Copyright by Yi-Shao Lai 2002

Copyright

by

Yi-Shao Lai

2002

The Dissertation Committee for Yi-Shao LaiCertifies that this is the approved version of the following dissertation:

A Study of Three-Dimensional Cracks

Committee:

Gregory J. Rodin, Supervisor

Richard A. Schapery

Mark E. Mear

Alexander B. Movchan

K. Ravi-Chandar


by

Yi-Shao Lai, B.S.; M.S.

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

August 2002

Dedicated to my parents with love.

Acknowledgments

First of all, I would like to thank my advisor, Prof. G. J. Rodin, for his

guidance, patience and care throughout my Ph.D. degree course. No words

of “Thank-You” will sufficiently encapsulate this mentor–protege relationship

during my entire stay in the States. Prof. Rodin introduced me to Prof. A. B.

Movchan and Prof. K. Ravi-Chandar and together we achieved novel fruition.

For this, I profusely thank them for all the discussions and discourses we had.

I am gratefully indebted to Prof. M. E. Mear for his generosity in letting

me use his numerical codes as well as his helpful, insightful and timely technical

advices and direction, and to Prof. R. A. Schapery for his useful opinions and

suggestions about the dissertation. Thanks are due to Prof. S. Kyriakides and

technicians in ASE/EM machine shops for their support in experiments.

My appreciation also goes to Dr. Y. Fu for her generous assistance

and guidance during my first two years of Ph.D. study. I owe special thanks

to Dr. J. R. Overfelt for his munificent help concerning the cryptic coding

conventions, styles and kinks.

Last but not least. I cherish the precious memories I have had with my

friends, in particular, S. J. Park, G. R. Kotamraju, L.-H. Lee, and members

of the Iguana softball team, as their support and constant encouragement is

always a guiding spirit in order to complete my Ph.D. program.

v


Publication No.

Yi-Shao Lai, Ph.D.

The University of Texas at Austin, 2002

Supervisor: Gregory J. Rodin

This dissertation presents three studies of three-dimensional cracks.

The first study is concerned with a fast boundary element method, which

allows one to solve problems involving hundreds and thousands of cracks of

arbitrary three-dimensional cracks. The new boundary element method is con-

structed as a merger between a conventional boundary element method and

a fast iterative method in which matrix-vector products are computed with

the fast multipole method. The accuracy and efficiency of the new method is

confirmed by numerical examples involving closely spaced cracks, large arrays

of cracks imbedded in an infinite solid, and periodic arrays of cracks. The

second study is concerned with regular and singular asymptotic solutions for

non-planar quasi-circular cracks. The regular first-order asymptotic solution

complements an existing asymptotic solution for planar quasi-circular cracks,

and thus, these two solutions constitute a complete first-order regular asymp-

totic solution for quasi-circular cracks. The singular first-order asymptotic

vi

solution is limited to axisymmetric problems only. The range of validity of the

regular and singular asymptotic solutions is evaluated upon comparing them

with detailed numerical and closed-form solutions. Also, the asymptotic solu-

tions are applied to stability analysis of growing cracks and crack-like cavities.

The third study is concerned with experimental characterization of mixed-

mode cracks in PMMA. The objective of this study is to identify a criterion

for crack growth initiation under truly three-dimensional mixed-mode condi-

tions. Based on macroscopic measurements and microscopic observations, it

is proposed that the Mode III loading component would affect the fracture

toughness KIC , but does not lead to crack growth initiation in the absence

of the Mode I loading component. This implies that one can develop three-

dimensional crack growth initiation criteria using existing two-dimensional cri-

teria as the basis – the key difference is that in three dimensions KIC must

be treated as a function of the Mode III stress intensity factor rather than a

constant.

vii

Table of Contents

Acknowledgments v

Abstract vi

List of Figures xi

List of Tables xiv

Chapter 1. Introduction 1

1.1 Fast Boundary Element Method . . . . . . . . . . . . . . . . . 2

1.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Crack Growth Initiation . . . . . . . . . . . . . . . . . . . . . 5

Chapter 2. Fast Boundary Element Method 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Governing Integral Equation . . . . . . . . . . . . . . . 11

2.2.3 Governing Algebraic Equation . . . . . . . . . . . . . . 12

2.3 Fast Matrix-Vector Multiplication . . . . . . . . . . . . . . . . 14

2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 Two nearly touching coplanar cracks . . . . . . . . . . . 18

2.4.2 A regular cubic array of circular cracks . . . . . . . . . . 21

2.4.3 A disordered cubic array of elliptical cracks . . . . . . . 22

2.4.4 A regular array of colinear circular cracks . . . . . . . . 24

2.4.5 A periodic array of non-aligned elliptical cracks . . . . . 25

2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 26

viii

Chapter 3. On Quasi-Circular Cracks 28

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Circular Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Asymptotic Expansion . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Evaluation of Asymptotic Solutions . . . . . . . . . . . . . . . 36

3.5.1 Boundary Element Analysis . . . . . . . . . . . . . . . . 37

3.5.2 Planar Cracks . . . . . . . . . . . . . . . . . . . . . . . 39

3.5.3 Non-Planar Cracks . . . . . . . . . . . . . . . . . . . . . 42

3.6 Quasi-Static Crack Growth . . . . . . . . . . . . . . . . . . . . 51

3.6.1 Mode I Planar Crack . . . . . . . . . . . . . . . . . . . 52

3.6.2 Axisymmetric Non-Planar Crack . . . . . . . . . . . . . 54

3.6.3 Axisymmetric Crack Opening . . . . . . . . . . . . . . . 59

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Chapter 4. Mixed-Mode Cracking 66

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.1 Edge Crack Specimen (mixed-mode II/III) . . . . . . . 71

4.2.2 Three-Point Bend Specimen (mixed-mode I/III) . . . . 71

4.2.3 Torsion Specimen (pure Mode III) . . . . . . . . . . . . 73

4.3 Experimental Results and Observations . . . . . . . . . . . . . 73

4.3.1 Results for Edge Crack Specimens . . . . . . . . . . . . 74

4.3.2 Results for Three-point Bend Specimens . . . . . . . . . 75

4.3.3 Results for Torsion Specimens . . . . . . . . . . . . . . . 82

4.4 Analysis of Experiments . . . . . . . . . . . . . . . . . . . . . 84

4.4.1 Analysis on Edge Crack Specimens . . . . . . . . . . . . 85

4.4.2 Analysis on Three-point Bend Specimens . . . . . . . . 87

4.4.3 Analysis on Torsion Specimens . . . . . . . . . . . . . . 90

4.5 Microcrack-Macrocrack Interaction . . . . . . . . . . . . . . . 91

4.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 91

4.5.2 Dipole Approximation . . . . . . . . . . . . . . . . . . . 93

4.6 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . 95

ix

Chapter 5. Closing Remarks 99

5.1 Fast Boundary Element Method . . . . . . . . . . . . . . . . . 99

5.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . 100

5.3 Mixed-Mode Cracking . . . . . . . . . . . . . . . . . . . . . . . 101

Appendix 102

Bibliography 104

Vita 113

x

List of Figures

2.1 The coarse and fine meshes for the circular crack. . . . . . . . 17

2.2 Comparison of BEM solution (curves) and Fabrikant’s solution(dots) for two nearly touching coplanar circular cracks. . . . . 19

2.3 Comparison of BEM solution with (lower curves) or without(upper curves) the singular integration schemes for two nearlytouching circular cracks. . . . . . . . . . . . . . . . . . . . . . 20

2.4 A regular cubic array of circular cracks: CPU time required toperform one matrix-vector multiplication. . . . . . . . . . . . . 22

2.5 A disordered cubic array of elliptical cracks: CPU time requiredto perform one matrix-vector multiplication. . . . . . . . . . . 23

2.6 A regular array of colinear circular cracks: CPU time requiredto perform one matrix-vector multiplication. . . . . . . . . . . 24

2.7 A periodic array of non-aligned elliptical cracks: CPU time re-quired to perform one matrix-vector multiplication. . . . . . . 25

3.1 Meshes for the circular crack. . . . . . . . . . . . . . . . . . . 38

3.2 A perturbed planar crack. . . . . . . . . . . . . . . . . . . . . 39

3.3 Meshes for planar wavy cracks. . . . . . . . . . . . . . . . . . 41

3.4 Comparison of asymptotic and boundary element solutions forplanar wavy cracks. . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Meshes for axisymmetric non-planar cracks. . . . . . . . . . . 44

3.6 Comparison of asymptotic and BEM solutions for non-planarcracks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.7 Meshes for non-axisymmetric non-planar cracks. . . . . . . . . 46

3.8 Comparison of asymptotic and BEM solutions for the non-axisymmetric R3 crack. . . . . . . . . . . . . . . . . . . . . . . 47

3.9 Comparison of asymptotic and BEM solutions for the non-axisymmetric R4 crack. . . . . . . . . . . . . . . . . . . . . . . 48

3.10 Geometry of the spherical cap crack. . . . . . . . . . . . . . . 49

3.11 Comparison of analytic, BEM, and asymptotic solutions for thespherical cap crack subjected to equal triaxial tensile load. . . 50

xi

3.12 Contours of a quasi-statically growing planar crack. . . . . . . 53

3.13 Local coordinate systems for characterization of the crack tipfields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.14 The function h0(r) characterizing the initial perturbation of thecrack surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.15 Crack surface deflection: (a) unstable crack surface for the caseof positive σrr and (b) the crack surface for the case of negativeσrr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.16 An axisymmetric opening for a three-dimensional crack. . . . . 61

4.1 Schematic diagram of the rectangular edge crack specimen (unit:cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Schematic diagram of the T-shaped edge crack specimen (unit:cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Schematic diagram of the mixed-mode I/III three-point bendspecimen (unit: cm). . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Schematic diagram of the torsion specimen (unit: cm). . . . . 74

4.5 Load-displacement curves for the rectangular and the T-shapededge crack specimens under monotonic loading. . . . . . . . . 75

4.6 Load-displacement curves for the mixed-mode I/III and thepure Mode I three-point bend specimens under monotonic loading. 76

4.7 Fractography of the mixed-mode I/III three-point bend PMMAspecimen subjected to the monotonic load. . . . . . . . . . . . 77

4.8 Needle-like crazing structures and stacked microcracks on thecrack front of the monotonically loaded mixed-mode I/III PMMAspecimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.9 Possible mechanism of microcrack nucleation. . . . . . . . . . 79

4.10 Fractography of the mixed-mode I/III three-point bend poly-carbonate specimen subjected to the monotonic load. . . . . . 80

4.11 Coarse-scale Fractography of the mixed-mode I/III three-pointbend specimen subjected to the fatigue load. . . . . . . . . . . 81

4.12 Fine-scale fractography of the mixed-mode I/III three-point bendspecimen subjected to the fatigue load. . . . . . . . . . . . . . 82

4.13 Load-rotation curve for the torsion specimen. . . . . . . . . . 83

4.14 Evolved cracks from the pre-cracked front of the torsion specimen. 84

4.15 Edge crack specimens: boundary element meshes for outer bound-aries and the crack face. . . . . . . . . . . . . . . . . . . . . . 85

xii

4.16 Boundary conditions for the BEM analysis of the edge crackspecimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.17 Stress intensity factors of edge crack specimens at their respec-tive peak loads. . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.18 Three-point bend specimens: boundary element meshes for outerboundaries and the crack face. . . . . . . . . . . . . . . . . . . 88

4.19 Stress intensity factors for mixed-mode I/III and pure Mode Ithree-point bend PMMA specimens under their peak loads. . . 89

4.20 Stress intensity factors for mixed-mode I/III three-point bendpolycarbonate specimen under the peak load. . . . . . . . . . 90

4.21 An inclined circular microcrack ahead of a flat semi-infinitemacrocrack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.22 Normalized microcrack nucleation angles and corrections to stressintensity factors at X0, 0, 0 for different I/III mode mix (ν =0.38 for PMMA). . . . . . . . . . . . . . . . . . . . . . . . . . 95

xiii

List of Tables

2.1 Solutions of KI/K0I at θ = 0 for two identical coplanar circular

cracks under uniform normal loading. . . . . . . . . . . . . . . 19

4.1 Mechanical properties of PMMA and polycarbonate. . . . . . 70

xiv

Chapter 1

Introduction

This dissertation presents three studies of three-dimensional cracks.

The first study is concerned with a fast boundary element method, which

allows one to solve problems involving hundreds and thousands of cracks of

arbitrary three-dimensional cracks. The new boundary element method is con-

structed as a merger between a conventional boundary element method and

a fast iterative method in which matrix-vector products are computed with

the fast multipole method. The accuracy and efficiency of the new method

is confirmed by numerical examples involving closely spaced cracks, large ar-

rays of cracks imbedded in an infinite solid, and periodic arrays of cracks.

The second study is concerned with regular and singular asymptotic solutions

for non-planar quasi-circular cracks. The regular first-order asymptotic so-

lution complements an existing asymptotic solution for planar quasi-circular

cracks, and thus, these two solutions constitute a complete first-order reg-

ular asymptotic solution for quasi-circular cracks. The singular first-order

asymptotic solution is limited to axisymmetric problems only. The range of

validity of the regular and singular asymptotic solutions is evaluated upon

comparing them with detailed numerical and closed-form solutions. Also, the

asymptotic solutions are applied to stability analysis of growing cracks and

1

crack-like cavities. The third study is concerned with experimental charac-

terization of three-dimensional cracks under mixed-mode loading conditions.

The objective of this study is to identify a criterion for crack growth initia-

tion under truly three-dimensional mixed-mode loading conditions. Based on

macroscopic measurements and microscopic observations, it is proposed that

the Mode III loading component affects the fracture toughness KIC , but does

not lead to crack growth initiation in the absence of the Mode I loading com-

ponent. This implies that one can develop a three-dimensional crack growth

initiation criterion using an existing two-dimensional criterion as the basis –

the key difference is that in three dimensions KIC must be treated as a function

of the Mode III stress intensity factor rather than a constant.

The dissertation structure is as follows. In the remainder of this chap-

ter, we provide brief descriptions of the three studies. Each of the next three

chapters is dedicated to one of the studies and has its own introduction and

conclusion sections. As a result the introduction and conclusion chapters of

the dissertation are rather short.

1.1 Fast Boundary Element Method

In this chapter, we present a new fast boundary element method, which

allows one to solve problems involving hundreds and thousands of arbitrary

three-dimensional cracks. We refer to this method as “fast” because in com-

parison to conventional methods, which require O(N2) memory and O(N3)

arithmetic operations, the fast method requires only O(N) memory and O(N)

2

arithmetic operations; here N is the problem size.

The proposed boundary element method is constructed as a merger

between a conventional boundary element method (Li and Mear; 1998; Li

et al.; 1998) and a fast iterative method in which matrix-vector products are

computed with the fast multipole method (Greengard and Rokhlin; 1987).

The accuracy and efficiency of the new method is confirmed by numerical

examples involving closely spaced cracks and large arrays of cracks imbedded

in an infinite solid.

The fast boundary element method is developed along the lines of the

approach proposed by Fu et al. (1998). These authors considered direct inte-

gral equations of linear elasticity for problems involving solids containing mul-

tiple inhomogeneities. For those integral equations, they developed decompo-

sitions of the integral equation kernels that allows one to recast the discretized

integral equation as a combination of (generalized) many-body electrostatics

problems, each of which can be solved with the fast multipole method. An

alternative approach to fast boundary element methods for linearly elastic

problems has been proposed by Yoshida et al. (2001a,b,c). Their method may

be more efficient than the present one. However, it is much more difficult to

implement.

1.2 Asymptotic Analysis

This study is concerned with the development, verification, and appli-

cation of asymptotic solutions for the stress intensity factors along the front

3

of a single quasi-circular crack. Such solutions are important because they

provide us with a qualitative understanding of cracks with arbitrary shapes.

Furthermore, asymptotic solutions could play a significant role in the analysis

of stationary and growing cracks, as they can partly replace detailed numerical

solutions. Existing asymptotic solutions for quasi-circular cracks are limited

to planar cracks (e.g., Gao and Rice; 1987; Gao; 1988). Moreover, the validity

of those solutions has not been evaluated.

Our objective is to develop asymptotic solutions for non-planar quasi-

circular cracks and to evaluate the validity of the existing and new asymp-

totic solutions by comparing them with detailed numerical solutions. We

analyze non-planar quasi-circular cracks following the approach of Movchan

et al. (1998) for non-planar perturbations of the semi-infinite three-dimensional

crack. Accordingly, we have to develop asymptotic solutions for effective trac-

tions applied to the faces of a reference circular crack. Once the effective

tractions are determined, we can obtain the stress intensity factors using the

weight functions (Bueckner; 1987; Fabrikant; 1989). The validity of asymptotic

solutions are evaluated with boundary element analysis for several benchmark

problems. Those problems are chosen such that they provide a quantitative

characterization of the range of validity of asymptotic solutions and a qualita-

tive understanding of how and why asymptotic solutions for three-dimensional

cracks fail.

4

1.3 Crack Growth Initiation

Three-dimensional cracks under mixed-mode loading conditions have

not received much attention in the literature, mostly due to enormous chal-

lenges associated with their experimental and analytical characterization. Well

recognized crack growth initiation criteria are limited to two-dimensional cracks

under Mode I and II loading conditions, and it is unclear how to extend them

to three-dimensional cracks under general mixed-mode loading conditions. Of

course in the absence of a criterion involving all three modes, one cannot relate

the stress intensity factors to the fracture behavior.

Our objective is to provide experimental results and observations for

three-dimensional cracks under mixed-mode loading conditions for construct-

ing a reliable crack growth initiation criterion. Towards this objective we tested

several specimens made of polymethyl methacrylate (PMMA) and polycarbon-

ate with cracks subjected to mixed-mode II/III, mixed-mode I/III and pure

Mode III loading conditions. In addition, using an optical microscope, we

studied microcracks near the main crack. The specimens were analyzed with

a boundary element method, and the effect of microcracks was analyzed using

a dipole approximation. Based on results of our experiments and calculations,

it appears that the Mode III component leads to nucleation of microcracks

rather than growth of the main crack. This implies the crack growth initiation

criterion for a small amount of KIII can be stated as

KI F

(KII

KI

)= KIC G

(KIII

KIC

),

5

where the constitutive function F can be adopted from a two-dimensional

criterion and G has to be determined from experiments involving mixed modes

I and III.

6

Chapter 2

Fast Boundary Element Method

In this chapter, we present a fast boundary element method for ana-

lyzing three-dimensional linearly elastic solids containing many cracks. The

method is constructed in a modular fashion, by combining existing conven-

tional boundary element method, iterative method, and fast summation method

for the many-body electrostatics problems. The fast boundary element method

preserves the advantageous features of the conventional boundary element

method – weakly-singular kernel, problem-specific approximations, and effec-

tive near-singular integration scheme. Numerical examples confirm the accu-

racy and efficiency of the new method for problems involving closely-spaced

cracks, large arrays of cracks imbedded in an infinite solid, and periodic prob-

lems.

2.1 Introduction

Like other fast boundary element methods, the present method relies on

an iterative solution method in which a dense matrix is multiplied by a vector

without constructing the matrix, so that each matrix-vector product requires

only O(N) or O(N log N) memory and arithmetic operations. Further, it

7

appears that arising linear algebraic equations are well conditioned, so that a

typical number of required iterations is independent of N , and as a result the

total arithmetic operation demand is O(N) or O(N log N). In comparison,

conventional boundary element methods, which rely on Gaussian elimination

solvers, require O(N2) memory and O(N3) arithmetic operations.

The present method is based on the idea central to other similar meth-

ods developed by Rodin and co-workers for analyzing large-scale problems

arising in micromechanical characterization of linearly elastic composites (Fu

et al.; 1998), dislocation dynamics (Rodin; 1998), and low-Reynolds-number

emulsions (Fu and Rodin; 2000; Overfelt; 2002). This idea is to represent the

matrix-vector product as a superposition of (generalized) electrostatics prob-

lems. Such a representation allows one to use fast summation methods for

electrostatics problems for solving integral equations arising in fluid and solid

mechanics. This approach is particularly useful for code development because

it allows one to treat the fast summation component of the code as a black

box. In developing the present code, we merged the boundary element code

developed by Mear and co-workers (Li and Mear; 1998; Li et al.; 1998) with

existing GMRES (Saad and Schultz; 1986) and FMM (Greengard and Rokhlin;

1987) codes (Fu et al.; 1998). Later, we replaced the FMM code with a mod-

ified version capable of handling periodic boundary-value problems (Overfelt;

2002).

Another approach to formulating fast boundary element methods for

multiple cracks has been pursued by Nishimura and co-workers (Yoshida et al.;

8

2001a,b,c). Their approach is based on a generalization of the FMM transla-

tion theorems to the classical kernels of linear elasticity and a similar kernel

appearing in a weakly-singular integral equation for crack analysis. Initially,

Nishimura and co-workers (Yoshida et al.; 2001a) developed their method using

the original translation theorems (Greengard and Rokhlin; 1987). In subse-

quent papers (Yoshida et al.; 2001b,c), they were able to extend their approach

to include the new translation theorems developed by Greengard and Rokhlin

(1997). We have not been able to compare our method with those of Nishimura

and co-workers. A simple comparison, which assumes that the original and

new translation theorems require the same number of operations, favors the

approach of Nishimura and co-workers. However, the validity of this assump-

tion has not been established and the comparison does not take into account

the time it takes to develop the codes. At any rate, considering that there are

very few fast boundary element codes available, we believe that, at this stage,

it is more important to present new codes rather than compare existing ones.

The rest of this chapter is organized as follows. In Section 2.2, we

formulate the governing integral and algebraic equations. In Section 2.3, we

recast the governing algebraic equations in a form such that the matrix-vector

product is represented as a superposition of generalized electrostatics prob-

lems. In Section 2.4, we present several numerical examples to demonstrate

the capability of the fast boundary element code. In Section 2.5, we summarize

the principal results of this work and discuss directions for future work.

9

2.2 Governing Equations

2.2.1 Problem Statement

Consider a finite number of non-intersecting (not necessarily planar)

cracks Ω embedded in an unbounded isotropic linearly elastic solid. The crack

faces are loaded such that the traction fields on the opposite faces are opposite

of each other. At infinity, the solid is neither loaded nor constrained.

The boundary-value problem for the solid is formulated in terms of the

displacement field ui(x), which must satisfy Navier’s equation in the exterior

of Ω,

(λ + µ)uk,ki(x) + µui,kk(x) = 0 x ∈ R3 \ Ω , (2.1)

where λ and µ are Lame’s elastic constants. The boundary conditions on the

crack faces are

(λuk,kni + µui,knk + µuk,ink)(x) = ti(x) x ∈ Ω+ (2.2)

and

(λuk,kni + µui,knk + µuk,ink)(x) = −ti(x) x ∈ Ω− . (2.3)

Here ti(x) is a prescribed traction field, and Ω± denote the crack faces. The

boundary conditions at infinity are prescribed in the form

ui(x) ∼ O(|x|−2) |x| → ∞ . (2.4)

This decay for large |x| is consistent with the fact that the prescribed load is

self-equilibrated.

10

2.2.2 Governing Integral Equation

For the boundary-value problem defined by (2.1-2.4), boundary ele-

ment methods are particularly attractive because they significantly reduce

the problem size and simplify meshing procedures. Among boundary element

methods, we prefer those that involve weakly-singular kernels, and in partic-

ular the method proposed by Mear and co-workers (Li and Mear; 1998; Li

et al.; 1998). This method involves not only a weakly-singular kernel but also

special approximation and integration schemes, which are particularly useful

for accurate analysis of closely-spaced cracks.

The integral equation in the method of Mear and co-workers has the

form∫

Ω

Dkvl(x)

∫

Ω

Cijkl(x− y)Dibj(y)dΩydΩx = −∫

Ω

tl(x)vl(x)dΩx . (2.5)

In (2.5) the following definitions have been used:

• The tangential operator Dk is defined as

Dk = eimk n+i ∂m ,

where eimk is the permutation symbol and n+i is the outward normal to

Ω+.

• The field vi is an element of the Galerkin projection space, which must

be sufficiently smooth, but arbitrary otherwise.

11

• The kernel Cijkl is defined as

Cijkl(x) =µ

4π(1− ν)

[(1− ν)δklδij + 2νδilδjk − δjlδik

|x| − δikxlxj

|x|3]

,

(2.6)

where ν denotes Poisson’s ratio.

• The field bi denotes the displacement jump across Ω:

bi ≡ u+i − u−i .

2.2.3 Governing Algebraic Equation

We convert the governing integral equation (2.5) into a system of alge-

braic equations following the standard finite element discretization technique.

In what follows, we use superscripts for discretization attributes and subscripts

for tensorial indices. Let

x =∑

k

xkφk(ξ) ≡ x(ξ) (2.7)

represent the map of the position vector xx1, x2, x3 on the master element

ξξ1, ξ2. Here xk is the position vector of a node with the local node number

k and φk(ξ) is a corresponding standard shape function. By adopting the

isoparametric map for b and t, we obtain

b =∑

k

bkφk(ξ) and t =∑

k

tkφk(ξ) . (2.8)

The projection basis v involves the head functions ψq(ξ) constructed from the

shape functions as

vqi = ψq(ξ) =

∑r

φr(ξ) , (2.9)

12

where r runs over the shape functions of elements adjacent to the node q that

satisfy the condition φr(ξq) = 1. For elements containing the crack front,

there are special shape functions (Li et al.; 1998), and the head functions are

generated according to (2.9).

The governing algebraic equation is obtained by substituting (2.7-2.9)

into (2.5) followed by numerical integration:

∑m

∑q,p

ωpJ [x(ξp)]Dk[x(ξp)]ψq(ξp)

∑n

∑r,s

ωsJ [x(ξs)]Cijkl[x(ξp)− y(ξs)]brjDi[y(ξs)]φr(ξs) (2.10)

= −∑m

∑q,r,p

ωpJ [x(ξp)]tql φq(ξp)ψr(ξp) .

Here m and n run over the elements, p and s run over the quadrature points,

and q and r run over the nodal points. Further, ω denotes the quadrature

weights, the Jacobian J is computed as

J [x(ξp)] =

∣∣∣∣∂x(ξ)

∂ξ1

× ∂x(ξ)

∂ξ2

∣∣∣∣ ,

and the operator D is computed as

Dk[x(ξ)] =1

J [x(ξ)]

[∂xk(ξ)

∂ξ2

∂

∂ξ1

− ∂xk(ξ)

∂ξ1

∂

∂ξ2

].

The algebraic equation in (2.10) represents a linear algebraic problem for the

nodal unknowns brj . The matrix for this problem is dense because the kernel

Cijkl(x−y) is not compactly supported. Alternatively, the density is due to the

fact that the inner and outer sums in (2.10) must be evaluated simultaneously

rather than separately.

13

2.3 Fast Matrix-Vector Multiplication

For fast evaluation of the sums in the left-hand side of (2.10) we repre-

sent the kernel Cijkl(x−y) in a form that separates the dependencies on x and

y, and therefore allows us to evaluate the inner and outer sums in (2.10) sep-

arately rather than simultaneously. This conceptual view of fast summation,

in general, and the fast multipole method (FMM) (Greengard and Rokhlin;

1987, 1997), in particular, was suggested by Greegard (1994). Rodin and

co-workers demonstrated that the concept of separation can be exploited for

integral equations with kernels other than |x − y|−1 (Fu et al.; 1998; Rodin;

1998; Fu and Rodin; 2000; Overfelt; 2002). To do this, one has to express the

kernel in the form∑

k

Fk(x) qk(y) |x− y|−1 ,

where Fk(x) is an operator defined strictly at x and qk(y) is a function defined

strictly at y. Using this form, the complete separation of the kernel can be

realized using FMM or any other method applicable to the kernel |x− y|−1.

To express Cijkl(x− y) in a separable form, we rely on the identity

(xi − yi)(xj − yj)

|x− y|3 = −xj∂i

(1

|x− y|)

+ ∂i

(yj

|x− y|)

,

to obtain

Cijkl(x− y) = Rijkl(x)|x− y|−1 + Sikl(x)yj|x− y|−1,

where

Rijkl(x) =µ

4π(1− ν)[(1− ν)δijδkl + 2νδjkδil − δjlδik + xj∂lδik]

14

and

Sikl(x) = − µ

4π(1− ν)∂lδik .

Then the governing integral equation (2.5) can be rewritten as

∫

Ω

Dkvl(x)Rijkl(x)

∫

Ω

Dibj(y)

|x− y| dΩydΩx

+

∫

Ω

Dkvl(x)Sikl(x)

∫

Ω

yjDibj(y)

|x− y| dΩydΩx = −∫

Ω

tl(x)vl(x)dΩx ,

Similarly, the governing algebraic equation (2.10) can be rewritten as

∑m

∑q,p

ωpJ [x(ξp)]Dk[x(ξp)]ψq(ξp)Rijkl[x(ξp)]

∑n

∑r,s

ωsJ [y(ξs)]brjDi[y(ξs)]φr(ξs)

|x(ξp)− y(ξs)|+

∑m

∑q,p

ωpJ [x(ξp)]Dk[x(ξp)]ψq(ξp)Sikl[x(ξp)] (2.11)

∑n

∑r,s

ωsJ [y(ξs)]brjyj(ξ

s)Di[y(ξs)]φr(ξs)|x(ξp)− y(ξs)|

= −∑m

∑q,r,p

ωpJ [x(ξp)]tql φq(ξp)ψr(ξp) .

The sums on the left-hand side of (2.11) have the same structure, which lends

itself to the following interpretation. The expressions in the curly brackets are

evaluated at the quadrature points y(ξ) of the inner integral, and they can be

regarded as (generalized) charges. Then the inner sum becomes the electro-

static potential induced by the charges at a quadrature point x(ξ) of the outer

integral, and the outer sum can be regarded as evaluation of generalized elec-

trostatic forces for all quadrature points x(ξ). The first nested sum involves

nine generalized charges and the second nested sum involves three generalized

15

charges. Therefore the entire summation on the left-hand side of (2.11) can be

performed via twelve-fold application of FMM, or any other method for fast

summation of |x− y|−1.

Special approximation and integration schemes can be incorporated fol-

lowing the approach presented in Fu et al. (1998) and Overfelt (2002). The key

idea is that the special approximation and integration schemes can be regarded

as corrections to the sums in (2.11) evaluated for regular approximation and

integration rules. Such corrections involve closely-spaced or coincident inte-

gration points, and therefore their evaluation is local and can be done in O(N)

operations.

2.4 Numerical Examples

In this section, we present several numerical examples obtained with a

fast boundary element code based on the boundary element method developed

by Mear and co-workers (Li and Mear; 1998; Li et al.; 1998), the generalized

minimum residual algorithm (GMRES) (Saad and Schultz; 1986), and an op-

timized version of the original FMM (Greengard and Rokhlin; 1987). All

computations were performed on a single processor of SGI Onyx2 equipped

with a 400 MHz CPU and 1 GB memory.

All examples involve cracks imbedded in an isotropic linearly elastic

solid with Young’s modulus E and Poisson’s ratio ν = 0.3; for our purpose,

the numerical value of E does not have to be specified. The solid is subjected

to remote uniaxial tensions.

16

Figure 2.1: The coarse and fine meshes for the circular crack.

For circular cracks, we used meshes shown in Figure 2.1. Each mesh

involved nine-node quadrilateral crack-tip elements (Li et al.; 1998) along the

crack front, and regular, either six-node triangular or eight-node quadrilat-

eral, elements. The coarse mesh contained 171 degrees of freedom and the fine

mesh contained 1851 degrees of freedom. The fine mesh was used in exam-

ples in which accuracy was of primary concern (Section 2.4.1), whereas the

coarse mesh in examples in which performance was of primary concern (Sec-

tions 2.4.2–2.4.5). For elliptical cracks, the meshes were obtained by affine

transformation of the circular meshes.

Numerical integration was carried out using Gauss quadrature rules.

For quadrilateral elements in the coarse mesh, we used sixteen and nine Gauss

quadrature points for the inner and outer integration, respectively. For the fine

mesh, the numbers increased to thirty-six and sixteen, respectively. For trian-

gular elements in both meshes, the quadrature rule was chosen to be twelve

and seven for the inner and outer integration, respectively. For coincident and

closely-located points, we used singular integration schemes (Xiao; 1998).

Fast matrix-vector multiplication was performed using the expansion

17

depth p = 10. Typically, in theory, this choice leads to errors of O(10−2)

(Greengard and Rokhlin; 1987) but in practice errors are of O(10−4) (Fu et al.;

1998). The original FMM was optimized by choosing an optimal value for the

tree depth (Singer; 1995; Greengard and Rokhlin; 1997).

2.4.1 Two nearly touching coplanar cracks

Consider two identical coplanar circular cracks of radius a whose centers

are (2+ε)a apart. The cracks are aligned perpendicular to the loading axis, so

that KII = KIII = 0. The purpose of this example is to demonstrate the need

for crack-tip elements and singular integration schemes for problems involving

nearly touching cracks.

We solved this problem using the fine mesh for ε = 0.1, 0.2 and 0.5.

The numerical solution for the stress intensity factor, normalized by the stress

intensity factor for a single circular crack, is presented in Figure 2.2. There

the normalized stress intensity factor, KI/K0I , where K0

I is the stress intensity

factor for an isolated crack, is plotted as a function of the angle θ, chosen such

that θ = 0 corresponds to the point closest to the other crack. For comparison,

we also plot the solution obtained by Fabrikant (1987a) using potential theory,

which we regard as the benchmark. Let us note that the smallest value for ε

chosen in this example was dictated by the size of the crack-tip element.

From Figure 2.2, it is clear that the two solutions are in very good

agreement. The results confirm the notion that crack interactions are signifi-

cant only near θ = 0 and small ε. The greatest discrepancy occurs at θ = 0. In

18

-180 -120 -60 0 60 120 1801.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

ε = 0.1 ε = 0.2 ε = 0.5

KI /

KI0

θ

Figure 2.2: Comparison of BEM solution (curves) and Fabrikant’s solution(dots) for two nearly touching coplanar circular cracks.

ε BEM Fabrikant Kushch Kachanov0.5 1.0665 1.06342 1.064 1.06390.2 1.1792 1.16785 1.177 1.17110.1 1.3199 1.28777 1.317 1.2964

Table 2.1: Solutions of KI/K0I at θ = 0 for two identical coplanar circular

cracks under uniform normal loading.

19

-180 -120 -60 0 60 120 1801.0

1.2

1.4

1.6

1.8

2.0

ε = 0.1 ε = 0.2 ε = 0.5

KI /

KI0

θ

Figure 2.3: Comparison of BEM solution with (lower curves) or without (up-per curves) the singular integration schemes for two nearly touching circularcracks.

Table 2.1 we compare our boundary element solution of normalized stress in-

tensity factor with all other available solutions provided by Fabrikant (1987a),

Kachanov and Laures (1989) and Kushch (1998).

The comparison shown in Figure 2.3 demonstrates that accurate so-

lutions require the crack-tip elements and singular integration schemes. The

smooth curves in Figure 2.3 coincide with the curves in Figure 2.2, whereas

the oscillatory curves were obtained using the crack-tip elements but without

the singular integration schemes. The latter curves are way off the benchmark

solution and exhibit numerical instability. When the crack-tip elements were

20

suppressed, the iterative method simply did not converge.

Other benchmark problems were considered by Li et al. (1998) and Lai

et al. (2002). In all cases, numerical solutions were in excellent agreement with

benchmark solutions.

2.4.2 A regular cubic array of circular cracks

Consider n3 circular cracks arranged such that all cracks are perpen-

dicular to the loading axis, and the crack centers form a cubic array with the

spacing 3a in each direction. The purpose of this example is to compare fast

and conventional matrix-vector multiplication.

Results of comparisons are shown in Figure 2.4, where the CPU time

required to perform one matrix-vector multiplication is plotted as a function

of the problem size. In the remainder of the chapter, we refer to such curves as

performance curves. Note that in this example we used the coarse mesh, and

therefore the problem size is determined as N = 171n3. The cross-over point

for the two multiplication methods is close to 1,500, which is typical for original

FMM codes. The wavy character of the performance curve corresponding to

fast matrix-vector multiplication is typical for FMM and similar tree codes

(Singer; 1995).

We do not present any comparisons of the iterative methods with direct

solvers. To a certain extent such a comparison would not be fair because direct

solvers require O(N2) memory, which, on our computer, limits the problem

size to about N ≈ 15, 000.

21

102 103 104 105 106

10-1

100

101

102

103

104

105

106

107

Fast Conventional

CP

U T

ime

(s)

Degrees of freedom

Figure 2.4: A regular cubic array of circular cracks: CPU time required toperform one matrix-vector multiplication.

2.4.3 A disordered cubic array of elliptical cracks

This example also involves a cubic lattice of n3 cracks, whose centers

form a cubic array with spacing 3a in each direction. However, now the cracks

have elliptical shapes and they are not aligned. The elliptical shapes are chosen

such that the large semi-axis is equal to a and the small semi-axis is a random

number between 0.3a and a. The normal for each crack is chosen by rotating

the original normal so that the first two Euler’s angles are chosen as random

numbers between zero and one. The purpose of the example is to demonstrate

22

102 103 104 105 106

10-1

100

101

102

103

104

105

106

107

Fast Conventional

CP

U T

ime

(s)

Degrees of freedom

Figure 2.5: A disordered cubic array of elliptical cracks: CPU time requiredto perform one matrix-vector multiplication.

that fast-matrix multiplication is not affected by the geometrical disorder.

The performance curves are shown in Figure 2.5. These curves are very

close to those shown in Figure 2.4, and actually fast matrix-vector multiplica-

tion is marginally improved with the disorder up to N ≈ 105. This behavior

is not surprising at all, once it is realized that the performance of FMM is

optimal if the charges are uniformly distributed in space. From this perspec-

tive, the disordered array of cracks may lead to a more uniform distribution of

charges than the ordered array. Further, as the number of cracks increases, the

23

102 103 104 105

10-1

100

101

102

103

104

105

106

CP

U T

ime

(s)

Degrees of freedom

Fast Conventional

Figure 2.6: A regular array of colinear circular cracks: CPU time required toperform one matrix-vector multiplication.

difference between two distributions should become negligible, and therefore

the two problems become indistinguishable as far as the FMM performance is

concerned.

2.4.4 A regular array of colinear circular cracks

This example involves n colinear circular cracks whose centers are 3a

apart. The purpose of the example is to demonstrate that fast matrix-vector

multiplication deteriorates if the problem geometry gives rise to a non-uniform

24

102 103 104 105 106

100

101

102

103

104

105

non-periodic periodic

CP

U T

ime

(s)

Degrees of Freedom

Figure 2.7: A periodic array of non-aligned elliptical cracks: CPU time re-quired to perform one matrix-vector multiplication.

distribution of the charges. This point is fully supported by the performance

curve shown in Figure 2.6.

2.4.5 A periodic array of non-aligned elliptical cracks

By replacing the FMM code with an FMM code for periodic gener-

alized electrostatics problems (Overfelt; 2002), we created a fast boundary

element code for solving periodic boundary-value problems. As an example,

we considered a periodic array whose unit cell is the disordered array of cracks

25

described in Section 2.4.3. The size of the unit cell was chosen such that cracks

did not intersect upon replication of the cell. The purpose of this example is

to demonstrate the applicability of fast boundary element methods to periodic

boundary-value problems.

In Figure 2.7, we compare the performance curves for fast matrix-vector

multiplication for the periodic and non-periodic problem, considered in Sec-

tion 2.4.3. The trend seen in Figure 2.7 is as expected – for small N , the

overhead of implementing the periodic boundary conditions is significant, but

its significance decreases as N increases.

2.5 Concluding Remarks

In this chapter, we presented a simple procedure of developing fast

boundary element methods and codes for problems involving multiple cracks

imbedded inside linearly elastic isotropic solids. This procedure is based on

the idea of representing the integral equation kernel as a linear combination

of kernels, such that each kernel in the combination can be treated with a fast

summation method for the (generalized) many-body electrostatics problem.

This idea was proposed in Fu et al. (1998) for linear elasticity and was applied

to problems in dislocation dynamics and Stokesian dynamics by Rodin and co-

workers (Rodin; 1998; Fu and Rodin; 2000; Overfelt; 2002). The key advantage

of the approach is its simplicity, especially in terms of coding. This degree

of coding simplicity allows one to experiment with various combinations of

boundary element, iterative, and fast summation methods.

26

Examples presented in Section 2.4 lend support to what is becoming

common knowledge about fast boundary element methods – once implemented

they significantly outperform conventional boundary elements for problems in-

volving large number of unknowns and relatively uniformly distributed nodes.

Such problems are commonly encountered in micromechanical simulations of

heterogeneous fluids and solids. For other classes of problems, one may need

to use conventional boundary elements or consider developing new problem-

specific fast boundary element methods.

27

Chapter 3

On Quasi-Circular Cracks

This chapter presents regular and singular asymptotic solutions for non-

planar quasi-circular cracks. Regular asymptotic solutions, which do not give

rise to hyper-singular stresses near the crack tip, are developed in the general

three-dimensional setting. Singular asymptotic solutions are considered for

axisymmetric problems only, because those problems do not involve hyper-

singularity. The validity of asymptotic solutions for planar and non-planar

cracks is investigated by comparing them with detailed numerical and analyt-

ical solutions. Also asymptotic solutions are applied to analysis of quasi-static

crack growth in three dimensions.

3.1 Introduction

This chapter is concerned with the development, verification, and ap-

plication of asymptotic solutions for the stress intensity factors along the front

of quasi-circular cracks. Such solutions are important because they provide us

with a qualitative understanding of cracks with arbitrary shapes. Furthermore,

asymptotic solutions could play a significant role in analysis of stationary and

growing cracks, as they can partly replace detailed numerical solutions.

28

Existing asymptotic solutions for quasi-circular cracks are limited to

planar cracks. Apparently the first asymptotic analysis for planar quasi-

circular cracks dates back to Panasyuk, whose work was first published in 1962

in Ukranian, and was later reviewed by Panasyuk et al. (1981) in English. Rice

(1985a) established a methodology which allows one to simplify asymptotic

analysis through the use of the weight functions (Bueckner; 1970; Rice; 1972;

Bueckner; 1987; Fabrikant; 1987b; Gao and Rice; 1989). Rice’s methodology

is central to asymptotic solutions for planar quasi-circular cracks presented

by Gao and Rice (1987) and Gao (1988). An alternative first-order solu-

tion method for planar quasi-circular cracks was developed by Martin (1996,

2000, 2001a), who formulated the problem in terms of a hyper-singular integral

equation, and regularized that equation using conformal mapping. Methods of

asymptotic analysis for quasi-circular cracks are similar to those for the semi-

infinite crack with a perturbed front, which, among others, were considered

by Rice (1985a,b); Gao and Rice (1989); Bower and Ortiz (1990); Xu et al.

(1994); Leblond and Mouchrif (1996); Lazarus and Leblond (1998); Movchan

et al. (1998).

In this chapter, we pursue three objectives. The first objective is to

develop asymptotic solutions for non-planar cracks with circular contours. For

the class of problems involving regular perturbation of the circle, we develop an

algorithm which includes both first- and second-order terms of the asymptotic

expansion, and this expansion is genuinely three-dimensional. For the class of

problems involving singular perturbation of the circle, the solution includes the

29

first-order terms only, and it is restricted to axisymmetric problems. Note that

the first-order solutions for planar and non-planar cracks can be superposed, so

that one can obtain the general first-order solution for quasi-circular cracks.

The second objective is to evaluate the range of validity of the asymptotic

solutions for planar and non-planar cracks by comparing their predictions with

those obtained with the boundary element method developed by Li and Mear

(1998) and Li et al. (1998). The third objective is to apply asymptotic solutions

to analysis of quasi-static growth of quasi-circular cracks and crack openings.

The remainder of this chapter is organized as follows. In Section 3.2,

we provide the mathematical formulation of the problem. In Section 3.3, we

summarize relevant results for the circular crack problem. In Section 3.4,

we develop asymptotic expansions for non-planar cracks. In Section 3.5, we

analyze several benchmark problems for planar and non-planar cracks using

asymptotic expansions developed in Section 3.4 and the boundary element

method developed by Li and Mear (1998) and Li et al. (1998). Comparison of

the asymptotic and detailed numerical solutions allows us to evaluate the range

of validity of asymptotic solutions. In Section 3.6, we consider asymptotic

analysis of quasi-static growth of cracks and crack openings. In Section 3.7,

we summarize the principal results of this work and discuss directions for

future work.

30

3.2 Problem Statement

Consider an elastic unbounded solid containing a crack Ωε with a quasi-

circular front Γε. We presume that in some sense this crack is close to a

circular crack Ω0 with contour Γ0. In particular, in cylindrical coordinates

(r, θ, z) with the origin at the center of Γ0 and the z-axis perpendicular to Γ0,

Ωε is prescribed as

Ωε = x : r ≤ a[1 + εϕ(θ)], 0 ≤ θ < 2π, x3 = εh(r, θ),

where a is the radius of Γ0, ε ¿ 1 is a small non-dimensional parameter, and

ϕ, h are smooth functions describing Ωε as a perturbation of Ω0. In what

follows, we refer to Ω0 as the reference crack.

The case of ϕ(θ) 6= 0 and h(r, θ) = 0, which represents planar perturba-

tions of Ω0, is well studied in the literature (see Section 3.1). In this chapter,

we are primarily concerned with the case ϕ(θ) = 0 and h(r, θ) 6= 0, which

represents non-planar perturbations of Ω0.

The principal boundary-value problem is formulated in the context of

classical linear elasticity for the solid containing Ωε, so that the displacement

field u(x) must satisfy the homogeneous Navier equations in the exterior of

Ωε:

Lu(x) = µ∇2u(x) + (λ + µ)∇∇ · u(x) = 0 ∀x ∈ R3 \ Ωε . (3.1)

The displacement field must also satisfy the traction boundary conditions on

the crack faces,

t(u, x) = 0 ∀x ∈ Ω±ε , (3.2)

31

and at infinity,

σ(u,x) → σ∞(x) as |x| → ∞ , (3.3)

where σ∞(x) is a prescribed remote stress field.

3.3 Circular Crack

In this chapter, we approximate the solution of the boundary-value

problem (3.1-3.3) using an asymptotic series in ε. Each coefficient of that

series is evaluated as the solution of a boundary-value problem defined for

the solid containing the reference crack Ω0. Therefore, in this section, we

summarize key solutions for the circular crack.

If the stress components σ13, σ23, and σ33 are prescribed on the crack

faces Ω±0 and σ∞ = 0, the stress intensity factors along the crack front can be

written in a compact form derived by Fabrikant (1989):

KI(θ) =1

π√

πa

∫ 2π

0

∫ a

0

σ33(r, φ)√

a2 − r2

a2 + r2 − 2ar cos(θ − φ)rdrdφ (3.4)

KII(θ) + iKIII(θ) =1

π√

πa

∫ 2π

0

∫ a

0

[σ13(r, φ) + iσ23(r, φ)√a2 − r2e−iθ

a2 + r2 − 2ar cos(θ − φ)

+ν

2− ν

σ13(r, φ)− iσ23(r, φ)√a2 − r23a− rei(θ−φ)eiθ

aa− rei(θ−φ)2

]rdrdφ

Here, the subscripts refer to Cartesian coordinates in the usual way related to

the earlier introduced cylindrical coordinates, and ν is Poisson’s ratio. Origi-

nally, the complete solution for the stress intensity factors along the crack front

was obtained by Bueckner (1987), but in a more complicated form compared

to (3.4).

32

For uniform tractions σ13 = σ23 = 0 and σ33 6= 0, the stress intensity

factors are

KI = 2σ33

√a

π, and KII = KIII = 0. (3.5)

The remaining stress components on the crack faces are

σ11 = σ22 = −1 + 2ν

2σ33 and σ12 = 0. (3.6)

3.4 Asymptotic Expansion

We seek the solution to the boundary-value problem formulated in Sec-

tion 3.2 in the form of the asymptotic series

u(x) = u(0)(x) + εu(1)(x) + ε2u(2)(x) + ... (3.7)

Therefore our task is to determine the functions u(0)(x), u(1)(x), and u(2)(x)

such that u(x) satisfies (3.1-3.3) up to order ε2. Each of these functions

is constructed as the solution of a boundary-value problem defined for an

elastic unbounded solid containing the reference crack Ω0. In what follows,

the functions u(0)(x), u(1)(x), and u(2)(x) are constructed for cracks with

circular fronts, that is ϕ(θ) = 0.

The boundary-value problem for u(0)(x) is identical to the principal

problem, except that the crack is Ω0 rather than Ωε. The boundary-value

problems for u(1)(x) and u(2)(x) involve certain tractions prescribed on Ω0

and a vanishing stress at infinity. In order to determine those tractions, we

need to develop the asymptotic expansion for the traction vector on Ωε, which

33

σ(1)13

σ(1)23

σ(1)33

|Ω0

=2∑

i=1

∂

∂xi

h

σ(0)1i

σ(0)2i

0

|Ω0

, (3.9)

and

σ(2)13

σ(2)23

σ(2)33

|Ω0

=2∑

i=1

∂

∂xi

h

σ(1)1i

σ(1)2i

σ(1)3i

+

h2

2

∂

∂x3

σ(0)1i

σ(0)2i

σ(0)3i

|Ω0

. (3.10)

In order to simplify the notation, in what follows we denote the traction vectors

in (3.9) and (3.10) by t(1) and t(2), respectively.

As far as linear fracture mechanics is concerned, the expressions for t(1)

and t(2) are meaningful as long as they do not lead to a hyper-singular stress

field near Γ0. To enforce this condition, one may require h(r, θ) to be such

that

h(r, θ) ∼ o[(a− r)1+α

]with α > 0, as r → a− . (3.11)

This choice of the exponent is sufficient for cancelling the stress-induced sin-

gularity near Γ0. The cracks that satisfy (3.11) are referred to as regular. All

other cracks are referred to as singular.

The stated restriction on h(r, θ) is sufficient but not necessary for elim-

inating the hyper-singularity. In particular, an important class of problems

for which the hyper-singularity due to t(1) is eliminated involves axisymmetric

problems for singular cracks. For those problems, the cylindrical components

of t(1) satisfy the conditions

t(1)z = t

(1)θ = 0

35

and

t(1)r =

∂

∂r

h(r)σrr(u

(0))|Ω0

.

The behavior of σrr(u(0))|Ω0 near Γ0 is governed by the standard asymptotic

solutions of linear elastic fracture mechanics (Williams; 1957). The structure

of those solutions is such that σrr induced by K(0)I on Ω0 is identically equal

to zero, and therefore

σrr(u(0))|Ω0 ∼ (a− r)−1/2 K

(0)II as r → a− and z = 0.

This relationship implies that for axisymmetric problems the hyper-singularity

is eliminated due to the condition K(0)II = 0.

Note that the component t(1)3 = 0, and therefore the first order asymp-

totic correction does not affect KI . This implies that non-planar perturba-

tions should be insignificant for initiation and growth of predominantly tensile

cracks. Nevertheless, such perturbations may play a significant role in deter-

mining the shape of growing cracks.

3.5 Evaluation of Asymptotic Solutions

In this section, we compare first-order asymptotic solutions for quasi-

circular cracks with the corresponding numerical solutions obtained with the

boundary element method of Li and Mear (1998) and Li et al. (1998). For

completeness, we consider both planar and non-planar cracks. Our evalua-

tion is restricted to predominantly tensile cracks under constant remote stress

σ∞ij = σ∞δ3iδ3j and solids with Poisson’s ratio ν = 0.3. The only exception is

36

the spherical cap crack in Section 3.5.3 where the remote stress is equal tri-

axial tension. To simplify our task, we selected the shape functions ϕ(θ) and

h(r, θ) such that the asymptotic solutions can be expressed in elementary func-

tions and accurate numerical solutions can be obtained with relatively coarse

meshes. For general shapes, the numerical integration schemes were suggested

by Kachanov and Laures (1989) and Karapetian and Kachanov (1998).

3.5.1 Boundary Element Analysis

In this chapter, all numerical solutions were obtained with the boundary

element method developed by Li and Mear (1998) and Li et al. (1998). This

method involves a symmetric Galerkin discretization of a boundary integral

equation with a weakly singular kernel.

The meshes for quasi-circular cracks were generated by mapping the

nodal coordinates of the six meshes shown in Figure 3.1. Those meshes are

formed by six-node triangular elements, eight-node quadrilateral elements, and

nine-node quadrilateral crack-tip elements. The meshes A through D were used

for analyzing planar cracks, and they involved 579, 1155, 1731, and 2307 de-

grees of freedom, respectively. The meshes E and F were used for analyzing

non-planar cracks, and they involved 3483 and 1035 degrees of freedom, re-

spectively. The principal drawback of the meshes A through D is the presence

of high-aspect-ratio triangular elements near the center. Those elements were

responsible for significant errors in analysis of non-planar cracks, and there-

fore, for those cracks, the meshes A through D were replaced with the meshes

37

meshes for planar cracks

meshes for non-planar cracks

Mesh A Mesh B

Mesh C Mesh D

Mesh E Mesh F

Figure 3.1: Meshes for the circular crack.

38

per tu rbedcrack f ron t

re fe rencec rack x 1

x 2

)(θa),(ˆ φθa φ

θ

Figure 3.2: A perturbed planar crack.

E and F.

3.5.2 Planar Cracks

Analysis of planar perturbations is particularly important for tensile

cracks since, unlike non-planar perturbations, they produce first order cor-

rections to KI . The first-order asymptotic solution for planar quasi-circular

tensile cracks has been obtained by Gao and Rice (1987). Their solution in-

volves θ-dependent reference cracks, such that Γε is prescribed as

a(θ, φ) = a(θ)1 + εϕ(θ, φ) ,

where the function ϕ(θ, φ) must satisfy ϕ(θ, θ) = 0 (Fig. 3.2).

With this notation, the asymptotic solution for the Mode I stress in-

39

tensity factor is (Gao and Rice; 1987, Eq. 15)

KI(θ) = K(0)I θ, a(θ)+

ε

8πP.V.

∫ 2π

0

K(0)I φ, a(θ)ϕ(θ, φ)

sin2 (θ − φ)/2 dφ (3.12)

Here K(0)I φ, a(θ) is the Mode I stress intensity factor at a location φ along

the front of the reference crack with radius a(θ); for uniform remote tension,

K(0)I φ, a(θ) can be computed from (3.5).

Gao and Rice (1987) have already evaluated their asymptotic solution

by applying it to elliptical cracks, for which KI(θ) can be expressed in terms

of elliptic integrals. Another asymptotic solution developed by Gao and Rice

(1987) is for wavy cracks whose front is described by the equation

a(θ) = a0(1 + ε cos nθ),

where a0 is a constant and n is an integer parameter. For such cracks, the

asymptotic solution is

KI(θ) = K(0)I θ, a(θ)

1− ε

n

2

a0

a(θ)cos nθ

.

From this expression one can conclude that small perturbations in geometry

can be significantly amplified by the wave number n, and therefore the validity

of the asymptotic solution should deteriorate as n increases. Also note that

problems involving large n pose difficulties for the boundary element method

because as n increases the local radius of curvature of the crack front decreases,

and therefore one is required to compute with fine meshes.

For comparison purposes, we set ε = 0.1 and n = 6 or n = 12. For

n = 6, the parameter εn/2 = 0.3, and for n = 12, εn/2 = 0.6. For n = 6,

40

wave number = 6

wave number =12

Mesh A Mesh B

Mesh C Mesh D

Mesh B Mesh D

Figure 3.3: Meshes for planar wavy cracks.

41

the numerical solutions were obtained with the meshes A through D, and for

n = 12, the numerical solutions were obtained with the meshes B and D

(Fig. 3.3). For n = 12, the meshes A and C involved concave finite elements,

which significantly lowered the solution accuracy. In Figure 3.4 we present

the asymptotic and numerical solutions for KI(θ) normalized by K0I . These

solutions are in excellent agreement for n = 6, except for the minimum values

of KI(θ), which occur at the finger tips. This tendency persists for n = 12,

but even in this case, the maximum discrepancy is only about ten percent.

An interesting feature of the solutions presented in Figure 3.4 is that

KI(θ) varies such that it reaches its maximum at amin = a0(1 − ε) and its

minimum at amax = a0(1 + ε). Thus in the absence of significant effects

involving anisotropy and/or inhomogeneity, a quasi-statically growing wavy

tensile crack should attain a circular shape.

3.5.3 Non-Planar Cracks

The problems considered in this section are divided into three groups.

The first group includes axisymmetric cracks, both regular and singular. For

those cracks KIII = 0 and therefore we compare the asymptotic and numerical

solutions for KII only. The second group includes non-axisymmetric cracks,

for which we compare the asymptotic and numerical solutions for both KII and

KIII . The third group includes spherical cap cracks, for which we compare the

asymptotic and numerical solutions with the analytical solution of Martynenko

and Ulitko (1978).

42

0 15 30 45 60 75 900.75

0.90

1.05

1.20

1.35

KI /

KI0

θ

Asymptotic Mesh A Mesh B Mesh C Mesh D

0 15 30 45 60 75 900.4

0.6

0.8

1.0

1.2

1.4

1.6

KI /

KI0

θ

Asymptotic Mesh B Mesh D

wave number =6

wave number=12

Figure 3.4: Comparison of asymptotic and boundary element solutions forplanar wavy cracks.

43

R1 R2

S1 S2

Figure 3.5: Meshes for axisymmetric non-planar cracks.

Axisymmetric Problems

We considered the following axisymmetric crack shapes:

R1 :h

a=

1

2

cos

(πr

a

)+ 1

,

R2 :h

a=

1

2

cos

(2πr

a

)− 1

,

S1 :h

a=

1−

(r

a

)2

,

S2 :h

a=

1−

(r

a

)4

.

The shapes R1 and R2 are regular perturbations and the shapes S1 and S2 are

singular perturbations. Each shape is chosen such that the maximum value of

h is equal to a. The meshes for each shape are shown in Figure 3.5; note that

there are two meshes per shape.

44

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

R2

R1

S1

S2

Fine Mesh Coarse Mesh Asymptotic

KII /

KI0

ε

Figure 3.6: Comparison of asymptotic and BEM solutions for non-planarcracks.

The asymptotic and numerical solutions are presented in Figure 3.6,

where KII normalized by K0I are plotted as functions of ε. These results show

that the asymptotic solutions are essentially indistinguishable from numerical

solutions for ε ≤ 0.1. For larger ε, the deviations become significant for S1 and

S2 but not for R1 and R2. It is conceivable that the second-order corrections

may virtually eliminate already small discrepancies between the asymptotic

and numerical solutions, but this is hardly worth the effort, at least for the

45

R3

R4

Figure 3.7: Meshes for non-axisymmetric non-planar cracks.

cases considered here.

Non-Axisymmetric Problems

We considered two non-axisymmetric regularly-perturbed crack shapes,

R3 :h

a=

1 +

(r

a

)2

cos2 θ

1−

(r

a

)22

and

R4 :h

a=

25√

5

32cos θ

1−

(r

a

)22

r

a.

Each shape is normalized such that the maximum height equals a. The bound-

ary element meshes for these shapes are shown in Fig. 3.7.

The asymptotic and numerical solutions are presented in Figure 3.8 for

R3 and in Figure 3.9 for R4. In these figures, KII and KIII normalized by

K0I are plotted as functions of θ for ε = 0.1 and ε = 0.25. Upon examining

46

0 60 120 180 240 300 3600.04

0.06

0.08

0.10

0.12

0.14


ε = 0.1

ε = 0.25

KII /

KI0

θ

0 60 120 180 240 300 360-6x10-3

-4x10-3

-2x10-3

0

2x10-3

4x10-3

6x10-3


ε = 0.1

ε = 0.25

KII

I / K

I0

θ

Figure 3.8: Comparison of asymptotic and BEM solutions for the non-axisymmetric R3 crack.

47

0 60 120 180 240 300 360-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

ε = 0.1

ε = 0.25 Fine Mesh Coarse Mesh Asymptotic

KII /

KI0

θ

0 60 120 180 240 300 360-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025


ε = 0.1

ε = 0.25

KII

I / K

I0

θ

Figure 3.9: Comparison of asymptotic and BEM solutions for the non-axisymmetric R4 crack.

48

D

5

K

α

ε D

U

α

Figure 3.10: Geometry of the spherical cap crack.

those curves, one can conclude that the difference between the asymptotic and

numerical solutions is less than 10% for ε = 0.1 and less than 20% for ε = 0.25.

Furthermore, the asymptotic solution correctly predicts the character of the

numerical solution for every curve.

Spherical Cap Crack

The spherical cap crack is a singularly perturbed axisymmetric crack.

The principal reason we singled out this shape is because we can compare

the asymptotic and numerical solutions with the exact solution of Martynenko

and Ulitko (1978), revisited recently by Martin (2001b), for the case of remote

equal triaxial tension. Further, for the sake of completeness, we compare the

solutions for KI and KII .

49

0 10 20 30 40 50 60 70 80 900.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Analytic Asymptotic Fine Mesh Coarse Mesh

KII

KI

K /

KI0

α (o)

Figure 3.11: Comparison of analytic, BEM, and asymptotic solutions for thespherical cap crack subjected to equal triaxial tensile load.

The spherical cap crack geometry is shown in Figure 3.10. With respect

to the sphere, the crack is prescribed by the sphere radius R and the angle

of the maximum chord contained by the crack; this angle is denoted by 2α.

Alternatively, the crack can be prescribed in terms of the base radius a and

the maximum height εa. These pairs of parameters are related as

a = R sin α and ε = tanα

2.

50

The solution of Martynenko and Ulitko (1978) is expressed as

KI = Φ(α, ν)σ∞√

πR

sin αand KII = Ψ(α, ν)σ∞

√πR

sin α,

where σ∞ is the magnitude of the triaxial remote stress and the functions

Φ(α, ν) and Ψ(α, ν) are defined in the Appendix. The corresponding asymp-

totic solution happens to be the same as for the shape S1 except that the

loading is triaxial instead of uniaxial. For this shape

KII =2 + 8ν

3εσ∞

√a0

π= ε

1 + 4ν

3K0

I .

The asymptotic, numerical and analytical solutions are plotted in Figure 3.11

for the normalized KI and KII as functions of the angle α. The plotted curves

indicate that the asymptotic solution for KII is very accurate for α ≤ 30o, and

the numerical solution obtained with the mesh F is very close to the analytical

solution for both KI and KII . Also, both numerical and analytical solutions

indicate that the slope of the KI-curve is zero for small α, which is consistent

with the fact that the correction for KI is O(ε2).

3.6 Quasi-Static Crack Growth

In this section, we apply asymptotic solutions to analysis of quasi-static

crack growth. Due to the absence of a well-accepted general criteria for mixed-

mode crack growth in three dimensions, our analysis is limited to planar Mode

I cracks and non-planar axisymmetric cracks, for which KIII = 0.

51

3.6.1 Mode I Planar Crack

Let us consider an initially circular crack of radius a0 subjected to a

remote uniaxial stress σ∞33 that varies linearly along the x1 axis:

σ∞33 = σ∞0

(1 +

x1

a0

)= σ∞0

(1 +

r cos θ

a0

).

We suppose that the crack grows such that the normal velocity of the points

along the front is prescribed as

vn(θ, t) =a0

τ

KI(θ, t)

K0I

, (3.13)

where t is time, τ is a material constant (time), and K0I = 2σ∞0

√a0/π. is

the stress intensity factor for the initial crack under remote uniaxial tension

σ∞0 . Our goal is to determine the crack shape a(θ, t) corresponding to the

prescribed initial geometry, loading, and growth law.

To compute a(θ, t) we have to combine equations (3.12) and (3.13); the

former allows us to evaluate the right-hand side for the current shape a(θ, t)

and the latter allows us to evaluate the shape increment corresponding to a

small time increment. To evaluate the integral in (3.12) we used the piece-wise

quadratic interpolation of a(θ, t) and the Gaussian quadrature rule with twelve

integration points. Integration with respect to time was performed with the

Euler forward scheme.

The function a(θ, t) normalized by a0 is plotted in Figure 3.12. The

final shape is significantly different from the original one as the maximum

a(θ, t) is almost 3a0. However, it appears that the evolution of a(θ, t) involves

52

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

x 2

x1

Figure 3.12: Contours of a quasi-statically growing planar crack.

two dominant modes: self-similar expansion of the initial circular crack and

translation of the circle center. Of course, a(θ, t) is not perfectly circular,

but it certainly can be regarded as quasi-circular, and therefore should lead

to accurate asymptotic solutions. To some extent, the tendency to maintain

quasi-circular shapes is not that surprising. For an elliptical crack under uni-

form remote tension, the stress intensity factors are maximum along the short

axis and minimum along the long axis. Thus such a crack should evolve into

a circular or quasi-circular shape. The prescribed non-uniform loading does

not necessarily favor cracks expansion along the x1-axis over that along the

53

x2-axis because KI(θ) is such that

KI(0) > KI(π/2) = KI(3π/2) > KI(π).

Instead the loading non-uniformity produces translation along the x1 axis,

driven by KI(0) −KI(π), but not along the x2 axis, since there is symmetry

about the x1 axis.

3.6.2 Axisymmetric Non-Planar Crack

Here we consider the axisymmetric problem for mixed-mode singular

cracks. In this problem, the original crack shape and a constant remote stress

are given, and the objective is to determine the shape function h(r) for the

crack growing along the path characterized by the criterion KI = KIC and

KII = 0. The determination of h(r) consists of two steps: (i) derivation of an

integro-differential equation for h(r) and (ii) solution of that equation.

In this section, the approach is somewhat different from that in Section

3.4 because the reference crack is always chosen in the plane z = 0, rather than

being shifted along the z-axis. Consequently, in general, h(a) 6= 0, and there-

fore the asymptotic expansions involve hyper-singular terms. Nevertheless,

those terms do not pose difficulties for the derivation of the integro-differential

equation for h(r), thanks to a regularization based on Betti’s reciprocal theo-

rem. Our presentation omits many details that can be found in Movchan et al.

(1998).

To derive the integro-differential equation, we introduce three local co-

54

η

η

ξ

ξ

φ

Figure 3.13: Local coordinate systems for characterization of the crack tipfields.

ordinate systems for characterization of the crack tip fields. All three co-

ordinate systems are two-dimensional and lie in the θ-planes (Figure 3.13):

Cartesian coordinates ξ = (ξ1, ξ2) are natural coordinates for the points on Γ0;

Cartesian coordinates η = (η1, η2) are natural coordinates for the points on Γε;

Polar coordinates (R, φ) are naturally related to ξ: ξ1 = R cos φ and ξ2 =

R sin φ.

To the first order, the ξ and η coordinates are related as

η =

(1 εh′(a)

−εh′(a) 1

)ξ −

(0

εh(a)

)

where the prime sign denotes differentiation with respect to r.

Also let us recall the well-known eigen-solutions (Williams; 1957) for

a semi-infinite traction-free crack in two dimensions. The angular function

corresponding to the displacement vector of the Mode II solution proportional

to R1/2 is denoted by ΦII(θ), and the angular functions corresponding to the

55

displacement vector of the Mode II solution proportional to R−1/2 is denoted

by ΨII(θ). Explicit expressions for the angular functions are given for the

polar components of the vectors:

ΦII(θ) =1

4µ√

2π

(−(2κ− 1) sin θ2

+ 3 sin 3θ2

−(2κ + 1) cos θ2

+ 3 cos 3θ2

),

ΨII(θ) =1

12µ√

2π

((2κ− 3) sin θ

2+ 5 sin 5θ

2

−(2κ + 3) cos θ2

+ 5 sin 5θ2

).

Here µ is the shear modulus, κ = 3− 4ν, and ν is Poisson’s ratio.

As in Section 3.4, the displacement field on Ω0 can be approximated

using the expansion

u(ξ, ε) = u(ξ, ε)|ε=0 + ε∂u(ξ, ε)

∂ε|ε=0 + . . .

= u(0) + ε∂

∂ε

(1 −εh′(a)

εh′(a) 1

)u(η)

ε=0

+ . . .

= u(0) + ε

(0 −h′(a)

h′(a) 0

)u(0)(η) +

∂

∂εu(η)

∣∣∣∣∣ε=0

+ . . .

= u(0)(η) + εu(1)(η) + . . . (3.14)

The function u(1) can be decomposed as

u(1) = u(1,s) + u(1,r),

where u(1,s) and u(1,r) denote the singular and regular components, respec-

tively.

Following the analysis of Movchan et al. (1998), we establish that

u(1,s) = −h(a)(1− ν)K

(0)I

µζII(R, φ),

56

where ζII(R, φ) denotes the Mode II weight function for the penny-shaped

crack (we do not need its explicit form). Near Γ0, this function has the asymp-

totic form

ζ(II)(R, φ) ∼ R−1/2ΨII(φ) as R → 0,

so that

u(1,s) ∼ −h(a)(1− ν)K

(0)I

µ√

RΨII(φ) as R → 0.

The regular component of u(1) has the form

u(1,r) =√

R

K

(1)II −

1

2h′(a)K

(0)I

ΦII(φ) .

This expression takes into account axial symmetry and uniformity of the re-

mote stress.

Note that the field u(1,s) has an inadmissible singularity at the edge of

the crack, and the elastic energy associated with this field is infinite. Never-

theless, since Lu(1,r) = 0, we can apply reciprocal theorem to the functions

ζII and u(1,r). The result can be written in the form

K(1)II −

1

2h′(a)K

(0)I = −2π

∫ a

0

rZ(a, r)h(r)σrr(u

(0), r)′

dr,

where Z is the trace of ζII on the upper face of the crack,

Z(a, r) =

(2

a

)3/2πr2

(2− ν)√

a2 − r2.

By requiring the crack to grow such that KI = KIC and KII = 0, we

obtain the integro-differential equation for the function h(r):

h′(a)KIC = 4π

∫ a

0

rZ(a, r)∂

∂rh(r)σrr(u

(0), r)′dr . (3.15)

57

This equation must be supplemented with the function h0(r), r ∈ [0, a0] that

describes the original shape of Ωε, and the fracture toughness functionKIC(a).

For numerical treatment, it is convenient to rewrite (3.15) in the form

h′(a)KIC − 4π

∫ a

a0

rZ(a, r)h(r)σrr(u(0)r)′dr

= 4π

∫ a0

0

rZ(a, r)h0(r)σrr(u(0), r)′dr ,

so that the right-hand side is computed only once, and the left-hand side is

approximated using techniques that respect the singularity of Z(a, r) at r = a.

As a demonstration, let us consider the case when

σ∞rr = σ∞ = const ,

the initial shape is

h0(r) = εa0 sin

(πr2

2a20

),

and

KIC/(|σ∞|√a0) = β = const ,

with ε = 10−3 and β = 103. Numerical values for the remaining parameters

do not need to be specified since they can be scaled. Note that in order

to maintain constant σrr and KI the stresses σ∞θθ and σ∞zz must be applied.

Nevertheless, these stresses are not required for computing h(r).

The graph of the function h0(r) normalized by εa0 is shown in Fig. 3.14.

Fig. 3.15(a) and Fig. 3.15(b) show the consequent crack path for σ∞rr > 0 and

σ∞rr < 0, respectively. These figures clearly demonstrate that σ∞rr > 0 favors

58

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

r/a0

h 0(r)/

(ε a

0)

Student Version of MATLAB

r/a0

0 0h (r)/( a )ε

Figure 3.14: The function h0(r) characterizing the initial perturbation of thecrack surface.

non-planar growth, whereas σ∞rr < 0 flattens the crack so that, eventually,

it grows in a horizontal plane. These stability conditions parallel those of

Cotterell and Rice (1980) for two-dimensional cracks.

3.6.3 Axisymmetric Crack Opening

Here we consider regularly perturbed cracks such that the functions

describing the perturbation of the upper and lower faces of the crack are dif-

ferent. An example of such a perturbation is shown in Fig. 3.16, where the

crack opening is defined as a thin void located between the two surfaces

−εh− ≤ x3 ≤ εh+.

It is meaningful to assume that h+ + h− ≥ 0.

59

r/a0

r/a00 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

r/a0

h(r)/(

ε a0)

(b)


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

r/a0

h(r)/(

ε a0)

(a)


0εh (r)/( a )

0h (r)/( a )ε

Figure 3.15: Crack surface deflection: (a) unstable crack surface for the caseof positive σrr and (b) the crack surface for the case of negative σrr.

60

3 x

2 x

1 x

o

) , ( 2 1 3 x x h x + =

) , ( 2 1 3 x x h x - =

Figure 3.16: An axisymmetric opening for a three-dimensional crack.

Asymptotic expansions for the crack opening are similar to those de-

veloped in Section 3.4, except that the effective traction boundary conditions

are governed by the difference between h+ and h−. Namely,

t(1)k = ±

2∑p=1

∂

∂xp

h±(x1, x2)σpk(u(0))

∣∣∣∣∣x3=±0

and

t(2)k = ±

2∑p=1

∂

∂xp

h±(x1, x2)σpk(u(1))∓ h2

±2

∂

∂x3

σpk(u(0))

∣∣∣∣∣x3=±0

.

Consider an axisymmetric problem such that

σ∞rr = σ∞θθ = σ∞ = const. > 0 and σ∞ij = 0 otherwise,

and h+ = h− = h(r). For this problem the stress intensity factors are given

by the formulae

KII = KIII = 0,

61

and

KI = ε2σ∞1√πa

∫ a

0

h(r)2(2a2 + r2)r

(a2 − r2)5/2dr + . . . , (3.16)

which correspond to symmetric loading of the faces with the opening traction

t(2)3 = σ∞

h

r(rh′)′ + h′2

.

Note that for regularly perturbed cracks, h(r) must be such that

h′(0) = h′(a) = 0 and h(a) = 0 .

Using (3.16), we can examine stability of a growing crack opening under

lateral loading. In particular, we show that stability is affected by the growth

pattern. To this end, we consider three examples that involve unstable, neu-

trally stable, and stable growth.

In the first example, the crack opening grows in a self-similar fashion.

It means that h(r) has the form

h(r) = ag(s) ,

where g is a smooth non-dimensional function of one variable s ∈ [0, 1]. In

this case, (3.16) can be rewritten as

KI = ε2σ∞√

a

π

∫ 1

0

g2(s)(2 + s2)s

(1− s2)5/2ds + . . .

From this equation, it is obvious that ∂KI/∂a > 0, and hence the growth is

unstable.

62

In the second example, the crack opening grows in a self-similar fashion

by maintaining a constant cohesive zone ahead of the tip. If the cohesive zone

length is denoted by c, then h(r) can be defined as

h(r) =

aαc1−αg (s) if 0 ≤ r ≤ a0 if a ≤ r ≤ a + c

Then (3.16) can be rewritten as

KI = ε2σ∞a2α−3/2c2(1−α)

√π(1 + c/a)

∫ 1

0

g2(s)2(1 + δ)2 + s2)s(1 + δ)2 − s2]5/2 ds + . . . ,

where δ = c/a. If α = 3/4 and δ ¿ 1, the expression for KI simplifies to

KI = ε2σ∞√

c

π

∫ 1

0

g2(s)(2 + s2)s

(1− s2)5/2ds + . . .

It is noted that this value is independent of a and hence the growth is neutrally

stable.

In the last example, the crack opening grows such that

h(r) =

h0(r) if 0 ≤ r ≤ a0

0 if a > a0

Then (3.16) can be rewritten as

KI = ε2σ∞1√πa

∫ a0

0

h20(r)(2a

2 + r2)r

(a2 − r2)5/2dr + . . . ,

and direct calculations show that ∂KI/∂a < 0, and hence the growth is stable.

3.7 Summary

In this chapter, we developed new asymptotic solutions for non-planar

quasi-circular cracks, and evaluated the validity of asymptotic solutions for pla-

nar and non-planar cracks by comparing them with numerical and analytical

63

solutions. For all benchmark problems considered in this chapter, the asymp-

totic solutions were able to match the numerical solutions well for ε ≤ 0.25.

This fact lends support to the notion that asymptotic solutions can be used

not only for qualitative but also for quantitative estimates.

Since this chapter is dedicated to the development and evaluation of

asymptotic solutions, we considered only relatively simple benchmark prob-

lems, for isolated cracks. For the majority of such problems, asymptotic anal-

ysis is unnecessary since they can be solved numerically to a high degree of

accuracy. However, for three-dimensional problems involving multiple cracks,

asymptotic solutions, combined with numerical methods, may be indispens-

able. Although such hybrid numerical methods may take on a variety of forms,

all of them should be similar in spirit to matched asymptotic expansions. For

example, one such method can be developed as a combination involving asymp-

totic solutions for isolated quasi-circular cracks and methods for computing

interactions among circular cracks (Kachanov; 1994). In particular, one can

determine the effective tractions on the reference crack by assuming that all

cracks are circular, and then compute the local corrections using asymptotic

solutions. Of course, asymptotic solutions can be also combined with bound-

ary element methods or even better, boundary element methods based on fast

iterative solvers, designed specifically for multiple crack problems (Yoshida

et al.; 2001a,b,c; Lai and Rodin; 2002).

Hybrid methods should be particularly useful for simulations of quasi-

static growth of multiple cracks because those problems are computationally

64

intensive and because both asymptotic analysis and iterative solvers can ex-

ploit the fact that each crack growth increment is associated with small geo-

metric changes. A major obstacle for carrying out such simulations is the lack

of a well-accepted criterion for crack growth in three dimensions.

65

Chapter 4

Mixed-Mode Cracking

In this chapter, we present a study of crack growth initiation in PMMA

under mixed-mode II/III, mixed-mode I/III and pure Mode III loading condi-

tions. This study combines macroscopic measurements leading to the critical

stress intensity factors, microscopic observations of microstructural features

near the crack front, boundary element analysis of the fracture specimens,

and micromechanical analysis of the interactions between the main crack and

microcracks. It is proposed that the Mode III loading component affects the

fracture toughness KIC , but does not lead to crack growth initiation in the

absence of the Mode I and II loading component. This implies that one can

develop a three-dimensional crack growth initiation criterion using an existing

two-dimensional criterion as the basis – the key difference is that in three di-

mensions KIC must be treated as a function of the Mode III stress intensity

factor rather than a constant.

4.1 Introduction

There is a significant body of literature concerned with criteria for

crack growth and initiation for two-dimensional cracks (e.g., Erdogan and Sih;

66

1963; Sih; 1974). Three-dimensional mixed-mode cracks have not received as

much attention as their two-dimensional counterparts in the literature, mostly

due to enormous challenges associated with their experimental and analytical

characterization.

The major distinction between two-dimensional and three-dimensional

crack growth arises from their geometry:

• For two-dimensional cracks, the crack front is a point moving along a

curve. In contrast, for three-dimensional cracks, the crack front is a curve

moving along a surface. This difference becomes particularly profound

if the curves and surfaces are non-smooth. Furthermore, rugged crack

fronts are common for mixed-mode three-dimensional cracks (e.g., Som-

mer; 1969; Knauss; 1970; Hull; 1995; Cooke and Pollard; 1996; Tanaka

et al.; 1998).

• To meet K-dominance conditions, effects of plasticity, heterogeneity, and

anisotropy must be confined to a small process zone ahead of the crack

front. For two-dimensional cracks, such conditions are well known in the

literature. For three-dimensional cracks, relationships between the plas-

tic zone size and length scales, associated with the waviness of the crack

front, are unclear. Apparently, if those scales are much smaller than the

plastic zone size, then the crack front can be regarded as straight. On the

other hand, if the waviness length scales are much larger than the plastic

zone size, then the crack should be regarded as wavy. Without the exact

67

definition of what is “much larger” and “much smaller” one may have

difficulties defining the crack front for three-dimensional cracks.

• Non-locality: For two-dimensional cracks, all criteria are local in the

sense that the crack-tip is just a point. For three-dimensional cracks,

non-locality is possible in the sense that crack growth at a reference

point along the front can depend on the stress intensity factors at this

point as well as points in the neighborhood of the reference point. The

non-locality becomes significant if the local radius of curvature is not

very much larger than the plastic zone size. Theoretically, this issue has

been considered by Hodgdon and Sethna (1993).

• Loss of local symmetry: Obvious generalizations of the two-dimensional

criterion of local symmetry cannot be applied for mixed-mode three-

dimensional cracks because one cannot enforce the absence of shear de-

formations simultaneously at all locations around the crack front and

still insist upon a continuous crack face evolution. This implies that the

criterion must involve at least two stress intensity factors, and therefore

it must involve at least a curve in the K-space.

• Loss of self-similarity: Since more than one stress intensity factor must

be involved in the criterion, the process zone ahead of the growing crack

front does not have to maintain self-similarity, which may significantly

complicate the situation. A similar situation exists for mixed-mode in-

terface cracks, for which the effective toughness strongly depends on the

68

mode mix.

• Identification of geometry: The process zone size is the fundamental

length scale in the problem. If the radius of curvature of the crack front

is significantly larger than the process zone size, then the crack may

be regarded as smooth. Otherwise, one must approximate the rugged

front with a smooth curve. This issue is important because the curve

selection affects the stress intensity factors, and consequently the fracture

criterion.

While each of these issues poses a significant intellectual challenge, we

believe that formulation of the three-dimensional crack growth criterion may

require all of them to be considered simultaneously. Analytical studies to a

certain extent can be found in the work of Gao and Rice (1986), Hodgdon and

Sethna (1993), Xu et al. (1994), Larralde and Ball (1995) and Lazarus et al.

(2001a,b). However, due to the limited amount of quantitative measurements,

there is no consensus yet on how a crack behaves in a general three-dimensional

mixed-mode situation. In the absence of a well developed crack growth cri-

terion, any numerical simulation of quasi-statically growing three-dimensional

cracks is futile.

Our goal in this chapter is to provide a qualitative understanding essen-

tial for formulating a growth initiation criterion for cracks under mixed-mode

loading. We conduct fracture experiments on specimens in different loading

configurations. The common features of our experiments are presented in Sec-

69

tion 4.2. In Section 4.3, we present experimental results and observations using

an optical microscope. A boundary element analysis of the test specimens is

presented in Section 4.4. In Section 4.5, we present a dipole approximation

for a nucleated microcrack and estimate its interaction with the main crack.

The conclusion and directions for future work are presented in Section 4.6.

4.2 Experimental Setup

In our studies we tested edge crack, three-point bend and torsion spec-

imens. These specimen configurations allow us to investigate cracking behav-

ior under mixed-mode II/III, mixed-mode I/III and pure Mode III loading

conditions. In most cases, the specimen were made of polymethyl methacry-

late (PMMA or Plexiglas). At room temperature and atmospheric pressure,

PMMA behaves as an isotropic, homogeneous, linearly elastic solid. Further,

the plastic zone size for PMMA is about 80 microns, so that conditions for

small-scale yielding are easy to satisfy. In one of the three-point bend experi-

ments, we used polycarbonate. Table 4.1 lists necessary mechanical properties

for these two materials.

E (GPa) ν σy (MPa)PMMA 2.71 0.38 45

polycarbonate 2.40 0.35 65

Table 4.1: Mechanical properties of PMMA and polycarbonate.

70

4.2.1 Edge Crack Specimen (mixed-mode II/III)

Two types of edge crack specimens, whose schematic diagrams are

shown in Figure 4.1 and 4.2, were made by slicing a flat notch from one edge

of the specimen block through the thickness. Around the pin holes, opposite

halves of the top and bottom wings were drilled off. Therefore as the pin and

grip fixtures were displaced away from each other, shear was applied along the

direction parallel to the crack front resulting in Mode III loading. A T-shaped

specimen was created to shorten the crack front in order to lower the energy

required to advance the crack. The tip was sharpened by driving a razor blade

about 1mm into the band-sawed notch.

4.2.2 Three-Point Bend Specimen (mixed-mode I/III)

The three-point bend specimen with an inclined slit was applied to in-

vestigate crack propagation under mixed-mode I/III loading for which KI is

usually larger than KIII . The configuration for the PMMA specimen is de-

picted in Figure 4.3. The polycarbonate specimen has identical dimensions

except that the thickness is 1.9cm. The 45-inclined center notch was intro-

duced into the specimen block using a 0.3mm slitting saw. The notch tip was

sharpened by drawing a razor blade back and forth along the slit. Following the

same procedures, we also made a specimen with a center notch perpendicular

to both edges in order to obtain the KIC under pure Mode I loading.

71

Figure 4.1: Schematic diagram of the rectangular edge crack specimen (unit:cm).

Figure 4.2: Schematic diagram of the T-shaped edge crack specimen (unit:cm).

72

F

F/2

Figure 4.3: Schematic diagram of the mixed-mode I/III three-point bend spec-imen (unit: cm).

4.2.3 Torsion Specimen (pure Mode III)

The cylindrical torsion specimen with a circumferential cut is capable

of generating any I/III load mix over the entire crack front by varying the

ratio of torsional and tensile load on the specimen. In our experiments, we

studied the pure Mode III loading condition only. The schematic diagram

of the specimen is shown in Figure 4.4. A 7.62mm-deep groove was drilled

around the circumferential surface of the cylinder. The circumferential notch

was then driven 1.28mm further into the cylindrical specimen using a 0.3mm

slitting saw, extending the total notch length to 8.9mm. The notch tip was

sharpened by drawing a razor blade along the slit.

4.3 Experimental Results and Observations

In this section, we summarize load-displacement data and present ob-

servations of microscopic features ahead of the main crack.

73

Figure 4.4: Schematic diagram of the torsion specimen (unit: cm).

4.3.1 Results for Edge Crack Specimens

Monotonic tests performed on edge crack specimens were displacement

controlled at a crosshead speed of 0.254mm/s. The load-displacement curves

for the rectangular and the T-shaped specimens are shown in Figure 4.5. Ap-

parently the T-shaped specimen is more compliant. Nevertheless, instead of

advancing the crack, both specimens failed at a location close to where the

crack front intersects the free boundary. One of the wings broke off, generat-

ing a cleavage normal to the crack faces, at about 1.62KN for the rectangular

specimen and 0.22KN for the T-shaped specimen.

To investigate any possible crack propagation with this specimen con-

figuration, a supplementary fatigue test was performed. A rectangular edge

crack specimen was repeatedly loaded at 1Hz to an initial maximum load of

1.1KN over a 0.635mm fixed displacement amplitude. After about 105 cycles

the specimen broke down at the pin and grip fixtures. The crack front still

did not propagate.

Overall, we were not able to observe any crack propagation under the

74

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

T-shaped specimen

Rectangular specimen

Lo

ad

(K

N)

Displacement (mm)

Figure 4.5: Load-displacement curves for the rectangular and the T-shapededge crack specimens under monotonic loading.

mixed-mode II/III loading condition generated by the current design of our

edge crack specimens. Before the crack front could propagate, the specimens

broke down at unexpected locations.

4.3.2 Results for Three-point Bend Specimens

The three-point bend specimens were used in both monotonic and fa-

tigue tests. The monotonic loading was displacement controlled at a crosshead

speed of 0.254mm/s. Figure 4.6 shows the load-displacement curves for the

mixed-mode I/III and the pure Mode I specimens. The mixed-mode I/III

75

0 10 20 30 40 50 60 70 80 90 1000.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3.0

PMMAmixed-I/III

polycarbonatemixed-I/III

PMMAMode-I

Lo

ad

(K

N)

Displacement (mm)

Figure 4.6: Load-displacement curves for the mixed-mode I/III and the pureMode I three-point bend specimens under monotonic loading.

curve depicts an unstable crack propagation whereas the Mode I curve implies

a crack arrest. For PMMA specimens, the peak load for the mixed-mode I/III

configuration was 1.31KN while that for the pure Mode I was 0.52KN. The

polycarbonate specimen shows a ductile cracking behavior with a peak load at

2.99KN. A fatigue test was conducted on a mixed-mode I/III specimen at a

fixed displacement amplitude of 5mm and a frequency of 1Hz. Since we were

unable to initiate crack propagation at a loading level lower than 60% of the

peak load, we switched the loading range several times during the test and

eventually settled at a maximum load of 0.9KN.

76

Figure 4.7: Fractography of the mixed-mode I/III three-point bend PMMAspecimen subjected to the monotonic load.

Figure 4.7 shows microscopic features of a central region of the mono-

tonically loaded mixed-mode I/III PMMA specimen. This and all other mi-

croscopic images were obtained with an optical microscope. Each triangular

structure represents a group of emerging cracks, each of which evolves in a

three-dimensional route and merges with neighbors. Pure Mode I propaga-

tion is achieved over the top of the array of triangles, about 7.5mm from the

pre-cracked front. Due to the shallow specimen depth, the global twisting of

evolved crack face could only be observed around the free boundaries implying

an ascending KII distribution.

A close-up view at the pre-cracked front, Figure 4.8, shows needle-

like crazing structures extending about 0.2mm from the main crack front.

A sequence of features can also be observed showing the crazes gradually

77

Figure 4.8: Needle-like crazing structures and stacked microcracks on the crackfront of the monotonically loaded mixed-mode I/III PMMA specimen.

78

crazing

turning

l inking

Figure 4.9: Possible mechanism of microcrack nucleation.

transform into microcracks. Fully developed microcracks are stacked one by

one, bisecting the main crack at a certain angle. These structures are trapped

in situ before evolving into macro-scale features as seen in Figure 4.7, providing

a good example to study the nucleation and interaction of microcracks under

mixed-mode loading conditions.

From the observation, we propose a possible mechanism of microcrack

nucleation in Figure 4.9. The needle-like crazing structures are first triggered

along the direction perpendicular to the Mode I stress. Each craze starts turn-

ing due to the Mode III shearing stress and eventually links with a neighboring

craze to form the microcrack front. It would be reasonable to suggest that the

mode mix ratio, KIII/KI , plays a dominant role in determining the size and

orientation of each microcrack.

The fractography of the mixed-mode I/III polycarbonate specimen un-

der monotonic loading is shown in Figure 4.10. The features on the surface

show periodically spaced facets aligned parallel at a certain angle. The spacing

of facets is about 0.6mm whereas the inclination angle is about 60. The mea-

sured angle is beyond the predicted range of 0 to 45 under mixed-mode I/III

79

Figure 4.10: Fractography of the mixed-mode I/III three-point bend polycar-bonate specimen subjected to the monotonic load.

loading (Pollard et al.; 1982; Cooke and Pollard; 1996), which may indicate

that these features are not anywhere close to the cracking at the initial stage.

The coarse and fine-scale fractographies of the central region of the

mixed-mode I/III specimen under fatigue loading are shown in Figure 4.11

and 4.12, respectively. Generally, the fatigue specimen shows features similar

to those of the monotonic specimen but in a finer scale. Moreover, the fea-

tures repeat themselves in a self-affine manner, which was studied by Chevrier

80

Figure 4.11: Coarse-scale Fractography of the mixed-mode I/III three-pointbend specimen subjected to the fatigue load.

81

Figure 4.12: Fine-scale fractography of the mixed-mode I/III three-point bendspecimen subjected to the fatigue load.

(1995). No trapped crazing structures can be found in the fatigue specimen.

This could possibly mean that each nucleated microcrack evolves and merges

with others during the course of fatigue load.

4.3.3 Results for Torsion Specimens

The torsion tests were performed on an MTS axial/torsional test ma-

chine. Each specimen was loaded in a fixed rotation rate of 0.125/s, resulting

in a pure Mode III loading on the entire crack front. It was also subjected to

a longitudinal compressive stress of 0.9Pa, which is negligible and remained

constant during the test. The experimental torque-rotation curve is shown

in Figure 4.13. The curve shows a peak torque of 0.44KNm, at which the

specimen broke down globally, forming a 45 cleavage with respect to the lon-

gitudinal axis due to tensile failure. However, before the peak torque was

82

0 1 2 3 4 5 6 7 8 9 10 11 120.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

To

rqu

e (

KN

m)

Rotation (o)

Figure 4.13: Load-rotation curve for the torsion specimen.

achieved, nonlinearity appeared on the curve when the applied torque was

larger than 0.375KNm. Since PMMA is a nominally brittle material, the tran-

sition from a linear structural behavior to a nonlinear one may well indicate

the nucleation of microcracks along the crack front. Therefore the critical

torque at this moment, rather than the peak torque, should be related to the

Mode III toughness, KIIIC .

Features of evolved cracks from the original crack front, shown in Figure

4.14 viewing along the direction of the longitudinal axis, are preserved in

unbroken parts of the specimen. On the left side of the figure, two major cracks

are about to merge with each other, resulting in an overlapped region which

83

Figure 4.14: Evolved cracks from the pre-cracked front of the torsion specimen.

will later become a river-line pattern. Detailed explanation of the process can

be found in the work of Hull (1993, 1995). Lateral view of the specimen reveals

that these cracks are periodically spaced and each one of them intersects the

original crack front at 45. This implies that cracks were nucleated normal to

the direction of the principal tensile stress.

4.4 Analysis of Experiments

In this section we present computational results for the stress intensity

factors for each specimen configuration. The analysis was performed with the

boundary element code developed by Li and Mear (1998) and Li et al. (1998).

84

Rectangular edge crack specimen

T-shaped edge crack specimen

Figure 4.15: Edge crack specimens: boundary element meshes for outer bound-aries and the crack face.

4.4.1 Analysis on Edge Crack Specimens

For the purpose of boundary element analysis, we mesh only a portion

of each edge crack specimen, for which the crack length equals the bonded

length, presuming that this segment is away from the pin and grip fixtures.

Figure 4.15 shows the boundary element meshes for the rectangular and the

T-shaped specimen. We assume that the wings act as beams and therefore the

boundary conditions, Figure 4.16, consist of a linear distribution of in-plane

85

σI I I

σ

y

x

Figure 4.16: Boundary conditions for the BEM analysis of the edge crackspecimen.

shear stress, σII , as well as a parabolic distribution of out-of-plane shear stress,

σIII . Both boundary conditions are imposed on the edge of the segment.

Figure 4.17 shows the stress intensity factors for the two edge crack

specimens at the moment when their respective peak loads are achieved.

Clearly from the figure, both specimens experience mixed-mode II/III load-

ing over the crack front except for a relatively small central region where KII

vanishes. The T-shaped specimen exhibits a higher KII/KIII mode mix ratio

because the two free boundaries that intersect the crack front are closer to

each other than those in the rectangular specimen. The negligible KI over the

crack front indicates that longitudinal twisting due to asymmetrically arranged

shearing fixtures is small. Although both specimens are different in geomet-

rical configurations and break down without advancing the preexisting crack

front, their stress intensity factors at their respective peak loads are quantita-

tively similar. Comparing with the experimental value of KIIC = 1.9MPa√

m

and KIIIC = 4.4MPa√

m for PMMA (Davenport and Smith; 1993), we no-

86

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

KII

KI

KIII

T-shaped Rectangular

SIF

(M

Pa

m1/

2 )

Normalized crack front coordinate

Figure 4.17: Stress intensity factors of edge crack specimens at their respectivepeak loads.

ticed that for both specimens at their peak loads, KIII reaches to about 12%

of the Mode III toughness while KII approaches the Mode II toughness at the

surface breaking points. The specimens may break down around these specific

locations due to large KII .

4.4.2 Analysis on Three-point Bend Specimens

Figure 4.18 shows the boundary element meshes for mixed-mode I/III

and pure Mode I three-point bend specimens. The boundary conditions, im-

posed on the crack face, consist of a linear distribution of the bending stress

87

mixed-mode I/III specimen

pure Mode I specimen

Figure 4.18: Three-point bend specimens: boundary element meshes for outerboundaries and the crack face.

and, for the mixed-mode I/III specimen, a parabolic distribution of the shear-

ing stress.

The distributions of stress intensity factors at peak loads are shown in

Figure 4.19 and 4.20 for the PMMA and the polycarbonate specimen, respec-

tively. For the PMMA specimen, the experimental KIC is found to be about

1.1MPa√

m, which is close to the empirical estimate of Tada et al. (1973) and

the experimental result obtained by Davenport and Smith (1993). For mixed-

mode I/III PMMA specimen, the mode mix ratio, KIII/KI , is about 0.5 over

the central region and KII is comparably negligible over the majority of the

crack front. The mixed-mode I/III polycarbonate specimen exhibits similar

88

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

I I/III

KI

KIII

KII

KI

SIF

(M

Pa

m1/

2 )

normalized crack front coordinate

Figure 4.19: Stress intensity factors for mixed-mode I/III and pure Mode Ithree-point bend PMMA specimens under their peak loads.

distributions of stress intensity factors while the mode mix ratio is about 0.45.

We did not perform any pure Mode I test using polycarbonate specimens, but

we note that the Mode I toughness for polycarbonate is about 2.0MPa√

m

(Ashby and Jones; 1980).

Figure 4.19 shows that the presence of Mode III loading depresses the

Mode I fracture toughness. Notice that the boundary element analysis con-

siders only the preexisting main crack. However, microcracks may have been

nucleated and interacted with the main crack long before the main crack starts

propagating. The mixed-mode I/III result implies that the existence of micro-

89

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

KII

KIII

KI

SIF

(M

Pa

m1/

2 )

Normalized crack front coordinate

Figure 4.20: Stress intensity factors for mixed-mode I/III three-point bendpolycarbonate specimen under the peak load.

cracks amplifies stress intensity factors over the front of the main crack. In

Section 4.5 we present a dipole approximation to analyze this effect.

4.4.3 Analysis on Torsion Specimens

Accurate asymptotic solutions for stress intensity factors induced by

a combination of tensile and torsional loading on a circumferentially-cracked

cylinder are available from Tada et al. (1973). Substituting geometrical pa-

rameters of our torsion specimen and the critical torque supposed to be related

to the microcrack nucleation, 0.375KNm, into the handbook solution, we ob-

90

tained KIIIC = 4.4MPa√

m for our torsion test. This value is identical to the

one reported by Davenport and Smith (1993).

4.5 Microcrack-Macrocrack Interaction

In the previous section, we notice from the boundary element analy-

sis that the mixed-mode I/III three-point bend specimen failed at the stress

intensity level below the Mode I fracture toughness. We assume that this is

due to the presence of microcracks, which amplifies the Mode I stress inten-

sity factor over the preexisting main crack front. The close-up view, Figure

4.8, shows that trapped crazing structures and stacked microcracks extend

about 0.2mm from the main crack front. This length scale is larger than the

estimated process zone size of 80 microns, from the mechanical properties of

PMMA in Table 4.1. Although this difference between the microcrack size

and plastic zone size may not be sufficient for valid linearly elastic fracture

mechanics, analyzing this scenario may provide us with qualitative results.

4.5.1 Problem Formulation

The geometrical settings of our analysis are shown in Figure 4.21, where

XX1, X2, X3 represents the global coordinates on the macrocrack front while

xx1, x2, x3 the local coordinates attached to the microcrack. Each micro-

crack is assumed circular with radius a and its center, a distance l from the

nearest point on the macrocrack front, lies on the X3 = 0 plane due to negli-

gible KII over the front. The orientation of each microcrack is characterized

91

l

2 a

1x

2x

3x

1X2X

3X

Figure 4.21: An inclined circular microcrack ahead of a flat semi-infinitemacrocrack.

by Euler angles θ1 and θ2, denoted according to the rotation with respect to

the two planar axes. We let θ3 = 0 because rotation with respect to the axis

normal to the microcrack face is trivial. Only one microcrack is taken into

account in the analysis. The interaction amongst microcracks is neglected.

Suppose that without microcracks, the macrocrack is subjected to a

mixed stress state characterized by K0I and K0

III . The stress field at the

location where the center of the microcrack is can be readily constructed using

asymptotic expansions (Williams; 1957). The traction vector on the imaginary

microcrack face under plain strain condition can therefore be written in local

coordinates as

t01 =cos θ1[2K

0III cos 2θ2 −K0

I (1− 2ν) sin 2θ2]

2√

2πl,

t02 =cos θ1 sin θ1 sin θ2[2K

0III cos θ2 −K0

I (1− 2ν) sin θ2]√2πl

, (4.1)

t03 =K0

I + cos2 θ1[K0III sin 2θ2 −K0

I (1− 2ν) sin2 θ2]√2πl

.

If it is assumed that the nucleation of microcracks seeks pure Mode I loading,

92

both t01 and t02 must vanish. In the case where K0III 6= 0, this assumption

suggests that either θ1 = 90, which rules out the effect of K0III completely, or

θ2 =1

2tan−1

(2

1− 2ν

K0III

K0I

)and θ1 = 0 . (4.2)

The criterion presented in (4.2) was suggested by Pollard et al. (1982) and

applied in the prediction of the “twist angle” of echelon fractures under mixed-

mode I/III loading (Pollard et al.; 1982; Cooke and Pollard; 1996). We notice

that echelon fractures are well developed features in a rather large scale. Dur-

ing the course of forming and propagating of echelon fractures, the stress field

around the macrocrack front must have been disturbed. This may explain

why (4.2) usually overestimates the angle (Pollard et al.; 1982). Here we use

the criterion to determine the orientation of microcracks, which are about to

nucleate ahead of the macrocrack. Note that θ2 is bounded by 0 for K0III → 0

and 45 for K0I → 0. For a specific θ2, components of the traction vector are

zero except that

t03 =K0

I + [K0III sin 2θ2 −K0

I (1− 2ν) sin2 θ2]√2πl

. (4.3)

4.5.2 Dipole Approximation

Let us estimate the effect of microcrack on the main crack by assuming

that the microcrack is a dipole. Although this approximation is valid only for

large separations between the microcrack and the macrocrack, it gives us a

qualitative description of the problem.

93

We assume that the dipole is induced by the stress field produced by

the macrocrack; then the dipole can be evaluated using equivalent inclusion

method (Eshelby; 1957). In the local coordinates the only non-zero component

of the dipole is

Q33 = 8(1− ν2) a3 t03 .

Now we can exploit the closed-form solution provided by Kachanov

and Karapetian (1997) to compute the corrections to the undisturbed stress

intensity factors:

∆KI =(1− 2ν)(1 + ν)

2

a3

√l5

t03π2

(2 + 3

√2 tan−1(1/

√2) sin2 θ2

),

∆KII = 0 , (4.4)

∆KIII = −2(1− ν2)(2− 3ν)

2− ν

a3

√l5

t03π2

cos θ2 sin θ2 ,

at X0, 0, 0. For pure Mode I loading, we have ∆KII = ∆KIII = 0 and

∆KI =(1− 2ν)(1 + ν)

π2√

2π

(a

l

)3

K0I ≡ ∆K0

I .

The corrections presented in (4.4) exhibit exponential decay with a/l,

which must be a small number for the dipole approximation to be valid. Nev-

ertheless, they provide qualitative descriptions of the cracking behavior. The

normalized microcrack nucleation angles and corrections to stress intensity

factors are plotted in Figure 4.22 for different I/III mode mix. The horizontal

axis of the figure is plotted for K0I /K

0III ≤ 1 as well as K0

III/K0I ≤ 1 and ar-

ranged in the way such that it covers the entire range of I/III mode mix. The

Poisson’s ratio is 0.38, standing for PMMA. From the figure we notice that

94

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

∆KI / ∆K

I

0

No

rma

lize

d F

act

ors

KIII

0 / KI

0

1.0 0.8 0.6 0.4 0.2 0.0

θ2 / 45o

- ∆KIII / ∆K

I

0

KI

0 / KIII

0

Figure 4.22: Normalized microcrack nucleation angles and corrections to stressintensity factors at X0, 0, 0 for different I/III mode mix (ν = 0.38 forPMMA).

∆KI is positive for all θ2. This implies that the microcrack leads to amplifi-

cation of K0I of the macrocrack, which is in agreement with the experimental

observations.

4.6 Conclusion and Future Work

In this chapter we present experimental results and analysis for frac-

ture specimens under mixed-mode II/III, mixed-mode I/III and pure Mode III

95

loading conditions. The boundary element analysis was applied to investigate

actual loading modes over the crack front and to provide quantitative value

of stress intensity factors for further analysis. Fractographic features were

observed using an optical microscope. Motivated by the observation of mi-

crocracking structures, we present a dipole approximation for the microcrack-

macrocrack interaction to explain the drop of Mode I fracture toughness under

mixed-mode I/III loading. The analysis shows that for any I/III mode mix,

the presence of microcracks amplifies the Mode I stress intensity factor. We

summarize future work in the following.

• Specimens with Imbedded Cracks

Since PMMA is thermoplastic, a specimen with imbedded cracks can

be made by inserting teflon sheets of required shapes in between the

specimen halves then heating the assembly up in a hard-press machine.

The two halves of the specimen except for the area barred by teflon sheets

would heal completely after the process. The required temperature and

curing duration has been studied experimentally by Jud et al. (1981).

• Crack Profile Measurement

One of the advantages of PMMA as a fracture experiment material is

its transparency, which allows one to observe the fracture propagation

easily. This advantage can be further utilized with the introduction of

the computer stereo vision system that measures crack face topography

quantitatively in real time. This system usually requires two or more

96

cameras placed in parallel or verged prospects to capture the same re-

gion of an object from different viewpoints. Each captured image records

a specific projection of the three-dimensional profile onto the camera de-

tector plane. The elevation data is acquired through correlation and

calibration of captured images using stereo imaging equations that in-

volve intrinsic and extrinsic parameters of the system (e.g., Chao and

Sutton; 1993; McNeill et al.; 1997). The technique is scale independent,

which makes measuring a micron-scale surface profile possible with the

help of a high magnification zooming lens.

Another candidate for crack profile measurement is the confocal micro-

scope. This equipment uses laser to provide the excitation light and scans

across the profiling surface through a complex reflecting mirror system,

rejecting out-of-focus reflecting light which is inevitable in an ordinary

optical microscope under high magnification. By scanning thin sections

through the specimen, a three-dimensional image can be rendered and

the quantitative data can be measured accordingly.

• At this stage, we would like to advance a criterion for crack growth

initiation under general mixed-mode conditions. The premise of this

criterion is that Mode I is essential for crack growth initiation, whereas

the shear modes alone lead to nucleation of the microcracks rather than

advance of the main crack. For a small amount of KIII , a functional

97

form consistent with existing criteria for two-dimensional cracks is

KI F

(KII

KI

)= KIC G

(KIII

KIC

),


criterion and G has to be determined from experiments involving mixed

modes I and III. Considering the limited amount information supporting

this form, it should be regarded as a working hypothesis for the future

work. Nevertheless, considering the fact that there is little known about

growth initiation for mixed-mode three-dimensional cracks, this equation

is essential to state.

98

Chapter 5

Closing Remarks

In this chapter, we summarize principal accomplishments of the pre-

sented work and propose directions for future work.

5.1 Fast Boundary Element Method

We developed and implemented a fast boundary element method in

Chapter 2 that allows one to analyze problems involving many interacting

cracks. The key feature of the new method is a fast iterative solver that

solves a problem with N unknowns using O(N) computer memory and O(N)

arithmetic operations. As a result, on current computers, one can analyze

problems involving hundreds and even thousands of three-dimensional cracks.

In contrast, conventional boundary element methods are limited to problems

involving several cracks. Also, the new method is more general than various

approximate methods (Kachanov; 1994), which are typically limited to circular

cracks.

The proposed boundary element method can be improved along several

directions. Those include development of pre-conditioners to reduce the itera-

tion count, parallelization of the code, and extension of the code to problems

99

involving finite rather than infinite solids. Nevertheless, we believe that the

most important direction for future work should be tightly connected to ap-

plications. In particular, through Monte-Carlo-like studies, one can pose and

answer many questions related to elastic and fracture properties of damaged

solids, macrocrack-microcrack interactions, and others.

5.2 Asymptotic Analysis

The asymptotic analysis presented in Chapter 3 accomplishes two main

objectives. First, it completes the first-order asymptotic analysis of regularly

perturbed quasi-circular cracks initiated by Gao and Rice (1987). Second, it

provides us with a qualitative understanding of irregularly shaped cracks. In

particular, we observed that, under reasonable growth criteria, quasi-circular

cracks subjected to remote Mode I loading conditions tend to regain the cir-

cular shape.

Besides the conventional uses of asymptotic solutions, we believe that

they can be very beneficial in the development of pre-conditioners for iterative

solvers. In essence, pre-conditioners should contain as many features of the in-

verse as possible, and those derived from asymptotic solutions certainly satisfy

this condition. Also, based on asymptotic solutions, one may consider extend-

ing approximate methods for interacting circular cracks (Kachanov; 1994) to

the general three-dimensional cracks.

100

5.3 Mixed-Mode Cracking

The principal accomplishment of the study presented in Chapter 4 is

a set of experimental measurements and observations on three-dimensional

cracks under general mixed-mode loading conditions and the conjecture that

the Mode III loading component leads to nucleation of microcracks rather

than to initiation of growth of the main crack. As a result, we proposed the

following functional form for the fracture criterion for three-dimensional cracks

under general mixed-mode loading conditions (KIII is small):

KI F

(KII

KI

)= KIC G

(KIII

KIC

),


criterion and G has to be determined from experiments involving mixed modes

I and III.

The work presented in Chapter 4 is a preliminary attempt towards

understanding the behavior of three-dimensional cracks under general loading

conditions. The obvious questions to be answered are as follows. First, is

the proposed criterion valid in a broad range of loading conditions? Does the

criterion hold for other materials? What is the mechanism of crack growth:

does the main crack advances toward the microcracks or the microcracks move

“backward” toward coalescence with the main crack? At this stage, it appears

that we have generated more questions than answers, but the issues raised in

this work are absolutely critical for further advances of fracture mechanics.

101

Appendix

The functions Φ(α) and Ψ(α) involved in the closed-form solution for

the spherical cap crack by Martynenko and Ulitko (1978) are

Φ(α) =1

2

[β sin

α

2+ δ1 sin

3α

2+ δ2 sinh(γα) cos(να) + δ3 cosh(γα) sin(να)

],

Ψ(α) =1

6

[−4δ1 cos

3α

2+ ς1 sinh(γα) sin(να) + ς2 cosh(γα) cos(να)

],

where

β =3

2π(1 + ν), γ =

√3− 4ν2

2, ξ = 3 + 2ν ,

ς1 = δ2ξ + 2δ3γ , ς2 = 2δ2γ − δ3ξ .

The coefficients are

δ1 = η1Ω1

Υ,

δ2 = η2($1 + ρ1)Ω3 + (κ1 + ρ3)Ω4 + λ1Ω2

Υ,

δ3 = η2($2 + ρ2)Ω3 + (κ2 + ρ4)Ω4 − λ2Ω2

Υ,

where

η1 = − 1

2π(1 + ν), η2 =

(1− ν)(7− 8ν)

12π(1 + ν) cos(α/2),

Ω1 = 4γ [4(−2 + ν) + (−5 + 4ν) cos α] cosh(2γα) sinα

2

+ 2γ

[4(1 + ν) sin

(3− 4ν)α

2− (7− 8ν) sin

(1 + 4ν)α

2+ (5− 4ν) sin

(1− 4ν)α

2

]

+ 2[−15 + 4ν + 16ν2 + (3 + 4ν − 8ν2) cos α

]sinh(2γα) cos

α

2,

102

Ω2 = 32(2− cos α) cos4 α

2sin

α

2, Ω3 = − 72

21− 24ν(1 + cos α) cos

α

2sin α ,

Ω4 = − 1

1− ν

[(3 cos

α

2− cos

3α

2

)cos

3α

2+ 3

(11 cos

α

2− cos

5α

2

)cos

α

2

],

Υ = 4γ [4(1− ν) + (3− 4ν) cos α] cosh(2γα) sinα

2

+ 2γ

[(3− 4ν) sin

(3− 4ν)α

2+ sin

(5− 4ν)α

2+ 4(1− ν) sin

(1− 4ν)α

2

]

+ 2[7− 8ν + (1− 8ν + 8ν2) cos α

]sinh(2γα) cos

α

2,

λ1 = −2ν cos(να) cosh(γα) + 2γ sin(να) sinh(γα) ,

λ2 = 2ν sin(να) sinh(γα) + 2γ cos(να) cosh(γα) ,

$1 = 2[−(6 + ν − 8ν2) cos α cos(να)− 6(1− ν) sin α sin(να)

]cosh(γα) ,

$2 = 2[−(6 + ν − 8ν2) cos α sin(να) + 6(1− ν) sin α cos(να)

]sinh(γα) ,

ρ1 = 2γ [6 sin α cos(να) + (1− 8ν) cos α sin(να)] sinh(γα) ,

ρ2 = 2γ [6 sin α sin(να)− (1− 8ν) cos α cos(να)] cosh(γα) ,

ρ3 = −2γ[3 cos

α

2cos(να) + (1− 4ν) sin

α

2sin(να)

]sinh(γα) ,

ρ4 = −2γ[3 cos

α

2sin(να) + (1− 4ν) sin

α

2cos(να)

]cosh(γα) ,

κ1 =[−(3 + 2ν − 8ν2) sin

α

2cos(να) + 3(1− 2ν) cos

α

2sin(να)

]cosh(γα) ,

κ2 =[−(3 + 2ν − 8ν2) sin

α

2sin(να)− 3(1− 2ν) cos

α

2cos(να)

]sinh(γα) .

103

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112

Vita

Yi-Shao Lai was born in Taipei, Taiwan on 21 November 1970, the son

of Chen-Chieh Lai and Li-Yu Chao. He received Bachelor of Science and Mas-

ter of Science degrees in Civil Engineering in 1993 and 1995, respectively, from

National Central University in Taiwan. His master’s thesis was supervised by

Prof. Chung-Yue Wang. From 1995 to 1997 he worked as a short-term research

fellow for Center for Bridge Engineering Research in the same institution. In

1997 he started his Ph.D. study in the Engineering Mechanics program at

The University of Texas at Austin, under the supervision of Prof. Gregory J.

Rodin.

Permanent address: 4F No. 6 Ln 59, Kuokuang Rd.Yungho 234, Taipei, Taiwan

This dissertation was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth’s TEX Program.

113

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