Note on the effective magnetic permeability of polycrystals

Many classical estimates for the effective behavior of heterogeneous materials can be reinterpreted in terms of inclusion problems. However, in the case of cubic polycrystals, a cubic permeability tensor for single crystals has to be written. In the framework of linear behavior, the description of the cubic symmetry reduces to isotropy. The heterogeneity of polycrystals, which results from single crystal anisotropy, cannot be described, and the classical estimates for the overall behavior of heterogeneous materials cannot be used. In this paper, we propose a particular description of the cubic symmetry for the magnetic permeability. We then derive estimates for the effective permeability of polycrystals from the solution of the basic inclusion problem, for both macroscopically isotropic and anisotropic polycrystals.