Conical surfaces and crescent singularities in crumpled sheets

We analyze the geometry and elasticity of the crescentlike singularity on a crumpled elastic sheet. We give a physical realization of this in terms of a free-boundary contact problem. An analytical solution is given for the universal shape of a developable cone that characterizes the singularity far from the tip, and some of its predictions are qualitatively verified experimentally. We also give a scaling relation for the core size, defined as the region close to the tip of the cone where the sheet is not developable.