Convergence of two-scale expansions for elastic heterogeneous plates

TL;DR

This paper proves strong convergence of two-scale expansions for solutions to highly oscillatory elastic problems in thin domains, extending classical homogenization techniques to complex elasticity cases with new proof strategies.

Contribution

It extends homogenization results to elastic thin plates, especially addressing the challenging bending problem with a novel proof approach.

Findings

Strong convergence results for diffusion equations in thin domains.

Convergence established for membrane problems under classical assumptions.

New proof strategy developed for the bending elasticity problem.

Abstract

The aim of this article is to prove strong convergence results on the difference between the solution to highly oscillatory problems posed in thin domains and its two-scale expansion. We first consider the case of the linear diffusion equation and establish such results in arbitrary dimensions, by using a straightforward adaptation of the classical arguments used for the homogenization of highly oscillatory problems posed on fixed (non-thin) domains. We next consider the linear elasticity problem, which raises challenging difficulties in its full generality. Under some classical assumptions on the symmetries of the elasticity tensor, the problem can be split into two independent problems, the membrane problem and the bending problem. Focusing on two-dimensional problems, we show that the membrane case can actually be addressed using a careful adaptation of classical arguments. In the bending case, the scheme of the proof used in the membrane and diffusion cases can however not be straightforwardly adapted. In that bending case, we establish the desired strong convergence results by using a different strategy of proof, which seems, up to our knowledge, to be new.