Instability of infinitesimal wrinkles against folding

TL;DR

This paper demonstrates that infinitesimal wrinkles on a floating membrane are inherently unstable to folding, revealing that localized folds can form even at very small displacements, challenging previous assumptions based on linear analysis.

Contribution

The study shows that wrinkles are unstable to localized folding at any small displacement, providing a new analytical approach and clarifying the energetic favorability of folds over wrinkles.

Findings

Wrinkles are unstable to folding for arbitrarily small displacements.

Localized boundary waves can be energetically favored over uniform wrinkles.

The surface pressure and decay length of folds are quantitatively characterized.

Abstract

We analyze the buckling of a rigid thin membrane floating on a dense fluid substrate. The interplay of curvature and substrate energy is known to create wrinkling at a characteristic wavelength $\lambda$, which localizes into a fold at sufficient buckling displacement $\Delta$. By analyzing the regime $\Delta<<\lambda$, we show that wrinkles are unstable to localized folding for {\em arbitrarily small} $\Delta$. After observing that evanescent waves at the boundaries can be energetically favored over uniform wrinkles, we construct a localized Ansatz state far from boundaries that is also energetically favored. The resulting surface pressure $P$ in conventional units is $2-(\pi^2/4)(\Delta/\lambda)^2$, in entire agreement with previous numerical results. The decay length of the amplitude is $\kappa^{-1}=(2/\pi^2)\lambda^2/\Delta$. This case illustrates how a leading-order energy expression suggested by the infinitesimal displacement can give a qualitatively wrong configuration.