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 . The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3.
Mots clés : 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}, pages = {493--505}, publisher = {EDP-Sciences}, volume = {17}, number = {2}, year = {2011}, doi = {10.1051/cocv/2010002}, mrnumber = {2801329}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2010002/} }
TY - JOUR AU - Lewicka, Marta TI - A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 493 EP - 505 VL - 17 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010002/ DO - 10.1051/cocv/2010002 LA - en ID - COCV_2011__17_2_493_0 ER -
%0 Journal Article %A Lewicka, Marta %T A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 493-505 %V 17 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010002/ %R 10.1051/cocv/2010002 %G en %F 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, Tome 17 (2011) no. 2, pp. 493-505. doi : 10.1051/cocv/2010002. http://www.numdam.org/articles/10.1051/cocv/2010002/
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