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

We study the linearized Fopl - von Karman theory of a long, thin rectangular elastic membrane that is bent through an angle $2\alpha$. We prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness $\sigma$ and the bending angle. We show that the minimum energy scales as ${\sigma }^{5\mathrm{/}3}{\alpha }^{7\mathrm{/}3}$. 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 $\sigma$.