An obstacle problem for conical deformations of thin elastic sheets

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 $ε$. Starting from a nonlinear model depending on the thickness $h > 0$ of the sheet, we prove a $Γ$-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 $ε$ is small. In particular, we rigorously justify previous results in the physics literature.

25 pages