Physics – Condensed Matter – Soft Condensed Matter
Scientific paper
2010-09-13
Physics
Condensed Matter
Soft Condensed Matter
8 pages; important comment added at the beginning
Scientific paper
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.
Diamant Haim
Witten Thomas A.
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