New phenomena concerning the effect of imperfect bonding on radial matrix cracking in fiber composites

International Journal of Engineering Science 39 (2001) 2033±2050

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New phenomena concerning the e€ect of imperfect bonding on radial matrix cracking in ®ber composites Y. Liu, C.Q. Ru, P. Schiavone *, A. Mioduchowski Department of Mechanical Engineering, 4-9 Mechanical Engineering Bldg., University of Alberta, Edmonton, Alta., Canada T6G 2G8 Received 22 January 2001; accepted 9 February 2001 (Communicated by I. STAKGOLD)

Abstract In this paper we study the e€ects of imperfect bonding on stress intensity factors (SIFs) calculated at a radial matrix crack in a ®ber (inclusion) composite subjected to various cases of mechanical loading. We use analytic continuation to adapt and extend the existing series methods to obtain series representations of deformation and stress ®elds in both the inclusion and the surrounding matrix in the presence of the crack. The interaction between the crack and the inclusion is demonstrated numerically for di€erent elastic materials, geometries and varying degrees of bonding (represented by imperfect interface parameters) at the interface. Some qualitatively new phenomena are predicted for radial matrix cracking, speci®cally the in¯uence of imperfect bonding at the inclusion±matrix interface on the direction of crack growth. For example, in the case of an inclusion perfectly bonded to the surrounding matrix, the SIF at the nearby crack tip is greater than that at the distant crack tip only when the inclusion is more compliant than the matrix. In contrast, the e€ects of imperfect bonding at the inclusion±matrix interface allow for the SIF at the nearby crack tip to be greater than that at the distant crack tip even when the inclusion is sti€er than the matrix. In fact, for any given case when the inclusion is sti€er than the matrix, we show that there is a corresponding critical value of the imperfect interface parameter below which a radial matrix crack grows towards the interface leading eventually to complete debonding. In particular, this critical value of the imperfect interface parameter tends to a non-zero ®nite value when the sti€ness of the inclusion approaches in®nity. To our knowledge, these results provide, for the ®rst time, a clear quantitative description of the relationship between interface imperfections and the direction of propagation of radial matrix cracks. Ó 2001 Elsevier Science Ltd. All rights reserved.

*

Corresponding author. Tel.: +1-780-492-3638; fax: +1-780-492-2200. E-mail address: [email protected] (P. Schiavone).

0020-7225/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 1 ) 0 0 0 4 9 - 0

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1. Introduction It is well known that the strength of metals and toughness of ceramics, as well as other mechanical properties of ductile or brittle materials, can be greatly improved by ®brous reinforcements. It is also known that the mechanical behavior of ®ber-reinforced composites is signi®cantly a€ected by the nature of the bond between ®bers and the surrounding matrix material. In addition, the study of the interaction between cracks in the surrounding matrix and any nearby ®bers (inclusions) is also extremely signi®cant in attempting to understand and predict the strengthening and hardening mechanisms of ®ber-reinforced composites. For example, thermal mismatch between the ®bers (inclusions) and surrounding matrix may lead to high residual stresses in the vicinity of the inclusion±matrix interface. These stresses can be tensile in nature and lead to matrix cracking or interface separation. Consequently, the study of matrix cracking in ®ber-reinforced composites has been an area of intense investigation in the literature concerning mechanical failure of ®ber-reinforced composite materials [2,3,6,7,13,14,16,18,19,25]. Almost all existing (fracture mechanics) models of ®ber-reinforced composites are based on the assumption that both displacements and tractions are continuous across the ®ber (inclusion)± matrix interface. This assumption is commonly referred to as `the perfect bonding assumption' or the `perfect interface'. In many practical problems however, various kinds of interfacial damage arising from, for example, microcracks or regions of partial debonding, make the perfect bonding assumption inadequate when modeling the inclusion±matrix interface. In these cases it becomes necessary to model the interface as an imperfectly bonded interface incorporating the in¯uence of interface imperfections on the mechanical behavior of the composite. The same is true when an interphase layer (i.e., a non-uniform, thin interfacial zone between the inclusion and the matrix) is created either intentionally, by coating the individual ®bers (for example, to improve adhesion) or inadvertently, during the manufacturing process, as a result of chemical reactions between the contacting ®ber and surrounding matrix materials. Although small in thickness, interphase layers can signi®cantly a€ect local stress ®elds and the subsequent analysis of the mechanical failure of the composite, [1±3,5,12,16,17,21,22,27]. Consequently, it becomes important to try to understand and predict the in¯uence of an imperfectly bonded interface on the mechanical failure of ®berreinforced composites. One particular case is the study of the e€ect of interface imperfections on matrix cracking, an area of great practical and theoretical interest. For example, one of the main results concerning an inclusion/crack interaction, in the case of a perfect interface, is that the stress intensity factor (SIF) at the nearby tip of a radial matrix crack is greater (smaller) than the SIF at the distant crack tip, if and only if the inclusion is more compliant (sti€er) than the matrix [4,15,26]. This result is of major importance since it determines whether the radial crack grows towards or away from the inclusion. In this paper, we show that this conclusion is qualitatively invalid when imperfect bonding is present at the inclusion/matrix interface. In fact, we show that the imperfect bonding allows for the SIF at the nearby crack tip to be greater than the SIF at the distant crack tip even when the inclusion is much sti€er than the matrix. Further, for any given case when the inclusion is sti€er than the matrix, we show that there is a corresponding critical value of the imperfect interface parameter below which the radial matrix crack will grow towards the interface leading eventually to interface debonding. In the case of the interaction between a circular inclusion and a radial matrix crack under the perfect interface assumption, the existing most popular method (dislocation-density method)

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assumes that the crack can be modeled as a distribution of dislocations with unknown density functions [4,8,11,15]. In this case, the derivation of the governing integral equations for the unknown density functions is based on the conditions that mechanical tractions vanish along the crack faces. Except for an earlier study concerning a slipping interface [9], to our knowledge, the inclusion/crack interaction remains to be investigated in the case of imperfect bonding between the inclusion and the matrix. This can be attributed to the fact that the extension of the dislocation-density method to an imperfect interface meets two major diculties: (1) the fundamental solution for the interaction between an isolated dislocation and a circular inclusion with a generally imperfect interface is not yet available; (2) numerical solution of the resulting singular integral equation for imperfect interfaces is extremely challenging. The main objective of this paper is to develop a simple series method (which makes use of analytic continuation) to study the interaction between a radial matrix crack and a circular inclusion under the assumption of imperfect bonding at the inclusion±matrix interface. The results obtained clearly demonstrate that the series method developed here is simple and e€ective in describing the in¯uence of interface imperfections on the radial matrix crack in a range of di€erent cases. 2. Formulation Consider an in®nite elastic plane (matrix) containing a circular inclusion centered at the origin. The circular inclusion is of radius R with shear modulus l2 and Poisson's ratio m2 . The surrounding matrix is characterized by shear modulus l1 and Poisson's ratio m1 . As illustrated in Fig. 1(a), a ®nite radial matrix crack L ˆ ‰a; bŠ with length 2l is located outside the inclusion at a distance d from the inclusion±matrix interface. Let S2 and S1 be the regions occupied by the inclusion and the cracked matrix (containing the crack L), respectively, and C the circular interface

Fig. 1. A circular inclusion with a radial matrix crack.

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separating S2 and S1 . In what follows, the subscripts 1 and 2 are used to identify quantities in S1 and S2 , respectively. In the case of plane deformations, the corresponding displacement and stress ®elds are given in terms of two complex potentials u…z† and w…z† as follows [10]: 2l…u ‡ iv† ˆ ju…z† zu0 …z† w…z†; h i 0 0 rxx ‡ ryy ˆ 2 u …z† ‡ u …z† ; rxx

irxy ˆ u0 …z† ‡ u0 …z†

zu00 …z†

…1† w0 …z†:

Here, z ˆ x ‡ iy is the complex coordinate, j ˆ …3 4m† in the case of plane strain (assumed henceforth in this paper) and …3 m†=…1 ‡ m† in the case of plane stress. If polar coordinates are introduced so that z ˆ x ‡ iy ˆ r eih , (1) takes the form: h i 2l…ur ‡ iuh † ˆ e ih ju…z† zu0 …z† w…z† ; h i rrr ‡ rhh ˆ 2 u0 …z† ‡ u0 …z† ; h i rrr irrh ˆ u0 …z† ‡ u0 …z† e2ih zu00 …z† ‡ w0 …z† :

…2†

One of the most widely used models of an imperfect interface [1±3,5,12,16,17,21] is based on the assumption that tractions are continuous but displacements are discontinuous across the interface. More precisely, jumps in the displacement components are assumed to be proportional, in terms of `spring-factor-type' interface parameters, to their respective interface traction components. This model of an imperfect interface is often referred to as a `spring-layer imperfect interface'. Assume that the circular inclusion is imperfectly bonded to the matrix along C by such an interface. The boundary value problem for the displacements in both the inclusion and the matrix can then be formulated in terms of the following interface conditions [22]: krrr

irrh k ˆ 0;

rrr ˆ mkur k

mu0r ;

rrh ˆ nkuh k

nu0h ;

z 2 C;

…3†

while the crack-face conditions are given by u1 …z† ‡ zu01 …z† ‡ w1 …z† ˆ 0;

z 2 L:

…4†

Here, m and n are two imperfect interface parameters (which are non-negative and constant along the entire interface), k  k ˆ …†1 …†2 denotes the jump across C and u0 is the additional displacement induced by the uniform eigenstrains fe0x ; e0y ; e0xy g prescribed within the inclusion. It is seen from (3) that in®nite values of the interface parameters m and n imply vanishing of displacement jumps and therefore correspond to perfect interface conditions. On the other hand, zero values of the interface parameters imply vanishing of the corresponding interface tractions which corresponds to complete debonding. Any ®nite positive values of the interface parameters

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de®ne an imperfect interface. As noted in Section 1, such interface imperfections may be due to the presence of an interphase layer or perhaps interface bond deterioration caused by, for example, fatigue damage or environmental and chemical e€ects. The uniform remote loading is described by the conditions: u1 …z†  Az ‡ O…1†;

w1 …z†  Bz ‡ O…1†;

j z j! 1;

…5†

where A (a given real number) and B (a given complex number) are determined by the uniform remote stresses. In the case of a uniaxial load r0 normal to the crack (see Fig. 1), A and B are given by: Aˆ

r0 ; 4

B ˆ B ˆ 2A ˆ

r0 : 2

For purely shear loading r0xy , we have A ˆ 0 and B ˆ ir0xy . Finally, for uniaxial loading r0 parallel to the crack (in the x-direction), we have A ˆ r0 =4 and B ˆ 2A ˆ r0 =2. The present single-inclusion model is suitable for modeling composites with low ®ber volume fraction [23]. Under this condition, the interaction among neighboring ®bers and its in¯uence on stress ®elds near the isolated ®ber are small and therefore negligible. A convenient method used to analyze the case of circular boundaries is the series method [10]. However, in the case of the present problem, the domain S1 contains a crack L so that u1 …z† and w1 …z† are not analytic outside the circle C. As a result, u1 …z† and w1 …z† cannot be expanded into standard Laurent series in S1 . To overcome this diculty, a method based on analytic continuation [10,26] is employed below to express u1 …z† and w1 …z† in terms of two new functions which are analytic outside the circle C where they can thus be expanded into standard Laurent series. To this end, denote by D the domain outside the circle C minus the matrix crack L (namely, D consists of the complex plane with the circular hole S2 , but without the matrix crack L). Clearly, u1 …z† and w1 …z† are analytic in S1 , but not in D. Thus, the crack-face conditions (4) along the upper and lower faces of L given by: u1 …z†‡ ‡ zu01 …z†‡ ‡ w1 …z†‡ ˆ 0;

z 2 L‡ ;

u1 …z† ‡ zu01 …z† ‡ w1 …z† ˆ 0;

z2L ;

can be written into the equivalent form: h i ‡ u1 …z† ‡ zu01 …z† ‡ w1 …z† ˆ 0; h i‡ u1 …z† ‡ zu01 …z† ‡ w1 …z† ˆ 0;

z 2 L;

…6†

z 2 L:

It thus follows that u1 …z†‡

h

zu01 …z† ‡ w1 …z†

i‡

ˆ u1 …z†

h

i zu01 …z† ‡ w1 …z† ;

z 2 L:

…7†

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Next, we de®ne an analytic function U…z† in S1 by h i zu01 …z† ‡ w1 …z† : U…z† ˆ u1 …z†

…8†

(Note that analytic continuation implies that both u1 …z† and w1 …z† are analytic in S1 .) The above continuity condition (7) implies that U…z† is continuous across L. Consequently, U…z† is analytic in D and can then be expanded into a Laurent series in D as U…z† ˆ

Bz ‡

1 X

ak z k ;

z 2 D;

…9†

kˆ1

where ak …k ˆ 1; 2 . . .† are unknown complex coecients. Furthermore, the remaining crack-face condition (6) can be rewritten as h i‡ h i u1 …z† ‡ zu01 …z† ‡ w1 …z† ‡ u1 …z† ‡ zu01 …z† ‡ w1 …z† ˆ 0; z 2 L:

…10†

We introduce another analytic function W…z† in S1 as follows: p h i …z a†…z b† W…z† ˆ u1 …z† ‡ zu01 …z† ‡ w1 …z† : z Then, since [10] q …z a†…z b†‡ ˆ

p …z a†…z b† ;

…11†

z 2 L ˆ ‰a; bŠ;

condition (11) can be written in terms of W…z† as W…z†‡

W…z† ˆ 0;

z 2 L:

…12†

It follows that W…z† is continuous across L and analytic in D and can therefore be expanded into a Laurent series in D as follows: W…z† ˆ …2A ‡ B†z ‡

1 X

bk z k ;

…13†

kˆ1

where bk …k ˆ 1; 2 . . .† are unknown complex coecients. Now, u1 …z† and w1 …z† can be expressed in terms of U…z† and W…z† as: w1 …z† ˆ u1 …z†

zu01 …z†

U…z†;

zW…z† U…z† u1 …z† ˆ p ‡ : 2 2 …z a†…z b†

…14† …15†

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On the other hand, u2 …z† and w2 …z† are analytic within the circular inclusion and can then be expanded into their respective Maclaurin's series in S2 : u2 …z† ˆ

1 X

dk zk ;

w2 …z† ˆ

kˆ0

1 X

ek zk ;

kˆ0

where dk and ek are complex coecients. Hence, the problem reduces to determining the four sets of unknown coecients ak ; bk ; dk ; and ek , such that the two interface conditions (3) can be satis®ed on C. The traction continuity conditions (3) give:  2  2 z2 0 R z2 0 R 0 00 0 00 0 0 w1 …z† ‡ u1 w2 …z† ‡ u2 u1 …z† zu1 …z† ˆ u2 …z† zu2 …z† : …16† 2 2 R z R z On the other hand, the remaining two displacement interface conditions (3) can be written in the complex form: rrr

irrh ˆ

m

n 2

kur ‡ iuh k ‡

m‡n kur 2

iuh k

‰mu0r

inu0h Š:

On using (2), the above displacement discontinuity conditions can be rewritten as follows:     2  2   2  2  m n R R R R R R w1 w2 j1 u1 …z† zu01 j2 u2 …z† zu02 2 2l1 z 2l2 z z z z z      2   2 m‡n z R z R 0 0 j1 u1 j2 u2 zu1 …z† w1 …z† zu2 …z† w2 …z† ‡ 2 2l1 R 2l2 R z z m n m ‡ n mRe1 R3 …e2 ‡ ie3 † …e2 ie3 †z2 2 2R 2z  2 z2 0 R 0 ˆ u02 …z† zu002 …z† w …z† ‡ u ; …17† 2 R2 2 z where e1 ˆ …e0x ‡ e0y †=2; e2 ˆ …e0x e0y †=2; and e3 ˆ e0xy (see, [22]). Hence, all unknown coecients ak ; bk ; dk ; and ek …k ˆ 1; 2 . . .† can be determined by the interface conditions (16) and (17). 3. Numerical results Here we compute (numerically) the unknown complex coecients ak ; bk ; dk ; and ek …k ˆ 1; 2 . . .† for a range of imperfect interface parameters m and n, modulus ratios l1 =l2 and crack distances d=R. The following three cases of remote mechanical loading are considered in the absence of any eigenstrain inside the inclusion: 1. uniaxial loading normal to the crack (of signi®cant practical interest); 2. purely shear loading and 3. uniaxial loading parallel to the crack.

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Noting that the interface model (3) can be used to represent an adhesive layer, we remark that the two interface parameters m and n depend on the thickness and modulus of the layer. If only a single adhesive material is assumed to make up the layer, the ratio of m=n is usually taken to be a material constant, independent of the thickness of the adhesive layer. For example, the ratio m=n can take the (approximate) value 1 for a layer modeled by a series of distributed springs [2,3], or the value 3 for an elastic interphase layer [17]. Consequently, in this paper, we analyze three di€erent cases: (1) m ˆ n; (2) m ˆ 3n; and (3) the case of a sliding interface (see [20]), which is de®ned by the values m ˆ 1 and n ˆ 0. Here, Poisson's ratios of both the inclusion and the matrix are held constant …m1 ˆ m2 ˆ 1=3†. For convenience, the imperfect interface parameters m and n will be characterized by the new parameters M and N , de®ned by M ˆ …m n†=2 and N ˆ …m ‡ n†=2. In addition, we de®ne a dimensionless interface parameter N 0 by N 0 ˆ N =…l1 =R†. To derive the formulas for the SIF, it is noted from (1) that the stresses in the matrix are given by: h i ryy ˆ Re u01 …z† ‡ u01 …z† ‡ zu001 …z† ‡ w001 …z† ; h i rxx ˆ Re u01 …z† ‡ u01 …z† zu001 …z† w001 …z† and h rxy ˆ Im u01 …z† ‡ u01 …z†

zu001 …z†

i w001 …z† :

On using w1 …z† ˆ u1 …z†

zu01 …z†

U…z†;

the above expressions can be re-written as: h i ryy ˆ Re u01 …z† ‡ u01 …z† ‡ …z z†u001 …z† U0 …z† ; h i rxx ˆ Re u01 …z† u01 …z† 2u01 …z† ‡ …z z†u001 …z† ‡ U0 …z† ; h i rxy ˆ Im u01 …z† u01 …z† ‡ 2u01 …z† ‡ …z z†u001 …z† ‡ U0 …z† ; where u1 …z† is given by (15). Since the leading-order singularity is the inverse square-root singularity, it follows that W…a† ˆ W…b† ˆ 0: As expected, these conditions are veri®ed by our numerical results (see later). Consider, for example, the case of a uniaxial load r0 normal to the crack. In this case, the stresses in the neighborhood of the crack tip z ˆ a are given by:

Y. Liu et al. / International Journal of Engineering Science 39 (2001) 2033±2050

rxx ˆ ryy ˆ rxy ˆ where z

2041

#

"

p  1 l 0a 3 h1 1 h1 1 X p r kbk‡1 a …k‡1† ‡ O…r10 †; sin ‡ sin 5 1 2ar0 kˆ1 2 4 2 2r1 l 4 # p  " 1 h1 1 h1 1 X l 0a 5 …k‡1† p r sin sin 5 ‡ O…r10 †; kbk‡1 a 1 2ar0 kˆ1 2 4 2 2r1 l 4 # p  " 1 h1 1 h1 1 X l 0a 1 …k‡1† p r cos cos 5 ‡ O…r10 †; kbk‡1 a 1 0 l 4 4 2ar 2 2 2r1 kˆ1 a ˆ r1 eih1 …0 6 h1 6 2p†. Similarly, the stress ®eld around the crack tip z ˆ b is given by:

p  h2 1 l 0b 3 cos ‡ rxx ˆ p r 2 4 2r2 l 4 p  b 5 h2 1 l cos ryy ˆ p r0 2 4 2r2 l 4 p  l 0b 1 h2   p sin r rxy ˆ l 4 2 2r2

h2 cos 5 2 h2 cos 5 2

" 1 " 1

1 h2 sin 5 4 2

" 1

#

1 1 X kbk‡1 a 2br0 kˆ1

…k‡1†

1 1 X kbk‡1 a 2br0 kˆ1

…k‡1†

‡ O…r20 †; #

1 1 X kbk‡1 a 2br0 kˆ1

‡ O…r20 †; #

…k‡1†

‡ O…r20 †;

where z b ˆ r2 eih2 … p 6 h1 6 p†. In the above equations, the ®rst term of each equation represents the leading-order singular stress near the crack tips, inversely proportional to the squareroot of the radial distance from the crack tip. The square-bracketed terms represent the in¯uence of the inclusion and the imperfect interface on the SIF. Similar expressions can be derived for other cases of remote loading. 3.1. Uniaxial loading perpendicular to the crack …A ˆ r0 =4, B ˆ 2A ˆ r0 =2† Figs. 2 and 3 show, for the case of m ˆ n and the ®xed crack length 2l …l ˆ R†, the SIF ratio at the nearby and crack tips. This ratio is de®ned by KI =KI0 (where KI is the actual mode-I SIF pdistant  0 0 and KI ˆ r pl is the mode-I SIF for the same crack in a homogeneous matrix material without the inclusion). Clearly, the inclusion has a signi®cant e€ect on KI …a† but not on KI …b†. On the other hand, the e€ect of the inclusion decreases with d=R. For the case of an inclusion sti€er than the surrounding matrix (Fig. 2), KI …a† increases as the imperfect interface parameter N 0 decreases. It is clear that the presence of the inclusion with imperfect interface can either increase the SIF at the 0:01, nearby crack tip (for example, when N 0 ˆ p  0.1, or 1), or reduce it signi®cantly (for example, when N 0 ˆ 10 or 100), as compared to r0 pl. For the case of an inclusion softer than the surrounding matrix (Fig. 3), the SIF at the nearby crack tip increases sharply when the crack approaches the inclusion. On the other hand, KI …a† increases when the imperfect interface parameter N 0 decreases and, most importantly, the SIF at the nearby crack tip is always larger than that at the distant crack tip. In particular, it is seen from Fig. 2 that, for a sti€er inclusion, the SIF at the nearby crack tip is smaller than that at the distant crack tip for a larger imperfect interface parameter but larger than

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Fig. 2. The SIFs of crack tips via interface parameter N 0 (uniaxial load perpendicular to crack, m ˆ n; u1 =u2 ˆ 0:5, I=R ˆ 1).

Fig. 3. The SIFs of crack tips via interface parameter N 0 (uniaxial load perpendicular to crack, m ˆ n; u1 =u2 ˆ 2, I=R ˆ 1).

the SIF at the distant crack tip for smaller values of the imperfect interface parameter. Hence, for any given inclusion composed of material sti€er than the surrounding matrix, there is a corresponding critical value of the imperfect interface parameter at which KI …a† ˆ KI …b†. Clearly, this critical value of the imperfect interface parameter depends on the ratio of shear moduli. For a ®xed value of d=R, Figs. 4 and 5 illustrate the shear moduli ratios of inclusion to matrix determined by the condition KI …a† ˆ KI …b† for di€erent imperfect interface parameters. In Fig. 4, four curves are drawn for di€erent d=R ratios. These curves correspond to the ®xed crack length l=R ˆ 1. Fig. 5 deals with the case of l=R ˆ 0:1. These curves show that, for the ®xed interface parameter N 0 , the ratio of l2 =l1 , determined by the condition KI …a† ˆ KI …b†, increases as d=R becomes larger. On the other hand, for the ®xed ratio l2 =l1 , the interface parameter N 0 , determined by the condition KI …a† ˆ KI …b†, becomes larger when d=R increases. In both Figs. 4 and 5, each curve de®nes a minimum critical value of N 0 , referred to as N  , below which the condition KI …a† ˆ KI …b† cannot be satis®ed by any ratio l2 =l1 (more precisely, below which KI …a† is always larger than KI …b† for any ratio l2 =l1 ). For example, the minimum critical values in Fig. 4 are N  ˆ 0:05, 0.1, 5.6, and 8.7 when d=R ˆ 0:01, 0.05, 0.5, and 1, respectively. In Fig. 5, the mini-

Y. Liu et al. / International Journal of Engineering Science 39 (2001) 2033±2050

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Fig. 4. The shear modulus ratio of inclusion to matrix determined by the condition KI …a† ˆ KI …b† for di€erent interface parameters …m ˆ n; I=R ˆ 1†.

Fig. 5. The shear modulus ratio of inclusion to matrix determined by the condition KI …a† ˆ KI …b† for di€erent interface parameters …m ˆ n; I=R ˆ 0:1†.

mum critical values are N  ˆ 0:01, 0.08, 5.2, and 6.9 when d=R ˆ 0:01, 0.05, 0.5, and 1, respectively. In any case, if the imperfect interface parameter is smaller than the corresponding critical value N  , we have KI …a† > KI …b† and the radial matrix crack will grow towards the interface leading to interface debonding for any ratio l2 =l1 . It is therefore possible to predict and control the direction of matrix cracking and therefore interfacial debonding by designing the inclusion± matrix interface accordingly. In the case of m ˆ 3n, our calculations indicate that there is no signi®cant di€erence from the case m ˆ n discussed above. This is probably due to the fact that the shear imperfect interface parameter n has a minor e€ect on stress ®elds subjected uniaxial loading. It should be noted that a consequence of the imperfect interface model employed in this paper is the prediction of possible overlapping of the two materials at the interface (a negative normal displacement jump across the interface; see [2,17]). As explained by Hashin [17], under the assumptions of this interface model, the matrix or the inclusion can be moved towards the interface by a small distance lesser than or equal to the initial interphase thickness without any physical overlapping of materials. With this interpretation, a small negative normal displacement jump is permissible so that the imperfect interface model used here does not, in fact, lead to any physical contradiction, at least for cases of relatively lower-order remote loading.

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An interesting special case of the imperfect interface is the so-called `sliding interface' characterized by m ˆ 1; n ˆ 0 [5,20]. In Fig. 6, the sliding interface is considered and the mode-I SIF ratio (de®ned as for Fig. 2) is drawn as a function of the distance d for both a softer inclusion …l1 =l2 ˆ 2† and a sti€er inclusion …l1 =l2 ˆ 0:5†. For the softer inclusion, the SIF at the nearby crack tip increases when the distance d becomes smaller. Fig. 6 shows that the SIF at the nearby crack tip is always larger than that at the distant crack tip, which implies that the crack grows towards the inclusion. On the other hand, for the sti€er inclusion, the SIF at the nearby crack tip increases monotonically if the distance d increases and the SIF at the nearby crack tip is always smaller than that at the distant crack tip. This means that the sti€er inclusion with sliding interface always resists crack propagation towards the inclusion. Clearly, the SIF at the distant crack tip is not sensitive to the distance d=R for both softer and sti€er inclusions. 3.2. Purely shear loading (A ˆ 0, B ˆ ir0xy ) First, we study the SIF ratio, which is de®ned by KII =KII0 (where KII is the actual modepmode-II  0 0 II SIF and KII ˆ rxy pl is the mode-II SIF for the same crack in a homogeneous matrix material), at the nearby crack tip (Fig. 7) for the case of a sliding interface (m ˆ 1; n ˆ 0) under purely shear loading. Compared with Fig. 6 which corresponds to the sliding interface under uniaxial

Fig. 6. The mode-I SIFs via the distance d=R for di€erent modulus ratio (sliding interface, uniaxial load perpendicular to the crack, I=R ˆ 1).

Fig. 7. The mode-II nearby SIFs via the distance d=R for di€erent modulus ratio (sliding interface, pure shear load, I=R ˆ 1).

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loading, Fig. 7 shows that the SIF at the nearby crack tip increases when the distance d decreases for both softer and sti€er inclusions. As shown in Fig. 7, inclusions sliding interfaces always pwith  have increasing SIFs at the nearby crack tip as compared to r0xy pl. Apparently, the sliding interface has a signi®cant e€ect on the SIF at the nearby crack tip. Figs. 8 and 9 show the in¯uence of the interface parameters …m ˆ 3n† on the behavior of modeII SIF ratios under pure shear loading, whereas Figs. 10 and 11 show the in¯uence on mode-I SIF ratios under the same loading conditions. In Figs. 8 and 9, the e€ects of the imperfect interface parameter N 0 on KII at the nearby crack tip are investigated for softer and sti€er inclusions. For the softer inclusion, as shown in Fig. 8, KII increases monotonically for the ®xed imperfect interface parameter N 0 when d=R becomes smaller, while KII decreases with increasing N 0 for the ®xed distance. On the other hand, for the sti€er inclusion, KII …a† decreases when d=R becomes larger, while KII …a† increases as the imperfect interface parameter N 0 decreases. When N 0 varies from 0.01 to 100, the inclusion with an imperfect interface (m ˆ 3n) can lead to either pan increase in the SIF at the nearby crack tip or a signi®cant reduction when compared to r0xy pl. Hence, in both cases, interface imperfections have a signi®cant e€ect on KII . In particular, Figs. 10 and 11 illustrate the e€ects of the imperfect interface parameters on KI at the nearby crack tip under pure shear loading. Owing to the presence of the inclusion and the imperfect interface, KI is non-zero even under pure shear loading and therefore cannot be ignored. It is clear that the curves in Figs. 10 and 11, for either softer or sti€er inclusions, are quite similar.

Fig. 8. Mode-II nearby crack tip SIF versus interface parameter (pure shear loading, m ˆ 3n; u1 =u2 ˆ 2; I=R ˆ 1).

Fig. 9. Mode-II nearby crack tip SIF versus interface parameter (pure shear loading, m ˆ 3n; u1 =u2 ˆ 0:5; I=R ˆ 1).

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Fig. 10. Mode-I nearby crack tip SIF versus interface parameter (pure shear loading, m ˆ 3n; u1 =u2 ˆ 2; I=R ˆ 1).

Fig. 11. Mode-I nearby crack tip SIF versus interface parameter (pure shear loading, m ˆ 3n; u1 =u2 ˆ 0:5; I=R ˆ 1).

For example, when the imperfect interface parameter N 0 is ®xed, KI increases monotonically when d=R becomes smaller while the SIF decreases with increasing N 0 when d=R is ®xed. 3.3. Uniaxial loading parallel to the crack (A ˆ r0 =4, B ˆ

2A ˆ

r0 =2)

Finally, we consider the case of uniaxial loading r0 parallel to the crack. Figs. 12±15 illustrate mode-I SIF ratios (de®ned as in Section 3.1) at the nearby crack tip for the case of m ˆ 3n. It is seen from Figs. 12±15 that imperfect bonding could make the SIF at the nearby tip signi®cantly large as compared to that corresponding to the perfect interface. Here, in order to have a clear idea of the e€ect of the interface imperfections on the SIF, four di€erent modulus ratios, l1 =l2 ˆ 10, 2, 0.5, and 0.1 are considered. It is clear from our results that the imperfect interface parameter has a signi®cant e€ect on the SIF. For the ®xed length d=R, when the imperfect interface parameter N 0 varies from 1000 (approximately perfect bonding) to 0.01 (approximately

Fig. 12. Nearby crack tip SIF versus interface parameter (uniaxial load parallel to crack, m ˆ 3n; u1 =u2 ˆ 0:5; I=R ˆ 1).

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Fig. 13. Nearby crack tip SIF versus interface parameter (uniaxial load parallel to crack, m ˆ 3n; u1 =u2 ˆ 2; I=R ˆ 1).

Fig. 14. Nearby crack tip SIF versus interface parameter (uniaxial load parallel to crack, m ˆ 3n; u1 =u2 ˆ 0:1; I=R ˆ 1).

Fig. 15. Nearby crack tip SIF versus interface parameter (uniaxial load parallel to crack, m ˆ 3n; u1 =u2 ˆ 10; I=R ˆ 1).

complete debonding), there is a signi®cant increase in the SIF for both softer and sti€er inclusions. Considering the in¯uence of the distance d, if we ®x the interface parameter N 0 , the SIF will decrease as d=R increases. 3.4. Special limiting cases As mentioned before, the present imperfect interface model describes perfect bonding when m ˆ n ˆ 1 and complete debonding when m ˆ n ˆ 0. Hence, for the sake of comparison, the SIFs calculated by the present method for very large or very small interface parameters are

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Table 1 Comparison between the calculated nearby SIF with very large interface parameter and the SIF corresponding to perfect bonding l2 =l1 ˆ 0:5

N 0 ˆ 100 Perfect bond

l2 =l1 ˆ 2

d=R ˆ 0:1

0.2

0.5

1

d=R ˆ 0:1

0.2

0.5

1

1.2876 1.30

1.2269 1.23

1.1154 1.12

1.0365 1.04

0.7546 0.73

0.8483 0.83

0.9435 0.94

0.9708 0.97

Table 2 Comparison between the calculated distant SIF with very large interface parameter and the SIF corresponding to perfect bonding l2 =l1 ˆ 0:5

N 0 ˆ 100 Perfect bond

l2 =l1 ˆ 2

d=R ˆ 0:1

0.2

1

d=R ˆ 0:1

0.2

1

1.0332 1.035

1.0267 1.028

1.0051 1.005

0.9402 0.93

0.9467 0.94

0.9801 0.98

Table 3 Comparison between the calculated nearby and distant SIFs and the SIF corresponding to traction-free hole l2 =l1 ˆ 0:5

N 0 ˆ 0:01 Traction-free hole

Nearby SIF

Distant SIF

d=R ˆ 0:1

0.2

0.5

1

d=R ˆ 0:1

0.2

1

1.8624 1.893

1.724 1.74

1.3434 1.35

1.1434 1.145

1.2069 1.22

1.1667 1.18

1.0551 1.055

compared with known results for the perfect interface and traction-free holes, respectively. For example, the calculated SIF at the nearby and distant crack tips under uniaxial loading normal to the crack …m ˆ n† for N 0 ˆ 100 are compared with SIFs corresponding to perfect bonding (see [15,26]). The results are shown in Tables 1 and 2, respectively. Similarly, in Table 3, we compare the SIFs for nearby and distant crack tips with very small interface parameter (N 0 ˆ 0:01) with the SIFs for the traction-free holes (see [24]). All comparisons show that the perfect interface can be described by very large imperfect interface parameters (for example, N 0 > 100). Similarly, traction-free debonding can be described by very small interface parameters (for example, N 0 < 0:01). In particular, these results con®rm the validity of the present series method. 4. Conclusions In this paper, the problem of the interaction between a circular inclusion with an imperfect interface and a radial matrix crack is studied using a spring-type interface model. Using analytic

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continuation, a novel series method is developed to ®nd the SIFs at the tips of the radial matrix crack. Numerical results show that the interface imperfection has a signi®cant e€ect on the SIFs, especially when the crack is close to the inclusion. For inclusions softer than the matrix, the radial matrix crack always propagates towards the inclusion regardless of the values of the imperfect interface parameters. However, for inclusions sti€er than matrix, the radial matrix crack can propagate either towards or away from the inclusion, depending on the values of the imperfect interface parameters. It is therefore crucial to quantify the e€ect of interface imperfection on the interaction between a circular inclusion and a radial matrix crack. In particular, for the special case of a perfectly bonded or entirely debonded interface, the present results are consistent with the known results in the literature. Remarkably, in contrast to the case of a perfect interface for which the SIF at the nearby crack tip is greater than the SIF at the distant crack tip only when the inclusion is more compliant than the matrix, the imperfect bonding condition allows for the possibility of a SIF at the nearby crack tip greater than that at the distant crack tip even when the inclusion is sti€er than the matrix. In fact, for any given inclusion sti€er than the surrounding matrix, there is a corresponding critical value of the imperfect interface parameter below which a radial matrix crack grows towards the interface leading to interface debonding. In particular, for given crack length and distance from the interface, there is a minimum critical value of the imperfect interface parameter (de®ned by N  in Section 3.1) below which (that is, when N 0 < N  ) the SIF in the nearby crack tip is always greater than that at the distant crack tip for any shear modulus ratio of the inclusion and the matrix. These results provide a clear quantitative relation between interface imperfection and the direction of growth of radial matrix cracks. All of the aforementioned results show that the present series method provides a simple and e€ective way of investigating an inclusion±crack interaction involving an imperfect interface.

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