A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, p. 493-505

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 $S\subset {ℝ}^{2}$. The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3.

DOI : https://doi.org/10.1051/cocv/2010002
Classification:  74K20,  74B20
Keywords: shell theories, nonlinear elasticity, gamma convergence, calculus of variations
@article{COCV_2011__17_2_493_0,
author = {Lewicka, Marta},
title = {A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {2},
year = {2011},
pages = {493-505},
doi = {10.1051/cocv/2010002},
mrnumber = {2801329},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_2_493_0}
}

Lewicka, Marta. A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, pp. 493-505. doi : 10.1051/cocv/2010002. http://www.numdam.org/item/COCV_2011__17_2_493_0/

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