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Energy release rate for an arbitrarily curved interface crack
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Mechanics of Materials 18 (1994) 195-204
Energy release rate for an arbitrarily curved interface crack H.G. Beom, Y.Y. Earmme *, S.Y. Choi Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-Dong, Yusung-Gu, Taejon, 305-701, South Korea Received 4 February 1993; revised version received 31 August 1993
Abstract
A new conservation integral, which includes path and area integrals, for an arbitrarily curved interface crack is proposed. The conservation integral is shown to have the physical meaning of energy release rate for the interracial crack. This makes it possible to compute easily the energy release rate for an arbitrarily curved interface crack through the finite element method. Some numerical examples are given; the interracial crack in the elliptical inclusion is chosen to compare the result with the existing solution and the kink from the straight crack with continuous slope in the homogeneous material is selected for comparison with the perturbation solution by Karihaloo et al.
I. Introduction
Since the work by Eshelby (1956) and Rice (1968), studies on conservation integrals have been widely performed. The vast amount of literature on this subject eludes the individual reference. In particular, Budiansky and Rice (1973) interpreted the conservation integrals J, M and L as the energy release rates associated with the translation, expansion and rotation of cavity or crack. Park and Earmme (1986) thoroughly investigated the properties of the conservation integrals for an interracial crack lying along the straight interface or circular arc-shaped interface. Curved cracks are frequently observed since the path of fracture is generally curved under mixed mode loading in homogeneous materials.
* Corresponding author.
Moreover when the crack lies along the curved bimaterial interface as is often the case for typical interfaces (Evans and Hutchinson, 1989), we have to deal with a curved interracial crack. As Hutchinson and Suo (1991) presented in their study of mixed mode cracking in a thin brittle adhesive layer joining two identical bulk solids, the crack propagates on a microscopic level with various local cracking morphologies such as an alternating crack or wavy crack, which can be regarded as curved cracks. The application of the conservation integrals such as J, M and L is, however, restricted to cracks having a straight or circular-arc shape in homogeneous or dissimilar materials. Due to the complexity of the problem, few curved cracks have been analyzed except circular cracks (England, 1966; Perlman and Sih, 1967; Toya, 1974), elliptic cracks (Toya, 1975; Karihaloo and Viswanathan, 1985) and curved cracks (Sendekyj, 1974; Cotterell and Rice, 1980;
0167-6636/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 6 3 6 ( 9 4 ) 0 0 0 1 0 - E
H.G. Beom et al. /Mechanics of Materials 18 (1994) 195-204
196
Karihaloo et al., 1981; Chen et al., 1991; Beom and Earmme, 1993). The purpose of this study is to propose a conservation integral having the physical meaning of the energy release rate for the interfacial crack with a curved surface in dissimilar elastic solids. The conservation integral, which includes path and area integrals, for the curved interface crack is derived and the relation of the integral to the energy release rate is examined next. With the help of a weighting function we recast the energy release rate including path and area integrals into integrals over finite domains around the crack tip, which is compatible with the finite element method. Some numerical computations are carried out for elliptic interface cracks and parabolic kinked cracks to verify the usefulness of the proposed conservation integral.
sian coordinates (x 1, X 2) and the repetition of an index in a term denotes a summation with respect to that index over its range 1 to 2. The strain energy density, denoted by W is defined as
W(E.ij ) = fo'Jo'ij dEij , where % is the strain tensor. Let us define the G* integral by
G* = f r o ( W n j - t,nUm,j)cl)j ds
+ fA(g,,.Um.j --
dA.
2.1. Curved crack in a homogeneous elastic solid
G* = 0.
We consider a homogeneous elastic body subjected to a two-dimensional small deformation field. In the absence of body forces, the equation of equilibrium is
(4)
In obtaining (4), the equation of equilibrium has been used and the area A enclosed by F 0 has been assumed to be free from any singularities. It is noted that for the case in which the function q~j is given as q~j = ~lj where 81j is the Kronecker delta, the G* integral is reduced to the J1 integral (Rice, 1968; Knowles and Sternberg, 1972) while it is reduced to the M integral (Knowles and Sternberg, 1972) if q~j = xj and W is a homo-
(1)
where ~,~ is the stress tensor and the subscript comma (,) denotes a partial derivative with respect to the Cartesian coordinates. In this paper, a roman letter subscript refers only to the Carte-
n
F
J
j
(a)
(3)
Here n i is the unit outward normal vector, u i is the displacement, t i is the surface traction and q~j is a function with continuous first partial derivatives in A to be exactly defined next. F 0 is the closed contour enclosing an area A and ds is an element of arc length along F 0 as shown in Fig. l(a). Applying the divergence theorem to the integral in (3), we can show that
2. G* integral
= 0,
(2)
(b)
-L
P (c)
Fig. 1. Integration path: (a) normal; (b) for a curved crack; (c) for a crack with a curvilinear kink.
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204 geneous function in the strain components. Also it is seen that if tbj e3jk~O,k , where e 3 j k is the permutation symbol and q~ is a twice differentiable function, the G* integral can be rewritten as =
G* =
Making use of e 3 j k n j q ~ , k 0 on the surfaces of the curved crack, it is seen that the contribution of the line integral over F + and F~- vanishes, therefore, (6) is reduced to =
G* = f e3j (Wn
ff3jkejt,~,k d A ,
197
- tmUm,j) ,
ds
(5)
where Pit is the static energy momentum tensor defined as Pjl = W 6 i t - °b,,u,,,,j. Furthermore, by choosing ~o,k= x k, G* is reduced to the L 3 integral (Knowles and Sternberg, 1972) for the case in which W depends only on the scalar invariants of the strain components. Now, suppose a curved crack, whose surface is described by ~o(x~, x 2) = 0, is embedded in the homogeneous elastic solid as shown in Fig. l(b). Here ~O(Xl, x 2) is assumed to have continuous second partial derivatives. Choosing the function q~i as q~j = e3jk~O,k, we define the G* integral for the curved crack as follows:
G* = fr+rc+ +rc_e3ik(Wni
-
tmUm,j)~, k ds
+ fAe3yktrlmUm,jq~,kt d A ,
(6)
where F is any path connecting any points on opposite sides of the crack and enclosing the crack tip, F + and F~- are the paths on the surfaces of the crack and A is the area enclosed by F + F + + F c . For simplicity, the surfaces of the curved crack are assumed to be traction free.
+ fAe3jktrlrnUm,j~O,kl d A = e__.OJF~ lira/" e 3.k(Wn` J \ s - tmUm,j)~, k ds,
Material 2 J
(a)
(7b)
where F, is the circular path enclosing the crack tip with a vanishingly small radius e. In deriving (7b), (4) has been used. Also, from (7b), the path-independency of G* as given by (Ta) is proved. In (7a), ~o is required to have continuous second derivatives in A. For the problem of kinking from the straight crack with continuous slope, which was dealt with by Cotterell and Rice (1980), the second derivatives are discontinuous at the point of kink, P (see Fig. 1(c)). We show that under certain conditions, G* can be defined and it is path-independent. Suppose each portion of the surfaces is described by ~(1)(XI, X2)= 0 and q~(Z)(xl, x 2) = 0, respectively. If the discontinuity in the second derivative at point P is finite and ¢~i)(xl, x 2) (i = 1, 2) can be written as ~pti) = x 2 - f ( i ) ( x 0 (i = 1, 2), i.e., the crack surface can be described uniquely as a function of xl, we divide A into A tl) and A (2) by drawing the vertical line through point P as in Fig. 1(c). It is easily
F
~...
(7a)
(b)
Fig. 2. Integration path: (a) enclosingan interface of the material 1 and material 2; (b) for an interfacial crack.
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204
198
seen that along line P'P", the jump in the second derivatives [q~,ii] = ~,ij .(1) _ ~,ij .(2) is finite while [q~] = 0 and [q~,i] = 0, therefore, G* in (7a) is uniquely determined and moreover path-independent. We will also show in the next section that (7a) is also path-independent for the interfacial crack, i.e., the G* integral is independent of path F and area A. As already seen in the case of homogeneous materials without crack, G* in (7a) is reduced to the J1 integral if q~(x~, x 2) = x2 = 0 (the straight crack) and the L 3 integral if ~ ( x 1, x 2) = 1/2(XmX m - R 2 ) = 0 (the circular-arc crack, R being the radius).
neous material with the curved crack, we can define
G* = f e3jk(Wn j +
tmUm,j)q~,k d s
fAe3jkO'lmUm,jqgkl dA
= lim [ e 3 . J W n - tmUma)q~,k ds, e~OJF,
) ~
J
(9a) (9b)
which is independent of path F and area A.
3. Energy release rate
2.2. Curved interface crack We now obtain the expressions of the G* integral for a two-dimensional interfacial crack in a curved interface. The procedure is similar to the case of the curved crack in a homogeneous elastic solid, which is briefly described as follows. Let us suppose a bonded interface, described by ~(x 1, x 2) = 0, between dissimilar elastic materials as shown in Fig. 2(a). Using (3) with qbj = e3jk~.k and (4), we get
G* = fFf 3jk(Wn j -- tmUm,j)q~ k ds q- f Ae3jkO'lmUm,jqg,kl d A
=f
JF
e3jktmUm,jq~kds,
(8a)
(8b)
/'i + + Fi
where F 0 is the closed path enclosing area A and containing the interface, and F~+ and F~- are the paths on the interface. In obtaining (8b), we use e3jk~,kn j = 0 on the interface. Introducing the curvilinear coordinates (~:1, ~ca) as shown in Fig. 2(a), it can be shown that the G* integral defined in (8b) vanishes identically for the case in which area A enclosed by F 0 includes the interface bonded perfectly. (See the Appendix for details.) Now consider an interracial crack with a curved interface described by ~o(x1, x 2 ) = 0. The surfaces of the crack are assumed to be traction free. In the same manner as the case of the homoge-
Consider an interfacial crack with a curved interface in dissimilar elastic materials. The curved crack in the homogeneous elastic solid can be regarded as a special case of this interfacial crack. The crack is assumed to grow along the interface. The energy release rate, denoted by G, due to the crack growth by d a is given from the energy balance by c
dui
G = 1 ti--ds is, da
d afA,w dA,
(10)
where A 0 is the total area of the cracked body and S t is the surface of A 0 where tractions are prescribed. Making use of the divergence theorem, it can be shown that the energy release rate is given as G = lim [ (Wni--tmUmj)C j ds, ~-)0 Jr,
(11)
where c~ is the unit tangent vector to the interface at the crack tip. It is worth noting that the above equation is valid only for elastic crack propagation along the interface described by ~(xl, x 2) ---0 (whose first derivative is continuous) but not in the case of a kinked crack. We now show that the G* integral can be interpreted as the energy release rate when the interfacial crack grows along the interface. The equation of the interface, ~p(x1, X 2) = 0 , is not unique in the sense that an arbitrary constant can be multiplied to q~(xl, x2). Hence, we normalize q~ so as to satisfy e3j~q~k = c j at the crack tip.
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204 That is, the magnitude of Vff at the crack tip is 1 and the direction of V4~ at the crack tip is along the x2-axis as shown in Fig. 2(b), where ff(x 1, x2) = 0 is the normalized equation of the interface. Replacing ~o(xl, x e) by q3(x1, x e) in (9a) and (9b), it is seen that the G* integral has the physical meaning of energy release rate. Thus, the energy release rate for the interfacial crack can be written as
G = fe3jk(Wnj
-
crack tip. Within the area A enclosed by F, q is an arbitrary smooth function of x a and x 2. Applying the divergence theorem, the expression of the energy release rate for the interfacial crack in (12) is rewritten as
G = fAe3jk(cr,,~U,~3(O,kZq- W(o,~q~ +~mUm,j¢,kq,t) d A .
(13)
It is clear that the value of integral G in (13) is independent of the size and shape of the domain.
tmUm,j)(~,k ds
+ fAe3jkO'lmUm,y~,kl d A .
199
(12) 5. Numerical example
The above expression for the energy release rate is less sensitive to numerical inaccuracies in the crack tip region than (11) since (12) does not require a knowledge of accurate crack tip fields. The area integral in (12) vanishes identically for the straight crack, and is reduced to the line integral for the straight crack.
4. D o m a i n integral expression for the energy release rate
In the analysis of a crack problem by means of a computational method, such as the finite element method, a fundamental difficulty is encountered in efforts to compute the values of field quantities for points close to the crack tip. The energy release rate obtained in the previous section can be evaluated along the contour remote from the crack tip where the numerical fields are more accurate and over the area. According to Li et al. (1985), however, the domain integral method is superior to the line integral method in calculating the energy release rate. Thus, we recast the energy release rate including path and area integrals into integrals over finite domains around the crack tip. We introduce a weighting function q which is defined over the domain of interest. The function q has a value of unity on the vanishingly small inner contour F~, and zero on the outer contour F which is any path connecting the same points on opposite sides of the crack and enclosing the
To illustrate the use of the domain integral expression (13) for the energy release rate and to examine the accuracy of the domain integral method, we consider here two examples. The first example is the problem of an interracial crack along an elliptic inclusion embedded in an elastic solid. The second example is the problem of a crack with a parabolic kink, which is an example for the case in which the equation of the crack surface has a discontinuity in the second derivative at a point while the first derivative is continuous as mentioned in Section 2.1.
5.1. Crack along the interface of an elliptic inclusion Let us consider a crack along the interface of an elliptic inclusion under uniform biaxial loads oas shown in Fig. 3. Tractions vanish on the interface crack surfaces. Two problems regarded as special cases of this elliptic-arc crack have been solved analytically. The first one is a circular interface crack in an infinite solid, which has been analyzed by Perlman and Sih (1967) and Toya (1974). The second one is an interface crack along a rigid elliptic inclusion in an infinite solid, which has been solved by Toya (1975). Although the energy release rates for the special cases mentioned above can be obtained from the known solution, it is intended to obtain numerically the energy release rates to ascertain the accuracy of the method proposed in this paper.
H.G. Beom et al. /Mechanics of Materials 18 (1994) 195-204
200
Adopting a similar method of the finite element implementation presented in Li et al. (1985), the discretized form of the domain expression for the energy release rate is obtained in the following form
ITT X2
(-a~cos6,, a2sir0,) IJ
(Y ¢'q
Mat. 2
a~
a =
Y'~
Y'~ e3j k O'lmUm,jgkl q -- W£kq j
elements a = 1
=
(16) Mat. 1
2W
(Y Fig. 3. Elliptic interface crack.
The elliptic crack and interface is described by
(.t ]
(o( x,, x2) = ~--~ [ \ a, ] + --a2
-1
.
(14)
Here a] and a 2 denote the major and minor semi-axes of the ellipse, respectively, and D t )2 = g'(cos Oo/a I + (sin 0o/a2) 2 where 0 o describes the Cartesian coordinates of the crack tip ( - a ~ cos 00, a z sin 00). The energy release rate is expressed from (13) and (14) as
where the quantities within [ ], are evaluated at the nine Gauss points, w, is the weight in Gauss numerical integration and det(Oxi/O %) is the Jacobian determinant. According to Shih et al. (1986), the value of the J integral obtained using the domain integral method is insensitive to the particular choice of the weight function q. We employed a practical scheme so that for a node inside the area, the nodal value of q is chosen by the linear interpolation with respect to its associated boundary nodes on F~ and F. We first consider a circular crack (ae/a I = 1). The FEM mesh for the circular crack with 974 nodes and 312 elements as shown in Fig. 4 is used in the computation. All elements are 8-noded
G = fAe3jk(O)mUmd~k,q- W2kqj +Orlmblm,j,~kq l ) d A ,
(15)
where 2 k = x J D a ~ and g~t = 6kt/Da2 (not summed on k). Elastic analyses are carried out with the A B A Q U S finite element program (Hibbit et al., 1984) to calculate the stresses and displacements. In order to compare the energy release rate obtained analytically with that evaluated numerically, the small contact zone at the crack tip is not taken into consideration in the F E M computation. The calculation for the energy release rate according to the domain formula (15) is carried out in a separate post-processing program.
,nte nterau°n O°mains X
Fig. 4. Mesh configuration used in F E M computation for a circular crack.
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204
isoparametric elements, and the singular crack tip element is not used. Four integration domains are chosen to compute the energy release rate numerically. The first integration domain consists of four elements adjoining the crack tip; the second domain, which includes the first domain and the adjoining layer of elements, has 16 elements. The third and fourth domains are chosen in this fashion. The numerical values used here for the circular crack are as follows: w = h = 15a l, ]2,1= 10 GPa, 1,'1 = v 2 = 0 . 3 and t r = 100 MPa where /z and v with subscript 1 and 2 are the shear modulus and Poisson's ratio of material 1 and material 2, respectively. According to Park and E a r m m e (1986), the close agreement between the case of a finite sized matrix, w / a 1 >1 10 and the case of an infinite matrix is observed for /Zl//Z 1 = 2. Thus w = h = 15a t is chosen to compare the numerical result for a finite sized matrix
201
with the existing solution for an infinite matrix. For the cases of /d,2///d, 1 -~-1 and /d, 2//jt/. l = 2 respectively, the normalized energy release rates G / G o for a plane strain condition are tabulated in Tables 1 and 2 for various 0 0 describing the coordinates of the crack tip, where G o = rr(1 - v 1) o-2a2 sin 00/2~1 is the energy release rate when a crack with half-length a 2 sin 00 is in the homogeneous material 1 and is subjected to a normal tension or, and G ~ is the energy release rate for an infinite plate (w = h = oo) obtained by Perlman and Sih (1967) and Toya (1974). The values of the energy release rate for each domain are almost constant within 1 percent. The results are also in good agreement with the analytic solution obtained by Perlman and Sih (1967) and Toya (1974). Next, suppose a crack along the interface of a rigid inclusion (/z 2 = oo). In a similar way, the energy release rates are evaluated for the elliptic
Table 1 Normalized energy release rate for the circular crack with w = h = 15a 1, p.2//Zl = 1 and v I = v 2 = 0.3. Here Go = ~r(1-/,,1 ) o2a2 sin 00/2/z t, and G ~ is the energy release rate for an infinite plate obtained by Toya (1974)
0 o (°) 45 60 75 90
G/G o
G=/Go
Domain 1
Domain 2
Domain 3
Domain 4
Average
0.7595 0.6401 0.5318 0.4430
0.7489 0.6338 0.5289 0.4425
0.7533 0.6376 0.5320 0.4451
0.7556 0.6395 0.5335 0.4463
0.7543 0.6378 0.5316 0.4442
Table 2 Normalized energy release rate for the circular crack with w = h = 15al, ~2//.t,1 = 2 and /"1 = or2a2 sin 00/2/zi, and G ® is the energy release rate for an infinite plate obtained by Toya (1974)
//'2 =
0 o (°)
G/G o Domain 1
Domain 2
Domain 3
Domain 4
Average
45 60 75 90
0.8027 0.7073 0.6161 0.5369
0.8003 0.7094 0.6210 0.5433
0.8045 0.7130 0.6240 0.5460
0.8063 0.7142 0.6250 0.5467
0.8035 0.7110 0.6215 0.5432
0.7608 0.6400 0.5323 0.4444
0.3. Here G o = lr(1 - v 1)
G=/Go 0.8132 0.7156 0.6240 0.5445
Table 3 Normalized energy release rate for the crack along a rigid elliptic inclusion with a2/a I = 2, w = h = 15a2, 00 = 90 °, P-1 = 10 GPa and v 1 = 0.3. Here G o = rr(1 - v l) or2a2 sin 00/2/.Ll, and G ® is the energy release rate for an infinite plate obtained by Toya (1975). G c and G represent respectively the numerical result with and without constraint condition
G/G o GC/Go
Domain 1
Domain 2
Domain 3
Domain 4
Average
G~/Go
0.7920 0.8228
0.8017 0.8330
0.8062 0.8377
0.8089 0.8406
0.8022 0.8335
0.8293 0.8293
H.G. Beom et al. / Mechanics o f Materials 18 (1994) 195-204
202
crack in the case of a2/a 1 = 2, w = h = 15a 2, 00 = 90 °,/.t 1 = 10 GPa, v I = 0.3 and cr = 100 MPa. The result is tabulated in Table 3. Here G c represents the numerical result under the constraint condition that the tangential derivatives of the normal displacement at the outer boundaries vanish. The result shows that the values of the energy release rate with or without the constraint at outer boundaries for each domain except the first domain (the nearest to the crack tip) are almost constant within 1 percent, however, the energy release rate obtained numerically without the constraint does not approach G = closely in contrast to the case of /x2/tx I = 1 or 2 for the circular crack (see Tables 1 and 2). As noted by Choi and Earmme (1992) for circular cracks, the effect of w / a 2 on the energy release rate may be significant if the difference of the shear moduli is large. For the case of a2/a 1 = 2, w = h = 15a 2, #l = 10 GPa, v~ = 0.3 and cr = 100 MPa, the normalized energy release rate for a plane strain condition is plotted in Fig. 5 as 00 varies. The solid line represents the result obtained analytically by Toya (1975) while the open circles represent the result computed numerically from F E M using (16) and averaged for the four domains with the constraint as explained above. Close agreement between them is observed.
1.2-
1.0-
G/G0
(Y
tt
tt X2
2a•C
0
B2 ~ a 0
O
~x(x''x2)
q)"~ --0
(pl2~ =0 2w
(Y Fig. 6. G e o m e t r y o f a c r a c k with a p a r a b o l i c kink.
5.2. Crack with a parabolic kink As another example, a crack with a segment at the crack tip in the form of a parabolic arc subjected to biaxial loads ~r as shown in Fig. 6 is considered. Tractions vanish on the crack surfaces. Cracks with the portion of a curvilinear pattern in the vicinity of the crack tip are frequently observed since the path of the fracture is generally curved under loading of mixed mode type in a homogeneous material. Specifically the parabolic pattern has been chosen here as a typical curvilinear crack path. The crack surface is described by
O.80.60.4-
q3=
Analytic Solution (Toya)
0.20.0
30°
i
i
r
i
45°
60°
75°
90°
x2-b
Xl 2 -D -1,
x~>0.
(17)
a 0
F.E.M. (Present Method)
o
~(2~=
105°
0o Fig. 5. N o r m a l i z e d e n e r g y r e l e a s e r a t e f o r t h e c r a c k a l o n g t h e i n t e r f a c e o f a rigid i n c l u s i o n as a f u n c t i o n o f 0 0 d e s c r i b i n g t h e c o o r d i n a t e s o f t h e c r a c k tip. H e r e a 2 / a I = 2, w = h = 15a 2, tx~ = 10 G P a a n d u I = 0.3 a r e u s e d .
Here D = ~ / ( 2 b x C / a 2 ) 2 + 1 where x~: denotes the x~ coordinate of the crack tip C. The energy release rates are calculated numerically from FEM using (13) with (17). The numerical values used in the computation are as follows: /.L = 10 GPa, u = 0.3, ~r= 100 MPa, a 0 = 1 cm, b = l / 3 a 0 and w = h = 6 0 a 0. The normalized energy release rate obtained from the numerical computation for a plane strain condition
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204
1.4 1.2G/G 0
1.0- . . . . . . . . . . . . . . .
0.80.6-
[] .......
0.4o 0.2- ~
0.0 0o
Crack Tip B F.E.M. (Present Method) Second Order Solution (Karihaloo et al.) Crack Tip C F.E.M. (Present Method) Second Order Solution (Karih~doo et al.) I i I 5° 10 o 15 °
20 °
O Fig. 7. Normalized energy release rates for the crack with a parabolic kink. at crack tips B and C is plotted in Fig. 7 as t9 varies, i.e., as the length of the parabolic segment varies. Here a9 denotes the angle between the x~ axis and BC, and is equal to O = tan-l[xC/(x c + 2a0)]. In particular, v~ = 0 represents the case of the straight crack of length 2a 0. The squares and circles represent the results at crack tips B and C, respectively, from the FEM using the method proposed in this paper, while the dotted and solid lines represent the second order solutions for the parabolic-kinked crack at crack tips B and C, respectively, which are evaluated from the solution for the stress intensity factors of a curved crack obtained by Karihaloo et al. (1981). G O ( = r r ( 1 - v)o'2a/2~) is the energy release rate for a crack with half-length a ( = l ~ ( x C + 2a0)Z + (x2C) z) in the elastic solid. The result for the parabolic-kinked crack is well compared with the second order perturbation solution for small O.
6. C o n c l u s i o n
A new conservation integral, which includes path and area integrals, for an interracial crack with a curved surface in dissimilar elastic solids is proposed. The conservation integral is shown to
203
have the physical meaning of energy release rate for the curved crack in a homogeneous material or in an interface. With the help of a function q we recast the expression of energy release rate to combine the path and area integrals into the domain integral over a finite domain around the crack tip. This domain integral expression is less sensitive to numerical inaccuracies in the crack tip region since the expression does not require a knowledge of accurate crack tip fields. The conventional elements, thus, can be adopted to evaluate the energy release rate by the finite element method, which is confirmed by the examples presented here.
Acknowledgement
The authors are grateful to Professor S. Nemat-Nasser of The University of California for helpful comments in finalizing Fig. 7.
References Beom, H.G. and Y.Y. Earmme (1993), Energy release rate for a curved crack in an elastic solid, Trans. Korean Soc. Mech. Engrs. 17, 543, (in Korean). Budiansky, B. and J.R. Rice (1973), Conservation laws and energy release rates, ASME J. Appl. Mech. 40, 201. Chen, Y.Z., D. Gross and Y.J. Huang (1991), Numerical solution of the curved crack problem by means of polynomial approximation of the dislocation distribution, Eng. Fract. Mech. 39, 791. Choi, N.Y. and Y.Y. Earmme (1992), Evaluation of stress intensity factors in circular arc-shaped interfacial crack using L integral, Mech. Mater. 14, 141. Cotterell, B. and J.R. Rice (1980), Slightly curved or kinked cracks, Int. J. Fract. 16, 155. England, A.H. (1966), An arc crack around a circular elastic inclusion, ASME J. Appl. Mech. 33, 637. Eshelby, J.D. (1956), The continuum theory of lattice defects, in: F. Seitz and D. Turnbull, eds., Solid State Physics 3, Academic Press, New York, p. 79. Evans, A.G. and J.W. Hutchinson (1989), Effects of nonplanarity on the mixed mode fracture resistance of bimaterial interfaces, Acta Metall. Mater. 37, 909. Hibbit, H.D., B. Karlsson and E.P. Sorensen (1984) ABAQUS User's Manual, Hibbit, Karlsson and Sorensen Inc., Providence, RI.
204
H.G. Beomet al./ Mechanicsof Materials18 (1994)195-204
Hutchinson, J.W. and Z. Suo (1991), Mixed mode cracking in layered materials, in: J.W. Hutchinson and T.Y. Wu, eds., Advances in Applied Mechanics, Vol. 29, Academic Press, New York. Karihaloo, B.L., L.M. Keer, S. Nemat-Nasser and A. Oranratnachai (1981), Approximate description of crack kinking and curving, ASME J. Appl. Mech. 48, 515. Karihaloo, B.L. and K. Viswanathan (1985), Elastic field of a partially debonded elliptic inhomogeneity in an elastic matrix (plane-strain), ASME J. Appl. Mech. 52, 835. Knowles, J.K. and E. Sternberg (1972), On a class of conservation laws in linearized and finite elastostatics, Arch. Rat. Mech. Anal, 44, 187. Li, F.Z., C.F. Shih and A. Needleman (1985), A comparison of methods for calculating energy release rates, Eng. Fract. Mech. 21, 405. Park, J.H. and Y.Y. Earmme (1986), Application of conservation integrals to interfacial crack problems, Mech. Mater. 5, 261. Perlman, A.B. and G.C. Sih (1967), Elastostatic problems of curvilinear cracks in bonded dissimilar materials, Int. J. Eng. Sci. 5, 845. Rice, J.R. (1968), A path independent integral and approximate analysisof strain concentration by notches and cracks, ASME J. Appl. Mech. 35, 379. Sendeckyj, S.D. (1974), Debonding of rigid curvilinear inclusions in longitudinal shear deformation, Eng. Fract. Mech. 6, 33. Shih, C.F., B. Moran and T. Nakamura (1986), Energy release rate along a three-dimensional crack front in a thermally stressed body, Int. J. Fract. 30, 79. Toya, M. (1974), A crack along the interface of a circular inclusion embedded in an infinite solid, J. Mech. Phys. Solids 22, 325. Toya, M. (1975), Debonding along the interface of an elliptic rigid inclusions, Int. Z Fract. 19, 989.
A p p e n d i x - E v a l u a t i o n o f (8b)
W e i n t r o d u c e the c u r v i l i n e a r c o o r d i n a t e s (sct, sc2) d e f i n e d by scj =~'(X1, X2) a n d ~:2= ~ ( x l , x 2) as shown in Fig. 2(a). D e n o t i n g the covariant base vectors a n d c o n t r a v a r i a n t base vectors for the curvilinear c o o r d i n a t e s by g,~ a n d g'~ ( a = 1, 2), respectively, it can be w r i t t e n that t = t ~ g a,
u ® V =
c3~ + u~
g~ ® g~'
V~ = g 2 ,
(A1)
w h e r e F ~ is the E u c l i d e a n Christoffel symbol a n d the G r e e k letter subscript a n d superscript d e n o t e the covariant a n d c o n t r a v a r i a n t compon e n t , respectively. U s i n g the relation g'~ x g t3 = e~t~g~, where E~t~ is the p e r m u t a t i o n t e n s o r in the curvilinear coordinates, it can be shown that
f+
F~ + F i-
e3jktmllm,j@,kd s [ 0u ~
= fri*
+Fi
Ea12t~t ~
+ U~F;1) ds.
On the interface bonded perfectly, u,,+
=t/a,
t+ =-t~(a= 1, 2), where the superscripts + and - denote the values on Fi+ a n d Fi-, respectively. It is readily seen that the integral in (A2) vanishes identically. (Ou~/O~l)+=(Ou~/O~l)
-
and
(A2)
.
.~L~ ''~
OF IIJtlrl!lllltlJ ELSEVIER
Mechanics of Materials 18 (1994) 195-204
Energy release rate for an arbitrarily curved interface crack H.G. Beom, Y.Y. Earmme *, S.Y. Choi Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-Dong, Yusung-Gu, Taejon, 305-701, South Korea Received 4 February 1993; revised version received 31 August 1993
Abstract
A new conservation integral, which includes path and area integrals, for an arbitrarily curved interface crack is proposed. The conservation integral is shown to have the physical meaning of energy release rate for the interracial crack. This makes it possible to compute easily the energy release rate for an arbitrarily curved interface crack through the finite element method. Some numerical examples are given; the interracial crack in the elliptical inclusion is chosen to compare the result with the existing solution and the kink from the straight crack with continuous slope in the homogeneous material is selected for comparison with the perturbation solution by Karihaloo et al.
I. Introduction
Since the work by Eshelby (1956) and Rice (1968), studies on conservation integrals have been widely performed. The vast amount of literature on this subject eludes the individual reference. In particular, Budiansky and Rice (1973) interpreted the conservation integrals J, M and L as the energy release rates associated with the translation, expansion and rotation of cavity or crack. Park and Earmme (1986) thoroughly investigated the properties of the conservation integrals for an interracial crack lying along the straight interface or circular arc-shaped interface. Curved cracks are frequently observed since the path of fracture is generally curved under mixed mode loading in homogeneous materials.
* Corresponding author.
Moreover when the crack lies along the curved bimaterial interface as is often the case for typical interfaces (Evans and Hutchinson, 1989), we have to deal with a curved interracial crack. As Hutchinson and Suo (1991) presented in their study of mixed mode cracking in a thin brittle adhesive layer joining two identical bulk solids, the crack propagates on a microscopic level with various local cracking morphologies such as an alternating crack or wavy crack, which can be regarded as curved cracks. The application of the conservation integrals such as J, M and L is, however, restricted to cracks having a straight or circular-arc shape in homogeneous or dissimilar materials. Due to the complexity of the problem, few curved cracks have been analyzed except circular cracks (England, 1966; Perlman and Sih, 1967; Toya, 1974), elliptic cracks (Toya, 1975; Karihaloo and Viswanathan, 1985) and curved cracks (Sendekyj, 1974; Cotterell and Rice, 1980;
0167-6636/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 6 3 6 ( 9 4 ) 0 0 0 1 0 - E
H.G. Beom et al. /Mechanics of Materials 18 (1994) 195-204
196
Karihaloo et al., 1981; Chen et al., 1991; Beom and Earmme, 1993). The purpose of this study is to propose a conservation integral having the physical meaning of the energy release rate for the interfacial crack with a curved surface in dissimilar elastic solids. The conservation integral, which includes path and area integrals, for the curved interface crack is derived and the relation of the integral to the energy release rate is examined next. With the help of a weighting function we recast the energy release rate including path and area integrals into integrals over finite domains around the crack tip, which is compatible with the finite element method. Some numerical computations are carried out for elliptic interface cracks and parabolic kinked cracks to verify the usefulness of the proposed conservation integral.
sian coordinates (x 1, X 2) and the repetition of an index in a term denotes a summation with respect to that index over its range 1 to 2. The strain energy density, denoted by W is defined as
W(E.ij ) = fo'Jo'ij dEij , where % is the strain tensor. Let us define the G* integral by
G* = f r o ( W n j - t,nUm,j)cl)j ds
+ fA(g,,.Um.j --
dA.
2.1. Curved crack in a homogeneous elastic solid
G* = 0.
We consider a homogeneous elastic body subjected to a two-dimensional small deformation field. In the absence of body forces, the equation of equilibrium is
(4)
In obtaining (4), the equation of equilibrium has been used and the area A enclosed by F 0 has been assumed to be free from any singularities. It is noted that for the case in which the function q~j is given as q~j = ~lj where 81j is the Kronecker delta, the G* integral is reduced to the J1 integral (Rice, 1968; Knowles and Sternberg, 1972) while it is reduced to the M integral (Knowles and Sternberg, 1972) if q~j = xj and W is a homo-
(1)
where ~,~ is the stress tensor and the subscript comma (,) denotes a partial derivative with respect to the Cartesian coordinates. In this paper, a roman letter subscript refers only to the Carte-
n
F
J
j
(a)
(3)
Here n i is the unit outward normal vector, u i is the displacement, t i is the surface traction and q~j is a function with continuous first partial derivatives in A to be exactly defined next. F 0 is the closed contour enclosing an area A and ds is an element of arc length along F 0 as shown in Fig. l(a). Applying the divergence theorem to the integral in (3), we can show that
2. G* integral
= 0,
(2)
(b)
-L
P (c)
Fig. 1. Integration path: (a) normal; (b) for a curved crack; (c) for a crack with a curvilinear kink.
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204 geneous function in the strain components. Also it is seen that if tbj e3jk~O,k , where e 3 j k is the permutation symbol and q~ is a twice differentiable function, the G* integral can be rewritten as =
G* =
Making use of e 3 j k n j q ~ , k 0 on the surfaces of the curved crack, it is seen that the contribution of the line integral over F + and F~- vanishes, therefore, (6) is reduced to =
G* = f e3j (Wn
ff3jkejt,~,k d A ,
197
- tmUm,j) ,
ds
(5)
where Pit is the static energy momentum tensor defined as Pjl = W 6 i t - °b,,u,,,,j. Furthermore, by choosing ~o,k= x k, G* is reduced to the L 3 integral (Knowles and Sternberg, 1972) for the case in which W depends only on the scalar invariants of the strain components. Now, suppose a curved crack, whose surface is described by ~o(x~, x 2) = 0, is embedded in the homogeneous elastic solid as shown in Fig. l(b). Here ~O(Xl, x 2) is assumed to have continuous second partial derivatives. Choosing the function q~i as q~j = e3jk~O,k, we define the G* integral for the curved crack as follows:
G* = fr+rc+ +rc_e3ik(Wni
-
tmUm,j)~, k ds
+ fAe3yktrlmUm,jq~,kt d A ,
(6)
where F is any path connecting any points on opposite sides of the crack and enclosing the crack tip, F + and F~- are the paths on the surfaces of the crack and A is the area enclosed by F + F + + F c . For simplicity, the surfaces of the curved crack are assumed to be traction free.
+ fAe3jktrlrnUm,j~O,kl d A = e__.OJF~ lira/" e 3.k(Wn` J \ s - tmUm,j)~, k ds,
Material 2 J
(a)
(7b)
where F, is the circular path enclosing the crack tip with a vanishingly small radius e. In deriving (7b), (4) has been used. Also, from (7b), the path-independency of G* as given by (Ta) is proved. In (7a), ~o is required to have continuous second derivatives in A. For the problem of kinking from the straight crack with continuous slope, which was dealt with by Cotterell and Rice (1980), the second derivatives are discontinuous at the point of kink, P (see Fig. 1(c)). We show that under certain conditions, G* can be defined and it is path-independent. Suppose each portion of the surfaces is described by ~(1)(XI, X2)= 0 and q~(Z)(xl, x 2) = 0, respectively. If the discontinuity in the second derivative at point P is finite and ¢~i)(xl, x 2) (i = 1, 2) can be written as ~pti) = x 2 - f ( i ) ( x 0 (i = 1, 2), i.e., the crack surface can be described uniquely as a function of xl, we divide A into A tl) and A (2) by drawing the vertical line through point P as in Fig. 1(c). It is easily
F
~...
(7a)
(b)
Fig. 2. Integration path: (a) enclosingan interface of the material 1 and material 2; (b) for an interfacial crack.
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204
198
seen that along line P'P", the jump in the second derivatives [q~,ii] = ~,ij .(1) _ ~,ij .(2) is finite while [q~] = 0 and [q~,i] = 0, therefore, G* in (7a) is uniquely determined and moreover path-independent. We will also show in the next section that (7a) is also path-independent for the interfacial crack, i.e., the G* integral is independent of path F and area A. As already seen in the case of homogeneous materials without crack, G* in (7a) is reduced to the J1 integral if q~(x~, x 2) = x2 = 0 (the straight crack) and the L 3 integral if ~ ( x 1, x 2) = 1/2(XmX m - R 2 ) = 0 (the circular-arc crack, R being the radius).
neous material with the curved crack, we can define
G* = f e3jk(Wn j +
tmUm,j)q~,k d s
fAe3jkO'lmUm,jqgkl dA
= lim [ e 3 . J W n - tmUma)q~,k ds, e~OJF,
) ~
J
(9a) (9b)
which is independent of path F and area A.
3. Energy release rate
2.2. Curved interface crack We now obtain the expressions of the G* integral for a two-dimensional interfacial crack in a curved interface. The procedure is similar to the case of the curved crack in a homogeneous elastic solid, which is briefly described as follows. Let us suppose a bonded interface, described by ~(x 1, x 2) = 0, between dissimilar elastic materials as shown in Fig. 2(a). Using (3) with qbj = e3jk~.k and (4), we get
G* = fFf 3jk(Wn j -- tmUm,j)q~ k ds q- f Ae3jkO'lmUm,jqg,kl d A
=f
JF
e3jktmUm,jq~kds,
(8a)
(8b)
/'i + + Fi
where F 0 is the closed path enclosing area A and containing the interface, and F~+ and F~- are the paths on the interface. In obtaining (8b), we use e3jk~,kn j = 0 on the interface. Introducing the curvilinear coordinates (~:1, ~ca) as shown in Fig. 2(a), it can be shown that the G* integral defined in (8b) vanishes identically for the case in which area A enclosed by F 0 includes the interface bonded perfectly. (See the Appendix for details.) Now consider an interracial crack with a curved interface described by ~o(x1, x 2 ) = 0. The surfaces of the crack are assumed to be traction free. In the same manner as the case of the homoge-
Consider an interfacial crack with a curved interface in dissimilar elastic materials. The curved crack in the homogeneous elastic solid can be regarded as a special case of this interfacial crack. The crack is assumed to grow along the interface. The energy release rate, denoted by G, due to the crack growth by d a is given from the energy balance by c
dui
G = 1 ti--ds is, da
d afA,w dA,
(10)
where A 0 is the total area of the cracked body and S t is the surface of A 0 where tractions are prescribed. Making use of the divergence theorem, it can be shown that the energy release rate is given as G = lim [ (Wni--tmUmj)C j ds, ~-)0 Jr,
(11)
where c~ is the unit tangent vector to the interface at the crack tip. It is worth noting that the above equation is valid only for elastic crack propagation along the interface described by ~(xl, x 2) ---0 (whose first derivative is continuous) but not in the case of a kinked crack. We now show that the G* integral can be interpreted as the energy release rate when the interfacial crack grows along the interface. The equation of the interface, ~p(x1, X 2) = 0 , is not unique in the sense that an arbitrary constant can be multiplied to q~(xl, x2). Hence, we normalize q~ so as to satisfy e3j~q~k = c j at the crack tip.
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204 That is, the magnitude of Vff at the crack tip is 1 and the direction of V4~ at the crack tip is along the x2-axis as shown in Fig. 2(b), where ff(x 1, x2) = 0 is the normalized equation of the interface. Replacing ~o(xl, x e) by q3(x1, x e) in (9a) and (9b), it is seen that the G* integral has the physical meaning of energy release rate. Thus, the energy release rate for the interfacial crack can be written as
G = fe3jk(Wnj
-
crack tip. Within the area A enclosed by F, q is an arbitrary smooth function of x a and x 2. Applying the divergence theorem, the expression of the energy release rate for the interfacial crack in (12) is rewritten as
G = fAe3jk(cr,,~U,~3(O,kZq- W(o,~q~ +~mUm,j¢,kq,t) d A .
(13)
It is clear that the value of integral G in (13) is independent of the size and shape of the domain.
tmUm,j)(~,k ds
+ fAe3jkO'lmUm,y~,kl d A .
199
(12) 5. Numerical example
The above expression for the energy release rate is less sensitive to numerical inaccuracies in the crack tip region than (11) since (12) does not require a knowledge of accurate crack tip fields. The area integral in (12) vanishes identically for the straight crack, and is reduced to the line integral for the straight crack.
4. D o m a i n integral expression for the energy release rate
In the analysis of a crack problem by means of a computational method, such as the finite element method, a fundamental difficulty is encountered in efforts to compute the values of field quantities for points close to the crack tip. The energy release rate obtained in the previous section can be evaluated along the contour remote from the crack tip where the numerical fields are more accurate and over the area. According to Li et al. (1985), however, the domain integral method is superior to the line integral method in calculating the energy release rate. Thus, we recast the energy release rate including path and area integrals into integrals over finite domains around the crack tip. We introduce a weighting function q which is defined over the domain of interest. The function q has a value of unity on the vanishingly small inner contour F~, and zero on the outer contour F which is any path connecting the same points on opposite sides of the crack and enclosing the
To illustrate the use of the domain integral expression (13) for the energy release rate and to examine the accuracy of the domain integral method, we consider here two examples. The first example is the problem of an interracial crack along an elliptic inclusion embedded in an elastic solid. The second example is the problem of a crack with a parabolic kink, which is an example for the case in which the equation of the crack surface has a discontinuity in the second derivative at a point while the first derivative is continuous as mentioned in Section 2.1.
5.1. Crack along the interface of an elliptic inclusion Let us consider a crack along the interface of an elliptic inclusion under uniform biaxial loads oas shown in Fig. 3. Tractions vanish on the interface crack surfaces. Two problems regarded as special cases of this elliptic-arc crack have been solved analytically. The first one is a circular interface crack in an infinite solid, which has been analyzed by Perlman and Sih (1967) and Toya (1974). The second one is an interface crack along a rigid elliptic inclusion in an infinite solid, which has been solved by Toya (1975). Although the energy release rates for the special cases mentioned above can be obtained from the known solution, it is intended to obtain numerically the energy release rates to ascertain the accuracy of the method proposed in this paper.
H.G. Beom et al. /Mechanics of Materials 18 (1994) 195-204
200
Adopting a similar method of the finite element implementation presented in Li et al. (1985), the discretized form of the domain expression for the energy release rate is obtained in the following form
ITT X2
(-a~cos6,, a2sir0,) IJ
(Y ¢'q
Mat. 2
a~
a =
Y'~
Y'~ e3j k O'lmUm,jgkl q -- W£kq j
elements a = 1
=
(16) Mat. 1
2W
(Y Fig. 3. Elliptic interface crack.
The elliptic crack and interface is described by
(.t ]
(o( x,, x2) = ~--~ [ \ a, ] + --a2
-1
.
(14)
Here a] and a 2 denote the major and minor semi-axes of the ellipse, respectively, and D t )2 = g'(cos Oo/a I + (sin 0o/a2) 2 where 0 o describes the Cartesian coordinates of the crack tip ( - a ~ cos 00, a z sin 00). The energy release rate is expressed from (13) and (14) as
where the quantities within [ ], are evaluated at the nine Gauss points, w, is the weight in Gauss numerical integration and det(Oxi/O %) is the Jacobian determinant. According to Shih et al. (1986), the value of the J integral obtained using the domain integral method is insensitive to the particular choice of the weight function q. We employed a practical scheme so that for a node inside the area, the nodal value of q is chosen by the linear interpolation with respect to its associated boundary nodes on F~ and F. We first consider a circular crack (ae/a I = 1). The FEM mesh for the circular crack with 974 nodes and 312 elements as shown in Fig. 4 is used in the computation. All elements are 8-noded
G = fAe3jk(O)mUmd~k,q- W2kqj +Orlmblm,j,~kq l ) d A ,
(15)
where 2 k = x J D a ~ and g~t = 6kt/Da2 (not summed on k). Elastic analyses are carried out with the A B A Q U S finite element program (Hibbit et al., 1984) to calculate the stresses and displacements. In order to compare the energy release rate obtained analytically with that evaluated numerically, the small contact zone at the crack tip is not taken into consideration in the F E M computation. The calculation for the energy release rate according to the domain formula (15) is carried out in a separate post-processing program.
,nte nterau°n O°mains X
Fig. 4. Mesh configuration used in F E M computation for a circular crack.
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204
isoparametric elements, and the singular crack tip element is not used. Four integration domains are chosen to compute the energy release rate numerically. The first integration domain consists of four elements adjoining the crack tip; the second domain, which includes the first domain and the adjoining layer of elements, has 16 elements. The third and fourth domains are chosen in this fashion. The numerical values used here for the circular crack are as follows: w = h = 15a l, ]2,1= 10 GPa, 1,'1 = v 2 = 0 . 3 and t r = 100 MPa where /z and v with subscript 1 and 2 are the shear modulus and Poisson's ratio of material 1 and material 2, respectively. According to Park and E a r m m e (1986), the close agreement between the case of a finite sized matrix, w / a 1 >1 10 and the case of an infinite matrix is observed for /Zl//Z 1 = 2. Thus w = h = 15a t is chosen to compare the numerical result for a finite sized matrix
201
with the existing solution for an infinite matrix. For the cases of /d,2///d, 1 -~-1 and /d, 2//jt/. l = 2 respectively, the normalized energy release rates G / G o for a plane strain condition are tabulated in Tables 1 and 2 for various 0 0 describing the coordinates of the crack tip, where G o = rr(1 - v 1) o-2a2 sin 00/2~1 is the energy release rate when a crack with half-length a 2 sin 00 is in the homogeneous material 1 and is subjected to a normal tension or, and G ~ is the energy release rate for an infinite plate (w = h = oo) obtained by Perlman and Sih (1967) and Toya (1974). The values of the energy release rate for each domain are almost constant within 1 percent. The results are also in good agreement with the analytic solution obtained by Perlman and Sih (1967) and Toya (1974). Next, suppose a crack along the interface of a rigid inclusion (/z 2 = oo). In a similar way, the energy release rates are evaluated for the elliptic
Table 1 Normalized energy release rate for the circular crack with w = h = 15a 1, p.2//Zl = 1 and v I = v 2 = 0.3. Here Go = ~r(1-/,,1 ) o2a2 sin 00/2/z t, and G ~ is the energy release rate for an infinite plate obtained by Toya (1974)
0 o (°) 45 60 75 90
G/G o
G=/Go
Domain 1
Domain 2
Domain 3
Domain 4
Average
0.7595 0.6401 0.5318 0.4430
0.7489 0.6338 0.5289 0.4425
0.7533 0.6376 0.5320 0.4451
0.7556 0.6395 0.5335 0.4463
0.7543 0.6378 0.5316 0.4442
Table 2 Normalized energy release rate for the circular crack with w = h = 15al, ~2//.t,1 = 2 and /"1 = or2a2 sin 00/2/zi, and G ® is the energy release rate for an infinite plate obtained by Toya (1974)
//'2 =
0 o (°)
G/G o Domain 1
Domain 2
Domain 3
Domain 4
Average
45 60 75 90
0.8027 0.7073 0.6161 0.5369
0.8003 0.7094 0.6210 0.5433
0.8045 0.7130 0.6240 0.5460
0.8063 0.7142 0.6250 0.5467
0.8035 0.7110 0.6215 0.5432
0.7608 0.6400 0.5323 0.4444
0.3. Here G o = lr(1 - v 1)
G=/Go 0.8132 0.7156 0.6240 0.5445
Table 3 Normalized energy release rate for the crack along a rigid elliptic inclusion with a2/a I = 2, w = h = 15a2, 00 = 90 °, P-1 = 10 GPa and v 1 = 0.3. Here G o = rr(1 - v l) or2a2 sin 00/2/.Ll, and G ® is the energy release rate for an infinite plate obtained by Toya (1975). G c and G represent respectively the numerical result with and without constraint condition
G/G o GC/Go
Domain 1
Domain 2
Domain 3
Domain 4
Average
G~/Go
0.7920 0.8228
0.8017 0.8330
0.8062 0.8377
0.8089 0.8406
0.8022 0.8335
0.8293 0.8293
H.G. Beom et al. / Mechanics o f Materials 18 (1994) 195-204
202
crack in the case of a2/a 1 = 2, w = h = 15a 2, 00 = 90 °,/.t 1 = 10 GPa, v I = 0.3 and cr = 100 MPa. The result is tabulated in Table 3. Here G c represents the numerical result under the constraint condition that the tangential derivatives of the normal displacement at the outer boundaries vanish. The result shows that the values of the energy release rate with or without the constraint at outer boundaries for each domain except the first domain (the nearest to the crack tip) are almost constant within 1 percent, however, the energy release rate obtained numerically without the constraint does not approach G = closely in contrast to the case of /x2/tx I = 1 or 2 for the circular crack (see Tables 1 and 2). As noted by Choi and Earmme (1992) for circular cracks, the effect of w / a 2 on the energy release rate may be significant if the difference of the shear moduli is large. For the case of a2/a 1 = 2, w = h = 15a 2, #l = 10 GPa, v~ = 0.3 and cr = 100 MPa, the normalized energy release rate for a plane strain condition is plotted in Fig. 5 as 00 varies. The solid line represents the result obtained analytically by Toya (1975) while the open circles represent the result computed numerically from F E M using (16) and averaged for the four domains with the constraint as explained above. Close agreement between them is observed.
1.2-
1.0-
G/G0
(Y
tt
tt X2
2a•C
0
B2 ~ a 0
O
~x(x''x2)
q)"~ --0
(pl2~ =0 2w
(Y Fig. 6. G e o m e t r y o f a c r a c k with a p a r a b o l i c kink.
5.2. Crack with a parabolic kink As another example, a crack with a segment at the crack tip in the form of a parabolic arc subjected to biaxial loads ~r as shown in Fig. 6 is considered. Tractions vanish on the crack surfaces. Cracks with the portion of a curvilinear pattern in the vicinity of the crack tip are frequently observed since the path of the fracture is generally curved under loading of mixed mode type in a homogeneous material. Specifically the parabolic pattern has been chosen here as a typical curvilinear crack path. The crack surface is described by
O.80.60.4-
q3=
Analytic Solution (Toya)
0.20.0
30°
i
i
r
i
45°
60°
75°
90°
x2-b
Xl 2 -D -1,
x~>0.
(17)
a 0
F.E.M. (Present Method)
o
~(2~=
105°
0o Fig. 5. N o r m a l i z e d e n e r g y r e l e a s e r a t e f o r t h e c r a c k a l o n g t h e i n t e r f a c e o f a rigid i n c l u s i o n as a f u n c t i o n o f 0 0 d e s c r i b i n g t h e c o o r d i n a t e s o f t h e c r a c k tip. H e r e a 2 / a I = 2, w = h = 15a 2, tx~ = 10 G P a a n d u I = 0.3 a r e u s e d .
Here D = ~ / ( 2 b x C / a 2 ) 2 + 1 where x~: denotes the x~ coordinate of the crack tip C. The energy release rates are calculated numerically from FEM using (13) with (17). The numerical values used in the computation are as follows: /.L = 10 GPa, u = 0.3, ~r= 100 MPa, a 0 = 1 cm, b = l / 3 a 0 and w = h = 6 0 a 0. The normalized energy release rate obtained from the numerical computation for a plane strain condition
H.G. Beom et al. / Mechanics of Materials 18 (1994) 195-204
1.4 1.2G/G 0
1.0- . . . . . . . . . . . . . . .
0.80.6-
[] .......
0.4o 0.2- ~
0.0 0o
Crack Tip B F.E.M. (Present Method) Second Order Solution (Karihaloo et al.) Crack Tip C F.E.M. (Present Method) Second Order Solution (Karih~doo et al.) I i I 5° 10 o 15 °
20 °
O Fig. 7. Normalized energy release rates for the crack with a parabolic kink. at crack tips B and C is plotted in Fig. 7 as t9 varies, i.e., as the length of the parabolic segment varies. Here a9 denotes the angle between the x~ axis and BC, and is equal to O = tan-l[xC/(x c + 2a0)]. In particular, v~ = 0 represents the case of the straight crack of length 2a 0. The squares and circles represent the results at crack tips B and C, respectively, from the FEM using the method proposed in this paper, while the dotted and solid lines represent the second order solutions for the parabolic-kinked crack at crack tips B and C, respectively, which are evaluated from the solution for the stress intensity factors of a curved crack obtained by Karihaloo et al. (1981). G O ( = r r ( 1 - v)o'2a/2~) is the energy release rate for a crack with half-length a ( = l ~ ( x C + 2a0)Z + (x2C) z) in the elastic solid. The result for the parabolic-kinked crack is well compared with the second order perturbation solution for small O.
6. C o n c l u s i o n
A new conservation integral, which includes path and area integrals, for an interracial crack with a curved surface in dissimilar elastic solids is proposed. The conservation integral is shown to
203
have the physical meaning of energy release rate for the curved crack in a homogeneous material or in an interface. With the help of a function q we recast the expression of energy release rate to combine the path and area integrals into the domain integral over a finite domain around the crack tip. This domain integral expression is less sensitive to numerical inaccuracies in the crack tip region since the expression does not require a knowledge of accurate crack tip fields. The conventional elements, thus, can be adopted to evaluate the energy release rate by the finite element method, which is confirmed by the examples presented here.
Acknowledgement
The authors are grateful to Professor S. Nemat-Nasser of The University of California for helpful comments in finalizing Fig. 7.
References Beom, H.G. and Y.Y. Earmme (1993), Energy release rate for a curved crack in an elastic solid, Trans. Korean Soc. Mech. Engrs. 17, 543, (in Korean). Budiansky, B. and J.R. Rice (1973), Conservation laws and energy release rates, ASME J. Appl. Mech. 40, 201. Chen, Y.Z., D. Gross and Y.J. Huang (1991), Numerical solution of the curved crack problem by means of polynomial approximation of the dislocation distribution, Eng. Fract. Mech. 39, 791. Choi, N.Y. and Y.Y. Earmme (1992), Evaluation of stress intensity factors in circular arc-shaped interfacial crack using L integral, Mech. Mater. 14, 141. Cotterell, B. and J.R. Rice (1980), Slightly curved or kinked cracks, Int. J. Fract. 16, 155. England, A.H. (1966), An arc crack around a circular elastic inclusion, ASME J. Appl. Mech. 33, 637. Eshelby, J.D. (1956), The continuum theory of lattice defects, in: F. Seitz and D. Turnbull, eds., Solid State Physics 3, Academic Press, New York, p. 79. Evans, A.G. and J.W. Hutchinson (1989), Effects of nonplanarity on the mixed mode fracture resistance of bimaterial interfaces, Acta Metall. Mater. 37, 909. Hibbit, H.D., B. Karlsson and E.P. Sorensen (1984) ABAQUS User's Manual, Hibbit, Karlsson and Sorensen Inc., Providence, RI.
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Hutchinson, J.W. and Z. Suo (1991), Mixed mode cracking in layered materials, in: J.W. Hutchinson and T.Y. Wu, eds., Advances in Applied Mechanics, Vol. 29, Academic Press, New York. Karihaloo, B.L., L.M. Keer, S. Nemat-Nasser and A. Oranratnachai (1981), Approximate description of crack kinking and curving, ASME J. Appl. Mech. 48, 515. Karihaloo, B.L. and K. Viswanathan (1985), Elastic field of a partially debonded elliptic inhomogeneity in an elastic matrix (plane-strain), ASME J. Appl. Mech. 52, 835. Knowles, J.K. and E. Sternberg (1972), On a class of conservation laws in linearized and finite elastostatics, Arch. Rat. Mech. Anal, 44, 187. Li, F.Z., C.F. Shih and A. Needleman (1985), A comparison of methods for calculating energy release rates, Eng. Fract. Mech. 21, 405. Park, J.H. and Y.Y. Earmme (1986), Application of conservation integrals to interfacial crack problems, Mech. Mater. 5, 261. Perlman, A.B. and G.C. Sih (1967), Elastostatic problems of curvilinear cracks in bonded dissimilar materials, Int. J. Eng. Sci. 5, 845. Rice, J.R. (1968), A path independent integral and approximate analysisof strain concentration by notches and cracks, ASME J. Appl. Mech. 35, 379. Sendeckyj, S.D. (1974), Debonding of rigid curvilinear inclusions in longitudinal shear deformation, Eng. Fract. Mech. 6, 33. Shih, C.F., B. Moran and T. Nakamura (1986), Energy release rate along a three-dimensional crack front in a thermally stressed body, Int. J. Fract. 30, 79. Toya, M. (1974), A crack along the interface of a circular inclusion embedded in an infinite solid, J. Mech. Phys. Solids 22, 325. Toya, M. (1975), Debonding along the interface of an elliptic rigid inclusions, Int. Z Fract. 19, 989.
A p p e n d i x - E v a l u a t i o n o f (8b)
W e i n t r o d u c e the c u r v i l i n e a r c o o r d i n a t e s (sct, sc2) d e f i n e d by scj =~'(X1, X2) a n d ~:2= ~ ( x l , x 2) as shown in Fig. 2(a). D e n o t i n g the covariant base vectors a n d c o n t r a v a r i a n t base vectors for the curvilinear c o o r d i n a t e s by g,~ a n d g'~ ( a = 1, 2), respectively, it can be w r i t t e n that t = t ~ g a,
u ® V =
c3~ + u~
g~ ® g~'
V~ = g 2 ,
(A1)
w h e r e F ~ is the E u c l i d e a n Christoffel symbol a n d the G r e e k letter subscript a n d superscript d e n o t e the covariant a n d c o n t r a v a r i a n t compon e n t , respectively. U s i n g the relation g'~ x g t3 = e~t~g~, where E~t~ is the p e r m u t a t i o n t e n s o r in the curvilinear coordinates, it can be shown that
f+
F~ + F i-
e3jktmllm,j@,kd s [ 0u ~
= fri*
+Fi
Ea12t~t ~
+ U~F;1) ds.
On the interface bonded perfectly, u,,+
=t/a,
t+ =-t~(a= 1, 2), where the superscripts + and - denote the values on Fi+ a n d Fi-, respectively. It is readily seen that the integral in (A2) vanishes identically. (Ou~/O~l)+=(Ou~/O~l)
-
and
(A2)