A ZenerÂ±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-re**in**forced **composite** **material**Z.M. Xiao * , B.J. Chen, H. FanSchool of Mechanical and Production Eng**in**eer**in**g, Nanyang Technological University, Nanyang Avenue, S**in**gapore 639798, S**in**gaporeReceived 15 November 1999AbstractA Zener±Stroh **crack** is formed by coalesc**in**g a dislocation pileup and therefore is a net dislocation loaded **crack**. Inthis paper, the **in**teraction between a Zener±Stroch **crack** and surround**in**g cyl**in**drical ®bers **in** a ®ber-re**in**forced**composite** **material** has been **in**vestigated. 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** cyl**in**der model. Us**in**g the solution of stress ®eld fora s**in**gle dislocation **in** a three-phase cyl**in**der as the GreenÕs function, the **crack** is represented by a cont**in**uous distributionof dislocations. A set of s**in**gular **in**tegral equations are thus formulated and solved numerically. The stress**in**tensity factor (SIF) on the **crack** tip due to **crack**±**in**clusions **in**teraction 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 **in**tensity factor; Inclusion; Interaction; Self-consistent three-phase model; Composite **material**1. IntroductionMatrix **crack****in**g is a major pattern of failure of **composite** **material**s. Crack can form **in** the matrixdur**in**g manufactur**in**g, or be produced dur**in**g load**in**g, especially dur**in**g cyclic load**in**g. To accuratelypredict the fracture behavior of a **crack** **in** a **composite**, the **in**teraction of the **crack** with the surround**in**g**in**clusions should be fully **in**vestigated. In the past three decades, quite a number of research work on theabove-mentioned topic have been carried out. To name a few, the **in**teraction between a **crack** and a circular**in**clusion **in** a sheet under tension was studied by Tamate (1968). An elastic circular **in**clusion **in**teract**in**gwith two symmetrically placed coll**in**ear **crack**s was **in**vestigated by Hsu and Shivakumar (1976). Sendeckyj(1974) studied the problem of a **crack** located between two rigid **in**clusions. The **in**vestigation for a **crack**near an elliptic **in**clusion was carried **in** terms of body force method by Nisitani et al. (1996).Erdogan et al. (1974) ®rst considered the **in**teraction between an isolated circular **in**clusion and a l**in**e**crack** embedded **in** **in**®nite matrix. As commented by Erdogan et al. (1974), their model is applicable to the**composite** **material**s which conta**in** sparsely distributed **in**clusions. For **composite**s ®lled with ®nite concentrationof **in**clusions, it is commonly understood that the stress and stra**in** ®elds near the **crack** dependconsiderably on the microstructure around it. However, to treat the problem based on all details of the* Correspond**in**g 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** study**in**ge€ective moduli of **composite** **material**s. Unlike the two-phase model where the **in**clusion is embedded **in**the **in**®nitely extended pure matrix, the three-phase model considers that **in** the immediate neighborhood ofthe **in**clusion there is a layer of matrix **material** act**in**g on, but at certa**in** distance the average **in**¯uence of theheterogeneous medium can be represented by the smeared property of the **composite**. Thus, for theproblems of which the **in**terest is **in** the ®eld near the **in**clusion and **crack**, it can reasonably be accepted as agood model.Hav**in**g 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 gra**in**boundary (Fig. 1(a)) could coalesce **in**to a **crack** nucleus. Besides ZenerÕs mechanism of micro**crack** **in**itiation,there are some other variants. One was presented by Cottrell (1958) as shown **in** Fig. 1(b), wherepiled-up dislocations on two **in**tersect**in**g slip planes can coalesce **in**to a micro**crack**. The other variantproposed by Kikuchi et al. (1981) is given **in** Fig. 1(c) which shows a particle at a gra**in** 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, leav**in**g stationary dislocations of the oppositesign beh**in**d 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. For**in**stance, 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 antisymmetric**crack** plane traction stress, which arises from a symmetric **crack** plane dislocation distribution.Stress, displacement, dislocation density and stress **in**tensity factor (SIF) were compared between Grith**crack** and Zener±Stroh **crack** **in** a homogeneous **material** by Weertman (1986). Due to the displacementload**in**g mechanism, the total sum of the Burgers vectors of the dislocations b T with**in** a Zener±Stroh **crack**Fig. 1. (a) ZenerÕs mechanism of **crack** **in**itiation. (b) CottrelÕs model of **crack** **in**itiation. (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 **in**itiated from the sharp tip. Thisk**in**d of micro**crack** was often observed **in** metal matrix **composite** **material**s.2. The physical problem and formulationThe physical problem to be **in**vestigated **in** the current paper is shown **in** Fig. 2(a), where a Zener±Stroh**crack** is **in**itiated near a circular **in**clusion (a ®ber) **in** a ®ber-re**in**forced **composite**. For the case shown **in**Fig. 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 surround**in**g **in**clusions (other ®bers) on the **crack**, the three-phase**composite** model discussed **in** the previous section is employed, so the orig**in**al problem is made to beequivalent to Fig. 2(b), where the two-dimensional version of the three-phase model consists of threeconcentrically cyl**in**drical layers with the outer one be**in**g extended to **in**®nity. The **in**ner region r 6 a (Phase1) is the ®ber **in**clusion 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 **in**termediate 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 **in**ternal and external radii of the **in**termediatematrix-phase a and b be**in**g 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 **in**volved **in** discussion, weconcentrate on the **in**¯uence of the net dislocations **in**side the **crack** on the SIF at **crack** tip, so it is assumedthat the elastic system is free of external mechanical load**in**g. It is worth to mention that the problem can besolved similarly even with external mechanical load**in**g.The boundary conditions of the problem are:In the far ®eld:r xy ˆ 0;…1a†r yy ˆ 0:…1b†At the **in**terface:‰r xy Šˆ0; …2a†‰r yy Šˆ0: …2b†Fig. 2. A Zener±Stroh **crack** **in** a ®ber-re**in**forced **composite** **material** and the three-phase cyl**in**der 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 glid**in**g and climb**in**g dislocation densities along the **crack** l**in**e …t 1 ; t 2 †. Us**in**g thestress ®eld solution of a s**in**gle 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 **in**side the Zener±Stroh **crack**. Eqs (5a) and (5b) are the standard s**in**gular 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 follow**in**g 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±606**in** 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 obta**in**ed from Eqs. (4a) and (4b). Thus, the SIF of the **crack**can be evaluated accord**in**gly.3. Solution procedures of the **in**tegral equationsTo shift the **in**tegral **in**terval from …t 1 ; t 2 † to …1; 1† **in** the **in**tegral 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†S**in**ce the whole **crack** is located **in** Phase 2 (the pure matrix **material**), the s**in**gularities on the both **crack**tips should be 1=2. As a result, the two dislocation density functions B x …x† and B y …x† must have 1=2s**in**gular 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†Follow**in**g 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 l**in**ear algebraic equations to determ**in**e F y …s 1 †; ...; F y …s n †; while(18b) and (18d) provide another system of n l**in**ear algebraic equations to determ**in**e 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 obta**in**ed, 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 **material**s, respectively. As the **in**teraction between a **crack**and a s**in**gle **in**clusion has been studied by many researchers as discussed **in** Section 1, **in** our current examplecalculations, we concentrate on study**in**g the **in**¯uence of the volume fraction of **in**clusions on thebehavior of **crack**.Two typical types of **composite** **material**s 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 certa**in** voids(to represent a **composite** with ``soft'' **in**clusions), while the second one is a real ®ber-re**in**forced **material** (a**composite** with ``harder'' **in**clusions). 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†Correspond**in**g to di€erent volume fraction of **in**clusions c ˆ…a=b† 2 , the e€ective moduli of the **composite**(Phase 3) is evaluated accord**in**g 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 **in**clusions **in** both types of **material**s 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 forgo**in**g numerical discussion, we chie¯yconcentrated ourselves on study**in**g 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-re**in**forced **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-re**in**forced **composite** **material** (Case I).parison, the results calculated based on the two-phase model (a **crack** **in**teract**in**g with a s**in**gle **in**clusion) 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 the**in**clusion concentration **in**creases, the SIF **in**crease 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-re**in**forced **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 **in**terchange 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 **in**terest**in**g 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 the**in**clusions **in**creases, not only the elastic properties of the **composite** phase (Phase 3) changes, but also thedistance between the **in**clusion 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 **material**s (i.e., Phase 3 is not the **composite** phase, butthe third k**in**d of **material**). It is of **in**terest 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-re**in**forced **composite** **material** (Case II).Fig. 10. Normalized Mode II SIF at the sharp tip versus c **in** a ®ber-re**in**forced **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 follow**in**g 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 **material**s.Fig. 12. Normalized Mode II SIF at the sharp tip versus l 3 =l 2 (Case I) **in** a solid with three disimilar **material**s.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 follow**in**gparameters: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 **in**tersect at l 3 =l 2 ˆ 1. As l 3 =l 2 **in**creases **in**®nitely, the solution of thethree-phase model has an asymptotically value. In Figs. 13 and 14, it is observed that the thickness of the**in**termediate 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 **in**termediate 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 **material**s.Fig. 14. Normalized Mode II SIF at the sharp tip versus b=a (Case I) **in** a solid with three disimilar **material**s.ReferencesChristenson, R.M., Lo, K.H., 1979. Solution for e€ective shear properties **in** three-phase sphere and cyl**in**der 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 s**in**gular **in**tegral equations. Quarterly of Applied Mathematics 30, 525±534.Erdogan, F., Gupta, G.D., Ratwani, M., 1974. Interaction between a circular **in**clusion and an arbitrarily oriented **crack**. ASMEJournal of Applied Mechanics 41, 1007±1013.Hsu, Y.C., Shivakumar, V., 1976. Interaction between an elastic circular **in**clusion and two symmetrically placed coll**in**ear **crack**s.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** cyl**in**der model. ASME Journal of Applied Mechanics 58,75±86.Luo, H.A., Weng, G.J., 1987. On Eshelby's **in**clusion 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 cyl**in**drically concentric solids, and the elastic moduli of ®berre**in**forced**composite**s. Mechanics of Materials 8, 77±88.Nisitani, H., Chen, D.H., Saimoto, A., 1996. Interaction between an elliptic **in**clusion and a **crack**. In: Proceed**in**gs of the InternationalConference on Computer-Aided Assessment and Control, vol. 4, Computational Mechanics, Billerica, MA, USA, pp. 325±332.

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