J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 E, r/e h=0.01 T/E a=+0.5 是| Ratio of the substrate strength to interface strength, a/o Fig 8. A failure mechanism map showing the effects of modulus mismatch. These calculations were done for r/Gh=0.01,B=0,/Eh=1.0×10-6,=rm=r,i=t,andd/h=10. 2.3. Effects of modulus mismatch on crack deflection To explore the effects of modulus mismatch on crack deflection, the calculations were repeated for non-zero values of a, but leaving B=0. Some failure-mechanism maps for 990 are shown in Fig. 8. The general features of the transition curves do not appear to lepend on the modulus mismatch. However, an interesting feature of the plots is that the tendency for penetration seems to be particularly pronounced when the cracked layer and substrate have comparable values of stiffness. Further investigations showed that crack deflection is very sensitive to mixed-mode effects at the interface, and that the particular behavior indicated in Fig. 8 can be attributed to the assumption of a Griffith failure criterion(Tni/TI=1). As discussed earlier, the magnitude of the mode-II effects is quantified by the ratio of Gu/g, for the interface when the crack is just about to propagate. a detailed analysis of the present results showed that mode- ll effects become dominant as o increases(as the cracked layer becomes stiffer). This observation is consistent with the results of He and Hutchinson(1989), He et al. (1994) that indicate the phase angle at the tip of an interface kink increases as the cracked layer becomes much stiffer than the substrate. In this regime, a Griffith criterion might be expected to give significantly different predictions from mode-I dominated fracture criterion. Conversely, as the cracked layer becomes more compliant than the substrate, the interfacial fracture becomes increasingly dominated by ode-I deformation. In this regime, the distinction between a Griffith criterion and a mode-I dominated criterion becomes less critical. In this context, it is of interest to note that the deflection/penetration criterion of Gupta et al. (1992)was based on a comparison of the normal stresses in the penetrating and deflecting directions with the normal interface2.3. Effects of modulus mismatch on crack deflection To explore the effects of modulus mismatch on crack deflection, the calculations were repeated for non-zero values of a, but leaving b ¼ 0. Some failure-mechanism maps for aa0 are shown in Fig. 8. The general features of the transition curves do not appear to depend on the modulus mismatch. However, an interesting feature of the plots is that the tendency for penetration seems to be particularly pronounced when the cracked layer and substrate have comparable values of stiffness. Further investigations showed that crack deflection is very sensitive to mixed-mode effects at the interface, and that the particular behavior indicated in Fig. 8 can be attributed to the assumption of a Griffith failure criterion (GIIi=GIi ¼ 1). As discussed earlier, the magnitude of the mode-II effects is quantified by the ratio of GII=GI for the interface when the crack is just about to propagate. A detailed analysis of the present results showed that mode-II effects become dominant as a increases (as the cracked layer becomes stiffer). This observation is consistent with the results of He and Hutchinson (1989), He et al. (1994) that indicate the phase angle at the tip of an interface kink increases as the cracked layer becomes much stiffer than the substrate. In this regime, a Griffith criterion might be expected to give significantly different predictions from a mode-I dominated fracture criterion. Conversely, as the cracked layer becomes more compliant than the substrate, the interfacial fracture becomes increasingly dominated by mode-I deformation. In this regime, the distinction between a Griffith criterion and a mode-I dominated criterion becomes less critical. In this context, it is of interest to note that the deflection/penetration criterion of Gupta et al. (1992) was based on a comparison of the normal stresses in the penetrating and deflecting directions with the normal interface ARTICLE IN PRESS 0 5 10 15 20 01234567 8 Ratio of the substrate strength to interface strength, σs / σi ^ ^ Ratio of the substrate toughness to interface toughness, Γs / Γi penetration into substrate deflection along interface E f Γi /σi 2 h = 0.01 Γi /Ef h = 1.0 x 10-6 -0.95 ^ α = +0.5 0 -0.50 +0.95 Fig. 8. A failure mechanism map showing the effects of modulus mismatch. These calculations were done for E¯ f Gi=s^ 2 i h ¼ 0:01, b ¼ 0, Gi=E¯ f h ¼ 1:0 10
6 , GIi ¼ GIIi ¼ Gi, s^i ¼ t^i, and d=h ¼ 10. 278 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287