A Riemannian Approach to Reduced Plate, Shell, and Rod Theories

TL;DR

This paper develops a unified Riemannian framework for deriving reduced theories of elastic bodies, including plates, shells, and rods, using $ ext{Gamma}$-convergence, applicable to both compatible and incompatible elasticity.

Contribution

It introduces a Riemannian approach to dimension reduction in nonlinear elasticity, encompassing classical and non-Euclidean theories within a single unified framework.

Findings

Derives a limit theory for slender elastic bodies as a $k$-dimensional Riemannian manifold.

Includes classical plate, shell, and rod theories as special cases.

Extends to incompatible elasticity, covering non-Euclidean elastic models.

Abstract

We derive a dimensionally-reduced limit theory for an $n$-dimensional nonlinear elastic body that is slender along $k$ dimensions. The starting point is to view an elastic body as an $n$-dimensional Riemannian manifold together with a not necessarily isometric $W^{1,2}$-immersion in $n$-dimensional Euclidean space. The equilibrium configuration is the immersion that minimizes the average discrepancy between the induced and intrinsic metrics. The dimensionally reduced limit theory views the elastic body as a $k$-dimensional Riemannian manifold along with an isometric $W^{2,2}$-immersion in $n$-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 $\Gamma$-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.