J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 interface fails under mixed-mode conditions, and a mixed-mode analysis is required for crack propagation along the interface. The mode-I cohesive parameters of the interface are designated as TI and di, while the mode-II cohesive parameters of the interface are designated as Tui and ti 2. 2. Effects of fracture length-scales on crack deflection In this section, a general study of the effect of fracture parameters on crack deflection presented A macroscopic crack is assumed to impinge directly on an interface as shown in Fig. 4. There are no pre-existing kinks or flaws ahead of this main crack, and there is an axis of symmetry along the plane of the crack. Two rows of cohesive elements are placed at the tip of the crack; one row along the interface, and the other row in the substrate. A uniform tensile displacement is applied to the ends of the specimen. The required mesh density for the numerical calculations was determined by selecting several meshes, running the simulations, and analyzing the results to verify what mesh density gave consistent solutions that were within an acceptable range for the uncertainty of the results The transition between crack deflection and crack penetration for the geometry shown in Fig. 4 depends on the following material and geometrical parameters Ef,Es,vs Ts, Gi, ti, as, h, d Using the dundurs result for the effects of mismatched moduli across plane interfaces (Dundurs, 1969), these parameters can be re-expressed in the following non-dimensional groups for the plane-strain conditions considered in this paper: o,B, T:/Erh, E Ti/ah, 6s/G, T/Ti, Tui/Ti, t:/oi, d/h The additional non-dimensional parameters 81/8c and 82/8 which describe the shape of the traction-separation laws(Fig. 3)were kept at constant values of 0.01 and 0.75 throughout the paper. These parameters do not play a significant role in the fracture process and transition in failure mechanism The quantity Ti/Eh was fixed at 1.0 x 10- for all the results presented in this paper This is a physically reasonable value for the parameter, and numerical studies indicated that even fairly significant changes around this level had a negligible influence on the failure transition. The initial studies were conducted with a=0, and B=0(so that Ef=Es=E, and vf=vs=v), and with IIi= TIli=Ii and ti=dj. With these parameters fixed, a series of calculations was performed holding the non-dimensional interface fracture- length scale at a constant value of ET ah=0.01. After running umerical calculations with a given set of as/ai and Is/Ti, a note was made as to whether the crack first began to grow along the interface or through the substrate. In these calculations, the onset of crack growth along either plane was defined as occurring when the first element in a cohesive zone failed. 3 A systematic exploration of how the failure mechanism depended on the magnitude of as/Gi and Ts/ri allowed a failure-mechanism map to be plotted, as shown in Fig. 5. In this figure, which has axes of as/a; and Ts /Ti, constant values of the non-dimensional substrate SThe issue of whether a crack might kink off an interface, after first deflecting along it, was not explored here However, crack deflection off interfaces has been studied experimentally and modeled using a cohesive-zone approach in a paper by Li et al. (2005a)interface fails under mixed-mode conditions, and a mixed-mode analysis is required for crack propagation along the interface. The mode-I cohesive parameters of the interface are designated as GIi and s^i, while the mode-II cohesive parameters of the interface are designated as GIIi and t^i. 2.2. Effects of fracture length-scales on crack deflection In this section, a general study of the effect of fracture parameters on crack deflection is presented. A macroscopic crack is assumed to impinge directly on an interface as shown in Fig. 4. There are no pre-existing kinks or flaws ahead of this main crack, and there is an axis of symmetry along the plane of the crack. Two rows of cohesive elements are placed at the tip of the crack; one row along the interface, and the other row in the substrate. A uniform tensile displacement is applied to the ends of the specimen. The required mesh density for the numerical calculations was determined by selecting several meshes, running the simulations, and analyzing the results to verify what mesh density gave consistent solutions that were within an acceptable range for the uncertainty of the results. The transition between crack deflection and crack penetration for the geometry shown in Fig. 4 depends on the following material and geometrical parameters: Ef ; Es; nf ; ns;GIi; GIIi;Gs; s^i; t^i; s^s; h; d. Using the Dundurs result for the effects of mismatched moduli across plane interfaces (Dundurs, 1969), these parameters can be re-expressed in the following non-dimensional groups for the plane-strain conditions considered in this paper: a; b;Gi=E¯ f h;E¯ f Gi=s^ 2 i h; s^s=s^i;Gs=GIi; GIIi=GIi; t^i=s^i; d=h. The additional non-dimensional parameters d1=dc and d2=dc which describe the shape of the traction-separation laws (Fig. 3) were kept at constant values of 0.01 and 0.75 throughout the paper. These parameters do not play a significant role in the fracture process and transition in failure mechanism. The quantity Gi=Eh¯ was fixed at 1:0 10
6 for all the results presented in this paper. This is a physically reasonable value for the parameter, and numerical studies indicated that even fairly significant changes around this level had a negligible influence on the failure transition. The initial studies were conducted with a ¼ 0, and b ¼ 0 (so that Ef ¼ Es ¼ E, and nf ¼ ns ¼ n), and with GIi ¼ GIIi ¼ Gi and t^i ¼ s^i. With these parameters fixed, a series of calculations was performed holding the non-dimensional interface fracture-length scale at a constant value of E¯ Gi=s^ 2 i h ¼ 0:01. After running numerical calculations with a given set of s^s=s^i and Gs=Gi, a note was made as to whether the crack first began to grow along the interface or through the substrate. In these calculations, the onset of crack growth along either plane was defined as occurring when the first element in a cohesive zone failed.3 A systematic exploration of how the failure mechanism depended on the magnitude of s^s=s^i and Gs=Gi allowed a failure-mechanism map to be plotted, as shown in Fig. 5. In this figure, which has axes of s^s=s^i and Gs=Gi, constant values of the non-dimensional substrate ARTICLE IN PRESS 3 The issue of whether a crack might kink off an interface, after first deflecting along it, was not explored here. However, crack deflection off interfaces has been studied experimentally and modeled using a cohesive-zone approach in a paper by Li et al. (2005a). 274 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287