Crescent singularities and stress focusing in a developable cone

TL;DR

This paper studies developable cones (d-cones), revealing that their singularities are crescent-shaped rather than pointlike, with detailed analysis of curvature, deformation, and energy involved in their formation and buckling behavior.

Contribution

It introduces the concept of crescent-shaped singularities in finite-thickness d-cones and analyzes their curvature, deformation transition, and energy characteristics.

Findings

D-cone singularities are crescent-shaped, not pointlike.

Curvature scales differently during buckling.

Energy required for singularity formation is quantified.

Abstract

we investigate developable cones (d-cones) topology and mechanical properties. We found that for a sample of a finite thickness the singularity is never pointlike but has a spatial extension in form of a crescent. The variations of the sheet local curvature versus the distance from the d-cone tip for all deformations showed that the d-cone tip is rejected by a distance which depends on the deformation and on the singularity extension. A further deformation of the d-cone leads to a transition to a plastic deformation equivalent to a decrease in the singularity size characterized from curvature and profile analysis. The crescent radius of curvature is measured both at small deformations and at large deformations. It is found that during the buckling process, the curvature of the crescent exhibit two different scaling versus the deformation.From the cone profile, we measured the reaction force of the plate to deformation and from force measurements, the energy necessary to create the singularity is characterized.