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 → 0
to the critical points of the von
Kármán functional on S, recently proposed in [Lewicka et al.,
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)].
This result extends the statement in [Müller and Pakzad, Comm. Part. Differ.
Equ.33 (2008) 1018–1032], derived for the case
of plates when $S\subset\mathbb{R}^2$
.
The convergence holds provided the elastic energies of the 3d deformations scale
like h4 and the external body forces scale like h3.