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 much less than X has a width w similar to h(1/3)X(2/3) and a total energy E similar to mu h(M)(X/h)(M-5/3), where mu is a stretching modulus. 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. (C) 1997 American Institute of Physics.