J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 leflection-penetration criterion being expressed in terms of the relative toughnesses of the interface and second phase. At the present time, there is no crack deflection analysis that bridges these two historically distinct views of fracture. It is this gap in the understanding of the mechanics of interfaces that motivated the present stud The cohesive-zone view provides a coherent analytical framework for fracture that naturally incorporates both strength and energy criteria. Cohesive-zone modeling has its origins in the early models of Dugdale(1960)and Barenblatt (1962) that considered the effects of finite stresses at a crack tip. A cohesive-zone model incorporates a region of material ahead of the crack (the"cohesive zone") having a characteristic traction separation law that describes the fracture process. In a typical traction-separation law, the tractions across the crack plane increase with displacement up to a maximum cohesive strength, and then decay to zero at a critical opening displacement. When the critical displacement is reached, the material in the cohesive zone is assumed to have failed, and the crack advances. This approach to modeling fracture became particularly useful with the advent of sophisticated computational techniques, since it allowed crack propagation to be predicted for different geometries(Hillerborg et al., 1976: Needleman, 1987, 1990; Tvergaard and Hutchinson, 1992; Ungsuwarungsri and Knauss, 1987). The fracture behavior in a single mode of deformation tends to be dominated by two characteristic quantities of the traction-separation law-a characteristic toughness(the area under the curve),I, and a characteristic strength(closely related to the cohesive strength for many traction-separation laws), a. Cohesive-zone models provide a particularly powerful approach for analyzing fracture since their predictions appear to be fairly insensitive to the details of the traction-separation law, being dependent only on these two characteristic The dependence of cohesive models on both strength and toughness parameters makes them a natural bridge between the two traditional views of fracture(Parmigiani and Thouless, 2006). By varying the parameters of a cohesive model it is possible to move from a regime in which fracture is controlled only by the toughness, through a regime in which both toughness and strength control fracture, to a regime in which only strength dominates fracture. The relative importance of these two parameters is indicated by comparing the fracture-length scale, Er/a(where E is the modulus of the material) to the appropriate characteristic length, L of the geometry(Suo et al., 1993). When the fracture- length scale is relatively small, i. e, the non-dimensional group Er/GL is very small, the toughness controls fracture; when the fracture-length scale is relatively large, the strength controls fracture. In the intermediate range, both parameters are important. Consideration of the fracture-length scale immediately highlights an inherent problem with energy-based analyses of crack deflection at interfaces. These models invoke a pre- existing kink along the interface. This kink has to be very small in comparison to any other characteristic dimension of the problem, so that asymptotic solutions for the crack-tip stress field can be used. However, the length of the kink then becomes the characteristic dimension that the fracture-length scale must be compared to, in order to determine whether fracture is controlled by energy or stress. Therefore, the kink has to be short compared to any other dimensions of the problem, for crack-tip asymptotic solutions to be valid; but, simultaneously, the kink has to be long compared to the fracture length scale, so 2The shape of the traction-separation curves can occasionally affect fracture. For example, there are laws in hich the characteristic strength is not related to the cohesive strength(Li et al., 2005a, b)deflection–penetration criterion being expressed in terms of the relative toughnesses of the interface and second phase. At the present time, there is no crack deflection analysis that bridges these two historically distinct views of fracture. It is this gap in the understanding of the mechanics of interfaces that motivated the present study. The cohesive-zone view provides a coherent analytical framework for fracture that naturally incorporates both strength and energy criteria. Cohesive-zone modeling has its origins in the early models of Dugdale (1960) and Barenblatt (1962) that considered the effects of finite stresses at a crack tip. A cohesive-zone model incorporates a region of material ahead of the crack (the ‘‘cohesive zone’’) having a characteristic traction￾separation law that describes the fracture process. In a typical traction-separation law, the tractions across the crack plane increase with displacement up to a maximum cohesive strength, and then decay to zero at a critical opening displacement. When the critical displacement is reached, the material in the cohesive zone is assumed to have failed, and the crack advances. This approach to modeling fracture became particularly useful with the advent of sophisticated computational techniques, since it allowed crack propagation to be predicted for different geometries (Hillerborg et al., 1976; Needleman, 1987, 1990; Tvergaard and Hutchinson, 1992; Ungsuwarungsri and Knauss, 1987). The fracture behavior in a single mode of deformation tends to be dominated by two characteristic quantities of the traction-separation law—a characteristic toughness (the area under the curve), G, and a characteristic strength (closely related to the cohesive strength for many traction–separation laws), s^. Cohesive-zone models provide a particularly powerful approach for analyzing fracture since their predictions appear to be fairly insensitive to the details of the traction-separation law, being dependent only on these two characteristic parameters.2 The dependence of cohesive models on both strength and toughness parameters makes them a natural bridge between the two traditional views of fracture (Parmigiani and Thouless, 2006). By varying the parameters of a cohesive model it is possible to move from a regime in which fracture is controlled only by the toughness, through a regime in which both toughness and strength control fracture, to a regime in which only strength dominates fracture. The relative importance of these two parameters is indicated by comparing the fracture-length scale, EG=s^ 2 (where E is the modulus of the material) to the appropriate characteristic length, L of the geometry (Suo et al., 1993). When the fracture-length scale is relatively small, i.e., the non-dimensional group EG=s^ 2 L is very small, the toughness controls fracture; when the fracture-length scale is relatively large, the strength controls fracture. In the intermediate range, both parameters are important. Consideration of the fracture-length scale immediately highlights an inherent problem with energy-based analyses of crack deflection at interfaces. These models invoke a pre￾existing kink along the interface. This kink has to be very small in comparison to any other characteristic dimension of the problem, so that asymptotic solutions for the crack-tip stress field can be used. However, the length of the kink then becomes the characteristic dimension that the fracture-length scale must be compared to, in order to determine whether fracture is controlled by energy or stress. Therefore, the kink has to be short compared to any other dimensions of the problem, for crack-tip asymptotic solutions to be valid; but, simultaneously, the kink has to be long compared to the fracture length scale, so ARTICLE IN PRESS 2 The shape of the traction–separation curves can occasionally affect fracture. For example, there are laws in which the characteristic strength is not related to the cohesive strength (Li et al., 2005a, b). 270 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287