Limits of elastic models of converging Riemannian manifolds

TL;DR

This paper establishes the stability of elastic energy models for converging sequences of non-Euclidean manifolds, with implications for defect modeling and approximation of elastic bodies.

Contribution

It proves the $ ext{Gamma}$-convergence of elastic energies for sequences of converging non-Euclidean manifolds, extending the understanding of elastic models with defects and approximations.

Findings

$ ext{Gamma}$-convergence of elastic energies for converging manifolds

Limit models are insensitive to torsion fields in dislocation densities

Applicable to bodies with dense edge-dislocations and piecewise-affine approximations

Abstract

In non-linear incompatible elasticity, the configurations are maps from a non-Euclidean body manifold into the ambient Euclidean space, $\mathbb{R}^k$. We prove the $\Gamma$-convergence of elastic energies for configurations of a converging sequence, $\mathcal{M}_n\to\mathcal{M}$, of body manifolds. This convergence result has several implications: (i) It can be viewed as a general structural stability property of the elastic model. (ii) It applies to certain classes of bodies with defects, and in particular, to the limit of bodies with increasingly dense edge-dislocations. (iii) It applies to approximation of elastic bodies by piecewise-affine manifolds. In the context of continuously-distributed dislocations, it reveals that the torsion field, which has been used traditionally to quantify the density of dislocations, is immaterial in the limiting elastic model.