Derivation of analytical results from Stability and post-bifurcation of film-substrate systems

Morphological instabilities in soft solids with free surfaces lead to an array of deformation modes including wrinkling, creasing, folding and ridge localization. While homogeneous systems tend to form creases, stiff films over soft substrates usually exhibit surface waves. Here, we look to analytically investigate this transition through the effects of film stiffness and finite thickness on the post-bifurcation stability of these surface waves. By considering both the film and substrate as compressible Neo-Hookean solids, we apply bifurcation theory and Lyaponov–Schmidt–Koiter asymptotics to produce a phase diagram of the surface wave stability over the parameter space. While earlier works have studied the effect of film-to-substrate stiffness ratios for thin films on deep substrates in the incompressible setting, we consider the additional effects of both finite film thickness and Poisson ratio. To investigate the further evolution of these surface waves, we turn to computational methods through finite-element simulations with bifurcation branch-following techniques. We see that as the unstable surface waves evolve, they eventually lead to the beginnings of crease formation. Thus, when the surface waves are unstable, we would expect snap-back or snap-through behaviour leading to creases.

具有自由表面的软固体的形貌失稳会诱发一系列变形模式，涵盖起皱、折痕、折叠与脊线局域化。尽管均质体系通常会形成折痕，但附着于软基底的刚性薄膜通常会表现出表面波。本文旨在通过解析探究薄膜刚度与有限厚度对这类表面波分叉后稳定性的影响，研究这一转变过程。本文将薄膜与基底均视为可压缩新胡克（Neo-Hookean）固体，结合分叉理论与李雅普诺夫-施密特-科特（Lyaponov–Schmidt–Koiter）渐近分析法，构建了参数空间内表面波稳定性的相图。尽管已有研究针对不可压缩条件下深基底上的薄薄膜，探究了薄膜与基底的刚度比的影响，但本文同时考虑了有限薄膜厚度与泊松比的额外效应。为探究这类表面波的后续演化过程，本文采用结合分叉分支追踪技术的有限元模拟等计算方法。研究发现，不稳定的表面波在演化过程中最终会催生折痕的初始形成。因此，当表面波处于失稳状态时，预计会出现回弹或跳越行为，进而诱发折痕形成。