Wrinkling of an elastic sheet floating on a liquid sphere

TL;DR

This paper analyzes the wrinkling behavior of a thin elastic sheet floating on a liquid sphere by expanding the energy minimization problem in terms of sheet thickness, providing bounds and generalizing previous models.

Contribution

It introduces a variational framework for thin sheets on liquid spheres, expanding the energy in terms of thickness and establishing bounds, extending prior work to liquid substrates.

Findings

Derived the leading-order energy term from relaxed 1D problems

Established lower and upper bounds for the next-order energy term

Generalized previous elastic sheet wrinkling models to liquid substrates

Abstract

A thin circular elastic sheet floating on a drop-like liquid substrate is deformed due to incompatibility between the curved substrate and the planar sheet. We adopt a variational viewpoint by minimizing the non-convex membrane energy together with a higher-order convex bending energy. Focusing on thin sheets, we expand the minimum of the energy in terms of a small thickness ratio $h$, and identify the first two terms of this expansion. The leading-order term arises from the minimization of a family of one-dimensional relaxed problems, while for the next-order term we establish lower and upper bounds. This generalizes the previous work [P. Bella and R.V. Kohn. Wrikling of a thin circular sheet bonded to a spherical substrate, Philos. Trans. Roy. Soc. A, 375(2017). arXiv:1611.01781] to the physically relevant case of a liquid substrate.