Crack deflection/penetration criteria for fiber-reinforced ceramic matrix composites 1777 demonstrate the insensitivity of the deflection criterion substrate cracking. Their result showed that thermal on B. At finite ad, ap and Vf, we find that the ratio of Gal expansion misfit can be significant in systems with pla G, decreases well below the HH limit for a>0, imply nar interfaces such as layered materials and thin film ing that deflection is more difficult than previously structures, but in fiber-reinforced composites the effect anticipated in this regime. In fact, we find Ga/Gp-0 as of misfit is expected minimal because of the coupling a-1 for fixed fiber and matrix Poisson,s ratios. between axial and radial residual stresses. Of some Relaxing the assumption of ad=ap we find that, at importance and relevance to the present work, HEH V=1%, the ratio Ga/Gp depends weakly on the ratio demonstrated that when residual stresses are present of aa ap with a scaling close to the asymptotic behavior the ratio of Ga/Gp is always dependent on ad and ap; the (ad/ap) predicted from eqn(2), while at a more rea- convenient cancellation obtained in the absence of resi- listic value of V=40% the dependence is even weaker. dual stresses does not occur. Thus, the deflection criter- Finally, we compare various predictions of deflection ion is an explicit function of both ad and ap. In the and penetration to experimental results on model com- present work we show that, for realistic volume frac- posites with well-established constitutive properties For tions and small to moderate crack extensions, the ratio these systems, the hh result predicts crack deflection Ga Gp depends on the crack extension even in the whereas the present results predict penetration over a absence of residual stresses. Furthermore, these finite wide range of crack extension lengths. In the experi length effects are probably larger than the effects ments, the real materials exhibit clear penetration, indi- obtained from realistic residual stress levels obtainable cating that the hh criterion overestimates the prospects in ceramic composites, particularly for the axisymmetric for crack deflection. The present results are consistent geometr with the data, indicating that finite volume fraction and The remainder of this paper is organized as follows finite ad and ap may play a key role in determining In Section 2, we define the specific problem to be solved deflection versus penetration. Ambiguity in the correct and briefly introduce the basic concepts of Pagano's choice for ad=ap, however, precludes any ability to Axisymmetric Damage Model and Reissner's varia- draw more definite conclusions on this issue until fur- tional principal. Section 3 contains detailed results on ther work is done and possibly new ideas are developed. deflection versus penetration. Comparison of the var Before proceeding with the present work, we close the ious deflection criteria with new experimental data is introduction with a brief presentation of recent impor- presented in Section 4. In Section 5, we summarize and tant work on this problem. Gupta et al. extended discuss directions for future work and Hutchinson,s work to the area of anisotropic materials for the case of a crack approaching perpend cular to the interface. They also derived a strength cri 2 MODELING OF FIBER/MATRIX INTERFACE terion for crack deflection, and confirmed their analysis CrACKs by using laser spallation experiment. Gupta et al.con- cluded that it is impossible to provide generalized dela- 2.1 Problem description mination charts as a function of a alone. Instead, they In order to predict the crack path at the fiber/matrix have tabulated the required values of the interface nterface, an axisymmetric microcomposite consisting of strength and fracture toughness for delamination in a isotropic, elastic fiber and matrix is considered. Fig. 2 number of composite materials. A later work by Marti- shows the fiber /matrix interface model with the appro nez and Gupta(MG) showed in detail that the energy priate boundary conditions. The main matrix crack lies criterion for deflection is sensitive to the material ani- in a plane perpendicular to the fiber and extending to sotropy and has corrected the previous work by HH. In the interface, where it may subsequently deflect along contrast to He and Hutchinson,s results, their calcula- the interface or penetrate into the fiber. Energy release tions show that the GaGp for the doubly deflected crack rates for deflection(Ga)and penetration(Gp) are calcu- higher than Ga Gp for singly deflected crack when lated and the ratio of Ga/Gp as a function of Dundurs Dundurs' parameter a is larger than zero. That is, the parameters a and B is determined. Detailed description doubly deflected crack is the dominating crack mode. on the boundary conditions in Fig. 2 is presented later Martinez and Gupta also examined the effect of aniso- in this section. tropy on the crack deflection by manipulating the ani To obtain energy release rates requires three calcula sotropy-related parameters including the other tions. First, with only the initial matrix crack extending Dundurs' parameter, B. They showed that B fiber /matrix interface, we calculate the elastic assumption may overestimate the Ga G, versus a beha- potential energy in the system at a fixed remote applied vior by 20-25% over the range B=-0-202 load. This gives the initial reference energy Wr. Second Following the work of Martinez and Gupta, He we advance the crack into the fiber by an amount ap and al.(HEH)provided a corrected result for the ratio G calculate the potential energy for this longer, penetrated Gp for the doubly deflected crack. More importantly, crack, Wp. Third, we advance the initial crack at a righ HEH investigated the influence of the residual stre angle along the interface by an equal amount ad to form n the competition between interface cracking the incipient deflected crack, and calculate the potentialdemonstrate the insensitivity of the de¯ection criterion on . At ®nite ad, ap and Vf, we ®nd that the ratio of Gd/ Gp decreases well below the HH limit for  > 0, imply￾ing that de¯ection is more dicult than previously anticipated in this regime. In fact, we ®nd Gd=Gp ! 0 as  ! 1 for ®xed ®ber and matrix Poisson's ratios. Relaxing the assumption of ad ˆ ap we ®nd that, at Vf ˆ 1%, the ratio Gd/Gp depends weakly on the ratio of ad/ap with a scaling close to the asymptotic behavior …ad=ap† 1ÿ2l predicted from eqn (2), while at a more rea￾listic value of Vf ˆ 40% the dependence is even weaker. Finally, we compare various predictions of de¯ection and penetration to experimental results on model com￾posites with well-established constitutive properties. For these systems, the HH result predicts crack de¯ection whereas the present results predict penetration over a wide range of crack extension lengths. In the experi￾ments, the real materials exhibit clear penetration, indi￾cating that the HH criterion overestimates the prospects for crack de¯ection. The present results are consistent with the data, indicating that ®nite volume fraction and ®nite ad and ap may play a key role in determining de¯ection versus penetration. Ambiguity in the `correct' choice for ad ˆ ap, however, precludes any ability to draw more de®nite conclusions on this issue until fur￾ther work is done and possibly new ideas are developed. Before proceeding with the present work, we close the introduction with a brief presentation of recent impor￾tant work on this problem. Gupta et al.3 extended He and Hutchinson's work2 to the area of anisotropic materials for the case of a crack approaching perpendi￾cular to the interface. They also derived a strength cri￾terion for crack de¯ection, and con®rmed their analysis by using laser spallation experiment. Gupta et al. con￾cluded that it is impossible to provide generalized dela￾mination charts as a function of  alone. Instead, they have tabulated the required values of the interface strength and fracture toughness for delamination in a number of composite materials. A later work by Marti￾nez and Gupta (MG)4 showed in detail that the energy criterion for de¯ection is sensitive to the material ani￾sotropy and has corrected the previous work by HH. In contrast to He and Hutchinson's results, their calcula￾tions show that the Gd/Gp for the doubly de¯ected crack is higher than Gd/Gp for singly de¯ected crack when Dundurs' parameter  is larger than zero. That is, the doubly de¯ected crack is the dominating crack mode. Martinez and Gupta also examined the e€ect of aniso￾tropy on the crack de¯ection by manipulating the ani￾sotropy-related parameters including the other Dundurs' parameter, . They showed that  ˆ 0 assumption may overestimate the Gd/Gp versus  beha￾vior by 20ÿ25% over the range =ÿ0.20.2. Following the work of Martinez and Gupta,4 He et al.5 (HEH) provided a corrected result for the ratio Gd/ Gp for the doubly de¯ected crack. More importantly, HEH investigated the in¯uence of the residual stresses on the competition between interface cracking and substrate cracking. Their result showed that thermal expansion mis®t can be signi®cant in systems with pla￾nar interfaces such as layered materials and thin ®lm structures, but in ®ber-reinforced composites the e€ect of mis®t is expected minimal because of the coupling between axial and radial residual stresses. Of some importance and relevance to the present work, HEH demonstrated that when residual stresses are present, the ratio of Gd/Gp is always dependent on ad and ap; the convenient cancellation obtained in the absence of resi￾dual stresses does not occur. Thus, the de¯ection criter￾ion is an explicit function of both ad and ap. In the present work we show that, for realistic volume frac￾tions and small to moderate crack extensions, the ratio Gd/Gp depends on the crack extension even in the absence of residual stresses. Furthermore, these ®nite length e€ects are probably larger than the e€ects obtained from realistic residual stress levels obtainable in ceramic composites, particularly for the axisymmetric geometry. The remainder of this paper is organized as follows. In Section 2, we de®ne the speci®c problem to be solved and brie¯y introduce the basic concepts of Pagano's Axisymmetric Damage Model and Reissner's varia￾tional principal. Section 3 contains detailed results on de¯ection versus penetration. Comparison of the var￾ious de¯ection criteria with new experimental data is presented in Section 4. In Section 5, we summarize and discuss directions for future work. 2 MODELING OF FIBER/MATRIX INTERFACE CRACKS 2.1 Problem description In order to predict the crack path at the ®ber/matrix interface, an axisymmetric microcomposite consisting of isotropic, elastic ®ber and matrix is considered. Fig. 2 shows the ®ber/matrix interface model with the appro￾priate boundary conditions. The main matrix crack lies in a plane perpendicular to the ®ber and extending to the interface, where it may subsequently de¯ect along the interface or penetrate into the ®ber. Energy release rates for de¯ection (Gd) and penetration (Gp) are calcu￾lated and the ratio of Gd/Gp as a function of Dundurs parameters  and  is determined. Detailed description on the boundary conditions in Fig. 2 is presented later in this section. To obtain energy release rates requires three calcula￾tions. First, with only the initial matrix crack extending to the ®ber/matrix interface, we calculate the elastic potential energy in the system at a ®xed remote applied load. This gives the initial reference energy Wr. Second, we advance the crack into the ®ber by an amount ap and calculate the potential energy for this longer, penetrated crack, Wp. Third, we advance the initial crack at a right angle along the interface by an equal amount ad to form the incipient de¯ected crack, and calculate the potential Crack de¯ection/penetration criteria for ®ber-reinforced ceramic matrix composites 1777