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Strong interactions of morphologically complex cracks
Engineering Fracture Mechanics Vol. 51, No. 6, pp. 665-687, 1991 0 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain
PII: soo13-7944(97)ooo47-7
STRONG
0013-7944/97 $17.00+ 0.00
INTERACTIONS OF MORPHOLOGICALLY COMPLEX CRACKS JIAN NIU and MAO S. WU
Department of Engineering Mechanics, University of Nebraska-Lincoln, NE 68588-0347, U.S.A.
212 Bancroft Hall, Lincoln,
Abstract-Previous works on crack morphology have focused on such cracks as a kinked crack, a branched crack, and an inclined array of identical branched cracks. In this paper, the strong interactions between two cracks in two-dimensional solids under remote tension are investigated. Three morphological types are considered: kinks, branches and zigzags. The method of analysis follows the singular integral equation approach in which the deviations from the main cracks are modeled by distributions of dislocations. Investigations are made on the dependence of the stress intensity factors on the asymmetry of the crack configuration, the crack separation, and the shape of the cracks. The results show that (i) strong interactions can have significant effects on the mode mixity of the stress intensity factors, (ii) a small asymmetry of the crack configuration can cause significant changes to the stress intensity factors, and (iii) zigzag cracks with rectangular steps reduce the stress intensity factors more efficiently than those with triangular or trapezoidal steps. 0 1997 Elsevier Science Ltd
1. INTRODUCTION of morphologically complex cracks has received wide attention since the 1970 s, e.g. refs [l-14]. In homogeneous isotropic solids, the crack types considered include kinked or bent cracks (cracks which kink out of plane), e.g. refs [l-7], branched or forked cracks (cracks which bifurcate), e.g. refs [3,4,7,8], and curved cracks, e.g. refs [3,5]. Other geometries such as the star [9] and cruciform [lo] shapes have also been considered. Kinked, branched and curved cracks in homogeneous anisotropic solids [ 11,121, and branched interface cracks between dissimilar isotropic[l3] and anisotropic [14] media, have also received attention. The motivation of such investigations stems from the observations that cracks are often not straight and indeed may contain multiple zigzags or branches, e.g. stress corrosion cracks and fatigue cracks under mixed mode conditions. Three methods for analyzing cracks of complex shapes are widely used: the conformal mapping method, the perturbation method, and the dislocation modeling method. In the conformal mapping method, a mapping function maps the crack shape into a unit circle and the stress functions are found by techniques such as the polynomial approximation method of Muskhelishvili [15]. Numerical difficulties may be encountered in the conformal mapping method, e.g. refs [2,3,9]. The perturbation method, based on a perturbation analysis of the deviation of a crack from a straight line, has been used for studying slightly curved cracks and kinked cracks with finite or infinitesimal kink lengths. It, however, suffers from a loss of accuracy at large kink angles [5,12]. The method of dislocations, in which the crack deviations are modeled by distributions of dislocations, can in principle be used for kinks of various lengths and angles, e.g. refs [l, 4,6-8,10,11,13,14], although it is also susceptible to numerical inaccuracies resulting from the solution of singular integral equations. Most previous works, e.g. refs [l-5,7-14], focus on a single kinked, branched, or curved crack. The problem of an inclined array of infinitely many identical branched cracks in compression has also been solved using the dislocation modeling method [6], but the formulation essentially reduces to that of a single crack since each interacting crack in the array is identical to any other. To our best knowledge, little work has been done on the analysis of the strong interactions of unequal cracks of complex shapes. On the other hand, interactions of straight cracks in isotropic, anisotropic, finite or three-dimensional solids have been extensively investigated, e.g. refs[15-181, to name just a few. Consequently, it is desirable to study the interactions of cracks of unequal size and complex shapes.
THE ANALYSIS
665
666
J. NIU and M. S. WU
N2k crack 1
4 x2
/
1\
crack 2
,.,,.fk
=
a1
(a)
Problem 1
Problem 2
Problem 3
Cc) Fig. 1. The problem of interacting kinked cracks: (a) original problem, (b) modeling of kinks by distributions of dislocations, (c) decomposition of original problem into three straight crack problems cracked body subjected to remote principal stresses, the stress field of dislocations on the left kink, and the stress field of dislocations on the right kink, respectively.
This paper considers two strongly interacting kinked, branched and zigzag cracks in a twodimensional homogeneous isotropic medium subjected to remote tension. For simplicity, the main cracks are taken to be collinear. Both small and large deviations from the main cracks are studied. The method of dislocations is adopted, since it permits a relatively easy treatment of complex crack morphology and multiple collinear cracks. It is intended to study how the mode I and mode II stress intensity factors (SIFs) of interacting cracks depend on the crack morphology, the asymmetry of the crack configurations, and the crack separation. The formulation is described in Section 2, and the integral equations for kinked, branched and zigzag cracks are developed in Sections 3 and 4. The solution method for integral equations is outlined in Section 5. Numerical results are presented in Section 6, and conclusions in Section 7.
2. FORMULATION:
COMPLEX
STRESS FUNCTIONS
Consider two kinked, branched, or zigzag cracks in an infinite body containing a rectangular coordinate frame x1--x2. Figure l(a) shows two kinked cracks (labelled 1 and 2) under biaxial loading as given by the principal stresses Ni and Nz with Ni at an angle of /I to the horizontal. The main cracks, which are collinear and lie on the negative and positive xl-axes, respectively, have their tips at xi = al, a2, u3, a4 where a4 > a3 > ~22> al. The deviations from the main cracks are modeled by initially unknown continuous distributions of infinitesimal edge dislocations, Fig. l(b). Figure l(b) can be decomposed into three straight crack problems, Fig, l(c). Problem 1 consists of the two main cracks subjected to remote loading. Problems 2 and 3 consist of the two main cracks subjected to the stress field of dislocations associated with the deviations of cracks 1 and 2, respectively. The traction-free boundary conditions on the two main cracks in all problems will be satisfied exactly, while the traction-free conditions on the crack deviations will be satisfied by the integral equations to be derived.
Morphologically complex cracks
The problem of multiple, straight problems [15]. The complex stress z = xl + ix*, where i = fl, are the denote the normal and shear tractions
667
and collinear cracks can be reduced to a pair of Hilbert functions d’(z) and o’(z) of the complex variable solutions of the Hilbert problems. If u&, 0; and a2;, a; on the upper and lower faces of the cracks, then[l5,16]
1 X(tlp0 w’(z) = 2niX(z) s L t-z
d+&
c
G-9
sL
where
(4) F’ = -t
(N, - N2)e2@,
(5)
and for the case of two cracks X(z) = J(z -
Ul)(Z
P,(z) =
-
u2)(z
coz2+ ClZ
-
a3>(z
+
-
a4)v
c,.
(6)
(7)
The overbar in eq. (5) denotes the complex conjugate. The integrations in eqs (1) and (2) are along t on the union L of the two crack lines. The branch of X(z) is taken to be that for which z-‘X(z) -+ 1 as z ---, 00. Furthermore, the constant C0 in eq. (7) is given by (8) with
where p is the shear modulus, K = 3 - 4u for plane strain, K = (3 - u)/(l + u) for plane stress (u is Poisson’s ratio), and coo is the rotation at infinity, taken to be zero in this formulation. The remaining constants C1 and CZ of eq. (7) are to be found from the condition that the displacement when taken around an appropriate contour yields either zero, in which case there is no net dislocation content within the contour, or m times the Burger’s vector, which represents the net dislocation content within the contour, i.e.
(10) where & (k = 1, 2) is a closed contour, Z is the complex conjugate of z, e is the exponential, and [u,.], [ue] denote the displacement jumps in the radial (r) and circumferential (0) directions across the dislocation line. 2.1. Problem I: two straight cracks subjected to remote loading In this case, cr& = cr& = 0% = a; = 0, since the cracks are traction-free. Thus P,(z)
m=X(z)-T’
F’
(11)
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J. NIU and M. S. WU
w ‘(4
P,(z) r =X(z)+7
(12)
Using eqs (lo)-( 12), the unknowns Ci and C2 can be determined from the condition of a zero net displacement around each crack. By shrinking each contour onto the crack faces, the following simultaneous equations arise: Cot2
2(K+ 1)
(t-
2(K+ 1)
a3
(f
-
Cl
c2
t +
az)(t
-
dt = 2(K+ l)(coll+ cl11 + C&J)=
a3)(t
-
-
al)(t
az)(t
-
-
dt = a3)(l
-
0,
(13)
a4)
Cor2+Clt+C2
a4
JJ
al)@
+
2(K
+
1)(coJ2
+
ClJl
+
c2.&3)
=
0.
(14)
a4)
In terms of the complete elliptic integrals of the first, second and third kinds, K(k), E(k) and ZZ({‘,k), the integrals lo, Ii and Z2can be written as 2
ZI-J = (a4
i
12 = i
(u4
_ .:,(.,
+ @(a4
_ u,)
-
a2)(a3
[(a4
-
-
(15)
K(k),
al)
-
a2)(03
-
GW)
+
(a4@4
-
al)(al
+
a2 + u3
+
u4)n(t2,
ul
-
a2))W)
(a4
k2 =
(a4
-
a3)(a2
-
ad
(a4
-
a2)(a3
-
al)
+
02)
k)],
(17)
where e2 _ ’
-
‘2
a4 -
a2
(18)
Similarly, Jo, Ji and 52 are given by eqs (15)-(18) with ul, u2, u3, u4 replaced by u3, u4, al, ~2, respectively. In particular, Z. = Jo. Solutions of eqs (13) and (14) are then
c, =
eco, I
1
c2=
z2J1 - J2z1 Co, ZoUl- Jd
Co
=i((Nl+N2) -
(Nl - N2)eZiB).
(19)
The expressions in eq. (19) reduce to those given in ref. [16] for a pair of identical cracks located symmetrically about the x2-axis. 2.2. Problem 2 or 3: two straight cracks subjected to the stress field of edge dislocations First consider an isolated edge dislocation at the location z. on the trace of the crack deviations in the infinite solid containing the two straight cracks. In analogy to the treatment of a single kinked crack in ref. [4], this problem is itself decomposed into two subproblems: (i) an untracked infinite solid subjected to the stress field of the dislocation, and (ii) an infinite solid containing the two straight cracks. In Subproblem 2, the two cracks are subjected to the negative of the normal and shear stresses acting on the trace of the two main cracks in Subproblem 1. In Subproblem 1, the stress functions untracked solid are given as (e.g. ref.[4])
(pd’ and $d ’ due to an edge dislocation
(pd’(z)= 01,
(20)
(z - zo)
+d ‘(z)
=-+(z -
a! zo)
CYi5J (z -
in an
zoj2
(21) ’
669
Morphologically complex cracks
where a = ,u([u,] + i[ue])eie/(ni(K + 1)) characterizes the dislocation. The normal and shear stresses on the trace of the two main cracks are then given by
t$@ + tid'(t)v
022(t) - iol2(t) = pd'(t) + bDd'(t)+
(22)
where t is as before the variable on the cracks and @j”(f) is the derivative of %‘(t) with respect to t. Proceeding to Subproblem 2, the normal and shear tractions on the cracks are obtained from eqs (20)+22), so that eqs (3) and (4) yield
P(f) = -
qzo-
(
A+%+
20)
Z())2> ’
(t -
q(t) = 0.
(23)
Also, by eq. (7), P,(z) = Cr ‘z + C2’ with CO’= 0, since remote loading is absent in this subproblem. The function X(z) remains as in eq. (6). The stress functions &‘(z) and w,‘(z) for this subproblem are then, using eqs (1) and (2) (24) Substituting eq. (23) in eq. (24) results in the following terms, apart from the factor -1/(2rriX(z)) to be integrated: a! --=-
X(t)
o
X(t)
---
t - zo (t - z) X(t) --=-a! t - ,5-J(t - z)
qzo -
20) X(t)
-=_
(t - zl)y (t - 4
a
(25)
t--z0 > ’ a
z-z, - 1
a.50[ z-20
X(t)
X(t) ---
(
X(0
X(0 --( t-z
Equations (25)-(27) show that the integrations
(26)
t-.20 > ’
t-z
t-50
ii -5,). )I
can all be reduced to integrations
(27)
similar
to X(z)-z’++z+$-:
,
(28)
>
where A = al + u2 + a3 + u4, B = ulu2 + ~23~4+ (aI + ~2)(~3 + ~4). The integration result in eq. (28) is obtained using the method explained in ref. [15]. Using eqs (25)-(28), the stress functions of eq. (24) become
@c’(z)= o,‘(z) = -(aF(z, zo) + aF(z, 20) + Cr(zo- &)G(z, 50)) +
(Cl
’ + w$2
+
C3’)
)
(29)
where (30)
Jqz
zo)
=
1- X(z0)I-w)= 2(z- zo) G(z, 5,) =
1 - N~OYW =
2(z - ZlJ)2
1 2(z -
am
20)
azo
X(Z0)
_ 4(Z
- Fl (z, zo), zo>
’
4
1
- 50)X(Z) ci=l 20 -
Ui ’
(31)
J. NIU and M. S. WU
670
1 = 2(z - &)*
(32)
- GI (z, 50)
and similarly for F(z, 50). Note that F(z,ze), F(z, 20) and (ZO- &)G(z, 5s) are not singular at z = za, even when z = zo = 50, provided that z. does not coincide with al, a2, a3 or u4. Also, the new functions Fi(z,za) and Gi(z, 50) have been introduced in eqs (31) and (32) respectively. Equation (10) with m = 1 and eq. (29) are used to determine the constants Ci ’ and Cz’. For Problem 2 in which z,-,is the location of the dislocation associated with crack 1, the contour A1 encloses crack 1 and the dislocation at zo, whereas the contour A2 encloses only crack 2: aF(z,
--K
zo) + aF(z,
Zo) + i(zo
- fo)G(z,
(cl’+++(c2’+c3’)
Zo) -
X(z)
f[ A!+
cS(zo dz+ aF(Z,zo)+ f[
(Cl ’+ a)? + (C2’ + C3’)
.&)G(Z,Zo) -
aF(T,Zo)+
WI
A,,
dz
2~([u~l + ibd)eie = hi(K +
=
1 I
for k = 1,
l)Cr
for k = 2.
0
(33)
By shrinking Ai to the contour of the kinked, branched, or zigzag crack, and A2 to crack 2, the integrations in eq. (33) can be evaluated. Denoting the elliptic integrals as AI
-
=
F
;
zo)
dt,
&(zo)
=
s
a2
Az(50) =
F
;‘
zo)
dt,
a3
(34)
a4
J
- G ;(t, Zo) dt, &(Zo) =
al
- G ;(t, 20) dt,
(35)
J 03
and similarly for A,(.&), B~(.?o), where F r(t, zo), etc. denote the values on the upper faces of the main cracks, the solutions of eq. (33) are
c, ’+ a!
=
P,,a + P,*(zo - z&i!,
C*’+ C3’ = -I$,(2
(36)
- P**(zo - ‘?o)&,
(37)
where Pll
=
Ai
+ AI
+ Jri -
Bl(ZO)
-
&(~o)
(38)
9
11 - JI
p,2 = A2(50) - B2(50) II - JI
p21 =
(Al(zo)
+ AI
+ ni)J,
(39)
’
- (&(zo)
+ B,(zo))Z,
Zo(Z, - Jl)
p22 = -42(5o)J1 - Bz(zo)Z1 Zo(Z1 - JI)
7
(40)
(41)
’
In Problem 3, where z. is the dislocation position associated with crack 2, Cl’ and Cz ’ are also given by eqs (36) and (41), except for eqs (38) and (40) which should be replaced by Aleo) PI1
=
+ 4@0)
-
&(zo)
11 - JI
-
&Go)
-
Ji
,
(42)
671
Morphologically complex cracks
(b)
crack 2
._-..+
4
a4
Fig. 2. The geometry of (a) kinked, (b) branched and (c) zigzag cracks. The main parts of the cracks have their tips at al, a~, q and ~4. The lengths and angles of the kinks, branches, or zigzags are denoted respectively by I, and 0,. The angle 0, is measured from the positive x,-direction. The two cracks 1 and 2 are located on the negative and positive xl-axes, respectively.
p21
=
(AI
+ ~l(~OM
~OVl
(Bl(ZO) -
+ Bl(~O)
+ iv1
(43)
A>
The stress functions of eq. (29) can be rewritten as &‘(z) = o,‘(z) = -a[F(z, G(z, 50) + a!
PllZ
-
zo) + F(z, z-j)] - ci(zo - 20) p21
+ cr(z0
-
-Jw
50)
p12z
-
-w)
p22 .
(44)
The stress functions for the problem of an edge dislocation in a body containing two collinear straight cracks can now be written consistently in terms of 4’(z) and $‘(z) by superposition of the two subproblems, i.e. 4’(z) =
(od’tz)
+ boc’(z)t
(45)
+‘(z) =
‘kd’I(z)
+ @c’(z),
(46)
where @j’(z), &i’(z), (PC’(Z)are given by eqs (20), (21) and (44) respectively, and &‘(z) by the formula in ref. [ 151: lcIc’(z)= $%‘(z)- (oc’(z) - %“(Z). (47) The function am” is the derivative of am’ with respect to z. Moreover, bpc’(z)= (PC’@). Finally, it is noted that the stress functions for a distribution of dislocations can be obtained from eqs (45) and (46) by integration. The variable a is then interpreted as a complex dislocation density function.
3. INTEGRAL
EQUATIONS
FOR KINKED CRACKS
Consider a pair of kinked cracks with the main parts collinear (Fig. 2(a)). Let the lengths of kinks 1 and 2 be Ii and 12, respectively. The stress functions derived above for Problems 1, 2 and 3 are exact, i.e. the traction-free conditions on the two main cracks are satisfied exactly. To satisfy the traction-free conditions on the kinks, the normal and shear tractions acting along the kinks in the three problems are superimposed and set equal to zero. The normal and shear stresses in polar coordinates (r,@ are given by[15]
672
J. NIU and M. S. WU
iafl = v’(z) + q’(z) + ezie(&Y(z) + q’(z)).
UM +
(48)
For Problem 1, substituting eqs (11) and (12) in eq. (48) and noting also that q’(z) = F’(z) - p’(z) - z#’ as in eq. (47) the normal and shear stresses og, crz on the kink locations due to the remotely applied principal stresses are given by
ug+ iaz
= 2Re
‘Oz2 + clz + C2 + e2iB co‘f*+c,z+c2 X(z) wz? 1 K
- coz2+c~z+c2 X(z) 2coz + Cl X(z)
[
_ (l -2e2’eJ(rf + F/J +
)
,2iO(,- _ z)
1
coz2 + ciz + cz 4 2X(z) ($&J]~ -
(49)
where I’, CO, Ci, Cz are given by eqs (5), (8) and (19), and X(z) by eq. (6). Also, the positions z along kinks 1 and 2 are written as zi = u2 + seiel, z2 = u3seiB’,
(50)
where 0 I s I 1i denotes the position variable along kink 1 with origin at (az,O), and 0 IS I 1, the position along kink 2 with origin at (as,O). Furthermore, 81 and 62 denote the angles of kinks 1 and 2, which are positive if measured counterclockwise from the positive xi-direction. For Problem 2 or 3, the normal and shear stresses in the cracked plane due to the edge dislocation at z. can be written in the polar frame oriented at angle 8 to the xi-axis by substituting eqs (45) and (46) into eq. (48): (51) where iY(z,
ZO,
1
PllZ - p21
0) = (e2ie - 1) F(z, zo) + F(z, 20) -
E
X(Z)
PiI F’(z, zo) + F’(z, 20) - X(z)
PllZ +
-
p21
2X(z)
1
+ e2ie(z- 2)
4 1 + (zo - Zo) i=, z - ai 1 c (52)
V(z, zo,
e>= -e2ie
[
F(z, zo) +
F(z,
-
(‘“&y
F(,?, ZO)+
F(Z,
-
(p”~~p~~)]+tzo-io)
[tz-,e2ie(G~tz,io)-~+p’~~z~~~)] Pl2Z
-
p22
X(z)
(53)
)I *
In eqs (52) and (53) F’(z, zo) =
wz, zo)= az
- 1 + mzo)Ixt4 + 2(z - zo)2
Wzo)
4
c- Z4(Z-ZO)X(Z)i=l
1 Ui'
(54)
Morphologically complex cracks
G'(z, 20) =
+
X(&l) 4(Z -
ZO)~X(Z)
aG(z, 20) az =
c4 i=l
-
1 - -w00)/w +
-
4
X(Z0) 4(Z - Z~J)~X(Z)
2(2 - Zo)3
1 20
613
c+l
1 Z -
Qi
-wo) Ui
+
8(Z
-
zO)x(z)
(55)
(~L)(g&J, i=l ’ -
ai
and similarly for F’(z, 20). Other functions F and G are defined in eqs (31) and (32). The positions z. of the edge dislocations on kinks 1 and 2 are distinguished by writing zoi = a2 + teiel, 202 =
(56)
a3 + teib,
where the definition of t is analogous to that of s. Similarly, the dislocation density distributions on the two kinks are distinguished by the subscripts in a,(t) and aZ(t). Due to the dislocation distribution on kink 1, the normal and shear stresses on kink 1 are +~l(f)U(z~,zo~,~~)+~~(t)V(z~,zol,~l)
I
dt,
and on kink 2
+~1W(z,,zo1,f322)
+~1(OJ'(z~,zo1~~2)
(58)
dt, >
where use has been made of eq. (51). Similarly, due to the dislocation distribution on kink 2, the stresses on kinks 2 and 1 are given by eqs (57) and (58) with the subscripts 1 and 2 interchanged. Superimposing the stresses on kinks 1 and 2 in the two problems and setting them to zero, two coupled singular integral equations in the dislocation densities al and a2 are obtained. These equations are of the Cauchy type. For kink 1, the equation is
+c$(z~,
el) + ioz(zi, el) = 0.
(59)
For kink 2, the equation is +
+~dW(~2,~01~
Q~~(z~,
zo2, e2)+
~2w(~2,
e2) +~l(0v(~2~~01,e22)
zo2,
e2) dt
dt+$g(z2,
e2) +i0,"(z2,e2)
=
0.
(60)
>
In writing the above equations, the singular terms involving l/(s - t) have been isolated. The second set of integrals, which represent the stresses on one kink due to the other kink, are non-singular. EFM 57/6-D
614
J. NIU and M. S. WU
4. INTEGRAL EQUATIONS
FOR BRANCHED OR ZIGZAG CRACKS
Consider a pair of branched cracks 1 and 2 with the main parts collinear (Fig. 2(b)). Each crack has two branches so that the total number of branches K is 4. The positions z and z. on the two branches j = 1, 3 of crack 1 can be distinguished by zj and zoj, where Zj =
Zoj =
U2 +
a2 +
se’+ (0 5 S _( lj),
te’@(0 5
t (
(61)
Zj).
(62)
Similarly, for the two branches j = 2 and 4 of crack 2 Zj =
Zoj =
a3 + se’4 (0 5 s < b),
(63)
te’4 (0 I: t 5 4).
(64)
U3 +
For a pair of zigzag cracks with a total of K zigzags (Fig. 2(c)), the positions z and z. on the various zigzags of crack 1 can be written as zj and zoj (j odd): Zj =
Zoj =
a2 +
a2
+
Iie’@+ ZseiB3 + . . . + se’4 (0 5 s 5 Zj),
(65)
Zleiei + Z3eie3+ . . . + te’4 (0 5 t I: 4).
(66)
In eqs (65) and (66), Ii, 6, .... 4 denote the lengths of the zigzags, and 8i, es, .... 13,denote the angles of the zigzags with respect to the positive xi-direction. For the zigzags of crack 2, zj and zoj 0’even) are given by Zj =
Zoj =
a3 +
a3
lzeiBz+ laeie4+ . . . + se’3 (0 5 s ( Zj),
(67)
Z2eie2+ ZJeie4+ . . . + te’+ (0 5 t ( lj).
(68)
+
The K integral equations are obtained by the superposition of the normal and shear stresses at zj in the following (1 + K) problems: (i) two straight cracks in an infinite body subjected to remote loading, and (ii) two straight cracks in an infinite body subjected to the dislocation distributions on each of the K zigzags. Use of eqs (49) and (51) and appropriate zj and Zoj leads to
dt +
z,, 6,) + ak(t)V(zj, -$ak(t)(i( zok,
k=l
ZOO,
0,)) dt + a$$(zj,$I+
iu,W(zj,$1 = 0,
(69)
o
where the K singular integral equations of the Cauchy type are obtained by setting j = 1, 2, 3, . ..) K. Setting K = 2 reduces eq. (69) to eqs (59) and (60) for a pair of kinked cracks.
5. SOLUTIONS
OF INTEGRAL
EQUATIONS
AND STRESS INTENSITY
FACTORS
The method proposed by Gerasoulis[lO] for solving integral equations is used in this paper and other works (e.g. refs [13, 141). The dislocation density a(t) is expressed as a product of a weight function w(t) and a continuous function 4(t). The latter is approximated by piecewise quadratic polynomials. The coupled singular equations reduce to a system of linear algebraic equations after appropriate substitutions. Additional conditions are required, as discussed in the following outline of the solution procedure. First, the limits of the integrals in eqs (59) and (60) or eq. (69) are normalized to -1 and + 1 using the transformations
615
Morphologically complex cracks
where 1 equals Ii, I*, etc. as appropriate. branch, or zigzag is written as
The dislocation
density function
for the jth kink,
cXj(j(t’ =) w(t’)(p#)
(71)
where w(t’) =
1 (1 - f’)“2(1 + f’)V’
(72)
and 0 I q I l/2. Furthermore, the l/2 and q singularities at t’ = 1 and t’ = - 1 are built into the definition of a,(P) through w(t’). In the solutions to follow, r~is taken as l/2 since for comparable levels of accuracy the assumption of an incorrect exponent requires less numerical computations than the use of the correct value [13]. This is because a large number of numerical integrations are needed for r~lying between 0 and l/2, whereas the integrals can be evaluated in closed forms for q = l/2. With q = l/2, the additional conditions required for solving the integral equations are obtained by setting qJ{-1) = 0, as in refs [4, 11, 13, 141. In the case of the kinked or branched cracks, these conditions are imposed at the points where the kinks or branches join the main cracks. In the case of the zigzag cracks, they are imposed at the points where the zigzags join the main cracks as well as where the zigzags join each other. After substituting eq. (71) into the integral equations, the equations are discretized and the Lagrange interpolation formula for three points is used to approximate the functions appearing in them. See ref. [lo] for details. The resulting expressions can be integrated exactly for the form of w(t’) chosen in eq. (72) with rl = l/2. The use of a set of R collocation points and the additional conditions generates a system of linear algebraic equations with q,{t’) at various values of t’ as unknowns. After solving the algebraic equations, the mode I and mode II SIFs (Kr, Kn) are easily calculated from the stress field around the crack tip. This requires the value #j(l) at the crack tip. The formula used for kinked cracks (e.g. ref.[4]) is also valid for branched and zigzag cracks: Kt + i&I = (2Jr)3’2eie1(pj(l)$‘2/2,
(73)
where 0, is the orientation of the kink, branch, or zigzag at whose tip the SIFs are computed.
6. NUMERICAL
RESULTS
In this section, the geometrical parameters 2c and 1 denote respectively the length of the main crack and the length of the kink, branch, or zigzag. For kinked or branched cracks, the parameter 2 d denotes the horizontal distance between the inner tips of the main cracks. For zigzag cracks, 2 d denotes the horizontal distance between the knees of the last zigzags of the two cracks. Unless otherwise stated, the two cracks are arranged symmetrically about the x*-axis. When the cracks are unequal, subscripts are used to distinguish between them, e.g. 2ct, Ii, etc. All solutions are obtained with a Cray supercomputer and six IBM RISC/6000 workstations. Only the SIFs at the tips of the kinks, branches, or zigzags are computed. Except for the solutions in Test Case 2 below, they are generated with values of R between 64 and 160. A solution generated with a given R is accepted when it is practically indistinguishable, on the scale of the figure, from another generated with an R twice as large. Also, the sign of the mode II SIFs is defined with respect to a local rectangular frame attached to the knee of the kink, branch, or zigzag such that the local positive xl-direction points from the knee towards the tip. Negative mode I SIFs indicating the compressive nature of the local stress state are predicted for certain kink or branch angles, as is also the case for single cracks in refs [5,8], but are not shown due to the possible complication of crack face contact and numerical inaccuracy.
676
J. NIU and M. S. WU
I””
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Fig. 3. Comparison of the SIFs of two identical kinked cracks located remotely from each other with the single crack solutions reported in ref. [4].
6.1. Test cases 6.1.1. Case 1: two identical kinked cracks at a large distance from each other. Consider two identical kinked cracks such that the inner tips of the main cracks are separated by a distance equal to at least 198 times the main crack length. A remote tensile stress 0 is applied in the vertical direction. Figure 3 plots the normalized SIFs & = K~/a,/ifZ and &;I = Kr~/afi of the kinks against logie(c/l). The symbols in the figure indicate the results for various kink angles computed using the current model. The full lines indicate the results computed from Lo’s[4] model for a single kinked crack. The two sets of results are in excellent agreement. Furthermore, it should be mentioned that the SIFs for two identical branched cracks separated by a large distance and subjected to a remote tensile stress also agree very well with those reported in ref. [4] for a single branched crack. 6.1.2. Case 2: two identical straight cracks with strong or weak interactions. Since to the best of our knowledge no SIF solutions are available in the literature for strongly interacting kinked, branched or zigzag cracks, we investigate the case of two main cracks each joined by a “kink” of 0” or 180” angle (see the diagram included in Table 1). The cracks are subjected to the tensile stress o in the vertical direction. Table 1 compares three sets of solutions for K~/q/m of the inner tips as the crack separation becomes smaller: (i) the exact solutions of Erdogan [16] obtained by treating each crack as an uninterrupted straight crack, (ii) the approximate
Morphologically complex cracks
611
Table 1. Comparison of the mode I SIFs, K&,/m, of two cracks adjoined with 0” or 180” kinks with the exact solutions for straight cracks. Acceptable accuracy is obtained for 6 2 0.01 even for R = 20. For S < 0.01, increasing R improves the solutions c/l = 1 J=d-’ ?Cfd 0.0001
0.001 0.01 0.02 0.05 0.1 0.2 0.99
Exact solution Erdogan[l6] 13.347 5.395 2.372 1.905 1.473 1.255 1.112 1.000
c/r = 100
R = 20
64
160
20
64
160
5.714 4.568 2.356 1.900 1.471 1.254 1.111 0.999
8.592 5.204 2.367 1.904 1.473 1.254 1.112 1.OOo
10.889 5.351 2.371 1.904 1.413 1.255 1.112 1.000
13.153 5.372 2.363 1.898 1.468 1.251 1.109 0.991
13.301 5.382 2.367 1.901 1.471 1.253 1.111 0.999
13.321 5.384 2.368 1.902 1.471
1.254 1.112 0.999
obtained by treating each crack as a main crack joined by a kink with c/Z = 1, and (iii) the approximate solutions obtained as in (ii) but with c/l = 100. All the approximate solutions agree well with the exact solutions when the separation parameter 6 = (d- 1)/(2c + 6) is greater than or equal to 0.01. At 6 = 0.01, the worst approximate solution, i.e. that computed with c/l = 1 and the polynomial degree of R = 20, has an error of less than 0.67%. At 6 = 0.001 and 0.0001, the approximate solutions tend towards the exact solutions of 5.395 and 13.347 (to three decimal places) as R increases. The best approximate solution at 6 = 0.0001, computed using c/f = 100 and R = 160, is 13.321. This represents an error of less than 0.2%. It may also be noted that the value of q = l/2 used to obtain the approximate solutions is at the opposite extreme of the correct value of q = 0 for a zero degree kink. This suggests that the approximate method of using q = l/2 and co,i-1) = 0 yields reasonably accurate solutions even in the case of strong interactions. solutions
6.2. Kinked cracks 6.2.1. Two identical kinked cracks. For two identical kinked cracks subjected to vertical remote tension Q, the dependence of ir and &;I on the kink angle 8 is shown in Fig. 4 for c/l = 1 and in Fig. 5 for c/l = 100. The SIFs plotted are those of the left kink, as indicated by a dot at the tip. Furthermore, four cases corresponding to d/l = 1.05, 1.10, 1.25 and 1.50 are investigated, with d/l = 1.05 representing the smallest separation between the cracks. One interpretation of these ratios is that for the same c, the kink length I and the crack separation 2 d in Fig. 5 are two orders of magnitude smaller than those in Fig. 4. For a given d/l, the mode I SIF is largest at 8 = 0” and decreases as 0 increases. In the case of c/l = 1, the mode I SIFs become negative as 8 increases beyond -85”. Negative mode I SIFs are also predicted in the case of a single kinked crack, as inferred from Fig. 5 of ref. [2]. The decrease of Kt with tl can be explained by the reduction of normal load on the kinks, the increasing separation between the kink tips, and the increasing shielding between each kink and the adjoining main crack at large 8. For a given 19, the mode I SIFs also decrease as d/l increases, as expected. At the larger angles, however, the dependence on d/l in the range between 1.05 and 1.50 is small. Comparison of Figs 4 and 5 shows that in the stronger interaction case of c/l = 100, the sensitivity of the mode I SIFs to d/l is greater at a given 8, and the SIFs remain generally large for 8 < 90”. For instance, at c/l = 100, & >- 1 for 8 < -9O”, whereas at c/ 1 = 1, k* >- 1 only for 8< - 40”. The mode II SI_Fs display maxima at 8 between 40” and 70”, depending on the value of c/l and d/l. Note that Kn = 0 at 8 = 0” for both ratios of c/l, but is -0.5 for c/l = 1 and -1 for c/ 1 = 100 at 8 = 90”. At both 0” and 90”, no remote shear stress acts on the kinks, so the rather large values at 0 = 90” must arise from the action of the main cracks and the interaction effects. Comparison of Figs 4 and 5 also shows that the sensitivity of & to d/l at a given 0 is greater in the case of c/l = 100 than in the case of c/l = 1. To show how the mode mixity varies with the kink angle, the parameter y = tan-‘(&/Ki) is plotted against 8 for c/l = 1 and 100 in Fig. 6. At 0 = 0”, y = 0”, i.e. a pure mode I state. As 0 increases, y increases towards 90”, a pure mode II state. In the case of c/l = 1, y reaches 90” at B - 83”. In the case of c/l = 100, y never reaches 90” within the range of 0 considered. Two
678
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important characteristics associated with increasing the interaction effect are: (i) increasing the ratio c/l from 1 to 100 at a constant value of d/l suppresses the transition of mode mixity towards pure mode II as 8 increases, (ii) decreasing d/l from 1.5 to 1.05 at the constant value of c/l = 1 or 100 does not change the mode mixity significantly. The distance between the kink tips changes by two orders of magnitude in the former case, while it changes by half an order of magnitude in the latter case. Thus, how interaction affects mode mixity depends on the details of the crack geometry and configuration. 6.2.2. Two kinked cracks with unequal main parts. In Fig. 7, asymmetry of the crack configuration is introduced by changing the length of the left main crack, 2~. The two kinked cracks are otherwise symmetrical about the vertical axis at the origin. Specifically, c/l = 1, f3 = 15”, d/ 1 = 1.1. The main cracks are subjected to a remote tension 6. It can be seen that as q/c changes from 0 to 2, there is a significant change of & and a perceptible change of & for both the left and right kinks. For both kinks, the slope of ii at cl/ c = 1 implies that a small perturbation from symmetry leads to small but not insignificant changes in $. It is also clear that & of the kink adjoined to the longer main crack is larger. In fact, a closer examination of the lower plot of Fig. 7 shows that the same conclusion can be made for the absolute value of &.
679
Morphologically complex cracks 10.0
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Kink Angle 8 Fig. 5. Variation of the SIFs of identical kinked cracks with the kink angle 0 for different separation parameters d/l. The ratio c/l = 100.
6.2.3.Two kinked cracks with unequal kink lengths. In Fig. 8, asymmetry of the crack configuration is introduced by changing the length of the left kink, 1i. The two kinked cracks are otherwise symmetrical about the +-axis. Specifically, c/l = 1, 8 = 15”, d/l = 1.l. The main cracks are subjected to a remote tension rr. In the following, amplification or shielding of SIFs is discussed in relation to the perfectly symmetric _case of [i/l = 1. A strong amplification of both Kr and Kn occurs around 1,/l = 1.26 due_ to the very close proximity of the two kink tips. The amplification is greater in it than in Kii. In this case, a small perturbation of the symmetry of the crack configuration can lead to significant changes of the SIFs. The amplification is followed by a shielding of &, with & of the left tip reaching a minimum close to zero at 1i/1 = 2.6. The shielding is due_ to the overshooting of the left kink over the right kink tip. It is interesting to observe that KI of the overshooting and longer left kink is smaller than that of the shorter right kink, suggesting that the & reduction is greater on the kink whose central portion is shielded than the one whose tip is shielded. As 11/l increases beyond 2.6, however, & of the right tip continues to decrease while it of the left tip increases and tends to the value of an isolated crack of length Ii and orientation 15”. On the other hand, shielding of &r is less strong, since the in values for 1,/l> 2 do not differ greatly from the Ku value at 1,/l = 1.
680
J. NIU and M. S. WU
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6.3.Branched cracks
6.3.1. Two identical branched cracks. In complete analogy to Fig. 4 for kinked cracks, Fig. 9 shows a pair of identical branched cracks subjected to the remote tension a perpendicular to the main cracks. All four branches are equal in length and c/l = 1. The crack configuration is symmetric about the xi- and x2-axes. The kr-0 and kn-0 curves are generated using d/l = 1.05, 1.lO, 1.25 and 1.50. The SIFs of the upper left branch are plotted against the common branch angle 0 in the range 2” I ~9I 90”. Solutions for 0 < 2” obtained using R < 160 are unreliable and are omitted. The trend of each SIF curve depends on the remote loading resolved on the branches and the complex interactions among the four branches. In particular, shielding of the branches of the same crack occurs in both modes I and II as 0 decreases. Consequently, & and &;I of the branches are smaller than those of the kinks, which do not shield each other. Furthermore, & of the branched cracks reaches a maximum not at 8 = 0” but at some 8 less than 15”, in contrast to kr of the kinked cracks. This suggests that as 8 decreases below -15”, the mode I shielding between the branches of the same crack increasingly dominates over the mode I
681
Morphologically complex cracks
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C,IC Fig. 7. Dependence of the SIFs on the asymmetry of kinked cracks as measured by the ratio of the lengths of the left and right main cracks, q/c. The kink lengths and angles are identical. The kink with the longer main crack has the larger mode I SIF as well as the larger absolute value of the mode II SIF. A small perturbation of symmetry about q/c = 1 leads to small but not insignificant changes of the SIFs.
amplification resulting from (i) the shorter distance between the branches of different cracks and (ii) the increased remote load normal to the branches. It also explains why the angle at which & is maximum decreases with decrease of d/f, since the increased amplification associated with the decrease of d/l can compensate more effectively the shielding effect of the branches of the same crack. Finally, zeroes of &r occur at angles below -15”, in conjunction with the existence of maxima for Et. 6.3.2. Two unequal branched cracks. Asymmetry is introduced by letting 1112= 10, keeping the left branch angle B = 30” and varying the other branch angle 0, = 180” - 0, from 0” to 90”, as can be seen in Fig. 10. The crack configuration remains symmetric about the xi-axis. Also, c/ I = 1 and d/l = 0.3, so that the short right branches lie within the region enclosed by the long left branches (the inset to Fig. 10 shows the short branches outside the region of the long branches for clarity). A remote tension ~7is applied in the x*-direction. The SIFs plotted are those of the upper left and right branches (A and B). Although the mode I and mode II SIFs of the crack with long branches do not change greatly as the angle of the short branches changes, they represent changes of roughly 50% com-
J. NIU and M. S. WU
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1,/l Fig. 8. Dependence of the SIFs on the asymmetry of kinked cracks as measured by the ratio of the lengths of the left and right kinks, It/l. The main crack lengths are identical. Because of the proximity of the two kinks, a small perturbation of symmetry about It/l = 1 leads to large changes in the SIFs. The interaction effects are complex: there is significant shielding of the mode I SIF of the longer left kink for 1.6 < 1,/l < 3.6, but only a more gradual decrease of the mode I SIF of the shorter right kink over this range.
pared to the SIFs of the same crack existing in isolation (see the case of d/l = 1.50 and 8 = 30” in Fig. 9). Thus, the presence of the right branched crack, although not the orienmtion of the right branches, does have a significant effect on the long braAnches. Note also that Kn becomes zero and & attains a maximum at approximately the angle 02 = 18”. If the left branched crack exists in the presence of a right crack that has not yet bifurcated, then branches may nucleate at approximately this angle from the right crack under the maximum j?t condition. 6.4. Zigzag cracks 6.4.1. Dependence of SIFs on the shape of two identical zigzag cracks. The inset to Fig. 11 shows two identical zigzag cracks each containing a trapezoidal step. The length of each main crack is 2c, and the two horizontal zigzags farthest from the main cracks have length c/100. The spacing between the tips of these two zigzags is set equal to O.O4c, i.e. d = O.O3c, to obtain strong interaction effects. For comparison, the case of practically no interaction is also studied with the spacing set equal to 599.98c, i.e. d = 300~. The base length and height of each trapezoid are 2c and
683
Morphologically complex cracks
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1/(2&)c(0.289c), respectively. Keeping the above parameters fixed, the length 1s = 1, of the uppermost horizontal zigzags is changed from 0 to 2c in such a manner that each trapezoid always possesses a vertical axis of symmetry. This alters the shape from triangular through trapezoidal to rectangular. A r_emote tension r~is applied in the vertical direction. Figure 11 shows that Kt of the strongly interacting zigzag cracks decreases slowly as the triangle evolves into a trapezoid, but decreases rapidly when the trapezoid evolves into a rectangle. The same is true in the case of no interaction, although the decrease of ir is considerably smaller. The percentage reduction of -40% due to the change of a triangle to a rectangle is approximately the same for both cases of strong and no interaction. In summary, the results suggest that zigzag cracks with triangular or trapezoidal steps are less efficient in reducing the mode I SIFs compared to zigzag cracks with rectangular steps. The values of Z?u of the left zigzag tip are also shown in Fig. 11. The absolute values are small in comparison to &. In the case of strong interaction, &i increases with 1s after 1s> 1SC, in contrast to the decrease of & with 1s.
684
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6.4.2. Dependence of SIFs on the step height of two identical zigzag cracks. In Fig. 12, each zigzag crack has a rectangular step of width 2c. The height h of the step is varied from zero to 0.5~. All other dimensions remain as in Fig. 11. A remote tension Q is also applied vertically. It is seen that increasing the height of the rectangular steps reduces Ki very effectively. In fact, Kr reduces from the value of 4.36 at h = 0 (straight cracks) to the value of 2.27 ath = c/2 when d k 0.03~. The reduction from 1.41 to 0.88 is considerably less when d = 300~. The percentages of reduction are -4_8% and -38%, respectively. The rate of reduction decreases with increase in h. In contrast, &I increases from zero to a maximum at small h and then decreases with increase in h. As in Fig. 11, the absolute values of & are small in comparison to &. Solutions for very small h, i.e. 0 < h < O.O2c, are omitted in Fig. 12 due to insufficient accuracy. When h is very small, the small distance between the ends of the very short vertical zigzags appears to play a negative role in the solution accuracy, since an incorrect exponent of the stress singularity is assumed at both ends. In the limiting case of h/c = 0, the solution is obtained by solving the problem of four horizontal zigzags of lengths 0.5c, c, 0.5~ and 0.01~.
685
Morphologically complex cracks
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The solution of 4.36 obtained using R = 128 is within 2.6% of the exact solution of 4.25 obtained from the direct analysis of straight cracks. For finite heights h/c > 0.2, the solutions obtained using R = 64 and 128 are practically indistinguishable. 7. CONCLUSIONS AND CLOSURE REMARKS The strong interactions of two kinked, branched and zigzag cracks are studied in this paper using the method of continuous dislocation distributions. The traction-free boundary conditions on the collinear main cracks are satisfied automatically, while imposing the same condition on the kinks, branches, or zigzags yields a system of coupled singular integral equations. The equations are solved by the use of piecewise quadratic polynomials, leading to a system of linear algebraic equations. The crack configurations are subjected to remote tension perpendicular to the main cracks, and the SIFs are calculated at the tips of the kinks, branches, or zigzags. The results for kinked cracks show that strong interactions can retard the transition of the mode mixity from mode I to mode II as the kink angle increases. A small but finite perturbation to the symmetry of the configuration of the kinked cracks, e.g. unequal kinks, can cause significant changes to the SIFs
686
J. NIU and M. S. WU
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A
’
’
0.30
’
’
’
’
’
’
0.40
’
’
0.50
hlC
1 .oo c/loo
L
ii
0.60
? d?
2c’2c’2d~2c’2c 000
_
0
0
0.40
0 0 0
0.20
h
fb
0.80
A
A
o.ool:a ” 0.00
”
0.10
A
”
A
”
”
0.20
A
”
0
A
”0.30”
0
0
A
A
”
”
0.40
0
-I
A
”
I’ 0.50
hlC Fig. 12. Dependence of the SIFs of zigzag cracks on the height h of rectangular steps. Increasing the step height reduces the mode I SIFs at the tips of the zigzags effectively. The reduction in the case of strong interaction (d = 0.03~) is greater than that in the case of no interaction (d = 300~).
of the kinks. Also, branched cracks display more complex interaction effects than kinked cracks due to the amplification and shielding of the four branches and the main cracks. Finally, zigzag cracks with rectangular steps reduce the mode I SIFs more than those with triangular steps. Increasing the height of the rectangular steps also reduces the mode I SIFs significantly. Whether the shape changes from a triangle to a rectangle or whether the step height of the rectangle increases, the reduction of the SIFs is greater in the case of strong interaction than in the case of no interaction. This paper focuses on the computation of the SIFs of complex cracks. The stability of the crack path is not addressed. Crack path instability has been used to explain the experimentally observed phenomenon of why originally collinear straight cracks avoid each other. For instance, Melin [19] suggested that the straight crack path is unstable. The analysis assumes that the cracks are initially slightly curved. First order solutions to the governing integral equations are given. Further investigation, however, is necessary to determine whether the initially assumed
Morphologically complex cracks
crack geometry and crack configuration the crack growth.
687
have an effect on the subsequent directional stability of
Acknowledgements-The authors gratefully acknowledge the support of the Office of Naval Research in this investigation (Grant No. N00014-92-J-1203). The computational part of this work was made possible by the use of a Cray supercomputer at the College of Engineering and Technology, University of Nebraska-Lincoln, and six IBM RISC/6000 workstations purchased with an earlier grant of the National Science Foundation (Grant No. MSS-9111895).
REFERENCES I. Bilby, B. A. and Cardew, G. E., The crack with a kinked tip. International Journal of Fracture, 1975, 11, 708-711. 2. Chatterjee, S. N., The stress field in the neighborhood of a branched crack in an infinite sheet. International Journal of Solids and Structures, 1975, 11, 521-538. 3. Kitagawa, H., Yuuki, R. and Ohira, T., Crack-morphological aspects in fracture mechanics. Engineering Fracture Mechanics, 1975, I, 515-529. 4. Lo, K. K., Analysis of branched cracks. Journal of Applied Mechanics, 1978,45, 797-802. 5. Cotterell, B. and Rice, J. R., Slightly curved or kinked cracks. International Journal of Fracture, 1980, 16, 155-169. 6. Horii, H. and Nemat-Nasser, S., Brittle failure in compression: splitting, faulting and brittle-ductile transition. Philosophical Transactions of the Royal Society, London, Series A, 1986, 319, 337-374. 7. Vitek, V., Plane strain stress-intensity factors for branched cracks. International Journal of Fracture, 1977, 13, 481501. 8. Theocaris, P. S., Asymmetric branching of cracks. Journal of Applied Mechanics, 1977,44, 611-618. 9. Andersson, H., Stress intensity factors at the tips of a star-shaped contour in an infinite tensile sheet. Journal of Mechanics and Physics of Solids, 1969, 17,405-417. 10. Gerasoulis, A., The use of quadratic polynomials for the solution of singular integral equations of Cauchy type. Computers & Mathematics with Applications, 1982, 8, 15-22. 11. Obata, M., Nemat-Nasser, S. and Goto, Y., Branched cracks in anisotropic elastic solids. Journal of Applied Mechanics, 1989, 56, 858-864. 12. Gao, H. J. and Chiu, C. H., Slightly curved or kinked cracks in anisotropic elastic solids. Inrernafionai Journal of Solids and Structures, 1992, 29, 947-972. 13. He, M. Y. and Hutchinson, J. W., Kinking of a crack out of an interface. Journal of Appfied Mechanics, 1989, 56, 270-278. 14. Miller, G. R. and Stock, W. L., Analysis of branched interface cracks between dissimilar anisotropic media. Journal of Applied Mechanics, 1989, 56, 844-849. 15. Muskhelishvili, N. I. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Leiden, 1953. 16. Erdogan, F., On the stress distribution in plates with collinear straight cuts under arbitrary loads. Proceedings of the Fourth U.S. Naiional Congress of Applied Mechanics, 1962, 1, 547-554.
17. Kachanov, M. and Laures, J.-P., Three-dimensional problems of strongly interacting arbitrarily located pennyshaped cracks. International Journal of Fracture, 1989, 41, 289-313. 18. Wu, M. S., Analysis of finite anisotropic media containing multiple cracks using superposition. Engineering Fracture Mechanics, 1993, 45, 159-175.
19. Mehn, S., Why do cracks avoid each other? International Journal of Fracture, 1983, 23, 3745. (Received
11November 1996, in final form 15 March 1997, accepted 15 March 1997)
PII: soo13-7944(97)ooo47-7
STRONG
0013-7944/97 $17.00+ 0.00
INTERACTIONS OF MORPHOLOGICALLY COMPLEX CRACKS JIAN NIU and MAO S. WU
Department of Engineering Mechanics, University of Nebraska-Lincoln, NE 68588-0347, U.S.A.
212 Bancroft Hall, Lincoln,
Abstract-Previous works on crack morphology have focused on such cracks as a kinked crack, a branched crack, and an inclined array of identical branched cracks. In this paper, the strong interactions between two cracks in two-dimensional solids under remote tension are investigated. Three morphological types are considered: kinks, branches and zigzags. The method of analysis follows the singular integral equation approach in which the deviations from the main cracks are modeled by distributions of dislocations. Investigations are made on the dependence of the stress intensity factors on the asymmetry of the crack configuration, the crack separation, and the shape of the cracks. The results show that (i) strong interactions can have significant effects on the mode mixity of the stress intensity factors, (ii) a small asymmetry of the crack configuration can cause significant changes to the stress intensity factors, and (iii) zigzag cracks with rectangular steps reduce the stress intensity factors more efficiently than those with triangular or trapezoidal steps. 0 1997 Elsevier Science Ltd
1. INTRODUCTION of morphologically complex cracks has received wide attention since the 1970 s, e.g. refs [l-14]. In homogeneous isotropic solids, the crack types considered include kinked or bent cracks (cracks which kink out of plane), e.g. refs [l-7], branched or forked cracks (cracks which bifurcate), e.g. refs [3,4,7,8], and curved cracks, e.g. refs [3,5]. Other geometries such as the star [9] and cruciform [lo] shapes have also been considered. Kinked, branched and curved cracks in homogeneous anisotropic solids [ 11,121, and branched interface cracks between dissimilar isotropic[l3] and anisotropic [14] media, have also received attention. The motivation of such investigations stems from the observations that cracks are often not straight and indeed may contain multiple zigzags or branches, e.g. stress corrosion cracks and fatigue cracks under mixed mode conditions. Three methods for analyzing cracks of complex shapes are widely used: the conformal mapping method, the perturbation method, and the dislocation modeling method. In the conformal mapping method, a mapping function maps the crack shape into a unit circle and the stress functions are found by techniques such as the polynomial approximation method of Muskhelishvili [15]. Numerical difficulties may be encountered in the conformal mapping method, e.g. refs [2,3,9]. The perturbation method, based on a perturbation analysis of the deviation of a crack from a straight line, has been used for studying slightly curved cracks and kinked cracks with finite or infinitesimal kink lengths. It, however, suffers from a loss of accuracy at large kink angles [5,12]. The method of dislocations, in which the crack deviations are modeled by distributions of dislocations, can in principle be used for kinks of various lengths and angles, e.g. refs [l, 4,6-8,10,11,13,14], although it is also susceptible to numerical inaccuracies resulting from the solution of singular integral equations. Most previous works, e.g. refs [l-5,7-14], focus on a single kinked, branched, or curved crack. The problem of an inclined array of infinitely many identical branched cracks in compression has also been solved using the dislocation modeling method [6], but the formulation essentially reduces to that of a single crack since each interacting crack in the array is identical to any other. To our best knowledge, little work has been done on the analysis of the strong interactions of unequal cracks of complex shapes. On the other hand, interactions of straight cracks in isotropic, anisotropic, finite or three-dimensional solids have been extensively investigated, e.g. refs[15-181, to name just a few. Consequently, it is desirable to study the interactions of cracks of unequal size and complex shapes.
THE ANALYSIS
665
666
J. NIU and M. S. WU
N2k crack 1
4 x2
/
1\
crack 2
,.,,.fk
=
a1
(a)
Problem 1
Problem 2
Problem 3
Cc) Fig. 1. The problem of interacting kinked cracks: (a) original problem, (b) modeling of kinks by distributions of dislocations, (c) decomposition of original problem into three straight crack problems cracked body subjected to remote principal stresses, the stress field of dislocations on the left kink, and the stress field of dislocations on the right kink, respectively.
This paper considers two strongly interacting kinked, branched and zigzag cracks in a twodimensional homogeneous isotropic medium subjected to remote tension. For simplicity, the main cracks are taken to be collinear. Both small and large deviations from the main cracks are studied. The method of dislocations is adopted, since it permits a relatively easy treatment of complex crack morphology and multiple collinear cracks. It is intended to study how the mode I and mode II stress intensity factors (SIFs) of interacting cracks depend on the crack morphology, the asymmetry of the crack configurations, and the crack separation. The formulation is described in Section 2, and the integral equations for kinked, branched and zigzag cracks are developed in Sections 3 and 4. The solution method for integral equations is outlined in Section 5. Numerical results are presented in Section 6, and conclusions in Section 7.
2. FORMULATION:
COMPLEX
STRESS FUNCTIONS
Consider two kinked, branched, or zigzag cracks in an infinite body containing a rectangular coordinate frame x1--x2. Figure l(a) shows two kinked cracks (labelled 1 and 2) under biaxial loading as given by the principal stresses Ni and Nz with Ni at an angle of /I to the horizontal. The main cracks, which are collinear and lie on the negative and positive xl-axes, respectively, have their tips at xi = al, a2, u3, a4 where a4 > a3 > ~22> al. The deviations from the main cracks are modeled by initially unknown continuous distributions of infinitesimal edge dislocations, Fig. l(b). Figure l(b) can be decomposed into three straight crack problems, Fig, l(c). Problem 1 consists of the two main cracks subjected to remote loading. Problems 2 and 3 consist of the two main cracks subjected to the stress field of dislocations associated with the deviations of cracks 1 and 2, respectively. The traction-free boundary conditions on the two main cracks in all problems will be satisfied exactly, while the traction-free conditions on the crack deviations will be satisfied by the integral equations to be derived.
Morphologically complex cracks
The problem of multiple, straight problems [15]. The complex stress z = xl + ix*, where i = fl, are the denote the normal and shear tractions
667
and collinear cracks can be reduced to a pair of Hilbert functions d’(z) and o’(z) of the complex variable solutions of the Hilbert problems. If u&, 0; and a2;, a; on the upper and lower faces of the cracks, then[l5,16]
1 X(tlp0 w’(z) = 2niX(z) s L t-z
d+&
c
G-9
sL
where
(4) F’ = -t
(N, - N2)e2@,
(5)
and for the case of two cracks X(z) = J(z -
Ul)(Z
P,(z) =
-
u2)(z
coz2+ ClZ
-
a3>(z
+
-
a4)v
c,.
(6)
(7)
The overbar in eq. (5) denotes the complex conjugate. The integrations in eqs (1) and (2) are along t on the union L of the two crack lines. The branch of X(z) is taken to be that for which z-‘X(z) -+ 1 as z ---, 00. Furthermore, the constant C0 in eq. (7) is given by (8) with
where p is the shear modulus, K = 3 - 4u for plane strain, K = (3 - u)/(l + u) for plane stress (u is Poisson’s ratio), and coo is the rotation at infinity, taken to be zero in this formulation. The remaining constants C1 and CZ of eq. (7) are to be found from the condition that the displacement when taken around an appropriate contour yields either zero, in which case there is no net dislocation content within the contour, or m times the Burger’s vector, which represents the net dislocation content within the contour, i.e.
(10) where & (k = 1, 2) is a closed contour, Z is the complex conjugate of z, e is the exponential, and [u,.], [ue] denote the displacement jumps in the radial (r) and circumferential (0) directions across the dislocation line. 2.1. Problem I: two straight cracks subjected to remote loading In this case, cr& = cr& = 0% = a; = 0, since the cracks are traction-free. Thus P,(z)
m=X(z)-T’
F’
(11)
668
J. NIU and M. S. WU
w ‘(4
P,(z) r =X(z)+7
(12)
Using eqs (lo)-( 12), the unknowns Ci and C2 can be determined from the condition of a zero net displacement around each crack. By shrinking each contour onto the crack faces, the following simultaneous equations arise: Cot2
2(K+ 1)
(t-
2(K+ 1)
a3
(f
-
Cl
c2
t +
az)(t
-
dt = 2(K+ l)(coll+ cl11 + C&J)=
a3)(t
-
-
al)(t
az)(t
-
-
dt = a3)(l
-
0,
(13)
a4)
Cor2+Clt+C2
a4
JJ
al)@
+
2(K
+
1)(coJ2
+
ClJl
+
c2.&3)
=
0.
(14)
a4)
In terms of the complete elliptic integrals of the first, second and third kinds, K(k), E(k) and ZZ({‘,k), the integrals lo, Ii and Z2can be written as 2
ZI-J = (a4
i
12 = i
(u4
_ .:,(.,
+ @(a4
_ u,)
-
a2)(a3
[(a4
-
-
(15)
K(k),
al)
-
a2)(03
-
GW)
+
(a4@4
-
al)(al
+
a2 + u3
+
u4)n(t2,
ul
-
a2))W)
(a4
k2 =
(a4
-
a3)(a2
-
ad
(a4
-
a2)(a3
-
al)
+
02)
k)],
(17)
where e2 _ ’
-
‘2
a4 -
a2
(18)
Similarly, Jo, Ji and 52 are given by eqs (15)-(18) with ul, u2, u3, u4 replaced by u3, u4, al, ~2, respectively. In particular, Z. = Jo. Solutions of eqs (13) and (14) are then
c, =
eco, I
1
c2=
z2J1 - J2z1 Co, ZoUl- Jd
Co
=i((Nl+N2) -
(Nl - N2)eZiB).
(19)
The expressions in eq. (19) reduce to those given in ref. [16] for a pair of identical cracks located symmetrically about the x2-axis. 2.2. Problem 2 or 3: two straight cracks subjected to the stress field of edge dislocations First consider an isolated edge dislocation at the location z. on the trace of the crack deviations in the infinite solid containing the two straight cracks. In analogy to the treatment of a single kinked crack in ref. [4], this problem is itself decomposed into two subproblems: (i) an untracked infinite solid subjected to the stress field of the dislocation, and (ii) an infinite solid containing the two straight cracks. In Subproblem 2, the two cracks are subjected to the negative of the normal and shear stresses acting on the trace of the two main cracks in Subproblem 1. In Subproblem 1, the stress functions untracked solid are given as (e.g. ref.[4])
(pd’ and $d ’ due to an edge dislocation
(pd’(z)= 01,
(20)
(z - zo)
+d ‘(z)
=-+(z -
a! zo)
CYi5J (z -
in an
zoj2
(21) ’
669
Morphologically complex cracks
where a = ,u([u,] + i[ue])eie/(ni(K + 1)) characterizes the dislocation. The normal and shear stresses on the trace of the two main cracks are then given by
t$@ + tid'(t)v
022(t) - iol2(t) = pd'(t) + bDd'(t)+
(22)
where t is as before the variable on the cracks and @j”(f) is the derivative of %‘(t) with respect to t. Proceeding to Subproblem 2, the normal and shear tractions on the cracks are obtained from eqs (20)+22), so that eqs (3) and (4) yield
P(f) = -
qzo-
(
A+%+
20)
Z())2> ’
(t -
q(t) = 0.
(23)
Also, by eq. (7), P,(z) = Cr ‘z + C2’ with CO’= 0, since remote loading is absent in this subproblem. The function X(z) remains as in eq. (6). The stress functions &‘(z) and w,‘(z) for this subproblem are then, using eqs (1) and (2) (24) Substituting eq. (23) in eq. (24) results in the following terms, apart from the factor -1/(2rriX(z)) to be integrated: a! --=-
X(t)
o
X(t)
---
t - zo (t - z) X(t) --=-a! t - ,5-J(t - z)
qzo -
20) X(t)
-=_
(t - zl)y (t - 4
a
(25)
t--z0 > ’ a
z-z, - 1
a.50[ z-20
X(t)
X(t) ---
(
X(0
X(0 --( t-z
Equations (25)-(27) show that the integrations
(26)
t-.20 > ’
t-z
t-50
ii -5,). )I
can all be reduced to integrations
(27)
similar
to X(z)-z’++z+$-:
,
(28)
>
where A = al + u2 + a3 + u4, B = ulu2 + ~23~4+ (aI + ~2)(~3 + ~4). The integration result in eq. (28) is obtained using the method explained in ref. [15]. Using eqs (25)-(28), the stress functions of eq. (24) become
@c’(z)= o,‘(z) = -(aF(z, zo) + aF(z, 20) + Cr(zo- &)G(z, 50)) +
(Cl
’ + w$2
+
C3’)
)
(29)
where (30)
Jqz
zo)
=
1- X(z0)I-w)= 2(z- zo) G(z, 5,) =
1 - N~OYW =
2(z - ZlJ)2
1 2(z -
am
20)
azo
X(Z0)
_ 4(Z
- Fl (z, zo), zo>
’
4
1
- 50)X(Z) ci=l 20 -
Ui ’
(31)
J. NIU and M. S. WU
670
1 = 2(z - &)*
(32)
- GI (z, 50)
and similarly for F(z, 50). Note that F(z,ze), F(z, 20) and (ZO- &)G(z, 5s) are not singular at z = za, even when z = zo = 50, provided that z. does not coincide with al, a2, a3 or u4. Also, the new functions Fi(z,za) and Gi(z, 50) have been introduced in eqs (31) and (32) respectively. Equation (10) with m = 1 and eq. (29) are used to determine the constants Ci ’ and Cz’. For Problem 2 in which z,-,is the location of the dislocation associated with crack 1, the contour A1 encloses crack 1 and the dislocation at zo, whereas the contour A2 encloses only crack 2: aF(z,
--K
zo) + aF(z,
Zo) + i(zo
- fo)G(z,
(cl’+++(c2’+c3’)
Zo) -
X(z)
f[ A!+
cS(zo dz+ aF(Z,zo)+ f[
(Cl ’+ a)? + (C2’ + C3’)
.&)G(Z,Zo) -
aF(T,Zo)+
WI
A,,
dz
2~([u~l + ibd)eie = hi(K +
=
1 I
for k = 1,
l)Cr
for k = 2.
0
(33)
By shrinking Ai to the contour of the kinked, branched, or zigzag crack, and A2 to crack 2, the integrations in eq. (33) can be evaluated. Denoting the elliptic integrals as AI
-
=
F
;
zo)
dt,
&(zo)
=
s
a2
Az(50) =
F
;‘
zo)
dt,
a3
(34)
a4
J
- G ;(t, Zo) dt, &(Zo) =
al
- G ;(t, 20) dt,
(35)
J 03
and similarly for A,(.&), B~(.?o), where F r(t, zo), etc. denote the values on the upper faces of the main cracks, the solutions of eq. (33) are
c, ’+ a!
=
P,,a + P,*(zo - z&i!,
C*’+ C3’ = -I$,(2
(36)
- P**(zo - ‘?o)&,
(37)
where Pll
=
Ai
+ AI
+ Jri -
Bl(ZO)
-
&(~o)
(38)
9
11 - JI
p,2 = A2(50) - B2(50) II - JI
p21 =
(Al(zo)
+ AI
+ ni)J,
(39)
’
- (&(zo)
+ B,(zo))Z,
Zo(Z, - Jl)
p22 = -42(5o)J1 - Bz(zo)Z1 Zo(Z1 - JI)
7
(40)
(41)
’
In Problem 3, where z. is the dislocation position associated with crack 2, Cl’ and Cz ’ are also given by eqs (36) and (41), except for eqs (38) and (40) which should be replaced by Aleo) PI1
=
+ 4@0)
-
&(zo)
11 - JI
-
&Go)
-
Ji
,
(42)
671
Morphologically complex cracks
(b)
crack 2
._-..+
4
a4
Fig. 2. The geometry of (a) kinked, (b) branched and (c) zigzag cracks. The main parts of the cracks have their tips at al, a~, q and ~4. The lengths and angles of the kinks, branches, or zigzags are denoted respectively by I, and 0,. The angle 0, is measured from the positive x,-direction. The two cracks 1 and 2 are located on the negative and positive xl-axes, respectively.
p21
=
(AI
+ ~l(~OM
~OVl
(Bl(ZO) -
+ Bl(~O)
+ iv1
(43)
A>
The stress functions of eq. (29) can be rewritten as &‘(z) = o,‘(z) = -a[F(z, G(z, 50) + a!
PllZ
-
zo) + F(z, z-j)] - ci(zo - 20) p21
+ cr(z0
-
-Jw
50)
p12z
-
-w)
p22 .
(44)
The stress functions for the problem of an edge dislocation in a body containing two collinear straight cracks can now be written consistently in terms of 4’(z) and $‘(z) by superposition of the two subproblems, i.e. 4’(z) =
(od’tz)
+ boc’(z)t
(45)
+‘(z) =
‘kd’I(z)
+ @c’(z),
(46)
where @j’(z), &i’(z), (PC’(Z)are given by eqs (20), (21) and (44) respectively, and &‘(z) by the formula in ref. [ 151: lcIc’(z)= $%‘(z)- (oc’(z) - %“(Z). (47) The function am” is the derivative of am’ with respect to z. Moreover, bpc’(z)= (PC’@). Finally, it is noted that the stress functions for a distribution of dislocations can be obtained from eqs (45) and (46) by integration. The variable a is then interpreted as a complex dislocation density function.
3. INTEGRAL
EQUATIONS
FOR KINKED CRACKS
Consider a pair of kinked cracks with the main parts collinear (Fig. 2(a)). Let the lengths of kinks 1 and 2 be Ii and 12, respectively. The stress functions derived above for Problems 1, 2 and 3 are exact, i.e. the traction-free conditions on the two main cracks are satisfied exactly. To satisfy the traction-free conditions on the kinks, the normal and shear tractions acting along the kinks in the three problems are superimposed and set equal to zero. The normal and shear stresses in polar coordinates (r,@ are given by[15]
672
J. NIU and M. S. WU
iafl = v’(z) + q’(z) + ezie(&Y(z) + q’(z)).
UM +
(48)
For Problem 1, substituting eqs (11) and (12) in eq. (48) and noting also that q’(z) = F’(z) - p’(z) - z#’ as in eq. (47) the normal and shear stresses og, crz on the kink locations due to the remotely applied principal stresses are given by
ug+ iaz
= 2Re
‘Oz2 + clz + C2 + e2iB co‘f*+c,z+c2 X(z) wz? 1 K
- coz2+c~z+c2 X(z) 2coz + Cl X(z)
[
_ (l -2e2’eJ(rf + F/J +
)
,2iO(,- _ z)
1
coz2 + ciz + cz 4 2X(z) ($&J]~ -
(49)
where I’, CO, Ci, Cz are given by eqs (5), (8) and (19), and X(z) by eq. (6). Also, the positions z along kinks 1 and 2 are written as zi = u2 + seiel, z2 = u3seiB’,
(50)
where 0 I s I 1i denotes the position variable along kink 1 with origin at (az,O), and 0 IS I 1, the position along kink 2 with origin at (as,O). Furthermore, 81 and 62 denote the angles of kinks 1 and 2, which are positive if measured counterclockwise from the positive xi-direction. For Problem 2 or 3, the normal and shear stresses in the cracked plane due to the edge dislocation at z. can be written in the polar frame oriented at angle 8 to the xi-axis by substituting eqs (45) and (46) into eq. (48): (51) where iY(z,
ZO,
1
PllZ - p21
0) = (e2ie - 1) F(z, zo) + F(z, 20) -
E
X(Z)
PiI F’(z, zo) + F’(z, 20) - X(z)
PllZ +
-
p21
2X(z)
1
+ e2ie(z- 2)
4 1 + (zo - Zo) i=, z - ai 1 c (52)
V(z, zo,
e>= -e2ie
[
F(z, zo) +
F(z,
-
(‘“&y
F(,?, ZO)+
F(Z,
-
(p”~~p~~)]+tzo-io)
[tz-,e2ie(G~tz,io)-~+p’~~z~~~)] Pl2Z
-
p22
X(z)
(53)
)I *
In eqs (52) and (53) F’(z, zo) =
wz, zo)= az
- 1 + mzo)Ixt4 + 2(z - zo)2
Wzo)
4
c- Z4(Z-ZO)X(Z)i=l
1 Ui'
(54)
Morphologically complex cracks
G'(z, 20) =
+
X(&l) 4(Z -
ZO)~X(Z)
aG(z, 20) az =
c4 i=l
-
1 - -w00)/w +
-
4
X(Z0) 4(Z - Z~J)~X(Z)
2(2 - Zo)3
1 20
613
c+l
1 Z -
Qi
-wo) Ui
+
8(Z
-
zO)x(z)
(55)
(~L)(g&J, i=l ’ -
ai
and similarly for F’(z, 20). Other functions F and G are defined in eqs (31) and (32). The positions z. of the edge dislocations on kinks 1 and 2 are distinguished by writing zoi = a2 + teiel, 202 =
(56)
a3 + teib,
where the definition of t is analogous to that of s. Similarly, the dislocation density distributions on the two kinks are distinguished by the subscripts in a,(t) and aZ(t). Due to the dislocation distribution on kink 1, the normal and shear stresses on kink 1 are +~l(f)U(z~,zo~,~~)+~~(t)V(z~,zol,~l)
I
dt,
and on kink 2
+~1W(z,,zo1,f322)
+~1(OJ'(z~,zo1~~2)
(58)
dt, >
where use has been made of eq. (51). Similarly, due to the dislocation distribution on kink 2, the stresses on kinks 2 and 1 are given by eqs (57) and (58) with the subscripts 1 and 2 interchanged. Superimposing the stresses on kinks 1 and 2 in the two problems and setting them to zero, two coupled singular integral equations in the dislocation densities al and a2 are obtained. These equations are of the Cauchy type. For kink 1, the equation is
+c$(z~,
el) + ioz(zi, el) = 0.
(59)
For kink 2, the equation is +
+~dW(~2,~01~
Q~~(z~,
zo2, e2)+
~2w(~2,
e2) +~l(0v(~2~~01,e22)
zo2,
e2) dt
dt+$g(z2,
e2) +i0,"(z2,e2)
=
0.
(60)
>
In writing the above equations, the singular terms involving l/(s - t) have been isolated. The second set of integrals, which represent the stresses on one kink due to the other kink, are non-singular. EFM 57/6-D
614
J. NIU and M. S. WU
4. INTEGRAL EQUATIONS
FOR BRANCHED OR ZIGZAG CRACKS
Consider a pair of branched cracks 1 and 2 with the main parts collinear (Fig. 2(b)). Each crack has two branches so that the total number of branches K is 4. The positions z and z. on the two branches j = 1, 3 of crack 1 can be distinguished by zj and zoj, where Zj =
Zoj =
U2 +
a2 +
se’+ (0 5 S _( lj),
te’@(0 5
t (
(61)
Zj).
(62)
Similarly, for the two branches j = 2 and 4 of crack 2 Zj =
Zoj =
a3 + se’4 (0 5 s < b),
(63)
te’4 (0 I: t 5 4).
(64)
U3 +
For a pair of zigzag cracks with a total of K zigzags (Fig. 2(c)), the positions z and z. on the various zigzags of crack 1 can be written as zj and zoj (j odd): Zj =
Zoj =
a2 +
a2
+
Iie’@+ ZseiB3 + . . . + se’4 (0 5 s 5 Zj),
(65)
Zleiei + Z3eie3+ . . . + te’4 (0 5 t I: 4).
(66)
In eqs (65) and (66), Ii, 6, .... 4 denote the lengths of the zigzags, and 8i, es, .... 13,denote the angles of the zigzags with respect to the positive xi-direction. For the zigzags of crack 2, zj and zoj 0’even) are given by Zj =
Zoj =
a3 +
a3
lzeiBz+ laeie4+ . . . + se’3 (0 5 s ( Zj),
(67)
Z2eie2+ ZJeie4+ . . . + te’+ (0 5 t ( lj).
(68)
+
The K integral equations are obtained by the superposition of the normal and shear stresses at zj in the following (1 + K) problems: (i) two straight cracks in an infinite body subjected to remote loading, and (ii) two straight cracks in an infinite body subjected to the dislocation distributions on each of the K zigzags. Use of eqs (49) and (51) and appropriate zj and Zoj leads to
dt +
z,, 6,) + ak(t)V(zj, -$ak(t)(i( zok,
k=l
ZOO,
0,)) dt + a$$(zj,$I+
iu,W(zj,$1 = 0,
(69)
o
where the K singular integral equations of the Cauchy type are obtained by setting j = 1, 2, 3, . ..) K. Setting K = 2 reduces eq. (69) to eqs (59) and (60) for a pair of kinked cracks.
5. SOLUTIONS
OF INTEGRAL
EQUATIONS
AND STRESS INTENSITY
FACTORS
The method proposed by Gerasoulis[lO] for solving integral equations is used in this paper and other works (e.g. refs [13, 141). The dislocation density a(t) is expressed as a product of a weight function w(t) and a continuous function 4(t). The latter is approximated by piecewise quadratic polynomials. The coupled singular equations reduce to a system of linear algebraic equations after appropriate substitutions. Additional conditions are required, as discussed in the following outline of the solution procedure. First, the limits of the integrals in eqs (59) and (60) or eq. (69) are normalized to -1 and + 1 using the transformations
615
Morphologically complex cracks
where 1 equals Ii, I*, etc. as appropriate. branch, or zigzag is written as
The dislocation
density function
for the jth kink,
cXj(j(t’ =) w(t’)(p#)
(71)
where w(t’) =
1 (1 - f’)“2(1 + f’)V’
(72)
and 0 I q I l/2. Furthermore, the l/2 and q singularities at t’ = 1 and t’ = - 1 are built into the definition of a,(P) through w(t’). In the solutions to follow, r~is taken as l/2 since for comparable levels of accuracy the assumption of an incorrect exponent requires less numerical computations than the use of the correct value [13]. This is because a large number of numerical integrations are needed for r~lying between 0 and l/2, whereas the integrals can be evaluated in closed forms for q = l/2. With q = l/2, the additional conditions required for solving the integral equations are obtained by setting qJ{-1) = 0, as in refs [4, 11, 13, 141. In the case of the kinked or branched cracks, these conditions are imposed at the points where the kinks or branches join the main cracks. In the case of the zigzag cracks, they are imposed at the points where the zigzags join the main cracks as well as where the zigzags join each other. After substituting eq. (71) into the integral equations, the equations are discretized and the Lagrange interpolation formula for three points is used to approximate the functions appearing in them. See ref. [lo] for details. The resulting expressions can be integrated exactly for the form of w(t’) chosen in eq. (72) with rl = l/2. The use of a set of R collocation points and the additional conditions generates a system of linear algebraic equations with q,{t’) at various values of t’ as unknowns. After solving the algebraic equations, the mode I and mode II SIFs (Kr, Kn) are easily calculated from the stress field around the crack tip. This requires the value #j(l) at the crack tip. The formula used for kinked cracks (e.g. ref.[4]) is also valid for branched and zigzag cracks: Kt + i&I = (2Jr)3’2eie1(pj(l)$‘2/2,
(73)
where 0, is the orientation of the kink, branch, or zigzag at whose tip the SIFs are computed.
6. NUMERICAL
RESULTS
In this section, the geometrical parameters 2c and 1 denote respectively the length of the main crack and the length of the kink, branch, or zigzag. For kinked or branched cracks, the parameter 2 d denotes the horizontal distance between the inner tips of the main cracks. For zigzag cracks, 2 d denotes the horizontal distance between the knees of the last zigzags of the two cracks. Unless otherwise stated, the two cracks are arranged symmetrically about the x*-axis. When the cracks are unequal, subscripts are used to distinguish between them, e.g. 2ct, Ii, etc. All solutions are obtained with a Cray supercomputer and six IBM RISC/6000 workstations. Only the SIFs at the tips of the kinks, branches, or zigzags are computed. Except for the solutions in Test Case 2 below, they are generated with values of R between 64 and 160. A solution generated with a given R is accepted when it is practically indistinguishable, on the scale of the figure, from another generated with an R twice as large. Also, the sign of the mode II SIFs is defined with respect to a local rectangular frame attached to the knee of the kink, branch, or zigzag such that the local positive xl-direction points from the knee towards the tip. Negative mode I SIFs indicating the compressive nature of the local stress state are predicted for certain kink or branch angles, as is also the case for single cracks in refs [5,8], but are not shown due to the possible complication of crack face contact and numerical inaccuracy.
676
J. NIU and M. S. WU
I””
I””
,
0
e=o”
9
0
t.)=15’
A
log
I
8=450 8=75’
-
’ 11
1
Lo[4]
,(@m
Fig. 3. Comparison of the SIFs of two identical kinked cracks located remotely from each other with the single crack solutions reported in ref. [4].
6.1. Test cases 6.1.1. Case 1: two identical kinked cracks at a large distance from each other. Consider two identical kinked cracks such that the inner tips of the main cracks are separated by a distance equal to at least 198 times the main crack length. A remote tensile stress 0 is applied in the vertical direction. Figure 3 plots the normalized SIFs & = K~/a,/ifZ and &;I = Kr~/afi of the kinks against logie(c/l). The symbols in the figure indicate the results for various kink angles computed using the current model. The full lines indicate the results computed from Lo’s[4] model for a single kinked crack. The two sets of results are in excellent agreement. Furthermore, it should be mentioned that the SIFs for two identical branched cracks separated by a large distance and subjected to a remote tensile stress also agree very well with those reported in ref. [4] for a single branched crack. 6.1.2. Case 2: two identical straight cracks with strong or weak interactions. Since to the best of our knowledge no SIF solutions are available in the literature for strongly interacting kinked, branched or zigzag cracks, we investigate the case of two main cracks each joined by a “kink” of 0” or 180” angle (see the diagram included in Table 1). The cracks are subjected to the tensile stress o in the vertical direction. Table 1 compares three sets of solutions for K~/q/m of the inner tips as the crack separation becomes smaller: (i) the exact solutions of Erdogan [16] obtained by treating each crack as an uninterrupted straight crack, (ii) the approximate
Morphologically complex cracks
611
Table 1. Comparison of the mode I SIFs, K&,/m, of two cracks adjoined with 0” or 180” kinks with the exact solutions for straight cracks. Acceptable accuracy is obtained for 6 2 0.01 even for R = 20. For S < 0.01, increasing R improves the solutions c/l = 1 J=d-’ ?Cfd 0.0001
0.001 0.01 0.02 0.05 0.1 0.2 0.99
Exact solution Erdogan[l6] 13.347 5.395 2.372 1.905 1.473 1.255 1.112 1.000
c/r = 100
R = 20
64
160
20
64
160
5.714 4.568 2.356 1.900 1.471 1.254 1.111 0.999
8.592 5.204 2.367 1.904 1.473 1.254 1.112 1.OOo
10.889 5.351 2.371 1.904 1.413 1.255 1.112 1.000
13.153 5.372 2.363 1.898 1.468 1.251 1.109 0.991
13.301 5.382 2.367 1.901 1.471 1.253 1.111 0.999
13.321 5.384 2.368 1.902 1.471
1.254 1.112 0.999
obtained by treating each crack as a main crack joined by a kink with c/Z = 1, and (iii) the approximate solutions obtained as in (ii) but with c/l = 100. All the approximate solutions agree well with the exact solutions when the separation parameter 6 = (d- 1)/(2c + 6) is greater than or equal to 0.01. At 6 = 0.01, the worst approximate solution, i.e. that computed with c/l = 1 and the polynomial degree of R = 20, has an error of less than 0.67%. At 6 = 0.001 and 0.0001, the approximate solutions tend towards the exact solutions of 5.395 and 13.347 (to three decimal places) as R increases. The best approximate solution at 6 = 0.0001, computed using c/f = 100 and R = 160, is 13.321. This represents an error of less than 0.2%. It may also be noted that the value of q = l/2 used to obtain the approximate solutions is at the opposite extreme of the correct value of q = 0 for a zero degree kink. This suggests that the approximate method of using q = l/2 and co,i-1) = 0 yields reasonably accurate solutions even in the case of strong interactions. solutions
6.2. Kinked cracks 6.2.1. Two identical kinked cracks. For two identical kinked cracks subjected to vertical remote tension Q, the dependence of ir and &;I on the kink angle 8 is shown in Fig. 4 for c/l = 1 and in Fig. 5 for c/l = 100. The SIFs plotted are those of the left kink, as indicated by a dot at the tip. Furthermore, four cases corresponding to d/l = 1.05, 1.10, 1.25 and 1.50 are investigated, with d/l = 1.05 representing the smallest separation between the cracks. One interpretation of these ratios is that for the same c, the kink length I and the crack separation 2 d in Fig. 5 are two orders of magnitude smaller than those in Fig. 4. For a given d/l, the mode I SIF is largest at 8 = 0” and decreases as 0 increases. In the case of c/l = 1, the mode I SIFs become negative as 8 increases beyond -85”. Negative mode I SIFs are also predicted in the case of a single kinked crack, as inferred from Fig. 5 of ref. [2]. The decrease of Kt with tl can be explained by the reduction of normal load on the kinks, the increasing separation between the kink tips, and the increasing shielding between each kink and the adjoining main crack at large 8. For a given 19, the mode I SIFs also decrease as d/l increases, as expected. At the larger angles, however, the dependence on d/l in the range between 1.05 and 1.50 is small. Comparison of Figs 4 and 5 shows that in the stronger interaction case of c/l = 100, the sensitivity of the mode I SIFs to d/l is greater at a given 8, and the SIFs remain generally large for 8 < 90”. For instance, at c/l = 100, & >- 1 for 8 < -9O”, whereas at c/ 1 = 1, k* >- 1 only for 8< - 40”. The mode II SI_Fs display maxima at 8 between 40” and 70”, depending on the value of c/l and d/l. Note that Kn = 0 at 8 = 0” for both ratios of c/l, but is -0.5 for c/l = 1 and -1 for c/ 1 = 100 at 8 = 90”. At both 0” and 90”, no remote shear stress acts on the kinks, so the rather large values at 0 = 90” must arise from the action of the main cracks and the interaction effects. Comparison of Figs 4 and 5 also shows that the sensitivity of & to d/l at a given 0 is greater in the case of c/l = 100 than in the case of c/l = 1. To show how the mode mixity varies with the kink angle, the parameter y = tan-‘(&/Ki) is plotted against 8 for c/l = 1 and 100 in Fig. 6. At 0 = 0”, y = 0”, i.e. a pure mode I state. As 0 increases, y increases towards 90”, a pure mode II state. In the case of c/l = 1, y reaches 90” at B - 83”. In the case of c/l = 100, y never reaches 90” within the range of 0 considered. Two
678
J. NIU and M. S. WU
,
3.0
,
1
I
I
I,
1
0-OooOO 2.0'%.,ooo
0
00
“~~OOOo
?n00? 0 0
I
I
0
d/l= 1.05
0
d/l= 1.10
0
d/l= 1.25
A
d/l = 1.50
I
-
?a0 oooo ?o0
r’AAAAAAAAAA~qo “h 1.0
-
‘.:?11111111**~*
%** 0.0
I 0.0
I
I,
I1
1
I
I
30.0
I,
60.0
‘fifi,,. 1
90.0
Kink Angle 8
0.6
0.6 t
0.4
G? 0.2
0.0 1 0.0
60.0
30.0
90.0
Kink Angle 8 Fig. 4. Variation of the SIFs of identical kinked cracks with the kink angle 0 for different separation parameters d/l. The ratio c/l = I.
important characteristics associated with increasing the interaction effect are: (i) increasing the ratio c/l from 1 to 100 at a constant value of d/l suppresses the transition of mode mixity towards pure mode II as 8 increases, (ii) decreasing d/l from 1.5 to 1.05 at the constant value of c/l = 1 or 100 does not change the mode mixity significantly. The distance between the kink tips changes by two orders of magnitude in the former case, while it changes by half an order of magnitude in the latter case. Thus, how interaction affects mode mixity depends on the details of the crack geometry and configuration. 6.2.2. Two kinked cracks with unequal main parts. In Fig. 7, asymmetry of the crack configuration is introduced by changing the length of the left main crack, 2~. The two kinked cracks are otherwise symmetrical about the vertical axis at the origin. Specifically, c/l = 1, f3 = 15”, d/ 1 = 1.1. The main cracks are subjected to a remote tension 6. It can be seen that as q/c changes from 0 to 2, there is a significant change of & and a perceptible change of & for both the left and right kinks. For both kinks, the slope of ii at cl/ c = 1 implies that a small perturbation from symmetry leads to small but not insignificant changes in $. It is also clear that & of the kink adjoined to the longer main crack is larger. In fact, a closer examination of the lower plot of Fig. 7 shows that the same conclusion can be made for the absolute value of &.
679
Morphologically complex cracks 10.0
4
8.0 -
0
0
‘.O~o Lz
? s
6.0 -
0 0
00
OrI
0 0
000 0 0 Qcl O
d/1=1.05
??
d/f = 1.10 :
o
dll= 1.25
:
A
dll= 1.50
-
:
on00
-~oooooo 4.01% ! AAAAAAAAAA~~
000 ““QO 00
Q*fi, ‘AAAAA
2.0 -
0.0' 0.0
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‘A:::jilrilttrlrrrrt,,,ti
'
'
'
' 30.0 '
L
,
I
I
I L 60.0
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I
I
90.0
Kink Angle 8
Kink Angle 8 Fig. 5. Variation of the SIFs of identical kinked cracks with the kink angle 0 for different separation parameters d/l. The ratio c/l = 100.
6.2.3.Two kinked cracks with unequal kink lengths. In Fig. 8, asymmetry of the crack configuration is introduced by changing the length of the left kink, 1i. The two kinked cracks are otherwise symmetrical about the +-axis. Specifically, c/l = 1, 8 = 15”, d/l = 1.l. The main cracks are subjected to a remote tension rr. In the following, amplification or shielding of SIFs is discussed in relation to the perfectly symmetric _case of [i/l = 1. A strong amplification of both Kr and Kn occurs around 1,/l = 1.26 due_ to the very close proximity of the two kink tips. The amplification is greater in it than in Kii. In this case, a small perturbation of the symmetry of the crack configuration can lead to significant changes of the SIFs. The amplification is followed by a shielding of &, with & of the left tip reaching a minimum close to zero at 1i/1 = 2.6. The shielding is due_ to the overshooting of the left kink over the right kink tip. It is interesting to observe that KI of the overshooting and longer left kink is smaller than that of the shorter right kink, suggesting that the & reduction is greater on the kink whose central portion is shielded than the one whose tip is shielded. As 11/l increases beyond 2.6, however, & of the right tip continues to decrease while it of the left tip increases and tends to the value of an isolated crack of length Ii and orientation 15”. On the other hand, shielding of &r is less strong, since the in values for 1,/l> 2 do not differ greatly from the Ku value at 1,/l = 1.
680
J. NIU and M. S. WU
0.0 0.0
1
I
I,
I,
I,
I
30.0
11,
60.0
90.0
Kink Angle 0
120.0
I, -
c/l=
I1
1
I,
r
I1
1
100
90.0 g 'i
d/l = 1 .OS
60.0 -
3 II +
30.0 -
0.0 0.0
I, 30.0
I
I
I
I I 60.0
I
I
I 90.0
Kink Angle 8 Fig. 6. Dependence of the mode mixity y, defined as the inverse tangent of the ratio of the mode II to mode I SIF, on the kink angle 0 and the strength of the crack interactions. Increasing the interaction by decreasing c/l from 1 to 100 substantially retards the transition from mode I (7 = 0”) to mode II (y = 90”) as 0 increases from 0” towards 90”.
6.3.Branched cracks
6.3.1. Two identical branched cracks. In complete analogy to Fig. 4 for kinked cracks, Fig. 9 shows a pair of identical branched cracks subjected to the remote tension a perpendicular to the main cracks. All four branches are equal in length and c/l = 1. The crack configuration is symmetric about the xi- and x2-axes. The kr-0 and kn-0 curves are generated using d/l = 1.05, 1.lO, 1.25 and 1.50. The SIFs of the upper left branch are plotted against the common branch angle 0 in the range 2” I ~9I 90”. Solutions for 0 < 2” obtained using R < 160 are unreliable and are omitted. The trend of each SIF curve depends on the remote loading resolved on the branches and the complex interactions among the four branches. In particular, shielding of the branches of the same crack occurs in both modes I and II as 0 decreases. Consequently, & and &;I of the branches are smaller than those of the kinks, which do not shield each other. Furthermore, & of the branched cracks reaches a maximum not at 8 = 0” but at some 8 less than 15”, in contrast to kr of the kinked cracks. This suggests that as 8 decreases below -15”, the mode I shielding between the branches of the same crack increasingly dominates over the mode I
681
Morphologically complex cracks
t3 \, k
A~&" 1.5 -
A~AA
&AA
A~AAoOo ~~ 00 00
bA
L
00 00 00 00
2c,1
L 2dl
2c+-
_ O0
1.0 1 0.0
I
01
I I 0.5
*
1
I
I1 1.0
11
A
I 1.5
2.0
C,IC
1.0
0.5
L
E
?
0.0
is=
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA -0.5
-1.0 0.0
0.5
1.0
1.5
2.0
C,IC Fig. 7. Dependence of the SIFs on the asymmetry of kinked cracks as measured by the ratio of the lengths of the left and right main cracks, q/c. The kink lengths and angles are identical. The kink with the longer main crack has the larger mode I SIF as well as the larger absolute value of the mode II SIF. A small perturbation of symmetry about q/c = 1 leads to small but not insignificant changes of the SIFs.
amplification resulting from (i) the shorter distance between the branches of different cracks and (ii) the increased remote load normal to the branches. It also explains why the angle at which & is maximum decreases with decrease of d/f, since the increased amplification associated with the decrease of d/l can compensate more effectively the shielding effect of the branches of the same crack. Finally, zeroes of &r occur at angles below -15”, in conjunction with the existence of maxima for Et. 6.3.2. Two unequal branched cracks. Asymmetry is introduced by letting 1112= 10, keeping the left branch angle B = 30” and varying the other branch angle 0, = 180” - 0, from 0” to 90”, as can be seen in Fig. 10. The crack configuration remains symmetric about the xi-axis. Also, c/ I = 1 and d/l = 0.3, so that the short right branches lie within the region enclosed by the long left branches (the inset to Fig. 10 shows the short branches outside the region of the long branches for clarity). A remote tension ~7is applied in the x*-direction. The SIFs plotted are those of the upper left and right branches (A and B). Although the mode I and mode II SIFs of the crack with long branches do not change greatly as the angle of the short branches changes, they represent changes of roughly 50% com-
J. NIU and M. S. WU
682
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8= W,dll=
Q
1.1
& %
2c
%2d%
2c
3-
+ 00
00
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0 0
AA 0
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0.0 0.0
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1
1
I
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oo” I
$2 AA 1
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I
1
2.0
4.0
6.0
2.0
4.0
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1,/l Fig. 8. Dependence of the SIFs on the asymmetry of kinked cracks as measured by the ratio of the lengths of the left and right kinks, It/l. The main crack lengths are identical. Because of the proximity of the two kinks, a small perturbation of symmetry about It/l = 1 leads to large changes in the SIFs. The interaction effects are complex: there is significant shielding of the mode I SIF of the longer left kink for 1.6 < 1,/l < 3.6, but only a more gradual decrease of the mode I SIF of the shorter right kink over this range.
pared to the SIFs of the same crack existing in isolation (see the case of d/l = 1.50 and 8 = 30” in Fig. 9). Thus, the presence of the right branched crack, although not the orienmtion of the right branches, does have a significant effect on the long braAnches. Note also that Kn becomes zero and & attains a maximum at approximately the angle 02 = 18”. If the left branched crack exists in the presence of a right crack that has not yet bifurcated, then branches may nucleate at approximately this angle from the right crack under the maximum j?t condition. 6.4. Zigzag cracks 6.4.1. Dependence of SIFs on the shape of two identical zigzag cracks. The inset to Fig. 11 shows two identical zigzag cracks each containing a trapezoidal step. The length of each main crack is 2c, and the two horizontal zigzags farthest from the main cracks have length c/100. The spacing between the tips of these two zigzags is set equal to O.O4c, i.e. d = O.O3c, to obtain strong interaction effects. For comparison, the case of practically no interaction is also studied with the spacing set equal to 599.98c, i.e. d = 300~. The base length and height of each trapezoid are 2c and
683
Morphologically complex cracks
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’
’
’
’
30.0
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’
’
’
’
60.0
’
’
’
’
’
90.0
Branch Angle 8 Fig. 9. Variation of the SIFs of identical branched cracks with the branch angle 0 for different separation parameters d/l. The ratio c/l = 1 for all four branches. Note the occurrence of the maximum mode I SIFs at branch angles between 0” and 15”. This arises from the increased shielding between the branches of the same crack.
1/(2&)c(0.289c), respectively. Keeping the above parameters fixed, the length 1s = 1, of the uppermost horizontal zigzags is changed from 0 to 2c in such a manner that each trapezoid always possesses a vertical axis of symmetry. This alters the shape from triangular through trapezoidal to rectangular. A r_emote tension r~is applied in the vertical direction. Figure 11 shows that Kt of the strongly interacting zigzag cracks decreases slowly as the triangle evolves into a trapezoid, but decreases rapidly when the trapezoid evolves into a rectangle. The same is true in the case of no interaction, although the decrease of ir is considerably smaller. The percentage reduction of -40% due to the change of a triangle to a rectangle is approximately the same for both cases of strong and no interaction. In summary, the results suggest that zigzag cracks with triangular or trapezoidal steps are less efficient in reducing the mode I SIFs compared to zigzag cracks with rectangular steps. The values of Z?u of the left zigzag tip are also shown in Fig. 11. The absolute values are small in comparison to &. In the case of strong interaction, &i increases with 1s after 1s> 1SC, in contrast to the decrease of & with 1s.
684
J. NIU and M. S. WU
I,
iz
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6.4.2. Dependence of SIFs on the step height of two identical zigzag cracks. In Fig. 12, each zigzag crack has a rectangular step of width 2c. The height h of the step is varied from zero to 0.5~. All other dimensions remain as in Fig. 11. A remote tension Q is also applied vertically. It is seen that increasing the height of the rectangular steps reduces Ki very effectively. In fact, Kr reduces from the value of 4.36 at h = 0 (straight cracks) to the value of 2.27 ath = c/2 when d k 0.03~. The reduction from 1.41 to 0.88 is considerably less when d = 300~. The percentages of reduction are -4_8% and -38%, respectively. The rate of reduction decreases with increase in h. In contrast, &I increases from zero to a maximum at small h and then decreases with increase in h. As in Fig. 11, the absolute values of & are small in comparison to &. Solutions for very small h, i.e. 0 < h < O.O2c, are omitted in Fig. 12 due to insufficient accuracy. When h is very small, the small distance between the ends of the very short vertical zigzags appears to play a negative role in the solution accuracy, since an incorrect exponent of the stress singularity is assumed at both ends. In the limiting case of h/c = 0, the solution is obtained by solving the problem of four horizontal zigzags of lengths 0.5c, c, 0.5~ and 0.01~.
685
Morphologically complex cracks
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Fig. 11. Dependence of the SIFs of zigzag cracks on the shape of the steps. Rectangular steps (/s/1 = 2) reduce the mode I SIFs at the tips of the zigzags more significantly than triangular (Zs/l = 0) or trapezoidal steps (0 < Is/l < 2). The reduction in the case of strong interaction (d = 0.03~) is greater than that in the case of no interaction (d = 300~).
The solution of 4.36 obtained using R = 128 is within 2.6% of the exact solution of 4.25 obtained from the direct analysis of straight cracks. For finite heights h/c > 0.2, the solutions obtained using R = 64 and 128 are practically indistinguishable. 7. CONCLUSIONS AND CLOSURE REMARKS The strong interactions of two kinked, branched and zigzag cracks are studied in this paper using the method of continuous dislocation distributions. The traction-free boundary conditions on the collinear main cracks are satisfied automatically, while imposing the same condition on the kinks, branches, or zigzags yields a system of coupled singular integral equations. The equations are solved by the use of piecewise quadratic polynomials, leading to a system of linear algebraic equations. The crack configurations are subjected to remote tension perpendicular to the main cracks, and the SIFs are calculated at the tips of the kinks, branches, or zigzags. The results for kinked cracks show that strong interactions can retard the transition of the mode mixity from mode I to mode II as the kink angle increases. A small but finite perturbation to the symmetry of the configuration of the kinked cracks, e.g. unequal kinks, can cause significant changes to the SIFs
686
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of the kinks. Also, branched cracks display more complex interaction effects than kinked cracks due to the amplification and shielding of the four branches and the main cracks. Finally, zigzag cracks with rectangular steps reduce the mode I SIFs more than those with triangular steps. Increasing the height of the rectangular steps also reduces the mode I SIFs significantly. Whether the shape changes from a triangle to a rectangle or whether the step height of the rectangle increases, the reduction of the SIFs is greater in the case of strong interaction than in the case of no interaction. This paper focuses on the computation of the SIFs of complex cracks. The stability of the crack path is not addressed. Crack path instability has been used to explain the experimentally observed phenomenon of why originally collinear straight cracks avoid each other. For instance, Melin [19] suggested that the straight crack path is unstable. The analysis assumes that the cracks are initially slightly curved. First order solutions to the governing integral equations are given. Further investigation, however, is necessary to determine whether the initially assumed
Morphologically complex cracks
crack geometry and crack configuration the crack growth.
687
have an effect on the subsequent directional stability of
Acknowledgements-The authors gratefully acknowledge the support of the Office of Naval Research in this investigation (Grant No. N00014-92-J-1203). The computational part of this work was made possible by the use of a Cray supercomputer at the College of Engineering and Technology, University of Nebraska-Lincoln, and six IBM RISC/6000 workstations purchased with an earlier grant of the National Science Foundation (Grant No. MSS-9111895).
REFERENCES I. Bilby, B. A. and Cardew, G. E., The crack with a kinked tip. International Journal of Fracture, 1975, 11, 708-711. 2. Chatterjee, S. N., The stress field in the neighborhood of a branched crack in an infinite sheet. International Journal of Solids and Structures, 1975, 11, 521-538. 3. Kitagawa, H., Yuuki, R. and Ohira, T., Crack-morphological aspects in fracture mechanics. Engineering Fracture Mechanics, 1975, I, 515-529. 4. Lo, K. K., Analysis of branched cracks. Journal of Applied Mechanics, 1978,45, 797-802. 5. Cotterell, B. and Rice, J. R., Slightly curved or kinked cracks. International Journal of Fracture, 1980, 16, 155-169. 6. Horii, H. and Nemat-Nasser, S., Brittle failure in compression: splitting, faulting and brittle-ductile transition. Philosophical Transactions of the Royal Society, London, Series A, 1986, 319, 337-374. 7. Vitek, V., Plane strain stress-intensity factors for branched cracks. International Journal of Fracture, 1977, 13, 481501. 8. Theocaris, P. S., Asymmetric branching of cracks. Journal of Applied Mechanics, 1977,44, 611-618. 9. Andersson, H., Stress intensity factors at the tips of a star-shaped contour in an infinite tensile sheet. Journal of Mechanics and Physics of Solids, 1969, 17,405-417. 10. Gerasoulis, A., The use of quadratic polynomials for the solution of singular integral equations of Cauchy type. Computers & Mathematics with Applications, 1982, 8, 15-22. 11. Obata, M., Nemat-Nasser, S. and Goto, Y., Branched cracks in anisotropic elastic solids. Journal of Applied Mechanics, 1989, 56, 858-864. 12. Gao, H. J. and Chiu, C. H., Slightly curved or kinked cracks in anisotropic elastic solids. Inrernafionai Journal of Solids and Structures, 1992, 29, 947-972. 13. He, M. Y. and Hutchinson, J. W., Kinking of a crack out of an interface. Journal of Appfied Mechanics, 1989, 56, 270-278. 14. Miller, G. R. and Stock, W. L., Analysis of branched interface cracks between dissimilar anisotropic media. Journal of Applied Mechanics, 1989, 56, 844-849. 15. Muskhelishvili, N. I. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Leiden, 1953. 16. Erdogan, F., On the stress distribution in plates with collinear straight cuts under arbitrary loads. Proceedings of the Fourth U.S. Naiional Congress of Applied Mechanics, 1962, 1, 547-554.
17. Kachanov, M. and Laures, J.-P., Three-dimensional problems of strongly interacting arbitrarily located pennyshaped cracks. International Journal of Fracture, 1989, 41, 289-313. 18. Wu, M. S., Analysis of finite anisotropic media containing multiple cracks using superposition. Engineering Fracture Mechanics, 1993, 45, 159-175.
19. Mehn, S., Why do cracks avoid each other? International Journal of Fracture, 1983, 23, 3745. (Received
11November 1996, in final form 15 March 1997, accepted 15 March 1997)