J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 significant effect on fracture. The strength and toughness(area under the curve) are the two dominant parameters that control fracture, and cracks can propagate only if both the ode-I and energy criteria are met. As discussed above, the use of separate mode-I and II laws allows for a general investigation of fracture, encompassing problems in which shear fracture has physical significance and problems in which pure shear only results in slip, not fracture The cohesive-zone modeling was implemented within the commercial finite-element package ABAQUS(version 6.3-1), as described by Yang(2000 ). Three- and four-node linear, plane-strain elements were used for the continuum elements. The elements for the cohesive zone were defined using the ABAQUS UEL feature, the traction-separation laws of Fig 3, and the failure criterion of Eq(2). These were implemented in a FORTRAN subroutine. An example of the code used is given in Parmigiani(2005) While several different geometries could have been used to study the problem of crack deflection, the work in this paper focuses on a laminated system subject to a uniform tensile displacement, as shown in Fig. 4. A layer of thickness h, with an elastic modulus of Ef and a Poissons ratio of v/, is bonded to a substrate of thickness d. The layer of thickness h has a crack that extends from the free surface to the interface. and that is normal to the interface. The substrate has an elastic modulus of e and a poisson 's ratio of Vs. For all the calculations reported in this paper, the substrate is ten times thicker than the cracked layer, so that d= 10h. Plane-strain conditions are assumed, so that the two Dundurs parameters can be defined as(Dundurs, 1969) Ef-es +e 8=E(1-2)/(1-y)-E(1-2)/(1-) 2(Er+es) where,E=E/(1-12). If the substrate cracks, it will do so under pure mode-I conditions therefore, only the mode-I fracture properties of the substrate are required. The mode-I substrate toughness is designated as Is, and the mode-I strength is designated asas. The crack impinging on interface cohesive-zone eleme The laminated geometry used to study crack deflection in this paper. A layer h and with an modulus of Er and a Poissons ratio of v is bonded to a substrate of thickne d=10. The ubstrate has an elastic modulus of e and a poisson's ratio of v. There is a crack surface to the interface and is normal to the interface Sets of cohesive elements exist e crack in the ubstrate and along the interface. There is a plane of symmetry along the crack, and is loaded by a uniform displacement applied to the ends of the specimen.significant effect on fracture. The strength and toughness (area under the curve) are the two dominant parameters that control fracture, and cracks can propagate only if both the stress and energy criteria are met. As discussed above, the use of separate mode-I and mode-II laws allows for a general investigation of fracture, encompassing problems in which shear fracture has physical significance and problems in which pure shear only results in slip, not fracture. The cohesive-zone modeling was implemented within the commercial finite-element package ABAQUS (version 6.3-1), as described by Yang (2000). Three- and four-node, linear, plane-strain elements were used for the continuum elements. The elements for the cohesive zone were defined using the ABAQUS UEL feature, the traction-separation laws of Fig. 3, and the failure criterion of Eq. (2). These were implemented in a FORTRAN subroutine. An example of the code used is given in Parmigiani (2005). While several different geometries could have been used to study the problem of crack deflection, the work in this paper focuses on a laminated system subject to a uniform tensile displacement, as shown in Fig. 4. A layer of thickness h, with an elastic modulus of Ef and a Poisson’s ratio of nf , is bonded to a substrate of thickness d. The layer of thickness h has a crack that extends from the free surface to the interface, and that is normal to the interface. The substrate has an elastic modulus of Es and a Poisson’s ratio of ns. For all the calculations reported in this paper, the substrate is ten times thicker than the cracked layer, so that d ¼ 10h. Plane-strain conditions are assumed, so that the two Dundurs parameters can be defined as (Dundurs, 1969) a ¼ E¯ f
E¯ s E¯ f þ E¯ s , (4) and b ¼ E¯ f ð1
2nsÞ=ð1
nsÞ
E¯ sð1
2nf Þ=ð1
nf Þ 2ðE¯ f þ E¯ sÞ , (5) where, E¯ ¼ E=ð1
n2Þ. If the substrate cracks, it will do so under pure mode-I conditions; therefore, only the mode-I fracture properties of the substrate are required. The mode-I substrate toughness is designated as Gs, and the mode-I strength is designated as s^s. The ARTICLE IN PRESS d = 10 h h 2L = 220 h Ef ,νf Es,νs crack impinging on interface cohesive-zone elements Fig. 4. The laminated geometry used to study crack deflection in this paper. A layer of thickness h and with an elastic modulus of Ef and a Poisson’s ratio of nf is bonded to a substrate of thickness d, where d ¼ 10h. The substrate has an elastic modulus of Es and a Poisson’s ratio of ns. There is a crack that extends from the top surface to the interface, and is normal to the interface. Sets of cohesive elements exist ahead of the crack in the substrate and along the interface. There is a plane of symmetry along the crack, and the system is loaded by a uniform displacement applied to the ends of the specimen. J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 273