Crescent singularities and stress focusing in a buckled thin sheet: mechanics of developable cones

The localization of deformation is a simple consequence of the fact that bending a thin sheet is energetically cheaper than stretching it. In this paper we investigate conical singularities that appear on a crumpled sheet and called developable cones (d cones). We found that for a sample of a finite thickness the singularity is never pointlike but has a spatial extension in the form of a crescent. 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 exhibits two different scalings versus the deformation. From the cone profile, we measured the reaction force of the plate to deformation; and from force measurements, the energy that is necessary to create the singularity is characterized.