Crack deflection/penetration criteria for fiber-reinforced ceramic matrix composites 1783 the HEh criterion would be expected to hold after explicit accounting for the marginally different lengths ad,ap at the atomic scale. While perhaps an appealing approach, the experimental data presented here would not be consistent with the HEH criterion. The length scales could be larger, and controlled by characteristic structural defects at larger length scales such as misfit dislocations or grain boundary orientation. The appar ently obvious candidates for ad and ap are pre-existing crack-like flaws along the interface and in the fibers Even if true. however. such flaw sizes could not be determined experimentally. Although'typical flaw sizes in the constituent fibers can be assessed by measuring fiber strength, such flaws are the largest flaws existing Fig 10. Fracture surface of an SCS-0/F-glass composite along a macroscopic length of fiber; they do not reflect the typical flaws present in the fiber at much smaller length scales. Measuring flaw sizes in the interface is because of the uncertainty in the physical meaning and even more difficult and suffers from similar interpreta values of ad and ap, the present predictions are not defi- tion problems. Furthermore, as a matrix crack approa nitive but merely illustrative of the deleterious effects ches a flawed fiber and/or interface, the problem to associated with finite crack extensions solve is quite different than the approach taken here and in the previous work, 2-5 where there is only one crack A first approach to this problem is to study the energy 5 SUMMARY release rates for each of two coexisting crack tips branching off from the main matrix crack. with one We have developed an energy-based criterion for pre- branch along the interface and one along the fiber dicting crack deflection versus penetration as a function While we will pursue this in the near future, the solution of the fiber and matrix elastic mismatch, fiber volume of such a problem still does not complete the space of fraction, and crack extension length for an axisym- possible cracking modes. For instance, macroscopic metric geometry Using a new numerical technique, we evidence of interfacial crack growth ahead of an have reproduced quite accurately the analytic HEH impinging matrix crack has been presented recently by results which apply in the limit of both Vr and ad=ap Lee et al. 3 and analogous behavior could occur at the approaching zero. For finite volume fractions and finite fiber/matrix interface on scales orders of magnitude but small, crack extension lengths, we find that the ten- smaller. This would suggest the applicability of a dency for crack penetration is enhanced, i.e. the ratio strength-based criterion using the stress field due to the Ga/Gp decreases as Vrand ad and ap increase. In the limit matrix crack acting on pre-existing flaws ad,a of a-1 we predict Ga/Gp=0 so that deflection although still presuming the existence of flaws becomes impossible. We also find a weak dependence on The issue of the physical nature of ad ap is clearly the Dundurs' parameter B and a weak dependence on critical, and the prospects for general engineering design the ratio ad ap at realistic volume fractions. Experiments rules which involve only macroscopic constitutive n model composite systems with convenient inter- properties, such as moduli and toughness, hinges on this mediate ratios of Ti/T show crack penetration, con- issue. If ad and ap are flaw-related, then composite per- sistent with the present predictions over a range of crack formance will ultimately be controlled by processing extensions and in contrast to the prediction of deflection details rather than constitutive material properties using the HEH criterion. Finite values of V and ad, ap which does not bode well for the application of general can thus play an important role in determining the ten- design guidelines. The present results begin to demon dency for interfacial crack deflection, which in turn then strate, within the existing energy-based framework of controls the toughness of the entire composite. The pre- deflection versus penetration, the sensitivity of the engi sent results provide a guideline for estimating deflection neering design curve(GaGp versus a)to the incipient or versus penetration for various elastic parameters, inter- notional crack extension lengths ad, ap in realistic(axi ace toughness, volume fraction, and crack extension symmetric and high volume fraction) fiber-reinforced The major unsolved issue in this entire field remains ceramics. the interpretation of the crack extension lengths ad and ap. Taken separately e en ease unphysical in the limit of ad, ap -0 and so there must ACKNOWLEDGEMENTS be some intrinsic length scales at which the continuum limits apply. These length scales could be atomic, in the The authors gratefully acknowledge the support for this case of perfect interfaces, in which case ad, ap <<ry and work provided by the US Air Force Office of Scientificbecause of the uncertainty in the physical meaning and values of ad and ap, the present predictions are not de®- nitive but merely illustrative of the deleterious e€ects associated with ®nite crack extensions. 5 SUMMARY We have developed an energy-based criterion for pre￾dicting crack de¯ection versus penetration as a function of the ®ber and matrix elastic mismatch, ®ber volume fraction, and crack extension length for an axisym￾metric geometry. Using a new numerical technique, we have reproduced quite accurately the analytic HEH results which apply in the limit of both Vf and ad ˆ ap approaching zero. For ®nite volume fractions and ®nite, but small, crack extension lengths, we ®nd that the ten￾dency for crack penetration is enhanced, i.e. the ratio Gd/Gp decreases as Vf and ad and ap increase. In the limit of  ! 1 we predict Gd=Gp ˆ 0 so that de¯ection becomes impossible. We also ®nd a weak dependence on the Dundurs' parameter  and a weak dependence on the ratio ad/ap at realistic volume fractions. Experiments on model composite systems with convenient inter￾mediate ratios of ÿi=ÿf show crack penetration, con￾sistent with the present predictions over a range of crack extensions and in contrast to the prediction of de¯ection using the HEH criterion. Finite values of Vf and ad, ap can thus play an important role in determining the ten￾dency for interfacial crack de¯ection, which in turn then controls the toughness of the entire composite. The pre￾sent results provide a guideline for estimating de¯ection versus penetration for various elastic parameters, inter￾face toughness, volume fraction, and crack extension. The major unsolved issue in this entire ®eld remains the interpretation of the crack extension lengths ad and ap. Taken separately, the energy release rates are unphysical in the limit of ad; ap ! 0 and so there must be some intrinsic length scales at which the continuum limits apply. These length scales could be atomic, in the case of perfect interfaces, in which case ad; ap << rf and the HEH criterion would be expected to hold after explicit accounting for the marginally di€erent lengths ad, ap at the atomic scale. While perhaps an appealing approach, the experimental data presented here would not be consistent with the HEH criterion. The length scales could be larger, and controlled by characteristic structural defects at larger length scales such as mis®t dislocations or grain boundary orientation. The appar￾ently obvious candidates for ad and ap are pre-existing crack-like ¯aws along the interface and in the ®bers. Even if true, however, such ¯aw sizes could not be determined experimentally. Although `typical' ¯aw sizes in the constituent ®bers can be assessed by measuring ®ber strength, such ¯aws are the largest ¯aws existing along a macroscopic length of ®ber; they do not re¯ect the typical ¯aws present in the ®ber at much smaller length scales. Measuring ¯aw sizes in the interface is even more dicult and su€ers from similar interpreta￾tion problems. Furthermore, as a matrix crack approa￾ches a ¯awed ®ber and/or interface, the problem to solve is quite di€erent than the approach taken here and in the previous work,2±5 where there is only one crack. A ®rst approach to this problem is to study the energy release rates for each of two coexisting crack tips branching o€ from the main matrix crack, with one branch along the interface and one along the ®ber. While we will pursue this in the near future, the solution of such a problem still does not complete the space of possible cracking modes. For instance, macroscopic evidence of interfacial crack growth ahead of an impinging matrix crack has been presented recently by Lee et al.13 and analogous behavior could occur at the ®ber/matrix interface on scales orders of magnitude smaller. This would suggest the applicability of a strength-based criterion using the stress ®eld due to the matrix crack acting on pre-existing ¯aws ad, ap, although still presuming the existence of ¯aws. The issue of the physical nature of ad, ap is clearly critical, and the prospects for general engineering design rules which involve only macroscopic constitutive properties, such as moduli and toughness, hinges on this issue. If ad and ap are ¯aw-related, then composite per￾formance will ultimately be controlled by processing details rather than constitutive material properties, which does not bode well for the application of general design guidelines. The present results begin to demon￾strate, within the existing energy-based framework of de¯ection versus penetration, the sensitivity of the engi￾neering design curve (Gd/Gp versus ) to the incipient or notional crack extension lengths ad, ap in realistic (axi￾symmetric and high volume fraction) ®ber-reinforced ceramics. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support for this work provided by the US Air Force Oce of Scienti®c Fig. 10. Fracture surface of an SCS-0/F-glass composite. Crack de¯ection/penetration criteria for ®ber-reinforced ceramic matrix composites 1783