On Asymptotic Rigidity and Continuity Problems in Nonlinear Elasticity on Manifolds and Hypersurfaces

TL;DR

This paper advances the understanding of nonlinear elasticity on manifolds by establishing geometric rigidity estimates, proving asymptotic rigidity of elastic membranes, and extending continuity results to higher dimensions and co-dimensions.

Contribution

It introduces the first geometric rigidity estimate for mappings from Riemannian manifolds to spheres in the non-Euclidean setting and extends continuity results to arbitrary dimensions.

Findings

Established a geometric rigidity estimate for Riemannian manifolds to spheres.

Proved asymptotic rigidity of elastic membranes under geometric conditions.

Extended the Ciarlet-Mardare theorem to higher dimensions and co-dimensions.

Abstract

Intrinsic nonlinear elasticity deals with the deformations of elastic bodies as isometric immersions of Riemannian manifolds into the Euclidean spaces (see Ciarlet [9,10]). In this paper, we study the rigidity and continuity properties of elastic bodies for the intrinsic approach to nonlinear elasticity. We first establish a geometric rigidity estimate for mappings from Riemannian manifolds to spheres (in the spirit of Friesecke-James-M\"{u}ller [23]), which is the first result of this type for the non-Euclidean case as far as we know. Then we prove the asymptotic rigidity of elastic membranes under suitable geometric conditions. Finally, we provide a simplified geometric proof of the continuous dependence of deformations of elastic bodies on the Cauchy-Green tensors and second fundamental forms, which extends the Ciarlet-Mardare theorem in [18] to arbitrary dimensions and co-dimensions.