13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- paper, we present a study on nucleation and subsequent growth of wrinkle-induced delamination using a cohesive zone model. 2. Wrinkling Analysis Consider an elastic thin film on an elastic compliant substrate, subject to lateral compression. Both the film and the substrate are taken to be linear elastic and isotropic, restricted to small, plane-strain deformation for the present study. Let ε be the nominal compressive strain, relative to the stress-free state. When ε is relatively small, the film/substrate bilayer is uniformly compressed and the surface is flat. When the strain exceeds a critical value, the film buckles and the substrate deforms coherently, forming surface wrinkles (Fig. 1a). The interface between the film and the substrate is assumed to be perfectly bonded in this section. Let h be the thickness of the film, while the substrate is considered infinitely thick. A well-known analytical solution predicts the critical strain for onset of wrinkling [9]: 2 3 3 4 1 = f s c E E ε , (1) where ) (1 2 ν = − E E is the plane-strain modulus with E for Young’s modulus and ν for Poisson’s ratio, and the subscripts f and s denote the film and substrate, respectively. The corresponding wrinkle wavelength is 1 3 3 2 = s f E E hπ λ . (2) In deriving the above analytical solution, the shear traction at the film/substrate interface was assumed to be zero. Alternatively, by assuming zero tangential displacement at the interface, a similar analytical solution can be obtained [8, 13]. The two solutions are identical if the substrate is incompressible (ν s = 0.5), in which case both the shear traction and tangential displacement are zero. However, when the substrate is compressible (ν s < 0.5), neither the shear traction nor the tangential displacement is zero at the interface as the film wrinkles. As a result, neither analytical solution accurately accounts for the effect of Poisson’s ratio of the substrate [15]. By taking into account both the shear traction and the tangential displacement at the interface, a more accurate analytical solution was developed [26], giving that 2 3 2 2 3 * 1 1 2 4 1 1 3 4 1 − − − − = s s f s c E E ν ν ε , (3) and 1 3 2 1 3 * 1 1 2 4 1 1 3 2 − − − = s s s f E E h ν ν π λ . (4) It was shown that, for a compressible substrate (ν s < 0.5), Eq. (1) underestimates the critical strain and Eq. (2) overestimates the wrinkle wavelength. The difference can be significant, up to about 20% for the critical strain and nearly 10% for the wavelength [26]. Beyond the critical strain, the wrinkle amplitude grows as a function of the nominal strain ε. An approximate solution for the wrinkle amplitude was obtained previously by a nonlinear approach that minimizes the strain energy in the film and the substrate [8, 9], namely
RkJQdWJsaXNoZXIy MjM0NDE=