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 R(3), 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.