J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 r/o E998958最标号6°标工 Er/o defleraclong Er2h=000 Ratio of the substrate strength to interface strength, a /o Fig. 5. The results of a set of calculations for Er/a h=0.01, a=B=0, Ti/Eh= 1.0 x 10-, Tu=lmi=r ii, and d/h= 10. The plot shows the regimes in which crack penetration or crack deflection will occur in /ri and a./@i space. The error bars indicate the range of uncertainty of the transition. fracture-length scale follow a parabolic form. The error bars on this figure are associated with numerical considerations, and show the uncertainty with which the boundary of the transition could be determined. In this context, it should be noted that, occasionally, the demands of the geometry created numerical difficulties close to the transition. In the worst cases the numerical simulation failed before the onset of crack growth. When this happened, the transition was quantified by examining the values of p/Ts and sd/ri at the point of the numerical instability, where p is the energy-release rate for penetration of the crack into the substrate and d is the energy-release rate for deflection of the crack along the interface. These ratios measure the extent of the appropriate traction-separation law that has been traversed and indicate how close the elements of the cohesive zone are to failure As might be intuitively expected, the failure-mechanism map of Fig. 5 shows that crack deflection is promoted by high values of both as ai and Ts/Ti. Conversely, crack enetration is promoted by low values of these two ratios. At larger values of the non- dimensional substrate fracture-length scale, the failure mechanism is controlled by the strength ratio; at smaller values, it becomes more sensitive to the toughness ratio However, even in this latter range, there is no indication that only toughness controls the failure mechanism. Indeed, if there is a lower bound on the toughness ratio required to guarantee crack penetration, it is much lower than the range that could be explored by the present calculations. Conversely, there does appear to be a vertical asymptote representing a critical value of the strength ratio below which crack penetration is guaranteed, irrespective of the toughness ratio. This implies that crack penetration will always occur if the substratefracture-length scale follow a parabolic form. The error bars on this figure are associated with numerical considerations, and show the uncertainty with which the boundary of the transition could be determined. In this context, it should be noted that, occasionally, the demands of the geometry created numerical difficulties close to the transition. In the worst cases, the numerical simulation failed before the onset of crack growth. When this happened, the transition was quantified by examining the values of Gp=Gs and Gd =Gi at the point of the numerical instability, where Gp is the energy-release rate for penetration of the crack into the substrate and Gd is the energy-release rate for deflection of the crack along the interface. These ratios measure the extent of the appropriate traction-separation law that has been traversed, and indicate how close the elements of the cohesive zone are to failure As might be intuitively expected, the failure-mechanism map of Fig. 5 shows that crack deflection is promoted by high values of both s^s=s^i and Gs=Gi. Conversely, crack penetration is promoted by low values of these two ratios. At larger values of the non￾dimensional substrate fracture-length scale, the failure mechanism is controlled by the strength ratio; at smaller values, it becomes more sensitive to the toughness ratio. However, even in this latter range, there is no indication that only toughness controls the failure mechanism. Indeed, if there is a lower bound on the toughness ratio required to guarantee crack penetration, it is much lower than the range that could be explored by the present calculations. Conversely, there does appear to be a vertical asymptote representing a critical value of the strength ratio below which crack penetration is guaranteed, irrespective of the toughness ratio. This implies that crack penetration will always occur if the substrate ARTICLE IN PRESS 0 10 20 30 40 50 60 70 80 01234567 Ratio of the substrate strength to interface strength, σs / σi ^ ^ Ratio of the substrate toughness to interface toughness, Γs / Γi deflection along interface penetration into substrate E Γi /σi 2 h = 0.01 Γi /Eh = 1.0 x 10-6 ^ E Γs /σs 2 h = 0.1 E Γs /σs 2 h = 0.01 E Γs /σs 2 h = 0.001 ^ ^ ^ Fig. 5. The results of a set of calculations for E¯ Gi=s^ 2 i h ¼ 0:01, a ¼ b ¼ 0, Gi=Eh¯ ¼ 1:0 10
6 , GIi ¼ GIIi ¼ Gi, s^i ¼ t^i, and d=h ¼ 10. The plot shows the regimes in which crack penetration or crack deflection will occur in Gs=Gi and s^s=s^i space. The error bars indicate the range of uncertainty of the transition. J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 275