Buckling and post-buckling of a stiff film resting on an elastic graded substrate

Systems consisting of a hard layer resting on an elastic graded soft substrate are frequently encountered both in nature and industry. In this paper, we study the surface wrinkling problem of such a composite system subjected to in-plane compression. The Young’s modulus of the elastic substrate is assumed to vary along its depth direction. In particular, we investigate two typical variations in the modulus, described by a power function and an exponential function, respectively. Analytical solutions which permit to determine the critical compressive strain for the onset of wrinkling and the wrinkling wavelength are derived. A series of finite element simulations are performed to validate the theoretical solutions and demonstrate the prominent features of the postbuckling evolution of the system. The results may not only find applications in thin-film metrology and surface patterning but also provide insight into the wrinkling phenomena of various living tissues.