13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Figure 4 shows an example of the numerical simulation. Start from an elastic film perfectly bonded to a compliant substrate (with a small initial perturbation). As the applied compressive strain increases, the film wrinkles first (Fig. 4a), followed by nucleation of an interfacial crack at one of the wrinkle crests (Fig. 4b). Upon subsequent growth of the delamination, buckling of the film becomes localized while the neighboring wrinkles are flattened (Fig. 4c). In Section 2, initiation of wrinkle-induced delamination is predicted by comparing the maximum normal traction at the interface with the strength of the interface. This strength-based criterion for crack initiation is found to be in good agreement with the numerical results by the cohesive zone modeling approach. With the cohesive interface elements, a nonlinear post-buckling analysis is performed to simulate progressive wrinkling and wrinkle-induce delamination, as shown in Fig. 4. The amplitude of wrinkling or buckling is plotted as a function of the nominal strain in Fig. 5(a) for three different values of the interfacial strength, whereas the interfacial toughness is taken to be a constant, 5 10− Γ = E h f . For comparison, the wrinkle amplitude without delamination is plotted as a continuous solid line. For each value of the interfacial strength, the wrinkle amplitude follows the solid line before the initiation of delamination. At a critical nominal strain that depends on the interfacial strength, a delamination crack is nucleated and the buckle amplitude at the location of delamination increases abruptly, deviating from the solid line. Thus, the initiation of delamination may be readily observable in experiments by measuring the buckle amplitude. Figure 5. (a) Buckle amplitude as a function of the nominal strain, for three different values of interfacial strength. The solid line represents the wrinkle amplitude by the analytical solution in Eq. (5). (b) The critical strain for initiation of winkle-induced delamination as a function of the interfacial strength. The critical strain for initiation of wrinkle-induced delamination can be determined from Fig. 5(a). Alternatively, by the cohesive zone model, each cohesive element is characterized by a damage parameter (D), which varies between 0 and 1 during the separation process [28]. Thus, the critical strain for crack nucleation can be determined numerically when the damage parameter reaches 1 for at least one of the cohesive elements. Figure 5(b) plots the critical strain for wrinkle-induced delamination as a function of the interfacial strength, for three different values of interface toughness. For comparison, the critical strain predicted by the analytical formula in Eq. (8) is plotted as the solid line. Clearly, the critical strain is nearly independent of the interface toughness, justifying the strength-based criterion. The numerical results agree closely with the analytical prediction for relatively low interfacial strengths (e.g., 4 0 10 − < f E σ ). For higher interfacial strengths, however, Eq. (8) underestimates the critical strain. Apparently, since the critical strain increases with increasing interfacial strength, the linear approximation of the normal traction that leads to Eq. (8) becomes increasingly inaccurate as the strain goes beyond a few per cent. (a) (b)
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