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

Lewicka, Marta

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 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.

Classification : 74K20, 74B20
Keywords: shell theories, nonlinear elasticity, gamma convergence, calculus of variations

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