A Zener±Stroh crack in a fiber-reinforced composite material

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A Zener±Stroh crack in a fiber-reinforced composite material

Mechanics of Materials 32 (2000) 593±606www.elsevier.com/locate/mechmatA Zener±Stroh crack in a ®ber-reinforced composite materialZ.M. Xiao * , B.J. Chen, H. FanSchool of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, SingaporeReceived 15 November 1999AbstractA Zener±Stroh crack is formed by coalescing a dislocation pileup and therefore is a net dislocation loaded crack. Inthis paper, the interaction between a Zener±Stroch crack and surrounding cylindrical ®bers in a ®ber-reinforcedcomposite material has been investigated. The crack is located in the matrix and near a ®ber, while the in¯uence of other®bers on the crack is considered through the three-phase composite cylinder model. Using the solution of stress ®eld fora single dislocation in a three-phase cylinder as the GreenÕs function, the crack is represented by a continuous distributionof dislocations. A set of singular integral equations are thus formulated and solved numerically. The stressintensity factor (SIF) on the crack tip due to crack±inclusions interaction is calculated. The results show that the volumeconcentration of ®bers has non-negligible in¯uence on the SIF of the crack. Ó 2000 Elsevier Science Ltd. All rightsreserved.Keywords: Crack; Stress intensity factor; Inclusion; Interaction; Self-consistent three-phase model; Composite material1. IntroductionMatrix cracking is a major pattern of failure of composite materials. Crack can form in the matrixduring manufacturing, or be produced during loading, especially during cyclic loading. To accuratelypredict the fracture behavior of a crack in a composite, the interaction of the crack with the surroundinginclusions should be fully investigated. In the past three decades, quite a number of research work on theabove-mentioned topic have been carried out. To name a few, the interaction between a crack and a circularinclusion in a sheet under tension was studied by Tamate (1968). An elastic circular inclusion interactingwith two symmetrically placed collinear cracks was investigated by Hsu and Shivakumar (1976). Sendeckyj(1974) studied the problem of a crack located between two rigid inclusions. The investigation for a cracknear an elliptic inclusion was carried in terms of body force method by Nisitani et al. (1996).Erdogan et al. (1974) ®rst considered the interaction between an isolated circular inclusion and a linecrack embedded in in®nite matrix. As commented by Erdogan et al. (1974), their model is applicable to thecomposite materials which contain sparsely distributed inclusions. For composites ®lled with ®nite concentrationof inclusions, it is commonly understood that the stress and strain ®elds near the crack dependconsiderably on the microstructure around it. However, to treat the problem based on all details of the* Corresponding author. Tel.: +65-799-4726; fax: +65-791-1859.E-mail address: mzxiao@ntu.edu.sg (Z.M. Xiao).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 0 021-1


594 Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606microstructure of a real composite would be too complicated. One notable simpli®ed model is the selfconsistentthree-phase model used by Christensen and Lo (1979), Luo and Weng (1987, 1989) in studyinge€ective moduli of composite materials. Unlike the two-phase model where the inclusion is embedded inthe in®nitely extended pure matrix, the three-phase model considers that in the immediate neighborhood ofthe inclusion there is a layer of matrix material acting on, but at certain distance the average in¯uence of theheterogeneous medium can be represented by the smeared property of the composite. Thus, for theproblems of which the interest is in the ®eld near the inclusion and crack, it can reasonably be accepted as agood model.Having a di€erent mechanism from the Grith crack, the Zener±Stroh crack was ®rst proposed byZener (1948). In his model, a pileup of edge dislocations that were stopped at an obstacle, such as a grainboundary (Fig. 1(a)) could coalesce into a crack nucleus. Besides ZenerÕs mechanism of microcrack initiation,there are some other variants. One was presented by Cottrell (1958) as shown in Fig. 1(b), wherepiled-up dislocations on two intersecting slip planes can coalesce into a microcrack. The other variantproposed by Kikuchi et al. (1981) is given in Fig. 1(c) which shows a particle at a grain boundary in astressed solid, concentrated stress ®elds exist near the ends of the particle, slip is thus nucleated in theseregions. Dislocations of one sign move away from the region, leaving stationary dislocations of the oppositesign behind to form a crack near the particle. This case is the experimental background for the currentstudy.Zener±Stroh crack is the crack complementary to the well-known Grith crack. Physical parametersthat are symmetric for the Grith crack are anti-symmetric for the Zener±Stroh crack and vice versa. Forinstance, a Grith crack dislocation distribution along the crack plane is anti-symmetric. This dislocationdistribution gives rise to a symmetric crack plane traction stress. The Zener±Stroh crack has an antisymmetriccrack plane traction stress, which arises from a symmetric crack plane dislocation distribution.Stress, displacement, dislocation density and stress intensity factor (SIF) were compared between Grithcrack and Zener±Stroh crack in a homogeneous material by Weertman (1986). Due to the displacementloading mechanism, the total sum of the Burgers vectors of the dislocations b T within a Zener±Stroh crackFig. 1. (a) ZenerÕs mechanism of crack initiation. (b) CottrelÕs model of crack initiation. (c) Anti-Zener±Stroh crack model.


Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606 595does not equal to zero. The crack tip where the dislocation enters the Zener±Stroh crack is called blunt tip,while the other tip is called sharp tip. The crack propagation is always initiated from the sharp tip. Thiskind of microcrack was often observed in metal matrix composite materials.2. The physical problem and formulationThe physical problem to be investigated in the current paper is shown in Fig. 2(a), where a Zener±Strohcrack is initiated near a circular inclusion (a ®ber) in a ®ber-reinforced composite. For the case shown inFig. 2, the sharp crack tip is on the right-hand side and the left crack tip is blunt. We denote this case asCase I. While when the sharp tip is on the right, the problem is denoted as Case II. Both cases have beenconsidered in our numerical examples shown in Section 5. The formulations are parallel to both cases.In order to consider the e€ect of the surrounding inclusions (other ®bers) on the crack, the three-phasecomposite model discussed in the previous section is employed, so the original problem is made to beequivalent to Fig. 2(b), where the two-dimensional version of the three-phase model consists of threeconcentrically cylindrical layers with the outer one being extended to in®nity. The inner region r 6 a (Phase1) is the ®ber inclusion with radius a and elastic properties j 1 (j 1 ˆ 3 4m 1 with m 1 the Poisson's ratio) andl 1 is the intermediate layer (Phase 2) is the pure matrix material with elastic properties j 2 and l 2 , occupiesthe annular region a 6 r 6 b. The outside layer (Phase 3) is the composite material with e€ective elasticmoduli j 3 and l 3 , occupies the outer in®nite region r 6 b. The internal and external radii of the intermediatematrix-phase a and b being related to each other by …a=b† 2 ˆ c, where c is the volume concentration of the®bers in composite. The e€ective moduli of the composite j 3 and l 3 are calculated by the method ofChristensen and Lo (1979). The Zener±Stroh matrix crack is along the radial direction and located in Phase2 (the pure matrix material). In order to reduce the number of parameters involved in discussion, weconcentrate on the in¯uence of the net dislocations inside the crack on the SIF at crack tip, so it is assumedthat the elastic system is free of external mechanical loading. It is worth to mention that the problem can besolved similarly even with external mechanical loading.The boundary conditions of the problem are:In the far ®eld:r xy ˆ 0;…1a†r yy ˆ 0:…1b†At the interface:‰r xy Šˆ0; …2a†‰r yy Šˆ0: …2b†Fig. 2. A Zener±Stroh crack in a ®ber-reinforced composite material and the three-phase cylinder model.


596 Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606At the crack surface:r xy ˆ 0; r yy ˆ 0; t 1 6 x 6 t 2 ; …3a†‰r xy Šˆ0; ‰r yy Šˆ0; x < t 1 or x > t 2 ; …3b†where [f] is the jump of the function f.In order to solve the problem, the equivalence between a crack and a pile-up of dislocations is employed.Let B x …x† and B y …x† be the gliding and climbing dislocation densities along the crack line …t 1 ; t 2 †. Using thestress ®eld solution of a single edge dislocation in a three-phase composite (1991) as Green functions, thetractions at …x; 0† due to those dislocations are written asr yy …x; 0† ˆr xy …x; 0† ˆ2l 2…j 2 ‡ 1†p2l 2…j 2 ‡ 1†pZ t2t 1Z t2t 1B y …n†n x dn ‡B x …n†n x dnZ t2t 1k 1 …x; n†B y …n† dn ; …4a†Z t2‡ k 2 …x; n†B x …n† dn : …4b†t 1With the aid of the above equations, the traction free conditions on the upper and lower crack surfaces (i.e.,boundary conditions (3a) and (3b)) are written in terms of B x ; B y as1pZ t2t 1B y …n†n x dn ‡ Z t2t 11p k 1…x; n†B y …n† dn ˆ 0; t 1 6 x 6 t 2 ; …5a†Z1t2ZB x …n†t2p t 1n x dn ‡ 1t 1p k 2…x; n†B y …n† dn ˆ 0; t 1 6 x 6 t 2 : …5b†Moreover, the dislocation densities B x …x† and B y …x† must satisfyZ t2B y …n† dn ˆ b T y ; …6a†t 1Z t2t 1B x …n† dn ˆ b T x ; …6b†where b T x and bT yare the total sums of Burgers vectors of the net distributed dislocations inside the Zener±Stroh crack. Eqs (5a) and (5b) are the standard singular equations with Cauchy type kernels k 1 …x; n† andk 2 …x; n† which are given by (Luo and Chen, 1991):k 1 …x; n† ˆ A ‡ B2…x a 2 =n† ‡ A n2 a 2 a 2 n 2 n2 a 2 A ‡ B 1 a 2n 3 …x a 2 =n† 2 a 2 xn a 2 2x 2n x 2 A 2 n 2 1 ‡ B 2N A a2a 2 x Nb a3 0x 2 1 X 1 x n 2b n ‡ b n n x n x n2 a n ‰ bn …n 1†b n Š 2Abn2a a aaxnˆ1‡ Ab n n a n a n‡2‡ ‰ Abn …n ‡1†‡Bb n Š…7a†xx


andZ.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606 597k 2 …x; n† ˆ A ‡ B2…x a 2 n† ‡ A a 2 x n n2 a 2…x a 2 =n† 2 n 2 xn a 2 1 X 1 b 0 n2an x n‰b 0 na…n 1†‡b0 n Š xanˆ1‡‰Ab 0 n …n ‡ 1†Bb0 n Šax A ‡ B2x n2 B A 2nwhereA ˆ 1 m ; B ˆ j1 mj 21 ‡ mj 2 j 1 ‡ m ; C ˆ j3 lj 2; D ˆ 1 l ;j 3 ‡l 1 ‡ lj 2m ˆ l 1 =l 2 ; l ˆ l 3 =l 2 ; N ˆ j1 1 m…j 2 1†j 1 1 ‡ 2m‡ A a2x 3a 2x 2 Ab0 n na nx n‡2; …7b†and a n ; b n ; a 0 n ; b0 nare calculated from the following two sets of equations:a n ‡ 1 D b n ˆ 1b d n0 ‡ b n‡1 a 2n‡2 ‡…n ‡ 1†…b 2 a 2 1†b n1 a 2n ; n P 0; …9a†1A a n b n ˆ d n1 b 1n ‡…n 1†…1 b 2 †b 1n ; n P 1; …9b†…8†D…n 1†a n ‰Ca 2n ‡…1 C†d n0 Ša n ‰…n 1†…N 1†d n0 Ša 2 b n ‡‰1 ‡…N 1†d n0 Ša 2 b nˆ …D C† 1 b ‡ 2N b a2 d n0 ‡…C ‡ 2Db 2 a 2 †b n1 a 2n ‡ 3D…n ‡ 1†…b 2 a 2 1†b n1 a 2n D…n ‡ 1†b n‡1 a 2n‡2 2…1† n D…b 2 a 2 1†b n1 a 2n C n 3‡D…1 C†1 D a2 d n1 ; n P 0; …9c†and…n ‡ 1†a n ‡ a 2n a n A…n ‡ 1†a 2 b n Ba 2 b nˆA…n ‡ 1†b 1n a 2 Bb …n‡1† a 2 ‡ 3A…n 1†…b 2 1†b …n‡1† a 2 ‡ 2A…1† n‡1…b 21 1†b …n‡1† a 2 …1 d n1 †…1 d n2 †C n33 ‡ 2Aa2 d n1 ; n P 1 …9d†a 0 n ‡ 1 D b0 n ˆ 1b d n0 b n‡1 a 2n‡2 ‡…n ‡ 1†…b 2 a 2 1†b n1 a 2n ; n P 0; …10a†1A a0 n b0 n ˆd n1 ‡ b 1n ‡…n 1†…1 b 2 †b 1n ; n P 1; …10b†D…n 1†a 0 n ‡‰Ca2n ‡…1 C†d n0 Ša 0 n ‰…n 1†…N 1†d n0Ša 2 b 0 n ‰1 ‡…N 1†d n0Ša 2 b 0 nˆ…D C† 1 b d n0 ‡…C 2Db 2 a 2 †b n1 a 2n ‡ 3D…n ‡ 1†…b 2 a 2 1†b n1 a 2n ‡ D…n ‡ 1†b n‡1 a 2n‡2 2…1† n D…b 2 a 2 1†b n1 a 2n C n 3‡D…1 C†1 D a2 d n1 ; n P 0; …10c†


598 Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606in which…n ‡ 1†a 0 n a2n a 0 n A…n ‡ 1†a2 b 0 n ‡ Ba2 b 0 nˆ A…n ‡ 1†b 1n a 2 Bb …n‡1† a 2 ‡ 3A…n 1†…b 2 1†b …n‡1† a 2 ‡ 2A…1† n‡1…b 21 1†b …n‡1† a 2 …1 d n1 †…1 d n2 †C n33 2Aa2 d n1 ; n P 1 …10d†a ˆ ab ; b ˆ n=a; …3†…4†…3 n ‡ 1†Cn 3 ˆ …11†n!and d n1 is the Kroneker delta.Once the dislocation densities B x …x† and B y …x† have been solved from Eqs. (5a)±(6b) together withEqs. (7a)±(8), the stress ®elds in Phase 2 can be obtained from Eqs. (4a) and (4b). Thus, the SIF of the crackcan be evaluated accordingly.3. Solution procedures of the integral equationsTo shift the integral interval from …t 1 ; t 2 † to …1; 1† in the integral Eqs. (5a)±(6b), letx ˆ t2 t 12t ‡ t 2 ‡ t 12; n ˆ t2 t 12Eqs. (5a) and (5b) are rewritten in terms of s; t aswhere1pZ 11ZB y …s†1s t ds ‡1s ‡ t 2 ‡ t 1: …12†21p k 11…t; s†B y …s† ds ˆ 0; 1 6 s; t 6 1; …13a†Z1 1ZB x …s†1p 1 s t ds ‡ 11 p k 22…t; s†B x …s† ds ˆ 0; 1 6 s; t 6 1; …13b†k 11 …t; s† ˆt2 t 12k 22 …t; s† ˆt2 t 12t 2 t 1k 12t 2 t 1k 22tt‡ t 2 ‡ t 12‡ t 2 ‡ t 12; t 2 t 12; t 2 t 12and Eqs. (6a) and (6b) are similarly rewritten ass ‡ t 2 ‡ t1; …14a†2s ‡ t 2 ‡ t12…14b†Z 11B y …s† ds ˆ2bT yt 2 t 1;…15a†Z 11B x …s† ds ˆ2bT xt 2 t 1:…15b†Since the whole crack is located in Phase 2 (the pure matrix material), the singularities on the both cracktips should be 1=2. As a result, the two dislocation density functions B x …x† and B y …x† must have 1=2singular behaviors on the both crack tips. Let


Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606 599B x …x† ˆw…x†F x …x†; B y …x† ˆw…x†F y …x†; …16†where F x …x† and F y …x† are bounded functions in t 1 6 x 6 t 2 , andw…x† ˆ…x t 1 † 1=2 …t 2 x† 1=2 :…17†Following the method developed by Erdogan and Gupta (1972), we get the discretized forms of (13a) and(13b) and (15a) and (15b):X n 1n F 1y…s k † ‡ k 11 …u r ; s k † ˆ 0;…18a†s k u rkˆ1X nkˆ11n F 1x…s k †s k u r‡ k 22 …u r ; s k † ˆ 0;…18b†X nkˆ11n F y…s k †ˆbT yp ;…18c†X nkˆ11n F x…s †ˆbT xkp…18d†in whichs k ˆ cos p …2k 1†;2n ur ˆ cos pr ; k ˆ 1; ...; n; r ˆ 1; ...; n 1: …18e†nEqs. (18a) and (18c) provide a system of n linear algebraic equations to determine F y …s 1 †; ...; F y …s n †; while(18b) and (18d) provide another system of n linear algebraic equations to determine F x …s 1 †; ...; F x …s n †. Oncethe functions F x …x† and F y …x† are solved, the dislocation density functions B x …x† and B y …x† can be evaluatedfrom Eqs. (14a) and (14b).With the solution of the dislocation density functions obtained, the Mode I and Mode II SIF on theblunt (left) and sharp (right) crack tips as shown in Fig. 2 (Case I) are then given by (Weertman, 1996)pK B 2lIˆlim 2 2p pp2ln t 1 B y …n† ˆ2 ppF y …1†; …19a†n!t1 j 2 ‡ 1…j 2 ‡ 1† …t 2 t 1 †=2pK B II ˆlim 2l 2 2p pp2ln t 1 B x …n† ˆ2 ppF x …1†; …19b†n!t 1 j 2 ‡ 1…j 2 ‡ 1† …t 2 t 1 †=2K S Iˆ limn!t22l 2p2p pp2lt 2 nB y …n† ˆ2 ppF y …1†; …19c†j 2 ‡ 1…j 2 ‡ 1† …t 2 t 1 †=2pK S II ˆ lim 2l 2 2p pp2lt 2 nB x …n† ˆ2 ppF x …1†; …19d†n!t 2 j 2 ‡ 1…j 2 ‡ 1† …t 2 t 1 †=2where B and S represent the left and right crack tips, respectively.


600 Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±6064. Example calculations and discussionsFor the numerical examples given in this section, the SIFs of the crack are normalized byK 0 I ˆ2l 2 b T yp ; K 0 II…j 2 ‡ 1† p…t 2 t 1 †=2ˆ 2l 2 b T xp …j 2 ‡ 1† p…t 2 t 1 †=2…20†they are the Mode I and Mode II SIFs at the sharp crack tip of the same size Zener±Stroh crack with thesame Berbers vectors (b T x , bT y) in a homogeneous materials, respectively. As the interaction between a crackand a single inclusion has been studied by many researchers as discussed in Section 1, in our current examplecalculations, we concentrate on studying the in¯uence of the volume fraction of inclusions on thebehavior of crack.Two typical types of composite materials are chosen for the examples:Type I : m ˆ l 1 =l 2 ˆ 0; j 1 ˆ 1:8; j 2 ˆ 1:6: …21a†Type II : m ˆ l 1 =l 2 ˆ 23; j 1 ˆ 1:8; j 2 ˆ 1:6 …21b†in which j i ˆ 3 4m i …i ˆ 1; 2†. The ®rst composite material is actually an elastic matrix with certain voids(to represent a composite with ``soft'' inclusions), while the second one is a real ®ber-reinforced material (acomposite with ``harder'' inclusions). The size and location of the Zener±Stroh matrix crack is de®ned bythe parameters t 1 and t 2 ast 1 ˆ 1:05a; t 2 ˆ 1:35a: …22†Corresponding to di€erent volume fraction of inclusions c ˆ…a=b† 2 , the e€ective moduli of the composite(Phase 3) is evaluated according to Christensent and Lo (1979).The normalized Mode I and Mode II SIFs at the blunt (left-handed) crack tip versus the volume fractionc of the inclusions in both types of materials are listed in Table 1. It is found that the SIFs at the blunt tipare always negative. This result is consistent to our discussion in Section 1, i.e., the propagation of a Zener±Stroh crack is always along the sharp crack tip. Therefore, in the forgoing numerical discussion, we chie¯yconcentrated ourselves on studying the SIFs at the sharp crack tip (the right-handed crack tip in Fig. 2(b)).For Case I (sharp crack tip at right), the normalized Mode I and II SIFs at the sharp tip in the elasticmatrix with voids (Type I material) are depicted in Figs. 3 and 4, while Figs. 5 and 6 show the Mode I and IISIFs of the sharp crack tip when the crack is in ®ber-reinforced material (Type II material). For com-Table 1Normalized SIFs at the blunt tip (Case I)c Type I material: l 1 ˆ 0 Type II material: l 1 ˆ 23l 2K B I =K0 IK B II =K0 IIK B I =K0 IK B II =K0 II0.00 )1.461 )1.550 )0.690 )0.6490.05 )1.420 )1.545 )0.691 )0.6550.10 )1.372 )1.540 )0.692 )0.6630.15 )1.314 )1.533 )0.693 )0.6730.20 )1.243 )1.525 )0.694 )0.6850.25 )1.155 )1.515 )0.696 )0.6990.30 )1.045 )1.502 )0.697 )0.7150.35 )0.905 )1.486 )0.701 )0.7330.40 )0.729 )1.464 )0.710 )0.7520.45 )0.504 )1.438 )0.721 )0.7720.50 )0.213 )1.404 )0.739 )0.791


Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606 601Fig. 3. Normalized Mode I SIF at the sharp tip versus c in an elastic matrix with voids (Case I).Fig. 4. Normalized Mode II SIF at the sharp tip versus c in an elastic matrix with voids (Case I).Fig. 5. Normalized Mode I SIF at the sharp tip versus c in a ®ber-reinforced composite material (Case I).parison, the results calculated based on the two-phase model (a crack interacting with a single inclusion) arealso displayed in the ®gures. It can be clearly observed that as the ®ber (or void) concentration is very dilute(small volume fraction c), the present solutions approach to those of the two-phase model. When theinclusion concentration increases, the SIF increase for the void case and decrease for the ®ber case.


602 Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606Fig. 6. Normalized Mode II SIF at the sharp tip versus c in a ®ber-reinforced composite material (Case I).Fig. 7. Normalized Mode I SIF at the sharp tip versus c in an elastic matrix with voids (Case II).All the calculations mentioned above are for Case I shown in Fig. 2(b), i.e., the sharp crack tip is on theright. For the case when the sharp crack tip is towards to left (Case II), the calculations can be similarlyperformed. The SIF can also be evaluated from Eqs. (19a)±(19d) by interchange F y …1† and F x …1† in (19a)and (19b) with F y …1† and F x …1† in (19c) and (19d). The SIFs when the sharp crack tip is at the left-hand sideare shown in Figs. 7±10. An interesting phenomenon is that the variation trend of the SIFs in Figs. 7±10 istotally opposite to that shown in Figs. 3±6. This is due to the fact that when the volume fraction c of theinclusions increases, not only the elastic properties of the composite phase (Phase 3) changes, but also thedistance between the inclusion to the left crack tip, and the distance between the composite phase to theright crack tip change. All these parameter changes will in¯uence the behavior of the crack.5. Some other related resultsAs a ``by-product'' of the current study, our solution can be applied to the physical problem of a Zener±Stroh crack in a real case of three dissimilar isotropic materials (i.e., Phase 3 is not the composite phase, butthe third kind of material). It is of interest to study the respective in¯uence of the elastic property and thegeometric parameters on the behavior of the crack. The geometric con®guration is set to be the same as that


Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606 603Fig. 8. Normalized Mode II SIF at the sharp tip voids versus c in an elastic matrix with (Case II).Fig. 9. Normalized Mode I SIF at the sharp tip versus c in a ®ber-reinforced composite material (Case II).Fig. 10. Normalized Mode II SIF at the sharp tip versus c in a ®ber-reinforced composite material (Case II).in Fig. 2(b) (the sharp crack tip is on the right). Figs. 11 and 12 plot the normalized Mode I and Mode IISIFs at the sharp tip versus the modulus ratio l ˆ l 3 =l 2 for the following parameters:m ˆ l 1 =l 2 ˆ 20; j 1 ˆ j 2 ˆ j 3 ˆ 1:8; a=b ˆ 1=8; …23†


604 Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606Fig. 11. Normalized Mode I SIF at the sharp tip versus l 3 =l 2 (Case I) in a solid with three disimilar materials.Fig. 12. Normalized Mode II SIF at the sharp tip versus l 3 =l 2 (Case I) in a solid with three disimilar materials.t 1 ˆ 3:5a; t 2 ˆ 4:5a: …24†Figs. 13 and 14 show the normalized Mode I and Mode II SIFs at the sharp tip versus b=a for the followingparameters:m ˆ l 1 =l 2 ˆ 20; j 1 ˆ j 2 ˆ j 3 ˆ 1:8; l ˆ l 3 =l 2ˆ 10; …25†t 1 ˆ 3:5a; t 2 ˆ 4:5a: …26†In both cases, the results calculated based on the two-phase model is also displayed. From Figs. 11 and 12,it can be seen that the two curves intersect at l 3 =l 2 ˆ 1. As l 3 =l 2 increases in®nitely, the solution of thethree-phase model has an asymptotically value. In Figs. 13 and 14, it is observed that the thickness of theintermediate matrix phase a€ects slowly on the result. This is because the thickness of the outer phase(Phase 3) is always in®nitely large, and this phase is ``harder'' than the intermediate phase for the currentcalculation.


Z.M. Xiao et al. / Mechanics of Materials 32 (2000) 593±606 605Fig. 13. Normalized Mode I SIF at the sharp tip versus b=a (Case I) in a solid with three disimilar materials.Fig. 14. Normalized Mode II SIF at the sharp tip versus b=a (Case I) in a solid with three disimilar materials.ReferencesChristenson, R.M., Lo, K.H., 1979. Solution for e€ective shear properties in three-phase sphere and cylinder models. Journal ofMechanics and Physics of Solids 27, 315±330.Cottrell, A.H., 1958. Theory of brittle fracture in steel and similar metals. Transaction of the Metallurgical Society of the AIME 212,192±203.Erdogan, F., Gupta, G.D., 1972. On the numerical solution of singular integral equations. Quarterly of Applied Mathematics 30, 525±534.Erdogan, F., Gupta, G.D., Ratwani, M., 1974. Interaction between a circular inclusion and an arbitrarily oriented crack. ASMEJournal of Applied Mechanics 41, 1007±1013.Hsu, Y.C., Shivakumar, V., 1976. Interaction between an elastic circular inclusion and two symmetrically placed collinear cracks.International Journal of Fracture Mechanics 12, 619±630.Kikuchi, M., Shiozawa, K., Weertman, J., 1981. Void nucleation in astrology. Acta Mechanics 29, 1747±1758.Luo, H.A., Chen, Y., 1991. An edge dislocation in a three-phase composite cylinder model. ASME Journal of Applied Mechanics 58,75±86.Luo, H.A., Weng, G.J., 1987. On Eshelby's inclusion problem in a three-phase spherically concentric solids, and a modi®cation ofMori±Tanaka's method. Mechanics of Materials 6, 347±361.Luo, H.A., Weng, G.J., 1989. On Eshelby's S-tensor in a three-phase cylindrically concentric solids, and the elastic moduli of ®berreinforcedcomposites. Mechanics of Materials 8, 77±88.Nisitani, H., Chen, D.H., Saimoto, A., 1996. Interaction between an elliptic inclusion and a crack. In: Proceedings of the InternationalConference on Computer-Aided Assessment and Control, vol. 4, Computational Mechanics, Billerica, MA, USA, pp. 325±332.


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