J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 that an energy criterion for fracture is appropriate. If it is assumed that an interface can support singular stresses, the kink can be taken to the limit of zero length with no conceptual difficulty. However, if an interface with a finite cohesire strength does not contain a physical kink of a finite length, both energy and stress are expected to play a role in initiating fracture along the interface In this paper, a cohesive-zone model is used to analyze the problem of crack deflection at interfaces. Of major concern are(i) an elucidation of the roles of the interfacial strength, the interfacial toughness, the substrate strength and the substrate toughness on crack deflection, and (i) an understanding of the conditions under which any of these parameters might dominate design considerations. These issues are addressed by using a cohesive-zone analysis to look at the general problem of crack deflection at different fracture-length scales, in the absence of any pre-existing kinks. The results of the calculations are presented in non-dimensional terms for a wide range of parameter space, so that the effects of different strength and toughness values on the transition are fully explored. The roles of mixed-mode failure criteria and modulus mismatch across the interface are also explored nally, in the appendix, cohesive-zone models are used to look at kinked cracks. The results of these calculations are used to make a connection with existing energy-based analyses of crack deflection, and to show that the numerical approach used in this paper can accurately capture the classical energy-based criteria for this phenomenon, provided the fracture-length scales are small enough, and that appropriate assumptions about the kinks are made 2. Numerical results 2. Cohesive- zone model A cohesive-zone model was used to analyze crack deflection at interfaces. This problem requires a mixed-mode implementation of the model. Often, mixed-mode effects are modeled by combining normal and shear displacements into a single parameter that is used in a traction-separation law to indicate overall load-carrying ability(Tvergaard and Hutchinson, 1993). However, an alternative approach is to use separate and independent laws for mode I and mode Il, each being functions of only the normal and shear displacements, respectively. The ability to specify the mode-I and mode-II strength and toughness values independently appears to be necessary to capture some experimental results (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006). Since the traction-separation laws are prescribed independently, they need to be coupled through a mixed-mode failure criterion Such a failure criterion relates the normal and shear placements at which the load-bearing capability of the cohesive-zone elements fail. In this work. a linear failure criterion of the form 1/1+m/u=1 was used, where gI is the mode-I energy-release rate, TI is the mode-I toughness, n is the mode-II energy release rate, and Tu is the mode-II toughness. In this formulation, the toughness is defined as the total area under the traction-separation law, and the energy release rate is defined as the area under the traction-separation law at any particular instant of interest (Yang and Thouless, 2001). While simple, this linear criterion allows a fairly rich range of mixed-mode behavior to be mimicked, from what we will call athat an energy criterion for fracture is appropriate. If it is assumed that an interface can support singular stresses, the kink can be taken to the limit of zero length with no conceptual difficulty. However, if an interface with a finite cohesive strength does not contain a physical kink of a finite length, both energy and stress are expected to play a role in initiating fracture along the interface. In this paper, a cohesive-zone model is used to analyze the problem of crack deflection at interfaces. Of major concern are (i) an elucidation of the roles of the interfacial strength, the interfacial toughness, the substrate strength and the substrate toughness on crack deflection, and (ii) an understanding of the conditions under which any of these parameters might dominate design considerations. These issues are addressed by using a cohesive-zone analysis to look at the general problem of crack deflection at different fracture-length scales, in the absence of any pre-existing kinks. The results of the calculations are presented in non-dimensional terms for a wide range of parameter space, so that the effects of different strength and toughness values on the transition are fully explored. The roles of mixed-mode failure criteria and modulus mismatch across the interface are also explored. Finally, in the appendix, cohesive-zone models are used to look at kinked cracks. The results of these calculations are used to make a connection with existing energy-based analyses of crack deflection, and to show that the numerical approach used in this paper can accurately capture the classical energy-based criteria for this phenomenon, provided the fracture-length scales are small enough, and that appropriate assumptions about the kinks are made. 2. Numerical results 2.1. Cohesive-zone model A cohesive-zone model was used to analyze crack deflection at interfaces. This problem requires a mixed-mode implementation of the model. Often, mixed-mode effects are modeled by combining normal and shear displacements into a single parameter that is used in a traction-separation law to indicate overall load-carrying ability (Tvergaard and Hutchinson, 1993). However, an alternative approach is to use separate and independent laws for mode I and mode II, each being functions of only the normal and shear displacements, respectively. The ability to specify the mode-I and mode-II strength and toughness values independently appears to be necessary to capture some experimental results (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006). Since the traction–separation laws are prescribed independently, they need to be coupled through a mixed-mode failure criterion. Such a failure criterion relates the normal and shear displacements at which the load-bearing capability of the cohesive-zone elements fail. In this work, a linear failure criterion of the form GI=GI þ GII=GII ¼ 1 (2) was used, where GI is the mode-I energy-release rate, GI is the mode-I toughness, GII is the mode-II energy release rate, and GII is the mode-II toughness. In this formulation, the toughness is defined as the total area under the traction-separation law, and the energy￾release rate is defined as the area under the traction-separation law at any particular instant of interest (Yang and Thouless, 2001). While simple, this linear criterion allows for a fairly rich range of mixed-mode behavior to be mimicked, from what we will call a ARTICLE IN PRESS J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 271