Homogenization of Composite Ferromagnetic Materials

TL;DR

This paper rigorously derives the effective magnetic energy functional for periodic ferromagnetic composites using Gamma-convergence, aiding the prediction of their magnetic behavior in applications.

Contribution

It provides a novel mathematical framework for homogenizing the Gibbs-Landau free energy in periodic ferromagnetic composites using Gamma-convergence and two-scale convergence.

Findings

Derived the Gamma-limit of the Gibbs-Landau free energy functional.

Established a rigorous homogenization process for periodic ferromagnetic materials.

Provided insights into the effective magnetic properties of composite ferromagnets.

Abstract

Nowadays, nonhomogeneous and periodic ferromagnetic materials are the subject of a growing interest. Actually such periodic configurations often combine the attributes of the constituent materials, while sometimes, their properties can be strikingly different from the properties of the different constituents. These periodic configurations can be therefore used to achieve physical and chemical properties difficult to achieve with homogeneous materials. To predict the magnetic behavior of such composite materials is of prime importance for applications. The main objective of this paper is to perform, by means of Gamma-convergence and two-scale convergence, a rigorous derivation of the homogenized Gibbs-Landau free energy functional associated to a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the ferromagnetic media. We thus describe the Gamma-limit of the Gibbs-Landau free energy functional, as the period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to zero.