Lower bounds for the energy in a crumpled elastic sheet—a minimal ridge

We study the linearized Föppl–von Karman theory of a long, thin rectangular elastic membrane that is bent through an angle 2 α. We prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness, σ, and the bending angle. We show that the minimum energy scales as σ 5/3 α 7/3 . This scaling is in sharp contrast with previously obtained results for the linearized theory of thin sheets with isotropic compression boundary conditions, where the energy scales as σ.