J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 1993: Tu et al, 1996). For example, wood consists of aligned long, hollow cylindrical cells (Ashby and Jones, 1998; Wainwright et al., 1982), and common experience shows that attempts to fracture wood perpendicular to its grain are hindered by crack deflection along the grains. Shells of many animals provide examples of materials with exceptional toughness created by combining a hard, brittle, inorganic mineral with a compliant protein (Kessler et al., 1996). The protein exists at the interfaces between the mineral components, and provides a bonding agent that can delaminate and dissipate energy when an attempt is made to fracture the shell. It is clear that the balance between the organic interface and inorganic matrix is highly optimized for the evolutionary purposes of the shell 1. 2. Previous analyses of crack deflection The optimization of composites that exhibit crack deflection and interfacial delamina- tion requires an understanding of how the interfacial and bulk properties affect the mechanics of the problem. The role of crack deflection at interfaces was first recognized and analyzed about forty years ago by Cook and Gordon (1964). Their analysis used a trength-based fracture criterion. They considered a matrix crack perpendicular to a fiber having identical elastic properties as the matrix. The matrix crack was modeled as an ellipse with a very high aspect ratio, and the results of Inglis(1913)were used to investigate the stresses around the crack tip. Cook and Gordon(1964)noted that the maximum normal stress ahead of, and co-planar with, the crack is about five times greater than the maximum normal stress perpendicular to the crack tip. Based on this observation, they suggested that a fiber needs to be about five times stronger than the interface between it and the matrix to prevent fiber fracture, and to allow crack deflection to occur This concept was extended many years later by Gupta et al.(1992), who used earlier work(Zak and Williams, 1963: Williams, 1957, Swenson and Rau, 1970) on the stress field around a sharp crack at bimaterial interfaces, to look at the criterion for determining whether a crack at normal incidence to a bimaterial interface would deflect or not Comparisons between the maximum normal stress across the interface and the maximum normal stress ahead of the crack allowed predictions to be made about whether defection or penetration should occur. For example, in an elastically homogeneous system, the results indicated that crack deflection should occur if the material ahead of the crack is more than about three and a half times stronger than the interface. While giving a slightly different value for the ratio of the two strengths required for deflection, this result is consistent with the earlier work of Cook and Gordon(1964). Furthermore, this work showed that crack deflection along the interface becomes much less likely if the cracked matrix is stiffer than the second phase, with crack deflection becoming essentially impossible if there is a compliant second phase embedded in a rigid matrix. Conversely, the tendency for crack deflection increases slightly when the second phase is stiffer than the matrix These analyses follow an Inglis(1913) or strength-based approach to fracture. Both nalyses lead to design criteria for composites and laminates that are based on the ratio of the strengths of the interface and second phase. An alternative approach using interfacial fracture mechanics Rice(1988)follows that of Griffith(1920)and others(Irwin, 1957; Kies and Smith, 1955: Orowan, 1949), and is based on an energy criterion. Many authors have ed linear-elastic fracture mechanics to look at crack deflection from an en perspective(He and Hutchinson, 1989: Thouless et al., 1989: Martinez and gupta, I1993; Tu et al., 1996). For example, wood consists of aligned long, hollow cylindrical cells (Ashby and Jones, 1998; Wainwright et al., 1982), and common experience shows that attempts to fracture wood perpendicular to its grain are hindered by crack deflection along the grains. Shells of many animals provide examples of materials with exceptional toughness created by combining a hard, brittle, inorganic mineral with a compliant protein (Kessler et al., 1996). The protein exists at the interfaces between the mineral components, and provides a bonding agent that can delaminate and dissipate energy when an attempt is made to fracture the shell. It is clear that the balance between the organic interface and inorganic matrix is highly optimized for the evolutionary purposes of the shell. 1.2. Previous analyses of crack deflection The optimization of composites that exhibit crack deflection and interfacial delamina￾tion requires an understanding of how the interfacial and bulk properties affect the mechanics of the problem. The role of crack deflection at interfaces was first recognized and analyzed about forty years ago by Cook and Gordon (1964). Their analysis used a strength-based fracture criterion. They considered a matrix crack perpendicular to a fiber having identical elastic properties as the matrix. The matrix crack was modeled as an ellipse with a very high aspect ratio, and the results of Inglis (1913) were used to investigate the stresses around the crack tip. Cook and Gordon (1964) noted that the maximum normal stress ahead of, and co-planar with, the crack is about five times greater than the maximum normal stress perpendicular to the crack tip. Based on this observation, they suggested that a fiber needs to be about five times stronger than the interface between it and the matrix to prevent fiber fracture, and to allow crack deflection to occur. This concept was extended many years later by Gupta et al. (1992), who used earlier work (Zak and Williams, 1963; Williams, 1957; Swenson and Rau, 1970) on the stress field around a sharp crack at bimaterial interfaces, to look at the criterion for determining whether a crack at normal incidence to a bimaterial interface would deflect or not. Comparisons between the maximum normal stress across the interface and the maximum normal stress ahead of the crack allowed predictions to be made about whether deflection or penetration should occur. For example, in an elastically homogeneous system, the results indicated that crack deflection should occur if the material ahead of the crack is more than about three and a half times stronger than the interface. While giving a slightly different value for the ratio of the two strengths required for deflection, this result is consistent with the earlier work of Cook and Gordon (1964). Furthermore, this work showed that crack deflection along the interface becomes much less likely if the cracked matrix is stiffer than the second phase, with crack deflection becoming essentially impossible if there is a compliant second phase embedded in a rigid matrix. Conversely, the tendency for crack deflection increases slightly when the second phase is stiffer than the matrix. These analyses follow an Inglis (1913) or strength-based approach to fracture. Both analyses lead to design criteria for composites and laminates that are based on the ratio of the strengths of the interface and second phase. An alternative approach using interfacial fracture mechanics Rice (1988) follows that of Griffith (1920) and others (Irwin, 1957; Kies and Smith, 1955; Orowan, 1949), and is based on an energy criterion. Many authors have used linear-elastic fracture mechanics to look at crack deflection from an energy perspective (He and Hutchinson, 1989; Thouless et al., 1989; Martinez and Gupta, 1993; ARTICLE IN PRESS 268 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287