arxiv-nlin:

Authors: Raz Kupferman, Jake P. Solomon

We derive a dimensionally-reduced limit theory for an -dimensional nonlinear elastic body that is slender along dimensions. The starting point is to view an elastic body as an -dimensional Riemannian manifold together with a not necessarily isometric -immersion in -dimensional Euclidean space. The equilibrium configuration is the embedding that minimizes the average discrepancy between the induced and intrinsic metrics. The dimensionally-reduced limit theory views the elastic body as a -dimensional Riemannian manifold along with an isometric -immersion in -dimensional Euclidean space and linear data in the normal directions. The equilibrium configuration minimizes a functional depending on the average covariant derivatives of the linear data. The dimensionally-reduced limit is obtained using a -convergence approach. The limit includes as particular cases plate, shell, and rod theories. It applies equally to “standard” elasticity and to “incompatible” elasticity, thus including as particular cases so-called non-Euclidean plate, shell, and rod theories.