ(x)

=

∫

ΓS

Dkij(X′,x)tk(x)dΓ(x) (2.7)

where

Skij(X′,x) =

2µ

r2

2∂r

∂n

[

(1 − 2ν)δij∂r

∂xk

+ ν

(

δik∂r

∂xj

+ δjk∂r

∂xi

)

− 4∂r

∂xi

∂r

∂xj

∂r

∂xk

]

+ 2ν

(

ni∂r

∂xj

∂r

∂xk

+ nj∂r

∂xi

∂r

∂xk

)

+ (1 − 2ν)

(

2nk∂r

∂xi

∂r

∂xj

+ njδik + niδjk

)

− (1 − 4ν)nkδij

1

4π(1 − ν)(2.8)


and

Dkij(X′,x) =

1

r

(1 − 2ν)

[

δki∂r

∂xj

+ δkj∂r

∂xi

− δij∂r

∂xk

]

+ 2∂r

∂xi

∂r

∂xj

∂r

∂xk

1

4π(1 − ν)

(2.9)

In equation (2.7), Skij and Dkij are linear combinations of the derivatives of Tij and

Uij, respectively. Again, by moving the source point X′ to the upper crack boundary

x′, and using tj = σijni, where ni denotes the ith component of the outward normal

to the boundary at point x′ , we obtain the traction integral equation

1

2tj(x

′) + ni(x′)

∫

ΓS

Skij(x′,x)uk(x)dΓ(x) + ni(x

′)N∑

n=1

∫

Γ+n

= Skij(x′,x+)∆uk(x)dΓ(x)

= ni(x′)

∫

ΓS

Dkij(x′,x)tk(x)dΓ(x) (2.10)

where∫

= represents the Hadamard [50] principal value integral. Both Cauchy and

Hadamard principal-value integrals in equations (2.6) and (2.10) are finite parts of

improper integrals [60, 62]. The treatment of these finite part integrals follows the

method in Portela et al. [80].

Equations (2.6) and (2.10) are the governing equations to be solved in terms of the

displacement on the outer boundary and the relative displacement on the crack faces.

Equation (2.6) is applied for collocation on the outer boundary where continuous

quadratic elements are used, whereas equation (2.10) is applied on the upper crack

faces which are modelled by discontinuous quadratic elements.

By taking all the discretised nodes on the outer boundary ΓS and upper crack

surfaces∑N

n=1 Γ+n as the source point x′, the system of equations (2.6) and (2.10) for

the multiple cracks problem can be written in a matrix form as

[

T1 T2 0

S1 S2 I

]

uS

∆uc

t+c

=

[

U1

D1

]

[tS] (2.11)


where T1, T2, U1 and S1, S2, D1 are the corresponding assembled matrices from

equations (2.6) and (2.10). uS is the displacement vector and tS is the traction

vector on the outer boundary ΓS. ∆uc is the relative displacement vector of the

crack surfaces and t+c is the traction vector on the upper crack face.

2.3 Stress Intensity Factor Calculation

The ultimate task in fracture mechanics analysis is the calculation of the stress inten-

sity factor which is a local parameter. The most common methods of evaluation are

the J-integral method and the near tip displacement method. The later is much pre-

ferred computationally since the calculation is straight forward. However, to obtain

accurate results, the singular nature of the crack tip displacement has to be modelled

correctly. The required singularity can be achieved by placing special elements at

crack tips [44, 71]. Discontinuous quarter-point crack tip elements are used in the

present formulation. The stress intensity factors are calculated as

KI,II =µ

κ + 1

√

2π

r∆un,t(r) (2.12)

where κ = 3 − 4ν for plane strain problems, r is the distance from crack tip to the

nearest node on the upper crack face. ∆un(r) and ∆ut(r) denote the relative normal

and tangential displacement at r. The results show that the inclusion of the special

crack tip elements led to improved accuracy and efficiency in the SIF calculation.


2.4 Numerical Examples

To demonstrate the accuracy and efficiency of the present formulation, three cases are

considered, where a finite plate with different crack profiles is subjected to uniform

tension in one direction. In all the calculations, plane strain is assumed with a

Poisson’s ratio ν = 0.3.

First, consider a rectangular plate with a central slant crack subject to a uniform

traction T as shown in Figure 2.1. This is an example from Portela et al. [80].

Figure 2.1: Rectangular plate with a central slant crack (h/w = 2, θ = 45)

For the comparison purpose, the parameters are kept the same here. In solving this

problem, 30 quadratic elements were used compared to 36 in Portela et al. [80].

The stress intensity factors were calculated using the near tip relative displacements,

which requires very little calculation, while 30 internal points were needed in the

Chapter 2: Dual Boundary Integral Formulation 24 J – integral approach [80] for a similar degree of accuracy. As can be seen in Figures 2.2 and 2.3,

the present results match those of References [73] and [80] within two decimal places.

Figure 2.2: Normalised mode I SIF for the rectangular plate with a central slant crack: (a) the present method, (b) Reference [80], and (c) Reference [73]

Please see print copy for Figure 2.2


Figure 2.3: Normalised mode II SIF for the rectangular plate with a central slant crack: (a) the present method, (b) Reference [80], and [c] Reference [73]


Chapter 2: Dual Boundary Integral Formulation 26 For a/w = 0.1, the results of normalized mode I and mode II SIF for different θ are plotted in

Figures 2.4 and 2.5, respectively. The results obtained are remarkably accurate with the difference

between the present and the Reference [73] within three decimal places.

Figure 2.3: Normalised mode II SIF for the rectangular plate with a central slant crack: (a) the present method, (b) Reference [80], and [c] Reference [73]



Figure 2.5: Normalised mode II SIF for the rectangular plate with a central slant crack (a/w = 0.1): (a) the present method, and (b) Reference [73]



The second example is a rectangular plate with an internal kinked crack as shown

in Figure 2.6. The loading condition and other physical characters are again kept

Figure 2.6: Rectangular plate with an internal kinked crack (h/w = 2, a/w = 0.1)

the same as in Portela et al. [80]. The stress intensity factors at both tips A and

B were calculated and compared with the results published by Murakami [73] and

those obtained by the J−integral approach [80]. Again, excellent agreement is found

among the results as shown in Tables (2.1–2.2), where 2c = a +√

22

b.


Table 2.1: Mode I SIF for an internal kinked crack in a rectangular plate

KI/T√

πctip A tip B

b/a 0.2 0.4 0.6 0.2 0.4 0.6Present 0.996 0.991 0.988 0.600 0.575 0.569

Reference [73] 0.995 0.990 0.986 0.598 0.574 0.568Reference [80] 0.993 0.989 0.987 0.604 0.576 0.570

Table 2.2: Mode II SIF for an internal kinked crack in a rectangular plate

KII/T√

πctip A tip B

b/a 0.2 0.4 0.6 0.2 0.4 0.6Present 0.027 0.034 0.031 0.568 0.609 0.630

Reference [73] 0.028 0.033 0.030 0.557 0.607 0.627Reference [80] 0.030 0.036 0.032 0.556 0.603 0.624

The fracture condition for mixed mode is obtained by [9]

K = KI cos3 φ

2− 3KII cos2 φ

2sin

φ

2(2.13)

where

φ = 2 tan−1

1

4

KI

KII

± 1

4

√

(

KI

KII

)2

+ 8

While K reaches the critical value of the stress intensity factor, the crack will grow.

Using the results in Tables (2.1–2.2) we find that tip B is the worse case.


The following figures show the effect of crack ratio b/a on the stress intensity

factor. Different angle cases of the kinked crack are plotted. The results are shown

in Figures 2.7 and 2.8 for tip A, and Figures 2.9 and 2.10 for tip B.

b/a

Nor

mal

ised

mod

eI

SIF

attip

A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05(a)(b)(c)

Figure 2.7: Normalised mode I SIF vs. crack ratio b/a at tip A. The angle of thekinked crack is (a) 30, (b) 45, (c) 60


b/a

Nor

mal

ised

mod

eII

SIF

attip

A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05(a)(b)(c)

Figure 2.8: Normalised mode II SIF vs. crack ratio b/a at tip A. The angle of thekinked crack is (a) 30, (b) 45, (c) 60


b/a

Nor

mal

ised

mod

eI

SIF

attip

B

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1(a)(b)(c)

Figure 2.9: Normalised mode I SIF vs. crack ratio b/a at tip B. The angle of thekinked crack is (a) 30, (b) 45, (c) 60


b/a

Nor

mal

ised

mod

eII

SIF

attip

B

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1(a)(b)(c)

Figure 2.10: Normalised mode II SIF vs. crack ratio b/a at tip B. The angle of thekinked crack is (a) 30, (b) 45, (c) 60


The third example is a finite plate with two inclined cracks as shown in Figure

2.11, where d/a = 1.1. It was discussed in Chen & Chang [24] and Chen & Chen

Figure 2.11: Rectangular plate with two inclined cracks (h/w = 2, a/W = 0.25)

[25], and all the physical parameters are kept the same for comparison. The outer

boundary is modelled with 20 quadratic elements and the crack surface is discretised

with only 6 discontinuous quadratic elements compared with 18 constant elements

in Chen & Chen [25]. Our results were compared with those of Chen & Chang [24],

because the numerical results in Chen & Chen [25] are unavailable, but were shown

to be in excellent agreement with Chen & Chang’s. As shown in Table 2.3, the

correlation between the two sets of results is excellent with the maximum difference

being less than 2 per cent. The results for an infinite plate were also calculated and

compared with analytical results [73] with a maximum error of about 0.1 per cent.


Table 2.3: Comparison of normalised stress intensity factors

KI,II/T√

πaPresent Reference [24]

mode I mode II mode I mode IIθ A B A B A B A B45 0.596 0.520 0.595 0.518 0.513 0.557 0.511 0.55260 0.981 0.846 0.976 0.834 0.487 0.516 0.482 0.51175 1.405 1.128 1.394 1.113 0.316 0.319 0.313 0.31490 1.636 1.247 1.628 1.227 0 0 0 0


Figures 2.12 and 2.13 show the effects of the interaction of the cracks on stress

intensity factors. Because of the interaction of the cracks, the stress intensity factors

at A are greater than those at B. The results for an infinite plate are also plotted in

the same figures to show the effect of finite size.

θ

Nor

mal

ised

mod

eI

SIF

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8(a)(b)(c)(d)

Figure 2.12: Normalised mode I SIF vs. inclined angle θ: (a) Finite region (b) Infiniteregion at tip A, and (c) Finite region (d) Infinite region at tip B


θ

Nor

mal

ised

mod

eII

SIF

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6(a)(b)(c)(d)

Figure 2.13: Normalised mode II SIF vs. inclined angle θ: (a) Finite region (b) Infiniteregion at tip A, and (c) Finite region (d) Infinite region at tip B


The final example is a rectangular plate with two parallel cracks, as shown in Fig-

ure 2.14. The continuous and discontinuous quadratic elements are used to discretise

Figure 2.14: Rectangular plate with two parallel cracks (h/w = 2, a/w = 0.025)

the outer boundary and the crack faces, respectively. There are 36 elements used, in

which there are 6 elements on each crack face. The normalised mode I SIF is plotted

in Figure 2.15, where s = a/(a+d). The results agree well with the published results

[93] with a maximum difference of about 1.5 per cent. Also, the results at different

ratio a/w are plotted on figures 2.16 and 2.17.


Figure 2.15: Normalised mode I SIF for the rectangular plate with two parallel cracks: (a) the present method, and (b) Reference [93]



s

Nor

mal

ised

mod

eI

SIF

0.1 0.2 0.3 0.4 0.50.8

0.85

0.9

0.95

1

1.05

1.1(a)(b)(c)(d)

Figure 2.16: Normalised mode I SIF vs. s, with a/w given by (a) 0.025, (b) 0.05, (c)0.1, (d) 0.2


s

Nor

mal

ised

mod

eII

SIF

0.1 0.2 0.3 0.4 0.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07(a)(b)(c)(d)

Figure 2.17: Normalised mode II SIF vs. s, with a/w given by (a) 0.025, (b) 0.05, (c)0.1, (d) 0.2


2.5 Conclusion

In this chapter, an efficient boundary integral equation formulation was presented

to solve linear elastic crack problems. The formulation incorporates two boundary

integral equations; the displacement equation on the outer boundary and the traction

equation on one of the crack faces. General crack problems can be solved in a single

region formulation. The formulation is specially useful in the analysis of cracks of

arbitrary geometry, as well as structures with multi-cracks. It has the advantage over

the existing dual boundary element methods on saving of computing time because

fewer elements are needed. The use of discontinuous quarter-point elements at the

crack tips correctly describes the r1/2 behaviour of the near tip displacement, and

therefore leads to increased accuracy and efficiency.

Chapter 3

An Efficient Dual Boundary

Element Method for Crack

Problems with Anti-plane Shear

Loading

This chapter is concerned with an efficient dual boundary element method for 2-d

crack problems under anti-plane shear loading. The dual equations are the displace-

ment and the traction boundary integral equations. When the displacement equation

is applied on the outer boundary and the traction equation on one of the crack sur-

faces, general crack problems with anti-plane shear loading can be solved with a sin-

gle region formulation. The outer boundary is discretised with continuous quadratic

elements, however, only one of the crack surfaces needs to be discretised with dis-

continuous quadratic elements. Highly accurate results are obtained, when the stress

intensity factor is evaluated with the discontinuous quarter point element method.

Numerical examples are provided to demonstrate the accuracy and efficiency of the

present formulation.

43

Chapter 3: Anti-plane Shear Loading 44

3.1 Introduction

The problem of a cracked body subjected to an anti-plane shear loading had been

studied extensively. Sih [87] provided analytical solutions for mode III cracks in

infinite regions by using Westergaard stress functions and Muskhelishvili’s method.

Chiang [26] presented analytical solutions for slightly curved cracks in anti-plane

strain in infinite regions using perturbation procedures similar to those carried out

for in-plane loading cases by Cotterell & Rice [30]. Zhang [105, 106] and Ma & Zhang

[70] gave analytical solutions for a mode III stress intensity factor considering a fi-

nite region with an eccentric straight crack. Ma [69] provided analytical solutions for

mode III straight cracks in finite regions using Fourier transforms and Fourier series.

However, their solutions were concerned with specified geometries or boundary condi-

tions. To deal with the complexities of general geometries and boundary conditions,

an accurate and efficient numerical method is essential [48, 52, 100, 107].

Several numerical solutions had been devised for anti-plane crack problems. Paulino

et al. [77] provided numerical solutions for a curved crack subjected to an anti-plane

shear loading in finite regions by using the boundary integral equation method. Ting

et al. [94] provided numerical solutions for mode III crack problems by using the

boundary element alternating method. Liu & Altiero [63] provided numerical solu-

tions for mode III crack problems using the boundary integral equation with linear

approximation on displacements and stresses. Mews & Kuhn [72] provided numerical

solutions for the traction free central crack problem by using Green’s function, instead

of the usual fundamental solution. Sadegh & Altiero [5] used the indirect boundary

integral equation method to solve traction problems, using displacement-based for-

mulations. In general, the boundary element method (BEM) is a well established


numerical technique for the analysis of linear fracture mechanics problems. However,

the solution of general crack problems cannot be achieved with the direct application

of the BEM, because the coincidence of the crack surfaces gives rise to a singular

system of algebraic equations.

To overcome this shortcoming, we provide an efficient numerical procedure, based

on the dual boundary element method (DBEM), for anti-plane shear loading prob-

lems. The dual boundary element method seems to have certain apparent advantages

for in-plane loading problems with a single region formulation [66]. This method incor-

porates two independent boundary integral equations, the displacement and traction

equations. Portela et al. [80] considered the effective numerical implementation of

the two dimensional DBEM for solving general in-plane fracture mechanics problems.

Chen & Chen [25] proposed a different DBEM formulation for in-plane crack prob-

lems. Chen & Chen suggested the use of the displacement integral equation applied

only on the outer boundary and the traction integral equation on one of the crack

surfaces. In Chen & Chen’s formulation, relative displacement of crack surfaces was

used instead of the displacement. This reduces the degrees of freedom and hence the

computational effort. This study uses an integral equation formulation that combines

Lu & Wu’s [66] approach with the crack modelling strategy of quadratic boundary

elements for anti-plane crack problems. The stress intensity factor is calculated based

on the near tip displacement method. More accurate results are obtained by placing

discontinuous quarter point elements at crack tips [85], which correctly model the

behaviour of the crack tip displacement. This is a similar technique to that used for

continuous quarter point elements [71]. Numerical examples are provided to demon-

strate the accuracy and efficiency of the present formulation.


3.2 The Dual Boundary Integral Equation

for Anti-plane Problems

Consider a finite domain subjected to an arbitrary anti-plane shear loading, where

the only nonzero displacement component uz in the z direction may be specified as

follows [69]:

∇2uz = 0 (3.1)

The Laplace equation (3.1) can be transformed into a boundary integral equation,

as is typical with the BEM. The boundary integral formulation of the displacement

component, uz, at an internal point X′, is given by [18]

uz(X′) +

∫

Γ

H(X′,x)uz(x)dΓ(x) =

∫

Γ

G(X′,x)tz(x)dΓ(x) (3.2)

where H(X′,x) and G(X′,x) represent the fundamental traction and displacement

solutions, respectively, at a boundary point x, which are given as

H(X′,x) = − 1

2πr

∂r

∂n(3.3)

and

G(X′,x) =1

2πµln

(

1

r

)

(3.4)

where µ is the shear modulus, r is the distance between X′ and x, and n denotes the

outward normal unit vector at the point x on the boundary Γ.


If we consider a finite body with L cracks, equation (3.2) can be written as

uz(X′) +

∫

ΓS

H(X′,x)uz(x)dΓ(x) +L∑

l=1

∫

Γ+

l

H(X′,x+)uz(x

+)dΓ(x)

+L∑

l=1

∫

Γ−

l

H(X′,x−)uz(x

−)dΓ(x)

=

∫

ΓS

G(X′,x)tz(x)dΓ(x) +L∑

l=1

∫

Γ+

l

G(X′,x+)tz(x

+)dΓ(x)

+L∑

l=1

∫

Γ−

l

G(X′,x−)tz(x

−)dΓ(x) (3.5)

where x+ and x− are the field points located on upper and lower crack surfaces,

respectively. Note that ΓS denotes the outer boundary of the body, Γ+l the lth upper

crack boundary, Γ−l the lth lower crack boundary, and Γ = ΓS+

∑Ll=1(Γ

+l +Γ−

l ). Using

the fact that H(X′, x+)|Γ+ = H(X′,x

−)|−Γ− and G(X′,x

+)|Γ+ = G(X′,x

−)|−Γ− ,

equation (3.5) can be simplified to

uz(X′) +

∫

ΓS

H(X′,x)uz(x)dΓ(x) +L∑

l=1

∫

Γ+

l

H(X′,x+)∆uz(x)dΓ(x)

=

∫

ΓS

G(X′,x)tz(x)dΓ(x) +L∑

l=1

∫

Γ+

l

G(X′,x+)∆tz(x)dΓ(x) (3.6)

where ∆uz = uz(x+)−uz(x

−) and ∆tz = tz(x+)− tz(x

−), however ∆tz is always zero

on the crack faces. As the internal point approaches the outer boundary, that is, as

X′ → x′, the displacement equation becomes

c(x′)uz(x′) +

∫

ΓS

− H(x′,x)uz(x)dΓ(x) +L∑

l=1

∫

Γ+

l

H(x′,x+)∆uz(x)dΓ(x)

=

∫

ΓS

G(x′,x)tz(x)dΓ(x) (3.7)

where∫

− represents the Cauchy principle value integral and c(x′) = 12, given a smooth

boundary at the point x′.


The stress components σiz are obtained from differentiation of equation (3.6),

followed by the application of Hooke’s law. At an internal point X′, these components

are given by

σiz(X′) +

∫

ΓS

Si(X′,x)uz(x)dΓ(x) +

L∑

l=1

∫

Γ+

l

Si(X′,x)∆uz(x)dΓ(x)

=

∫

ΓS

Di(X′,x)tz(x)dΓ(x) (3.8)

where Si(X′,x) and Di(X

′,x) are linear combinations of derivatives of H(X′,x) and

G(X′,x) in the i direction, which are given as

Si(X′,x) =

µ

2πr2

[

∂r

∂xi

∂r

∂n−(

δij −∂r

∂xj

∂r

∂xi

)

nj

]

(3.9)

and

Di(X′,x) = − 1

2πr

∂r

∂xi

(3.10)

where ni denotes the ith component of the outward normal to the boundary at point

x, and δij is the Kronecker delta. Again, by moving the source point X′ to the upper

crack boundary x′, and using tz = σizni, we obtain the traction integral equation

1

2tz(x

′) +

∫

ΓS

ni(x′)Si(x

′,x)uz(x)dΓ(x) +L∑

l=1

∫

Γ+

l

= ni(x′)Si(x

′,x)∆uz(x)dΓ(x)

=

∫

ΓS

ni(x′)Di(x

′,x)tz(x)dΓ(x) (3.11)

where∫

= represents the Hadamard principal value integral. Both Cauchy and Hadamard

principal-value integrals in equations (3.7) and (3.11) are finite parts of improper in-

tegrals. The treatment of the finite part integrals follows the method in Portela et

al. [80].

The displacement integral equation (3.7) and the traction integral equation (3.11)

are the governing equations to be solved for the displacement of the outer bound-

ary and the relative displacement of the crack faces. Equation (3.7) is applied for


collocation on the outer boundary where continuous quadratic elements are used,

whereas equation (3.11) is applied on the upper crack faces which are modelled by

discontinuous quadratic elements.

By taking all the discretised nodes on the outer boundary ΓS and upper crack

surfaces∑L

l=1 Γ+l as the source point x′, the system of equations (3.7) and (3.11) for

the multiple cracks problem can be written in a matrix form as

[

H1 H2 0

S1 S2 I

]

uz,S

∆uz,c

tz,c+

=

[

G1

D1

]

[tz,S] (3.12)

where H1, H2, G1 and S1, S2, D1 are the corresponding assembled matrices from

equations (3.7) and (3.11), respectively. The uz,S and tz,S are the displacement and

traction vectors on the outer boundary ΓS, respectively. ∆uz,c and tz,c+ are the

relative displacement vector and the traction vector on the upper crack faces.

3.3 Calculation of the Mode III Stress Intensity

Factor

Near tip displacement extrapolation is used to evaluate the numerical values of the

stress intensity factor. The relative displacements of the crack surfaces are calculated

using the DBEM and are used in the near crack tip stress field equations to obtain

the stress intensity factor. Due to the singular behaviour of the stress around the

crack tip, it is reasonable to expect a better approximation by replacing the normal

discontinuous quadratic element with a transition element possessing the same order

of singularity at the crack tip. The discontinuous quarter point element method


is used in the present formulation [42, 85]. The mode III stress intensity factor is

evaluated as:

KIII =µ

4

√

2π

r∆uz(r) (3.13)

where r is the distance from the crack tip to the nearest node on the upper crack

face, and ∆uz(r) denotes the relative displacement in the anti-plane direction.


In order to demonstrate the accuracy and efficiency of the technique previously de-

scribed, and to illustrate possible applications, we now consider several examples. In

all the numerical tests, the outer boundary is modelled by 24 continuous quadratic

elements, and each crack discretisation is carried out with three different meshes of

6, 8, and 10 discontinuous quadratic elements, respectively. The best accuracy is

achieved with 6 elements, in which the crack discretisation is graded, towards the tip,

with ratios 0.25, 0.15, and 0.1. The plate is subjected to a uniform anti-plane shear

loading τ , the stress intensity factor is normalised with respect to

K0 = τ√

πa (3.14)

where a defines the half length of the crack. All computations are carried out under

the condition of plane strain.

Firstly, consider a rectangular plate containing a central slant crack as shown

in Figure 3.1. The crack has length 2a and makes an angle θ with the horizontal

direction. For a horizontal crack (θ = 0), the normalised mode III stress intensity

factor is calculated for various ratios of a/h and a/w, and compared to those given

in References [63] and [68] (see Table 3.1). The largest difference between these does


Figure 3.1: Rectangular plate with a central slant crack

not exceed 1.65 per cent. Further, the normalised mode III stress intensity factor is

calculated for h/w = 2, while the crack is slanted an angle θ with the various ratios

of a/w. Three cases are considered, where θ = 30, 45, and 60, respectively. The

results obtained are presented in Figure 3.2. As it can be seen, when the ratio of a/w

increases, the stress intensity factor increases due to edge effect.


Table 3.1: Normalised mode III stress intensity factor for a straight central cracka : h 1 : 0.25 1 : 0.5 1 : 1 1 : 2 1 : 4

a : w Present 1.909 1.724 1.689 1.688 1.6611:1.2 Reference [63] 1.897 1.723 1.689 1.686 1.686

Reference [68] 1.900 1.725 1.691 1.689 1.689a : w Present 1.796 1.467 1.371 1.361 1.3611:1.4 Reference [63] 1.780 1.460 1.369 1.359 1.358



Reference [68] 1.772 1.379 1.178 1.130 1.128


a/w

Nor

mal

ised

mod

eII

IS

IF

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.5

0.6

0.7

0.8

0.9

1

1.1

1.2(a)(b)(c)

Figure 3.2: Normalised mode III stress intensity factor (SIF) for the rectangular platewith a central slant crack (a) θ = 30, (b) θ = 45, and (c) θ = 60


For the case where a/w = 1/50, which could be considered as the case of infinite

geometry since a << w, we compare the results with the analytical results for the

latter as given in Reference [93]. The results are plotted in Figure 3.3. Excellent

agreement is observed; the maximum error is around 0.02 per cent.

θ

Nor

mal

ised

mod

eII

IS

IF

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2(a)(b)

Figure 3.3: Normalised mode III SIF for the infinite plate with a central slant crack(a) the analytical solutions, and (b) the present method


As shown in Figure 3.4, the second example is a rectangular plate containing

two identical collinear cracks. 2a is the length of the inclined crack and 2d is the

Figure 3.4: A finite plate with two collinear cracks

distance between the centre of the cracks. The geometric parameters are h/w = 2 and

a/w = 1/50. Figure 3.5 displays the variations of normalised mode III stress intensity

factors at tip A and tip B versus different ratios of a/d. Due to the interaction between

the two cracks, the computed normalised mode III stress intensity factor at tip A is

always larger then that at tip B. Hence, as the crack centre distance d decreases, the

difference of stress intensity factor increases. There is excellent correlation between

the computed results using the present method and those from analytical solutions;

the difference between these results does not exceed 0.03 per cent at tip A, or 0.09

per cent at tip B.


a/w

Nor

mal

ised

mod

eII

IS

IF

0.1 0.2 0.3 0.4 0.5 0.6 0.71

1.02

1.04

1.06

1.08

1.1

1.12

1.14(a)(b)(c)(d)

Figure 3.5: Normalised mode III SIF for the rectangular plate with two identicalcollinear cracks at tip A: (a) the analytical results, (b) the present method and attip B: (c) the analytical solutions, (d) the present method


The third example is an infinite plate (h/w = 2, a/w = 1/50) containing two

parallel cracks, as shown in Figure 3.6. 2a is the length of the two identical cracks

Figure 3.6: A finite plate with two parallel cracks

and 2d is the distance between the cracks. The computed results are compared with

the published results in Reference [93]. The results of normalised mode III stress

intensity factor for different s are plotted in Figure 3.7, where s = a/(a + d). The

effect of the interaction of cracks on the mode III stress intensity factor is observed.

The largest difference between the present and the published results does not exceed

0.65 per cent.


Figure 3.7: Normalised mode III SIF for the rectangular plate with two parallel cracks: (a) Reference [93] (b) the present method



3.5 Conclusions

An efficient and accurate dual boundary element technique has been successfully es-

tablished for the analysis of two dimensional cracks subjected to an anti-plane shear

loading. The dual boundary equations are the usual displacement boundary integral

equation and the traction boundary integral equation. When the displacement equa-

tion is applied on the outer boundary and the traction equation is applied on one of

the crack surfaces, a general crack problem can be solved in a single region formu-

lation. The discontinuous quarter point elements are used for evaluating the mode

III stress intensity factor, which correctly describes the r1/2 behaviour of the near tip

displacements. This allows accurate results for mode III stress intensity factors to be

calculated.

Chapter 4

The Evaluation of Stress Intensity

Factors

The present chapter is concerned with developing a new technique for calculating the

stress intensity factor in two-dimensional linear elastic crack problems. This technique

is combined with the dual boundary element method to solve in-plane tensile loading

and anti-plane shear loading crack problems. Because quadratic boundary elements,

commonly used in boundary element method analysis, do not correctly describe the

behaviour of displacement near the crack tips, special crack tip elements are used to

model the displacement in the vicinity of crack tips. Numerical examples are provided

to demonstrate the accuracy, efficiency and stability of the present technique.

4.1 Introduction

The application of boundary element method (BEM) to fracture mechanics is now

well established and widely used in engineering. One of the main reasons is the

possibility of evaluating the stress intensity factors accurately and efficiently. The

stress intensity factors incorporate applied stress levels, geometry and crack size in a

60

Chapter 4: The Evaluation of Stress Intensity Factors 61

systematised manner and may be evaluated from the elastic stress analysis of cracked

structures. Irwin [57] showed that the stresses and displacements near the crack tip

could be described by a single constant that was related to the energy release rate.

This crack tip characterising parameter is known as the stress intensity factor. There

are three independent movements of the upper and lower crack surfaces with respect

to each other: (1) the crack opening mode (mode I), (2) the crack in-plane shear

mode (mode II) or sliding mode, and (3) the crack out-of-plane shear mode (mode

III) or tearing mode, as illustrated in Figure 1.1. The fracture in a cracked body

can be in any of these modes, or a combination of two or three modes.

Consider a crack problem in an infinite domain, Williams [102] had shown that

the stresses for an in-plane traction free crack may be written in terms of an infinite

series expansion with respect to the polar coordinates r and θ. Substituting Irwin’s

definition of the stress intensity factors into Williams’ expansion, we obtain the stress

components (see Figure 4.1) which are singular at the crack tip. As r → 0 the leading

Figure 4.1: Crack in an infinite plane


terms in the stresses approach infinity, while the other terms remain finite or approach

zero. Consider only the first term of the Williams’ expression, the stress components

are expressed as [22]

σ11 =KI√2πr

cosθ

2

(

1 − sinθ

2sin

3θ

2

)

− KII√2πr

sinθ

2

(

2 + cosθ

2cos

3θ

2

)

(4.1)

σ22 =KI√2πr

cosθ

2

(

1 + sinθ

2sin

3θ

2

)

+KII√2πr

sinθ

2cos

θ

2cos

3θ

2(4.2)

σ12 =KI√2πr

sinθ

2cos

θ

2cos

3θ

2+

KII√2πr

cosθ

2

(

1 − sinθ

2sin

3θ

2

)

(4.3)

where KI and KII are the stress intensity factors corresponding to the opening mode

and the in-plane shear mode, respectively, and the size of r is much smaller than

the crack length. Integrating equations (4.1), (4.2), and (4.3) using the strain-

displacement and stress-strain relations, the displacement components in the vicinity

of the crack tip are

u1 =1

4µ

√

r

2π

KI

[

(2κ − 1) cosθ

2− cos

3θ

2

]

+KII

[

(2κ + 3) sinθ

2+ sin

3θ

2

]

(4.4)

u2 =1

4µ

√

r

2π

KI

[

(2κ − 1) sinθ

2− sin

3θ

2

]

−KII

[

(2κ − 3) cosθ

2+ cos

3θ

2

]

(4.5)

where µ is the shear modulus, κ = 3 − 4ν for plane strain, and ν the Poisson’s ratio.

Similarly, the anti-plane displacement component near the crack tip is expressed as

u3 = 4KIII

µ

√

r

2πsin

(

θ

2

)

(4.6)

where KIII is the stress intensity factor corresponding to the out-of-plane shear mode.

It is obvious from the above expressions that the stress and displacement fields vary


with O(1/√

r) and O(√

r), respectively, in the vicinity of the crack tip, where r is the

distance to the crack tip.

4.2 The Numerical Evaluation of Stress Intensity

Factors

The standard BEM elements are continuous quadratic elements with the collocation

points at ξ = −1, 0, and 1. The interpolating polynomial (shape function) is

fi = φ1f 1i + φ2f 2

i + φ3f 3i (4.7)

where φ1 = 12ξ(ξ − 1), φ2 = 1 − ξ2, and φ3 = 1

2ξ(ξ + 1), see Figure 4.2.

Figure 4.2: General continuous quadratic element

The displacement and traction components may be approximated as

ui = a0 + a1

(r

l

)

+ a2

(r

l

)2

(4.8)

ti = b0 + b1

(r

l

)

+ b2

(r

l

)2

(4.9)

where aj and bj are constants, r denotes the distance to the crack tip and l denotes the

length of the element. Since the displacement and stress fields obtained do not contain


O(1/√

r) and O(√

r), the standard continuous quadratic element does not model the

crack tip behaviours correctly and hence it may not lead to accurate results near

crack tip. There are several methods, such as the J-integral method [7, 9, 80, 81] or

the near tip displacement methods [16, 71, 74], to overcome this problem.

4.2.1 J-integral

The stress intensity factor can be related to a path independent integral, termed the

J-integral, described by Rice [81]. This integral is independent of a contour path

surrounding the crack tip chosen, as shown in Figure 4.3. The J-integral is defined

Figure 4.3: Coordinate reference system and contour path for J-integral

as

J =

∫

S

(

Wn1 − tj∂uj

∂n1

)

dS (4.10)

where S is an arbitrary contour surrounding the crack tip; W is the strain energy

density; tj are the traction components and n1 is the x1-component of the unit outward


normal to the contour path. The relationship between the J-integral and the stress

intensity factors is given by [81]

J =K2

I + K2II

E ′(4.11)

where E ′ = E for plane stress conditions, and E ′ = E/(1 − ν2) for plane strain

conditions, while E is the elastic modulus. In order to decouple the stress intensity

factors in equation (4.11), we consider two points P (x1, x2) and P′

(x1,−x2) that are

symmetric relative to the axis along the crack surface, as shown in Figure 4.3. At

these points the displacement and stress fields can be expressed as a combination of

symmetric and antisymmetric components:

u1

u2

=

uI1 + uII

1

uI2 + uII

2

,

u′

1

u′

2

=

uI1 − uII

1

−uI2 + uII

2

(4.12)

and

σ11

σ22

σ12

=

σI11 + σII

11

σI22 + σII

22

σI12 + σII

12

,

σ′

11

σ′

22

σ′

12

=

σI11 − σII

11

σI22 − σII

22

−σI12 + σII

12

(4.13)

When equations (4.12) and (4.13) are substituted into equation (4.10), the J-integral

is represented by the sum of two integrals as follows [7]:

J = J I + J II (4.14)

where the superscripts indicate the deformation mode. Consequently, the J-integral

can be decoupled into mode I and mode II components, hence KI and KII can be

evaluated separately. Since the J-integral uses the potential energy theorem, the

elaborate representation of the crack tip singular fields is not necessary. However,

more source points surrounding the crack tip are required in the calculation of stress

intensity factors, because more points are needed for the contour integral calculation.


4.2.2 The Discontinuous Quarter Point Element Method

The displacement and traction in a given crack problem can be expressed in terms

of singular and regular terms. The displacement and traction in the vicinity of the

crack tip are dominated by the singular terms which could be represented as unknown

stress intensity factors. The quarter point element methods [75, 79, 92, 103] are

used to model analytically the dominant√

r and 1/√

r behaviour exhibited by the

displacements and tractions in the vicinity of the crack tip. This can be achieved by

placing special elements at crack tips.

By shifting the midpoint node of a continuous quadratic element to the quarter

point position (close to the crack tip, see Figure 4.4), the displacement and traction

Figure 4.4: Singular quarter-point boundary elements

components are now expressed as follows

ui = A0 + A1

√

r

l+ A2

(r

l

)

(4.15)

ti = B0 + B1

√

r

l+ B2

(r

l

)

(4.16)

where Aj and Bj are constants. The displacement field in the vicinity of the crack is

correctly modelled to represent O(√

r) behaviour, but the traction in equation (4.16)

does not possess the correct 1/√

r singularity. However, in the boundary element


method, the displacements and the tractions are represented independently. The

inclusion of the singularity can be accomplished by multiplying the right-hand side

of this equation by√

l/r [16], which then yields

ti =

√

l

r

[

B0 + B1

√

r

l+ B2

(r

l

)

]

= B0

√

l

r+ B1 + B2

√

r

l(4.17)

The above expression corresponds to the correct singular traction field O(1/√

r) in the

vicinity of the crack tip. Using the equations (4.15) and (4.17), both displacements

and tractions will be correctly represented.

In the boundary element method, subregion method is generally used in order to

avoid the problem that the coincidence of the crack surfaces gives rise to a singular

system of algebraic equations. The subregion method introduces artificial boundaries

into the cracked body, which connect the cracks to the boundaries. The prescribed

tractions and displacements along the crack surfaces are independently represented

in each domain. Then, the stress intensity factors can be computed by using the

equation (4.15) on the crack surface and equation (4.17) on the artificial boundary

around the crack tip elements. However, the selected artificial boundaries are not

unique and the method generates a large system of algebraic equations.

To overcome those shortcomings, the dual boundary element method was devel-

oped, which can solve the crack problem by a single region formulation. In this

method, both the displacement and the traction boundary integral equations are ap-

plied on the crack surfaces. An essential ingredient of the dual boundary element

formulation is the evaluation of the singular integrals. This feature requires a special

integration around the singular point (crack tip). Consider the collocation nodes at

the boundary elements, the displacement equation always satisfies the requirement of

the first order finite-part integrals (continuity of the displacement components at the


nodes). However, the traction equation is required to satisfy the second order finite-

part integrals (continuity of the displacement derivatives at the nodes on a smooth

boundary). The discontinuous boundary elements implicitly provide the necessary

smoothness required, since the nodes are internal points of the element. Therefore,

the discretisation of the crack is best done with discontinuous boundary elements. In

addition, the problem of collocation at crack tips, crack kinks and crack-edge corners

is automatically circumvented by the use of discontinuous elements.

In order to keep the feature of the quarter point element method (shifting the

nodes to correctly model the behaviours of the displacement and traction on the crack

tip), we consider a discontinuous quadratic element, where the collocation points are

at ξ = −23, 0, and 2

3, respectively. By using the shape functions given in Figure 4.4,

the nodes of a discontinuous quadratic element are placed at r/l = 136

, 14, and 25

36,

respectively, where r denotes the distance to the crack tip and l denotes the length of

the element (see Figure 4.5). Thus, the displacement and traction variations for the

elements adjoining the crack tip are given by equations (4.15) and (4.17), respectively.

Figure 4.5: Modeling of the quarter point boundary element


4.2.3 The Special Crack Tip Element Method

In this chapter, a particular method to evaluate the stress intensity factors is pre-

sented. Instead of using the interpolating polynomial functions on the near crack tip

element, we let the displacement be represented by

ui ≈ Bi

√r (4.18)

because of the O(√

r) behaviour of the near tip displacement field. The other bound-

aries are still discretised by using the standard interpolating polynomial functions.

Besides the unknown displacements of the collocation nodes, an unknown constant

Bi needs to be obtained. Let r and θ be a polar coordinate system with origin at the

crack tip, such that θ = ±π defines the crack faces. The relative displacement near a

traction free crack tip from equations (4.4), (4.5), and (4.6) can be written as

∆ux = ux(θ = π) − ux(θ = −π) =κ + 1

µKII

√

r

2π(4.19)

∆uy = uy(θ = π) − uy(θ = −π) =κ + 1

µKI

√

r

2π(4.20)

∆uz = uz(θ = π) − uz(θ = −π) =4

µKIII

√

r

2π(4.21)

where µ is the shear modulus, ν is the Poisson’s ratio and κ = 3−4ν for plane strain.

The stress intensity factors are given by

KI = Byµ

κ + 1

√2π (4.22)

KII = Bxµ

κ + 1

√2π (4.23)

KIII = Bzµ

4

√2π (4.24)

As mentioned in the previous section, in order to correctly model the behaviours of

the displacement and the traction in the vicinity of the crack tip, the collocation points


of the near crack tip element need to be shifted to particular locations (quarter point

element). The locations of the nodal points depend on the choice of the quadratic

shape functions. On the contrary, the special crack tip element method is a relatively

straight-forward technique to follow, which correctly models the distribution of the

displacement and the traction in the near crack tip element.


In order to demonstrate the accuracy and efficiency of the proposed special crack tip

element method described in the previous section, the same examples as shown in

previous chapters, are calculated and the results are compared with those obtained

from other methods. All of the cases are under plane strain condition, and Poisson’s

ratio is taken to be ν = 0.3. The stress intensity factors are obtained from near

crack tip displacement calculations by using the discontinuous quarter point element

method (QPE) and the proposed special crack tip element method (SCT). The nu-

merical examples will show the results of the comparison of the stability and efficiency

between these methods. In each case the structure is subjected to a uniform traction

t or an anti-plane shear loading τ . The stress intensity factors are normalised with

respect to

K0 = t√

πa (4.25)

or

K0 = τ√

πa (4.26)

respectively, where a defines the half length of the crack.


4.3.1 A Plate with a Central Slant Crack

When the crack length is small in comparison with plate dimensions (a/w = 140

and

h/w = 2), we could consider it as the case of an infinite region with a central crack.

So that we could compare our results with its analytical solutions. The geometry

of the present problem is shown in Figure 2.1 for in-plane problem or Figure 3.1 for

anti-plane problem. The analytical results for the stress intensity factors (SIF) are

given in Reference [93]:

KI = t cos2 θ√

πa (4.27)

KII = t cos θ sin θ√

πa (4.28)

KIII = τ cos θ√

πa (4.29)

The study is carried out with three different meshes of 6, 8, and 10 discontinuous

quadratic elements on each crack surface, respectively, and the outside boundary is

discretised with 24 quadratic elements. The crack discretisation is graded, towards

the tip, with the ratios 0.25, 0.15, and 0.1 for 6 elements, 0.2, 0.15, 0.1, and 0.0.5

for 8 elements, and equal size for 10 elements. The results obtained are shown in

Figures 4.6, 4.7 and 4.8. It can be seen that the special crack tip element method

is more stable than the discontinuous quarter point element method for different

crack discretisations. The results obtained with 6 crack elements show a higher level

of accuracy, the largest difference between the calculated results and the analytical

results does not exceed 0.08%.


Figure 4.6: Relative error of Mode I SIF for the infinite plate with a central slantcrack from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elements and the SCTwith (d) 6 elements, (e) 8 elements, and (f) 10 elements


Figure 4.7: Relative error of Mode II SIF for the infinite plate with a central slantcrack from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elements and the SCTwith (d) 6 elements, (e) 8 elements, and (f) 10 elements


Figure 4.8: Relative error of Mode III SIF for the infinite plate with a central slantcrack from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elements and the SCTwith (d) 6 elements, (e) 8 elements, and (f) 10 elements


The special crack tip element method can also produce comparably accurate re-

sults as that of the discontinuous quarter point element method for the finite region

problem. By using similar boundary discretisation, 24 quadratic element on the out-

side boundary and 6 on the crack face, three cases (a/w = 23, 1

2, 1

5) are studied, the

results are shown in Figures 4.9, and 4.10.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 10 20 30 40 50 60 70 80 90

Mod

e I S

IF

Theta

QPE(a)(b)(c)

SCT(d)(e)(f)

Figure 4.9: Mode I SIF of the finite plate with a central slant crack from the QPEwith a

w= (a) 2

3, (b) 1

2, (c) 1


3, (e) 1

2, (f) 1

5


Figure 4.10: Mode II SIF of the finite plate with a central slant crack from the QPEwith a

w= (a) 2

3, (b) 1

2, (c) 1


3, (e) 1

2, (f) 1

5


4.3.2 Infinite Plate with Two Inclined Cracks

Consider, now, the analysis of two inclined cracks in an infinite plate (h/w = 2,

a/w = 140

). The plate is subjected to a normal loading t as shown in Figure 2.11 or

an anti-plane shear loading τ as in Figure 3.4. The crack has the length 2a and makes

an angle of θ = 90 with the vertical direction. The distance between the centre of

the cracks is 2d. The analytical results are given below [93]

KI,A =d − a

2a

√

d − a

d

[

(

d + a

d − a

)2E(k)

K(k)− 1

]

t√

πa (4.30)

KI,B =d + a

2a

√

d + a

d

(

1 − E(k)

K(k)

)

t√

πa (4.31)

for an in-plane normal loading and

KIII,A =d − a

2a

√

d − a

d

[

(

d + a

d − a

)2E(k)

K(k)− 1

]

τ√

πa (4.32)

KIII,B =d + a

2a

√

d + a

d

(

1 − E(k)

K(k)

)

τ√

πa (4.33)

for an anti-plane shear loading, where

k =

√

1 −(

d − a

d + a

)2

(4.34)

and

K(k) =

∫ π/2

0

1√

1 − k2 sin2 θdθ (4.35)

E(k) =

∫ π/2

0

√

1 − k2 sin2 θdθ (4.36)

To calculate the stress intensity factors with the discontinuous quarter point element

method and the special crack tip element method, the outside boundary was discre-

tised with 24 continuous quadratic elements and three different meshes of 6, 8, and


10 discontinuous quadratic boundary elements were set up on each crack surface; the

elements are graded towards the tips, with the same ratios as mentioned in previous

example. The stress intensity factors were obtained for both tip A and tip B. The

error percentages for the methods are compared for the various crack discretisations,

the results are shown in Figures 4.11 and 4.12 for the mode I SIF and in Figures 4.13

and 4.14 for the mode III SIF. With such coarse meshes, the results obtained with

these methods are remarkably close; the results obtained by using the special crack

tip element method with 6 crack elements differ from the analytical results within

0.1% at tip A and 0.3% at tip B.

Figure 4.11: Relative error of Mode I SIF for the infinite plate with two inclinedcracks at tip A from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elementsand the SCT with (d) 6 elements, (e) 8 elements, and (f) 10 elements


Figure 4.12: Relative error of Mode I SIF for the infinite plate with two inclinedcracks at tip B from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elementsand the SCT with (d) 6 elements, (e) 8 elements, and (f) 10 elements


Figure 4.13: Relative error of Mode III SIF for the infinite plate with two inclinedcracks at tip A from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elementsand the SCT with (d) 6 elements, (e) 8 elements, and (f) 10 elements


Figure 4.14: Relative error of Mode III SIF for the infinite plate with two inclinedcracks at tip B from the QPE with (a) 6 elements, (b) 8 elements, (c) 10 elementsand the SCT with (d) 6 elements, (e) 8 elements, and (f) 10 elements


4.3.3 Infinite Plate with Two Parallel Cracks

As a final test, consider an infinite plate (h/w = 2, a/w = 140

) with two parallel cracks,

as shown in Figure 2.14 for in-plane problem or Figure 3.6 for anti-plane problem,

each crack has the length 2a and the distance between the cracks is 2d. The plate

is subjected to, at the ends, a uniform traction t for the in-plane problem or with

an anti-plane shear τ for the anti-plane problem. The stress intensity factors were

obtained by using three different boundary element meshes of quadratic elements as

mentioned in previous examples. The results are presented in Figures 4.15, 4.16, and

4.17. Comparing with the published results [93]

FI ≈ 1 − 0.293s[

1 − (1 − s)4]

(4.37)

and

FIII ≈ 1 − 0.293s2 (4.38)

where s = a/(a + d), our results showed excellent agreement with a maximum differ-

ence less than 1%; again, convergence was achieved with coarse meshes.


0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Nor

mal

ised

mod

e I S

IF

s

IsidaQPE(06)QPE(08)QPE(10)SCT(06)SCT(08)SCT(10)

Figure 4.15: Mode I SIF for the infinite plate with two parallel cracks from (a) theReference [93], the QPE with (b) 6 elements, (c) 8 elements, (d) 10 elements and theSCT with (e) 6 elements, (f) 8 elements, and (g) 10 elements


0

0.01

0.02

0.03

0.04

0.05

0.06

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Nor

mal

ised

mod

e II

SIF

s

QPE(06)QPE(08)QPE(10)SCT(06)SCT(08)SCT(10)

Figure 4.16: Mode II SIF for the infinite plate with two parallel cracks from the QPEwith (a) 6 elements, (b) 8 elements, (c) 10 elements and the SCT with (d) 6 elements,(e) 8 elements, and (f) 10 elements


0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Nor

mal

ised

mod

e III

SIF

s

IsidaQPE(06)QPE(08)QPE(10)SCT(06)SCT(08)SCT(10)

Figure 4.17: Mode III SIF for the infinite plate with two parallel cracks from (a) theReference [93], the QPE with (b) 6 elements, (c) 8 elements, (d) 10 elements and theSCT with (e) 6 elements, (f) 8 elements, and (g) 10 elements


4.4 Conclusion

Due to the O(1/√

r) and O(√

r) terms contained in the traction and displacement

fields, the standard interpolating polynomial functions can not accurately model dis-

placement and traction behaviours near crack tip. In order to correctly describe

the near crack tip behaviours, we presented the discontinuous quarter point element

method and the special crack tip element method for the evaluation of stress intensity

factors in two dimensional crack problems. In the discontinuous quarter point element

method, the locations of the collocation points needed to be chosen specially, which

depend on the shape functions at the crack tip element. However, in the special crack

tip element method it does not require shifting of the collocation points at the crack

tip element. This feature shows that it is easier to formalise the problem by using the

special crack tip element method. Furthermore, the special crack tip element method

slightly improves the performance because each crack tip element is treated as a sin-

gle unknown parameter only. Several examples indicate that the crack discretisation

has some effect on the accuracy of the numerical results of stress intensity factors.

Both methods obtained accurate results. The special crack tip element method has

a better stability than the discontinuous quarter point element method.

Chapter 5

A New Subregion Boundary

Element Technique

A new subregion boundary element technique is presented in this chapter. This

technique is applicable to the stress analysis of multi-region elastic media, such as

layered-materials. The technique is more efficient than traditional methods because

it significantly reduces the size of the final matrix. This is advantageous when a large

number of elements need to be used, such as in crack analysis. Also, as the system

of equations for each subregion is solved independently, parallel computing can be

utilized. Further, if the boundary conditions are changed the only equations required

to be recalculated are the ones related to the regions where the changes occur. This is

very useful for cases where crack extension is modelled with new boundary elements or

where crack faces come to contact. Numerical examples are presented to demonstrate

the accuracy and efficiency of the method.

87

Chapter 5: A New Subregion Boundary Element Technique 88

5.1 Introduction

Composite materials are increasingly used in various engineering structures, such as

in the aerospace and automotive industries. One of the advantages of these materials

is their ability to be tailored for individual applications. The use of composites could

be potentially limited by the lack of efficient methods to evaluate the strength and life

expectation of composite structures. While defects or micro-cracks are unavoidable,

they do have significant influence on the load transfer behaviour within the com-

posite. Due to the fact that composite materials are made of regions or zones with

different material properties, it is not always possible to utilise the general method for

homogenous materials. Therefore, it is crucial to develop accurate and efficient tech-

niques for numerical analysis of such materials, in case of fracture mechanics analysis,

calculating the stress intensity factor in layered materials with cracks.

A wide variety of analytical and numerical methods have been used to solve the

fracture problems of layered materials [13, 39, 53, 58, 82, 91]. If a straightforward

analytical solution is not possible, numerical procedures must be used. The finite

element method (FEM) is one of the most popular techniques to analyse fracture

problems in composite materials. The interior points have mesh connectivity to the

boundary points and extensive remeshing is required for crack propagation problems.

However, FEM remeshing for each crack length tends to be time consuming. In gen-

eral, the boundary element method (BEM) together with a subregion technique is

widely considered to be a very accurate numerical tool for the analysis of problems

where the materials consist of several homogeneous zones [18, 20]. All the bound-

aries of the body have to be discretised, including internal boundaries that separate

homogeneous zones. The BEM equations, constructed from all homogeneous zones


combined with the interface traction and displacement continuity conditions, produce

a global matrix system. The numerical solution of this matrix system is the most

time consuming step of the numerical method, and hence can be the bottleneck for

the method being applied to problems that require a large number of elements.

Kita & Kamiya [59] presented a special method for the subregion boundary ele-

ment analysis to overcome this disadvantage. The linear system for each subregion

is transformed into equations similar to the stiffness equations of the FEM, and then

the global matrix equation is constructed by superposition of those equations for the

subregions. The matrix equation for each subregion is derived using the algorithm

in Brebbia & Georgiou [19]. This algorithm can be applied easily to objects divided

into subregions. The interface traction components are not obtained in the resulting

matrix system, but can be calculated from the equations for the subregions. The

technique has the advantage that the global coefficient matrix can be constructed

easily and a smaller system of algebraic equations is obtained. This method is more

effective for objects with multiple internal boundaries. However, a relatively large

global coefficient matrix is still needed.

Furthermore, for Kita & Kamiya’s method, in order to deduce the global matrix

system a matrix inversion for each subregion is required, which further increases com-

putation time. The number of numerical operations required for solving a system of

n linear equations and that in finding the inverse of a n × n matrix are of the same

order, n3, so even a slight increase in n increases computational time significantly.

Therefore, the reduction of computing time is an important task in practical cases.

High performance computing techniques, including parallel computing, are now being

applied in many engineering and scientific applications [86]. As a result, developing


efficient numerical algorithms specifically aimed at high performance computers be-

comes a challenging issue.

In this chapter, an efficient technique for multi layer crack problems is proposed.

The matrices of all subregions are used to assemble the final interface traction ma-

trix, which is solved for the unknown interface traction components. Unlike other

methods which solve the displacement and traction components on the boundaries

and interfaces at the same time, the distribution of traction on the interfaces is ob-

tained first. The displacement components can then be calculated from the equations

associated with the corresponding subregions. Initially, extra numerical steps maybe

needed to set up the final interface matrix equation. However, our final matrix sys-

tem is significantly smaller than the final matrix systems obtained by other methods.

If the boundary conditions are changed, only the equations for the subregions con-

cerned need to be recalculated. Therefore, it greatly reduces computational time, and

provides overall efficiency.

The effects of crack size, layer size, and the material properties of the composite

on the stress intensity factor are studied using the proposed numerical technique to

demonstrate its accuracy and efficiency. The dual boundary element method (DBEM)

[25, 80, 66] is incorporated into the present method to overcome the singularity in

crack analysis. Further, in order to improve accuracy in the stress intensity factor

calculation, discontinuous quarter point elements [42, 85] are used to model the near

tip elements.


5.2 The Multi Region Technique of Boundary El-

ement Method

Consider a two dimensional body consisting of several subregions. For any subre-

gion that contains no cracks, the displacement formulation of the boundary integral

equation, at a boundary point x′, is written in the form (the body force term is

neglected)

cij(x′)uj(x

′) +

∫

Γ

− Tij(x′,x)uj(x)dΓ(x) =

∫

Γ

Uij(x′,x)tj(x)dΓ(x) (5.1)

where∫

− stands for the Cauchy principal value integral. uj(x) and tj(x) are dis-

placement and traction components in the j direction, respectively. If the boundary

is smooth, cij(x′) = 1

2δij, where δij is the Kronecker delta. The kernel functions

Tij(x′,x) and Uij(x

′,x) represent the Kelvin traction and displacement fundamental

solutions, respectively, at the boundary point x. For any subregion containing cracks,

the DBEM is employed. The dual equations of the DBEM are the displacement and

the traction boundary integral equations. The traction equation, which is applied on

the crack surfaces, is obtained by differentiation of the displacement equation (5.1),

and followed by the application of Hooke’s law. It is written as

1

2tj(x

′) + ni(x′)

∫

Γ

= Skij(x′,x)uk(x)dΓ(x)

= ni(x′)

∫

Γ

− Dkij(x′,x)tk(x)dΓ(x) (5.2)

where∫

= stands for the Hadamard principal value integral, ni denotes the ith compo-

nent of the unit outward normal to the boundary at a boundary point x′. Skij(x′,x)


and Dkij(x′,x) are linear combinations of derivatives of Tij(x

′,x) and Uij(x′,x), re-

spectively. The displacement integral equation (5.1) and the traction integral equa-

tion (5.2) are the governing equations to be solved for the displacement on the outer

boundary and the relative displacement on the crack faces.

We consider a three-subregion problem shown in Figure 5.1. In order to solve the

integral equations numerically, the boundary is discretised into a series of elements

on which displacement and traction components are written in terms of their values

at the nodal points. There are n1, n2 and n3 nodes placed on outer boundaries of the

subregions, m12 and m23 nodes on the interface between subregions, and nc nodes on

the crack face. Let ui and ti denote the nodal displacement and traction vectors on

Figure 5.1: A three subregion medium

boundary Γi respectively. Then, for the non-cracked subregions Ω1 and Ω3, the BEM


equations can be written together in matrix form:(

h11 h12

h21 h22

)(

u1

u2

)

=

(

g11 g12

g21 g22

)(

t1

t2

)

(5.3)

and(

h77 h78

h87 h88

)(

u7

u8

)

=

(

g77 g78

g87 g88

)(

t7

t8

)

(5.4)

For the cracked subregion Ω2, referring to [66], the DBEM equations can be written

together in matrix form:

h33 h34 h35 h36

h43 h44 h45 h46

h53 h54 h55 h56

h63 h64 h65 h66

u3

u4

u5

∆u6

=

g33 g34 g35 g36

g43 g44 g45 g46

g53 g54 g55 g56

g63 g64 g65 g66

t3

t4

t5

t+6

(5.5)

where ∆u6 is the relative displacement vector on the crack surfaces and t+6 is the

traction vector on the upper crack surface.

The interface traction and displacement continuity conditions are

t2 = −t3, u2 = u3 (5.6)

between Ω1 and Ω2, and

t5 = −t7, u5 = u7 (5.7)

between Ω2 and Ω3.

In each subregion, we have more unknowns than the number of equations. To

solve the problem in the subregion level, we will have to use iterative methods which

are time consuming. A common practice would be to form a global matrix system

using the interface boundary conditions, such as in [18, 20, 59]. However, the resulting

matrix system is large that it is not cost effective. We propose a new subregion BEM

to form a smaller global matrix system. The differences between the current approach

and other approaches are discussed in the next section.


5.3 Comparison of Subregion BEM Techniques

5.3.1 The Traditional Method

The traditional subregion BEM generate an N × N final matrix system based upon

equations (5.3), (5.4) and (5.5), and the interface conditions (5.6) and (5.7).

Au = Bt (5.8)

where

A =

h11 h12 0 0 0 0 −g12 0

h21 h22 0 0 0 0 −g22 0

0 h33 h34 h35 h36 0 g33 −g35

0 h43 h44 h45 h46 0 g43 −g45

0 h53 h54 h55 h56 0 g53 −g55

0 h63 h64 h65 h66 0 g63 −g65

0 0 0 h77 0 h78 0 g77

0 0 0 h87 0 h88 0 g87

, B =

g11 0 0 0

g21 0 0 0

0 g34 g36 0

0 g44 g46 0

0 g54 g56 0

0 g64 g66 0

0 0 0 g78

0 0 0 g88

u =(

u1 u2 u4 u5 ∆u6 u8 t2 t5

)T

, t =(

t1 t4 t+6 t8

)T

and N = 2(n1 + n2 + n3 + 2m12 + 2m23 + nc). The above matrix system may be

solved once the boundary conditions are prescribed. However, much computing time

and memory cost are required.

5.3.2 Kita & Kamiya’s Method

Both sides of equations (5.3), (5.4), and (5.5) are multiplied by the inverses of right

hand side matrices, respectively, to obtain the following boundary matrix equations(

a11 a12

a21 a22

)(

u1

u2

)

=

(

t1

t2

)

(5.9)


(

a77 a78

a87 a88

)(

u7

u8

)

=

(

t7

t8

)

(5.10)

and

a33 a34 a35 a36

a43 a44 a45 a46

a53 a54 a55 a56

a63 a64 a65 a66

u3

u4

u5

∆u6

=

t3

t4

t5

t+6

(5.11)

where aij are the corresponding assembled matrices. Once the interface conditions

(5.6) and (5.7) are applied, Kita & Kamiya’s method then generates a N × N matrix

system

a11 a12 0 0 0 0

a21 a22 + a33 a34 a35 a36 0

0 a43 a44 a45 a46 0

0 a53 a54 a55 + a77 a56 a78

0 a63 a64 a65 a66 0

0 0 0 a87 0 a88

u1

u2

u4

u5

∆u6

u8

=

t1

0

t4

0

t+6

t8

(5.12)

where N = 2(n1 + n2 + n3 + m12 + m23 + nc). Once the interface displacement

components are solved from (5.12), the traction components can be calculated from

the systems of equations (5.9), (5.10) and (5.11). A smaller matrix system needs to

be solved compared with the traditional subregion BEM. However, three matrices of

the sizes N1 × N1, N2 × N2 and N3 × N3 need to inverted before the final matrix

can be established, where N1 = 2(n1 + m12), N2 = 2(n2 + m12 + m23 + nc) and

N3 = 2(n3 + m23). Since the number of operations required for inverting an n × n

matrix is of order n3, it offsets the savings provided by having a smaller global matrix.

That is the drawback of this method.


5.3.3 The Proposed Method

In this section we propose a method that allows us to solve most of the problem at

the subregion level. In each subregion, we use the outer boundary conditions and

interface traction to represent the interface displacement. Upon applying the interfa-

cial equilibrium and compatibility conditions, we obtain a global matrix equation. In

this equation the interface traction is the only unknown. Once the matrix equation

is solved, the rest of the unknowns can then be solved at the subregion level on a

need to know basis. The interface traction matrix equation can be formed by a direct

inverse matrix method or a domain domain decomposition method.

The Direct Inverse Matrix Method

In the direct inverse matrix method [64], we multiply both sides of equations (5.3),

(5.4), and (5.5) by the inverse matrices of left hand side matrices to obtain

(

u1

u2

)

=

(

c11 c12

c21 c22

)(

t1

t2

)

(5.13)

(

u7

u8

)

=

(

c77 c78

c87 c88

)(

t7

t8

)

(5.14)

u3

u4

u5

∆u6

=

c33 c34 c35 c36

c43 c44 c45 c46

c53 c54 c55 c56

c63 c64 c65 c66

t3

t4

t5

t+6

(5.15)

where cij are the corresponding assembled matrices. Combing equations (5.13), (5.14)

and (5.15) with the equilibrium and compatibility conditions (5.6) and (5.7) on the

interface between the subregions, we obtain a 2(m12 + m23)× 2(m12 + m23) interface


matrix equation

(

c22 + c33 c35

c53 c55 + c77

)(

t3

t5

)

=

(

c21 −c34 −c36 0

0 −c54 −c56 c78

)

t1

t4

t+6

t8

(5.16)

Once the interface traction components t3 and t5 are solved from (5.16), the displace-

ments can be calculated from equations (5.13), (5.14) and (5.15).

If part of the crack faces come to contact, instead of recalculating equation (5.5),

we only need to add an extra contact boundary condition on the final matrix equation

(5.16). Let ∆u6 = ∆uc6 + ∆uo

6, where ∆uc6 is the relative displacement vector of the

contact part on the crack faces and ∆uo6 is that of the open part on the crack faces.

Then the extra boundary condition is ∆uc6 = 0. For example, if the crack is fully

closed, the extra contact boundary condition can be obtained from (5.15), which is

0 = ∆u6 = ( c63 c64 c65 c66 )( t3 t4 t5 t+6 )T (5.17)

Therefore, the modified final matrix equation is

c22 + c33 c35 c36

c53 c55 + c77 c56

−c63 −c65 −c66

t3

t5

t+6

=

c21 −c34 0

0 −c54 c78

0 c64 0

t1

t4

t8

(5.18)

The Domain Decomposition Method

Instead of finding the inverse matrices directly, we can use the domain decomposition

method [51, 54, 83, 95] to form the interface traction matrix equation. In (5.3) and

(5.4), the outside boundary displacements can be expressed as

ui = h−1ii (giiti + gijtj − hijuj) (5.19)


The displacements and tractions on the interface can then be easily written as

uj = Djjtj + Qj (5.20)

where

C−1jj = (hjj − hjih

−1ii hij)

−1

Djj = C−1jj (gjj − hjih

−1ii gij)

Qj = C−1jj (gji − hjih

−1ii gii)ti

and i = 1, j = 2 in Ω1, and i = 8, j = 7 in Ω3.

Rearrange (5.5) to allow terms (u4, ∆u6) together, these vectors can be expressed

as

(

u4

∆u6

)

= A−1

(

g43t3 + g44t4 + g46t+6 + g45t5 − h43u3 − h45u5

g63t3 + g64t4 + g66t+6 + g65t5 − h63u3 − h65u5

)

(5.21)

where

A−1 =

(

h44 h46

h64 h66

)−1

Let

B =

(

g44 g46

g64 g66

)

The unknown interface terms, i = 3 on the interface Ω1

⋂

Ω2 and i = 5 on the

interface Ω2

⋂

Ω3, can then be written as

Ci3u3 − Di3t3 + Ci5u5 − Di5t5 = Qi (5.22)

where

Ci3 = hi3 − (hi4,hi6)A−1

(

h43

h63

)


Di3 = gi3 − (hi4,hi6)A−1

(

g43

g63

)

Ci5 = hi5 − (hi4,hi6)A−1

(

h45

h65

)

Di5 = gi5 − (hi4,hi6)A−1

(

g45

g65

)

Qi = [(gi4,gi6) − (hi4,hi6)A−1B]

(

t4

t+6

)

Upon applying the interface conditions (5.6) and (5.7) and substituting (5.20) into

(5.22), we obtain a 2(m12 + m23) × 2(m12 + m23) interface matrix equation,

(

C33D22 + D33, C35D77 + D35

C53D22 + D53, C55D77 + D55

)(

t2

t7

)

=

(

Q3 − C33Q2 − C35Q7

Q5 − C53Q2 − C55Q7

)

(5.23)

Once the interface tractions t2(= −t3) and t7(= −t5) are solved from (5.23), the

displacements can be calculated from the systems of equations (5.19), (5.20), and

(5.21).

The domain decomposition method has the advantage for cases where crack exten-

sion is modelled with new boundary elements. When boundary conditions changed

only part of the coefficient matrices need to be recalculated.

Both methods, the direct inverse matrix method and the domain decomposition

method, will derive the same final matrix equations. The direct inverse matrix method

is straightforward in formulation. It is useful for analyses when boundary conditions

remain the same. It is also useful when crack faces come to partial or full contact

because only the final matrix system needs to be modified. On the other hand,

in using the domain decomposition method, one does not need to solve as many

inverse matrices compared to the use of the direct inverse matrix method. It would


provide further savings in computing time. Therefore, it is more efficient under general

boundary conditions.

The proposed method is suitable for parallel computation because the coefficient

matrix for each subregion can be calculated independently. This implementation

achieves a perfect scaling of the memory consumed per process, since each process

needs to access only a dedicated subregion of the entire domain. It enables us to

refine the boundary conditions in any subregion without having to recalculate others.

This feature is very useful for numerical analysis of crack extension, and cases where

crack faces come into contact.

Our method provides significant savings in computational time and memory usage.

Unlike other methods, which calculate the unknown components on the boundaries

and interfaces at the same time, only traction components on the interfaces are cal-

culated in the final matrix. As a result, the size of our final matrix is the number

of nodal degrees of freedom over the interfaces, and it is much smaller than those of

other methods.

Above all, our method has the advantage over the others because the reduction of

final matrix size and the ability of accessing each subregion independently. It results

in savings of the computing time and memory usage.

5.4 Numerical Results

In order to demonstrate the accuracy and efficiency of the proposed method and to

show its possible applications, several examples are presented here. All the calcula-

tions are carried out under plane strain conditions with a tensile loading T .

In the first example, consider the stress analysis of perfectly bonded dissimilar


elastic semi-strips, as illustrated in Figure 5.2, where h1 = h2 = 2w. 30 elements

Figure 5.2: A perfectly bonded dissimilar elastic semi-strip

are placed on the outer boundary of each semi-stripe, and 20 elements on the inter-

face. Two cases are considered, with (E2/E1, ν1, ν2) = (9.0, 0.5, 0.5) and (3.0, 0.5, 0.5),

where E1 and E2 are the Young’s moduli, and ν1 and ν2 are the Poisson’s ratios, re-

spectively. The normalised normal stress distribution on the interface are shown in

Figure 5.3. The results agree well with published results [17].


x/w

Nor

mal

ised

norm

alst

ress

dist

ribu

rtio

non

inte

rfac

e

0.5 0.6 0.7 0.8 0.9 10.9

1

1.1

1.2

1.3

1.4

1.5(a)(b)(c)(d)

Figure 5.3: Normalised normal stress distribution on the interface(E2/E1, ν1, ν2) = (9.0, 0.5, 0.5): (a) Ref. [17], (b) the present method(E2/E1, ν1, ν2) = (3.0, 0.5, 0.5): (c) Ref. [17], (d) the present method


Consider, now, a three layered plate with a crack in the middle layer as shown

in Figure 5.4, where h1 = h2 = h3 = 0.5w, a/w = 0.1. The middle layer with

Figure 5.4: A three-layer plate with a centre crack

shear modulus µ2 and Poisson’s ration ν2 is perfectly bonded between two layers

having identical elastic properties µ1 = µ3 and ν1 = ν3. A crack of length 2a is

located l distance away from the top layer. Discontinuous quadratic elements are

used to discretise the boundaries, 36 elements on each subregion and 6 elements on

the crack surface. The stress intensity factor is normalised with respect to K0 =

T√

πa, calculated for various ratios of µ1/µ2 while ν1 = ν2 = ν3 = 0.3. Figure

5.5 shows the normalised mode I stress intensity factor versus the ratio of the shear

moduli (µ1/µ2) for various values of l/h2. The stress intensity factor increases as the

shear modulus ratio (µ1/µ2) decreases. It should be mentioned that the calculated

normalised stress intensity factor approaches 1 when the ratio 2a/h2 is small enough.


It is expected, as this case is equivalent to an infinite homogeneous plate with a

central crack. Compared with the results calculated using other methods, the present

results match those of the traditional BEM method within eight decimal places.

µ1/µ2

Nor

mal

ised

mod

eI

SIF

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6(a)(b)(c)(d)(e)

Figure 5.5: Normalised mode I stress intensity factor (SIF) on the three-layer platewith a centre crack: (a) l/h2 = 0.1, (b) l/h2 = 0.2, (c) l/h2 = 0.3, (d) l/h2 = 0.4, (e)l/h2 = 0.5


Finally, consider a three layered plate with two identical co-linear cracks of length

2a located symmetrically in the middle layer as shown in Figure 5.6, where h1 =

h2 = h3 = 0.5w, a/w = 0.1. The distance between the centres of the cracks is

2d = 2.4a. The elastic properties are taken as µ1 = µ3 and ν1 = ν2 = ν3 = 0.3.

36 discontinuous boundary elements are placed on each subregion boundary, and 6

Figure 5.6: A three-layer plate with two identical co-linear cracks

discontinuous elements on each crack. Again the normalised mode I stress intensity

factor is calculated for various ratios of the shear moduli (µ1/µ2), the results are

shown in Figure 5.7 for tip A and Figure 5.8 for tip B. It is noted that the stress

intensity factor increases as the shear modulus ratio (µ1/µ2) decreases. Further, due

to the interaction between the two cracks, the mode I stress intensity factor at the

crack tip A is always smaller than that at the crack tip B.


µ1/µ2

Nor

mal

ised

mod

eI

SIF

attip

A

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.1

1.2

1.3

1.4

1.5

1.6

1.7(a)(b)(c)(d)(e)

Figure 5.7: Normalised mode I SIF at tip A on the three-layer plate with two co-linearcracks: (a) l/h2 = 0.1, (b) l/h2 = 0.2, (c) l/h2 = 0.3, (d) l/h2 = 0.4, (e) l/h2 = 0.5


µ1/µ2

Nor

mal

ised

mod

eI

SIF

attip

B

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.1

1.2

1.3

1.4

1.5

1.6

1.7(a)(b)(c)(d)(e)

Figure 5.8: Normalised mode I SIF at tip B on the three-layer plate with two co-linearcracks: (a) l/h2 = 0.1, (b) l/h2 = 0.2, (c) l/h2 = 0.3, (d) l/h2 = 0.4, (e) l/h2 = 0.5


5.5 Conclusion

Three examples were examined using the new subregion boundary element technique.

Compared with other methods, our method was shown to be very effective and ac-

curate for the boundary element analysis of an object composed of subregions. Our

method achieved a substantial reduction in the number of numerical computations.

For example, for a problem involving three subregions, only a small number of ele-

ments are used on the boundary of each subregion, the number of computations is

reduced by 43%. This shows that our method has the advantages of significantly

reducing computing time and memory usage. Furthermore, it would be advantageous

to use the method for solving problems with changing boundary conditions such as

those encountered in crack growth analysis. Because each calculation in each sub-

region is independent, only those equations related to the regions where changes of

crack boundary conditions occur need to be recalculated.

Chapter 6

A Subregion DRBEM Formulation

for the Dynamic Analysis of Two

Dimensional Cracks

The dual reciprocity boundary element method employing the step by step time inte-

gration technique is developed to analyse two-dimensional dynamic crack problems.

In this method the equation of motion is expressed in boundary integral form us-

ing elastostatic fundamental solutions. In order to transform the domain integral

into an equivalent boundary integral, a general radial basis function is used for the

derivation of the particular solutions. The dual reciprocity boundary element method

is combined with an efficient subregion boundary element method to overcome the

difficulty of a singular system of algebraic equations in crack problems. Dynamic

stress intensity factors are calculated using the discontinuous quarter point elements.

Several examples are presented to show the formulation details and to demonstrate

the computational efficiency of the method.

109

Chapter 6: A Subregion DRBEM Formulation for Dynamic Analysis 110

6.1 Introduction

Stresses and displacements in dynamic loading can be greatly different to those as-

sociated with static loading. The equation of motion is also more complex than the

corresponding equilibrium equation for the same configuration and material class.

Therefore, structures with arbitrary shapes and time dependent boundary conditions

need to be analysed using numerical methods. One numerical method which has

had great success in solving elastodynamic problems is the boundary element method

(BEM) [34]. Various numerical techniques using BEM are usually implemented by

using either the time domain method, the Laplace transform method or the dual

reciprocity boundary element method (DRBEM).

The time domain method combined with the dual boundary element method was

used by Fedelinski et al. [42, 45]. A hypersingular formulation for time domain anti-

plane elastodynamic problems was developed by Gallego and Dominguez [46]. And a

three dimensional time domain boundary integral equation method was presented by

Zhang and Gross [104] for transient elastodynamic crack analysis. All of these cur-

rent methods possess advantages and disadvantages. For example, the time domain

method requires time and space integrations. These integrations have a complicated

form, resulting in the time domain method being difficult to implement. Furthermore,

this method is computationally intensive, as it requires a large amount of computation

time because of its double multiplication.

The Laplace transform method was used by Sladek and Sladek [88, 89] to solve

elastodynamic problems. The boundary value problems were formulated by the

boundary integral element method in the Laplace transform domain. This method

was combined with the numerical Laplace inversion algorithm of Durbin [37]. The


Laplace transform method combined with the dual boundary element method was

used by Fedelinski et al. [44] to solve two-dimensional dynamic crack problems. The

dynamic problems can then be solved with a single region formulation in the Laplace

transform domain. Three-dimensional dynamic problems using a similar method were

also solved by Wen et al. [98]. In the Laplace transform method, only the space in-

tegration is required, although this transform depends on complex Bessel functions.

The generation of equations in this method is time consuming because of the com-

plexity of the transformed fundamental solutions. The accuracy and efficiency of the

Laplace transform method depend on the level of difficulty of the numerical inversion

of the Laplace transform.

A considerable improvement on the time domain and Laplace transform methods

is the dual reciprocity boundary element method (DRBEM). In DRBEM, the equa-

tion of motion is expressed as a boundary integral equation by using the fundamental

solutions of elastostatics and the approximation of the acceleration. The fundamental

solutions of elastostatics are time independent, thus only space integration is required.

The acceleration can be obtained by approximation functions multiplied by unknown

time dependent coefficients. The approximation functions do not require integration.

Radial basis functions are chosen for the approximation functions and used for deter-

mining particular solutions of the displacement and traction vectors in DRBEM. This

method has an easier formulation with direct application to dynamic problems. The

DRBEM combined with the dual boundary element method was used by Fedelinski et

al. [40, 41]. Various radial basis functions were implemented by Agnantiris et al. in

the solutions of the two dimensional and three dimensional symmetric elastodynamic

problems using the DRBEM [3, 4]. The vibration problem of a thin plate was solved


by Davies and Moslehy [33] using the DRBEM. However, the coincidence nodes on

the crack surfaces give rise to a singular system of algebraic equations. The unknown

time dependent coefficients cannot be calculated by the direct solution of this sys-

tem. This difficulty is usually overcome by using either the subregion BEM or the

dual boundary element method, see Fedelinski et al. [40].

Although DRBEM possesses certain obvious advantages over the other methods,

the algebraic singularities present in the method have not yet been adequately dealt

with. An efficient subregion BEM is generally needed to overcome the singular system

of algebraic equations on the coincidence crack nodes. In this chapter, we propose a

DRBEM formulation based on an advanced subregion boundary element technique

[64, 67] to address the singularity problem. A structure is divided into sub-domains

along crack surfaces. The interface equations are assembled using the interface equi-

librium and compatibility conditions. The treatment of time dependent functions

with the DRBEM is dealt directly in the time domain. The Houbolt method [78, 12]

will be used for the direct step by step time integration procedures. Since the system

of equations for each subregion are pre-solved independently, only the time depen-

dent equations need to be recalculated. This formulation is very effective due to the

reduction in the size of final matrix and in the number of intermediate steps. Numer-

ical examples are provided to demonstrate the accuracy and efficiency of the present

formulation. The effect of internal collocation points on numerical accuracy is also

examined.


6.2 Dual Reciprocity Boundary Element Method

Consider an elastic isotropic homogeneous body occupying any two dimensional do-

main Ω, enclosed by a boundary Γ. The differential equation of motion can be

expressed as follows:

σjk,k + bj − ρuj = 0 (6.1)

where σjk is the cartesian stress tensor, uj is the displacement vector, bj is the body

force, ρ denotes the mass density, subscript k proceeded by a comma denotes differen-

tiation with respect to the coordinate xk and dots denote differentiation with respect

to time. The equation can also be expressed in terms of the displacement

µuj,kk +µ

1 − 2νuk,kj + bj − ρuj = 0 (6.2)

where µ is the shear modulus and ν is the Poisson’s ratio.

In the absence of body forces, the boundary element formulation for elastodynamic

problems can be expressed as

ui(X′, τ) −

∫

Γ

Uij(X′,x)tj(x, τ)dΓ(x) +

∫

Γ

Tij(X′,x)uj(x, τ)dΓ(x)

= −ρ

∫

Ω

Uij(X′,X)uj(X, τ)dΩ(X) (6.3)

where ui is the displacement component in the i-direction at internal point X′ and

time τ , Uij(X′,x) and Tij(X

′,x) represent the static Kelvin displacement and traction

fundamental solutions, respectively. For a boundary point (i.e. X′ → x′), equation

(6.3) can be written as

cij(x′)uj(x

′, τ) −∫

Γ

Uij(x′,x)tj(x, τ)dΓ(x) +

∫

Γ

− Tij(x′,x)uj(x, τ)dΓ(x)

= −ρ

∫

Ω

Uij(x′,X)uj(X, τ)dΩ(X) (6.4)


where∫

− stands for a Cauchy principal value integral. Coefficient cij(x′) depends on

boundary point x′, on a smooth boundary cij(x′) = 1

2δij. The DRBEM will be used

to transform the domain integral in equation (6.4) into a boundary integral.

The acceleration uj at a point X and time τ is approximated as a sum of m

coordinate functions fn(X′′,X) multiplied by unknown time-dependent coefficients

αnj (τ):

uj(X, τ) =m∑

n=1

αnj (τ)fn(X′′,X) (6.5)

Equation (6.5) is valid over the whole domain. X′′ can be a boundary or a domain

point (see Figure 6.1). Using (6.5), the domain term of the displacement integral

Figure 6.1: Boundary and internal nodes

equation (6.4) can be written as

ρ

∫

Ω

Uij(x′,X)uj(X, τ)dΩ(X) = ραn

l (τ)

∫

Ω

Uij(x′,X)δljf

n(X′′,X)dΩ(X) (6.6)

where αnj (τ) = αn

l (τ)δlj, and δlj is the Kronecker delta. In order to transform the

domain integral into equivalent boundary integral, we need to find particular solutions


unlj and tnlj of the following differential equation:

σljk,k − δljfn = 0, (6.7)

in the absence of body forces. The domain integral in equation (6.6) is rewritten as

ραnl (τ)

∫

Ω

Uij(x′,X)δljf

n(X′′,X)dΩ(X)

= −ραnl (τ)

[

cij(x′)un

lj(X′′,x′) −

∫

Γ

Uij(x′,x)tnlj(X

′′,x)dΓ(x)

+

∫

Γ

− Tij(x′,x)un

lj(X′′,x)dΓ(x)

]

(6.8)

where the particular displacement unlj(X

′′,x) and traction tnlj(X′′,x) are given in Sec-

tion 6.3. Substituting equation (6.8) into equation (6.4), the displacement boundary

equation can be written as

cij(x′)uj(x

′, τ) −∫

Γ

Uij(x′,x)tj(x, τ)dΓ(x) +

∫

Γ

− Tij(x′,x)uj(x, τ)dΓ(x)

= ραnl (τ)

[

cij(x′)un

lj(X′′,x′) −

∫

Γ

Uij(x′,x)tnlj(X

′′,x)dΓ(x)

+

∫

Γ

− Tij(x′,x)un

lj(X′′,x)dΓ(x)

]

(6.9)

In order to obtain a solution of the elastodynamic problem, the boundary is divided

into small boundary elements. We obtain a set of equations written in matrix form

as

Hu − Gt = ρ(

Hu − Gt)

α (6.10)

The relationship between u and α is established by applying equation (6.5) to every

boundary and domain node. The resulting set of equations can be written in matrix

form:

u(x, τ) = F(X′′,x)α(τ) (6.11)


where the elements of the matrix F(X′′,x) are the values of function fn(X′′,x) at all

m nodes. The unknown coefficients α can be expressed as

α(τ) = F−1(X′′,x)u(x, τ) (6.12)

If the coincident nodes of the crack are used in the approximation of the acceleration

field, the system of equations (6.11) will be singular. Therefore, the coefficients α

cannot be calculated by the direct solution of this system. A new subregion BEM

[67] is used to overcome this difficulty (see Chapter 5). Substituting equation (6.12)

into equation (6.10), we obtain

Hu − Gt = ρ(

Hu − Gt)

F−1u (6.13)

Equation (6.13) can be written in the compact form as

Mu + Hu = Gt (6.14)

where

M = −ρ(

Hu − Gt)

F−1 (6.15)

The system of equations (6.14) is modified, according to the boundary conditions,

and can be solved using a direct integration method.

The Houbolt integration scheme [12, 78] is used to approximate the acceleration

components in terms of the displacement components. The time span under consid-

eration, T , is divided into N equal time intervals ∆τ (T = N · ∆τ). Assuming that

the solution of (6.14) is known at 0, ∆τ, 2∆τ, . . . , T , the following algorithm is used

to obtain the acceleration at time τ +∆τ , and is derived by using the finite difference

expression

uτ+∆τ =1

(∆τ)2(2uτ+∆τ − 5uτ + 4uτ−∆τ − uτ−2∆τ ) (6.16)


The general system of equation (6.14) at time τ + ∆τ can be written as

Muτ+∆τ + Huτ+∆τ = Gtτ+∆τ (6.17)

Upon substituting (6.16) into (6.17) one obtains

[

2

(∆τ)2M + H

]

uτ+∆τ = Gtτ+∆τ + M1

(∆τ)2(5uτ − 4uτ−∆τ + uτ−2∆τ ) (6.18)

or

Auτ+∆τ = Gtτ+∆τ + M (a1uτ + a2uτ−∆τ + a3uτ−2∆τ ) (6.19)

where A = 2/(∆τ)2M + H, a1 = 5/(∆τ)2, a2 = −4/(∆τ)2, and a3 = 1/(∆τ)2.

The resulting boundary integral equations on a subregion at time τ+∆τ , rewritten

from equation (6.19), are

A+u+τ+∆τ = G+t+

τ+∆τ + M+(

a1u+τ + a2u

+τ−∆τ + a3u

+τ−2∆τ

)

(6.20)

on the subregion with the upper crack face, Ω+ and

A−u−τ+∆τ = G−t−τ+∆τ + M−

(

a1u−τ + a2u

−τ−∆τ + a3u

−τ−2∆τ

)

(6.21)

on the subregion with the lower crack face, Ω− (see Figure 6.2). Using the new

subregion boundary element method in Chapter 5, we combine equations (6.20) and

(6.21) with the interface equilibrium and compatibility conditions

t+2 = −t−1 , t+

4 = −t−3 and u+2 = u−

1 , u+4 = u−

3 (6.22)

to obtain the following interface traction matrix

[

b21 b22

b41 b42

][

t+2

t+4

]

τ+∆τ

=

[

b23 b24 b25 b26

b43 b44 b45 b46

]

t+1

t+3

t−2

t−4

τ+∆τ

+

[

w1

w2

]

τ+∆τ

(6.23)


Figure 6.2: Crack on the body for subregion method

where bij, and wi are matrix coefficients from A+,−, G+,−, M+,−, and ai. Once

the unknown interface traction components are solved from equation (6.23), the dis-

placement components can be obtained from equations (6.20) and (6.21) subject to

boundary conditions.

6.3 Derivation of Particular Solutions

In this section, a general method for analytically obtaining a particular solution of

equation (6.7) for two dimensional problems is described as in [3, 4]. A polynomial

radial basis function is used for deriving the solutions of the particular displacement

and traction vectors. The expression for the approximating radial basis function

fn(X′′,x) is

fn(X′′,x) =

q∑

p=0

Ap [r(X′′,x)]p

(6.24)


where r(X′′,x) is the distance between the point X′′ and the boundary point x.

The point X′′ can be located in the domain or at the boundary. The particular

displacement solution unlj(X

′′,x) is obtained from equation (6.7) using equation (6.24)

unlj(X

′′,x) =1

µ

q∑

p=0

Ap

[

(4p + 14) − 4(p + 4)ν

4(1 − ν)(p + 2)2(p + 4)rp+2δlj

− 1

2(1 − ν)(p + 2)(p + 4)rp+2r,lr,j

]

(6.25)

The corresponding expression for the particular traction solution tnlj(X′′,x) is

tnlj(X′′,x) =

q∑

p=0

Ap

[

1

2(1 − ν)

(

1 − 2ν

p + 2+

1

p + 4

)

rp+1 (r,lnj + r,knkδlj)

− prp+1

(1 − ν)(p + 2)(p + 4)r,knkr,lr,j

+1

2(1 − ν)

(

−1 − 2ν

p + 2+

1

p + 4

)

rp+1nlr,j

]

(6.26)

6.4 The Dynamic Stress Intensity Factors

One of the important parameters in dynamic fracture mechanics analysis is the dy-

namic stress intensity factor (DSIF) since it characterizes the stress field in the vicinity

of the crack and controls crack growth. In the present approach it is calculated di-

rectly from the displacement components near the crack tip. In order to improve the

accuracy in calculating the dynamic stress intensity factors, the discontinuous quarter

point element method is applied [16, 71]. The dynamic stress intensity factors have

the same definition as in a static problem except that they are time dependent. The

dynamic stress intensity factors for mode I and mode II can be calculated from the

crack opening displacement as follows

KI,II =µ

κ + 1

√

2π

r∆un,t(r, τ) (6.27)


where κ = 3−4ν for plane strain, r is the distance from crack tip to the nearest node

on the crack face. ∆un(r, τ) and ∆ut(r, τ) denote the relative normal and tangential

displacement at a distance r to the crack tip at time τ respectively.


In order to demonstrate the accuracy and efficiency of the present method, we present

three examples. The first example shows an application to a rectangular plate with

a central crack. The second example considers the analysis of a rectangular plate

with a central slant crack. The solutions of these two examples are compared with

other published solutions [7, 8, 43, 76]. The third example shows the analysis of a

rectangular plate with an internal kinked crack. The dynamic stress intensity factors

are obtained from crack opening displacement calculations by using the discontinuous

quarter point element method. In each case the structure is subjected to a uniform

tension

σ(τ) = σ0H(τ) (6.28)

where H(τ) is the Heaviside step function. The plate is under plane strain condition.

The dynamic stress intensity factors are normalised with respect to

K0 = σ0

√πa (6.29)

where a defines the half cracklength, except in the third example, where a different

normalising factor is more suitable.


6.5.1 A Rectangular Plate with a Central Crack

Consider a rectangular plate of length 2h = 40 mm and width 2w = 20 mm containing

a central crack. The crack has length 2a = 4.8 mm, as shown in Figure 6.3. The

Figure 6.3: Rectangular plate with a central crack

material properties are: the shear modulus µ = 7.692 × 1010Pa, Poisson’s ratio

ν = 0.3, and density ρ = 5 × 103kg · m−3. Discontinuous quadratic elements are

used to discritise the boundary. There are 30 elements and 21 additional domain

points used on each subregion, in which 6 elements are on the crack face. The time

step ∆τ = 0.2 µs is used here. f(X′′,x) = 1 + r(X′′,x) is used as the approximation

function of the particular solutions. The mode I dynamic stress intensity factor vs.

time τ is plotted in Figure 6.4. It is seen that at time τ = 6.8 µs, the first peak

value of the normalised mode I dynamic stress intensity factor is 2.52, and compares

well with the results published in Reference [6], where the peak values are 2.54 at

Chapter 6: A Subregion DRBEM Formulation for Dynamic Analysis 122 Time 6.6 µs for the dual reciprocity method and 2.67 at time 6.4 µs for the Laplace transform

method, respectively.

Figure 3.7: Normalised mode I DIF for the rectangular plate with a central crack (a) the present method, (b) the dual reciprocity method [6] and (c) the Laplace transform method [6]



6.5.2 A Rectangular Plate with a Central Slant Crack


a central slant crack. The crack has length 2a = 14.14 mm slanted with an angle

θ = 45 from the horizontal direction, as shown in Figure 6.5. The material properties

Figure 6.5: Rectangular plate with a central slant crack

are: the shear modulus µ = 7.692 × 1010Pa, Poisson’s ratio ν = 0.3, and density

ρ = 5× 103kg ·m−3. The discontinuous quadratic elements are used to discretise the

boundary. There are 36 elements and 25 additional domain points on each subregion,

in which 6 elements are on the crack face. The time step ∆τ = 0.3 µs is used

here. f(X′′,x) = 1 + r(X′′,x) is used as the approximation function of the particular

solutions. The mode I dynamic stress intensity factor vs. time τ is plotted in Figure

6.6. The first peak value of the normalised mode I dynamic stress intensity factor

is 1.26 at τ = 10.8 µs, and compares well with the dual reciprocity and the Laplace


transform methods whose peak values are 1.27 at time 11.6 µs and 1.31 at time 11.3 µs,

respectively. The mode II dynamic stress intensity factor vs. time τ is plotted in Figure 6.7. At τ =

10.2 µs, the first peak value of the normalized mode II dynamic stress intensity factor is 1.50, and

compares well with the results published in Reference [6], where the peak values are 1.44 at time

10.3 µs for the dual reciprocity method and 1.51 at time 10.2 µs for the Laplace transform method,

respectively.

Figure 6.6: Normalised mode I DSIF for the rectangular plate with a central slant crack (a) the present method, (b) the dual reciprocity method [6] and (c) the Laplace transform method [6]



Figure 6.7: Normalised mode II DSIF for the rectangular plate with a central slant crack (a) the present method, (b) the dual reciprocity method [6] and (c) the Laplace transform method [6]



6.5.3 A Rectangular Plate with an Internal Kinked Crack


an internal kinked crack, shown in Figure 6.8. One of the segments of the crack is

Figure 6.8: Rectangular plate with an internal kinked crack

horizontal with length a while the other segment makes an angle of 45 with the

horizontal and has a length b. The horizontal projection of the total crack is given

by 2c = a +√

2b/2. The kink of the crack is at the centre of the plate. The material

properties are: the shear modulus µ = 7.692 × 1010Pa, Poisson’s ratio ν = 0.3, and

density ρ = 5 × 103kg · m−3. The time step ∆τ = 0.3 µs is used here. f(X′′,x) =

1 + r(X′′,x) is used as the approximation function of the particular solutions. Three

cases were considered, b/a = 0.2, 0.4, and 0.6, with a/w = 0.1. The dynamic stress


intensity factors are normalised with respect to

K0 = σ0

√πc (6.30)

They were obtained for both tips A and B, with a boundary element mesh of 45

discontinuous quadratic elements and 19 additional domain points on each subregion,

in which the horizontal and the inclined segments of the crack were discretised with 5

and 4 discontinuous quadratic elements on each crack face, respectively. The results

are presented in Figure 6.9 for tip A. The first peak value of the normalised mode

I dynamic stress intensity factor is 2.48, 2.47, and 2.47 for b/a = 0.2, 0.4, and 0.6,

respectively, at τ = 6.6 µs. The results for tip B are presented in Figure 6.10 and

6.11. The first peak values of the normalised mode I dynamic stress intensity factor

are 1.53 at τ = 6.6 µs for b/a = 0.2, 1.32 at τ = 6.9 µs for b/a = 0.4 and 1.26 at

τ = 6.9 µs for b/a = 0.6; the peak value decreases when the ratio b/a increases. The

first peak value of the normalised mode II dynamic stress intensity factor is −1.74,

−1.81, and −1.85 for b/a = 0.2, 0.4, and 0.6, respectively, at τ = 6.3 µs; the peak

value increases when the ratio b/a increases.


τ(µs)

Nor

mal

ised

mod

eI

DS

IFat

tipA

0 2 4 6 8 10 12 14-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6(a)(b)(c)

Figure 6.9: Normalised mode I DSIF for the rectangular plate with an internal kinkedcrack at tip A (a) b/a = 0.2, (b) b/a = 0.4, and (c) b/a = 0.6


τ(µs)

Nor

mal

ised

mod

eI

DS

IFat

tipB

0 2 4 6 8 10 12 14-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8(a)(b)(c)

Figure 6.10: Normalised mode I DSIF for the rectangular plate with an internalkinked crack at tip B (a) b/a = 0.2, (b) b/a = 0.4, and (c) b/a = 0.6


τ(µs)

Nor

mal

ised

mod

eII

DS

IFat

tipB

0 2 4 6 8 10 12 14-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2(a)(b)(c)

Figure 6.11: Normalised mode II DSIF for the rectangular plate with an internalkinked crack at tip B (a) b/a = 0.2, (b) b/a = 0.4, and (c) b/a = 0.6


6.6 Conclusion

In this chapter the dual reciprocity boundary element method employing the step

by step time integration technique is developed to analyse two-dimensional dynamic

crack problems. The dual reciprocity boundary element method was used to trans-

form the domain integrals into equivalent boundary integrals in the two dimensional

boundary element analysis. An efficient subregion technique is used to overcome the

difficulty of a singular system of algebraic equations in crack problems. The system

of ordinary differential equations in time is solved using the step by step finite differ-

ence method. The proposed method is more efficient than traditional BEM because

it significantly reduces the size of the final matrix system. Further, as the system

of equations for each subregion is pre-solved independently, only the time dependent

equations need to be recalculated, which is advantageous when using the step by step

time integration approach. The dynamic stress intensity factors are calculated from

the crack tip opening displacement by using the discontinuous quarter point method.

The calculated results for the examples compared well with those of published solu-

tions. It is, therefore, shown that the present formulation can be effectively applied

to study elastodynamic fracture problems.

Chapter 7

Conclusion

In this thesis, we studied the advanced boundary element method for fracture mechan-

ics, including static and dynamic problems. Static problems were solved by using the

dual boundary element method, which based on the works in [66]. Normally, the stress

intensity factor is obtained by using discontinuous quarter point element method. In

this work, we presented a special crack tip element method, which provides similar

accuracy as that of quarter point element method, but a much easier discritisation of

the crack face for evaluating stress intensity factors.

Further, a new subregion boundary element method was presented to solve com-

posite material problems, which based on the work in [64]. Similar composite prob-

lems are also solved by using domain decomposition method, which based on the

work in [67]. Dynamic fracture mechanics problems were solved by using the dual

reciprocity boundary element method, which based on the work in [65].

In chapters 2 and 3, we provided a detailed study of two dimensional linear elas-

tic crack problems under in-plane tensile and anti-plane shear loadings, respectively.

The difficulty in using the boundary element method directly is that the upper crack

132

Chapter 7: Conclusion 133

surface and lower crack surface are both located in the same mathematical posi-

tion. However, they have different physical boundary conditions. The dual boundary

element method was used to overcome this difficulty. In order to obtain accurate

results, the dual boundary element method was combined with the discontinuous

quarter point element method to correctly model the characteristics of displacements

near the crack tip. The improved efficiency of this method was due to the fact that the

crack problems could be solved with a single region formulation. The accuracy of the

method was also improved compared with published results. The error percentages

of the present numerical results were less than 1 per cent.

In Chapter 4, we concerned with a special crack tip element method to evalu-

ate stress intensity factors. The calculated results were compared with those of the

discontinuous quarter point element method. Using the discontinuous quarter point

element method we need to choose the locations of the collocation points which de-

pend on the shape functions at the crack tip element. However, the special crack tip

element method does not require shifting of the collocation points at the crack tip

element. Therefore, the presented method is a more straight-forward technique than

discontinuous quarter point element method. Furthermore, both methods produced

results with similar accuracy.

In Chapter 5, we presented a new subregion boundary element method. Existing

methods generate a large system of equations, the solution process is time consuming

and computationally intensive. In the new method, only a relatively smaller matrix

system, the interface traction matrix system, needs to be solved. Therefore, a much

better computational efficiency is achieved. The embedded crack problem was inves-

tigated by combining the dual boundary element method with this new technique.

Chapter 7: Conclusion 134

In Chapter 6, dynamic fracture problems were studied using the dual reciprocity

boundary element method (DRBEM). The DRBEM is used to transform the domain

integral term, and polynomial radial basis functions provided the particular solutions

of both displacement and traction kernels. The study also employed the subregion

boundary element method, which was presented in Chapter 5. The time-dependent

differential equation was solved using a direct step-by-step procedure, and the inter-

face traction matrix system were solved by using an iterative procedure that updates

the estimate at each time step.

The work in this thesis could be extended to three-dimensional crack analysis.

However, due to the complexity of the three-dimensional geometry, it requires more

effective modelling strategy to achieve better computational efficiency. We could

combine the boundary element method (BEM) and the finite element method (FEM)

in future study. FEM models are very good at representing global behaviours, but

difficult to model small details and features such as cracks, whereas BEM can capture

local behaviours and small geometry details. By combining the techniques we can

reduce the computation time involved in crack modelling and obtain more accurate

predictions.

The idea of combining the BEM and FEM is to decompose the domain of interest

into sub-domains and to use the most appropriate technique for each sub-domain.

The key point in doing so is the coupling of the systems of equations on the inter

domain boundaries [1, 2, 14, 47]. The equations produced by the BEM and FEM

are expressed in terms of different variables and cannot be linked as they stand [38].

Conventional coupling methods may destroy the desirable features of the BEM and

FEM. Hence future work should include finding a suitable coupling method.

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