Crack deflection/penetration criteria for fiber-reinforced ceramic matrix composites 1781 nergy released due to relaxation of the tensile fiber fiber volume fraction the effect of B becomes essentially stress and no energy released by the matrix. For a negligible. These results are in keeping with the general penetrating crack, there is certainly energy released in results of the previous work the fiber. Therefore, Ga should approach zero while Gp Finally, since we have all of the individual values for is still finite, and hence Ga G, approaches zero. Another Ga and Gp at the various crack extension lengths, we can way to consider this limit is that at a= l there is no real consider(to a limited extent) the variations in Ga/, for matrix material and so the 'interface is already essen- the situation adap. Figure 8(a)shows the ratio Ga/Gp tially a free boundary and so a 'crack along this versus a for V=1% and ap=0-0lry with variable boundary makes no change to the stress field and no ad=0-002r/, 0-01 ry and 0-025 r. The variations in G change to the energy. The behavior for a- I is easier Gp are generally as expected from the analytic results of to observe in a composite of a large fiber volume frac- eqn(2): as aa increases, the ratio and hence the tendency tion(small matrix volume fraction) because the total for deflection increase for 2< 1/2(a>0)but they strain energy in the matrix is smaller than for a larger decrease for i>1/2(a<0. In fact, the analyti matrix volume fraction. Although this limit is not gen- dependence of Ga/G, a(ad/a,)-closely describes the erally attained in CMCs it could have implications for variations found here over the modest range of aa/ap matrix crack growth in polymer matrix composites studied ( versus a, B is given in Ref. 2). Figure 8(b) (PMCS), for which a can approach unity shows the ratio Ga/Gp versus a for V=40% and We now return to consider the effect of the Dundurs ap=0-0lry with variable ad=0./, 0-01 rr and 0-025 parameter A. As noted earlier, the original HH work ro. In this case, the dependence on ad/, is rather smaller and HEH work correspond to B=0 and Martinez and than at the lower volume fraction and the variations at Gupta showed relatively small changes for B#0 over a larger a are somewhat weaker than the asymptotic narrow range. Here we explicitly consider these differ- behavior(aalap)-. In general, for the higher vol ences within our computational model. To fix B=0 fractions, the equal case ad=ap can be used with rea requires choosing certain special combinations of the sonable accuracy to approximate modest variations in Poissons ratios of the fiber and matrix. To demonstrate aalap the b effect we studied several values of a: a=0. 333 0-5,0-667 and 0-75, and fixed Em. By varying E, vand Vm at the same time using math Cadm, we found sets of(a)1.2 underlying material parameters for which B=0, as shown in Table 1. Note that to obtain B=0 requires 10 that one of the poissons ratios must become rather Although the values in Table I are not unique, we have B% large, and in the limit of a= l one must have Vm=0.5. found that the results for Ga/Gp at fixed a and B=0 as0.6 obtained from different sets of the remaining material properties Es vy and vm are very close. The values in Table I thus suffice for investigating some effects of B Figure 7(a) shows our results for GaGp with V/=1% At the smallest crack extension of ad=p=a=0.00 our predictions with B=0 are now in extremely good 00806-04020002040.6081.0 agreement with the heh results over the full range of a, although the differences with our previous results are b)1.2 quite small. At larger crack extensions, the difference B=0 and B+0(fixed vr and vm) increases slightly with increasing a, with the ratio GaGp increas- ing for the b=0 case. Thus, deflection is slightly more difficult under the realistic conditions of fixed Poissons ratios rather than fixed Figure 7(b) shows our results for V=40%, and it appears that with the high Table 1. Combinations of material properties used to obtained 2-27 0-111 0-500 0-179 0-667 0.750 0.473 Fig. 7(a)GaGp versus a with p=0 assumption for V= b)GaGp versus a with B=0 assumption for V=40%energy released due to relaxation of the tensile ®ber stress and no energy released by the matrix. For a penetrating crack, there is certainly energy released in the ®ber. Therefore, Gd should approach zero while Gp is still ®nite, and hence Gd/Gp approaches zero. Another way to consider this limit is that at  ˆ 1 there is no real matrix material and so the `interface' is already essen￾tially a free boundary and so a `crack' along this boundary makes no change to the stress ®eld and no change to the energy. The behavior for  ! 1 is easier to observe in a composite of a large ®ber volume frac￾tion (small matrix volume fraction) because the total strain energy in the matrix is smaller than for a larger matrix volume fraction. Although this limit is not gen￾erally attained in CMCs it could have implications for matrix crack growth in polymer matrix composites (PMCs), for which  can approach unity. We now return to consider the e€ect of the Dundurs parameter . As noted earlier, the original HH work and HEH work correspond to  ˆ 0 and Martinez and Gupta showed relatively small changes for  6ˆ 0 over a narrow range. Here we explicitly consider these di€er￾ences within our computational model. To ®x  ˆ 0 requires choosing certain special combinations of the Poisson's ratios of the ®ber and matrix. To demonstrate the  e€ect we studied several values of :  ˆ 0
333, 0.5, 0.667 and 0.75, and ®xed Em. By varying Ef, vf and vm at the same time using MathCadTM, we found sets of underlying material parameters for which  ˆ 0, as shown in Table 1. Note that to obtain  ˆ 0 requires that one of the Poisson's ratios must become rather large, and in the limit of  ˆ 1 one must have vm ˆ 0
5. Although the values in Table 1 are not unique, we have found that the results for Gd/Gp at ®xed  and  ˆ 0 as obtained from di€erent sets of the remaining material properties Ef, vf and vm are very close. The values in Table 1 thus suce for investigating some e€ects of . Figure 7(a) shows our results for Gd/Gp with Vf ˆ 1%. At the smallest crack extension of ad ˆ ap ˆ a ˆ 0
002rf, our predictions with  ˆ 0 are now in extremely good agreement with the HEH results over the full range of , although the di€erences with our previous results are quite small. At larger crack extensions, the di€erence between  ˆ 0 and  6ˆ 0 (®xed vf and vm) increases slightly with increasing a, with the ratio Gd/Gp increas￾ing for the  ˆ 0 case. Thus, de¯ection is slightly more dicult under the realistic conditions of ®xed Poisson's ratios rather than ®xed  ˆ 0. Figure 7(b) shows our results for Vf ˆ 40%, and it appears that with the high ®ber volume fraction the e€ect of  becomes essentially negligible. These results are in keeping with the general results of the previous work.2345 Finally, since we have all of the individual values for Gd and Gp at the various crack extension lengths, we can consider (to a limited extent) the variations in Gd/Gp for the situation ad 6ˆ ap. Figure 8(a) shows the ratio Gd/Gp versus  for Vf ˆ 1% and ap ˆ 0
01rf with variable ad ˆ 0
002rf, 0.01 rf and 0.025 rf. The variations in Gd/ Gp are generally as expected from the analytic results of eqn (2): as ad increases, the ratio and hence the tendency for de¯ection increase for l < 1=2… > 0† but they decrease for l > 1=2… < 0†. In fact, the analytic dependence of Gd=Gp / …ad=ap† 1ÿ2l closely describes the variations found here over the modest range of ad/ap studied (l versus ,  is given in Ref. 2). Figure 8(b) shows the ratio Gd/Gp versus  for Vf ˆ 40% and ap ˆ 0
01rf with variable ad ˆ 0
002rf, 0.01 rf and 0.025 rf. In this case, the dependence on ad/ap is rather smaller than at the lower volume fraction and the variations at larger  are somewhat weaker than the asymptotic behavior …ad=ap† 1ÿ2l . In general, for the higher volume fractions, the equal case ad=ap can be used with rea￾sonable accuracy to approximate modest variations in ad/ap. Fig. 7. (a) Gd/Gp versus  with  ˆ 0 assumption for Vf ˆ 1%; (b) Gd/Gp versus  with  ˆ 0 assumption for Vf ˆ 40%. Table 1. Combinations of material properties used to obtained  ˆ 0  Ef/Em Vf Vm 0.333 2.27 0.111 0.360 0.500 3.54 0.179 0.425 0.667 5.81 0.311 0.471 0.750 8.60 0.220 0.473 Crack de¯ection/penetration criteria for ®ber-reinforced ceramic matrix composites 1781