A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry

TL;DR

This paper demonstrates that critical points of 3D nonlinear elasticity functionals for thin shells of arbitrary geometry converge to the von Kármán functional's critical points as thickness approaches zero, extending previous plate results.

Contribution

It extends convergence results of critical points from plates to shells of arbitrary geometry, including weak solutions, under specific energy and force scalings.

Findings

Critical points of 3D elasticity converge to von Kármán critical points as thickness tends to zero.

Convergence holds for shells of arbitrary geometry, not just flat plates.

Results include convergence of weak solutions to static equilibrium equations.

Abstract

We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness $h$ and around the mid-surface $S$ of arbitrary geometry, converge as $h\to 0$ to the critical points of the von K\'arm\'an functional on $S$, recently derived in \cite{lemopa1}. This result extends the statement in \cite{MuPa}, derived for the case of plates when $S\subset\mathbb{R}^2$. We further prove the same convergence result for the weak solutions to the static equilibrium equations (formally the Euler- Lagrange equations associated to the elasticity functional). The convergences hold provided the elastic energy of the 3d deformations scale like $h^4$ and the external body forces scale like $h^3$.