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The stress intensity factors and interaction between cylindrical cracks in fiber-matrix composites
Damage and Interfacial Debonding in Composites G.Z. Voyiadjis and D.H. Allen 9 1996 Elsevier Science B.V. All rights reserved.
The Stress Intensity Factors and Interaction Between Cylindrical Cracks in Fiber1Vfatrix Composites S. Close and H.M. Zbib School of Mechanical and Materials Engineering Washington State University, Pullman, WA 99164-2920, USA
The elastic interaction between two cylindrical cracks in an infinite, homogeneous, isotropic, elastic medium is investigated. The cylindrical cracks represent a case of fiber-matrix debonding. We examine the effect of the cracks spacing and size on the stress intensity factors, K I and KII, which result from a pressure loading. Each crack is modeled as a pile-up of Somigliana ring dislocations. The solution is based on analytical expressions obtained earlier for the ring dislocation. Continuous distributions of dislocation densities, modeling the two cracks, are obtained numerically using a piecewise quadratic approximation and an iterative scheme to evaluate the interaction between the two cracks. The analysis provides estimates for the stress intensity factors and their relation to the cracks spacing and size. The analysis also reveals that each crack can be represented by a pair of superdislocations, which leads to the analytical solution. The interaction among the superdislocations also provides a closed form expression for the stress intensity factor.
I. INTRODUCTION Recent advances in the field of material science have led to the development of a class of unconventional materials, such as fibrous composites. In general, composites are composed of strong, lightweight fibers embedded in a matrix. As the development of composites has progressed, the utilization of these materials in industry has become increasingly more common. As with any solid material, there are unavoidable stress raisers present, due to internal defects, which have important implications on the mechanical behavior of the material. Some of the most common defects which have been studied extensively include crystal defects and planar cracks. The development of composite materials has given rise to a series of internal defects which have not been thoroughly investigated. These defects arise from the characteristic, geometric structure of fiber-matrix composites, and include matrix cracking, broken fibers, fiber pull-out, and fibermatrix debonding as shown in Figure 1. Stress singularities caused by cracks, voids or inclusions, may lead to structural failure at stress levels far below the limits estimated using a macromechanical analysis. Therefore, it is necessary to have a comprehensive understanding of fracture initiation and growth, the effects of voids and small inclusions, and their interaction with
each other. In this instance, we examine the interaction between cylindrical cracks; a defect type which might occur in the case of fiber-matrix debonding, as shown in Figure ld.
b2
i Li i I .......
b2
.........
Figure 1. Defects in fibrous composite materials: (a) matrix cracking, (b) broken fibers, (c) fiber pull-out, (d) fiber-matrix debonding.
---
~th
Figure 2. The Somigliana ring dislocation.
The fiber-matrix debonding problem has been previously addressed by a number of investigators. When debonding occurs, cylindrical cracks are formed at the fiber-matrix interface. The overall strength of the composite becomes dependent upon the sizes and geometries of these cracks. A review of the interface crack can be found in the recent work of Rice [1]. The most common method of modeling interfacial cracks is the dislocation approach, in which the crack is represented by a pile:up of dislocations. The work of Erdogan [2] gives a comprehensive review of fracture problems in composite materials with special emphasis on the linear elastic fracture mechanics models. The theory of dislocations has become an increasingly useful tool for modeling many mechanical properties. Dislocation theory arose in an attempt to explain why the observed yield stresses of crystals are much lower than the theoretical yield stresses. Dislocations were first considered as singularities in continuous media and then later as crystal imperfections. A review of the early works and developments of the concepts of dislocations can be found in [3-5]. The original purpose for developing the theory of dislocations was to model singularities in a continuum, but later, the concepts and the,ones were adapted to a variety of problems in continuum mechanics. Today, the theory of dislocations is an important tool for modeling the continuum elastic-plastic description of deformable sofids. By combining large numbers of dislocations in various ways, it is possible to model many different defects in both homogeneous and nonhomogeneous media. An introduction to the mathematical theory of dislocations can be found in [6]. The first dislocation models utilized straight dislocations of the edge and screw type. The Burgers vector of a straight dislocation remains constant and fixed at all positions along the dislocation line. Dislocations of this type are called Voltera dislocations. Later development led to the introduction of dislocations where the Burgers vector changed in magnitude and/or
direction along the dislocation line. Dislocations of this type are called Somigliana dislocations [7]. We define a special type of Somigliana dislocation where the two ends of the dislocation are joined together to form a circular loop, as shown in Figure 2. This type of dislocation is called a Somigliana ring dislocation, and the stress and displacement fields associated with it are given in [8]. In continuum mechanics, dislocations are used to model internal defects. The defect which is most commonly modeled by dislocations is the planar crack. The planar crack is modeled as a pile-up of straight dislocations, and the macroscopic effects of the crack can be determined by summing the effects of the individual dislocations. The procedure for modeling planar cracks is thoroughly established [9], but this practice is not only limited to planar crack problems. The theory of dislocation pile-ups can also be applied to the cylindrical cracks which may occur in a fiber-matrix debonding problem. Cylindrical cracks may propagate along the fiber-matrix interface in composite materials. Since excessive crack propagation may ultimately lead to failure of the structure, one is very interested in the stress state in the neighboring region surrounding the crack. This problem has been investigated by a number of people who considered interfacial cracks between two isotropic materials [10,11], homogeneous transversely isotropic materials [12], and nonhomogeneous anisotropic materials [13]. These studies modeled the cylindrical crack as a pile-up of Somigliana ring dislocations. Approximate solutions for the stress fields near the crack tip were achieved by numerically solving a set of integral equations. This problem was recently re-examined by Demir et al. [14], who modeled the cylindrical crack as a pile-up of ring dislocations, but also utilized an earlier result they obtained for a single ring dislocation [8]. Demir et al. were able to achieve numerical solutions for the dislocation distributions, the extended stress field, and the stress intensity factors associated with a cylindrical crack. In addition, they were able to show that the pile-up of dislocations can be approximately represented by an equivalent pair of superdislocations, with magnitudes and positions determined toproduce a similar stress field. Since the solution for the single dislocation was already given in [8], and the authors additionally provided an exact expression for the interaction between two Somigliana ring dislocations in [ 15], the superdislocation representation provided a closed form solution for the extended stress field of a cylindrical crack. The next logical step is to analyze a crack-crack interaction problem, establishing the framework for examining a multiple crack problem. The two-crack problem shown in Figure 3 is proposed to investigate the macroscopic effects of the interaction between two collinear cylindrical cracks. The purpose of this study is to determine the total stress field and the stress intensity factors which arise from the interaction between the stress fields of the two cracks. Each crack is represented by a distribution of dislocation loops. From these distributions, we are able to numerically calculate the stress field and stress intensity factors resulting from applied stress in the presence of two cracks. After the final dislocation distributions are determined, we replace the continuous distributions by sets of discrete superdislocations with magnitudes and positions calculated to produce similar extended stress fields. Based on these results, we then propose a simplified procedure to determine the extended stress field surrounding a pair of coupled cylindrical cracks. This procedure involves a series of graphs from which one can select the magnitudes and positions for the sets of superdislocations necessary to produce the extended stress field. Once the stress field has been established, the calculations for the stress intensity factors are performed by summing the
interaction between all superdisloeations representing the cracks. Furthermore, from the superdisloeation representation, we then propose an approximate analytical model to calculate the magnitudes and positions for the sets of superdislocations. Once these expressions are established, they can be used to obtain an approximate expression for the stress intensity factors near the crack tips.
/ Figure 3. The dual, collinear cylindrical crack problem.
2. COLLINEAR CRACKS We consider the case of two cylindrical cracks, both with radius R and collinear axes of syrmnetry as shown in Figure 3. The longer of the two cracks is designated the alpha crack and the remaining crack is designated the beta crack. The length of the alpha crack is 2a and the length of the beta crack is 2h. The distance separating the two inner crack tips is d We define two local cylindrical coordinate systems. The first coordinate system is defined with the origin at the center of the alpha crack, and the second is defined with the origin at the center of the beta crack. It is important to note that the z-axes of both coordinate systems are coincident with each other. The same method that is used by Demir et. al. [14] to model the single cylindrical crack is utilized to model each of the cylindrical cracks in the two-crack problem. In an actual composite, the fiber and surrounding matrix are composed of two dissimilar materials. However, in this model, we consider the case of similar material because it can be treated analytically, which makes it possible to establish a framework for the treatment of the more complicated case of fiber-matrix problems where the solution must be carried out numerically. Therefore, although this case does
not correspond to an actual interracial crack, the method and concepts developed in this paper can be easily applied for the interracial crack case. Since each cylindrical crack induces both an opening mode and a shearing mode [ 10,11], the pile-up must consist of ring dislocations with both radial and axial components for the Burgers vectors. The stress and displacement fields for ring dislocations of this type are given by Demir et ai. [8] to be
{r = {bl[Ai(s, {U}a- {b,[Bi(s,
r)]+ b2[A~_(s,z, R, r)]}{E}+ bl{cr~ z, R, r)]+ b~.[B~_(s,z, R, r)]}{E} + bl {U~
z, R,
(1)~
(1)2
where {r : {err' Cro, ~,, Cr,~} are the non-zero stress components, { U } r : {u, w} are the displacement components, b1 and b2 are the radial and axial components of the Burgers vector, [A~], [A2], [B~], and [B2] are the geometric matrices listed in the Appendix, {E}r= {K(k), E(k), rI(k)} are the elliptic integrals of the first, second and third kind, with
k - 2dr-Rig and g= ~ ( z - s ) ' + ( r + R)', {~r~ is the Lain6 solution also listed in the Appendix, and s is the location of the dislocation along the z-axis. The subscript d indicates quantities corresponding to a single dislocation. The stress-displacement fields for each individual crack are determined by assuming a continuous distribution of dislocations and integrating (1) over the entire distribution, leading to
-a a
-a
h
{o'}~t~ : f({bl(S')~t~ [A~(s' , z', R, r)]+ b2(s')~t~[A2 (s', z', R, r)]}{E} + b~(s')beta{O'~}) ds',
(3) l
-h h
{U}~. : f({b,(s')~to[B,(s', z', R, r)]+ b2(s')~t:[B2(s', z', R, r)l}{E} + b,(s')~t: {U~
(3),
-h
where the subscripts alpha and beta correspond to the alpha and beta crack respectively. The variables s and z are related to the coordinate system with its origin at the center of the alpha crack, and the variables s' and z' are related to the coordinate system with its origin at the center of the beta crack. The problem is now to find the appropriate dislocation distributions that will accurately model the cracks.
2.1. Superposition Principle Because we are dealing with an elastic, homogeneous, isotropic medium, we can use the superposition principle to simplify the current two-crack problem. As shown in Figure 4, the problem of two collinear cylindrical cracks imbedded in an infinite medium under an external stress, p, can be considered to be the sum of three eases.
p
(a)
(b)
(c)
(d)
Figure 4. The superposition principle.
The first ease is that of a single cylindrical crack embedded in an infinite medium which is under no external stress, as shown in Figure 4b. This single crack has the same size and location as the alpha crack and is subject to an internal pressure equal to the magnitude of the original applied external stress, p, plus the radial stress induced by the presence of the beta crack. Additionally, there is a shear stress on the surface of the crack which is equal in magnitude but opposite in sign to the shear stress induced by the beta crack. The second ease is that of a single cylindrical crack embedded in an infinite medium which is subject to no external stress, as shown in Figure 4c. This crack has the same size and location as the beta crack and is subject to an internal pressure equal in magnitude to the original, externally applied stress, p, plus the radial stress induced by the presence of the alpha crack. Additionally, there is a shear stress acting on the surface of the crack which is equal in magnitude but opposite in sign to the shear stress induced by the alpha crack. The third case is that of a continuous medium subject to an external pressure of magnitude p, as shown in Figure 4d. The axisymmetry of the problem is not destroyed by this external loading. When these three situations are superimposed on each other, the result is the original two-crack problem shown in Figure 4a. Since the third ease is simply a homogeneous state of stress, we will only focus on solving the first two problems.
2.2. Boundary and Closure Conditions The traction boundary conditions for the pressurized cylindrical cracks are given by
:-po.(n,
,,.(n,
IzI < a,
Izl < a,
(4)1
oJ,~(R, Z)ot~o : o~,~(R, z)o,p~, ~(R,
Z)al~~ :
~(R,
(4)2
z)ap~,
for the alpha crack, and try(R, z')~,~ : - p - cry(R, Z):iph~,
z' I < h,
tr,~(R, z')~,. = - tr,~(R, z):,~,
Iz'l < h,
(5),
oJ=(R, Z')b.~ = ~ ( R , Z')b,,:,
(sh
oJ,.(n, z')~, = ~ ( R , z')~.,
for the beta crack. It is important to point out that (4)1 and (5)1 state that the radial stress along the surface of either crack must be equal and opposite to the stress caused by the externally applied pressure and the stress induced by the second crack. Similarly, the shear stress on the surface of either crack must be equal and opposite to the shear stress induced by the second crack in order to maintain a crack surface that is free of stress when the three eases are superimposed on one another. The superscripts 1 and 2 indicate quantities in the regions r <_ R and r > R, respectively. Since the fundamental solution (1) satisfies the conditions (4)2 and (5) 2 pointwise, the integral solutions (2)~ and (3)1 also satisfy (4)2 and (5)2. By applying the conditions for crr and cr,~ given in (4)1 and (5)1, to (2)1 and (3)1 we obtain the following set of integral equations
i -a a
b~(s)~s~,[Ll(s, z, R)] ds+ Ib,_(s),~,[/a(s, z, R)] ds=[-p - tr,(R, z')~t:] 2n(1- v) a -a
G
Ib,(~),,,,,,,[,~(~,:, ~)1 d~+ ib~(s)<,,,,,:[,,(~,:, ~)1 d~:[-
--a
-a
<>-,.,(~,z,)~] ~"('-G ") '
(6), (6)2
h
Ib,(s')~,,,[L',(s' z' R)]ds'+ b,_(s')~,,,[L',.(s',z' R)]ds'=[-p-tr,(R, z),~p~,]2n(1- it) ~h
'
"
'
ih
G
(7h
h
I~b,(~),,,,,:['s.,,~(~,, :,, R)] a~,+.f b:(~'),,,,,,,[s_,'4(~', :', R)] ,~'= [- <,-,.,(R,z)<,,,,,:]2,,(1-~ ,,), -h
-h
where
z~ = 4,,(~, z, R)K(k)+ A,,,,~(~, z, R)E(k), I~, = A,.,,(~, z, R)X(k)+ A,,,(~, ~, R)E(k), L~ = A,,(~, z, R)x(k)+ A,~(s, ~, R)E(k), L,: A,.,,(s, z, R)K(k)+ A,.,,(s, z, n)E(k),
(7h
l0
and
L',= A,,,,(~', z', m/C(k)+ A,,,,_(~', ~', R)E(k), L'2 -- A2,11(s', z', R)K(k)-.]-A2,12(st, z', R)E(k), L'~- A~,,,(~', z', R)X(k)+ A~,,~(s', ~', R)E(k), L'4= A4,,,(s' , z', R)K(k)+ A4,,2($', z', R)E(k),
(8h
and the A terms are given in the Appendix. The continuity of the displacements outside of the crack requires that the net dislocation content is zero. This closure condition is given below. a
o
]bl($)alv~
( i S : O,
-a
]'b2(s)atp m d s =
O,
(9)1
~ b2CS')beta ds' -- O.
(9).
-a
h
h
b l (S')beta ds'-- O, -h
-h
2.3. Nondimensionalization
The integral equations (6) and (7) can be nondimensionalized by defining the new variables S a
g= - -
~-
g a
--,
-
r a
r = --,
b]: l{~},
G
b,(s)~,~ :
2n(1- v)pa b ` ( s ) ~ '
~'=~, ~'=~, ~'=~, st
-
(lo)~
,
z t
P
~,= h a
(1oh
G
b~(s)~ = 2n(1- v)ph b~(s')~. For the sake of simplicity, the bars on the symbols are dropped for the remainder of this paper. However, the quantities hereafter should be assumed to be nondimensional. Substitution of the nondimensionalized terms into (5) and (7) results in the set of integral equations shown below.
a
a
fb~(s),,,:,.,[La(s, z, R)] ds+ j'b2(s).~p~[L2(s, z, R)] ds = -1 a
1-
6r(R, z')~t.,,
(11)1
-a a
~b,(s),,,ph,,[La(s,z, R)] ds+
~b~(s)~,,h,,[L4(s, z, R)] ds = -
-a
-a
1
6=(R, z')uu,,
(11)2
1
ha(St)be.taLz t a ,( s[ t ,L
R)] ds'+ fb2(s)~u[/a(s',' z', R)] ds'=- 1-
-a a
f bl(S')~t,,[La(s', z',
6,.(R, z),ap~,,
(12)1
-a a
R)] ds'+ f
-1
b2(s')~ta[La(s', z',
R)] ds ' = -
6=(R, Z),aph,,,
(12)2
-a
where 8r (R, z ' ) ~ is the additional normalized radial stress induced by the beta crack acting on the surface of the alpha crack, 8,, (R, z')~, is the normalized shear stress induced by the beta crack acting on the surface of the alpha crack, 6r (R, z)~pu is the additional normalized radial stress induced by the alpha crack acting on the surface of the beta crack, and 8,, (R, z)o~phais the normalized shear stress induced by the alpha crack acting on the surface of the beta crack. Additionally, the substitution of (10) into (9) results in the nondimensionalized closure conditions a
a
j'ba(s),a~a d s - 0,
j'b2(s),a~, ' d s - 0,
-1 a
-a 1
~b,(s')~ ds'= 0,
fb2( St)betads'- 0.
-a
-a
(13)1 (13)2
Now, the problem is to determine the dislocation distributions ba(s)~pu, b2(S)aph~, b I ( S ?)/~ta' and b2(s')~, which will satisfy (11-13). Since the dislocation distribution for the alpha crack is affected by the induced stress field from the beta crack, and the stress field of the beta crack is affected by the induced stress field from the alpha crack, there is no discrete solution for the dislocation distribution of the alpha crack. 3. SOLUTION TECHNIQUE AND RESULTS The solution to (11-13) is obtained numerically. There are several different numerical procedures which can be performed to determine ba(s) and b2(s), including the Gauss-Chebyshev technique [16] and the piecewise polynomial technique [ 17]. For the single crack problem, Demir et al. [14] utilized a collocation method to determine the dislocation distributions. This collocation method proved to be effective for three collocation points but became increasingly unstable at higher numbers of collocation points due to the high order polynomials used to approximate the distributions. Additionally, the results were very sensitive to the choice of
12 collocation points and required a search routine to determine the optimum points. For these reasons, it was decided that a piecewise quadratic method would be a more effective means of approximating the dislocation distributions. The piecewise quadratic method assumes a quadratic distribution of dislocations over any given interval along the crack. Additionally, the dislocation densities are known to have the following form [ 17]
bi(s)= w(s)B~(s),
(14)
where w(s) is the weight function and B,(s) is a continuous function in the interval weight function w(s), for this problem is given in [ 17] as w ( s ) : (1+ s)-l/2+a(1 - S)I/2-p,
a:
O,
f l : 1.
[-1,1].
The
(15)
A rough analysis of the integral equations (11,12) reveals that B~(s) should be odd functions and B2(s) should be even functions. Each crack is divided into N segments of equal length. The points along each crack where two segments join, and the crack tips, are node points at which we determine the magnitude of the functions B~(s) and B2(s). For a crack, consisting of N segments, there are N + 1 node points which are labeled si for the points corresponding to the alpha crack and s'i for the points corresponding to the beta crack, where i ranges from 1 to N + 1. For each node point, there are associated two unknowns, B~(Si)ot~h~ and B2(s~),dph~, or B~(s'~)b~t~, and B,(s'i)~t ~. Therefore, for each crack, there are 2N + 2 unknowns to solve for. The kernels/~_ 4 and L'~_4 are evaluated at some point along the alpha and beta cracks respectively. The set of integral equations (11-13)1 must be satisfied at any given point, z, along the length of the alpha crack, and the integral equations (11-13)2 must be satisfied at any given point, z', along the length of the beta crack. A set of Gauss points zj, along the alpha crack, and z'j, along the beta crack, wherej ranges from 1 to N, is defined to be the mid-point of each interval. The integral equations (11,12)1 and (11,12)2 are evaluated at each Gauss point zj and z'j respectively, to provide 2N equations per crack which must be satisfied by the functions B~(s~)ozp~, B, (s~)o,p~, B~(s'~)beta' and B2(s'~)b~t~" The last four equations which have to be satisfied are the closure conditions (13). We now have a system of 2N + 2 equations and 2N + 2 unknowns, per crack. For each crack, the set of linear equations are of the form (16)1
[ C ( z j ) L p ~ - { B } ~ = { D } ~ - {8}b,to,
(16)2 who, o
.he
ma
oos,
v
tors
the
magnitudes of the functions B~(s)~ph~ and B2(s)~ph~b~ at each node point, and {D}~r~b~ are
13 the vectors containing the right-hand sides of equations (11,12) without the additional stress terms induced by the opposing crack. The vectors (6}~t~,Qtpha contain the additional stresses induced by the opposing crack. The coefficient matrices have the form --%-l
sl+ 1
si-t
si_l
sl+ 1
st+ l
s~_l
sj_l
~L~(zj)w(s)E.ds ~L2(zj)w(s)E.ds ~L3(zj)w(s)~kds f L4(zj)w(s)7~kds
9 9
dk 0
... ...
. ~
. .
0 ~k
... "'"
(17)
where s~_ ~---}s~+~ is the section of the distribution that we integrate over, zj is the Gauss point for the four kernels, d k is the Lagrange interpolation formula given by [ 17], evaluated for the section of the distribution which has a central point s~, and k indicates which interpolation formula is used (k = 1,2,3). The vectors ( B } o ~ p ~ have the form "
',
(18)
1
where B1(s 1) is the magnitude of the radial component of the distribution function at point sl, and B2(sl) is the magnitude of the axial component of the distribution function at point s~, etc. Due to the interdependence of the stress fields of the two cracks, the solutions for the distributions can not be achieved discretely. An iterative technique is utilized to arrive at the otherwise indeterminate distributions. The iterative solution procedure is as follows: All of the terms in the vector { 8 ) ~ are initially set to zero. A current solution for the vector { B } . ~ in (16)1 is determined by inverting the eoett~cient
matrix [dZj)Lpha and multiplying by the vector {D}ayh~.
14
c.
d.
A new alpha coefficient matrix is constructed, but instead of evaluating the kernels L 1, L~,/~, and L4, at the set of Gauss points zj, we evaluate the kernels at each of the Gauss points z'j on the beta crack, where j ranges from 1 to N. The vector {8}=!~, containing the additional stresses induced on the surface of the beta crack by the alpha crack, is determined by multiplying the new coefficient matrix, C 1+ d + h + z j h by the current solution for the vector {B},I~,.
[(
e. f.
g.
,)]
The current solution for the vector {B}~= in (16)2 is determined. A new beta coefficient matrix is constructed, but instead of evaluating the kernels L'1, L ' , , L'3, and L'4 at the set of Gauss points z~ on the beta crack, we evaluate the kernels at each of the Gauss points z s on the alpha crack, wherej ranges from 1 to N. The vector (~}beto, containing the additional stresses induced on the surface of the alpha crack by the beta crack, is determined by multiplying the new coefficient matrix, [C(-1-d/~h-
/1/~h+ zJ/~h)]
by thecurrent solution for the vector {B}be~. beta
h.
Steps b-g are repeated until convergence is achieved.
The above iteration technique is repeated until the distribution functions for both cracks converge to the point where a complete iteration does not produce any significant change in the distributions. Iteration is continued until 16~]_< 10-4,
e, =
(B~)~,,,,t - (B~)~o~, (B,)r~o~ ~
(19),
i - 1,...2N + 2,
(19)2
where 6i is the percent difference between the magnitude of the distribution Bi at the position z = s,, for the current iteration and the magnitude of the distribution B~ at the position z = s~, for the previous iteration. When the magnitude of 6~ is less than or equal to 10-4 for all of the distribution magnitudes B~, the solution for the distribution function is considered to have converged. Sample results are shown in Figures 5a-5c. A brief examination of the dislocation densities reveals an obvious effect of the second crack. For the single crack case, the dislocation densities, B~(s) and B2(s), are antisymmetric and symmetric with respect to s, respectively [14]. In the presence of the second crack, the dislocation densities lose this syrmnetry due to the interaction between the two stress fields.
15
0.4
0.04
0.2
0.03
~
0
0.02
-0.2
0.01
~
|
!
|
!
|
|
R/a--0.25
|
|
~::_~:0 _.___.
R/a=l.0
','~.
i
j'"
/ l~.a=5.0
-0.4 -0.01
-0.6
h/a=l.0 -0.8
h/a=0.5 ...... 1
i
i
-0.02
I
-0.03
-I -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
I
i
i
i
i
i
i
i
i
-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
S
]
S
(b) 0.3
.
.
.
.
.
.
.
.
0.04
.
.
.
.
.
.
O. 1
.
.
h/a,..-0.5 ...... 0.02
o
....
-0.1
0.01
-0.2
0
-0.3 -0.01 -0.4
h/a=l.O h/a=0.5 ......
-0.02
-0.5
-0.6
. . . . . . . . -1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
-0.03 1
-1 -0.8-0.6-0.4-0.2
S
(c)
0 0.2 0.4 0.6 0.8
1
S
(d)
Figure 5. Regular part of the dislocation densities: (a) radial component (d / a = O.01), (b) axial component (d / a = 0. 01 ), (c) radial component (d / a = 0.1), (d) axial component (d / a = 0.1 ).
By substituting the distribution functions, B , ( s ) ~ ~
and B~ ( s ) ~ ~ ,
into (14) and then
substituting the resulting distributions into (2,3)1, a numerical solution for the extended stress field is obtained. The stress intensity factors K~ and K n near the crack tip are d c f n ~ as
16
:),
r, = Z--I,g
(20)
K H = ~'un~2a(z- a)cr,.,(R, z), where cr~(R, z) and o,.,(R, z) are the normal radial and shear axial stress components of the stress field. The stress intensity factors, K/and KH, normalized by p~/7ra are shown in Figure 6, illustrating the effect of interaction between the two cracks for different values of spacing d / a . For d / a --, oo, the results compare well with those given by Demir et aL [14] for a single crack ~ as R / a becomes large, the crack geometries approach that of two pressurized planar cracks. As R / a approaches infinity, the K: stress intensity factor approaches that of a planar crack and the Ku stress intensity factor approaches zero.
A
3.5
|
!
!
!
i
!
!
|
|
0.02
- h/a=l.O ~ / '~
2.5
o
d/a=O.O1 d/a=O.l d/a=l.O
/ /
....... .......
2 !
1.5
~
1
";
0.5
|
!
|
!
!
!
|
!
.
0.01 0 -0.01
/
-0.02
7/
-0.03 w~
=
-0.04
--=
-0.05
g
-o.o6
"'"
""-"--'--
j
J
/
: /
d/a--O.Ol
/i
: / -/
-0.07 !~[/
.....
o/~=l.O
d/a--infinity
......
h/a=l.O
-o.o8 0
m
m
m
I
1
2
3
4
I
I
5 6 R/a
I
I
I
7
8
9
l0
-0.09 0
1
2
3
4
5 6 R/a
7
8
9
I0
(b) Figure 6. Normalized stress intensity factors" (a) mode I, (b) mode ILl.
3.1. Superdisiocation Representation The continuous distributions of dislocations can be represented as a set of superdislocatio~ with magnitudes equal to the areas under the dislocation curves and at a distance from the center of the crack equal to the first moment of the areas. Justification for this representation is given by Dendr et al. [ 14]. The magnitude of the superdislocations and their respective positions along the z-axis, for pressure loading, are shown in the figure below. The method of representing a continuous distribution of dislocations as a set of d i ~ e superdislocations is just as applicable to the two-crack problem as it is to the sinOe crack problem [14]. As in [18], the magnitudes of bz(s) are very small compared to those of b~(s) and, thus, can
17
be neglected. Therefore, each crack in our two-crack problem can be modeled as a pair of radial superdislocations, as shown in Figure 7.
A
b! "r
beta crack
e
c ii
~i
~,
r
Figure 7. Superdisl~ation r e p r ~ o n 0.45
I
0.4 9~
-
I
I
of the cr~k-crack mteraction problem.
I
I
I
I
I
I
h/a=l.O
0.35
0.3 0
=
0.25
8
0.2
-
0.15 u~
0.1
"~
0.05
d/a=O.O1
~-
cUa--O.1
.......
d/a=l.O ....... d / a mFatity
0 0
I
I
I
I
I
I
I
I
I
1
2
3
4
5 R/a
6
7
8
9
0
Figure 8. Normalized superdislocation magnitudes.
The magnitudes and locations for these superdislocations are determined so as to produce stress fields which are similar to the stress fields produced by the cominuous distributions of
18
dislocations. Figure 8 plots the magnitudes of the two b1 superdislocations versus R / a for dilferent values of crack spacing. Figure 9 shows the positions of the two superdislocations versus R / a for several different values of crack spacing. In general, the magnitudes of the superdislocations increase as the crack spacing decreases and the positions of the superdislocations shift towards the inner crack tips as the crack spacing decreases.
i!
d/a--O.Ol
--
d/a=O.~ d/a=,.O
i
...... .......
d/afinfinity
!
|
|
i
i !
' i
i
i,
h/a=l.O -
4
~.ii
3L 2-
l \" [" '%
h/a=l.0
I ",~.
1 0
I
-l
I
i
I,Va--O.O~
/
/ d/a=O.l / d/a-l.O
[
".
--~
...... ......
~,d/a---infinity---
I
-0.95 -0.9 -0.85 -0.8 -0.75 s/a (position alongcrack)
-0.7
0.8
(a)
0.85 0.9 0.95 s/a (position along crack)
1
(b)
Figure 9. Superdislocation positions: (a) outer crack tip, (b) inner crack tip.
Now, the cylindrical cracks are represented as two sets of Somigliana ring dislocations, as shown in Figure 7. The strengths and locations for these superdislocations are known for given values of R / a and d / a . The interaction force per unit length for a pair of radial ring dislocations is given in [18]. Since we have four superdislocations to model our system, we need to include the interaction of all the superdislocations when we calculate the stress intensity factors. The total interaction force per unit length, acting on superdislocation C, in Figure 7, is determined by summing the interaction forces between C and the remaining three superdislocations. The resulting expression for the total interaction force acting on superdislocation C is given below
Fzc = 2 d l G - v) {b,:,.[A,,,rIk)+A,.,.E(k)L
"~"bI~ID[AI.IIK(I)Jr
+
AI.12E(/)]C-,D },
L (21)
19
where AI.,,, AL,2 are given in the Appendix and evaluated at r : R. The terms K(k ) and E(k) are the complete elliptic integrals of the first and second kind, and biA, b~n, b~c, and bm are the magnitudes of superdislocations A, B, C, and D, respectively. The subscripts CA, CB, and CD indicate that the quantities inside the brackets are evaluated for the corresponding superdislocation interactions. The J-integral force on the crack tip is identical to the PeachKoehler force on the superdislocation representation. Therefore, the interaction force Fzc per unit length is equal to the value of the J x-integral near the crack tip. Close to the crack tip, the behavior is similar to a plane crack with K2(1 - v) J~ =
(22)
2p
By substituting the interaction force Fzc in for ,/i in (22), the approximate stress intensity factor at the inner crack tip of the alpha crack is determined. Substituting (2 l) into (22) and then solving for KI results in the expression K 2 lc
--
G. .(i_
A..:(k) L +
.
(23)
+
+ A,.,:(k)L }.
A similar expression can be constructed for K~o, the stress intensity factor at the outer crack tip of the alpha crack. The mode I stress intensity factor at the inner crack tip of the alpha crack is plotted versus R / a for several different crack-tip spacings in Figure 10. Figure 10 shows that the superdislocation representation is reasonably accurate for crack radii R / a _< 1.0. ..,
2
,
,
,
~
d/a=O.O1 d/a--O.1 ..... 1.6 " d/affil,O ....... . d/a--infinity ...... 1.4 1.8
~
o
1.2 ~ -~.
1 0.8
~=~
0.6 ~
0.4
-~
o.2
~
o
o
0
I
I
I
I
I
I
i
I
I
1
2
3
4
5 R/a
6
7
8
9
10
Figure 10. Mode I stress intensity factors resulting from superdislocation representation.
20
3.2. Approximate Analytical Model As can be seen in Figure 8, the magnitudes of the superdislocations appear to be linear functions of the crack radii, for R / a <_ 1 0, and are not affected by the crack-tip spacing We assume the magnitudes of the b; superdislocations to be linear in R / a and obtain the approximation bt,,, = 0 . 1 5 1 ( R ) ,
1.0.
R/a<_
(24)
When the calculation of the stress intensity factors is performed, it will be necessary to place the proper sign in front of the superdislocation magnitudes. It is apparent from Figure 9, that the locations of the superdislocations along the cracks are not linear functions of R / a. Instead, the positions of the superdislocations near the outer crack tips are approximated by
z~ =
l+a 1
+a s
,
R / a _< 1.0,
(25)
and the positions of the superdislocations near the inner crack tips are approximated by zi = l+a 3
+a 4
,
1.0,
R/a<_
(26)
where al, az, a3, and a 4 are determined to approximate the curves in Figure 9. The values of the coefficients a 1, a2, a3, and a 4 are plotted versus d / a in Figure 11. 0.4
|
l
i
i
a
|
i
!
!
0.3 .o
0.2 a4 0.1 0
..................................................................................................................................................................................................
-0.1
a2
.,N
-0.2 -... ....
r~ -0.3
,
a3
I
I
I
I
I
i
i
!
I
1
2
3
4
5 d/a
6
7
g
9
l0
Figure 11 Coefficients for the analytical model of the superdislocation positions
21
We now have an analytical model to approximate the mode I stress intensity factors at the crack R / a less than or equal to 1.0. The analytical expression for the stress intensity factor at the inner crack tip of the alpha is obtained by substituting (24-26) into (23), resulting in the expression
tips for values of
0.0229
Gfl
g 2 ir
fr
,,
,,, A,.,,Etk)I
--
-
[A,.,,x(k) +
--[a4l.llK(k)-I-a41,12E(k)]CD}
(27)
The negative signs in the above expression are the result of the negative values of b~A and b~c. The expressions for the dislocation positions are embedded in the terms At, ' and A,.,z and therefore are not seen directly in (27). A similar expression can be constructed for the stress intensity factor at the outer crack tip of the alpha crack. Figure 12 compares the stress intensity factors from the analytical model to the results from the superdislocation representation and the exact numerical solution. The approximate analytical model provides the stress intensity factors at the outer crack tips with an error of 11-13% for R / a _< 1.0, d / a > 0.1. At the inner crack tips, the analytical model gives the stress intensity factors with an error of 3-6% for R / a < 1.0, d / a > 0.1.
0.7
h/a=l.O d/a=O.I
0.6 O
.~ ./-'"
o.s
.~
0.6
o~ "~
0.55 0.5 0.45
..,.,
0.35
. . . .
|
|,
!
,
!
!
!
!
|',
!
...]|
..-"I
h/a=1.0
";:~'S~;~-/I
0.4
._~
o.4
"
0.3
o ~perdislocau'on ::~
o
model
.......
0.2
0.2 o
i
0.1 0
i
i
i
i
i
i
i
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.3 0.25
0.15 0.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R/a
R/a
(b) Figure 12. Stress intensity factors resulting from analytical model: (a) d / a = O.1, (b) d / a = 1.O.
22 4. CONCLUSION The crack-crack interaction problem consists of two collinear cylindrical cracks, both with common radii but varying lengths, separated by a known distance between the inner crack tips. The problem is simplified by utilizing the superposition principle. Each crack is modeled as a pileup of Somigliana ring dislocations, where the boundary conditions on either crack are functions of the stress field of its partner. The dislocation distributions for the crack pair are obtained via an iterative solution technique. Once the dislocation distributions are obtained, the stress intensity factors at the crack tips are evaluated. A superdislocation representation of the two-crack problem is constructed and the stress intensity factors arising from this representation are determined. An approximate, analytical model is constructed to provide an explicit solution for the stress intensity factors. The solution to the crack-crack interaction problem suggests that when two cylindrical cracks are sufficiently close to one another, the coupling of the stress fields causes the mode I stress intensity factors to increase at the inner crack tips and the mode II stress intensity factors to decrease at the inner crack tips. The elevated K~ stress intensity factors tend to cause the inner crack tips to propagate towards each other. As the two inner tips grow closer together, the K~ stress intensity factors continue to increase in magnitude, perhaps causing continued crack growth. The superdisloeation representation suggests that replacing the pile-up of dislocations with a set of discrete superdislocations provides a reasonable approxa'mation for the two-crack problem at values of R / a < 1.0 and d / a > 0.1. When the crack radii are larger than 1.0, the results from the superdislocation representation diverge from the exact results. The analytical model provides an explicit expression for approximating the K~ stress intensity factors at the tips of the cracks. A comparison between the analytical model and the results from the superdisloeation and numerical solutions, reveal that the model provides stress intensity factors that are reasonably accurate. The error in the stress intensity factors at the outer crack tips ranges between 11% and 13%, while the error in the stress intensity factors at the inner crack tips is between 3% and 6%, for R / a < 1.0, d / a > 0.1. The analytical model does not accurately predict the stress intensity factors for crack tip spacings less than 0.1. The error in the stress intensity factors, at spacings less than 0.1, could probably be reduced by making the requirement for convergence more stringent, but this would increase the number of iterations and therefore, increase the time required to run the numerical routine. Additionally, at extremely dose spacings, the two dislocations at the inner crack tips may actually intrude on the core cutoff of the opposing dislocation. When the two core cutoffs overlap each other, the stress level near the crack tips increases dramatically. The superdislocation representation places the two inner superdislocations at distances far removed from the actual crack tips, and therefore, eliminates any increases in the stresses near the crack tips. Since the analytical model is based on the superdislocation representation, we would expect the predicted stress intensity factors at the inner crack tips for close crack spacings, to be lower than the actual stress intensity factors.
23 The present work provides a preliminary understanding to the complex problem of crack-crack interaction in cases that have significant potential applications. Further studies could address several, more complex, crack-crack interaction problems.
ACKNOWLEDGMENT
The support of the National Science Foundation under grant number MSS-9302327 is gratefully acknowledged. Special thanks to professors John P. Hirth and Ismail Demir for their valuable suggestions.
REFERENCES
[1] [2]
[3] [4] [5] [6] [7]
IS] [9] [10]
J. R. Rice : Elastic Fracture Mechanics Concepts for Interracial Cracks, Journal of Applied Mechanics 110 (1988) 98-103. F. Erdogan : Fracture Problems in Composite Materials, Engineering Fracture Mechanics, 4 (1972) 811-840. F. R. N. Nabarro : Theory of Crystal Dislocations, Oxford Univ. Press, London (1967). J. P. Hirth and J. Lothe : Theory of Dislocations, 2nd Edition, Wiley, New York (1982). T. Mura : Mechanics of Elastic and Inelastic Solids and Micromechanics of Defects in Solids, Martinus NijhoffPublishers (1982). J. P. Hirth : Introduction to the Mathematical Theory of Dislocations, In: T. Mura (ed.) Mathematical Theory of Dislocations, ASME (1969) 1-24. C. Somigliana : Sulla Teoria delle Distorsioni Elastiche, Atti Acad, naz. Lincii, Rend. CI. Sci. Fis. Mat. Natur 23 (1914) 463-472. I. Demir, J. P. Hirth, and H. M. Zbib : The Somigliana Ring Dislocation, Journal of Elasticity 28 (1992) 223-246. B. A. Bilby and J. D. Eshelby : Dislocations and the Theory of Fracture, In: H. Liebowitz (cd.) Fracture, Vol. I. Academic Press, New York (1968) 99-179. F. Erdogan and T. {)zbek 9 Stress in Fiber-Reinforced Composites with Imperfect Bonding, Journal of Applied Mechanics 36 (1969) 865-869. T. Ozbek and F. Erdogan 9 Some Elasticity Problems in Fiber-Reinforced Composites with Imperfect Bonds, International Journal of Engineering Science 7 (1969) 931-946. H. Kasano, H. Matsumoto and I. Nakahara : A Torsion-free Axisynunetric Problem of a Cylindrical Crack in a Transversely Isotropic Body, Bulletin of JSME 27 (1984) 1323-1332. H. Kasano, H. Matsumoto and I. Nakahara : A Cylindrical Interface Crack in a Nonhomogeneous Anisotropic Elastic Body, Bulletin of JSME 29 (1986) 1973-1981. I. Dcmir, J. P. Hirth, and H. M. Zbib : The Extended Stress Field Around a Cylindrical Crack Using the Theory of Dislocation Pile-ups, International Journal of Engineering Science 30 (1992) 829-845. ..
[ll] [12]
[13] [14]
24 I. Demir, J. P. Hirth, and H. M. Zbib : Interaction Between Two Interfaeial Circular Ring Dislocations, InternationalJournal of Engineering Science 31 (1993) 483-492. F. Erdogan and G. Gupta : On the Numerical Solution of Singular Integral Equations, Quarterly of AppliedMathematics 30 (1972) 525-534. A. Gerasoulis : The Use of Pieeewise Quadratic Polynomials for the Solution of Singular Integral Equations of the Cauehy Type, Comp. & Maths. with Appls. 8 (1982) 15-22. H. M. Zbib, J. P. Hirth, and I. Demir : The Stress Intensity Factor of Cylindrical Cracks, International Journal of Engineering Science 33 (1995) 247-253. C. Atkinson, J. Auila, E. Betz, and R. E. Smelser : Journal of Mechanics andPhysics of Solids 30 (1982) 97-120. I. N. Sneddon and M. Lowengrub: Crack Problems in the Classical Theory of Elasticity, Wiley, New York (1969).
[15] [16] [17]
[18] [19] [20]
APPENDIX
The stress field around the Somigliana ring dislocation is given by:
tr,
Aln Au2 A1~3)
_
0",
A,',,
ab2 I A2"2' Ai,32 A,'33~E(k)] 2~1- ,,) A~, A,,~, ~,~ L.A..o .,Llt"<*)J [A,,,
-2n(1-v)A,'3,
oo, _
[ a2,11 A2,12 A2,13
Gbl
11 for r ( R
2(1- v)R -1
~ 2(1- + "
for
r)R
where the coefficients a~ and A~ are given explicitly by
2b {pg2[b2_v(b2+r2+R2)]_r2R2b2}, A,,,,: i'-~3p A,,2,_-ri-~g-b.[3b2(r2 + R2l+(r 2- R212+2b4 -2 vpg2]- p--~[4b2bRr2/2[r +R2)+4(r2-R2)2-pg2],
25
b(r-R) 2 A'"3=
r'Rg
"
A121 = -b [2b2 + 3r2_ 2 v(b2 + R2 +r 2)], ' gr2R b {(R2_r2)2_r2(R2_r2)+b2(2b2+3R2_4r2)_2v[b4 +2b2(r2+R2)+r4+R4]},
A1,22=r2Rpg
bfr - R)(R' + r 2)
r2Rg(r + R)
A1'23 =
'
A,,, = b [2b2R, +( 1 _2v)pg2] ' ' pg3 R A1,32 :
Rpb2g3{4R2[p( r + R) 2 +g2(r-R)2]-pg2[b 2 +r25R2]},
Al,a3 = 2 v,
b(r-R)
Rg(r + R) ' =rpga[b2(3r2+R2)+(r2-R2)2]--~
+r2-2v(b2+r2+
At42 : -R [pg2(R2_r2) + 8r2b2(b2+r2 - R2)]+__~__l[b2(2b2 + 3r2+ 3R2) + (r2_ R2)2_2 vpg2], p2g3r rKpg t ~ 0) AI,43
A2,,2= ~-,,,:2,2 [ p g [R - )-862R 2<,,+R 2 -r2)]
b2(2b 2 + 3r 2 + 3 R 2 ) + ( r 2 -R2) 2 -2 vpg2],
A~:~ = 0,
A22, ,
:
-I[R'
r2gt
+
r2
R2
)],
1 {b2(2b 2 +3r 2 + 3R2)+(R2 _r2)2 + 2 v[-b2(b 2 +r 2 + 2R2)+ R2(r 2 R 2)]}, :1222 , = r2pg A~,~ = O,
Az,a, : -1 [b2(R2 +3r2+2b2)+(r2_R2)2],
pg3 [
A2,32 = p2-g2 {pg2( R2 -r2 - 2b2)+4b2R[p( r+ R)+ g2(R-r)]}, A~,~ - O, _
-b [b2(r2+R2)+(r2_R2)2],
~ , , , - rpg~
26
,4,.43 = 0
The displacement field is given by
{u} ~ [~:......2 w d = 2n(1-- V) B~2, B~,22
b,(~ 4(1 -2+{~o} - v)R 4(1 - v ) r
--
iI ! ".'3,[rI(,.k)j
62 [&,, 8,_,,,. ~""~ E(k)
- for r ( R, - forr(R,
where the coefficients BLo and B~o are given explicitly by
~,,= '
B,,~ ,
_b
2rRg
{262+3r2_2~b2+2r2+R2+3R2(b2-r2)]}
b 2rRgp[R4+2r4+2b2(b2+2r2)-2wg2+3R2(b2-r2)],
b(r-R) [(r2 + R 2 w2], Bm - 2rRg(r + R) )-2 B121 1 b2+ r2 + R2 -4vR2], , = 2Rg[
B~22 = -1 [b2(b2+2r2)+(r2_R2)2], '
2Rgp
Bl,z3 = 0 B2,,, = ~rg[2b2 + R2 +r2 - 2v(b2 + R2 +r2 )], 1 {(R,__r,~f + b,.[2b,~+3(r,+R,.)]_2vpg2}, B2'1~ = 2rgp B,_,. = 0,
B,~.~=-(l-Z~)~, ,
27 -b r2 - R2], 8~,~ =~[b~+
b(r-R)
B2,23-(1- V)g(R+r ) ,
b-s-z,
g-~/bZ +(r+R) z,
p-b2+(r-R) 2
The Stress Intensity Factors and Interaction Between Cylindrical Cracks in Fiber1Vfatrix Composites S. Close and H.M. Zbib School of Mechanical and Materials Engineering Washington State University, Pullman, WA 99164-2920, USA
The elastic interaction between two cylindrical cracks in an infinite, homogeneous, isotropic, elastic medium is investigated. The cylindrical cracks represent a case of fiber-matrix debonding. We examine the effect of the cracks spacing and size on the stress intensity factors, K I and KII, which result from a pressure loading. Each crack is modeled as a pile-up of Somigliana ring dislocations. The solution is based on analytical expressions obtained earlier for the ring dislocation. Continuous distributions of dislocation densities, modeling the two cracks, are obtained numerically using a piecewise quadratic approximation and an iterative scheme to evaluate the interaction between the two cracks. The analysis provides estimates for the stress intensity factors and their relation to the cracks spacing and size. The analysis also reveals that each crack can be represented by a pair of superdislocations, which leads to the analytical solution. The interaction among the superdislocations also provides a closed form expression for the stress intensity factor.
I. INTRODUCTION Recent advances in the field of material science have led to the development of a class of unconventional materials, such as fibrous composites. In general, composites are composed of strong, lightweight fibers embedded in a matrix. As the development of composites has progressed, the utilization of these materials in industry has become increasingly more common. As with any solid material, there are unavoidable stress raisers present, due to internal defects, which have important implications on the mechanical behavior of the material. Some of the most common defects which have been studied extensively include crystal defects and planar cracks. The development of composite materials has given rise to a series of internal defects which have not been thoroughly investigated. These defects arise from the characteristic, geometric structure of fiber-matrix composites, and include matrix cracking, broken fibers, fiber pull-out, and fibermatrix debonding as shown in Figure 1. Stress singularities caused by cracks, voids or inclusions, may lead to structural failure at stress levels far below the limits estimated using a macromechanical analysis. Therefore, it is necessary to have a comprehensive understanding of fracture initiation and growth, the effects of voids and small inclusions, and their interaction with
each other. In this instance, we examine the interaction between cylindrical cracks; a defect type which might occur in the case of fiber-matrix debonding, as shown in Figure ld.
b2
i Li i I .......
b2
.........
Figure 1. Defects in fibrous composite materials: (a) matrix cracking, (b) broken fibers, (c) fiber pull-out, (d) fiber-matrix debonding.
---
~th
Figure 2. The Somigliana ring dislocation.
The fiber-matrix debonding problem has been previously addressed by a number of investigators. When debonding occurs, cylindrical cracks are formed at the fiber-matrix interface. The overall strength of the composite becomes dependent upon the sizes and geometries of these cracks. A review of the interface crack can be found in the recent work of Rice [1]. The most common method of modeling interfacial cracks is the dislocation approach, in which the crack is represented by a pile:up of dislocations. The work of Erdogan [2] gives a comprehensive review of fracture problems in composite materials with special emphasis on the linear elastic fracture mechanics models. The theory of dislocations has become an increasingly useful tool for modeling many mechanical properties. Dislocation theory arose in an attempt to explain why the observed yield stresses of crystals are much lower than the theoretical yield stresses. Dislocations were first considered as singularities in continuous media and then later as crystal imperfections. A review of the early works and developments of the concepts of dislocations can be found in [3-5]. The original purpose for developing the theory of dislocations was to model singularities in a continuum, but later, the concepts and the,ones were adapted to a variety of problems in continuum mechanics. Today, the theory of dislocations is an important tool for modeling the continuum elastic-plastic description of deformable sofids. By combining large numbers of dislocations in various ways, it is possible to model many different defects in both homogeneous and nonhomogeneous media. An introduction to the mathematical theory of dislocations can be found in [6]. The first dislocation models utilized straight dislocations of the edge and screw type. The Burgers vector of a straight dislocation remains constant and fixed at all positions along the dislocation line. Dislocations of this type are called Voltera dislocations. Later development led to the introduction of dislocations where the Burgers vector changed in magnitude and/or
direction along the dislocation line. Dislocations of this type are called Somigliana dislocations [7]. We define a special type of Somigliana dislocation where the two ends of the dislocation are joined together to form a circular loop, as shown in Figure 2. This type of dislocation is called a Somigliana ring dislocation, and the stress and displacement fields associated with it are given in [8]. In continuum mechanics, dislocations are used to model internal defects. The defect which is most commonly modeled by dislocations is the planar crack. The planar crack is modeled as a pile-up of straight dislocations, and the macroscopic effects of the crack can be determined by summing the effects of the individual dislocations. The procedure for modeling planar cracks is thoroughly established [9], but this practice is not only limited to planar crack problems. The theory of dislocation pile-ups can also be applied to the cylindrical cracks which may occur in a fiber-matrix debonding problem. Cylindrical cracks may propagate along the fiber-matrix interface in composite materials. Since excessive crack propagation may ultimately lead to failure of the structure, one is very interested in the stress state in the neighboring region surrounding the crack. This problem has been investigated by a number of people who considered interfacial cracks between two isotropic materials [10,11], homogeneous transversely isotropic materials [12], and nonhomogeneous anisotropic materials [13]. These studies modeled the cylindrical crack as a pile-up of Somigliana ring dislocations. Approximate solutions for the stress fields near the crack tip were achieved by numerically solving a set of integral equations. This problem was recently re-examined by Demir et al. [14], who modeled the cylindrical crack as a pile-up of ring dislocations, but also utilized an earlier result they obtained for a single ring dislocation [8]. Demir et al. were able to achieve numerical solutions for the dislocation distributions, the extended stress field, and the stress intensity factors associated with a cylindrical crack. In addition, they were able to show that the pile-up of dislocations can be approximately represented by an equivalent pair of superdislocations, with magnitudes and positions determined toproduce a similar stress field. Since the solution for the single dislocation was already given in [8], and the authors additionally provided an exact expression for the interaction between two Somigliana ring dislocations in [ 15], the superdislocation representation provided a closed form solution for the extended stress field of a cylindrical crack. The next logical step is to analyze a crack-crack interaction problem, establishing the framework for examining a multiple crack problem. The two-crack problem shown in Figure 3 is proposed to investigate the macroscopic effects of the interaction between two collinear cylindrical cracks. The purpose of this study is to determine the total stress field and the stress intensity factors which arise from the interaction between the stress fields of the two cracks. Each crack is represented by a distribution of dislocation loops. From these distributions, we are able to numerically calculate the stress field and stress intensity factors resulting from applied stress in the presence of two cracks. After the final dislocation distributions are determined, we replace the continuous distributions by sets of discrete superdislocations with magnitudes and positions calculated to produce similar extended stress fields. Based on these results, we then propose a simplified procedure to determine the extended stress field surrounding a pair of coupled cylindrical cracks. This procedure involves a series of graphs from which one can select the magnitudes and positions for the sets of superdislocations necessary to produce the extended stress field. Once the stress field has been established, the calculations for the stress intensity factors are performed by summing the
interaction between all superdisloeations representing the cracks. Furthermore, from the superdisloeation representation, we then propose an approximate analytical model to calculate the magnitudes and positions for the sets of superdislocations. Once these expressions are established, they can be used to obtain an approximate expression for the stress intensity factors near the crack tips.
/ Figure 3. The dual, collinear cylindrical crack problem.
2. COLLINEAR CRACKS We consider the case of two cylindrical cracks, both with radius R and collinear axes of syrmnetry as shown in Figure 3. The longer of the two cracks is designated the alpha crack and the remaining crack is designated the beta crack. The length of the alpha crack is 2a and the length of the beta crack is 2h. The distance separating the two inner crack tips is d We define two local cylindrical coordinate systems. The first coordinate system is defined with the origin at the center of the alpha crack, and the second is defined with the origin at the center of the beta crack. It is important to note that the z-axes of both coordinate systems are coincident with each other. The same method that is used by Demir et. al. [14] to model the single cylindrical crack is utilized to model each of the cylindrical cracks in the two-crack problem. In an actual composite, the fiber and surrounding matrix are composed of two dissimilar materials. However, in this model, we consider the case of similar material because it can be treated analytically, which makes it possible to establish a framework for the treatment of the more complicated case of fiber-matrix problems where the solution must be carried out numerically. Therefore, although this case does
not correspond to an actual interracial crack, the method and concepts developed in this paper can be easily applied for the interracial crack case. Since each cylindrical crack induces both an opening mode and a shearing mode [ 10,11], the pile-up must consist of ring dislocations with both radial and axial components for the Burgers vectors. The stress and displacement fields for ring dislocations of this type are given by Demir et ai. [8] to be
{r = {bl[Ai(s, {U}a- {b,[Bi(s,
r)]+ b2[A~_(s,z, R, r)]}{E}+ bl{cr~ z, R, r)]+ b~.[B~_(s,z, R, r)]}{E} + bl {U~
z, R,
(1)~
(1)2
where {r : {err' Cro, ~,, Cr,~} are the non-zero stress components, { U } r : {u, w} are the displacement components, b1 and b2 are the radial and axial components of the Burgers vector, [A~], [A2], [B~], and [B2] are the geometric matrices listed in the Appendix, {E}r= {K(k), E(k), rI(k)} are the elliptic integrals of the first, second and third kind, with
k - 2dr-Rig and g= ~ ( z - s ) ' + ( r + R)', {~r~ is the Lain6 solution also listed in the Appendix, and s is the location of the dislocation along the z-axis. The subscript d indicates quantities corresponding to a single dislocation. The stress-displacement fields for each individual crack are determined by assuming a continuous distribution of dislocations and integrating (1) over the entire distribution, leading to
-a a
-a
h
{o'}~t~ : f({bl(S')~t~ [A~(s' , z', R, r)]+ b2(s')~t~[A2 (s', z', R, r)]}{E} + b~(s')beta{O'~}) ds',
(3) l
-h h
{U}~. : f({b,(s')~to[B,(s', z', R, r)]+ b2(s')~t:[B2(s', z', R, r)l}{E} + b,(s')~t: {U~
(3),
-h
where the subscripts alpha and beta correspond to the alpha and beta crack respectively. The variables s and z are related to the coordinate system with its origin at the center of the alpha crack, and the variables s' and z' are related to the coordinate system with its origin at the center of the beta crack. The problem is now to find the appropriate dislocation distributions that will accurately model the cracks.
2.1. Superposition Principle Because we are dealing with an elastic, homogeneous, isotropic medium, we can use the superposition principle to simplify the current two-crack problem. As shown in Figure 4, the problem of two collinear cylindrical cracks imbedded in an infinite medium under an external stress, p, can be considered to be the sum of three eases.
p
(a)
(b)
(c)
(d)
Figure 4. The superposition principle.
The first ease is that of a single cylindrical crack embedded in an infinite medium which is under no external stress, as shown in Figure 4b. This single crack has the same size and location as the alpha crack and is subject to an internal pressure equal to the magnitude of the original applied external stress, p, plus the radial stress induced by the presence of the beta crack. Additionally, there is a shear stress on the surface of the crack which is equal in magnitude but opposite in sign to the shear stress induced by the beta crack. The second ease is that of a single cylindrical crack embedded in an infinite medium which is subject to no external stress, as shown in Figure 4c. This crack has the same size and location as the beta crack and is subject to an internal pressure equal in magnitude to the original, externally applied stress, p, plus the radial stress induced by the presence of the alpha crack. Additionally, there is a shear stress acting on the surface of the crack which is equal in magnitude but opposite in sign to the shear stress induced by the alpha crack. The third case is that of a continuous medium subject to an external pressure of magnitude p, as shown in Figure 4d. The axisymmetry of the problem is not destroyed by this external loading. When these three situations are superimposed on each other, the result is the original two-crack problem shown in Figure 4a. Since the third ease is simply a homogeneous state of stress, we will only focus on solving the first two problems.
2.2. Boundary and Closure Conditions The traction boundary conditions for the pressurized cylindrical cracks are given by
:-po.(n,
,,.(n,
IzI < a,
Izl < a,
(4)1
oJ,~(R, Z)ot~o : o~,~(R, z)o,p~, ~(R,
Z)al~~ :
~(R,
(4)2
z)ap~,
for the alpha crack, and try(R, z')~,~ : - p - cry(R, Z):iph~,
z' I < h,
tr,~(R, z')~,. = - tr,~(R, z):,~,
Iz'l < h,
(5),
oJ=(R, Z')b.~ = ~ ( R , Z')b,,:,
(sh
oJ,.(n, z')~, = ~ ( R , z')~.,
for the beta crack. It is important to point out that (4)1 and (5)1 state that the radial stress along the surface of either crack must be equal and opposite to the stress caused by the externally applied pressure and the stress induced by the second crack. Similarly, the shear stress on the surface of either crack must be equal and opposite to the shear stress induced by the second crack in order to maintain a crack surface that is free of stress when the three eases are superimposed on one another. The superscripts 1 and 2 indicate quantities in the regions r <_ R and r > R, respectively. Since the fundamental solution (1) satisfies the conditions (4)2 and (5) 2 pointwise, the integral solutions (2)~ and (3)1 also satisfy (4)2 and (5)2. By applying the conditions for crr and cr,~ given in (4)1 and (5)1, to (2)1 and (3)1 we obtain the following set of integral equations
i -a a
b~(s)~s~,[Ll(s, z, R)] ds+ Ib,_(s),~,[/a(s, z, R)] ds=[-p - tr,(R, z')~t:] 2n(1- v) a -a
G
Ib,(~),,,,,,,[,~(~,:, ~)1 d~+ ib~(s)<,,,,,:[,,(~,:, ~)1 d~:[-
--a
-a
<>-,.,(~,z,)~] ~"('-G ") '
(6), (6)2
h
Ib,(s')~,,,[L',(s' z' R)]ds'+ b,_(s')~,,,[L',.(s',z' R)]ds'=[-p-tr,(R, z),~p~,]2n(1- it) ~h
'
"
'
ih
G
(7h
h
I~b,(~),,,,,:['s.,,~(~,, :,, R)] a~,+.f b:(~'),,,,,,,[s_,'4(~', :', R)] ,~'= [- <,-,.,(R,z)<,,,,,:]2,,(1-~ ,,), -h
-h
where
z~ = 4,,(~, z, R)K(k)+ A,,,,~(~, z, R)E(k), I~, = A,.,,(~, z, R)X(k)+ A,,,(~, ~, R)E(k), L~ = A,,(~, z, R)x(k)+ A,~(s, ~, R)E(k), L,: A,.,,(s, z, R)K(k)+ A,.,,(s, z, n)E(k),
(7h
l0
and
L',= A,,,,(~', z', m/C(k)+ A,,,,_(~', ~', R)E(k), L'2 -- A2,11(s', z', R)K(k)-.]-A2,12(st, z', R)E(k), L'~- A~,,,(~', z', R)X(k)+ A~,,~(s', ~', R)E(k), L'4= A4,,,(s' , z', R)K(k)+ A4,,2($', z', R)E(k),
(8h
and the A terms are given in the Appendix. The continuity of the displacements outside of the crack requires that the net dislocation content is zero. This closure condition is given below. a
o
]bl($)alv~
( i S : O,
-a
]'b2(s)atp m d s =
O,
(9)1
~ b2CS')beta ds' -- O.
(9).
-a
h
h
b l (S')beta ds'-- O, -h
-h
2.3. Nondimensionalization
The integral equations (6) and (7) can be nondimensionalized by defining the new variables S a
g= - -
~-
g a
--,
-
r a
r = --,
b]: l{~},
G
b,(s)~,~ :
2n(1- v)pa b ` ( s ) ~ '
~'=~, ~'=~, ~'=~, st
-
(lo)~
,
z t
P
~,= h a
(1oh
G
b~(s)~ = 2n(1- v)ph b~(s')~. For the sake of simplicity, the bars on the symbols are dropped for the remainder of this paper. However, the quantities hereafter should be assumed to be nondimensional. Substitution of the nondimensionalized terms into (5) and (7) results in the set of integral equations shown below.
a
a
fb~(s),,,:,.,[La(s, z, R)] ds+ j'b2(s).~p~[L2(s, z, R)] ds = -1 a
1-
6r(R, z')~t.,,
(11)1
-a a
~b,(s),,,ph,,[La(s,z, R)] ds+
~b~(s)~,,h,,[L4(s, z, R)] ds = -
-a
-a
1
6=(R, z')uu,,
(11)2
1
ha(St)be.taLz t a ,( s[ t ,L
R)] ds'+ fb2(s)~u[/a(s',' z', R)] ds'=- 1-
-a a
f bl(S')~t,,[La(s', z',
6,.(R, z),ap~,,
(12)1
-a a
R)] ds'+ f
-1
b2(s')~ta[La(s', z',
R)] ds ' = -
6=(R, Z),aph,,,
(12)2
-a
where 8r (R, z ' ) ~ is the additional normalized radial stress induced by the beta crack acting on the surface of the alpha crack, 8,, (R, z')~, is the normalized shear stress induced by the beta crack acting on the surface of the alpha crack, 6r (R, z)~pu is the additional normalized radial stress induced by the alpha crack acting on the surface of the beta crack, and 8,, (R, z)o~phais the normalized shear stress induced by the alpha crack acting on the surface of the beta crack. Additionally, the substitution of (10) into (9) results in the nondimensionalized closure conditions a
a
j'ba(s),a~a d s - 0,
j'b2(s),a~, ' d s - 0,
-1 a
-a 1
~b,(s')~ ds'= 0,
fb2( St)betads'- 0.
-a
-a
(13)1 (13)2
Now, the problem is to determine the dislocation distributions ba(s)~pu, b2(S)aph~, b I ( S ?)/~ta' and b2(s')~, which will satisfy (11-13). Since the dislocation distribution for the alpha crack is affected by the induced stress field from the beta crack, and the stress field of the beta crack is affected by the induced stress field from the alpha crack, there is no discrete solution for the dislocation distribution of the alpha crack. 3. SOLUTION TECHNIQUE AND RESULTS The solution to (11-13) is obtained numerically. There are several different numerical procedures which can be performed to determine ba(s) and b2(s), including the Gauss-Chebyshev technique [16] and the piecewise polynomial technique [ 17]. For the single crack problem, Demir et al. [14] utilized a collocation method to determine the dislocation distributions. This collocation method proved to be effective for three collocation points but became increasingly unstable at higher numbers of collocation points due to the high order polynomials used to approximate the distributions. Additionally, the results were very sensitive to the choice of
12 collocation points and required a search routine to determine the optimum points. For these reasons, it was decided that a piecewise quadratic method would be a more effective means of approximating the dislocation distributions. The piecewise quadratic method assumes a quadratic distribution of dislocations over any given interval along the crack. Additionally, the dislocation densities are known to have the following form [ 17]
bi(s)= w(s)B~(s),
(14)
where w(s) is the weight function and B,(s) is a continuous function in the interval weight function w(s), for this problem is given in [ 17] as w ( s ) : (1+ s)-l/2+a(1 - S)I/2-p,
a:
O,
f l : 1.
[-1,1].
The
(15)
A rough analysis of the integral equations (11,12) reveals that B~(s) should be odd functions and B2(s) should be even functions. Each crack is divided into N segments of equal length. The points along each crack where two segments join, and the crack tips, are node points at which we determine the magnitude of the functions B~(s) and B2(s). For a crack, consisting of N segments, there are N + 1 node points which are labeled si for the points corresponding to the alpha crack and s'i for the points corresponding to the beta crack, where i ranges from 1 to N + 1. For each node point, there are associated two unknowns, B~(Si)ot~h~ and B2(s~),dph~, or B~(s'~)b~t~, and B,(s'i)~t ~. Therefore, for each crack, there are 2N + 2 unknowns to solve for. The kernels/~_ 4 and L'~_4 are evaluated at some point along the alpha and beta cracks respectively. The set of integral equations (11-13)1 must be satisfied at any given point, z, along the length of the alpha crack, and the integral equations (11-13)2 must be satisfied at any given point, z', along the length of the beta crack. A set of Gauss points zj, along the alpha crack, and z'j, along the beta crack, wherej ranges from 1 to N, is defined to be the mid-point of each interval. The integral equations (11,12)1 and (11,12)2 are evaluated at each Gauss point zj and z'j respectively, to provide 2N equations per crack which must be satisfied by the functions B~(s~)ozp~, B, (s~)o,p~, B~(s'~)beta' and B2(s'~)b~t~" The last four equations which have to be satisfied are the closure conditions (13). We now have a system of 2N + 2 equations and 2N + 2 unknowns, per crack. For each crack, the set of linear equations are of the form (16)1
[ C ( z j ) L p ~ - { B } ~ = { D } ~ - {8}b,to,
(16)2 who, o
.he
ma
oos,
v
tors
the
magnitudes of the functions B~(s)~ph~ and B2(s)~ph~b~ at each node point, and {D}~r~b~ are
13 the vectors containing the right-hand sides of equations (11,12) without the additional stress terms induced by the opposing crack. The vectors (6}~t~,Qtpha contain the additional stresses induced by the opposing crack. The coefficient matrices have the form --%-l
sl+ 1
si-t
si_l
sl+ 1
st+ l
s~_l
sj_l
~L~(zj)w(s)E.ds ~L2(zj)w(s)E.ds ~L3(zj)w(s)~kds f L4(zj)w(s)7~kds
9 9
dk 0
... ...
. ~
. .
0 ~k
... "'"
(17)
where s~_ ~---}s~+~ is the section of the distribution that we integrate over, zj is the Gauss point for the four kernels, d k is the Lagrange interpolation formula given by [ 17], evaluated for the section of the distribution which has a central point s~, and k indicates which interpolation formula is used (k = 1,2,3). The vectors ( B } o ~ p ~ have the form "
',
(18)
1
where B1(s 1) is the magnitude of the radial component of the distribution function at point sl, and B2(sl) is the magnitude of the axial component of the distribution function at point s~, etc. Due to the interdependence of the stress fields of the two cracks, the solutions for the distributions can not be achieved discretely. An iterative technique is utilized to arrive at the otherwise indeterminate distributions. The iterative solution procedure is as follows: All of the terms in the vector { 8 ) ~ are initially set to zero. A current solution for the vector { B } . ~ in (16)1 is determined by inverting the eoett~cient
matrix [dZj)Lpha and multiplying by the vector {D}ayh~.
14
c.
d.
A new alpha coefficient matrix is constructed, but instead of evaluating the kernels L 1, L~,/~, and L4, at the set of Gauss points zj, we evaluate the kernels at each of the Gauss points z'j on the beta crack, where j ranges from 1 to N. The vector {8}=!~, containing the additional stresses induced on the surface of the beta crack by the alpha crack, is determined by multiplying the new coefficient matrix, C 1+ d + h + z j h by the current solution for the vector {B},I~,.
[(
e. f.
g.
,)]
The current solution for the vector {B}~= in (16)2 is determined. A new beta coefficient matrix is constructed, but instead of evaluating the kernels L'1, L ' , , L'3, and L'4 at the set of Gauss points z~ on the beta crack, we evaluate the kernels at each of the Gauss points z s on the alpha crack, wherej ranges from 1 to N. The vector (~}beto, containing the additional stresses induced on the surface of the alpha crack by the beta crack, is determined by multiplying the new coefficient matrix, [C(-1-d/~h-
/1/~h+ zJ/~h)]
by thecurrent solution for the vector {B}be~. beta
h.
Steps b-g are repeated until convergence is achieved.
The above iteration technique is repeated until the distribution functions for both cracks converge to the point where a complete iteration does not produce any significant change in the distributions. Iteration is continued until 16~]_< 10-4,
e, =
(B~)~,,,,t - (B~)~o~, (B,)r~o~ ~
(19),
i - 1,...2N + 2,
(19)2
where 6i is the percent difference between the magnitude of the distribution Bi at the position z = s,, for the current iteration and the magnitude of the distribution B~ at the position z = s~, for the previous iteration. When the magnitude of 6~ is less than or equal to 10-4 for all of the distribution magnitudes B~, the solution for the distribution function is considered to have converged. Sample results are shown in Figures 5a-5c. A brief examination of the dislocation densities reveals an obvious effect of the second crack. For the single crack case, the dislocation densities, B~(s) and B2(s), are antisymmetric and symmetric with respect to s, respectively [14]. In the presence of the second crack, the dislocation densities lose this syrmnetry due to the interaction between the two stress fields.
15
0.4
0.04
0.2
0.03
~
0
0.02
-0.2
0.01
~
|
!
|
!
|
|
R/a--0.25
|
|
~::_~:0 _.___.
R/a=l.0
','~.
i
j'"
/ l~.a=5.0
-0.4 -0.01
-0.6
h/a=l.0 -0.8
h/a=0.5 ...... 1
i
i
-0.02
I
-0.03
-I -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
I
i
i
i
i
i
i
i
i
-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
S
]
S
(b) 0.3
.
.
.
.
.
.
.
.
0.04
.
.
.
.
.
.
O. 1
.
.
h/a,..-0.5 ...... 0.02
o
....
-0.1
0.01
-0.2
0
-0.3 -0.01 -0.4
h/a=l.O h/a=0.5 ......
-0.02
-0.5
-0.6
. . . . . . . . -1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
-0.03 1
-1 -0.8-0.6-0.4-0.2
S
(c)
0 0.2 0.4 0.6 0.8
1
S
(d)
Figure 5. Regular part of the dislocation densities: (a) radial component (d / a = O.01), (b) axial component (d / a = 0. 01 ), (c) radial component (d / a = 0.1), (d) axial component (d / a = 0.1 ).
By substituting the distribution functions, B , ( s ) ~ ~
and B~ ( s ) ~ ~ ,
into (14) and then
substituting the resulting distributions into (2,3)1, a numerical solution for the extended stress field is obtained. The stress intensity factors K~ and K n near the crack tip are d c f n ~ as
16
:),
r, = Z--I,g
(20)
K H = ~'un~2a(z- a)cr,.,(R, z), where cr~(R, z) and o,.,(R, z) are the normal radial and shear axial stress components of the stress field. The stress intensity factors, K/and KH, normalized by p~/7ra are shown in Figure 6, illustrating the effect of interaction between the two cracks for different values of spacing d / a . For d / a --, oo, the results compare well with those given by Demir et aL [14] for a single crack ~ as R / a becomes large, the crack geometries approach that of two pressurized planar cracks. As R / a approaches infinity, the K: stress intensity factor approaches that of a planar crack and the Ku stress intensity factor approaches zero.
A
3.5
|
!
!
!
i
!
!
|
|
0.02
- h/a=l.O ~ / '~
2.5
o
d/a=O.O1 d/a=O.l d/a=l.O
/ /
....... .......
2 !
1.5
~
1
";
0.5
|
!
|
!
!
!
|
!
.
0.01 0 -0.01
/
-0.02
7/
-0.03 w~
=
-0.04
--=
-0.05
g
-o.o6
"'"
""-"--'--
j
J
/
: /
d/a--O.Ol
/i
: / -/
-0.07 !~[/
.....
o/~=l.O
d/a--infinity
......
h/a=l.O
-o.o8 0
m
m
m
I
1
2
3
4
I
I
5 6 R/a
I
I
I
7
8
9
l0
-0.09 0
1
2
3
4
5 6 R/a
7
8
9
I0
(b) Figure 6. Normalized stress intensity factors" (a) mode I, (b) mode ILl.
3.1. Superdisiocation Representation The continuous distributions of dislocations can be represented as a set of superdislocatio~ with magnitudes equal to the areas under the dislocation curves and at a distance from the center of the crack equal to the first moment of the areas. Justification for this representation is given by Dendr et al. [ 14]. The magnitude of the superdislocations and their respective positions along the z-axis, for pressure loading, are shown in the figure below. The method of representing a continuous distribution of dislocations as a set of d i ~ e superdislocations is just as applicable to the two-crack problem as it is to the sinOe crack problem [14]. As in [18], the magnitudes of bz(s) are very small compared to those of b~(s) and, thus, can
17
be neglected. Therefore, each crack in our two-crack problem can be modeled as a pair of radial superdislocations, as shown in Figure 7.
A
b! "r
beta crack
e
c ii
~i
~,
r
Figure 7. Superdisl~ation r e p r ~ o n 0.45
I
0.4 9~
-
I
I
of the cr~k-crack mteraction problem.
I
I
I
I
I
I
h/a=l.O
0.35
0.3 0
=
0.25
8
0.2
-
0.15 u~
0.1
"~
0.05
d/a=O.O1
~-
cUa--O.1
.......
d/a=l.O ....... d / a mFatity
0 0
I
I
I
I
I
I
I
I
I
1
2
3
4
5 R/a
6
7
8
9
0
Figure 8. Normalized superdislocation magnitudes.
The magnitudes and locations for these superdislocations are determined so as to produce stress fields which are similar to the stress fields produced by the cominuous distributions of
18
dislocations. Figure 8 plots the magnitudes of the two b1 superdislocations versus R / a for dilferent values of crack spacing. Figure 9 shows the positions of the two superdislocations versus R / a for several different values of crack spacing. In general, the magnitudes of the superdislocations increase as the crack spacing decreases and the positions of the superdislocations shift towards the inner crack tips as the crack spacing decreases.
i!
d/a--O.Ol
--
d/a=O.~ d/a=,.O
i
...... .......
d/afinfinity
!
|
|
i
i !
' i
i
i,
h/a=l.O -
4
~.ii
3L 2-
l \" [" '%
h/a=l.0
I ",~.
1 0
I
-l
I
i
I,Va--O.O~
/
/ d/a=O.l / d/a-l.O
[
".
--~
...... ......
~,d/a---infinity---
I
-0.95 -0.9 -0.85 -0.8 -0.75 s/a (position alongcrack)
-0.7
0.8
(a)
0.85 0.9 0.95 s/a (position along crack)
1
(b)
Figure 9. Superdislocation positions: (a) outer crack tip, (b) inner crack tip.
Now, the cylindrical cracks are represented as two sets of Somigliana ring dislocations, as shown in Figure 7. The strengths and locations for these superdislocations are known for given values of R / a and d / a . The interaction force per unit length for a pair of radial ring dislocations is given in [18]. Since we have four superdislocations to model our system, we need to include the interaction of all the superdislocations when we calculate the stress intensity factors. The total interaction force per unit length, acting on superdislocation C, in Figure 7, is determined by summing the interaction forces between C and the remaining three superdislocations. The resulting expression for the total interaction force acting on superdislocation C is given below
Fzc = 2 d l G - v) {b,:,.[A,,,rIk)+A,.,.E(k)L
"~"bI~ID[AI.IIK(I)Jr
+
AI.12E(/)]C-,D },
L (21)
19
where AI.,,, AL,2 are given in the Appendix and evaluated at r : R. The terms K(k ) and E(k) are the complete elliptic integrals of the first and second kind, and biA, b~n, b~c, and bm are the magnitudes of superdislocations A, B, C, and D, respectively. The subscripts CA, CB, and CD indicate that the quantities inside the brackets are evaluated for the corresponding superdislocation interactions. The J-integral force on the crack tip is identical to the PeachKoehler force on the superdislocation representation. Therefore, the interaction force Fzc per unit length is equal to the value of the J x-integral near the crack tip. Close to the crack tip, the behavior is similar to a plane crack with K2(1 - v) J~ =
(22)
2p
By substituting the interaction force Fzc in for ,/i in (22), the approximate stress intensity factor at the inner crack tip of the alpha crack is determined. Substituting (2 l) into (22) and then solving for KI results in the expression K 2 lc
--
G. .(i_
A..:(k) L +
.
(23)
+
+ A,.,:(k)L }.
A similar expression can be constructed for K~o, the stress intensity factor at the outer crack tip of the alpha crack. The mode I stress intensity factor at the inner crack tip of the alpha crack is plotted versus R / a for several different crack-tip spacings in Figure 10. Figure 10 shows that the superdislocation representation is reasonably accurate for crack radii R / a _< 1.0. ..,
2
,
,
,
~
d/a=O.O1 d/a--O.1 ..... 1.6 " d/affil,O ....... . d/a--infinity ...... 1.4 1.8
~
o
1.2 ~ -~.
1 0.8
~=~
0.6 ~
0.4
-~
o.2
~
o
o
0
I
I
I
I
I
I
i
I
I
1
2
3
4
5 R/a
6
7
8
9
10
Figure 10. Mode I stress intensity factors resulting from superdislocation representation.
20
3.2. Approximate Analytical Model As can be seen in Figure 8, the magnitudes of the superdislocations appear to be linear functions of the crack radii, for R / a <_ 1 0, and are not affected by the crack-tip spacing We assume the magnitudes of the b; superdislocations to be linear in R / a and obtain the approximation bt,,, = 0 . 1 5 1 ( R ) ,
1.0.
R/a<_
(24)
When the calculation of the stress intensity factors is performed, it will be necessary to place the proper sign in front of the superdislocation magnitudes. It is apparent from Figure 9, that the locations of the superdislocations along the cracks are not linear functions of R / a. Instead, the positions of the superdislocations near the outer crack tips are approximated by
z~ =
l+a 1
+a s
,
R / a _< 1.0,
(25)
and the positions of the superdislocations near the inner crack tips are approximated by zi = l+a 3
+a 4
,
1.0,
R/a<_
(26)
where al, az, a3, and a 4 are determined to approximate the curves in Figure 9. The values of the coefficients a 1, a2, a3, and a 4 are plotted versus d / a in Figure 11. 0.4
|
l
i
i
a
|
i
!
!
0.3 .o
0.2 a4 0.1 0
..................................................................................................................................................................................................
-0.1
a2
.,N
-0.2 -... ....
r~ -0.3
,
a3
I
I
I
I
I
i
i
!
I
1
2
3
4
5 d/a
6
7
g
9
l0
Figure 11 Coefficients for the analytical model of the superdislocation positions
21
We now have an analytical model to approximate the mode I stress intensity factors at the crack R / a less than or equal to 1.0. The analytical expression for the stress intensity factor at the inner crack tip of the alpha is obtained by substituting (24-26) into (23), resulting in the expression
tips for values of
0.0229
Gfl
g 2 ir
fr
,,
,,, A,.,,Etk)I
--
-
[A,.,,x(k) +
--[a4l.llK(k)-I-a41,12E(k)]CD}
(27)
The negative signs in the above expression are the result of the negative values of b~A and b~c. The expressions for the dislocation positions are embedded in the terms At, ' and A,.,z and therefore are not seen directly in (27). A similar expression can be constructed for the stress intensity factor at the outer crack tip of the alpha crack. Figure 12 compares the stress intensity factors from the analytical model to the results from the superdislocation representation and the exact numerical solution. The approximate analytical model provides the stress intensity factors at the outer crack tips with an error of 11-13% for R / a _< 1.0, d / a > 0.1. At the inner crack tips, the analytical model gives the stress intensity factors with an error of 3-6% for R / a < 1.0, d / a > 0.1.
0.7
h/a=l.O d/a=O.I
0.6 O
.~ ./-'"
o.s
.~
0.6
o~ "~
0.55 0.5 0.45
..,.,
0.35
. . . .
|
|,
!
,
!
!
!
!
|',
!
...]|
..-"I
h/a=1.0
";:~'S~;~-/I
0.4
._~
o.4
"
0.3
o ~perdislocau'on ::~
o
model
.......
0.2
0.2 o
i
0.1 0
i
i
i
i
i
i
i
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.3 0.25
0.15 0.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R/a
R/a
(b) Figure 12. Stress intensity factors resulting from analytical model: (a) d / a = O.1, (b) d / a = 1.O.
22 4. CONCLUSION The crack-crack interaction problem consists of two collinear cylindrical cracks, both with common radii but varying lengths, separated by a known distance between the inner crack tips. The problem is simplified by utilizing the superposition principle. Each crack is modeled as a pileup of Somigliana ring dislocations, where the boundary conditions on either crack are functions of the stress field of its partner. The dislocation distributions for the crack pair are obtained via an iterative solution technique. Once the dislocation distributions are obtained, the stress intensity factors at the crack tips are evaluated. A superdislocation representation of the two-crack problem is constructed and the stress intensity factors arising from this representation are determined. An approximate, analytical model is constructed to provide an explicit solution for the stress intensity factors. The solution to the crack-crack interaction problem suggests that when two cylindrical cracks are sufficiently close to one another, the coupling of the stress fields causes the mode I stress intensity factors to increase at the inner crack tips and the mode II stress intensity factors to decrease at the inner crack tips. The elevated K~ stress intensity factors tend to cause the inner crack tips to propagate towards each other. As the two inner tips grow closer together, the K~ stress intensity factors continue to increase in magnitude, perhaps causing continued crack growth. The superdisloeation representation suggests that replacing the pile-up of dislocations with a set of discrete superdislocations provides a reasonable approxa'mation for the two-crack problem at values of R / a < 1.0 and d / a > 0.1. When the crack radii are larger than 1.0, the results from the superdislocation representation diverge from the exact results. The analytical model provides an explicit expression for approximating the K~ stress intensity factors at the tips of the cracks. A comparison between the analytical model and the results from the superdisloeation and numerical solutions, reveal that the model provides stress intensity factors that are reasonably accurate. The error in the stress intensity factors at the outer crack tips ranges between 11% and 13%, while the error in the stress intensity factors at the inner crack tips is between 3% and 6%, for R / a < 1.0, d / a > 0.1. The analytical model does not accurately predict the stress intensity factors for crack tip spacings less than 0.1. The error in the stress intensity factors, at spacings less than 0.1, could probably be reduced by making the requirement for convergence more stringent, but this would increase the number of iterations and therefore, increase the time required to run the numerical routine. Additionally, at extremely dose spacings, the two dislocations at the inner crack tips may actually intrude on the core cutoff of the opposing dislocation. When the two core cutoffs overlap each other, the stress level near the crack tips increases dramatically. The superdislocation representation places the two inner superdislocations at distances far removed from the actual crack tips, and therefore, eliminates any increases in the stresses near the crack tips. Since the analytical model is based on the superdislocation representation, we would expect the predicted stress intensity factors at the inner crack tips for close crack spacings, to be lower than the actual stress intensity factors.
23 The present work provides a preliminary understanding to the complex problem of crack-crack interaction in cases that have significant potential applications. Further studies could address several, more complex, crack-crack interaction problems.
ACKNOWLEDGMENT
The support of the National Science Foundation under grant number MSS-9302327 is gratefully acknowledged. Special thanks to professors John P. Hirth and Ismail Demir for their valuable suggestions.
REFERENCES
[1] [2]
[3] [4] [5] [6] [7]
IS] [9] [10]
J. R. Rice : Elastic Fracture Mechanics Concepts for Interracial Cracks, Journal of Applied Mechanics 110 (1988) 98-103. F. Erdogan : Fracture Problems in Composite Materials, Engineering Fracture Mechanics, 4 (1972) 811-840. F. R. N. Nabarro : Theory of Crystal Dislocations, Oxford Univ. Press, London (1967). J. P. Hirth and J. Lothe : Theory of Dislocations, 2nd Edition, Wiley, New York (1982). T. Mura : Mechanics of Elastic and Inelastic Solids and Micromechanics of Defects in Solids, Martinus NijhoffPublishers (1982). J. P. Hirth : Introduction to the Mathematical Theory of Dislocations, In: T. Mura (ed.) Mathematical Theory of Dislocations, ASME (1969) 1-24. C. Somigliana : Sulla Teoria delle Distorsioni Elastiche, Atti Acad, naz. Lincii, Rend. CI. Sci. Fis. Mat. Natur 23 (1914) 463-472. I. Demir, J. P. Hirth, and H. M. Zbib : The Somigliana Ring Dislocation, Journal of Elasticity 28 (1992) 223-246. B. A. Bilby and J. D. Eshelby : Dislocations and the Theory of Fracture, In: H. Liebowitz (cd.) Fracture, Vol. I. Academic Press, New York (1968) 99-179. F. Erdogan and T. {)zbek 9 Stress in Fiber-Reinforced Composites with Imperfect Bonding, Journal of Applied Mechanics 36 (1969) 865-869. T. Ozbek and F. Erdogan 9 Some Elasticity Problems in Fiber-Reinforced Composites with Imperfect Bonds, International Journal of Engineering Science 7 (1969) 931-946. H. Kasano, H. Matsumoto and I. Nakahara : A Torsion-free Axisynunetric Problem of a Cylindrical Crack in a Transversely Isotropic Body, Bulletin of JSME 27 (1984) 1323-1332. H. Kasano, H. Matsumoto and I. Nakahara : A Cylindrical Interface Crack in a Nonhomogeneous Anisotropic Elastic Body, Bulletin of JSME 29 (1986) 1973-1981. I. Dcmir, J. P. Hirth, and H. M. Zbib : The Extended Stress Field Around a Cylindrical Crack Using the Theory of Dislocation Pile-ups, International Journal of Engineering Science 30 (1992) 829-845. ..
[ll] [12]
[13] [14]
24 I. Demir, J. P. Hirth, and H. M. Zbib : Interaction Between Two Interfaeial Circular Ring Dislocations, InternationalJournal of Engineering Science 31 (1993) 483-492. F. Erdogan and G. Gupta : On the Numerical Solution of Singular Integral Equations, Quarterly of AppliedMathematics 30 (1972) 525-534. A. Gerasoulis : The Use of Pieeewise Quadratic Polynomials for the Solution of Singular Integral Equations of the Cauehy Type, Comp. & Maths. with Appls. 8 (1982) 15-22. H. M. Zbib, J. P. Hirth, and I. Demir : The Stress Intensity Factor of Cylindrical Cracks, International Journal of Engineering Science 33 (1995) 247-253. C. Atkinson, J. Auila, E. Betz, and R. E. Smelser : Journal of Mechanics andPhysics of Solids 30 (1982) 97-120. I. N. Sneddon and M. Lowengrub: Crack Problems in the Classical Theory of Elasticity, Wiley, New York (1969).
[15] [16] [17]
[18] [19] [20]
APPENDIX
The stress field around the Somigliana ring dislocation is given by:
tr,
Aln Au2 A1~3)
_
0",
A,',,
ab2 I A2"2' Ai,32 A,'33~E(k)] 2~1- ,,) A~, A,,~, ~,~ L.A..o .,Llt"<*)J [A,,,
-2n(1-v)A,'3,
oo, _
[ a2,11 A2,12 A2,13
Gbl
11 for r ( R
2(1- v)R -1
~ 2(1- + "
for
r)R
where the coefficients a~ and A~ are given explicitly by
2b {pg2[b2_v(b2+r2+R2)]_r2R2b2}, A,,,,: i'-~3p A,,2,_-ri-~g-b.[3b2(r2 + R2l+(r 2- R212+2b4 -2 vpg2]- p--~[4b2bRr2/2[r +R2)+4(r2-R2)2-pg2],
25
b(r-R) 2 A'"3=
r'Rg
"
A121 = -b [2b2 + 3r2_ 2 v(b2 + R2 +r 2)], ' gr2R b {(R2_r2)2_r2(R2_r2)+b2(2b2+3R2_4r2)_2v[b4 +2b2(r2+R2)+r4+R4]},
A1,22=r2Rpg
bfr - R)(R' + r 2)
r2Rg(r + R)
A1'23 =
'
A,,, = b [2b2R, +( 1 _2v)pg2] ' ' pg3 R A1,32 :
Rpb2g3{4R2[p( r + R) 2 +g2(r-R)2]-pg2[b 2 +r25R2]},
Al,a3 = 2 v,
b(r-R)
Rg(r + R) ' =rpga[b2(3r2+R2)+(r2-R2)2]--~
+r2-2v(b2+r2+
At42 : -R [pg2(R2_r2) + 8r2b2(b2+r2 - R2)]+__~__l[b2(2b2 + 3r2+ 3R2) + (r2_ R2)2_2 vpg2], p2g3r rKpg t ~ 0) AI,43
A2,,2= ~-,,,:2,2 [ p g [R - )-862R 2<,,+R 2 -r2)]
b2(2b 2 + 3r 2 + 3 R 2 ) + ( r 2 -R2) 2 -2 vpg2],
A~:~ = 0,
A22, ,
:
-I[R'
r2gt
+
r2
R2
)],
1 {b2(2b 2 +3r 2 + 3R2)+(R2 _r2)2 + 2 v[-b2(b 2 +r 2 + 2R2)+ R2(r 2 R 2)]}, :1222 , = r2pg A~,~ = O,
Az,a, : -1 [b2(R2 +3r2+2b2)+(r2_R2)2],
pg3 [
A2,32 = p2-g2 {pg2( R2 -r2 - 2b2)+4b2R[p( r+ R)+ g2(R-r)]}, A~,~ - O, _
-b [b2(r2+R2)+(r2_R2)2],
~ , , , - rpg~
26
,4,.43 = 0
The displacement field is given by
{u} ~ [~:......2 w d = 2n(1-- V) B~2, B~,22
b,(~ 4(1 -2+{~o} - v)R 4(1 - v ) r
--
iI ! ".'3,[rI(,.k)j
62 [&,, 8,_,,,. ~""~ E(k)
- for r ( R, - forr(R,
where the coefficients BLo and B~o are given explicitly by
~,,= '
B,,~ ,
_b
2rRg
{262+3r2_2~b2+2r2+R2+3R2(b2-r2)]}
b 2rRgp[R4+2r4+2b2(b2+2r2)-2wg2+3R2(b2-r2)],
b(r-R) [(r2 + R 2 w2], Bm - 2rRg(r + R) )-2 B121 1 b2+ r2 + R2 -4vR2], , = 2Rg[
B~22 = -1 [b2(b2+2r2)+(r2_R2)2], '
2Rgp
Bl,z3 = 0 B2,,, = ~rg[2b2 + R2 +r2 - 2v(b2 + R2 +r2 )], 1 {(R,__r,~f + b,.[2b,~+3(r,+R,.)]_2vpg2}, B2'1~ = 2rgp B,_,. = 0,
B,~.~=-(l-Z~)~, ,
27 -b r2 - R2], 8~,~ =~[b~+
b(r-R)
B2,23-(1- V)g(R+r ) ,
b-s-z,
g-~/bZ +(r+R) z,
p-b2+(r-R) 2