On the simultaneous homogenization and dimension reduction in elasticity and locality of $\Gamma$-closure

TL;DR

This paper develops a framework for simultaneous homogenization and dimension reduction in elasticity, applicable to linear and nonlinear cases, including complex domains, and establishes the locality of the $mma$-closure property.

Contribution

It introduces a unified approach for homogenization and dimension reduction in elasticity, extending to perforated and oscillatory boundary domains, and proves the locality of $mma$-closure.

Findings

Framework applies to linear and nonlinear elasticity models.

Locality of $mma$-closure is rigorously established.

Applicable to perforated and oscillatory boundary domains.

Abstract

On the example of linearized elasticity we provide a framework for simultaneous homogenization and dimension reduction in the setting of linearized elasticity as well as non-linear elasticity for the derivation of homogenized von K\'arm\'an plate and bending rod models. The framework encompasses even perforated domains and domains with oscillatory boundary, provided that the corresponding extension operator can be constructed. Locality property of $\Gamma$-closure is established, i.e. every energy density obtained by the homogenization process can be in almost every point be obtained as the limit of periodic energy densities.