The von Karman equations, the stress function, and elastic ridges in high dimensions

The elastic energy functional of a thin elastic rod or sheet is generalized to the case of an M-dimensional manifold in N-dimensional space. We derive potentials for the stress field and curvatures and find the generalized von Karman equations for a manifold in elastic equilibrium. We perform a scaling analysis of an M-1 dimensional ridge in an M = N-1 dimensional manifold. A ridge of linear size X in a manifold with thickness h << X has a width w ~ h^{1/3}X^{2/3} and a total energy E ~ h^{M} (X/h)^{M-5/3}. We also prove that the total bending energy of the ridge is exactly five times the total stretching energy. These results match those of A. Lobkovsky [Phys. Rev. E 53, 3750 (1996)] for the case of a bent plate in three dimensions.