Compression induced folding of a sheet: an integrable system

TL;DR

This paper presents an exact solution for the shape of a fold in a compressed elastic film on a fluid substrate, revealing the buckling behavior and progression from sinusoidal waves to a large fold.

Contribution

It introduces an integrable system model that describes the entire buckling progression, including the shape and pressure characteristics, of a compressed elastic film on a fluid substrate.

Findings

Exact solution for fold shape in elastic films

Buckling occurs at a nonzero wavevector

Pressure decreases quadratically with displacement

Abstract

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 wavevector 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 wavevector whose magnitude remains invariant with compression.