J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 trength is less than about three times greater than the interface strength, which is very consistent with the results of strength-based criteria(Cook and Gordon, 1964; Gupta al.,1992 Non-dimensional fracture-length scales that are of the order of 0.01 for both the interface and substrate should be considered to represent linear-elastic conditions (Parmigiani and Thouless, 2006). It is therefore of interest to note that a deflection criterion based solely on energy considerations is not captured in this analysis. Even in the linear-elastic regime, both the strength and toughness ratios determine the failure mechanism. However, as demonstrated in the appendix the deflection criterion of He and Hutchinson(1989), He et al.(1994)can be reproduced by the cohesive-zone model if pre- existing kinks are included in the analysis, as they are in the classical energy-based analyses. In the complete absence of a physical kink at the interface, a finite interface conditions for only an energy-criterion to be valid. Therefore both strength lolate strength is always going to cause the interface fracture-length scale to violate the should be expected to play a role in initiating a crack along the interface from a crack that impinges the interface; and, hence, both strength and energy will determine the failure mechanisn The approach used to produce Fig. 5 was extended to look at the effect of increasing the fracture-length scale of the interface, away from linear-elastic conditions. The calculations were repeated for additional values of ET:a:h=0.1, 1.0 and 10. The corresponding failure- mechanism maps are plotted in Fig. 6. On this figure, the boundaries(with appropriate error bars) between deflection and penetration are drawn for the different values of ET/ah. The basic form of the failure-mechanism map does not change as this quantity is varied, but the tendency for crack deflection appears to be reduced as the interface fracture-length scale is increased. The apparent vertical asymptote observed in Fig. 5 appears to be a general phenomenon, with the precise value of the critical strength ratio depending on the interfacial fracture-length scale. In addition, the fact that crack deflection can occur with relatively tough interfaces, provided the interfacial strength is low enough, also appears to be a general feature. Furthermore, the form of Fig. 6 suggests that there may be an asymptotic curve for very low values of ET/ah. This asymptotic behavior was explored by keeping Ers/oh= Eri/ah, and systematically decreasing them both. The results, plotted in Fig. 7, clearly show an asymptotic value of as/a: A 3.2, below which crack penetration always occurs. It is interesting that this is very close to the strength-based, linear-elastic results of Gupta et al. (1992), which predict a value of approximately 3. 4 for the maximum ratio of the substrate to interface strength required for crack penetration in a homogeneous system If one assumes self-similar traction-separation laws for the substrate and interface, so that a, /a;=rs/ri. Fig. 5 indicates that the transition in failure mechanism occurs when both the strength and toughness ratios are equal to about four. If one focuses only on the toughness ratio, this might appear to match the criterion of Refs. THouless et al.. 1989: He and Hutchinson, 1989). as noted in an earlier cohesive-zone analysis in which the rength and toughness ratios were related in this fashion(Siegmund et al., 1997). However, this match should robably be considered to be coincidental, since Fig. 5 shows the failure mechanism actually depends on both the rength and toughness. Furthermore, there is probably no reason to expect such a relationship between the fracture parameters of the interface and substrate to be universally valid, especially when the interface consists ofstrength is less than about three times greater than the interface strength, which is very consistent with the results of strength-based criteria (Cook and Gordon, 1964; Gupta et al., 1992). Non-dimensional fracture-length scales that are of the order of 0.01 for both the interface and substrate should be considered to represent linear-elastic conditions (Parmigiani and Thouless, 2006). It is therefore of interest to note that a deflection criterion based solely on energy considerations is not captured in this analysis.4 Even in the linear-elastic regime, both the strength and toughness ratios determine the failure mechanism. However, as demonstrated in the appendix, the deflection criterion of He and Hutchinson (1989), He et al. (1994) can be reproduced by the cohesive-zone model if pre￾existing kinks are included in the analysis, as they are in the classical energy-based analyses. In the complete absence of a physical kink at the interface, a finite interface strength is always going to cause the interface fracture-length scale to violate the conditions for only an energy-criterion to be valid. Therefore, both strength and energy should be expected to play a role in initiating a crack along the interface from a crack that impinges the interface; and, hence, both strength and energy will determine the failure mechanism. The approach used to produce Fig. 5 was extended to look at the effect of increasing the fracture-length scale of the interface, away from linear-elastic conditions. The calculations were repeated for additional values of E¯ Gi=s^ 2 i h ¼ 0:1; 1:0 and 10. The corresponding failure-mechanism maps are plotted in Fig. 6. On this figure, the boundaries (with appropriate error bars) between deflection and penetration are drawn for the different values of E¯ Gi=s^ 2 i h. The basic form of the failure-mechanism map does not change as this quantity is varied, but the tendency for crack deflection appears to be reduced as the interface fracture-length scale is increased. The apparent vertical asymptote observed in Fig. 5 appears to be a general phenomenon, with the precise value of the critical strength ratio depending on the interfacial fracture-length scale. In addition, the fact that crack deflection can occur with relatively tough interfaces, provided the interfacial strength is low enough, also appears to be a general feature. Furthermore, the form of Fig. 6 suggests that there may be an asymptotic curve for very low values of E¯ Gi=s^ 2 i h. This asymptotic behavior was explored by keeping E¯ Gs=s2 s h ¼ E¯ Gi=s2 i h, and systematically decreasing them both. The results, plotted in Fig. 7, clearly show an asymptotic value of s^s=s^i 3:2, below which crack penetration always occurs.It is interesting that this is very close to the strength-based, linear-elastic results of Gupta et al. (1992), which predict a value of approximately 3.4 for the maximum ratio of the substrate to interface strength required for crack penetration in a homogeneous system. ARTICLE IN PRESS 4 If one assumes self-similar traction-separation laws for the substrate and interface, so that s^s=s^i ¼ Gs=Gi, Fig. 5 indicates that the transition in failure mechanism occurs when both the strength and toughness ratios are equal to about four. If one focuses only on the toughness ratio, this might appear to match the criterion of Refs. (Thouless et al., 1989; He and Hutchinson, 1989), as noted in an earlier cohesive-zone analysis in which the strength and toughness ratios were related in this fashion (Siegmund et al., 1997). However, this match should probably be considered to be coincidental, since Fig. 5 shows the failure mechanism actually depends on both the strength and toughness. Furthermore, there is probably no reason to expect such a relationship between the fracture parameters of the interface and substrate to be universally valid, especially when the interface consists of a bonding layer that is different from the substrate. 276 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287