Paper detail

Instability of infinitesimal wrinkles against folding

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 $λ$, which localizes into a fold at sufficient buckling displacement $Δ$. By analyzing the regime $Δ<<λ$, we show that wrinkles are unstable to localized folding for {\em arbitrarily small} $Δ$. 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 $2-(π^2/4)(Δ/λ)^2$, in entire agreement with previous numerical results. The decay length of the amplitude is $κ^{-1}=(2/π^2)λ^2/Δ$. This case illustrates how a leading-order energy expression suggested by the infinitesimal displacement can give a qualitatively wrong configuration.