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

TL;DR

This paper generalizes the von Karman equations to M-dimensional manifolds in N-dimensional space, deriving stress potentials, analyzing elastic ridges, and confirming scaling laws with previous 3D results.

Contribution

It extends elastic energy and von Karman equations to higher dimensions and analyzes ridge scaling, providing new theoretical insights.

Findings

Ridge width scales as h^{1/3}X^{2/3}

Total energy scales as h^{M}(X/h)^{M-5/3}

Bending energy is five times the stretching energy

Abstract

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.