Instability of infinitesimal wrinkles against folding

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.

8 pages; important comment added at the beginning

References in corpus (1)

• Unfolding the Sulcus