1780 B. K Ahn et al conditions. far-field stress in the axial direction is applied at the end of the composite (==L, but equivalent results for GaGp are obtained using a pres- sure load on the matrix crack surface. Readers inter- ested in the details of the ADM model are encouraged dispacement B to read the papers by Pagano.8-II 3 RESULTS AND DISCUSSION “高 rot To validate our numerical solutions, we first investigate the small VA small ad=ap limits and compare our 00.8060.40200020.40.60.810 results to the well-known analytical solution given by He and Hutchinson. Since the original work by he and Hutchinson(HH)has been corrected by Martinez and Fig. 6. GaGp versus a with various crack extensions for Gupta(MG), and later by He et al. (HEH), we com- pare to the result from HEH. Figure 5 shows our results for the energy release rate ratio, Ga/Gp, as a function of At this point, it is relevant to point out that the Dundurs' elastic mismatch parameter a along with the minimum crack extension of 0-002 r used here can HEH result In our calculations, we have fixed the fiber physically quite small. For the largest commonly used and matrix Poissons ratios at 0.2, and hence fibers, Textron SCS-6 SiC fibers, the radius is rf= 70um B=0.375a, rather than fixing B=0 as in HEH; we will and our minimum crack extension is 0.14 um. Fibers discuss the effects of B below. To simulate semi-infinite more typically used in most CMCs with commercial crack size, or equivalently zero' fiber volume fraction, potential have a much smaller diameter. The widel and the infinitesimal crack extension which are assumed used Nicalon SiC fiber has a radius of about 7 5um so in HEH, the results in Fig. 5 correspond to Vf=1% that our minimum crack extension is 0.015um or 15 nm and ad=ap=a=0. 002rf. The latter is the smallest Nextel alumina fibers are slightly smaller still, and many value permitted by the current ADM code. The agree- graphite fibers have radii around 3.5 um. The crack ment between our calculations and the heh analytic extensions used here are thus approaching the atomic result is excellent over the full range of a. This validates scale. Furthermore, the minimum crack extensions used the use of the ADM code for this particular problem, here are certainly much smaller than the typical flaw and also shows the equivalence of the plane-strain /pla- sizes in the fibers. Flaws in the interface are not expec nar-interface and the axisymmetric versions of this pro- ted to be much smaller. Thus, if the crack extension nterpreted as a flaw size in the fiber or interface to Figure 5 also shows Ga/Gp versus a for V= 1% but which the matrix crack can connect and propagate, then for increased values of the crack extension ad=ap=a the range of a used in this study is well within the range For modest increases in the crack extension, Ga/Gp of flaws in typical CMC systems. Since in many CMCs, drops substantially at higher a. At a=0.8 and the range of elastic mismatch a lies between a=0-025rf, Ga/Gp is only about 50% of the value at+ 8. the deviations from the heh result for a=0-002r. This result explicitly demonstrates the sen- Ga/Gp shown here can be significant in real material sitivity of the deflection criterion to the presumed crack systems extension. and moreover indicates that deflection is in Figure 6 also shows that Ga/ Gp is actually non-mono- the general case, more difficult than predicted by the tonic in a for finite crack extension and volume fraction HEH criterion At sufficiently high a, Ga Gp begins to decrease rapidly Figure 6 shows results for Ga Gp at V=40% and Furthermore, GaGp appears to drop to zero in the limit lite crack extensions ad=a,= a as used in Fig. 5. a=1.0 for all cases considered. This differs significantly Here, even at the smallest crack extension of 0-002 rr from the HEH result, for which GaGp appears to there is a reasonable difference when compared with the diverge as a approaches unity. The decrease in Ga/Gp at very low volume fraction result and again the ratio of very high a can be understood physically within the GalGp decreases relative to the HEH result. For larger framework of our calculations as follows. With the crack extensions, Ga G, decreases even further. At same Poissons ratios for both fiber and matrix, a is 0-8, Ga Gp for a=0-025ry is only 19% of HEH simply defined as(ErEm)/(Er+ Em) under the plane result and 35% of the result for a=0-002rg. In fact, for strain assumption. As a increases toward unity, the fiber the larger crack extensions the ratio is nearly constant become infinitely stiff relative to the matrix, i.e. over a wide range of a. While we expect that in the limit Er/Em+o0. Thus, all of the applied load is carried by of a-0 the ratio Ga/Gp would eventually increase up the fiber and all of the strain energy is stored in the to the HEh result, the crack extension must be much fiber; there is no strain energy in the matrix For a crack maller than 0-002 rf. which deflects along the interface, there is essentially noconditions, far-®eld stress in the axial direction is applied at the end of the composite …z ˆ L†, but equivalent results for Gd/Gp are obtained using a pres￾sure load on the matrix crack surface. Readers inter￾ested in the details of the ADM model are encouraged to read the papers by Pagano.8±11 3 RESULTS AND DISCUSSION To validate our numerical solutions, we ®rst investigate the small Vf, small ad ˆ ap limits and compare our results to the well-known analytical solution given by He and Hutchinson.2 Since the original work by He and Hutchinson (HH) has been corrected by Martinez and Gupta (MG),4 and later by He et al.5 (HEH), we com￾pare to the result from HEH. Figure 5 shows our results for the energy release rate ratio, Gd/Gp, as a function of Dundurs' elastic mismatch parameter  along with the HEH result. In our calculations, we have ®xed the ®ber and matrix Poisson's ratios at 0.2, and hence  ˆ 0
375, rather than ®xing  ˆ 0 as in HEH; we will discuss the e€ects of  below. To simulate semi-in®nite crack size, or equivalently `zero' ®ber volume fraction, and the in®nitesimal crack extension which are assumed in HEH, the results in Fig. 5 correspond to Vf ˆ 1% and ad ˆ ap ˆ a ˆ 0
002rf. The latter is the smallest value permitted by the current ADM code. The agree￾ment between our calculations and the HEH analytic result is excellent over the full range of . This validates the use of the ADM code for this particular problem, and also shows the equivalence of the plane-strain/pla￾nar-interface and the axisymmetric versions of this pro￾blem. Figure 5 also shows Gd=Gp versus  for Vf ˆ 1% but for increased values of the crack extension ad ˆ ap ˆ a. For modest increases in the crack extension, Gd/Gp drops substantially at higher . At  ˆ 0
8 and a ˆ 0
025rf, Gd/Gp is only about 50% of the value at a ˆ 0
002rf. This result explicitly demonstrates the sen￾sitivity of the de¯ection criterion to the presumed crack extension, and moreover indicates that de¯ection is, in the general case, more dicult than predicted by the HEH criterion. Figure 6 shows results for Gd/Gp at Vf ˆ 40% and ®nite crack extensions ad ˆ ap ˆ a as used in Fig. 5. Here, even at the smallest crack extension of 0.002 rf there is a reasonable di€erence when compared with the very low volume fraction result and again the ratio of Gd/Gp decreases relative to the HEH result. For larger crack extensions, Gd/Gp decreases even further. At  ˆ 0
8, Gd/Gp for a ˆ 0
025rf is only 19% of HEH result and 35% of the result for a ˆ 0
002rf. In fact, for the larger crack extensions the ratio is nearly constant over a wide range of . While we expect that in the limit of  ! 0 the ratio Gd/Gp would eventually increase up to the HEH result, the crack extension must be much smaller than 0.002 rf. At this point, it is relevant to point out that the minimum crack extension of 0.002 rf used here can be physically quite small. For the largest commonly used ®bers, Textron SCS-6 SiC ®bers, the radius is rf ˆ 70m and our minimum crack extension is 0
14m. Fibers more typically used in most CMCs with commercial potential have a much smaller diameter. The widely used Nicalon SiC ®ber has a radius of about 7
5m so that our minimum crack extension is 0
015m or 15 nm. Nextel alumina ®bers are slightly smaller still, and many graphite ®bers have radii around 3
5m. The crack extensions used here are thus approaching the atomic scale. Furthermore, the minimum crack extensions used here are certainly much smaller than the typical ¯aw sizes in the ®bers. Flaws in the interface are not expec￾ted to be much smaller. Thus, if the crack extension a is interpreted as a ¯aw size in the ®ber or interface to which the matrix crack can connect and propagate, then the range of a used in this study is well within the range of ¯aws in typical CMC systems. Since in many CMCs, the range of elastic mismatch  lies between +0.3+0.8, the deviations from the HEH result for Gd/Gp shown here can be signi®cant in real material systems. Figure 6 also shows that Gd/Gp is actually non-mono￾tonic in  for ®nite crack extension and volume fraction. At suciently high , Gd/Gp begins to decrease rapidly. Furthermore, Gd/Gp appears to drop to zero in the limit  ˆ 1:0 for all cases considered. This di€ers signi®cantly from the HEH result, for which Gd/Gp appears to diverge as  approaches unity. The decrease in Gd/Gp at very high  can be understood physically within the framework of our calculations as follows. With the same Poisson's ratios for both ®ber and matrix,  is simply de®ned as (EfÿEm)/(Ef+Em) under the plane strain assumption. As  increases toward unity, the ®ber become in®nitely sti€ relative to the matrix, i.e. Ef=Em ! 1. Thus, all of the applied load is carried by the ®ber and all of the strain energy is stored in the ®ber; there is no strain energy in the matrix. For a crack which de¯ects along the interface, there is essentially no Fig. 6. Gd/Gp versus  with various crack extensions for Vf ˆ 40%. 1780 B. K. Ahn et al