A Riemannian approach to the membrane limit in non-Euclidean elasticity

TL;DR

This paper develops a Riemannian-based dimension reduction model for non-Euclidean elasticity, specifically the membrane limit of thin incompatible bodies, generalizing classical techniques to arbitrary dimensions and metrics.

Contribution

It introduces a general theorem for dimension reduction in Riemannian elasticity, applicable to bodies of any dimension and metric, focusing on membrane energy without bending effects.

Findings

Derives a membrane energy functional depending only on the first derivative of the immersion.

Shows the limiting model minimizes an integral over immersions of a submanifold.

Extends classical dimension reduction techniques to Riemannian incompatible elasticity.

Abstract

Non-Euclidean, or incompatible elasticity is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally-reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limiting model implies the minimization of an integral functional defined over immersions of a limiting submanifold in Euclidean space. The limiting energy only depends on the first derivative of the immersion, and for frame-indifferent models, only on the resulting pullback metric induced on the submanifold, i.e., there are no bending contributions.