Plate theory as the variational limit of the complementary energy functionals of inhomogeneous anisotropic linearly elastic bodies

TL;DR

This paper investigates the asymptotic behavior of the complementary energy functionals of inhomogeneous, anisotropic elastic bodies as their thickness approaches zero, deriving a plate theory via Gamma-convergence.

Contribution

It provides a rigorous derivation of plate theory as the variational limit of the complementary energy for inhomogeneous, anisotropic elastic bodies using Gamma-convergence.

Findings

Identification of the limit functional as a dual problem for a 2D plate

Characterization of the convergence of stress fields

Establishment of a variational limit for complex elastic bodies

Abstract

We consider a sequence of linear hyper-elastic, inhomogeneous and fully anisotropic bodies in a reference configuration occupying a cylindrical region of height epsilon. We then study, by means of Gamma-convergence, the asymptotic behavior as epsilon goes to zero of the sequence of complementary energies. The limit functional is then identified as a dual problem for a two-dimensional plate. Our approach gives a direct characterization of the convergence of the equilibrating stress fields.