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Effect of multi-flat inclusions on stress intensity factor of a semi-infinite crack
EnginmingFractureMechanicsvol. 47, No. 2, QQ. tiQ,‘li&t
Pergamon
0
1994
ElscviW
157468, !ki-
1994 Ltd.
F’rinted in Great Britain. All rights rwcrvai 0013-7944/94 s6.00 + 0.00
EFFECT OF MULTI-FLAT INCLUSIONS ON STRESS INTENSITY FACTOR OF A SEMI-INFINITE CRACK CA0 WEN and K. Y. LAM Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511, Republic of Singapore Abstract--To investigate the plane interaction between a semi-infinite crack and two micro flat inclusions or an array of micro flat inclusions, a numerical technique based on the singular integral method with a mathematical model which represents the it inclusions (including cracks) by distributions of edge dislocations is presented. It is assumed that the flat inclusion is an elastic sheet with negligible bending rigidity and the thickness of the flat inclusion is small compared to its lateral dimensions, so that the thickness distributions of the stress and strain in the inclusion may be neglected. The stress intensity factors of the main crack tip are calculated for a variety of crack-inclusion geometries. Numerical results for some cases are compared with available data in the literature and are found to be very accurate.
1. INTRODUCTION CRACKS AND flat inclusions are two important classes of imperfections in brittle materials such as concretes, rocks and ceramics, and in welded joints of metals. The edges of these imperfections are lines of stress singularity. Hence, they are expected to be the locations around which the fracture of the medium would generally nucleate [l]. The problems of cracks and flat inclusions which exist separately in materials have been reported by many researchers [2-4], but the interaction problems between the cracks and flat inclusions have not been widely studied in the literature. This interaction problem will be very important in studying, for example, the micromechanics of fatigue, the fracture in welded joints and the toughness of brittle materials. Liu and Erdogan gave a solution of a crack and a flat inclusion in an infinite plane in 1986 [5]. Cao and Lam recently studied the problem of a semi-infinite crack and a flat inclusion [6]. The present paper will investigate the interaction between multi-flat inclusions and an edge main crack in a semi-infinite plane. In this analysis, the plane interaction problem between a semi-infinite crack and two flat inclusions or an array of flat inclusions is proposed. The solution technique is based on the surface integral method using distributions of edge dislocations to represent the cracks [7,8] and flat inclusions. It is assumed that the flat inclusion is an elastic sheet with negligible bending rigidity and that the thickness of the flat inclusion is small compared to its lateral dimensions, so that the thickness distributions of the stress and strain in the inclusion may be neglected. As with most other analyses, the body is assumed to be in a state of plane strain. Relevant stress intensity factors of the semi-infinite main crack are calculated for various crack-inclusion geometries.
2. FORMULATIONS AND IMPLEMENTATIONS A brief outline of the formulations and numerical implementations is given here; for a fuller account see refs [a-S]. As a general formulation, we let a semi-infinite main flat inclusion lie on the axis y = 0 (x < 0) in the (x, JJ) plane (see Fig. 1). There are N - 1 micro flat inclusions located and oriented arbitrarily around the tip of the main flat inclusion. The flat inclusion could become a crack when its shear modulus is equal to zero; the formulations are suitable to the interaction between any flat inclusions, any cracks, and between any flat inclusions and cracks. As the problem is linearly elastic, the solution of the present problem can be obtained through the superposition of two problems. The first refers to the given medium subjected to a remote mode I loading without the flatinclusions. By using the transformation relationship [4], the normal and shear tractions along ‘the ith inclusion due to the external loading can be obtained. In the second problem, only the stress disturbance due to the existence of the flat inclusions in the medium is considered. Here, the flat inclusions are represented by distributions of edge dislocations. The 157
158
CA0 WEN and K. Y. LAM
normal and shear tractions along the ith flat inclusion surfaces due to the edge dislocations are [4,7]:
(1) where pc is the corresponding dislocation density at point (x,, y,); r is the relevant stress influence function, and b = 1 and 2 refers to the Burger vector in the x and y directions, respectively. The sum of these two tractions should satisfy the boundary conditions along the ith inclusion. They are: (2)
{~;“d,j+~~~~:~~}={~~~~}~
where o&(x,) and ab(x,) are the boundary normal and shear tractions on the ith flat inclusions. Fand S refer to the tractions in the first and second problems of the superposition. To obtain a$(~,) and a;(~,), we define: b,(xi) = -$,
bm(xi)=
[u(xi, +O) -
-&
[#(xi,
O(Xi9
-o)]
(3)
+O)- u(x*, -o)],
I
where u and D are the displacement vectors in the xl and yi directions, respectively (see Fig. 1). As mentioned before, the thickness of the ith flat inclusion h(x,) is small, so that the thickness distributions of stress and strain in the flat inclusion can be neglected. From Hooke’s law for the plane strain case, we have: c;(q)
= --
ki + 1 k-
6~(Xi)
= -I
*-
l
G,
Xi
b,(t)
h(&)
G. h(xi)
dt (a, d xi < b,)
s aj
(5)
xi sq
bm(f) dt (ai G xi 6 b,),
where k, = 3 - 4Vi, G1, k, and Viare the elastic constants of the ith flat inclusion, a, and b, are two ends of the ith inclusion (see Fig. l), and: b, (Xi) = cos Oli* ~1 (x,) + sin ai * ~(2(xi)
I
1
1
I&
I
I
Fig. 1. Flat inclusion configuration.
(7)
159
Fig. 2. A flat inclusion near an equal length collinear crack in an infmite plane.
ktxi) = sin % *Ccl(xi) - cm ai*c(2(4),
(8)
where a, = 180” - !P,. Now we can obtain: cr;(q) = --
k, + 1
G,
*k - 1 4x,)
b$(X,)= -2
G.
[cos ai * pl (t) +
sin a, *pz(t)] dt
(9
XI
I
h(x,)
[sin a, * ,ul (t) - cos a, - p*(t)] dt.
(10)
a,
Using the method described in refs [4,8], the unknown dislocation density function can be expressed as: Ml) c(btti)
=
(11) 41
-
t:>
Table 1. Comparison of k;, at the crack tips due to a collinear tlat inclusion in an infinite plane [see Fig. 2; let c = -a, d = -b, 2u/(b -a) = 0.011 Zh/(b - a) 0.01
0.02
0.10
0.20
G = 0.0
LiuPlK,(c) Present 4(c) Liu PI KIVI Present 4(d)
2.9642 2.969 1.2063 1.206
2.9642 2.969 1.2063 1.206
2.9642 2.969 1.2063 1.206
2.9642 2.969 1.2063 1.206
G = 0.05
Liu [S] K,(c) r’resent K,(c) Liu M 4 (d) Present K,(d)
1.1795 1.172 1.0116 1.011
1.2952 1.282 1.0211 1.020
1.7825 1.753 1.0693 1.066
2.0764 2.044 1.1019 1.098
G=l.O
Liu [5] K,(c) Present K,(c) Liu [S] K,(d) f’resent K,(d)
1.0132 1.013 l.ooo7 1.001
1.0255 1.025 1.0014 1.002
1.1045 1.101 1.0063 1.006
1.1795 1.172 1.0116 1.011
G = 5.0
Liu (51K,(c) mt C(c)
1.0027 1.003 1.0001 1.001
1NJ54 1.006 1.003 1.001
1.0255 1.025 1.0014 1.002
1.0479 1.047 1.0027 1.003
fiu1514(4
Present
K,(d)
CA0 WEN and K. Y. LAM
160
Table 2. Comparison of K& at the crack tips due to a collinear flat inclusion in an infinite plane [see Fig. 2; let c = -a. d = 4, 2h/(b -0) = 0.04 h/(b
-a)
0.01
0.5
1.0
2.0
Liu [5] K,(c) Present K,(c) Liu [5] K,(d) Present K,(d)
2.9642 2.969 1.2063 1.206
1.1125 1.112 1.0517 1.052
1.0480 1.048
1.0280 1.028
1.0176 1.018 1.0125 1.013
Liu [5] K,(c) Present K,(c) Liu [5] K,(d) Present K,(d)
1.5321 1.510 1.041
1.0229 1.021 1.0104 1.010
1.0096 1.009 1.0056 1.006
1.0035 1.003 1.0025 1.003
G= 1.0
Liu [5] K,(c) Present K,(c) Liu [5] K,(d) Present K,(d)
1.0583 1.057 1.0033 1.003
1.0015 1.001 1.0007 1.001
1.006 1.001 1.0004 1.001
1.0002 1.000 1.0002 1.000
G = 5.0
Liu [5] K,(c) Present K,(c) Liu [5] K,(d) Present K,(d)
1.0132 1.013 1.0007 1.001
1.0003 1.000 1.0001 1.000
1.0001 1.000 l.OmIl 1.000
1.0000 1.000 1.0000 1.000
G = 0.0
G = 0.05
1.0432
and the integral can be discretized as follows:
(12) 1 n
’
s
h
(Ii)
-&-x)&l
dti
l
-tf)
2
htfij)
= Yi&, , ‘ii-xik’
(13)
where tij = cos[(2j - 1)7r/(2Mi)]
(j = 1,2,
X& = COS(h
(k=1,2,...,Mi-1);
/Ml)
. . . , Mi)
tij and xlt are called Chebyshev zeros of the first and second kind, respectively. Mi is the number of dislocation points we use to represent the ith flat inclusion. After solving for the unknown functions f, and f2, the mode I stress intensity factor of the main flat inclusion can be calculated as follows: K
=
I
_E&a)
4(1 _ vq Lfi(r) . cosY + f*(t) *sin Y],
(14)
where a is the half inclusion length, and E and v are the elastic constants of the matrix. As the main interest of this work is to solve the interaction problem between a semi-infinite crack and two flat inclusions or an array of flat inclusions, we let G, = v, = 0. The elastic constants of the matrix Go and the ith inclusion G, (i = 2,3, . . . , N) are assumed to be equal to 2. The results apply to the plane strain condition for v,,= vi = 0.25, where v0 and vi are the Poisson’s ratios of the matrix and the ith flat inclusion, respectively. It is also assumed that the thickness of the ith inclusion is a constant. In the calculation, the mode I stress intensity factors $ are normalized with respect to K,,,, where K,, is the mode I stress intensity factor for the case of no interaction effect. The notation used is:
3. EXAMPLES AND DISCUSSION In order to verify the correctness and accuracy of the present method, the problem of a crack and an equal length collinear flat inclusion in an infinite plane under a remote mode I loading is considered (see Fig. 2). The results obtained by the present method are shown in Tables 1 and 2,
Multi-&it inclusions
I
t
161
f i t I t
Fig. 3. Two flat inclusions near the main crack tip in a semi-infinite plane.
together with those values given by Liu and Erdogan [5]. It is shown that good agreement is achieved between these two solutions with a difference of less than 1.7% when we use 15 dislocation points to represent one crack or one flat inclusion. Tables 1 and 2 indicate that the present method is efficient and accurate. In the following section, the crack-inclusion configuration consists of a semi-infInite crack which intersects the free edge of a semi-infinite medium and two micro flat inclusions which are located and oriented symmetrically to the main crack or an array of flat inclusions under a remote mode I loading. It should be noted that the thickness of the inclusion is much smaller than the length of the inclusion (let h/u = 0.15). The effective stress intensity factor K, may be either increased (enhancement, & > 1) or decreased (shielding, Kdf < l), depending on the locations (di/a and a,), orientations (/II) and the non-dimensional&d thickness (hi/U) of the flat inclusions, and depending on the modulus ratios of the flat inclusions to matrix (G = G,/G,,), where G, and Gi are the shear moduli of the matrix and the ith flat inclusion, respectively. 3.1. Two micro j?at inclusions In this section, the crack-inclusion configuration consists of a main crack which intersects the free edge of a semi-infinite medium and two micro flat inclusions which are located and oriented
0.4-I
0
20
40
80
80 loo a (DEG)
I
120
140
160
t
180
Fig. 4. The effect on stress intensity factor due to two “stacked” flat inclusions (d/a = 1.1).
CA0 WEN and K. Y. LAM
162
0.8
0
I 20
I 40
1 60
I 80
I
I
I
I
6
100
120
140
160
160
a (DEG) Fig. 5. The effect on stress intensity factor due to two “stacked” flat inclusions (d/a = 1.5).
symmetrically to the main crack (see Fig. 3). The effective stress intensity factors are calculated for three cotigurations: (a) two “stacked” flat inclusions; (b) two inclined flat inclusions; (c) two symmetrically oriented flat inclusions. 3.1.1. Two “stacked”Jat inclusions. Consider two “stacked” micro flat inclusions which are parallel to the semi-infinite crack (Fig. 3; let fi, = j?*= 0). The effect on the stress intensity factor at the main crack tip is studied by varying the modulus ratios of the flat inclusions to matrix (G = G,/G, = GJG,,), where Go is the shear modulus of the matrix, and G, and G2 are the shear modulus of the first and second flat inclusions respectively, the location angle (a = a, = ar) and the non-dimensional&d distance (d/a = d,/a = d,/a; Fig. 4 for d/a = 1.1 and Fig. 5 for d/a = 1.5). The values of G that are taken are 0.00,0.02, 0.05,0.10,0.20 and 0.50. The case of G = 0.00 refers to a crack (the flat inclusion becomes a crack). From Figs 4 and 5, the following can be observed that. (1) Enhancement occurs in front of the main crack tip at about 120-150”, beyond which shielding occurs. (2) For any tixed angle a, the interaction effect increases for decreasing values of G. When G = 0, this effect is maximum. (3) When the modulus ratios of the flat inclusions to matrix G decrease, the minimum K& increases and the angles which correspond to Kdf = 1 decrease. This indicates that the interaction effect becomes more significant and the region which yields an enhancement effect diminishes as the value of G decreases. (4) When the flat inclusions move away from the main crack tip, the interaction effect will be less significant, and the angles which 2.0-
1.a-
1.6&4i 1.4-
1.0+ 0
30 a(DEG)
Fig. 6. The effect on stress intensity factor due to two inclined flat inclusions (d/a = 1.1).
Multi-flat inclusions
163
a(DEG) Fig. 7. The effect on stress intensity factor due to two inched
llat inclusions (d/u = 1.5).
correspond to maximum K& and & = 1 increase. It should be noted that the & lines in these two figures are not extended to the locations of a = 0” and a = 180” because the micro flat inclusions in these cases are considered to be located too close to each other or to the main crack, and the numerical results obtained may not be accurate and realistic in these regions. 3.1.2. Two inched @zt inclusions. The interaction effects between two inclined micro flat inclusions and a semi-infinite main crack under a remote mode I loading (see Fig. 3; let a, = a2 = /3, = f12) are studied. The effective stress intensity factor of the main crack tip is plotted vs the location angle (a = a, = az) for various values of G(G = G,/G, = G,/G,) and d/u (d/u = d, /a = dJa; Fig. 6 for d/a = 1.1 and Fig. 7 for d/a = 1.5). It can be shown from these two figures that enhancement always occurs when a < 150” and the shielding effect can be ignored in this configuration. For any fixed value of a (0“ < a < 150’), & increases with decreasing values of G. The maximum & increases and the angle at which the maximum lu, occurs decreases as the value of G decreases; when the ratio d/a increases, the maximum & decreases and the angle at which the maximum A& occurs decreases. From this, we can observe that the interaction effect will be less significant when the inclined flat inclusions move away from the main crack tip or the flat inclusions become harder (0 < G < 0.5). 3.1.3. Two symmetrically orientedPat inclusions. To investigate further the interaction effect between the main crack and two flat inclusions due to the orientations of these two flat inclusions, the problem which consists of a semi-infinite crack and two symmetrically oriented flat inclusions is studied. The effective stress intensity factor at the main crack tip is plotted vs the orientation angle (B = j?, = &) for various locations of these two flat inclusions (x,/u = x&r = 1.5, 1.0, 0.0,
1.2-
1.1-
0.9-
bd...-1.5 0.8
I
0
I
I
I
I
I
20
40
60
80
1w
I
120
I
140
I
180
I
180
B PEG) Fig. 8. The e!Tect on stress intensity factor due to two symmetrically oriented flat inclusions (G = 0.02).
CA0 WEN and K. Y. LAM
164
0.90 0
20
40
60
80
loo
120
140
160
180
PK'W
Fig. 9. The effect on stress intensity factor due to hvo symmetrically oriented flat inclusions (G = 0.05).
-1.0, -1.5; y,/a = 1.5, JJJa = - 1.5; where x, and y, are the central points of the first flat inclusion, x, and yZ are the central points of the second flat inclusion) and values of G(G = G,/Go = G,/G,; Fig. 8 for G = 0.02, Fig. 9 for G = 0.05). These two figures show that the shielding effect occurs even with the flat inclusions present in front of the main crack tip (for example, see the lines of x/a = 1.5), and the enhancement effect occurs even with the flat inclusions present in the flanks of the main crack (for example, see the lines of x/a = 0.0) due to the orientations of the flat inclusions. It should be noted that the angle which corresponds to maximum lu, decreases when the coordinate x increases (from - 1.5 to 1.5). It should also be noted that the interaction effect becomes more significant for decreasing values of G. The results in these two figures show that orientations of the flat inclusions indeed play an important role in the crack-inclusion
interaction.
3.2. An array of micro flat inclusions In this section, the crack-inclusion configuration consists of a semi-infinite crack and a micro flat inclusion array. The effective stress intensity factors are calculated for three configurations: (a) an array of collinear flat inclusions; (b) an array of “stacked” flat inclusions (odd); (c) an array of “stacked” flat inclusions (even). 3.2.1. An array of collinear fiat inclusions. In the case of a semi-infinite crack and an array of collinear flat inclusions, the effective stress intensity factor at the main crack tip is studied by varying the non-dimensional&d size d/a and the ratio G (G = Gi/Go, i = 1,2, . . . , where Go is the 20G.O.00 G-O.02
1.8-
G=O.O5
G.O.lO
1.6-
6.010
Kefl
G-Oil
1.41.2-
1.0 1.0
1.5
20
2.5
3.0
d/a Fig. 10. The effect on stress intensity factor due to an array of collinear tlat inclusions.
Multi-flat illclusi0M
165
II1.b t.5 IA-
bfl
1.3-
1.21.1-
l.O!
1.0
I
I
I
I
1
I
1.2 I.4 I.6 1.8 2.0 22
I
,
2.4 26
1
28
,
3.0
xla Fig. Il. The effect on stress intensity factor due to an array of “stacked” flat inclusions (odd, p = a).
shear modulus of the matrix and Gi is the shear modulus of the ith micro flat inclusion). The numerical results are presented in Fig. 10. It can be seen that for any fixed value of d/a, the stress intensity factor increases with decreasing values of G. When G = 0, lu, is maximum. It should be noted that K& is quite high when the array of collinear flat inclwions is near the main crack tip. If the ratio d/a increztsm, & will diminish for any certain value of G. In the case of d/a > 3, the interaction effect is not significant (with accuracy of up to 5%). 3.2.2. An array of “stacked”jZat incZwions (odd). Now consider a semi-infinite crack and an array of “stacked” flat inclusions (odd). This array of inclusions is symmetrical with respect to the x-axis, equally spaced with a pitchp and has a flat inclusion on the x-axis (see Fig. 11). The effective stress intensity factors are plotted vs x/a for various values of G (G = GJG,) and p (p = a in Fig. 11, p = 2a in Fig. 12 and p = 4a in Fig. 13). We compare results obtained here for the array with the data given for a micro single collinear crack near the main crack tip in these three figures (see one-crack lines). From Figs 1l-13, the following can be seen. (1) When the array of “stacked” flat inclusions is near the main crack tip, K& is quite high. The interaction effect will diminish as the array of the flat inclusions moves away from the main crack tip. (2) For any fixed value of x/a, & will increase when the value of G decreases. (3) Generally, with increasing value of p, lyeff will decrease except when both x/a and G are very small. (4) In the case of p/a = 1, krrf due to the array of cracks (G = 0) will be larger than lu, due to a single collinear crack when the value of x/a is greater than 1.17; in the cases when p/a = 2 and 4, the corresponding values of x/a are 1.5 and 2.2, respectively. This is expected as the effective stress intensity factor due to an array of “stacked” cracks will be smaller than K;B due to the corresponding single crack when the array 1.71.6G.O.00
1.5 IAbfl
1% 1.2i.l-
1.0
1.2
1.4
1.6
1.8
2:O 22
2.4
iS
28
3.0
x/a Fig. 12. The effect on stress intensity factor due to an array of “stacked” flat inclusions (odd, p = 20).
CA0 WEN and K. Y. LAM
166 1.71.6-
1.0 1.0
1.2
1.4
1.6
1.6
2.0
2.2
2.4
26
2.6
3.0
x/a Fig. 13. The effect on stress intensity factor due to an array of “stacked” flat inclusions (odd, p = 4a).
cracks are closed to the main crack tip. When the array of cracks moves away from the main crack tip, the interaction effect due to the array of cracks will be stronger than the effect due to the corresponding single cracks. If p is very big, J& due to the array of cracks will always be smaller than or equal to & due to a single crack. 3.2.3. An array of “stacked” j7at inclusions (men). In this section, the effect on the stress intensity factor at the main crack tip due to an array of “stacked” flat inclusions (even) is presented. This array of inclusions is also symmetric with respect to the x-axis, equally spaced with a pitch p, but does not have a flat inclusion on the x-axis (see Fig. 14). k;B of the main crack tip is plotted vs x/a for various values-of G (G = GJG,) and p (p = a in Fig. 14, p = 2a in Fig. 15 and p = 4a in Fig. 16). We compare the results obtained here for the array with the data given for two “stacked” cracks near the main crack tip (p is the distance between these two cracks) in these three figures (see two-crack lines). From Figs 14-16, the following can be seen. (1) Stress amplification changes to shielding at about x/a = 0.4-0.9 (the point which corresponds to lu, = 1 is called the neutral point). When the array of inclusions moves away from the main crack (from the neutral point), & at first increases for a short distance of x/a, and then decreases. When the value of x/a decreases (from the value of the neutral point), at first the shielding effect increases, but then this effect diminishes. The trends of the curves indicate that II: ,+l as x/a-&m. (2) For any lixed value of x/a, the interaction effect increases with decreasing value of G. (3) With increasing value of G, the minimum & increases and shifts to the left, and the value of x/a which corresponds to the neutral point decreases. (4) The interaction effect will be decreased with increasing value of p. l.61
P
1.4
I
1 ==g
:
1.2
:
1.0
Kefl
0.6 0.6 0.40.2 -2.0
I -1.5
I -1.0
I -0.5
I 0
I 0.5
I 1.0
I 1.5
I 2.0
x/a Fig. 14. The effect on stress intensity factor due to an array of “stacked” flat inclusions (even, p = a).
167
Multi-flat inclusions
Fig. 15. The effect on stress intensity factor due to an array of “stacked” flat inclusions (even, p = 2a).
It should be noted that the present results in some configurations for cracks agree well with the data of Rubinstein and Choi [9]. For example, in the case of two “stacked” cracks (even, p = a) approaching the semi-infinite main crack, according to our method the maximal shielding effect is 76% (see Fig. 14); the corresponding value in ref. [9] is 77%. The interaction between microcracks and a main semi-infinite crack has been investigated by Lam and Cao [lo, 111. 4. CONCLUSIONS An effective numerical technique based on the singular integral method with a mathematical model which represents inclusions (including cracks) by distributions of edge dislocations was applied to the interaction problem between a semi-infinite crack and two micro flat inclusions or an array of micro inclusions. The method and numerical formulation in this paper were carefully examined in some cases with known solutions in the literature and were found to be very accurate. The results illustrate that the effective interaction due to micro flat inclusions occurs only at very small distances from the flat inclusions to the main crack tip and the flat inclusions can yield either a stress enhancement or a stress shielding effect to the main crack tip depending upon the locations and orientations of the flat inclusions and also upon the modulus ratios of the flat inclusions to matrix.
.I
bG.O.50
ic---Go.O.20
G-0.10 0.0.05 G -0.02 G=O.OO
x/a Fig. 16. The effect on stress intensity factor due to an array of “stacked” flat inclusions (even, p = 40).
168
CA0 WEN and K. Y. LAM
RJ!TERENCFS (11 F. Erdo8an and C. D. Gupta. Stresses near a Bat inclusion in bonded dissimilar materials. Inr. J. Solids Srrucrures 8, 533-547 (1972). [2] A. B. Doyum and A. Gursoy, An arbitrarily oriented inclusion a half-space subjected to an inclined surface load. Engng Fracfare Me& 48, 609-625 (1991). [3] C. Atkinson, Some ribbon-like inclusion problems. Inr. J. Engng Sci. 11, 243-266 (1973). [4] K. Y. Lam and S. P. Phua, Multiple crack interaction and its cffcct on stress intensity factor. Theor. appf. Fracture Mech. (in press). [S] Liu XucHui and F. Erdo8an, The crack-inclusion interaction problem. Engng Fracture Mech. 23, 821-832 (1986). [6] Cao Wcn and K. Y. Lam, Effect of a flat inclusion on stress intensity factor of a sami-i&rite crack. 7’heor. appl. Fracture Mech. (in press). [7] K. Y. Lam and M. P. Clcary, Slippa8c and m-initiation of (hydraulic) fracture at frictional intcrfaccs. Int. J. namer. Anal. Meth. Geomech. 8, 589-604 (1984). [8] K. Y. Lam, B. Cottcre.11and S. P. Phua, Statistics of 8aw interaction in brittle materials. J. Am. Cerum. Sot. 74,352-357 (1991). [9] A. A. Rubinstcin and H. C. Choi, Microcrack interaction with transverse array of microcracks. Int. J. Fracture 36, 15-26 (1988). (lo] K. Y. Lam and Cao Wen, Interaction between microcracks and a main crack in a semi-infinite medium. Engng Fracture Mech. 44, 753-761 (1993). [ll] Cao Wcn and K. Y. Lam, Use of dislocation based method for microcracks interaction. Proc. Inf. Conf. on Computational Methodr in Engineering: Advances & Applications (Edited by A. A. 0. Tay and K. Y. Lam), pp. 1081-1086. World Scientific, Singapore. (Received 19 Februurv 1993)
Pergamon
0
1994
ElscviW
157468, !ki-
1994 Ltd.
F’rinted in Great Britain. All rights rwcrvai 0013-7944/94 s6.00 + 0.00
EFFECT OF MULTI-FLAT INCLUSIONS ON STRESS INTENSITY FACTOR OF A SEMI-INFINITE CRACK CA0 WEN and K. Y. LAM Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511, Republic of Singapore Abstract--To investigate the plane interaction between a semi-infinite crack and two micro flat inclusions or an array of micro flat inclusions, a numerical technique based on the singular integral method with a mathematical model which represents the it inclusions (including cracks) by distributions of edge dislocations is presented. It is assumed that the flat inclusion is an elastic sheet with negligible bending rigidity and the thickness of the flat inclusion is small compared to its lateral dimensions, so that the thickness distributions of the stress and strain in the inclusion may be neglected. The stress intensity factors of the main crack tip are calculated for a variety of crack-inclusion geometries. Numerical results for some cases are compared with available data in the literature and are found to be very accurate.
1. INTRODUCTION CRACKS AND flat inclusions are two important classes of imperfections in brittle materials such as concretes, rocks and ceramics, and in welded joints of metals. The edges of these imperfections are lines of stress singularity. Hence, they are expected to be the locations around which the fracture of the medium would generally nucleate [l]. The problems of cracks and flat inclusions which exist separately in materials have been reported by many researchers [2-4], but the interaction problems between the cracks and flat inclusions have not been widely studied in the literature. This interaction problem will be very important in studying, for example, the micromechanics of fatigue, the fracture in welded joints and the toughness of brittle materials. Liu and Erdogan gave a solution of a crack and a flat inclusion in an infinite plane in 1986 [5]. Cao and Lam recently studied the problem of a semi-infinite crack and a flat inclusion [6]. The present paper will investigate the interaction between multi-flat inclusions and an edge main crack in a semi-infinite plane. In this analysis, the plane interaction problem between a semi-infinite crack and two flat inclusions or an array of flat inclusions is proposed. The solution technique is based on the surface integral method using distributions of edge dislocations to represent the cracks [7,8] and flat inclusions. It is assumed that the flat inclusion is an elastic sheet with negligible bending rigidity and that the thickness of the flat inclusion is small compared to its lateral dimensions, so that the thickness distributions of the stress and strain in the inclusion may be neglected. As with most other analyses, the body is assumed to be in a state of plane strain. Relevant stress intensity factors of the semi-infinite main crack are calculated for various crack-inclusion geometries.
2. FORMULATIONS AND IMPLEMENTATIONS A brief outline of the formulations and numerical implementations is given here; for a fuller account see refs [a-S]. As a general formulation, we let a semi-infinite main flat inclusion lie on the axis y = 0 (x < 0) in the (x, JJ) plane (see Fig. 1). There are N - 1 micro flat inclusions located and oriented arbitrarily around the tip of the main flat inclusion. The flat inclusion could become a crack when its shear modulus is equal to zero; the formulations are suitable to the interaction between any flat inclusions, any cracks, and between any flat inclusions and cracks. As the problem is linearly elastic, the solution of the present problem can be obtained through the superposition of two problems. The first refers to the given medium subjected to a remote mode I loading without the flatinclusions. By using the transformation relationship [4], the normal and shear tractions along ‘the ith inclusion due to the external loading can be obtained. In the second problem, only the stress disturbance due to the existence of the flat inclusions in the medium is considered. Here, the flat inclusions are represented by distributions of edge dislocations. The 157
158
CA0 WEN and K. Y. LAM
normal and shear tractions along the ith flat inclusion surfaces due to the edge dislocations are [4,7]:
(1) where pc is the corresponding dislocation density at point (x,, y,); r is the relevant stress influence function, and b = 1 and 2 refers to the Burger vector in the x and y directions, respectively. The sum of these two tractions should satisfy the boundary conditions along the ith inclusion. They are: (2)
{~;“d,j+~~~~:~~}={~~~~}~
where o&(x,) and ab(x,) are the boundary normal and shear tractions on the ith flat inclusions. Fand S refer to the tractions in the first and second problems of the superposition. To obtain a$(~,) and a;(~,), we define: b,(xi) = -$,
bm(xi)=
[u(xi, +O) -
-&
[#(xi,
O(Xi9
-o)]
(3)
+O)- u(x*, -o)],
I
where u and D are the displacement vectors in the xl and yi directions, respectively (see Fig. 1). As mentioned before, the thickness of the ith flat inclusion h(x,) is small, so that the thickness distributions of stress and strain in the flat inclusion can be neglected. From Hooke’s law for the plane strain case, we have: c;(q)
= --
ki + 1 k-
6~(Xi)
= -I
*-
l
G,
Xi
b,(t)
h(&)
G. h(xi)
dt (a, d xi < b,)
s aj
(5)
xi sq
bm(f) dt (ai G xi 6 b,),
where k, = 3 - 4Vi, G1, k, and Viare the elastic constants of the ith flat inclusion, a, and b, are two ends of the ith inclusion (see Fig. l), and: b, (Xi) = cos Oli* ~1 (x,) + sin ai * ~(2(xi)
I
1
1
I&
I
I
Fig. 1. Flat inclusion configuration.
(7)
159
Fig. 2. A flat inclusion near an equal length collinear crack in an infmite plane.
ktxi) = sin % *Ccl(xi) - cm ai*c(2(4),
(8)
where a, = 180” - !P,. Now we can obtain: cr;(q) = --
k, + 1
G,
*k - 1 4x,)
b$(X,)= -2
G.
[cos ai * pl (t) +
sin a, *pz(t)] dt
(9
XI
I
h(x,)
[sin a, * ,ul (t) - cos a, - p*(t)] dt.
(10)
a,
Using the method described in refs [4,8], the unknown dislocation density function can be expressed as: Ml) c(btti)
=
(11) 41
-
t:>
Table 1. Comparison of k;, at the crack tips due to a collinear tlat inclusion in an infinite plane [see Fig. 2; let c = -a, d = -b, 2u/(b -a) = 0.011 Zh/(b - a) 0.01
0.02
0.10
0.20
G = 0.0
LiuPlK,(c) Present 4(c) Liu PI KIVI Present 4(d)
2.9642 2.969 1.2063 1.206
2.9642 2.969 1.2063 1.206
2.9642 2.969 1.2063 1.206
2.9642 2.969 1.2063 1.206
G = 0.05
Liu [S] K,(c) r’resent K,(c) Liu M 4 (d) Present K,(d)
1.1795 1.172 1.0116 1.011
1.2952 1.282 1.0211 1.020
1.7825 1.753 1.0693 1.066
2.0764 2.044 1.1019 1.098
G=l.O
Liu [5] K,(c) Present K,(c) Liu [S] K,(d) f’resent K,(d)
1.0132 1.013 l.ooo7 1.001
1.0255 1.025 1.0014 1.002
1.1045 1.101 1.0063 1.006
1.1795 1.172 1.0116 1.011
G = 5.0
Liu (51K,(c) mt C(c)
1.0027 1.003 1.0001 1.001
1NJ54 1.006 1.003 1.001
1.0255 1.025 1.0014 1.002
1.0479 1.047 1.0027 1.003
fiu1514(4
Present
K,(d)
CA0 WEN and K. Y. LAM
160
Table 2. Comparison of K& at the crack tips due to a collinear flat inclusion in an infinite plane [see Fig. 2; let c = -a. d = 4, 2h/(b -0) = 0.04 h/(b
-a)
0.01
0.5
1.0
2.0
Liu [5] K,(c) Present K,(c) Liu [5] K,(d) Present K,(d)
2.9642 2.969 1.2063 1.206
1.1125 1.112 1.0517 1.052
1.0480 1.048
1.0280 1.028
1.0176 1.018 1.0125 1.013
Liu [5] K,(c) Present K,(c) Liu [5] K,(d) Present K,(d)
1.5321 1.510 1.041
1.0229 1.021 1.0104 1.010
1.0096 1.009 1.0056 1.006
1.0035 1.003 1.0025 1.003
G= 1.0
Liu [5] K,(c) Present K,(c) Liu [5] K,(d) Present K,(d)
1.0583 1.057 1.0033 1.003
1.0015 1.001 1.0007 1.001
1.006 1.001 1.0004 1.001
1.0002 1.000 1.0002 1.000
G = 5.0
Liu [5] K,(c) Present K,(c) Liu [5] K,(d) Present K,(d)
1.0132 1.013 1.0007 1.001
1.0003 1.000 1.0001 1.000
1.0001 1.000 l.OmIl 1.000
1.0000 1.000 1.0000 1.000
G = 0.0
G = 0.05
1.0432
and the integral can be discretized as follows:
(12) 1 n
’
s
h
(Ii)
-&-x)&l
dti
l
-tf)
2
htfij)
= Yi&, , ‘ii-xik’
(13)
where tij = cos[(2j - 1)7r/(2Mi)]
(j = 1,2,
X& = COS(h
(k=1,2,...,Mi-1);
/Ml)
. . . , Mi)
tij and xlt are called Chebyshev zeros of the first and second kind, respectively. Mi is the number of dislocation points we use to represent the ith flat inclusion. After solving for the unknown functions f, and f2, the mode I stress intensity factor of the main flat inclusion can be calculated as follows: K
=
I
_E&a)
4(1 _ vq Lfi(r) . cosY + f*(t) *sin Y],
(14)
where a is the half inclusion length, and E and v are the elastic constants of the matrix. As the main interest of this work is to solve the interaction problem between a semi-infinite crack and two flat inclusions or an array of flat inclusions, we let G, = v, = 0. The elastic constants of the matrix Go and the ith inclusion G, (i = 2,3, . . . , N) are assumed to be equal to 2. The results apply to the plane strain condition for v,,= vi = 0.25, where v0 and vi are the Poisson’s ratios of the matrix and the ith flat inclusion, respectively. It is also assumed that the thickness of the ith inclusion is a constant. In the calculation, the mode I stress intensity factors $ are normalized with respect to K,,,, where K,, is the mode I stress intensity factor for the case of no interaction effect. The notation used is:
3. EXAMPLES AND DISCUSSION In order to verify the correctness and accuracy of the present method, the problem of a crack and an equal length collinear flat inclusion in an infinite plane under a remote mode I loading is considered (see Fig. 2). The results obtained by the present method are shown in Tables 1 and 2,
Multi-&it inclusions
I
t
161
f i t I t
Fig. 3. Two flat inclusions near the main crack tip in a semi-infinite plane.
together with those values given by Liu and Erdogan [5]. It is shown that good agreement is achieved between these two solutions with a difference of less than 1.7% when we use 15 dislocation points to represent one crack or one flat inclusion. Tables 1 and 2 indicate that the present method is efficient and accurate. In the following section, the crack-inclusion configuration consists of a semi-infInite crack which intersects the free edge of a semi-infinite medium and two micro flat inclusions which are located and oriented symmetrically to the main crack or an array of flat inclusions under a remote mode I loading. It should be noted that the thickness of the inclusion is much smaller than the length of the inclusion (let h/u = 0.15). The effective stress intensity factor K, may be either increased (enhancement, & > 1) or decreased (shielding, Kdf < l), depending on the locations (di/a and a,), orientations (/II) and the non-dimensional&d thickness (hi/U) of the flat inclusions, and depending on the modulus ratios of the flat inclusions to matrix (G = G,/G,,), where G, and Gi are the shear moduli of the matrix and the ith flat inclusion, respectively. 3.1. Two micro j?at inclusions In this section, the crack-inclusion configuration consists of a main crack which intersects the free edge of a semi-infinite medium and two micro flat inclusions which are located and oriented
0.4-I
0
20
40
80
80 loo a (DEG)
I
120
140
160
t
180
Fig. 4. The effect on stress intensity factor due to two “stacked” flat inclusions (d/a = 1.1).
CA0 WEN and K. Y. LAM
162
0.8
0
I 20
I 40
1 60
I 80
I
I
I
I
6
100
120
140
160
160
a (DEG) Fig. 5. The effect on stress intensity factor due to two “stacked” flat inclusions (d/a = 1.5).
symmetrically to the main crack (see Fig. 3). The effective stress intensity factors are calculated for three cotigurations: (a) two “stacked” flat inclusions; (b) two inclined flat inclusions; (c) two symmetrically oriented flat inclusions. 3.1.1. Two “stacked”Jat inclusions. Consider two “stacked” micro flat inclusions which are parallel to the semi-infinite crack (Fig. 3; let fi, = j?*= 0). The effect on the stress intensity factor at the main crack tip is studied by varying the modulus ratios of the flat inclusions to matrix (G = G,/G, = GJG,,), where Go is the shear modulus of the matrix, and G, and G2 are the shear modulus of the first and second flat inclusions respectively, the location angle (a = a, = ar) and the non-dimensional&d distance (d/a = d,/a = d,/a; Fig. 4 for d/a = 1.1 and Fig. 5 for d/a = 1.5). The values of G that are taken are 0.00,0.02, 0.05,0.10,0.20 and 0.50. The case of G = 0.00 refers to a crack (the flat inclusion becomes a crack). From Figs 4 and 5, the following can be observed that. (1) Enhancement occurs in front of the main crack tip at about 120-150”, beyond which shielding occurs. (2) For any tixed angle a, the interaction effect increases for decreasing values of G. When G = 0, this effect is maximum. (3) When the modulus ratios of the flat inclusions to matrix G decrease, the minimum K& increases and the angles which correspond to Kdf = 1 decrease. This indicates that the interaction effect becomes more significant and the region which yields an enhancement effect diminishes as the value of G decreases. (4) When the flat inclusions move away from the main crack tip, the interaction effect will be less significant, and the angles which 2.0-
1.a-
1.6&4i 1.4-
1.0+ 0
30 a(DEG)
Fig. 6. The effect on stress intensity factor due to two inclined flat inclusions (d/a = 1.1).
Multi-flat inclusions
163
a(DEG) Fig. 7. The effect on stress intensity factor due to two inched
llat inclusions (d/u = 1.5).
correspond to maximum K& and & = 1 increase. It should be noted that the & lines in these two figures are not extended to the locations of a = 0” and a = 180” because the micro flat inclusions in these cases are considered to be located too close to each other or to the main crack, and the numerical results obtained may not be accurate and realistic in these regions. 3.1.2. Two inched @zt inclusions. The interaction effects between two inclined micro flat inclusions and a semi-infinite main crack under a remote mode I loading (see Fig. 3; let a, = a2 = /3, = f12) are studied. The effective stress intensity factor of the main crack tip is plotted vs the location angle (a = a, = az) for various values of G(G = G,/G, = G,/G,) and d/u (d/u = d, /a = dJa; Fig. 6 for d/a = 1.1 and Fig. 7 for d/a = 1.5). It can be shown from these two figures that enhancement always occurs when a < 150” and the shielding effect can be ignored in this configuration. For any fixed value of a (0“ < a < 150’), & increases with decreasing values of G. The maximum & increases and the angle at which the maximum lu, occurs decreases as the value of G decreases; when the ratio d/a increases, the maximum & decreases and the angle at which the maximum A& occurs decreases. From this, we can observe that the interaction effect will be less significant when the inclined flat inclusions move away from the main crack tip or the flat inclusions become harder (0 < G < 0.5). 3.1.3. Two symmetrically orientedPat inclusions. To investigate further the interaction effect between the main crack and two flat inclusions due to the orientations of these two flat inclusions, the problem which consists of a semi-infinite crack and two symmetrically oriented flat inclusions is studied. The effective stress intensity factor at the main crack tip is plotted vs the orientation angle (B = j?, = &) for various locations of these two flat inclusions (x,/u = x&r = 1.5, 1.0, 0.0,
1.2-
1.1-
0.9-
bd...-1.5 0.8
I
0
I
I
I
I
I
20
40
60
80
1w
I
120
I
140
I
180
I
180
B PEG) Fig. 8. The e!Tect on stress intensity factor due to two symmetrically oriented flat inclusions (G = 0.02).
CA0 WEN and K. Y. LAM
164
0.90 0
20
40
60
80
loo
120
140
160
180
PK'W
Fig. 9. The effect on stress intensity factor due to hvo symmetrically oriented flat inclusions (G = 0.05).
-1.0, -1.5; y,/a = 1.5, JJJa = - 1.5; where x, and y, are the central points of the first flat inclusion, x, and yZ are the central points of the second flat inclusion) and values of G(G = G,/Go = G,/G,; Fig. 8 for G = 0.02, Fig. 9 for G = 0.05). These two figures show that the shielding effect occurs even with the flat inclusions present in front of the main crack tip (for example, see the lines of x/a = 1.5), and the enhancement effect occurs even with the flat inclusions present in the flanks of the main crack (for example, see the lines of x/a = 0.0) due to the orientations of the flat inclusions. It should be noted that the angle which corresponds to maximum lu, decreases when the coordinate x increases (from - 1.5 to 1.5). It should also be noted that the interaction effect becomes more significant for decreasing values of G. The results in these two figures show that orientations of the flat inclusions indeed play an important role in the crack-inclusion
interaction.
3.2. An array of micro flat inclusions In this section, the crack-inclusion configuration consists of a semi-infinite crack and a micro flat inclusion array. The effective stress intensity factors are calculated for three configurations: (a) an array of collinear flat inclusions; (b) an array of “stacked” flat inclusions (odd); (c) an array of “stacked” flat inclusions (even). 3.2.1. An array of collinear fiat inclusions. In the case of a semi-infinite crack and an array of collinear flat inclusions, the effective stress intensity factor at the main crack tip is studied by varying the non-dimensional&d size d/a and the ratio G (G = Gi/Go, i = 1,2, . . . , where Go is the 20G.O.00 G-O.02
1.8-
G=O.O5
G.O.lO
1.6-
6.010
Kefl
G-Oil
1.41.2-
1.0 1.0
1.5
20
2.5
3.0
d/a Fig. 10. The effect on stress intensity factor due to an array of collinear tlat inclusions.
Multi-flat illclusi0M
165
II1.b t.5 IA-
bfl
1.3-
1.21.1-
l.O!
1.0
I
I
I
I
1
I
1.2 I.4 I.6 1.8 2.0 22
I
,
2.4 26
1
28
,
3.0
xla Fig. Il. The effect on stress intensity factor due to an array of “stacked” flat inclusions (odd, p = a).
shear modulus of the matrix and Gi is the shear modulus of the ith micro flat inclusion). The numerical results are presented in Fig. 10. It can be seen that for any fixed value of d/a, the stress intensity factor increases with decreasing values of G. When G = 0, lu, is maximum. It should be noted that K& is quite high when the array of collinear flat inclwions is near the main crack tip. If the ratio d/a increztsm, & will diminish for any certain value of G. In the case of d/a > 3, the interaction effect is not significant (with accuracy of up to 5%). 3.2.2. An array of “stacked”jZat incZwions (odd). Now consider a semi-infinite crack and an array of “stacked” flat inclusions (odd). This array of inclusions is symmetrical with respect to the x-axis, equally spaced with a pitchp and has a flat inclusion on the x-axis (see Fig. 11). The effective stress intensity factors are plotted vs x/a for various values of G (G = GJG,) and p (p = a in Fig. 11, p = 2a in Fig. 12 and p = 4a in Fig. 13). We compare results obtained here for the array with the data given for a micro single collinear crack near the main crack tip in these three figures (see one-crack lines). From Figs 1l-13, the following can be seen. (1) When the array of “stacked” flat inclusions is near the main crack tip, K& is quite high. The interaction effect will diminish as the array of the flat inclusions moves away from the main crack tip. (2) For any fixed value of x/a, & will increase when the value of G decreases. (3) Generally, with increasing value of p, lyeff will decrease except when both x/a and G are very small. (4) In the case of p/a = 1, krrf due to the array of cracks (G = 0) will be larger than lu, due to a single collinear crack when the value of x/a is greater than 1.17; in the cases when p/a = 2 and 4, the corresponding values of x/a are 1.5 and 2.2, respectively. This is expected as the effective stress intensity factor due to an array of “stacked” cracks will be smaller than K;B due to the corresponding single crack when the array 1.71.6G.O.00
1.5 IAbfl
1% 1.2i.l-
1.0
1.2
1.4
1.6
1.8
2:O 22
2.4
iS
28
3.0
x/a Fig. 12. The effect on stress intensity factor due to an array of “stacked” flat inclusions (odd, p = 20).
CA0 WEN and K. Y. LAM
166 1.71.6-
1.0 1.0
1.2
1.4
1.6
1.6
2.0
2.2
2.4
26
2.6
3.0
x/a Fig. 13. The effect on stress intensity factor due to an array of “stacked” flat inclusions (odd, p = 4a).
cracks are closed to the main crack tip. When the array of cracks moves away from the main crack tip, the interaction effect due to the array of cracks will be stronger than the effect due to the corresponding single cracks. If p is very big, J& due to the array of cracks will always be smaller than or equal to & due to a single crack. 3.2.3. An array of “stacked” j7at inclusions (men). In this section, the effect on the stress intensity factor at the main crack tip due to an array of “stacked” flat inclusions (even) is presented. This array of inclusions is also symmetric with respect to the x-axis, equally spaced with a pitch p, but does not have a flat inclusion on the x-axis (see Fig. 14). k;B of the main crack tip is plotted vs x/a for various values-of G (G = GJG,) and p (p = a in Fig. 14, p = 2a in Fig. 15 and p = 4a in Fig. 16). We compare the results obtained here for the array with the data given for two “stacked” cracks near the main crack tip (p is the distance between these two cracks) in these three figures (see two-crack lines). From Figs 14-16, the following can be seen. (1) Stress amplification changes to shielding at about x/a = 0.4-0.9 (the point which corresponds to lu, = 1 is called the neutral point). When the array of inclusions moves away from the main crack (from the neutral point), & at first increases for a short distance of x/a, and then decreases. When the value of x/a decreases (from the value of the neutral point), at first the shielding effect increases, but then this effect diminishes. The trends of the curves indicate that II: ,+l as x/a-&m. (2) For any lixed value of x/a, the interaction effect increases with decreasing value of G. (3) With increasing value of G, the minimum & increases and shifts to the left, and the value of x/a which corresponds to the neutral point decreases. (4) The interaction effect will be decreased with increasing value of p. l.61
P
1.4
I
1 ==g
:
1.2
:
1.0
Kefl
0.6 0.6 0.40.2 -2.0
I -1.5
I -1.0
I -0.5
I 0
I 0.5
I 1.0
I 1.5
I 2.0
x/a Fig. 14. The effect on stress intensity factor due to an array of “stacked” flat inclusions (even, p = a).
167
Multi-flat inclusions
Fig. 15. The effect on stress intensity factor due to an array of “stacked” flat inclusions (even, p = 2a).
It should be noted that the present results in some configurations for cracks agree well with the data of Rubinstein and Choi [9]. For example, in the case of two “stacked” cracks (even, p = a) approaching the semi-infinite main crack, according to our method the maximal shielding effect is 76% (see Fig. 14); the corresponding value in ref. [9] is 77%. The interaction between microcracks and a main semi-infinite crack has been investigated by Lam and Cao [lo, 111. 4. CONCLUSIONS An effective numerical technique based on the singular integral method with a mathematical model which represents inclusions (including cracks) by distributions of edge dislocations was applied to the interaction problem between a semi-infinite crack and two micro flat inclusions or an array of micro inclusions. The method and numerical formulation in this paper were carefully examined in some cases with known solutions in the literature and were found to be very accurate. The results illustrate that the effective interaction due to micro flat inclusions occurs only at very small distances from the flat inclusions to the main crack tip and the flat inclusions can yield either a stress enhancement or a stress shielding effect to the main crack tip depending upon the locations and orientations of the flat inclusions and also upon the modulus ratios of the flat inclusions to matrix.
.I
bG.O.50
ic---Go.O.20
G-0.10 0.0.05 G -0.02 G=O.OO
x/a Fig. 16. The effect on stress intensity factor due to an array of “stacked” flat inclusions (even, p = 40).
168
CA0 WEN and K. Y. LAM
RJ!TERENCFS (11 F. Erdo8an and C. D. Gupta. Stresses near a Bat inclusion in bonded dissimilar materials. Inr. J. Solids Srrucrures 8, 533-547 (1972). [2] A. B. Doyum and A. Gursoy, An arbitrarily oriented inclusion a half-space subjected to an inclined surface load. Engng Fracfare Me& 48, 609-625 (1991). [3] C. Atkinson, Some ribbon-like inclusion problems. Inr. J. Engng Sci. 11, 243-266 (1973). [4] K. Y. Lam and S. P. Phua, Multiple crack interaction and its cffcct on stress intensity factor. Theor. appf. Fracture Mech. (in press). [S] Liu XucHui and F. Erdo8an, The crack-inclusion interaction problem. Engng Fracture Mech. 23, 821-832 (1986). [6] Cao Wcn and K. Y. Lam, Effect of a flat inclusion on stress intensity factor of a sami-i&rite crack. 7’heor. appl. Fracture Mech. (in press). [7] K. Y. Lam and M. P. Clcary, Slippa8c and m-initiation of (hydraulic) fracture at frictional intcrfaccs. Int. J. namer. Anal. Meth. Geomech. 8, 589-604 (1984). [8] K. Y. Lam, B. Cottcre.11and S. P. Phua, Statistics of 8aw interaction in brittle materials. J. Am. Cerum. Sot. 74,352-357 (1991). [9] A. A. Rubinstcin and H. C. Choi, Microcrack interaction with transverse array of microcracks. Int. J. Fracture 36, 15-26 (1988). (lo] K. Y. Lam and Cao Wen, Interaction between microcracks and a main crack in a semi-infinite medium. Engng Fracture Mech. 44, 753-761 (1993). [ll] Cao Wcn and K. Y. Lam, Use of dislocation based method for microcracks interaction. Proc. Inf. Conf. on Computational Methodr in Engineering: Advances & Applications (Edited by A. A. 0. Tay and K. Y. Lam), pp. 1081-1086. World Scientific, Singapore. (Received 19 Februurv 1993)