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 $\epsilon$. Starting from a nonlinear model depending on the thickness $h>0$ of the sheet, we prove a $\Gamma$-convergence result as $h\to 0$ 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.

25 pages