The one-dimensional model for d-cones revisited

Abstract

AbstractA d-cone is the shape one obtains when pushing an elastic sheet at its center into a hollow cylinder. In a simple model, one can treat the elastic sheet in the deformed configuration as a developable surface with a singularity at the “tip” of the cone. In this approximation, the renormalized elastic energy is given by the bending energy density integrated over some annulus in the reference configuration. The thus defined variational problem depends on the indentation ${{h}}$ of the sheet into the cylinder. This model has been investigated before in the physics literature; the main motivation for the present paper is to give a rigorous version of some of the results achieved there via formal arguments. We derive the Gamma-limit of the energy functional as ${{h}}$ is sent to 0. Furthermore, we analyze the minimizers of the limiting functional, and list a number of necessary conditions that they have to fulfill.