Interfacial partial debonding and its in uence on the elasticity of a two-phase composite

Mechanics of Materials 32 (2000) 695±709 www.elsevier.com/locate/mechmat Interfacial partial debonding and its in uence on the elasticity of a two-phase composite S.F. Zheng, M. Denda, G.J. Weng * Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA Received 8 July 1999; received in revised form 4 May 2000 Abstract A dual e ective-medium and nite-element study is carried out to examine the in uence of interfacial partial debonding on the elastic sti ness of a two-phase composite containing aligned elliptic bers. In the e ective-medium approach double debonding on the top and bottom of the elliptic interface is considered, but in the nite-element approach both double and single debonding are examined. The e ective-medium approach makes use of the concept of a ctitious ber whose load-carrying capacity is taken to be lost in the debonding direction but remains intact in the transverse direction. The nite-element analysis allows one to examine further the in uence of debonding angle on the sti ness of the composite and it also provides the needed magnitude for this angle at which the e ective-medium approach applies. It is found that the angle has to be su ciently wide, but not so wide as to lead to a potential complete debonding. Such an angle also increases when the bers become more ribbon-like, but it decreases with increasing volume concentration. The basic assumption that the ctitious ber has zero, or very low, load-carrying capacity along the debonding direction is also veri ed by the nite-element results. For both double and single debonding the Young's modulus of the composite along the debonding direction can be signi cantly reduced as the debonding angle increases. The local stress distributions inside the ber and the matrix are also illustrated for both types of debonding. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Interfacial partial debonding; Elastic sti ness; Two-phase composites 1. Introduction In a recent paper Zhao and Weng (1997) developed a simple concept of ctitious inclusions to mimic the load-transfer capacity of two kinds of partially debonded inclusions in a composite. The rst kind involves partial debonding on the top and bottom of the oblate inclusions while the * Corresponding author. Tel.: +1-732-445-2223; fax: +1-732- 445-5313. E-mail address: weng@jove.rutgers.edu (G.J. Weng). second kind involves partial debonding on the lateral surface of the prolate ones. A ctitious inclusion was used to represent the smeared e ect of the original inclusion and the surrounding interface cracks. When debonding is su ciently wide, the load-carrying capacity of the ctitious inclusion was taken to be zero in the debonding direction, but its ability to carry the load in the bonded direction was taken to remain present. As such, the isotropic inclusion and the interfacial cracks together were replaced by the ctitious inclusion which now becomes perfectly bonded to the matrix but with transversely isotropic moduli. Following 0167-6636/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7-6 6 3 6 ( 0 0 ) 0 0 041-7

696 S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 this concept, the e ective moduli of the transversely isotropic composite with such partially debonded inclusions were estimated using a well established micromechanical model for a twophase composite containing aligned, transversely isotropic spheroidal inclusions. The advantage of such an approach is that it delivers simple ± albeit approximate ± explicit results for the e ective moduli of an otherwise rather complicated composite. What remains unclear is how wide the partial debonding has to be in order for the concept to apply. Apparently, the loadcarrying capacity of the ctitious inclusion can not be zero in the debonding direction if the debonding occurs only over a small portion of the interface, and that its load-transfer ability in the perpendicular direction can not be retained if the debonding extends over the entire interface. It is thus highly desirable to have this question answered. Moreover, the extent of the partial debonding for this concept to hold is also likely to depend on the inclusion shape as a platelet inclusion will need a wider debonding than a spherical one, and the overall sti ness of the debonded composite will also depend on the extent of debonding. The e ective-medium method as usual does not deliver this sort of information and it also gives no local stress and strain elds which can serve to detect the subsequent damage process. These questions can perhaps be best answered by a companion nite-element analysis. It is with this perspective that the present study was undertaken. Here we are concerned with a two-dimensional problem involving partially debonded, aligned elliptic bers. The unit cells are shown in Figs. 1(a) and (b) for double debonding on the top and bottom, and for single debonding on the top, respectively. Fibers are taken to be aligned along x 1, and the debonded direction along x 3. The cross-sectional aspect ratio or the thickness to the width ratio, t=w, is represented by a, so that by varying a the e ect of the ber shape can also be taken into account. In both cases the composite as a whole is orthotropic. The e ective-medium method based on the concept of ctitious bers can only be applied to con guration (a), for in (b) it is not realistic to expect the smeared region to lose its load-carrying capacity. Moreover, for the e ective-medium method to hold the cross-sectional aspect ratio should be limited to the range a < 1, as for a > 1 ± especially when it is very large, the shear stress (shear lag) may still transfer the load from the matrix to the ber. For the - nite-element method both con gurations will be examined so that a comparison between these two types of partial debonding can be made. In both gures the extent of partial debonding is speci ed by the debonding angle /, with / ˆ 0 corresponding to perfect bonding and / ˆ 90 to complete debonding. It is intuitively intriguing to see how the sti ness of the composite is weakened as the debonding angle increases. The outer cell of the unit cell is chosen to have an identical shape as the inclusion for the reason that the e ective-medium theory to be used corresponds to such a microgeometry. A rectangular outer cell would lead to a composite with a periodic structure and this is not the case for the model to be adopted. The radii of the inner and outer cells are chosen in accordance with the volume concentration of the bers, and when the aspect ratio of inclusions becomes one, the unit cell re- Fig. 1. Unit cells for: (a) double interfacial debonding; (b) single interfacial debonding.

S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 697 turns to the composite cylinder assemblage. The unit cell implies that every ber has an identical damage. This is a highly idealized situation and the results must be viewed in this light. We also note in passing that the elastic eld of an interfacial arc crack has been derived by England (1966) and Toya (1974) for a circular ber (a ˆ 1) in an in nitely extended matrix, and the result has been applied by Ju (1991) to develop a damage model for a ber-reinforced composite. For an elliptic interfacial crack in a nite matrix, however, the problem remains unsolved. In what follows the concept of zero load-transfer capacity in the debonding direction will be used to establish the orthotropic property of the ctitious ber. Then based on an e ective-medium method the nine e ective moduli of the partially debonded composite will be calculated as a function of the cross-sectional aspect ratio a and volume concentration c 1 of the elliptic bers. The nite-element study will subsequently be presented. Comparison between the two for double debonding, and the additional results by the nite element for single debonding, will be given at the end. 2. Properties of the ctitious bers and the e ectivemedium method In the two-phase composite the elliptic bers will be referred to as phase 1 and the matrix as phase 0. For simplicity both phases will be taken to be isotropic, with the bulk and shear moduli denoted by the Greek j r and l r, respectively, for phase r. Its plane-strain bulk modulus will be denoted by the Roman k r, and volume concentration by c r. The orthogonal symmetry of the debonding con guration as depicted in Fig. 1(a) necessitates that the ctitious ber be orthotropic. For an orthotropic material its elastic constitutive equations carry the form r 11 ˆ c 11 11 c 12 22 c 13 ; r 22 ˆ c 21 11 c 22 22 c 23 ; r ˆ c 31 11 c 32 22 c ; r 32 ˆ 2c 44 32 ; r 31 ˆ 2c 55 31 ; r 12 ˆ 2c 66 12 ; 1 in terms of stress r ij, strain ij, and elastic constants c ij ( ˆ c ji ). We shall denote the nine elastic constants of the ctitious ber by c 1 ij. When interfacial debonding or the debonding angle / is su ciently wide, we may assume that the load-carrying capacity of the ctitious ber be zero in the x 3 -direction so that c 1 ˆ c1 32 ˆ c1 31 ˆ 0: 2 No shear stress on the x 3 -surface of the ctitious ber is taken to endure either, with c 1 44 ˆ c1 55 ˆ 0: 3 The r 12 -loading is on the bonded edge of the elliptic cylinder and is taken to be una ected by the partial debonding c 1 66 ˆ l 1: 4 There remain c 1 11, c1 22, and c1 12 to be determined, all in the x 1 ±x 2 plane. These three constants could still be somewhat weakened by the partial loss of load-transfer ability along x 1 and x 2. We expect their values to be less than those of the original ber, which carries c 11 ˆ c 22 ˆ c ˆ k 1 l 1 and c 12 ˆ c 13 ˆ c 23 ˆ k 1 l 1. These values will be determined by arti cially imposing r ˆ 0 in the constitutive equations of the isotropic ber so that ˆ k 1 l 1 k 1 l 1 11 22 ; and consequently r 11 ˆ 4k 1l 1 k 1 l 1 11 2l 1 k 1 l 1 k 1 l 1 22 ; r 22 ˆ 2l 1 k 1 l 1 k 1 l 1 11 4k 1l 1 k 1 l 1 22 : The three constants are then identi ed as c 1 11 ˆ c1 22 ˆ 4k 1l 1 k 1 l 1 ; c 1 12 ˆ 2l 1 k 1 l 1 k 1 l 1 : 5 6 7 The reductions in c 1 11, c1 22, and c1 12 from the original k 1 l 1 and k 1 l 1 are found to be all k 1 l 1 2 = k 1 l 1.

698 S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 With this scheme, the partially debonded isotropic bers are replaced by the perfectly bonded orthotropic bers of the same cross-section. The original problem ± which in essence is a three-phase one involving bers, matrix, and interface cracks ± has been replaced by a two-phase problem with only the ctitious bers and the matrix. In this context it is important to recognize that the bers in the threephase problem are not the same as the ctitious bers in the two-phase problem. Rather, each ctitious ber now represents the smeared medium of the original ber and the two interface cracks. We shall represent these orthotropic constants by the tensor L 1. It is then straightforward to nd the e ective moduli of such a two-phase composite by some well established e ective-medium method. For the present problem the simplest one is likely to be Willis' (1977) approach, by taking the elastic moduli of the comparison material as those of the matrix, denoted by the tensor L 0. It follows that the e ective moduli tensor is given by L ˆ c 1 L 1 A 1 c 0 L 0 I c 1 A 1 c 0 I 1 ; 8 where I is the fourth-order identity tensor, and A 1 is the strain concentration tensor of a single ctitious ber embedded in the in nitely extended matrix. It is given by A 1 ˆ I SL 1 0 L 1 L 0 Š 1 ; 9 in terms of Eshelby's (1957) S-tensor for an elliptic inclusion. Its components can be found in Mura (1987). As proved in Weng (1992), these moduli are identical to those derived by the Mori and Tanaka (1973) method. Since Willis' model involves a distribution function that is identical to the inclusion shape, and the outer cell of the unit cell also represents the distribution function, its microgeometry is exactly what is depicted in Fig. 1. It then makes sense to compare the nite-element calculations based on such a unit cell with the theoretical results. In order to show how the double debonding on the top and bottom of the elliptic cylinders would a ect the overall sti ness of the composite, we have applied this approach to calculate the nine independent elastic constants of the partially debonded composite, using the Young's modulus and Poisson's ratio of silicon carbide and aluminum (Arsenault, 1984; Nieh and Chellman, 1984): Silicon carbide : E 1 ˆ 490 GPa; m 1 ˆ 0:17; Aluminum : E 0 ˆ 68:3 GPa; m 0 ˆ 0:: 10 The results are shown in Fig. 2, normalized by the properties of the matrix. Here the dependence of the three Young's moduli, three shear moduli, and three Poisson's ratios on the volume concentration c 1, of the debonded bers are illustrated. In each gure the in uence of the ber cross-sectional aspect ratio a, ranging from 1.0 to 0.01, are also given. For comparison, the corresponding moduli with perfectly bonded bers are plotted in Fig. 3. A quick glance over these gures indicates that E 11, E 22, l 12, and m 12 are almost una ected by the partial debonding, but E, l 13, and l 23 are signi cantly reduced. Partial debonding has also changed the Poisson's ratios m 13 and m 32, which now can go out of the range of the constituent phases. A closer examination on the in uence of the cross-sectional shape further reveals that, for the Young's modulus E and shear moduli l 13 and l 23, the reduction of the sti ness is far more signi cant when bers take the shape of thin ribbons (a ˆ 0.01). The moduli displayed here must be viewed only under the condition of a crack opening mode. Consequently E is a tensile Young's modulus, not a compressive one. 3. Description of the nite-element analysis As the most important property change is the transverse Young's modulus E, our nite-element investigation will focus on the tensile loading along x 3. We consider a plane-strain problem for the unit cell subjected to a boundary traction giving rise to a uniform stress r (an overbar signi es the volume, or rather the area, average over the unit cell here). The traction boundary condition gives rise to a lower bound value for the e ective moduli; it was chosen over the displacement boundary condition for the e ective-

S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 699 Fig. 2. The nine e ective moduli of the doubly debonded composite as calculated by the e ective-medium method. medium approach also provides a lower bound estimate (even though not always the same type of lower bound) when the matrix is softer than the inclusion in the perfectly bounded case. Since the direction of the loading is perpendicular to the debonding con guration, contact of the debonded surfaces will not occur when the crosssectional aspect ratio is small and the debonding

700 S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 Fig. 3. The nine e ective moduli of the perfectly bonded composite as calculated by the e ective-medium method. angle is far from 90. But in every step of the way we have checked for the contact condition and whenever such a contact was detected the calculation for the speci c debonding angle / was halted. Indeed such a contact tends to occur under the in-plane shear r 23. The treatment of such a contact problem is essentially nonlinear and is not included here.

S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 701 The transverse Young's modulus is calculated by E ˆ r ; 11 where is the overall average strain in the x 3 - direction. In a cracked body, the contributions to the overall strain ± as pointed out by Benveniste (1986) and others ± must include the displacement jump across the crack surfaces. For the present problem as depicted in Figs. 1(a) and (b), it comes from the average strains of matrix (phase 0) and the ber (phase 1), with their weighted sum now denoted by 0 1, and the displacement jump, u 3 Š, across the debonded surfaces in the x 3 -direction ˆ c 0 0 c 1 1 1 X Z u 3 Šn 3 ds k V k S k ˆ 0 1 uš ; 12 where n 3 is the component of the unit normal to the debonding surface in the x 3 -direction, S k the kth debonded surface and V is the volume of the unit cell. The summation is taken on all the debonded surfaces in the unit cell. The symbol uš represents the last term from the displacement jump across the interfaces. It turns out that, when the extent of debonding is small, uš contributes little to the overall strain, but when it is wide this term will dominate. Typical nite-element meshes used for both double and single debonding are shown in Figs. 4(a) and (b), respectively. Here the illustrations are given for the aspect ratio a ˆ 1/4, ber concentration c 1 ˆ 30%, and debonding angle / ˆ 67 (for reasons to become clear later). The debonding along the ber±matrix interface is simulated by a sharp crack-like notch introduced by removing a notched region from the matrix adjacent to the interface. The maximum opening of the notch d max is determined to be 1% of the half debonding length. Since our primary concern is with the overall modulus E ± not with the singular behavior of the stress eld at the crack tip ± the debonding area is represented by a sharp cracklike notch, rather than by a crack. To justify this approximation we have introduced an alternative representation of the debonding by a slit of constant width d max with rounded tips. This model eliminates the singular behavior at all. Our numerical results have shown that these two models di er only in the local stress distribution near the tips and practically have no e ects on the average strain of the unit cell. Even the local notch tip stress distributions do not di er much if the radius of the rounded notch is small enough, such as less than 1% of the half debonding length. We have used ANSYS nite-element program. Eight-node isoparametric quadratic elements have been used except at the debonding tip, where triangular quarter-node singular elements have been employed. Each of the singular element surrounding the debonding tip is a six-node triangular element with its radial length less than one hundredth of the half debonding length. The next layer of elements consists of quadratic elements, edge lengths of which are magni ed by 120% of those for the inner singular elements. The mesh, re ned in the debonding tip region, gradually becomes coarse as we move away from it. Strictly speaking, the quarter-node p singular element that numerically generates the 1= r singular stress behavior at the tip, is valid only for the crack tip in a homogeneous elastic material. Since the region of the oscillatory singularity is con ned to the near interface crack tip region and the envelope p of this singularity is still of the order of 1= r, use of the quarter-node is not a bad approximation even if we replace the crack by a notch whose singularity is slightly weaker than that of the crack. Exploiting the symmetry of the geometry and loading, we have analyzed one quarter of the unit cell for the double debonding case and the half for the single debonding. 4. Dependence of the tensile Young's modulus E on the debonding angle / The variations of tensile Young's modulus E as a function of the debonding angle / are shown in Figs. 5(a)±(c) for three cross-sectional aspect ratios a ˆ 1=10, 1=4, and 1, respectively. In each gure two volume concentrations of the bers c 1 ˆ 30% and 10% have been calculated. Fig. 5(a) indicates that, when the aspect ratio is low ± namely when

702 S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 Fig. 4. Typical nite-element mesh for the unit cell under: (a) double debonding (one quadrant); (b) single debonding (one half). the bers are ribbon-like ± the tensile modulus remains una ected when the debonding angle is small, but the reduction becomes very sharp after / reaches a certain critical angle. For instance with c 1 ˆ 30%, E remains at until the debonding angle is close to 30, and with c 1 ˆ 10% it remains almost una ected until it reaches 60. As the crosssectional aspect ratio increases, such a sustained region also shortens, and when a ˆ 1 the variation of E is almost instant as / increases. Such a dependence is easy to conceive as, when a ˆ 1=10, most portion of the ber would be between the perfectly bonded interfaces if the debonding angle is small, but such is not the case with circular bers. On each of these curves a heavy dot has also been marked; theses points correspond to the E calculated by the e ective-medium method, as shown in Fig. 2. The e ective-medium method is based on the assumption that the debonding angle / is su ciently wide, but it does not provide the information on how wide this angle has to be. These calculations indicate that, with a ˆ 1=10, this angle is about 72 if c 1 ˆ 30%, and it is close to 78 if c 1 ˆ 10%. With the more rounded a ˆ 1=4, these angles are somewhat lower, about 67 and 74, respectively (the choice of a ˆ 1=4, c 1 ˆ 30%, and / ˆ 67 in Fig. 4 is now explained). The greater angle needed for the more ribbon-like a ˆ 1=10 is understandable as, again, when the

S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 703 Fig. 5. Dependence of the transverse Young's modulus E on the debonding angle /.

704 S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 Fig. 6. The variation of the debonding angle / in the e ectivemedium theory as the ber concentration c 1 increases. angle is small the inclusion can still transfer substantial amount of stress to the matrix. The heavy dots in Fig. 5(c) lie on the extended dashed lines, which in essence can not be attained as our niteelement calculations indicated that, when / is so large, the surfaces of the circular cylinder and the matrix would come to contact under the tensile loading. Such a possibility has also been noted by Toya (1974). This implies that, for a ˆ 1, the concept of ctitious ber is not valid and the E -curve in Fig. 2 for a ˆ 1 should be discarded. This all the more reiterates that the concept of ctitious bers should be restricted to the range a < 1, as stated earlier. The fact that the two dotted points in (a) and (b) do not lie vertically implies that the debonding angle / associated with the e ective-medium theory is somewhat dependent upon the ber concentration. Such a dependence is shown in Fig. 6 over a continuous range of c 1. These results display an almost linear correlation, and further indicate that, for the e ective-medium theory to hold, the debonding angle indeed has to be su ciently wide, typically ranging around 60±80. The increasing signi cance of the displacement jump uš as the debonding angle increases is illustrated in Fig. 7, for a ˆ 1=10 and c 1 ˆ 30%. It is evident from Fig. 7(a) that when / > 57 the contribution to the overall strain by the displacement jump ± represented by the uš -curve ± has exceeded that by the weighted sum of the average strains of the bers and the matrix, and that at / ˆ 72 ± which corresponds to the e ective-medium theory ± the displacement jump accounts for 90% of the total strain. To verify the validity of the assumption that the load-carrying capacity of the ctitious ber is zero in the debonding direction, we have plotted in Fig. 7(b) the ratio of the jump strain uš to the inclusion strain 1. Here uš, written without an overbar, is calculated as uš in Eq. (12) but divided by the volume V 1 of the ber ( uš uš ˆ / c 1 ) and, together with 1, form the strain of the ctitious ber. Again, as / increases the contribution of the displacement jump to the strain of the smeared region also increases, and at / ˆ 72 where the e ective-medium theory is intended for application, the strain from the cracks is as high as 158-times of that of the ber. As the crack regions have zero moduli, this means that the sti ness of the ctitious ber in the debonding direction is only 1=158th, or less than 1%, of the sti ness of the real ber. As such, the basic assumption that the ctitious ber has no load-carrying capacity in the debonding direction is justi ed. With single debonding as sketched in Fig. 1(b), the reduction of the sti ness as a function of debonding angle / is shown in Fig. 8, for the same three aspect ratios. For comparison the corresponding results for double debonding are also depicted as dashed lines. Both classes of curves show similar characteristics, such as an incubation period for / when a is small. Even though the overall appearance of these two groups of curves does not seem to di er signi cantly, the magnitudes of E at a given / within the sharply a ected region can be quite di erent. For instance at / ˆ 60 and when a ˆ 1=10 and c 1 ˆ 30%, the overall sti ness under single debonding is 38% higher than that under double debonding. The local stress distributions for r x 2 ; x 3 under double debonding at c 1 ˆ 30% are shown for a ˆ 1=10 and 1=4 in Figs. 9(a) and (b), respectively. The debonding angles are deliberately chosen at 72 and 67 here so that the plotted results would also shed some light for the elds based on the e ective-medium method. The x 2 -x 3 plane at

S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 705 Fig. 7. The relative contribution of the displacement jump over the interface cracks to: (a) the overall strain of the composite; (b) the overall strain of the smeared ber and interface cracks region. the bottom of each gure re ects the geometry of the rst quadrant of Fig. 1(a), with the ber±matrix interface marked by the sign ``interface'' on both axes. Looking along the x 3 -axis, it is apparent that the stress in the ber is zero. Continuous increase along the x 3 -axis takes one into the matrix region and the stress begins to increase from zero towards a greater quantity. The stress along the x 2 - axis shows a sustained region of zero stress also inside the ber. Its value, however, begins to increase as one moves below the notch tip. Similar features are also present in Fig. 8(b), with a ˆ 1=4. It is evident that stress distribution in the ber and the matrix is quite heterogeneous due to the presence of the interface cracks and elastic inhomogeneity. It may be reminded that, while the stress in the ber is not zero, this is not to be interpreted as a direct violation to the basic concept of the ctitious ber. Instead as we did in Fig. 7(b) such a validity should be examined from the strain consideration. The local stress elds under single debonding are shown in Figs. 10(a) and (b), chosen for the same ber aspect ratios and debonding angles as in Figs. 9(a) and (b) so as to render a comparison. These illustrations are displayed for the right half of the con guration in Fig. 1(b) as the stress eld is no longer symmetric with respect to the x 2 -axis. The stress r x 2 ; x 3 along the x 3 -axis is seen to be zero on the top but nite at the bottom of the interface. The lower portion of the ber ± which remains bonded to the matrix ± now carries a noticeable magnitude of tensile stress everywhere, leading to a higher sti ness for the overall composite as re ected in Fig. 8. 5. Concluding remarks In this paper we have presented a dual e ectivemedium and nite-element study on the overall elastic behavior of a partially debonded ber-re-

706 S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 Fig. 8. Comparison between the e ects of single debonding and double debonding.

S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 707 Fig. 9. Local stress distribution in the unit cell for r x 2 ; x 3 under double debonding. inforced composite. The e ective-medium method is aimed at double debonding, while the nite-element analysis is concerned with both double and single debonding and it has been further used to examine how the debonding angle / a ects the reduction of the overall sti ness of the composite.

708 S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 Fig. 10. Local stress distribution in the unit cell for r x 2 ; x 3 under single debonding. The e ective-medium method is based on the concept of ctitious bers whose load-carrying capacity is taken to be lost in the debonding direction and whose orthotropic properties are then derived from the isotropic ones of the original - bers. The debonded composite is then treated as a perfectly bonded two-phase composite. The niteelement analysis on the other hand treats the problem as a three-phase material including the bers, matrix, and the interface cracks, the extent of the latter being represented by the debonding angle /. A detailed examination on the nite-element results indicates that, for ribbon-like bers, the debonding angle has very little e ect on the

S.F. Zheng et al. / Mechanics of Materials 32 (2000) 695±709 709 reduction of the transverse Young's modulus E until it reaches a su ciently large value, but that for circular bers its e ect is immediate. As the debonding angle increases, the interfacial displacement jump is found to contribute the major portion of the overall strain. The nite-element results also provide the precise / at which the effective-medium method can be best applied. For debonding angles signi cantly di erent from this one, the ctitious- ber approach is not recommended. The validity of the basic assumption that the ctitious ber has zero, or very low, load-carrying capacity in the debonding direction has also been veri ed. The results for single debonding suggest that, while the general trend of the E variation remains similar to the double debonding case, the di erence in their magnitudes can be signi cant for certain critical range of debonding angle. The local stress elds for both debonding types have also been illustrated. Acknowledgements This work was supported by the National Science Foundation, under Grant CMS-9625304, and by the O ce of Naval Research, under Grant N- 00014-91-J-1937. References Arsenault, R.J., 1984. The strengthening of aluminum alloy 6061 by ber and platelet silicon carbide. Mater. Sci. Eng. 64, 171±181. Benveniste, Y., 1986. On the Mori±Tanaka's method in cracked bodies. Mech. Res. Comm. 13, 193±201. England, A.H., 1966. An arc crack around a circular elastic inclusion. ASME J. Appl. Mech., 637±640. Eshelby, J.D., 1957. The determination of the elastic eld of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. London. Ser. A 241, 376±396. Ju, J.W., 1991. A micromechanical damage model for uniaxially reinforced composites weakened by interfacial arc microcracks. ASME J. Appl. Mech. 58, 923±930. Mori, T., Tanaka, K., 1973. Average stress in the matrix and average elastic energy of materials with mis tting inclusions. Acta Metall. 21, 571±574. Mura, T., 1987. Micromechanics of Defects in Solids, second ed. Martinus Niho, Dordrecht. Nieh, T.G., Chellman, D.J., 1984. Modulus measurements in discontinuous reinforced aluminum composites. Scr. Metall. 8, 925±928. Toya, M., 1974. A crack along the interface of a circular inclusion embedded in an in nite solid. J. Mech. Phys. Solids 22, 325±348. Weng, G.J., 1992. Explicit evaluation of Willis' bounds with ellipsoidal inclusions. Int. J. Eng. Sci. 30, 83±92. Willis, J.R., 1977. Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25, 185±202. Zhao, Y.H., Weng, G.J., 1997. Transversely isotropic moduli of two partially debonded composites. Int. J. Solids Struct. 34, 493±507.