Asymptotic Homogenization in the Determination of Effective Intrinsic Magnetic Properties of Composites

TL;DR

This paper develops a computational two-scale asymptotic homogenization method to accurately determine the effective magnetic permeability of composite materials, accounting for microstructural anisotropy and interaction effects.

Contribution

It introduces a homogenization algorithm using displacement method and proposes an anisotropy index based on permeability tensor eccentricity, enhancing prediction accuracy for complex microstructures.

Findings

Effective permeability bounds match Reuss and Voigt limits.

Periodic composites agree with analytical solutions.

Random microstructures converge quickly with volume size.

Abstract

We present a computational framework for two-scale asymptotic homogenization to determine the intrinsic magnetic permeability of composites. To this end, considering linear magnetostatics, both vector and scalar potential formulations are used. Our homogenization algorithm for solving the cell problem is based on the displacement method presented in Lukkassen et al. 1995, Composites Engineering, 5(5), 519-531. We propose the use of the meridional eccentricity of the permeability tensor ellipsoid as an anisotropy index quantifying the degree of directionality in the linear magnetic response. As application problems, 2D regular and random microstructures with overlapping and nonoverlapping monodisperse disks, all of which are periodic, are considered. We show that, for the vanishing corrector function, the derived effective magnetic permeability tensor gives the (lower) Reuss and (upper) Voigt bounds with the vector and scalar potential formulations, respectively. Our results with periodic boundary conditions show an excellent agreement with analytical solutions for regular composites, whereas, for random heterogeneous materials, their convergence with volume element size is fast. Predictions for material systems with monodisperse overlapping disks for a given inclusion volume fraction provide the highest magnetic permeability with the most increased inclusion interaction. In contrast, the disk arrangements in regular square lattices result in the lowest magnetic permeability and inadequate inclusion interaction. Such differences are beyond the reach of the isotropic effective medium theories, which use only the phase volume fraction and shape as mere statistical microstructural descriptors.