The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells

TL;DR

This paper investigates the asymptotic behavior of nonlinear elastic energy in thin elliptic shells as their thickness approaches zero, revealing that under certain energy scaling, the limit theory simplifies to linear pure bending.

Contribution

It establishes the matching property of infinitesimal isometries on elliptic surfaces and connects this to the elasticity of thin shells using Gamma-convergence.

Findings

Limit of elastic energy reduces to linear pure bending for specific scaling.

Density of smooth infinitesimal isometries in the space of $W^{2,2}$ infinitesimal isometries.

Existence of exact isometric immersions approximating smooth infinitesimal isometries on elliptic surfaces.

Abstract

Using the notion of Gamma-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like $h^\beta$ with $2<\beta<4$. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of $W^{2,2}$ first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.