Integral constraints in multiple scales problems with a slowly varying microstructure

TL;DR

This paper develops a homogenised model for electric potential in dielectric composites with slowly varying microstructure and integral constraints, using multiple scales analysis and flux transport theorem, validated against established methods.

Contribution

It introduces a novel form of integral constraint in multiple scales analysis for slowly varying microstructures, validated through comparison with existing approaches.

Findings

Derived a homogenised model incorporating integral constraints and microstructure variation.

Validated the model against established multiple scales analysis methods.

Demonstrated the approach with an application to dielectric composites.

Abstract

Asymptotic homogenisation is considered for problems with integral constraints imposed on a slowly-varying microstructure; an insulator with an array of perfectly dielectric inclusions of slowly varying size serves as a paradigm. Although it is well-known how to handle each of these effects (integral constraints, slowly-varying microstructure) independently within multiple scales analysis, additional care is needed when they are combined. Using the flux transport theorem, the multiple scales form of an integral constraint on a slowly varying domain is identified. The proposed form is applied to obtain a homogenised model for the electric potential in a dielectric composite, where the microstructure slowly varies and the integral constraint arises due to a statement of charge conservation. A comparison with multiple scales analysis of the problem with established approaches provides validation that the proposed form results in the correct homogenised model.