Compression Induced Folding of a Sheet: An Integrable System

The apparently intractable shape of a fold in a compressed elastic film lying on a fluid substrate is found to have an exact solution. Such systems buckle at a nonzero wave vector set by the bending stiffness of the film and the weight of the substrate fluid. Our solution describes the entire progression from a weakly displaced sinusoidal buckling to a single large fold that contacts itself. The pressure decrease is exactly quadratic in the lateral displacement. We identify a complex wave vector whose magnitude remains invariant with compression.

Figure 1

Schematic view of the system and its parametrization.

Figure 2

(a) Symmetric and (b) antisymmetric configurations of the sheet in the $xz$ plane as a function of decreasing pressure (increasing displacement) from a point close to the instability threshold ($P_c=2$) down to self-contact. The curves are vertically shifted by 3 from one another for clarity.

Figure 3

The fully “damped” self-intersecting configurations obtained for $P\to -2$. (a) Symmetric configuration for $P=-2$ [$\varphi (s)=4{tan}^{-1}(s/\cosh s)$]. (b) Antisymmetric configuration for $P=-1.9999$; as $P\to -2$, the two loops are pushed toward the boundaries, leaving a flat sheet ($\varphi =2\pi$) in between.