Magneto-resistance in three-dimensional composites

TL;DR

This paper investigates the magneto-resistance behavior of three-dimensional composite materials, extending previous two-dimensional models, and proves an inequality relating effective and phase-specific dissipation energies, validating the Kohler law.

Contribution

It extends the two-dimensional magneto-resistance analysis to three dimensions using H-convergence, establishing a key inequality and exploring the effects of magnetic field orientation.

Findings

Effective dissipation energy exceeds or equals the average of phase-specific energies.

The Kohler law holds for the composite, validating known behavior in homogeneous conductors.

The equality case is highly sensitive to magnetic field orientation.

Abstract

In this paper we study the magneto-resistance, i.e. the second-order term of the resistivity perturbed by a low magnetic field, of a three-dimensional composite material. Extending the two-dimensional periodic framework of [4], it is proved through a H-convergence approach that the dissipation energy induced by the effective magneto-resistance is greater or equal to the average of the dissipation energy induced by the magneto-resistance in each phase of the composite. This inequality validates for a composite material the Kohler law which is known for a homogeneous conductor. The case of equality is shown to be very sensitive to the magnetic fi eld orientation. We illustrate the result with layered and columnar periodic structures.