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A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry. (English) Zbl 1377.74013
Summary: 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 \(h \rightarrow 0\) to the critical points of the von Kármán functional on \(S\), recently proposed in [M. Lewicka et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 9, No. 2, 253–295 (2010; Zbl 1425.74298)]. This result extends the statement in [S. Müller and M. R. Pakzad, Commun. Partial Differ. Equations 33, No. 6, 1018–1032 (2008; Zbl 1141.74034)], 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 \(h^{4}\) and the external body forces scale like \(h^{3}\).

MSC:
74K25 Shells
74G65 Energy minimization in equilibrium problems in solid mechanics
74B20 Nonlinear elasticity
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