Theoretical and numerical analysis of period-doubling bifurcation in sandwich systems

Abstract

Period-doubling bifurcation will appear when a stiff film is embedded into a compliant matrix or constrained to a soft substrate and undergoes in-plane compression. Though period-doubling bifurcation in bilayer systems have received extensive investigation, the mechanism of intricate post-buckling phenomena in sandwich systems remains an open question due to the inherent high nonlinearity. In this paper, a theoretical model for period-doubling bifurcation in sandwich systems is established and verified by finite element simulations. The proposed theoretical model is generalized to neo-Hookean materials and transformed into an equivalent nonlinear oscillator through an internal-to-external force transformation. Theoretical analysis reveals that an increase in the degree of up-down symmetry suppresses period-doubling bifurcation. Especially, when both the substrate and superstrate are identical, the influence of nonlinear deformation partly reflected by even high terms on the pressure difference on the film disappears, as a result, a novel period-doubling-like pattern appears due to the competitive effects from both sides strike a balance.