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
PARTIAL COPYRIGHT LICENSE
I hereby g ran t t o Simon Fraser U n i v e r s i t y t he r i g h t t o lend
my t h e s i s , p r o j e c t o r extended essay ( t h e t i t l e o f which i s shown below)
t o users o f t he Simon Fraser U n i v e r s i t y L i b r a r y , and t o make p a r t i a l o r
s i n g l e copies o n l y f o r such users o r i n response t o a request f rom the
l i b r a r y o f any o t h e r u n i v e r s i t y , o r o t h e r educa t iona l i n s t i t u t i o n , on
i t s own beha l f o r f o r one o f i t s users . I f u r t h e r agree t h a t permiss ion
f o r m u l t i p l e copy ing o f t h i s work f o r s c h o l a r l y purposes may be g ran ted
by me o r the Dean o f Graduate Stud ies. I t i s understood t h a t copy ing
o r p u b l i c a t i o n o f t h i s work f o r f i n a n c i a l ga in s h a l l n o t be a l lowed
w i t h o u t my w r i t t e n permiss ion.
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 var- ious 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 var- ious 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 + $a) = cos n8 instead of cos no and sin no. Therefore the solutions corresponding to &, 4, and may be obtained by making the following replacements
cos n8 - sin no; sin n8 a cos no,
Substitution of (2.10)-(2.12) into the displacement field (2.4)-(2.6) gives the displacements in
00
ur (r, 8 , ~ ) = C U: (r, Z) C O S ( ~ ~ ) , n=O
where the Fourier coefficients ur(r, z), u;(r, Z) and u ~ ( T , Z) of the displacement components are
given by
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS 8
Here, of course, we have used the assumption that &(r, z ) = '$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.
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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 ) , $Jn ( s , z ) and Z n ( s , z ) are known functions of z and s and expressions of &(T,z ) ,
& ( T , Z ) and xn(r7 Z ) can be obtained by Hankel inverse transforms,
and so on. Substituting this into the displacement equations (2.16)-(2.18), we can rewrite the
displacements in terms of Z n ( s , z) , G n ( s , z ) and Z n ( s , z ) ,
00 azn -n azn up + US = - 1 [ ( I - 2v)+n + Z- - 2JIn + 2(1- v)zn + z - ] ~ ~ ~ ~ + ~ ( r s ) d s , a2 8~
Here we have used the following properties of Bessel functions
d n - [ J ~ ( s x ) ] - - J ~ ( s x ) = - S J ~ + ~ ( S X ) , dx x d n
- [ J ~ ( s x ) ] + - J ~ ( s x ) = s J ~ - ~ ( s x ) . dx x
Similarly we find that the stresses of interest take the form
agn ax';;" a2zn re", + rz = -2rl~[zg - - 8% +- dz + zT]s2 Jn+l(rs)ds, a2
-', = -2pJ00[ 2-+- a2En 8 K n +- ax';;" + z - ] s ~ J ~ - ~ ( T s ) ~ s , a2Zn (, a 2 2 az az a z 2
= 1 a2Ln a3Zn assn 1- a,a + z- a23 + z- 8 , IsJn (rs)ds,
from equations (2.19)-(2.21). In terms of An(s), Bn(s ) , Cn(s ) , &(s ) , &(s) and C n ( s ) , the
displacement field becomes
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS 10
+ lw [ ( I - 2u + S I ) & ( s ) + (2 - 2u + sr)& ( s ) + 2~ (s) ]s2 Jn- 1 (rs)e"dr, (2.26) 00
U: ( r , I ) = 1 [(2 - 2~ + sr)An ( s ) + ( 1 - 2~ + S Z ) B ~ ( s ) ]s2 Jn (rs)e-.'ds
and the stress field takes the form
It is essential for the method used in this thesis to find the displacement and stress expressions
in terms of another cylindrical co-ordinate system ( f , 8 , z ) . Let the origin 0 of co-ordinates
( T , 8, Z ) be point ( f , T , 0 ) in co-ordinate system ( f , 8, z ) and origin 0 of ( f , 8, z ) be point ( f , 0,O) in co-ordinates ( T , 0, z), where f is the distance between 0 and d. Let {u i ( f , 8, z ) , uB(f ,8 , z ) ,
placement and stress fields in terms of the new co-ordinates and { (u:(f , z ) , u$ ( f , z ) , ug( f , 2 ) ) )
and { @ ( f , I ) , o ; ( f , z ) , u:(f, z ) , T$(T, z ) , rFZ(F, z ) , rFB(f, z ) ) be the corresponding Fourier coef-
ficients. With the aid of the addition formula for Bessel functions as given in Watson[33],
we can express the three harmonic functions in terms of co-ordinates ( f , 8, z ) as follows:
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
with
Here the prime on the summation sign implies that the (-l)nJ,,+n(ds) terms do not appear
when n = 0. This shows that the potential functions take the same form in the two systems
of cylindrical co-ordinates ( T , 8 , z ) and ( F , 8 , z ) . Therefore it is clear that the displacement and stress components in terms of co-ordinates ( F , 8, z ) also take the same form as those in terms of
( ~ 7 '37 z ) as 10% as we replace An(s), Bn(s), Cn(s), An(s)7 Bn(s), Cn( s ) by A:(s)7 B:(s)7 C;t(s)7
En($), 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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS 15
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
(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"
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS 18
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS 19
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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 )
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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.
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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 ) ]
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS 24
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.
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS
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
CHAPTER 2. SOME THREE-DIMENSIONAL ELASTIC PROBLEMS 26
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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 coeffi- cients 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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. .
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ... 35
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) =
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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)
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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.
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. .
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ... 46
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ... 50
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
+ 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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. . 52
co(r) = 0
if n = 0 and
an(^) t
t
+ + -
+ +
+ cn (r ) + bn(r) +
+ -
+ +
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
when n 2 1. On crack c, they are
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
+ + + + +
+ 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 )
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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.
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. .
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.
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. .
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. .
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(.) +
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS ...
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.
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. .
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. . 62
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
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. . 63
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 .
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. .
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(
CHAPTER 3. INTERACTION OF PENNY-SHAPED CRACKS.. . 65
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