Geodesic
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

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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⊂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 : 10.1051/cocv/2010002
Classification : 74K20, 74B20
Keywords: shell theories, nonlinear elasticity, gamma convergence, calculus of variations 
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     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://geodesic.mathdoc.fr/articles/10.1051/cocv/2010002/}
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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

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