An obstacle problem for conical deformations of thin elastic sheets

TL;DR

This paper rigorously analyzes the limiting behavior of thin elastic sheets forming a developable cone when pressed into a cylinder, deriving a simplified obstacle problem and characterizing minimizers for small indentation.

Contribution

It establishes a b3-convergence result from a nonlinear elastic model to a fourth-order obstacle problem, providing a rigorous foundation for previous physics-based findings.

Findings

Derivation of a limit obstacle problem for conical deformations

Characterization of minimizers for small indentation b5

Rigorous justification of physics literature results

Abstract

A developable cone ("d-cone") is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance $\epsilon$. Starting from a nonlinear model depending on the thickness $h > 0$ of the sheet, we prove a $\Gamma$-convergence result as $h \rightarrow 0$ to a fourth-order obstacle problem for curves in $\mathbb{S}^2$. We then describe the exact shape of minimizers of the limit problem when $\epsilon$ is small. In particular, we rigorously justify previous results in the physics literature.