13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- width remains a constant (b/h ~ 40) during unloading. Meanwhile, the buckle amplitude decreases, following a different path from C to D. The buckle amplitude is nearly zero at D, with the nominal strain (ε ~ 0.000384) corresponding to the critical strain for onset of buckling with b/h ~ 40. Upon reloading, the buckle amplitude follows the same path of unloading from D to C, during which the delamination does not grow. Further increasing the nominal strain beyond C to point E (ε = 0.02), the delamination grows and the buckle amplitude increases. Apparently, the buckle amplitude during reloading follows a drastically different path compared to that for the first loading. Such a behavior qualitatively agrees with an experiment by Vella et al. [25]. However, in their experiment, a discontinuous jump of the buckle amplitude during the first loading was reported, presumably due to the unstable growth of the delamination. 0 0.005 0.01 0.015 0.02 0 50 100 150 Nominal strain, ε Delamination width, b/h L = 120h L = 220h L = 320h σ 0 /Ef = 5e-5 Γ/Ef h = 1e-5 Figure 8. Delamination width as a function of the nominal strain for different lengths used in the finite element model. It is noted that the growth of buckle-delamination is strongly influenced by the boundary condition. When the crack tip approaches one end of the model, where the symmetric boundary condition is assumed, the energy release rate drops rapidly and the crack is arrested when the energy release rate is less than the interface toughness. Figure 8 shows the delamination width as a function of the nominal strain by finite element models with three different lengths (L). Apparently, as L increases, the delamination width increases upon initiation, while the critical strain for initiation is insensitive to the model size. With a sufficiently large L/b, the localized buckle-delamination and periodic wrinkles may co-exist [26]. To simulate both initiation and co-evolution of wrinkling and buckle-delamination, the finite element model with cohesive elements is employed with L = 1000h and H = 200h. The bilinear traction-separation relation is assumed for the interface with 5 0 5 10 − = × f E σ and 5 5 10− Γ = × E h f . Figure 9 shows the evolution of deformation profiles of the film (solid lines) and the substrate surface (dashed lines) with increasing nominal strain. The film remains flat until the nominal strain reaches the critical strain for wrinkling ( 0.00556 = W ε ). Beyond the critical strain, periodic wrinkles form as shown in Fig. 9(a) at ε = 0.00758. As the nominal strain increases to the critical value for wrinkle-induced delamination ( 0.0076 = WID ε ), an interfacial crack initiates and grows rapidly, as shown in Figs. 9 (b) and (c) for two strains slightly above the critical strain. The growth of delamination leads to large, localized buckling, which relaxes the compressive stress in the film over a region close to the delamination. As a result, the wrinkles are flattened in the relaxation region. Further away from the delamination, the film remains wrinkled. The size of the relaxation region depends on the relative compliance of the substrate and the delamination width. With continuous growth of the delamination and a finite length (L = 1000h) in the finite element model, all the wrinkles are flatted eventually, as shown in Fig. 9(d).
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