New mountain ridge modes in a film/substrate bilayer

Abstract

This article is concerned with the buckling of a film-substrate bilayer in a state of plane strain when it is subjected to a uni-axial compression along its free surface. Previous numerical simulations have indicated that pre-stretching the substrate in such a bilayer may lead to the formation of a mountain ridge mode as a secondary bifurcation. We present a scenario in which such a localized mode is also possible as a first bifurcation. It is first shown through a linear bifurcation analysis that by applying a pre-compression to the substrate, the stretch λ versus wavenumber k may develop a local minimum in addition to the local maximum that already exists in the absence of a pre-compression when the film is stiffer than the substrate. As a result, the λ at k = 0 may become larger than the local maximum if the pre-compression exceeds a threshold value, and hence becomes the critical stretch for bifurcation. This case is considered in this article, and it is shown through a weakly nonlinear analysis that multiple long wavelength modes may bifurcate subcritically from the uniform solution and quickly localize in the form of a mountain ridge. The solutions thus found are probably unstable, but form an essential part in the understanding of the global bifurcation behavior. It is hoped that our analytical results will guide future numerical simulations and experimental studies.