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 be nding a thin sheet is energetically cheaper than stretching it. In this pap er 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 i n the form of a crescent. A further deformation of the d cone leads to a tr ansition to a plastic deformation equivalent to a decrease in the singulari ty size characterized from curvature and profile analysis. The crescent rad ius of curvature is measured both at small deformations and at large deform ations. 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; a nd from force measurements, the energy that is necessary to create the sing ularity is characterized. [S1063-651X(99)09310-1].