1776 B. K Ahn et al crack, and hence hold in the limit of ad, ap-0. For identical elastic properties, i=1/2 but for fibers stiffer than the matrix 2< 1/2 while for matrix stiffer than the fibers 1>1/2. Therefore, taking the limit of zero crack extension in either case leads to either (i) zero energy release rate (< 2)and hence no possible cracking or (ii) infinite energy release rate (i>1/2) ence cracking at any finite stress level To overcome the basic difficulties evident from the above results, He and Hutchinson(HH) proposed a nice a to consider the ratio Ga/Gp with ad=ap, in which ae e concept that led to an analytic and finite result for assessing deflection versus penetration. HH propose the crack extension length drops out of the problem Singly deflected crack Doubly deflected crack Penetrated crack Furthermore, HH proposed that crack deflection would Fig 1. Three potential fracture modes at a fiber/matrix inter- occur if Gd/Gp> Td/Tp. For a penetrating crack at the fiber surface, Tp= r where T, is the critical energy release rate or surface energy of the fiber, and for a To predict crack growth thus requires an ability to cal- deflecting crack at the interface, Id= ri where Ti is the culate the energy release rate, G, or elastic energy surface energy of the interface normally lower relieved per unit area of crack advance, and a knowl- than Im of matrix. Hence, the deflection criterion at the edge of the underlying surface fracture energy, T, cre- fiber/matrix interface is ated as the crack grows. Here, we denote by Ga and rd the energy release rate and surface energy for the case of Ga/Gp <Ti/rf deflection, and by Gp and Tp the corresponding quan- tities for penetration. If Gd> rd the crack can deflect while if Gp>Tp the crack can penetrate the fiber. It is This criterion was then studied in considerable detail by not clear which path is selected if both conditions are He and Hutchinson under certain conditions. They stu satisfied, and in fact other fundamental problems arise died a planar interface under plane strain and traction for elastically-mismatched materials, as discussed briefly boundary conditions, with isotropic'matrix'and'fiber below Their analysis implicitly assumed that the crack size in For a bi-material interface under plane strain condi- the 'matrix'is semi-infinite and, as noted above, the tions, the stresses in the system depend on two basic crack extensions are considered infinitesimal. The spe material combinations the dundurs parameters' cial case of B=0 was studied although limited result suggested that the deflection criterion was only weakly affected by the value of B relative to its dependence on E(1-m)-Em(1-v a. Singly and doubly deflected interface cracks were EA1-12+Em( considered within the limitations of plane strain. HH E/(l,m(1-2vm)-Em(1+v(1-2y () also considered cracks approaching the interface at E(1-m)+Em(1-v) oblique angles. The result of hH for the perpendicular, doubly deflected crack is shown as a solid line in Fig. 5. (The result shown here is adopted from Ref. 5 which provides the corrected result of the original work by HH For a crack perpendicular to the interface and under in Ref. 2.) applied load parallel to the interface, the energy release In the present paper, we adopt the hh deflection cri- rates as a function of crack extension ad along the terion based on energy and a ratio of energy release interface and a, into the fiber are well-known to be of rates. We then investigate, using a numerical technique the forms2 developed by Pagano that employs Reissners varia- tional principle, the dependence of the deflection criter- ion on crack extension lengths ad, ap and on fiber (2) volume fraction V for an axisymmetric fiber/matrix interface geometry. We restrict the problem to a per- pendicular matrix crack impinging onto the interface In eqn(2), kr is a Mode I stress-intensity-like factor, d and to the doubly-deflected crack case shown previously and c are complex functions of the Dundurs' para- to be the dominant fracture mode. In the limits of small neters, and the exponent i is also a function of the ad, ap and small V accessible numerically, we reproduce elastic mismatch between fiber and matrix. The above the corrected hh results which also validates the use of forms arise from the asymptotic near-tip field of the the relatively new numerical technique. We alsoTo predict crack growth thus requires an ability to cal￾culate the energy release rate, G, or elastic energy relieved per unit area of crack advance, and a knowl￾edge of the underlying surface fracture energy, ÿ, cre￾ated as the crack grows. Here, we denote by Gd and ÿd the energy release rate and surface energy for the case of de¯ection, and by Gp and ÿp the corresponding quan￾tities for penetration. If Gd  ÿd the crack can de¯ect while if Gp  ÿp the crack can penetrate the ®ber. It is not clear which path is selected if both conditions are satis®ed, and in fact other fundamental problems arise for elastically-mismatched materials, as discussed brie¯y below. For a bi-material interface under plane strain condi￾tions, the stresses in the system depend on two basic material combinations, the Dundurs parameters6  ˆ Ef…1 ÿ v2 m† ÿ Em…1 ÿ v2 f † Ef…1 ÿ v2 m ‡ Em…1 ÿ v2 f † ; 2 ˆ Ef…1vm†…1 ÿ 2vm† ÿ Em…1 ‡ vf†…1 ÿ 2vf† Ef…1 ÿ v2 m† ‡ Em…1 ÿ v2 f † …1† For a crack perpendicular to the interface and under applied load parallel to the interface, the energy release rates as a function of crack extension ad along the interface and ap into the ®ber are well-known to be of the forms2 Gd ˆ d…; †k2 Ia1ÿ2l d ; Gp ˆ c…; †kI 1a1ÿ2l p …2† In eqn (2), kI is a Mode I stress-intensity-like factor, d and c are complex functions of the Dundurs' para￾meters, and the exponent l is also a function of the elastic mismatch between ®ber and matrix. The above forms arise from the asymptotic near-tip ®eld of the crack, and hence hold in the limit of ad; ap ! 0. For identical elastic properties, l ˆ 1=2 but for ®bers sti€er than the matrix l < 1=2 while for matrix sti€er than the ®bers l > 1=2. Therefore, taking the limit of zero crack extension in either case leads to either (i) zero energy release rate …l < 1 =2† and hence no possible cracking or (ii) in®nite energy release rate …l > 1=2† and hence cracking at any ®nite stress level. To overcome the basic diculties evident from the above results, He and Hutchinson (HH) proposed a nice concept that led to an analytic and ®nite result for assessing de¯ection versus penetration.2 HH proposed to consider the ratio Gd=Gp with ad ˆ ap, in which case the crack extension length drops out of the problem. Furthermore, HH proposed that crack de¯ection would occur if Gd=Gp > ÿd=ÿp. For a penetrating crack at the ®ber surface, ÿp ˆ ÿf where ÿf is the critical energy release rate or surface energy of the ®ber, and for a de¯ecting crack at the interface, ÿd ˆ ÿi where ÿi is the surface energy of the interface which is normally lower than ÿm of matrix. Hence, the de¯ection criterion at the ®ber/matrix interface is Gd=Gp < ÿi=ÿf …3† This criterion was then studied in considerable detail by He and Hutchinson under certain conditions. They stu￾died a planar interface under plane strain and traction boundary conditions, with isotropic `matrix' and `®ber'. Their analysis implicitly assumed that the crack size in the `matrix' is semi-in®nite and, as noted above, the crack extensions are considered in®nitesimal. The spe￾cial case of  ˆ 0 was studied, although limited results suggested that the de¯ection criterion was only weakly a€ected by the value of  relative to its dependence on . Singly and doubly de¯ected interface cracks were considered within the limitations of plane strain. HH also considered cracks approaching the interface at oblique angles. The result of HH for the perpendicular, doubly de¯ected crack is shown as a solid line in Fig. 5. (The result shown here is adopted from Ref. 5 which provides the corrected result of the original work by HH in Ref. 2.) In the present paper, we adopt the HH de¯ection cri￾terion based on energy and a ratio of energy release rates. We then investigate, using a numerical technique developed by Pagano that employs Reissner's varia￾tional principle, the dependence of the de¯ection criter￾ion on crack extension lengths ad, ap and on ®ber volume fraction Vf for an axisymmetric ®ber/matrix interface geometry. We restrict the problem to a per￾pendicular matrix crack impinging onto the interface, and to the doubly-de¯ected crack case shown previously to be the dominant fracture mode. In the limits of small ad, ap and small Vf accessible numerically, we reproduce the corrected HH results, which also validates the use of the relatively new numerical technique. We also Fig. 1. Three potential fracture modes at a ®ber/matrix inter￾face. 1776 B. K. Ahn et al