Crescent Singularities in Crumpled Sheets

TL;DR

This paper investigates the crescent singularity in crumpled sheets, analyzing stress distributions, energy scaling, and core size, providing numerical evidence for a specific scaling law and discussing its limitations.

Contribution

It offers a detailed analysis of the core region in a developable cone, confirming a specific scaling law for core size and exploring stress and energy behaviors.

Findings

Core size scales as R_c ~ h^{1/3} R^{2/3}.

Outer region stresses scale differently with distance to the tip.

Numerical evidence supports the proposed scaling law.

Abstract

We examine the crescent singularity of a developable cone in a setting similar to that studied by Cerda et al [Nature 401, 46 (1999)]. Stretching is localized in a core region near the pushing tip and bending dominates the outer region. Two types of stresses in the outer region are identified and shown to scale differently with the distance to the tip. Energies of the d-cone are estimated and the conditions for the scaling of core region size R_c are discussed. Tests of the pushing force equation and direct geometrical measurements provide numerical evidence that core size scales as R_c ~ h^{1/3} R^{2/3}, where h is the thickness of sheet and R is the supporting container radius, in agreement with the proposition of Cerda et al. We give arguments that this observed scaling law should not represent the asymptotic behavior. Other properties are also studied and tested numerically, consistent with our analysis.