An asymptotic modeling and resolution framework for morphology evolutions of multiple-period post-buckling modes in bilayers

Abstract

A stiff thin film bonded to a compliant substrate initially buckles into wrinkles, following with some intricate advanced instability modes when the compression exceeds higher thresholds. Here, we present an asymptotic modeling and resolution framework to quantitatively predict and continuously trace secondary bifurcation transitions in the non-linear post-buckling region. An advantage of this framework, besides its applicability to finite-strain deformations, is its asymptotic consistency with the three-dimensional (3D) field equations and interface continuity conditions in a pointwise manner. Based on our model, we reveal intricate post-buckling responses involving successive advanced mode transitions, i.e., period-doubling, period-tripling, period-quadrupling, ridge, and hierarchical wrinkles in film–substrate bilayers upon excess compression. Apart from modulus ratio, pre-stretch and pre-compression of the substrate can alter surface morphology of film–substrate bilayers. With substrate pre-compression, a bilayer may eventually involve into a period-tripling or period-quadrupling mode. We observe hierarchical wrinkles and ridges in films attached on pre-stretched substrates. With high substrate pre-tension and modulus ratio, a novel pattern, namely, periodic ridges, appears at the secondary bifurcation. Fundamental understanding and quantitative prediction of non-linear morphology evolutions of soft bilayers play important roles in rational designs of wrinkle-tunable functional surfaces.