Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three

TL;DR

This paper derives homogenized laws for high-contrast two-phase conductivities affected by magnetic fields in two and three dimensions, revealing dimension-dependent behaviors and extending classical transformations.

Contribution

It extends the Dykhne transformation to non-periodic high conductivities in 2D and analyzes 3D fiber-reinforced media, highlighting differences in homogenized conductivities.

Findings

Homogenized conductivity depends on magnetic field parameter in 2D.

Extension of Dykhne transformation to non-periodic settings.

Distinct homogenized behaviors observed in 3D fiber-reinforced media.

Abstract

Homogenized laws for sequences of high-contrast two-phase non-symmetric conductivities perturbed by a parameter $h$ are derived in two and three dimensions. The parameter $h$ characterizes the antisymmetric part of the conductivity for an idealized model of a conductor in the presence of a magnetic field. In dimension two an extension of the Dykhne transformation to non-periodic high conductivities permits to prove that the homogenized conductivity depends on $h$ through some homogenized matrix-valued function obtained in the absence of a magnetic field. This result is improved in the periodic framework thanks to an alternative approach, and illustrated by a cross-like thin structure. Using other tools, a fiber-reinforced medium in dimension three provides a quite different homogenized conductivity.