# Pattern transitions in a compressible floating elastic sheet

## TL;DR

This paper extends the theory of pattern transitions in floating elastic sheets to include finite compressibility, analyzing how it affects wrinkling, folding, and the stability of various states.

## Contribution

It introduces a comprehensive model accounting for finite compressibility in floating elastic sheets, detailing the flat, wrinkled, and folded state transitions.

## Key findings

- Finite compressibility influences wrinkle and fold formation.
- The state diagram includes flat, wrinkled, and folded states.
- Compressibility affects the transition thresholds and pattern characteristics.

## Abstract

Thin rigid sheets floating on a liquid substrate appear, for example, in coatings and surfactant monolayers. Upon uniaxial compression the sheet undergoes transitions from a compressed flat state to a periodic wrinkled pattern to a localized folded pattern. The stability of these states is determined by the in-plane elasticity of the sheet, its bending rigidity, and the hydrostatics of the underlying liquid. Wrinkles and folds, and the wrinkle-to-fold transition, were previously studied for incompressible sheets. In the present work we extend the theory to include finite compressibility. We analyze the details of the flat-to-wrinkle transition, the effects of compressibility on wrinkling and folding, and the compression field associated with the pattern formation. The state diagram of the floating sheet including all three states is presented.