Conical singularities in thin elastic sheets

TL;DR

This paper investigates the energy scaling behavior of developable cones (d-cones) in thin elastic sheets, showing that deviations from expected logarithmic scaling are bounded by a double logarithm of the sheet's thickness.

Contribution

The authors improve existing results on the energy scaling of d-cones, providing tighter bounds on deviations from the logarithmic energy behavior.

Findings

Deviation from logarithmic energy scaling is bounded by a constant times the double logarithm of thickness.

The study refines the understanding of energy distribution in d-cones under boundary conditions.

Results contribute to the theoretical understanding of elastic energy in thin sheets with conical singularities.

Abstract

When one slightly pushes a thin elastic sheet at its center into a hollow cylinder, the sheet forms (to a high degree of approximation) a developable cone, or "d-cone" for short. Here we investigate one particular aspect of d-cones, namely the scaling of elastic energy with the sheet thickness. Following recent work of Brandman, Kohn and Nguyen we study the Dirichlet problem of finding the configuration of minimal elastic energy when the boundary values are given by an exact d-cone. We improve their result for the energy scaling. In particular, we show that the deviation from the logarithmic energy scaling is bounded by a constant times the double logarithm of the thickness.