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

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 $h^3$.