Wrinkles and folds in a fluid-supported sheet of finite size

TL;DR

This paper extends the theoretical understanding of wrinkle-to-fold transition in finite-sized fluid-supported elastic sheets, deriving exact and approximate solutions and identifying a critical confinement for the transition.

Contribution

It provides a new theoretical framework for finite sheets, including exact solutions for wrinkles and an approximate solution for folds, explaining the transition observed experimentally.

Findings

Exact solution for periodic wrinkle deformation

Approximate solution for localized fold

Critical confinement for transition: Δ_F = λ^2 / L

Abstract

A laterally confined thin elastic sheet lying on a liquid substrate displays regular undulations, called wrinkles, characterized by a spatially extended energy distribution and a well-defined wavelength $\lambda$. As the confinement increases, the deformation energy is progressively localized into a single narrow fold. An exact solution for the deformation of an infinite sheet was previously found, indicating that wrinkles in an infinite sheet are unstable against localization for arbitrarily small confinement. We present an extension of the theory to sheets of finite length $L$, accounting for the experimentally observed wrinkle-to-fold transition. We derive an exact solution for the periodic deformation in the wrinkled state, and an approximate solution for the localized, folded state. We show that a second-order transition between these two states occurs at a critical confinement $\Delta_F=\lambda^2/L$.