The one-dimensional model for d-cones revisited

TL;DR

This paper rigorously analyzes a simplified elastic sheet model for d-cones, deriving the Gamma-limit of the energy as indentation approaches zero and studying the properties of minimizers.

Contribution

It provides a rigorous mathematical derivation of the energy limit and necessary conditions for minimizers in the d-cone model, extending previous physics-based results.

Findings

Gamma-limit of the energy functional derived as indentation tends to zero

Necessary conditions for minimizers identified

Provides a rigorous foundation for the d-cone model analysis

Abstract

A 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 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 the indentation is sent to 0. Further, we analyze the minimizers of the limiting functional, and list a number of necessary conditions that they have to fulfill.