Crumpled paper

The crumpling of a piece of paper leaves permanent scars, showing a focusing of the stress. This is explained by looking at the geometry of the developable surfaces. According to Gauss, such a surface should have everywhere an infinite principal radius of curvature. The same condition holds when one minimizes the elastic energy of a bended plate; up to a small flexural part, this energy is minimum when the plate follows a developable surface. By considering the developable surfaces that are bounded by given closed curves in R3, we show that such a curve does not always bound a piece of developable surface. But one can find a special class of conical surfaces, the d–cones, that are still developable in the sense that they can be mapped on a plane by conserving the distances. This d–cone gives the outer solution of the elasticity equations, although the vicinity of the tip is described by the full equations, including the flexural term.