Almost conical deformations of thin sheets with rotational symmetry

TL;DR

This paper rigorously analyzes nearly conical deformations in thin elastic sheets with rotational symmetry, focusing on the von-Kármán limit, and establishes existence and shape properties of energy minimizers.

Contribution

It provides a mathematical proof of minimizer existence and shape characterization for radially symmetric deformations in the von-Kármán limit, including energy bounds.

Findings

Existence of energy minimizers under specified conditions

Lower bound for elastic energy in the context of d-cones

Shape of minimizers determined up to exponentially decaying terms

Abstract

It has been found in numerical experiments that when one removes a sector from an elastic sheet and glues the edges of the sector back together, the resulting configuration is radially symmetric and nearly conical. We make a rigorous analysis of this setting under two simplyfying assumptions: Firstly, we only consider radially symmetric configurations. Secondly, we consider the so-called von-K\'arm\'an limit, where the size of the removed region as well as the deformations are small. We choose free boundary conditions for a sheet of infinite size. We show existence of minimizers of the suitably renormalized free energy functional. As a by-product, we obtain a lower bound for the elastic energy that has been conjectured in the related context of d-cones. Moreover, we determine the shape of minimizers at infinity up to exponentially decaying terms.