Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures

TL;DR

This paper derives the effective conductivities of non-periodic, high-contrast cylindrical composites in three dimensions under magnetic fields, extending previous periodic models to more general geometries without assuming specific cross-sectional shapes.

Contribution

It introduces a homogenization method based on H-convergence for non-periodic, high-contrast cylindrical composites with magnetic fields, generalizing previous periodic results.

Findings

Effective conductivities characterized in 3D for non-periodic structures

Extension of periodic homogenization results to more general geometries

Method applicable to high-contrast, magnetic-field-affected composites

Abstract

In this paper we determine, in dimension three, the effective conductivities of non periodic high-contrast two-phase cylindrical composites, placed in a constant magnetic field, without any assumption on the geometry of their cross sections. Our method, in the spirit of the H-convergence of Murat-Tartar, is based on a compactness result and the cylindrical nature of the microstructure. The homogenized laws we obtain extend those of the periodic fibre-reinforcing case of [M. Briane and L. Pater. Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three. J. Math. Anal. Appl., 393 (2) (2012), 563 -589] to the case of periodic and non periodic composites with more general transversal geometries.