Numerical Investigation of Isolated Crescent Singularity

TL;DR

This study numerically investigates the scaling properties of an isolated crescent singularity in elastic sheets, revealing distinct scaling laws for its width and radius of curvature related to thickness and separation.

Contribution

It introduces a method to produce isolated crescent singularities without central pushing and identifies different scaling behaviors for two length scales of the crescent.

Findings

Crescent width scales with thickness and separation to the 1/2 power.

Radius of curvature scales with thickness to the 1/3 and separation to the 2/3 powers.

Findings align with previous observations of developable cone sizes.

Abstract

In this paper we examine numerically the properties, especially the scaling properties, of an isolated crescent singularity similar to that of a developable cone. The desired isolated crescent region is produced by applying six potential forces to an elastic sheet in a controlled way, for which no central pushing force is required. Two types of length scales of the crescent are identified and shown to scale differently with the thickness and the separation of potentials. It is found that in one direction, the width of the crescent scales with both thickness and separation to the 1/2 power. In the other direction, the radius of curvature of the crescent scales with thickness to the 1/3 power and separation to the 2/3 power, in agreement with previous observation for the crescent size of a developable cone. We expect our findings of the double features of the crescent singularity to have importance in understanding the puzzling scaling behavior of the crescent.