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.

Coexistence and Coevolution of Wrinkle and Ridge Patterns in the Film–Substrate System by Uniaxial Compression

“…Although the structure of alternating packets of large and small undulations has been observed in some experiments, ,, the mechanisms of formation and evolution of such pattern are still unknown. Recently, Xu’s group theoretically obtained similar alternating packet structures in the cases of large prestretch (∼1.8) and small film–substrate modulus ratio (45–200). , But these formation conditions are different from our experiment with comparatively small prestretch (1.3) and extremely large modulus ratio (above 10 4 ). Further theoretical and simulation research is needed to illuminate the mechanisms of spontaneous strain localization and heterogeneous patterns with oscillatory amplitude and wavelength in a real film–substrate system.…”

Building blocks that govern spontaneous and programmed pattern formation in pre-compressed bilayers

“…Computational results revealed that the strain energies of period tripling and period quadrupling modes are comparable in the post-wrinkling regime, and this was suggested as a possible explanation for the coexistence of these two wrinkling modes in experiments. Xu et al developed latticed models [33] and an asymptotic modelling and resolution framework [34] to trace the post-wrinkling behaviour of bilayers with pre-strained substrate. A period-tripling mode is observed when the substrate is highly pre-compressed.…”

In memory of Prof. Hui-Hui Dai

“…The power of this method can be seen in the recent paper by Wang et al [1] who illustrated that quite a few existing linear and nonlinear plate theories can be recovered by appropriate specialisations. The first five papers in this special issue [2–6] are all concerned with further extensions/applications/validations of this method. In particular, Wang et al [2], Fu et al [3], and Wang et al [4] study, respectively, the nonlinear deformation of a clamped hyperelastic plate, morphology evolutions of multiple-period post-buckling modes in bilayers, and shape-programming of thin hyperelastic plates through differential growth.…”

“…The first five papers in this special issue [2–6] are all concerned with further extensions/applications/validations of this method. In particular, Wang et al [2], Fu et al [3], and Wang et al [4] study, respectively, the nonlinear deformation of a clamped hyperelastic plate, morphology evolutions of multiple-period post-buckling modes in bilayers, and shape-programming of thin hyperelastic plates through differential growth. The method has recently been employed to derive a consistent rod theory for a linear elastic anisotropic material with circular cross-section [7].…”