An Asymptotic Variational Problem Modeling a Thin Elastic Sheet on a Liquid, Lifted at One End

TL;DR

This paper analyzes a 1D variational model of a thin elastic sheet on water, lifted at one end, revealing a simplified explicit solution in a specific inextensible regime where bending and weight are negligible.

Contribution

It introduces a Gamma-limit analysis to identify a parameter regime where the sheet's behavior simplifies significantly, providing explicit solutions.

Findings

Identified a parameter regime with inextensible sheets where bending and weight are negligible.

Derived a simple explicit solution in this asymptotic regime.

Provided a rigorous Gamma-limit analysis for the variational problem.

Abstract

We discuss a 1D variational problem modeling an elastic sheet on water, lifted at one end. Its terms include the membrane and bending energy of the sheet as well as terms due to gravity and surface tension. By studying a suitable Gamma-limit, we identify a parameter regime in which the sheet is inextensible, and the bending energy and weight of the sheet are negligible. In this regime, the problem simplifies to one with a simple and explicit solution.