An asymptotic analysis of the period-doubling secondary bifurcation in a film/substrate bilayer

Fu, YB; Cai, ZX

doi:10.1137/15M1027103

An asymptotic analysis of the period-doubling secondary bifurcation in a film/substrate bilayer

Fu, YB; Cai, ZX

Authors

Yibin Fu y.fu@keele.ac.uk

ZX Cai

Abstract

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 period-doubling 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 four-term 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.

Acceptance Date	Aug 25, 2015
Publication Date	Nov 3, 2015
Journal	SIAM Journal on Applied Mathematics
Print ISSN	0036-1399
Publisher	Society for Industrial and Applied Mathematics
Pages	2381-2395
DOI	https://doi.org/10.1137/15M1027103
Keywords	bilayer, period-doubling, nonlinear elasticity, tunable patterns
Publisher URL	http://epubs.siam.org/doi/10.1137/15M1027103