On a boundary value problem for conically deformed thin elastic sheets

Abstract

We consider a thin elastic sheet in the shape of a disk that is clamped at its boundary such that the displacement and the deformation gradient coincide with a conical deformation with no stretching there. These are the boundary conditions of a so-called “d-cone”. We define the free elastic energy as a variation of the von Kármán energy, which penalizes bending energy in $L^p$ with $p\in \left(2,\frac{8}{3}\right)$ (instead of, as usual, $p=2$). We prove ansatz-free upper and lower bounds for the elastic energy that scale like $h^{p∕\left(p-1\right)}$, where $h$ is the thickness of the sheet.