An Asymptotic Analysis of the Period-Doubling Secondary Bifurcation in a Film/Substrate Bilayer

It has previously been observed experimentally and simulated numerically that when a thin film bonded to a much softer substrate is subjected to a uni-axial compression parallel to the interface, the initial buckled pattern will suffer a secondary bifurcation that doubles the period of the original pattern when the compressive strain reaches a critical value. This perioddoubling phenomenon is analyzed in this paper using an asymptotically self-consistent approach based on the exact theory of nonlinear elasticity. The predicted critical strain based on a fourterm expansion shows good agreement with that obtained using fully numerical simulations, and it is demonstrated that four is the minimum number of terms that should be included in order to give realistic predications. Although our illustrative calculations are conducted for neo-Hookean materials, the proposed approach can deal with any material models and can be extended to higher orders.