Asymptotic rigidity for shells in non-Euclidean elasticity

Itai Alpern, Raz Kupferman, Cy Maor*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We consider a prototypical “stretching plus bending” functional of an elastic shell. The shell is modeled as a d-dimensional Riemannian manifold endowed, in addition to the metric, with a reference second fundamental form. The shell is immersed into a (d+1)-dimensional ambient space, and the elastic energy accounts for deviations of the induced metric and second fundamental forms from their reference values. Under the assumption that the ambient space is of constant sectional curvature, we prove that any sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell into ambient space, having the reference second fundamental form. In particular, if the ambient space is Euclidean space, then the reference metric and second fundamental form satisfy the Gauss-Codazzi-Mainardi compatibility conditions. This theorem can be viewed as a (manifold-valued) co-dimension 1 analog of Reshetnyak's asymptotic rigidity theorem. It also relates to recent results on the continuity of surfaces with respect to their fundamental forms.

Original languageAmerican English
Article number109575
JournalJournal of Functional Analysis
Volume283
Issue number6
DOIs
StatePublished - 15 Sep 2022

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

  • Non-Euclidean elasticity
  • Riemannian manifolds
  • Rigidity
  • Shell theory

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