Two-scale convergence for locally-periodic microstructures and homogenization of plywood structures

TL;DR

This paper introduces a locally-periodic two-scale convergence method that broadens the scope of microstructure analysis and applies it to derive macroscopic equations for plywood structures in linear elasticity.

Contribution

It develops a new locally-periodic two-scale convergence framework and characterizes its limits, extending homogenization techniques to more complex microstructures.

Findings

Established compactness theorem for locally-periodic two-scale convergence.

Derived macroscopic equations for plywood structures in linear elasticity.

Extended homogenization theory to microstructures with variable periodicity.

Abstract

The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in $H^1(\Omega)$ are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.