Calculus of Variations and Geometric Measure Theory

M. Lewicka - M. G. Mora - M. R. Pakzad

The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells

created by mora on 03 Dec 2008
modified on 04 Jul 2011


Published Paper

Inserted: 3 dec 2008
Last Updated: 4 jul 2011

Journal: Arch. Ration. Mech. Anal.
Volume: 200
Pages: 1023-1050
Year: 2011


Using the notion of Gamma-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like $h^\beta$ with $2<\beta<4$. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of $W^{2,2}$ first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.

Keywords: Gamma-convergence, nonlinear elasticity, shell theories, elliptic surfaces, isometric immersions