J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 Penetration Fig. 2. Details of the crack deflection problem modeled by He and Hutchinson(1989). The macroscopic view shown in(a). A comparison is made between the conditions for(b)a small kink to extend across the interface, an (c)a small kink to extend al He et al, 1994; Lu and Erdogan, 1983: Tullock et al., 1994). These generally follow the approach of Cotterell and Rice(1980), where the energy-release rates of kinks at different angles ahead of a main crack are considered The ratio of the energy-release rates in different directions is taken to be proportional to the critical ratio of the toughnesses required to trigger fracture in the different directions deflection or penetration occurs when a crack impinges a bimaterial interface in a normal direction was examined by comparing the energy-release rate at the tip of a small kink extending across the interface, p, with the energy-release rate at the tip of a small kink deflected along the interface d(Fig. 2). The condition for crack deflection along the interface can be written as where Ti is the toughness of the interface under the appropriate mixed-mode conditions, and Is is the toughness of the material (substrate) ahead of the interface. One particularly well-known result is that when the elastic properties across the interface are identical, and the kinks are vanishingly small, crack deflection occurs if the toughness of the material on the other side of the interface is more than approximately four times the mixed-mode toughness of the interface(He and Hutchinson, 1989; Thouless et al., 1989) 1.3. Problem addressed in the present work In the analyses described above, two different fracture criteria were used: a stress- base criterion and an energy-based criterion. These lead to two different types of material parameters forming the basis for design of interfaces. A stress-based fracture criterion leads to the deflection-penetration criterion being expressed in terms of the relative strengths of the interface and second phase. An energy-based fracture criterion leads to theHe et al., 1994; Lu and Erdogan, 1983; Tullock et al., 1994). These generally follow the approach of Cotterell and Rice (1980), where the energy-release rates of kinks at different angles ahead of a main crack are considered. The ratio of the energy-release rates in different directions is taken to be proportional to the critical ratio of the toughnesses required to trigger fracture in the different directions. Of particular note is the work by He and Hutchinson (1989), with corrections (He et al., 1994; Martinez and Gupta, 1993). In their work, the problem of determining whether crack deflection or penetration occurs when a crack impinges a bimaterial interface in a normal direction was examined by comparing the energy-release rate at the tip of a small kink extending across the interface, Gp, with the energy-release rate at the tip of a small kink deflected along the interface, Gd (Fig. 2). The condition for crack deflection along the interface can be written as Gi Gs o Gd Gp , (1) where Gi is the toughness of the interface under the appropriate mixed-mode conditions, and Gs is the toughness of the material (substrate) ahead of the interface. One particularly well-known result is that when the elastic properties across the interface are identical, and the kinks are vanishingly small, crack deflection occurs if the toughness of the material on the other side of the interface is more than approximately four times the mixed-mode toughness of the interface (He and Hutchinson, 1989; Thouless et al., 1989). 1.3. Problem addressed in the present work In the analyses described above, two different fracture criteria were used: a stress-based criterion and an energy-based criterion. These lead to two different types of material parameters forming the basis for design of interfaces. A stress-based fracture criterion leads to the deflection–penetration criterion being expressed in terms of the relative strengths of the interface and second phase. An energy-based fracture criterion leads to the ARTICLE IN PRESS f s f s k Penetration f s k Deflection (c) (b) (a) Fig. 2. Details of the crack deflection problem modeled by He and Hutchinson (1989). The macroscopic view is shown in (a). A comparison is made between the conditions for (b) a small kink to extend across the interface, and (c) a small kink to extend along the interface. J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 269