Some elastic multi-crack and multi-punch problemssummit.sfu.ca/system/files/iritems1/5039/b1485286x.pdf · Three-dimensional crack problems have received a great deal of attention

SOME ELASTIC MULTI-CRACK AND MULTI-PUNCH PROBLEMS

Qiang Lan

M.Sc., Simon Eraser University, Canada, 1991

M.Sc., Eudan University, Shanghai, China, 1987

A THESIS SUBMITTED IN PARTIAL FULFILLMENT

O F THE REQUIREMENTS FOR T H E DEGREE O F

DOCTOR OF PHILOSOPHY

in the Department of Mathematics & Statistics

@ Qiang Lan 1994

SIMON FRASER UNIVERSITY

June 1994

All rights reserved. This work may not be

reproduced in whole or in part, by photocopy

or other means, without the permission of the author.

APPROVAL

Name: Qiang Ilan

Dcgrce: Doctor of I'l~ilosopl~y

Title of thesis: Sornc Elas tic Multi-Crack a.rld Multi-Purlcli I'rohlclns

Examining Committee:

Chairman: Dr. S.K. 'I'honzcnson

Date Approved:

--

Dr. G.A.C. Gmhum

Dr. A. P.S. Selvudulni

I3xtcrna.l Esamincr Professor and Chair, Dcpartmcnt of Civil Er~ginccring and Applied M d i ~ n i c s McGill University, Montrcal

June 13, 1994

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T i t l e o f Thes is /Pro ject /Extended Essay

Author:

(s igna tu re )

(name)

(date)

Abstract

In this thesis, two types of problems involving a homogeneous isotropic linear elastic medium

are studied. The first problem is that of an elastic body, which could be an infinite solid or a

semi-infinite solid or a layer, containing two or more pardel penny-shaped cracks whose upper

and lower surfaces are loaded by equal and opposite arbitrary tractions. The second problem

is concerned with two or more circular punches, whose faces are of arbitrary shape, indenting

the surface of an elastic layer which rests on a rigid foundation. Both normal indentation and

tangential indentation are examined.

The method used in this thesis is based on a formulation, which is very similar to that given

by Muki[24], for general three-dimensional asymmetric elasticity problems and the superposition

principle of linear elasticity. For example, the solution to the problem of an elastic layer con-

taining two penny-shaped cracks under arbitrary loadings can be considered as a superposition

of the solutions to two layer problems, each containing one penny-shaped crack. Furthermore,

the solution to the problem of a layer weakened by one penny-shaped crack can be decomposed

into the sum of solutions to two basic problems, namely problem of an infinite solid containing

one crack and problem of a layer without any crack. With the aid of the solutions to the basic

problems and the addition formula for Bessel functions, we can superpose solutions in different

cylindrical co-ordinates and then reduce the original problem to a system of Redholm integral

equations of the second kind, which can be solved by iteration for some special cases.

All the solutions given in this thesis can be extended to the case where the elastic solid

is transversely isotropic. More important, the method used in this thesis applies to any other

problem if the problem can be decomposed into several basic problems and analytic solutions to

the basic problems in terms of some potential functions are available.

Acknowledgements

This work was initiated and completed under the supervision of Professor Cecil Graham. The

author wishes to express his deepest gratitude to Dr. Graham for his guidance, support and

encouragement.

The author would also like to thank Mrs. S. Holmes and Mrs. M. Fankboner for their help,

and Simon Fraser University for financial support during part of the writing of this thesis.

Dedication

To my wife and our parents

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements iv

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dedication v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction 1

2 Some Three-Dimensional Elastic Problems . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Half-Space Problems 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Crack problem 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Contact Problem 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Layer Problem 23 . . . . . . . . . . . . . . . 3 Interaction of Penny-Shaped Cracks in an Elastic Solid 27

3.1 Penny-shaped cracks in an infinite solid . . . . . . . . . . . . . . . . . . . 27 3.1.1 Derivation of the integral Equations . . . . . . . . . . . . . . . . 29 3.1.2 Some Asymptotic Solutions . . . . . . . . . . . . . . . . . . . . . 33

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Discussion 46 . . . . . . . . . . . . . . . . 3.2 Penny-Shaped Cracks in a Semi-Infinite Solid 48

3.2.1 Derivation of integral equations . . . . . . . . . . . . . . . . . . 48 3.2.2 Some Asymptotic Solutions . . . . . . . . . . . . . . . . . . . . . 55

. . . . . . . . . . . . . . . . . . . . . . . 3.3 Penny-Shaped Cracks in a Layer 69 . . . . . . . . . . . . . . . . 3.3.1 Derivation of the integral equations 70

3.3.2 Some asymptotic solutions . . . . . . . . . . . . . . . . . . . . . 75 . . . . . . . . . . . . . . . . . 4 Interaction of Circular Punches on an Elastic Layer 84

. . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Normal Indentation Problem 85 . . . . . . . . . . . . . . . . 4.1.1 Derivation of the integral equations 86

. . . . . . . . . . . . . . . . . . . . . 4.1.2 Some asymptotic solutions 87 . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Tangential indentation Problem 90

. . . . . . . . . . . . . . . . 4.2.1 Derivation of the integral equations 91 . . . . . . . . . . . . . . . . . . . . . 4.2.2 Some asymptotic solutions 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Kernels 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography 109

vii

List of Figures

Two parallel penny-shaped cracks in an infinite solid. . . . . . . . . . . . . . . . . Equal uniform normal loadings: variation of stress intensity factor 2 with angle B for c j = 0.2 and various f when a = ii

3.12 Two parallel penny-shaped cracks in a semi-infinite solid. . . . . . . . . . . . . . 3.13 Two co-axial cracks with equal uniform normal loadings: variation of the stress

- intensity factor & and & with respect to f for various €h. . . . . . . . . . . .

3.14 Two co-axial cracks with equal uniform normal loadings: variation of the stress -

intensity factor 2 and 2 with respect to 4 for various s. . . . . . . . . . . 3.15 Two co-axial cracks with equal uniform normal loadings: variation of the stress

- intensity factor & and 2 with respect to ch for various f . . . . . . . . . . .

3.16 Two co-axial cracks with equal uniform shear loadings: variation of the stress -

intensity factor & and $ with respect to f for various Q. . . . . . . . . . . . 3.17 Two co-axial cracks with equal uniform shear loadings: variation of the stress

- intensity factor 2 and $ with respect to 4 for various 4. . . . . . . . . . . .

3.18 Two co-planar cracks with equal uniform normal loadings: variation of the stress

intensity factor & with respect to : for various y. . . . . . . . . . . . . . 3.19 Two co-planar cracks with equal uniform normal loadings: variation of the stress

intensity factor & with respect to f; for various q. . . . . . . . . . . . . . . . . 3.20 Two co-planar cracks with equal uniform shear loadings: variation of the stress

intensity factor 2 with respect to : for various y. . . . . . . . . . . . . . . . . 3.21 Two co-planar cracks with equal uniform shear loadings: variation of the stress

intensity factor & with respect to f; for various U. . . . . . . . . . . . . . . . . 3.22 Two co-planar penny-shaped cracks in the mid-plane of an elastic layer. . . . . . 3.23 Normal loading: variation of the stress intensity factor &- with respect to 3 at

8 = 0 for various ~f when the boundaries are stress free. . . . . . . . . . . . . . . 3.24 Normal loading: variation of the stress intensity factor 2 with respect to f; at

8 = 0 for various ~h when the boundaries are stress free. . . . . . . . . . . . . . . 3.25 Normal loading: variation of the stress intensity factor 2 with respect to €1 at

8 = 0 for various 9 when the boundaries are stress free. . . . . . . . . . . . . . . 3.26 Shear loading: variation of the stress intensity factor $ with respect t o 9 at

8 = 0 for various ~f when the boundaries are stress free. . . . . . . . . . . . . . .

3.27 Shear loading: variation of the stress intensity factor 2 with respect to i at 8 = 0 for various rh when the boundaries are stress free. . . . . . . . . . . . . . . 81

3.28 Shear loading: variation of the stress intensity factor $ with respect to s j at 8 = 0 for various when the boundaries are stress free. . . . . . . . . . . . . . . 82

3.29 Normal loading: variation of the stress intensity factor 3 with respect to $ at 8 = 0 for various sf when the layer has fixed boundaries. . . . . . . . . . . . . . . 82

3.30 Normal loading: variation of the stress intensity factor & with respect to i at 8 = 0 for various rh when the layer has fixed boundaries. . . . . . . . . . . . . . . Normal loading: variation of the stress intensity factor & with respect to s j at 8 = 0 for various 4 when the layer has fixed boundaries. . . . . . . . . . . . . . . Variation of the nondimensional resultant force P, with respect t o $ for various s f . Variation of the nondimensional resultant moment M, with respect to 4 for various r j in the case where two punches undergo equal rotational displacements (in

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . same direction). Variation of the nondimensional resultant moment M, with respect to 4 for various sf in the case where two punches undergo equal rotational displacements (in

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . opposite directions). Variation of the nondimensional resultant force P, with respect to 9 for various tzj in the case where two punches undergo same x-direction displacemets. . . . . Variation of the nondimensional resultant force P, with respect to $ for various r j in the case where two punches undergo equal x-direction displacemets (in opposite

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . directions). Variation of the nondimensional resultant force P, with respect to $ for various €1 in the case where two punches undergo same y-direction displacemets. . . . . Variation of the nondimensional resultant force P, with respect to 3 for various r j in the case where two punches undergo equal y-direction displacemets (in opposite

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . directions).

Chapter 1

Introduction

In this thesis, two types of problems involving a homogeneous isotropic linear elastic medium

are studied. The first problem is that of an elastic body, which could be an infinite solid or a

semi-infinite solid or a layer, containing two or more parallel penny-shaped cracks whose upper

and lower surfaces are loaded by equal and opposite arbitrary tractions. We shall find the stress

intensity factors at the edge of each crack. The second problem deals with two or more circular

punches, whose faces are of arbitrary shape, indenting the surface of an elastic layer which rests

on a rigid foundation. We shall find how the existence of the other punches and the thickness of

the layer affect the total force applied on the punch if the indentations (normal or tangential)

are given.

Three-dimensional crack problems have received a great deal of attention in the literature.

A review with a comprehensive list of reference can be found in Panasyuk et al [25]. In most

of these works, there is only one crack involved. Problems of an elastic solid containing more

than one crack have received much less attention. Collins[4] considered the problem of an

infinite elastic solid containing two coplanar cracks which are subjected to normal loading only.

By equivalenting the problem t o a Neumann problem in potential theory, he showed that this

problem is governed by an infinite set of Fredholrn integral equations, which can be solved

approximately by iteration when the spacing between the circular regions is sufficiently large

compared with their radii. By using Westmann's[34] technique for solving simultaneous pair

CHAPTER 1 . INTRODUCTION 2

of dual integral equations, Fu and Keer[ll] extended Collins' solutions to the case in which

arbitrary shear loadings are applied on the crack faces. As pointed out by Fabrikant[8], some of

the formulas and results in [ l l ] are incorrect. Based on his earlier results in potential theory,

Fabrikant [7,8] studied these two loading cases by a different method. Instead of ending up with

an infinite set of integral equations, he reduced the problem to a finite set of integral equations

with the number of equations equal to the number of cracks. His method enables us to obtain

practically exact numerical solutions to the problem, since the integral equations involved are

non-singular and the iteration procedure is rapidly convergent. Recently, Xiao, Lin and Liew[35]

presented solutions to the problem of two coplanar penny-shaped cracks under uniaxial tension

by using the superposition principle of elasticity theory and Eshelby's [5] equivalent inclusion

method. All these methods fail to deal with off-set cracks, which will be considered in this

thesis. The only paper we find in the literature dealing with arbitrarily located penny-shaped

cracks was by Kachanov and Laures[20]. The problems are dramatically simplified in [20] by

the assumption that the non-uniformities of the traction(with zero average) on a crack have no

impact on the other cracks. This assumption can be considered as an instance of Saint-Venant's

principle and allows the authors to find only approximate solutions even for weak interactions.

The first solution to the problem involving two or more punches was given by Collins[4]

for the case where the frictionless punches are normally indenting an elastic half space. It is

shown that this contact problem is equivalent to a Dirichelet problem in potential theory and

the Dirichelet problem is governed by an infinite set of integral equations, which can be solved

approximately by iteration when the punches are far away. By using Galin's[l2] expression for

pressure under the punch caused by a concentrated load at another point of the half-space,

Gladwell and Fabrikant[l4] derived very simple approximate relationships among the forces,

moments, and indentations for a system of punches on the elastic half-space. Later Fabrikant[S]

extended this to elliptic punches. A similar problem (tangential indentation problem), in which

uniform tangential displacements instead of normal displacement are prescribed on several elliptic

domains, was solved by Fabrikant[G] using a method based on his previous result in potential

theory. By employing the mean value theorem, he related the resulting tangential forces acting

CHAPTER 1 . INTRODUCTION 3

on each domain to the given displacements through a system of linear algebraic equations.

Again Collins' and Fabrikant's methods fail when the half-space is an elastic layer, which will

be considered in this thesis.

The method used in this thesis is based on a formulation, which is very similar to that given

by Muki[24], for general three-dimensional asymmetric elasticity problems and the superposition

principle of linear elasticity. For example, the solution to the problem of an elastic layer con-

taining two penny-shaped cracks under arbitrary loadings can be considered as a superposition

of the solutions to two layer problems, each containing one penny-shaped crack. Furthermore,

the solution to the problem of a layer weakened by one penny-shaped crack can be decomposed

into the sum of solutions to two basic problems, which are problem of an infinite solid containing

one crack and problem of a layer without any crack. With the aid of the solutions to the basic

problems and the addition formula for Bessel functions, we can superpose solutions in different

cylindrical co-ordinates and then reduce the original problem to a system of Fredholm integral

equations of the second kind, which can be solved by iteration for some special cases.

All the solutions given in this thesis can be extended to the case where the elastic solid

is transversely isotropic. More important, the method used in this thesis applies to any other

problem if the problem can be decomposed into several basic problems and analytic solutions

to the basic problems in terms of some potential functions are available. We believe that this is

the first attempt to apply this method in three-dimensional crack and contact problems, even

though the very same idea has been used by Chen[2] and many others in two-dimensional fracture

mechanics.

This thesis contains four chapters. In chapter 2 a general formulation for three-dimensional

linear elasticity is presented. Unlike Muki's formulation, three harmonic functions instead of

one harmonic function and one biharmonic function are employed. The advantage of using three

harmonic functions is that the expressions of these functions in terms of the other cylindrical co-

ordinates can be obtained in a fairly easy way. Several basic problems involving three-dimensional

elastic solids are then solved by using this general formulation. These basic solutions will be

used later in chapter 3 and chapter 4.

CHAPTER 1. INTRODUCTION 4

In chapter 3 interaction of penny-shaped cracks in an elastic solid is examined. The elastic

solid involved can be an infinite elastic solid, or a semi-infinite elastic solid, or an elastic layer.

By using the basic solutions given in chapter 2 and the superposition principle, we reduce the

problem to an infinite set of Fredholm integral equations, which are then solved for some special

cases. Solutions of the problem for an infinite solid containing two parallel penny-shaped cracks

are presented in section 3.1, solutions for the semi-infinite solid case are given in section 3.2

while section 3.3 deals with the layer case. For all three cases, asymptotic solutions for stress

intensity factors in terms of some small parameters are obtained. Comparisons are made with

previous results whenever possible.

Chapter 4 is concerned with the interaction of circular punches on an elastic layer. Both nor-

mal indentation (normal displacement in the contact areas is prescribed) problem and tangential

indentation ( tangential displacements are given in the contact areas) problem are considered.

Again it is shown that this indentation problem is governed by an infinite set of Fredholm in-

tegral equations, which can be solved by iteration for some special cases. Asymptotic solutions

for resultant forces and moments are also presented in terms of some small parameters.

Chapter 2

Some Three-Dimensional Elastic

Problems

In this chapter a general formulation for three-dimensional linear elasticity is presented and then

solutions to several three-dimensional elastic problems are given. These basic solutions will be

used later in chapter 3 when we investigate interaction between penny-shaped cracks in an elastic

solid and in chapter 4 when interaction between circular punches on an elastic layer is examined.

General asymmetric three-dimensional linear elastic problems for a homogeneous, isotropic

half-space or elastic layer were first considered systematically and solved by Muki [24] using

Hankel transforms. Muki's solution like the symmetrical one, by Harding and Sneddon[l8], of

which it is a generalization, is based on the expression of the displacement field in terms of a

harmonic function and a biharmonic function. The simple partial differential equations for these

two functions are solved by the use of Hankel transforms. Muki's method applies to any three-

dimensional elastic problem involving an infinite solid, a semi-infinite solid or a layer. By using

Muki's formulation, the problem can be reduced to the determination of six functions, which are

to be found by the given boundary conditions of the problem. As pointed out by Sneddon[31]

that due to some reasons Muki's work does not seem to have attracted the attention it deserves.

In our formulation, the elastic field of the problem is expressed in terms of three harmonic

CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS 6

functions instead of one harmonic function and one biharmonic function. Otherwise the formu-

lation is very much like that given by Muki. The advantage of using three harmonic functions

becomes clear later when expressions of these functions in terms of the other cylindrical co-

ordinates are needed.

Problems in the theory of classical elasticity are governed by Navier equations. If we em-

ploy cylindrical co-ordinates (r, 8, a), the Navier equations in terms of displacements u, (r, 8, z ) ,

uB(r, 8, z), u,(T, 8, z) in the r, 8, z directions are

where v is Poisson's ratio, e stands for the dilatation

and V2, the Laplacian operator, takes the form

in the cylindrical co-ordinates. It may be verified[21] by direct substitution that

is a special solution for the Navier equations (2.1)-(2.3), if 4(r, 8, z), +(r, 0, z) and ~ ( r , 8, z)

are harmonic functions. Stress field {a,(r, 8, z), U@(T, 8, z), az(r, 8, a), T,,(T, 8, z), rgz(r, 8, z),

rre(r, 8, z) ) corresponding to the above displacement field can then be found. Among these

stresses, the ones of interest are

CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS

where p is the shear modulus of the elastic solid. We may write the harmonic functions in the

Fourier series of 8 as follows

00

s( r , 8, Z) = C [ d n ( r , z) cos n8 - &(r, Z) sin no], n=O

00

~ ( r , 8, Z) = C [xn(r7 z) cos n8 - xn(r, z) sin no], n=O 00

$(T, 8, Z) = x [$n(r, z) sin n8 + ( r , Z) cos no]. n=O

Here &(T, z) is the Fourier cosine coefficient of c$(r, 8, z) and so on. Without loss of generality, - -

we assume &(T, z) = $ n ( ~ , z) = j&(r, z) = 0. The formulation hereafter remain valid if we write

cos(n8 + $K) = - sin(n8) and sin(n8 + $a) = cos n8 instead of cos no and sin no. Therefore the solutions corresponding to &, 4, and may be obtained by making the following replacements

cos n8 - sin no; sin n8 a cos no,

Substitution of (2.10)-(2.12) into the displacement field (2.4)-(2.6) gives the displacements in

00

ur (r, 8 , ~ ) = C U: (r, Z) C O S ( ~ ~ ) , n=O

where the Fourier coefficients ur(r, z), u;(r, Z) and u ~ ( T , Z) of the displacement components are

given by


Here, of course, we have used the assumption that &(r, z ) = '$n(T, Z ) = f n ( r , Z ) = 0. In terms

of & ( T , z ) , $ n ( ~ , Z ) and xn(r , z ) , the corresponding stresses of interest can be written as

by using equations (2.7)-(2.9). Here the Fourier coefficients r:(r, z ) , T&(T, Z ) and o;(r, Z ) of the

stress components are given by

The requirements of $(r, 8, z ) to be harmonic function gives us the following partial differential

equation for c$,(T, z )

which can be solved by Hankel transforms. The kth order Hankel transform of a function f ( r )

is defined to be

P ( s ) = im 7 - f ( r ) J+)dr, where Jk(rs ) is the kth order Bessel function of the first kind. Taking the nth order Hankel

transform of the above partial differential equation for &(r, z ) , we obtain

which is an ordinary differential equation with parameter s and the general solution for this

equation is

K n ( s , I) = An(s)e-" + &(s)esz.


Similarly we obtain the general forms for K n ( s , Z ) and z n ( s 7 z ) ,

Here the arbitrary functions An(s) , A , ( s ) , Bn(s ) , B n ( s ) , Cn ( s ) and C n ( s ) are to be determined

from the given boundary conditions of the problem. Once these six functions have been found, -n

G n ( s , z ) , $Jn ( s , z ) and Z n ( s , z ) are known functions of z and s and expressions of &(T,z ) ,

& ( T , Z ) and xn(r7 Z ) can be obtained by Hankel inverse transforms,

and so on. Substituting this into the displacement equations (2.16)-(2.18), we can rewrite the

displacements in terms of Z n ( s , z) , G n ( s , z ) and Z n ( s , z ) ,

00 azn -n azn up + US = - 1 [ ( I - 2v)+n + Z- - 2JIn + 2(1- v)zn + z - ] ~ ~ ~ ~ + ~ ( r s ) d s , a2 8~

Here we have used the following properties of Bessel functions

d n - [ J ~ ( s x ) ] - - J ~ ( s x ) = - S J ~ + ~ ( S X ) , dx x d n

- [ J ~ ( s x ) ] + - J ~ ( s x ) = s J ~ - ~ ( s x ) . dx x

Similarly we find that the stresses of interest take the form

agn ax';;" a2zn re", + rz = -2rl~[zg - - 8% +- dz + zT]s2 Jn+l(rs)ds, a2

-', = -2pJ00[ 2-+- a2En 8 K n +- ax';;" + z - ] s ~ J ~ - ~ ( T s ) ~ s , a2Zn (, a 2 2 az az a z 2

= 1 a2Ln a3Zn assn 1- a,a + z- a23 + z- 8 , IsJn (rs)ds,

from equations (2.19)-(2.21). In terms of An(s), Bn(s ) , Cn(s ) , &(s ) , &(s) and C n ( s ) , the

displacement field becomes


+ lw [ ( I - 2u + S I ) & ( s ) + (2 - 2u + sr)& ( s ) + 2~ (s) ]s2 Jn- 1 (rs)e"dr, (2.26) 00

U: ( r , I ) = 1 [(2 - 2~ + sr)An ( s ) + ( 1 - 2~ + S Z ) B ~ ( s ) ]s2 Jn (rs)e-.'ds

and the stress field takes the form

It is essential for the method used in this thesis to find the displacement and stress expressions

in terms of another cylindrical co-ordinate system ( f , 8 , z ) . Let the origin 0 of co-ordinates

( T , 8, Z ) be point ( f , T , 0 ) in co-ordinate system ( f , 8, z ) and origin 0 of ( f , 8, z ) be point ( f , 0,O) in co-ordinates ( T , 0, z), where f is the distance between 0 and d. Let {u i ( f , 8, z ) , uB(f ,8 , z ) ,

placement and stress fields in terms of the new co-ordinates and { (u:(f , z ) , u$ ( f , z ) , ug( f , 2 ) ) )

and { @ ( f , I ) , o ; ( f , z ) , u:(f, z ) , T$(T, z ) , rFZ(F, z ) , rFB(f, z ) ) be the corresponding Fourier coef-

ficients. With the aid of the addition formula for Bessel functions as given in Watson[33],

we can express the three harmonic functions in terms of co-ordinates ( f , 8, z ) as follows:


with

Here the prime on the summation sign implies that the (-l)nJ,,+n(ds) terms do not appear

when n = 0. This shows that the potential functions take the same form in the two systems

of cylindrical co-ordinates ( T , 8 , z ) and ( F , 8 , z ) . Therefore it is clear that the displacement and stress components in terms of co-ordinates ( F , 8, z ) also take the same form as those in terms of

( ~ 7 '37 z ) as 10% as we replace An(s), Bn(s), Cn(s), An(s)7 Bn(s), Cn( s ) by A:(s)7 B:(s)7 C;t(s)7

En($), g n ( s ) , c n ( s ) and replace T by F . This argument will be used repeatly in later chapters.

2.1 Half-Space Problems

Problem of an elastic half-space was first considered by Boussinesq[l] and due to that reason the

half-space problem is sometimes referred to as Boussinesq problem. If the half-space is taken to

be z 2 0 and the boundary conditions are given on the plane z = 0, then all the displacement

and stress components approach zero as z + oo. Therefore An(s), &(s) and c n ( s ) have to be


zero and the potential functions $(T, 8, z), $(r, 8, z) and X ( T , 0, z) can be written as

The corresponding displacement and stress fields in the half-space can then be obtained

03

u:(r, Z ) - U; ( r , Z ) = 1 [ ( I - 2. - SZ)A. ( s ) + (2 - 2v - sz)Bn ( s ) + 2G'.(s)]s2 ~ ~ - ~ ( ~ s ) e - ' " d s , ( 2 . 4 4 )

and

roo

Now we have reduced the half-space problem to the determination of three functions A,(s),

B,(s) and C,(s), which will be found by the given boundary conditions of the problem. In this

section, we consider two different boundary conditions.

Half-space problem 1 (Problem HI)

Problem H1 is that of an elastic half-space with tractions applied on the surface of the

half-space. Boundary conditions of this problem can be written as

where F,(r), G,(r) and H,(r) are some given functions. Imposing these boundary conditions

on the stresses (2.46)-(2.48) and using the definitions of Hankel transform and its inverse, we


obtain three linear equations for An(s) , Bn ( s ) and Cn(s) . Solutions for these equations are

This enables us to obtain, from (2.46)-(2.48), the stresses in the half-space W n n -n+l n n - l

( T , Z ) = sJn (TS)e-" [(2 + 281) Fn ( s ) - szGn ( s ) - s1Hn ( s ) ] ~ s ,

which can be rewritten in a little shorter form

1 n n n n + l n n - 1 *"(s, Z ) = -e-" [(2 + 2sz)Fn ( s ) - S Z G ~ ( s ) - szHn ( s ) ] , (2.52)

2 n n + l n n + l 1 n+l n-1 T ( s , ) + r Z ( s , z ) = - ~ - " [ ~ s z ~ ~ ( s ) + ( ~ - s z ) G ~ ( s ) - s r H n ( s ) ] , (2.53)

2 n n - 1 n n - l 1 n n n n + l -n-1 re", ( s , z ) - T & ( s , z ) = - ~ - ~ ~ [ ~ s z F ~ ( s ) - s z G ~ ( ~ ) + ( 2 - S Z ) H ~ (s)]. (2.54)

2

These solutions will be used later in section 3.2 when we study interaction of penny-shaped

cracks in a semi-infinite elastic solid with stress free surface.

Half-space problem 2 (Problem H2)

Problem H2 is that of an elastic half-space with displacements given on the surface of the

half-space. Boundary conditions of this problem can be written as

where Un(r) , Vn ( r ) and W n ( r ) are some given functions. Similarly we can express An(s) , B,(s)

and Cn(s ) in terms of these functions and then obtain the stresses of interest in the half-space

-n n n + l u: ( s , z ) = -- ps e- lZ[(1 - 2v - sz )un ( s ) 3 - 4v

-n -n-l +(4 - 4 v + 2sz)Wn (s) - ( 1 - 2v - sz)Vn ( s ) ] ,

n n + l -n+l -n " e-Sz[(2sz + 2 - 4 v ) w n ( s ) T& (s,z)+r,", ( s , z ) = - - 3 - 4v


2.2 Crack problem

Now let us consider a problem, which will be referred to as problem C later on, of an infinite

elastic solid containing a penny-shaped crack C whose upper and lower surfaces are loaded by

equal and opposite arbitrary tractions. We choose a cylindrical co-ordinates (r, 8, z) such that

this crack is taken to occupy

Z = O , r < a , o < o < ~ T .

In order to have a stress field which is consistent with the symmetry of the potentials (2.40)-

(2.42), we assume that the arbitrary tractions on the crack face take the form CO

g.z ( r7 47 0) = C an ( r ) cos(n6) 7 (2.55) n=O

00

~ r r ( ~ 7 87 0) = C bn(r) cos(n8) (2.56) n=O 03

roz(r7 8,O) = C cn(r) sin(n8). (2.57) n=O

This crack problem has been studied by many researchers. Among them are Sneddon [30] for

the axisymmetric loading, Kassir and Sih[2l] and Guidera and Lardner[lS] for general loadings.

Kassir and Sih7s solutions are quoted here. Without loss in generality the arbitrary loadings

(2.55)-(2.57) applied on the crack surfaces may be resolved into symmetrical (corresponding to

normal loading) and skew-symmetrical (corresponding to shear loadings) parts with respect to

the crack plane. For clarity, these two parts are treated separately.

Normal Loadings If the crack faces are subject to normal loading only, then the crack problem

is symmetric with respect to the crack plane z = 0 and we need only to find the displacement

and stress distributions in the upper half-space z 2 0. Therefore formulas (2.43)-(2.48) for the


half-space problem can be used to solve this crack problem. The boundary conditions of the

problem can be written as

The first boundary condition gives us Bn( s ) = Cn(s ) = 0 , and the remaining boundary conditions

lead us to the following dual integral equations for An(s ) ,

If we assume that An( s ) is given by the following integral

where a l ( t ) is a function to be determined on interval 0 < t 5 a, then it is shown that integral equation (2.59) is automatically satisfied with the aid of identity

Substituting (2.60) into equation (2.58), we end up with an integral equation for @l(t)

after changing the order of integration. Solution of this integral equation is

Here we have used the following two formulas for the Bessel functions


Solution for a l ( t ) and the assumption (2.60) enable us to obtain

Therefore we can find the three harmonic functions and determine the displacement and stress

fields in the upper half-space of this problem by equations (2.43)-(2.48).

Shear loadings If the crack faces are subjected to shear loadings only, then the crack problem

is skew-symmetric with respect to the crack plane z = 0 . Again we only need to find the stress

and displacement field in the upper half-space z > 0 . In this case equations (2.43)-(2.48) are still valid. The boundary conditions of this problem can be written as

The first boundary condition gives us An(s ) = 0 and the remaining boundary conditions lead us

to Co(s) = 0 and the following dual integral equations for Bo(s )

when n = 0 and simultaneous pairs of dual integral equations for B n ( s ) and C n ( s )

when n > 1. Dual integral equations (2.64)-(2.65) can be solved by using the same technique as for the normal loading. It is seen, by comparing these equations with equations (2.58) and


(2.59), that Bo(s) is the same as Al (s) with al(x) replaced by -bl(x), and therefore

Solutions for the simultaneous pairs of dual integral equations (2.66)-(2.69) can be obtained by

using Westmann7s[34] technique. They are

with

Bn(s) and Cn(s) together with An(s) = 0 enable us to find the potential functions and then the

displacement and stress field in the upper half-space.

Superposing the solutions for the normal loading and shear loadings results the solutions

for the crack problem (Problem C ) under arbitrary loading (2.55)-(2.57). The displacements

and stresses of interest for the upper half-space(z 2 0) are given by (2.43)-(2.48) with An(s)

given by (2.63), Co(s) = 0, Bo(s) given by (2.70), and Bn(s) and Cn(s) given by (2.71) and

(2.72) respectively. By using the symmetry of the normal loading crack problem and the skew-

symmetry of the crack problem under shear loading, the displacements and stresses of interest

in the lower half-space ( z < 0) can also be found. In terms of the very same An(s), Bn(s) and Cn(s) (rather than An(s), &(s) and Cn(s)),

the elastic field in the lower half-space takes the following form

u:(r, Z ) + u ~ ( T , Z ) = [ ( 2 ~ - 1 - S Z ) A ~ ( S ) + ( 2 - 2~ + S Z ) Bn ( s ) - 2Cn(s)] Jn+l (rs)eSzds, (2.75) 1" u:(r, z ) - u;f(r, z ) = [ ( I - 2v + sz)An ( s ) + (2v - 2 - sz) Bn ( s ) - 2Cn(s)] J n - l ( ~ ~ ) e " d ~ s , (2.76) 1"


This solution together with the elastic field (2.43)-(2.48) in the upper half-space will be used

repeatly in chapter 3 when we examine the multi-crack problems. In deriving the integral

equations for the multi-crack problems, expressions for the elastic field in terms of another

cylindrical co-ordinates ( f , 8, z) are also required. This task can be accomplished by using the

arguments right before section 2.1.

In linear elasticity fracture mechanics, stresses have singularities of the form ( r - a)-'I2 along

edge of the crack. Stress singularity factors are the most important parameters and concepts in

linear fracture mechanics. If the stress intensity factors are defined by

kl = lim \/2(r - a) cz(r, 8, O), r+a

k2 = lim 4 2 ( r - a) r,,(r, 8, O), ?-+a

k3 = lirn 4 2 ( r - a) ss,(r, 8, O), r+a

then in terms of the given loadings applied on the crack faces they are

00

2~ 2 112 k3 = - -(-) [(1 - u)@2(a) f %(a)] sin(n8). a n=1

It is clear that the intensity factor kl of stress u,(r, 8, z) is dependent on an(x) only and that k2

and k3 have nothing to do with a,(x) due to the geometric symmetry of the problem.

2.3 Contact Problem

Contact problems or indentation problems are related to the name of Hertz, who first in 1882

successfully treated a static contact problem. This section is concerned with two indentation


problems, namely normal indentation problem and tangential indentation problem for an elastic

half-space. The normal indentation problem can be described as an rigid punch of arbitrary

shape indenting a lubricated elastic half-space z 2 0. The shear stresses on the contact plane

z = 0 are zero and the normal displacement in the contact region is prescribed. The tangential

indentation problem can be interpreted either as that of two elastic half-space interconnected

by the contact region and subjected to remote shear loadings, or as a contact problem with a

flexible punch connected to the half-space such that there is no normal stress on the contact

surface and the tangential displacements in the contact region are given.

Normal indentation problem (Problem 11)

Problem I1 is that of a half-space z > 0 indented by a circular cylinder punch with arbitrary preface. If the radius of the punch is a and the shape of its preface can be expressed by a function

with Fourier cosine coefficients fn(r), then boundary conditions for this problem can be written

as

This contact problem has been studied by many researchers, for example, Sneddon[BO] for the

axisymmetric case and Muki[24] for the asymmetric case. It can also be treated by formulas

(2.43)-(2.48). The shear stress boundary conditions give us Bn(s) = Cn(s) = 0 and the remaining

boundary conditions lead us to a dual integral equation for An(s)

If we assume that

Pl (t) Jn-1/2(~t)dt,

then integral equation (2.84) can be rewritten as


after using the following property for Bessel function

Now we can see, from identity (2.61), that integral equation (2 .84) is automatically satisfied and

equation (2.85) becomes to an Abel's integral equation for !PJ( t )

after changing the order of integration. Solution for this Abel's integral equation is

which enables us to find A n ( s ) from (2 .86) and then the displacement and stress fields from

equations (2.43)-(2.48).

Tangential indentation problem (Problem 12)

Problem I2 is that of a half-space z 1 0 indented by a flexible circular punch so that the

normal stress is zero on the whole contact surface z = 0 and the tangential displacements are

given in the contact region. If the radius of the punch is a and the tangential displacements can

be expanded as Fourier series, then boundary conditions for this problem can be written as

where gn(r) and h n ( r ) are given functions determined by the Fourier coefficients of the prescribed

displacements. Unlike the bonded punch problem considered by Gladwell[l3], the tangential

indentation problem defined here is very similar to that studied by Fabrikant[lO]. This problem

can be reduced to a system of a pair of integral equations by using formulas (2.43)-(2.48) for the

half-space problem. The normal stress free boundary condition gives us A n ( s ) = 0 and aJl the

other boundary conditions lead us to Co(s) = 0, a dual integral equations for Bo(s )


when n = 0 and a pair of simultaneous integral equations for Bn(s) and Cn(s)

when n 2 1. By comparing the dual integral equation for Bo(s) with the one for An(s) in

the normal indentation case, we find that Bo(s) is the same as Al(s) with fl(r) replaced by

-go(r). As for the simultaneous pair dual integral equations for Bn(s) and C,,,(s), again we use

Westmann's[34] method. If we assume that

then we can show that equation (2.90) is automatically satisfied with the aid of identity (2.61).

Replacing Jn-3/2(~t) and Jn+llz(st) in equation (2.91) by

and applying indentity (2.61) again, it is shown that integral equation (2.91) is also satisfied.

Substituting equations (2.92) and (2.93) into (2.89) and changing the order of integration lead

to a Abel's integral equation for Q2(t) with solution

Now it remains to satisfy equation (2.88), which can be rewritten as

la Q(t)dt J m Jn+l (rs) Jn-1/2(st)ds = gn(r> 0 2


by using the same technique applied for (2.91). Here

Making use of identity (2.61), we can rewrite (2.95) as an Abel integral equation for Q(t).

Solutions of this Abel equation for Q(t) in terms of gn(x) result

after some integrations. The result as well as the solution procedure presented here will be

used later in chapter 4 when we study the two punch tangential indentation for an elastic

layer. Reference for tangential indentation problem of an elastic half-space can also be made to

Fabrikant[lO] where a different method is used.

As an example, we consider the case where the flexible punch undergoes a unidirectional

displacement u,(T, 8,O) = A and U ~ ( T , 8,O) = 0. Therefore the displacement components in

the cylindrical co-ordinates are u,(T, 8,O) = A cos 8 and u ~ ( T , 8,O) = -A sin 8 and all gn(r) and

hn(r) are zero except hl(r) = 2A. Then it is not hard to find

Substituting these two fuctions into (2.92) and (2.93) and then using the stress field expressions

(2.46)-(2.48), we obtain

In terms of Cartesian co-ordintaes (x, y, z), the stresses are

Like the bonded punch problem[l3] and the rigid disk inclusion probelm[27], the stress has

singularity of the form (T - at the boundary of the contact region.


2.4 Layer Problem

In this section, several layer problems are examined by using the general formulation for three-

dimensional asymmetric elastic problems given in the first part of this chapter. Generally, in

this case, none of An(s), Bn(s), Cn(s) and &(s), Bn(s), C,(s) in equations (2.25)-(2.30) is zero.

Layer problems have been studied by many researches and references can be made to Lur7e[23]

and also to Muki[24].

Layer Problem l(Prob1em L1)

Problem L1 is that of an elastic layer - h < z < h with tractions prescribed on the layer surfaces. Boundary conditions for this problem can be written as

where the upper signs correspond to the upper layer surface z = h and the lower signs correspond

to z = -h and F,~(T), G,f(r) and H,~(T) are given functions determined by the stress boundary

conditions. By using equations (2.25)-(2.30), these six boundary conditions lead us to six linear

algebraic equations for An(s), Bn(s), Cn(s) and An(s), B,(s), Cn(s). Solutions for these six

equations enable us to find the stresses in the layer. In this thesis, it is only required the stresses

in the mid-plane of the layer. The Hankel transforms of these stresses take the form

-n sh sinh(sh) -n+l -n+l -n-1 e n - 1 "' O ) = 2[2sh + sinh (2sh)l x [G,f ( s ) - G; (a) + Hz ( s ) - HG ( s ) ] sh cosh (sh) + sinh (sh) -n

+ 2sh + sinh (2sh) x [F$ ( s ) + F: ( s ) ] , (2.101) n n + l n n + l -sh sinh(sh) -n -n .a", ( s , + = -2.h + sinh(2sh) x [F$ ( s ) - FL ( s ) ]

-sh C O S ~ ( S ~ ) + sinh(sh) -n+l -n-1 -n-1 + 2[-2sh + sinh(2sh)l x b,f ( s ) + G; ( s ) + Hz (4 + HG ( s ) ]

1 -n+l -n+l -n-1 -n-1

'4 cosh(sh) x [c,f ( s ) + G; ( 6 ) - Hz ( 6 ) - HL ( s ) ] , (2.102) -n-1 -n-1 -sh sinh(sh) -n -n re", (s,O) - T,", ( s , 0 ) = -2sh + sinh(2sh) x [FZ ( s ) - F; ( a ) ]

-sh cosh(sh) + sinh(sh) -n+l -n-1 -n-1 + 2[-2sh + sinh(2sh)l x b,f ( 8 ) + GZ ( s ) + Hz (8) + HG ( s ) ]


1 -n+l -n+l -n-1 -n-1 - 4 cosh(sh) X [ct (s) + C; (s) - HZ (s) - Xi (s)] . (2.103)

This solution will be used later in section 3.3 when we investigate interaction of penny-shaped

cracks in the mid-plane of a layer with stress free surfaces.

Layer Problem 2 (problem L2)

Problem L2 is that of a layer -h < z 5 h with displacements given on the two surfaces of the layer. Therefore the boundary conditions take the form

In the same way as for problem L1, the stresses in the mid-plane of the layer can be found

+ -n+l n n + l re", +T& (s,O) =

+

+ -n-1 -n-1 r& - T:~ (s, 0) =

+

- ps[(l - 2p) sinh(sh) - sh cosh(sh)] -2sh + (3 - 4v) sinh(2sh)

-n+l -n+l -n-1 -n-1 x [v.+ (s) - u (s) + v2 (s) - v; (s)] ps[2sh sinh (sh) + 4(1- v) cosh(sh)] -n

-2sh + (3 - 4p) sinh (2sh) ps[(-2 + 4v) sinh(sh) - 2sh cosh(sh)] -n

2sh + (3 - 4v) sinh(2sh) x [w.+ (s) + wi- (s)]

-n+l -n+l -n-1 -n-1 's X [.,+ (6) + u; (s) - v (6) - v; is)] , 2 sinh(sh)

p[(-2 + 4v) sinh(sh) - 2sh cosh(sh)] -n 2sh + (3 - 4v) sinh(2sh) x [w,+ (8) + w; (s)]

-n+l -n+l -n-1 -fa-1 x [u.f (s) - u; (8) + v2 (s) - v (s)]

We will need this solution later in section 3.3 when we examine interaction of cracks in the

mid-plane of a layer with fixed surfaces.


Layer Problem 3 (Problem L3)

Problem L3 is that of an elastic layer 0 5 z < h with given normal stress applied on the top surface z = 0 and the layer rests on a frictionless rigid foundation. Therefore boundary

conditions for the problem can be written as

on the top surface and

u:(r,h) = 0, r&(r,h) = r&(r,h) = 0,

on the bottom surface, where Pn(r) is the Fourier coefficients of the given normal stress. In

chapter 4 of this thesis, we are only interested in the relationship between the normal indentation

on surface z = 0 and the applied stress. In terms of An(s), B,(s) and &(s), Bn(s), this

relationship takes the form

Imposing the mixed boundary conditions on the elastic field equations (2.25)-(2.30), we again

get six linear algebraic equations for An(s), Bn(s), Cn(s) and &(s), B,(s), Cn(s). From these

equations, we find

where A,(s) is the nth order Hankel transform of the pressure Fourier coefficients P,(T) and

function Kl(x) is given by 1 + x - e-"

K1(x) = x + sinh(x) ' Therefore the relationship between the normal displacement on surface z = 0 and the applied

stress can be rewritten as


Considering that An(s) is determined by An(s) and &(s) only, the above expression in terms of

co-ordinates ( T , 8, Z) can also be written in terms of co-ordinates (F, 8,z) by using the arguments

right before section 2.1. It becomes to

where A*,(s) is determined by An(s) in the same way as A:(s) by An(s).

Layer Problem 4 (Problem L4)

Problem L4 can be described as: an elastic layer 0 5 z < h with given shear stresses applied on the top surface z = 0 and the layer is bonded with a rigid foundation on the bottom surface

z = h. Therefore boundary conditions of this problem are

Again we are only interested in the relationships between the resulting tangential displacements

on the surface z = 0 and the applied shear forces. In the same way as for the normal indentation

problem, we find

with

and functions K2(x) and K3(x) given by

In the same way as for equation (2.107), expressions for the above relationships in terms of

co-ordinates ( f , g, z ) can also be found. Solutions for problem L3 and problem L4 will be

used in chapter 4 when we investigate interaction of circular punches on an elastic layer.

Chapter 3

Interaction of Penny-Shaped Cracks

in an Elastic Solid

In this chapter, we shall study the problem of an elastic solid containing one or more parallel

circular cracks. The elastic solid involved can be an infinite elastic solid, or a semi-infinite elastic

solid, or an elastic layer and its boundary surfaces, if any, are parallel to the crack faces. It is

shown that the problem of an elastic solid containing two penny-shaped cracks can be reduced to

a system of Fredholm integral equations of the second kind by using some of the basic solutions

given in chapter 2 together with the superposition principle of the linear elasticity and the

addition formula for Bessel functions.

3.1 Penny-shaped cracks in an infinite solid

Problems concerning interaction between penny-shaped cracks in an infinite solid have only been

studied for the cases in which the cracks are either co-axial [3, 261 or co-planar [4, 7, 8, 111 with

only one exception by Kachanov and Laures[20] in which interaction of arbitrarily located penny-

shaped cracks is investigated. The problem is dramatically simplified in [20] by assuming that

the non-uniformities of the traction(with zero average) on a crack have no impact on the other

cracks. This assumption can be considered as an instance of Saint-Venant's principle and allows

CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...

the authors to find only approximate solutions even for weak interactions.

I . this section we shall study the interaction of two arbitrarily located parallel cracks C and

C in an infinite solid. It is assumed that radii of the two cracks involved are a and ii respectively. We choose two similarly oriented cylindrical co-ordinate systems ( r , 8, z ) and (F, 8, Z ) such that

cracks C and are taken to occupy

respectively. The origins 0 and 0 of these two coordinate systems are the points (f, 0 , g ) and (f, R, -g) in these respective sets of co-ordinates. Here f and g are the horizontal and vertical

distances between the centers of the cracks as shown in figure 3.1.

Figure 3.1: Two parallel penny-shaped cracks in an infinite solid.

On the crack faces tractions are prescribed and can be expanded in Fourier series. To be

compatible to the displacements given by (2.13)-(2.15), the tractions are assumed to take the


form

on crack face C and take the form

on crack face c. Here Rn(r) is the Fourier cosine coefficient of stress r,,(r, 8,O) on the crack face C and so on. We shall find the stress intensity factors at the perimeter of these two cracks.

3.1.1 Derivation of the integral Equations

Instead of solving this problem directly, we consider two easier problems: one is problem C

of an infinite solid containing just one crack C with arbitrary tractions (2.55)-(2.57) (they are

unknown at this time) loaded on the crack faces and the other is problem c of an infinite solid containing crack 6 with unknown tractions

applied on the crack faces 6. It is clear that solutions to the original problem can be represented as a superposition of the solutions for these two easier problems as long a s this superposition

CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. . 30

satisfies the given boundary conditions on the two crack faces. It will be shown that the original

problem can be reduced to a system of Fredholm integral equations for these Fourier coefficients

an(z), bn(z), cn(z) and Cn(z), L(x ) , En(z).

In deriving the integral equations, we require not only the expressions for the stress field

of problem C in terms of co-ordinates (r, 8, z) but also the expressions in terms of (F, 8 , ~ ) .

In order t o do this, we introduce an auxiliary co-ordinates system (F, 8, z). Let {u,(r, 8, z ) ,

U O ( ~ , 4 4, uz ( r , 8 , ~ ) ~ rrz(r, 8 , ~ ) ~ 7ez(r19,z), m ( r , 9 , ~ ) ) and {+, 8, z ) , u#(T, 8 , z ) , g Z ( ~ , g , ~ ) , T F ~ ( T , 8, z ) , r,-,(i;, 8, z ) , ri-,-(F, 8, a ) ) be the stress field of the problem C in terms of (T , 8, z) and (F, 8, z) re-

spectively and {u,"(r, z ) , u$(r, z ) , u,"(r, z ) , r,", ( r , z ) , r,",(r, z ) , r,",(r, z ) ) and {u;(F, z ) , g ; ( ~ , z ) , g , " ( ~ , z ) ,

r,?Z(F, z ) , rE(i;, z ) , rig(?, z ) ) be the corresponding coefficients. Furthermore, expressions for the

stress field of problem c in terms of (F, 8, t) and (r, 8, z) are also required. Introduce another auxiliary co-ordinates system (r, 8 , t ) and let { Z i , n ( ~ , z ) , ~ ; ( t , z ) , a ; ( ~ , z ) , ?$(F, Z ) , . f&(F, Z ) , ? i B ( F , 5) ) and {8F(r, Z ) , ar (r , z ) , Fg(r, Z ) , ?:Z(r, z ) , ?Fz(r, z ) , ~-2 ( r , 2)) be the corresponding stress coefficients in terms of (F7 8 , t ) and (r, 8 , t ) respectively. Then the stress boundary conditions can be

rewritten as

on crack C and

on crack face C. The expressions of the stresses of problem C in terms of (r, 9, z ) are given by (2.46)-(2.48)

in the upper half-space z > 0 and (2.78)-(2.80) in the lower half-space z 5 0. By using the arguments given in chapter 2 and the basic solutions (2.46)-(2.48), we find that the stress field

CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. .

in terms of ( f , 8, z ) takes the form

in the upper half-space z 2 0 . Similarly the expressions of the stress field of problem c in terms of both co-ordinates

( f , 8 , ~ ) and ( T , 8,Z) can be found. In terms of ( f , 8, E), the stresses are given again by (2.46)- (2.48) in the upper half-space E 2 0 and (2.78)-(2.80) in the lower half-space E 5 0 with T , 8 , z , An(s) , Bn(s ) , Cn( s ) replaced by f , 8, E, &(s ) , Bn(s ) , Cn( s ) . From (2.78)-(2.80), we obtain the

stress field in terms of co-ordinates (T , 8,Z)

Ttt(r , f ) + TFt(~ , f ) = 2p [ - s f $ , ( s ) + ( 1 + s f ) B ; ( S ) - C ; ( S ) ] S J ~ + ~ (r s )es zds , (3.19) 1" 03

( T , 2) = 2 1 [ ( I - s i ) A ; ( s ) + s ~ B ; ( s ) ] s J ~ (rs)es"ds, (3.21) in the lower half-space 5 0. Here r?;(s), B:(s) and R ( s ) are given by

and An(s) , Bn( s ) and Cn(s ) are determined by ~i , ( z ) , 6,(z) and Zn(x) in the same way as those

of An(s), Bn(s) and Cn(s) by an(z ) , bn(z), cn(z)-

Substituting of the above stress expressions into equations (3.10)-(3.15) and changing the

order of integrations lead us to a system of six integral equations, three on each crack. On the

face of crack C , they are

CHAPTER 3.

+ -

b o ( ~ ) -

+ +

~ o ( f ) =

if n = 0 and

INTERACTION OF PENNY-SHAPED CRACKS.. . 32

if n > 1. On the crack face C?, the same integral equations are found with a n ( x ) , b n ( x ) , c n ( x ) and i in(x) , &, (x) , En(x) interchanged, a and ii switched and (-1)" replaced by ( - l ) n . Kernels

of these integral equations are given in the Appendix. Now we have reduced the problem to a

system of six integral equations for a n ( x ) , bn(x) , c n ( x ) and i in (x) , 6 , (x ) , G ( x ) , the coefficients

of the stresses on the faces of the isolated crack in an infinite solid. Since the stress singularities

CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ... 33

for crack C are only dependent on the tractions applied on the faces of the isolated crack C in

an infinite solid, therefore these tractions will be called effective stresses and their coefficients

an(x), bn(x), cn(x), will be referred to as coefficients of the effective stresses for crack C from

now on and 7in(x), 6,(x), Zn(x) will be referred to as the effective stresses for crack c. The above integral equations, though formidable looking, can be solved by iteration for some special cases.

3.1.2 Some Asymptotic Solutions

In this subsection, we shall solve the above integral equations for some special cases. Without

loss of generality, we assume the two cracks have equal radii a and we will only consider two

cases. In the first, the horizontal distance f between these cracks is large compared with their

radii a while in the second the horizontal distance is zero and the vertical distance g is large

compared with a. In each case two different loadings, namely equal uniform normal loadings and

equal(in the same direction or the opposite direction) uniform shear loadings are considered.

CASEONEa=Si


for n 2 1.

Considering that the above integral equations (except equation (3.34)) are infinitely long for

each given n, and also that there are infinitely many these integral equations (n = 0,1,2, ...),

generally these integral equations cannot be solved by using numerical methods before some

significant simplifications.

From the kernels given in the Appendix, we can see that all the kernels can be expanded as

power series of e j = 4 , the ratio of the crack radii to the horizontal distance. It is also seen


that Knm33 is of order 0 ( € 7 m + n ) 7 Knmls and Knmll are of order O(rFm+"), Knml17 Knmz3

and Knm32 are of order o ( c ; + ~ + ~ ) , Knm12 and Knmzl are of order O ( C ? + ~ + "), and Knm22 is

of order 0 ( r 7 ~ + ~ ) when n 2 1. By using this observation, we obtain the following: to get

solutions correct to order O(ry), we can truncate all the terms with m 2 N - 1 to make each

equation finitely long, and neglect all the integral equations if n 2 N - 2. The resulting finite

set of Fredholrn integral equations, each of which are finitely long now, can be solved iteratively

by expanding all the kernels involved in terms of power series of ef and neglecting all the terms

with order higher than N.

In order to obtain the solutions correct to order 0($), we can neglect such terms that m 2 5

and only solve the first five equations. The non-zero terms of the solutions are

bo(xa) =

al(xa) =

c1 (xu) + b i (xu) = cl (xu) - b l (xu) =


which involve integrals 03

C = 1 y m ~ n ( y ) e - ~ d y . These integrals can be expressed in terms of Gamma functions and the hypergeometric function

as given in [29]

The integrals IF can be evaluated numerically either using the Gauss-Laguerre formula or using

the above formula and truncating the Hypergeometric function. The stress intensity factors can

then be found from (2.81)-(2.83)


The numerical results for these stress intensity factors are shown graphically in figures 3.2-3.8.

There klo is the intensity factor for stress o,, due to a single isolated crack subject to the uniform

normal loading P in an infinite solid. Later we will use k20 to express the stress intensity factor

for a penny-shaped crack in an infinite solid under uniform radial shear loading.

From figure 3.2 we can see that the effect of the angle 8 on the stress intensity factor kl is

fairly small when cf is s m d , say no larger than 0.25. Figure 3.3 (cf = 0.45) does indicate that

the angle 8 has a significant effect on the stress intensity factor when €3 is large.

Figure 3.2: Equal uniform normal loadings: variation of stress intensity factor & with angle 0 for cf = 0.2 and various f when a = ii


Figure 3.3: Equal uniform normal loadings: variation of stress intensity factor % with angle 9 for ef = 0.45 and various 7 when a = ii < f.

Figure 3.4: Equal uniform normal loadings: variation of the stress intensity factor with respect to c j at 9 = 0 for various 7 when a = ii < f .

Figure 3.4 shows the variation of the stress intensity factor kl with respect to the normalized

CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS. .. 39

crack size c j at 0 = 0 for some given values of 7. From this picture we can see that kl increases with c f for some values of f , say 0 or 0.25, and decreases for some other values of f , say 1. This

implies that increasing the size of the crack sometimes reduces the stress intensity factor kl and

sometimes increases it if the horizontal and vertical distances f and g are fixed.

Figure 3.5: Equal uniform normal loadings: variation of the stress intensity factor 5 with respect to f at 0 = 0 for various c f when a = 3 6 , the stress intensity factor is very close t o that for a single crack no matter what value of ~f is.


respect to f at 0 = 0 for various cf when a = ii.

CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ... 4 1

From the above formulas, it is seen that the non-dimensional stress intensity factors 2 and are dependent on v while & are not. Figures 3.6 and 3.7 show the behaviors of 2 at 8 = 0

kl0

and 2 at 8 = n/2. In these figures, v = 0.3. It is interesting to see that the behaviors of $ at 8 = 0 and -& at 8 = n/2 are very similar. They are heavily depend on 8, though they are very small (of the order O(E3f)). Also we note from figure 3.6 and figure 3.7 that they change signs

when f is about 0.3 and reach maximum absolute value around 1. Furthermore they reduce to

zero fairly quickly when f increases. Figure 3.8 shows that the variations of 2 with respect to the Poison's ratio v for some given values of €1 and they are very close to straight lines with very

s m d slopes. This implies that the effects of the Poison's ratio v on the stress intensity factor

& is not significant.

Figure 3.8: Equal uniform normal loadings: variation of the stress intensity factor with respect to v at 0 = 0 for 7 = 0.3 and various e j when a = Zi


B. Uniform shear loadings

In this case, the following unidirectional shears are applied on the crack faces

-@1(x) = R I ( x ) = S = f & ( x ) = T R ~ ( Z ) ,

& ( x ) = Z l ( x ) = O , (3.41)

On(.) = Rn(x) = On(. ) = R , ( x ) = Z n ( t ) = Z n ( x ) = 0, for n # 1,

where the upper signs correspond to the case in which shear loadings on both cracks are in

opposite directions and the lower signs correspond to loadings in the same direction. We can see

from (3.25) -(3.30) and their analogues on crack that

In the same way as for normal loading, we reduce the system of six integral equations to a system

of three integral equations and then solutions can found by iteration for the case when ef is very

small. By using these solutions, the stress intensity factors correct to order O(E6f) are found


2 * -[vI," 315n - 9c - (2 - V)I: + f&? cos(40)) , f k3 =

4S&(1- V) {U r 2 n(2 - v) 3n(2 - v)

1 9 5 5 *-[-(a + 3v)I; - 391: + (3v - 8)1: + 3-Io]rj

45n f f 8 3 2 6 4

(-I1 ) 'f + gr2(2 - v)2 + 9 r y 2 - u) f [UI; - 91; - (2 - U)I~ + ~ I ~ ] ~ c ~ } sin 6 f

2 9 Q 4 4 *{- gn(2 - u)

[VI; - -1: - (2 - U)I? + 711]rf f

2 + 3157r(2 - u)

[-7(2 - U)I: + 6 9 ~ + (7v - 2)1: - 69g] r?} sin(20) f f

2 9 5 5 - (2 - U)I; + -12]ef sin(38) r K[vI: - ?I4 f 2 r -[uI; - 9c - (2 - v)~: + p&F sin(46)) .

315n f

Figure 3.9: Equal uniform shear loadings: variation of the stress intensity factor f& with respect to €3 a t 0 = 0 for various 7 when a = ii


Figure 3.10: Equz d uniform shear loadings: variation of the stress intensity factor & a respect to f at B = 0 for various ef when a = ii < f.

Figure 3.11: Equal uniform shear loadings: variation of the stress intensity factor 2 with respect to f at B = 0 for various cf when a = ii < f.

AU -h and & are dependent on u when the cracks are subject to shear loadings and u k2o ' k2o


is taken to be 0.3 in all these figures.

Figure 3.9 shows that the presence of the second crack sometimes ( f = 0 or 0.25) reduces

the value of stress intensity and sometimes (f = 1) increases it. From figure 3.10 we can see

that 8 reaches the maximum value when f is around 0.6. Figure 3.11 shows that the behavior of 2 is very similar to those of & and 2 in the case of uniform normal loadings. CASETWO f =O; a = i i < < g

Since f = 0, all the kernels Kijn, equal to zero unless n = m and the infinite system of integral

equations decouple, which means that the nth Fourier coefficient is only related to other nth

terms. The integral equations (3.25)-(3.30), resulted from the boundary conditions on the crack

C, now become

if n = 0 and

for n 2 1. Similar integral equations can be obtained from the boundary conditions on c. Due to the geometric symmetry of the problem, each integral equation is finitely long, though there

are still infinitely many such kind of integral equations. For some special loadings, which can

be expressed by finite Fourier series, this infinite set of integral equations can be reduced to a

finite set and then can be solved by numerical methods. Asymptotic solutions are also available

when cg = :, the ratio of the radius to the vertical distance, is very s m d . Several asymptotic solutions are presented below.

A. Equal uniform normal loadings


If the cracks are opened by the equal uniform pressures (3 .31) , the only non-zero terms

of a n ( x ) , b n ( x ) , c n ( x ) and % ( x ) , b n ( x ) , E ~ ( x ) are a o ( x ) , bo(x) , and a o ( x ) , bo(x ) . Note that

a o ( x ) = go(,) and bo(x) = -bo(x) and therefore in this case we end up with just two integral

equations

Solutions to these equations accurate to order 0 ( e t ) are

Hence the stress intensity factors are

B. Uniform shear loadings

If the cracks are under equal uniform shear loadings (3 .41) , the only nono-zero terms are

a l ( x ) = k a l ( x ) , & ( x ) = r f b l ( x ) , el(%) = 7 c l ( x ) . In the same way as for the normal loading, we reduce the number of integral equations from six to three and obtain the stress intensity factors

3.1.3 Discussion

It is of interest to compare our results with some previous solutions for some particular cases.

Coplanar cracks

CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. . 4 7

Coplanar cracks opened by uniform pressures have been studied by Collins[4] and Fabrikant[7]

and expressions for the energy of each crack were given. In our notation, it is

which takes the following form

after substituting the solutions we obtained in last section. This agrees with that given by

Collins[4]. The series expansion of Fabrikant7s solution, obtained by using the mean value theo-

rem, in our notation is

which is slightly lower than ours. His method can be used to handle very close interactions but

failed in the non-coplanar case.

Coplanar cracks subjected t o uniform shear stresses have been studied by Fu and Keer[ll]

and Fabrikant[8]. As mentioned in 181, some formulas and results in [ll] are incorrect. In the

same way as for normal loadings we obtain the energy increase per crack under shear loadings

This corrects the results in [ l l ] and also confirms that (3.58) should be same as (3.56) when

v = 0, which had been observed by Fabrikant[8].

Co-axial cracks

If the cracks are opened by uniform pressures, the crack energy is

which confirms the result given by Collins [3]. For the case of shear loadings the results seem to

be new even for co-axial cracks.

A research paper1151 based on the work of this section is to be published in the Theoretical

and Applied fracture Mechanics.

CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS. ..

3.2 Penny-Shaped Cracks in a Semi-Infinite Solid

An elastic half-space containing one penny-shaped crack under uniform normal loading has been

studied by Srivastava and Singh[32]. The problem of a half-space containing one crack loaded

with arbitrary tractions and the problem of a half-space containing more than one are the

subjects of this section.

The problem to be considered in this section involves a semi-infinite elastic solid, with stress

free boundary, containing two arbitrarily located penny-shaped cracks C and c, which are parallel to the half-space surface. Like the infinite solid case, two local cylindrical co-ordinate

systems (r, 8, z) and ( F , 8 , ~ ) are employed such that cracks C and are taken to occupy

respectively and the stress free boundary is taken to be z = -h in the first co-ordinate and

5 = -6 in the second. The origins 0 and 0 of these two coordinate systems are points (f, 0, g) and (f,n, -g) in these respective sets of co-ordinates. Here f and g are the horizontal and

vertical distances between the centers of these cracks, respectively. The distances from the free

boundary to these two cracks are denoted by h and h respectively and therefore 6 - h = g, which is non-negative (figure 3.12).

Tractions on the crack faces are prescribed and can be expanded in Fourier series (3.1)-

(3.6). Besides these boundary conditions on the crack faces, stress free boundary conditions

are imposed on the surface z = - h of the half-space. In terms of co-ordinates ( r , 8 , z), these

boundary conditions can be written as

We s h d find the stress intensity factors for these two cracks.

3.2.1 Derivation of integral equations

The problem of an elastic half-space weakened by two parallel penny-shaped cracks can be de-

composed into two easier problems: problem a of a half-space containing crack C and problem


I I I I I I I

I

I I I I I

I I I I 2a - . I ,--Q--' ----,--- ,-,-,,, b' ------------ J ------- \

\ Stress free surface

Figure 3.12: Two parallel penny-shaped cracks in a semi-infinite solid.

b of a half-space containing c. Furthermore problem a can be decomposed into problem C of an infinite solid containing one crack with effective stresses an(x), bn(x) and cn(x) applied

on the crack face C and problem H1 of a half-space with properly chosen stresses applied on

the surface such that the sum of the solutions to these two problems satisfies the stress free

boundary conditions at z = -h. Solutions to problem b can be decomposed into two parts in

a similar fashion.

It will be shown that the original problem can be reduced to a system of Fredholm integral

equations for an(x), bn(x), cn(x) and &(z), Zn(x), En(x), where a,(x), bn(x), 4 ( x ) are the

corresponding effective stresses for crack c. The procedure of deriving these integral equations involves finding the stress fields of problem a and problem b in terms of the corresponding

effective stresses.

As mentioned before, the stress field of problem a can be considered as the sum of the


stress field of problem C and the stress field of problem HI. In order to satisfy the stress

free boundary conditions of problem a , the boundary conditions for problem H1 are chosen

so that

11 Fn ( s ) = 2p[An(s) ( l+ sh) - ~ ~ ( s ) s h ] e - " ~ , (3.61)

+I ( s ) = 2p[-shAn(s) - ( 1 - sh)Bn(s) + ~ n ( s ) ] e - " ~ , (3.62) -1 ( s ) = 2p[-shAn(s) - (1 - sh)Bn(s) - ~ , ( s ) ] e - " ~ . (3.63)

Here the basic solutions to problem C have been used. With those stresses applied on the

half-space surface of problem HI, we can find the stresses in the half-space by using the basic

solutions for problem H1. Adding these two stress fields results the stress field of problem a.

In terms of the effective stress coefficients a,(x), b,(x) and c,(x), the stresses on the crack C of

problem a can be written as

~,h, ( r , 0 ) = an ( T ) + 2p [ ( I + 2sh + 2s2h2)An(s) 1" - ~ s ~ ~ ~ B ~ ( s ) ] s ~ - ~ " ~ ~ ~ ( s r ) d s , (3.64)

Considering A, ( s ) , Bn ( s ) and Cn(s) are integrals of the effective stress coefficients a, ( x ) , cn(x) + bn(x) and cn(x) - bn(x), we now have expressed the stresses on the crack face by these stress coefficients. With the aid of the arguments given in chapter 2, expression of this stress field in

terms of ( F , # , z) can also mbe found. On the position where crack shall be, the stresses are

given by


+ 2p lm { ( sg + 2 s 2 h 6 ) ~ ; ( s ) - [ I - s(h + 6 ) + 2 s 2 h 4 ~ ; ( s ) - c;(s)} s~ , -1 (Fs )e - ' (~+~)ds .

Similarly we obtain the stress field of problem b in terms of both co-ordinate systems ( f , e , E )

and (r7 8,E).

Now return to the original problem. It is clear that the superposition of the above two

solutions automaticdy satisfies the stress free boundary conditions (3.60). Superposing these

two stress fields and substituting them into the crack face boundary conditions lead us to system

of Fredholm integral equations for the six effective stress coefficients after changing the order of

integration. On crack face C , the integral equations are


co(r) = 0

if n = 0 and

an(^) t

t

+ + -

+ +

+ cn (r ) + bn(r) +

+ -

+ +


when n 2 1. On crack c, they are


+ + + + +

+ e ( f ) =

if n = 0 and

a n ( f ) + + + +

+ + +

+ G ( f ) + in (? ) +

+ + +

1' ~ a n m l 3 ( ~ 7 X ) [ c m ( ~ ) - b m ( x ) ] d x ) = i n ( f ) 7 ( 3 . 7 6 )


when n 2 1. Kernels of these integral equations are given in the Appendix. Now we have reduced

the problem to a system of six integral equations for an($), bn(x), cn(x) and an(x), bn(x), z,(x).

These integral equations can be solved by iteration for some special cases. Once these effective

stress coefficients are known, stress intensity factors can be determined.

3.2.2 Some Asymptotic Solutions

In this subsection we shall solve the integral equations for three special cases, the single crack

case, the two co-axial crack case and two coplanar crack case. As for the infinite solid case, we

assume that the two cracks have equal radii a for the latter two cases. In the second and third

cases, we will consider two different loadings which are the equal uniform normal loadings and

equal uniform shear loadings.


CASE I SINGLE CRACK

If there is only one crack, the system of six integral equations becomes to a system of three

integral equations because of the absence of the second crack. Since all the kernels involving

parameters or g vanish, the system of integral equations decouple. This implies that the nth

coefficients of the effective stresses are only related to the other nth terms.

A. uniform normal loading

If the crack is under uniform normal loading P, all Zk(r), Rk(r) and Ok(r) equal to zero except

Zo(r). Therefore all ak(x), bk(x) and ck(x) vanish except ao(x) and bo(x). Hence the system of

integral equations becomes just two Fredholm integral equations for ao(x) and bo(x),

These integral equations can be solved for the case when r h = f , the ratio of the crack radius to

the distance between the crack and half-space surface, is very small, considering Kbnij(r, x) can

expanded as power series of r h . Solutions correct to O ( 4 ) are

Therefore the stress intensity factors are

These recover the results given by Srivastava and Singh[32] using a different method, noting that

the definitions of k2 differ by a constant.


B. unidirectional shear loading

If the crack is under uniform shear loading in the x direction,

then Z k ( r ) , R k ( r ) and O k ( r ) are all zero except - O 1 ( r ) = R l ( r ) = S and therefore all a k ( x ) ,

b k ( x ) and c k ( x ) are zero except a l ( x ) , b l ( x ) and c l ( x ) , which satisfy the following integral

equations

In the same way as for the normal loading, we obtain solutions correct to order O ( E ; )

these give us the stress intensity factors

kl = 10&S cos 8 21 3 r 2 ( 2 - v ) (4 + 3 3 7


CASE I1 TWO CO-AXIAL CRACKS

If the cracks share the same axis, then the system of six integral equations decouples too. Also

it is seen from the definitions of the kernels that K b n m i j ( y , x ) and Kbnmij(y, x ) can be expanded as power series in ~ h , K c n m i j ( y , x ) and K m m i j ( y , x ) can be expanded as power series in cg and

K a n m i j ( y , x ) and ~ a n m ; j ( ~ , x ) can either be expanded as power series in ~h or as power series

in cg = 5 Therefore different forms of solutions can be sought depending on which expressions 9 '

of K a n m i j ( ~ , x ) and K a n m ; j ( y , x ) we use. I f the ch power series of K a n m i j ( y , x ) and ~ , , , ; ~ ( y , x )

are employed, solutions take the following form

where a i k ( r ) itself is a power series of cg and so on. Examples given below show how this idea

works.

A. Equal normal loadings

When the cracks are opened by equal uniform pressure P as given by ( 3 . 3 1 ) , the only non-

zero terms are a o ( x ) , b o ( x ) and i i o (x ) , 6 0 ( x ) , which satisfy the following four Fredholm integral

equations,

a o ( r ) + + +

b o ( r ) + -

+ no(.) +


Substituting (3.86)-(3.88) into the above integral equations, we end up with system of integral

equations for a&, bkk and z$, 6kk by equaling the coefficients of 6; for k = 0,1,2, .... For each given k, the integral equations resulted can be solved by iteration when eg is small.

The integral equations for k = 0 are

which agree with the equations in the last section for two co-axial cracks in an infinite solid.

Solutions accurate to order 0($) are

It is not hard to find the solutions for the integral equations for k = 1 and k = 2

The integral equations for k = 3 are


where cl and c2 are constants given by

which involving integrals

The non-zero terms of the solutions of ab3(ax), bb3(az), ~ i b ~ ( a x ) , 6k3(ax) correct to order O ( { )

are

The stress intensity factors follow

It is worth mentioning that k2 and i2 are of order O ( $ ) + 0(&. From equation (3.50) in section 3.1 we know that the presence of the second co-axial crack

under the same uniform pressure as the first one reduces the stress intensity factor kl. Equation

(3.81) shows that the stress free boundary increases the value of kl when there is only one crack.

The variations of nondimensional kl and il with respect to f, and sh are shown in figures 3.13-3.15, respectively. From these figures we can see how the stress free boundary and the two

cracks effect each other.


1 .04 I 1 I I I

1.03 -

1.02 - -

1-01 - - -

rh = 0.15 -

-

0.98 I I I I I

1 2 3 4 5 6 7 9 h

Figure 3.13: Two co-axial cracks with equal uniform normal loadings: variation intensity factor & and 2 with respect to f for various Q.

of the stress

Figure 3.14: Two co-axial cracks with equal uniform normal loadings: variation of the stress intensity factor & and 2 with respect to a for various q.

Not surprisingly, we note from figure 3.13 that kl > El whatever rh is when f > 4, since


in this case the boundary plays the most important role. It is interesting to note that kl < i1 when < 3, even though C is the crack closer to the boundary. Figure 3.14 shows that kl and

El are monotonicdy decreasing with increasing for given E, and the effect of the boundary is ignorable when > 6 . Figure 3.15 tells us that increasing the size of the cracks sometimes

increases the stress intensity factor kl and sometimes reduces it when g and h are given.

Figure 3.15: Two co-axial cracks with equal uniform normal loadings: variation of the stress intensity factor & and & with respect to rh for various 9 .

B. Equal uniform shear loadings

When equal unidirectional shears are applied on the crack faces, the boundary conditions are

- & ( x ) = - R 1 ( x ) = -S = & ( x ) = - R 1 ( x ) ,

& ( x ) = Z l ( x ) = 0 , (3.91)

O ~ ( X ) = R n ( x ) = O ~ ( X ) = R n ( x ) = Zn(x) = Zn(x) = 0, for n # 1,

and the only non-zero solutions are a l ( x ) , b l ( x ) , c l ( x ) and i i l (x) , & ( x ) , El(x). The non-zero

terms of the solutions in the form of (3.86)-(3.88) accurate to O($)+O({ ) produce the following

stress intensity factors


and

Here cg and c4 are functions of R given by

Also it is worth mentioning that kl and i1 are of O(cj) + O ( E ~ ) .

Figure 3.16: Two co-axial cracks with equal uniform shear loadings: variation of the stress intensity factor 2 and $ with respect to f for various 4 .


Figure 3.17: Two co-axial cracks with equal uniform shear loadings: variation of the stress intensity factor 2 and & with respect to a for various cg.

From these formulas we can see that k2 and k2 are cosine functions of 8 and k3 and i3 are

sine functions of 8. Also we know that behaviors of kz at 0 = 0 and k3 at 8 = are the same

(correct to order O ( 4 ) + 0(


as power series of cf or power series of ch, we can find two different solutions depending on which

the expressions used. As an example, we use the ~f power series expression and seek solutions

of the following form

f where cnk(x) itself is a power series of ch and so on.

A. Equal normal loadings

If the cracks are opened by the equal uniform pressures we have

and thus the system of six integral equations can be reduced to a system of three integral

equations, which can be solved by the same procedure used for the co-axial case. Solutions

correct to O($) + O(4) lead to the following stress intensity factors

where c5 and c6 are functions of 3 given by

which involve the following integrals

Some elastic multi-crack and multi-punch problemssummit.sfu.ca/system/files/iritems1/5039/b1485286x.pdf · Three-dimensional crack problems have received a great deal of attention

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