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

Marta Lewicka, University of Minnesota, Department of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA. lewicka@math.umn.edu

Received: 10 December 2008
Revised: 3 December 2009

Abstract

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.