13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- where int ˆσ is the peak stress in the normal traction-separation relation for the interface as described by a cohesive zone model [27]. When ε > ε WID, the interfacial delamination grows concomitantly with buckling (Fig. 1b). 3. Wrinkle-Induced Delamination To simulate initiation and evolution of wrinkle-induced delamination, a two-dimensional finite element model was constructed in ABAQUS, as illustrated in Fig. 3(a). Both the film and the substrate were modeled by 2D quadrilateral elements (CPE8R). In addition, a layer of cohesive elements (COH2D4) was assigned along the interface to model the interaction between the film and the substrate. The bilinear traction-separation relation, as illustrated in Fig. 3(b), was used to describe the constitutive behavior of the cohesive elements. Figure 3. (a) Schematic illustration of the finite element model; (b) Illustration of the bilinear traction-separation law for the cohesive elements in ABAQUS. A delamination crack along an interface is typically under a mixed-mode condition due to elastic mismatch between the film and the substrate [20]. Consequently, the interfacial properties are required for both modes I and II conditions. For the bilinear traction-separation model, the interfacial properties to be specified include [28]: the initial elastic stiffness Kn and Ks, the cohesive strength σ n and τ s, and the interface toughness ΓI and ΓII. In this section, we ignore the effect of mode mix by taking 0 K K K s n = = , 0 σ τ σ = = s n and ΓI = Γ II = Γ for the interface. Previous studies have suggested that the initial elastic stiffness K0 plays a less important role in cohesive zone modeling. In the present study, the initial stiffness K0 is taken to be a constant (with 1 0 = f K h E ) in all simulations, while the strength (σ 0) and toughness (Γ) are varied. The elastic properties for the film and the substrate are such that Ef /Es = 1000 and vf = vs = 1/3. Figure 4. Numerical simulation of progressive wrinkling and interfacial delamination for an elastic thin film on a compliant substrate subject to increasing compression: (a) wrinkling with no delamination; (b) initiation of wrinkle-induced delamination; (c) subsequent growth of interfacial delamination. ε ε L h Cohesive elements K0 0 A B C c δ 0 σ δ σ (1-D)K0 Damage Initiation Final failure (a) (b) (a) (b) (c)
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