Exact relations between macroscopic moduli of composite media in three dimensions: Application to magnetoconductivity and magneto-optics of three-dimensional composites with related columnar microstructures

Generic three-dimensional (3D) exact relations are found between macroscopic or bulk effective moduli of composite systems with related microstructures. These relations can be established between effective values of material coefficients of the same type (e.g., conductivity and conductivity) as well as between different types of material coefficients (e.g., conductivity and permittivity). As example of possible application of these relations, a set of Keller-like quasi-3D relations are derived for the case of columnar-shaped parallel inclusions. The microstructure in the two samples can, in general, be different. In particular, exact relations between bulk effective magnetoconductivity tensor components of a pair of composite samples with interchanged constituents and different columnar microstructures (as well as between two composites with the same host but different inclusions, e.g., perfectly insulating and perfectly conducting) are found for general orientations of the applied magnetic field. Those relations are tested by comparing with a number of numerical calculations of macroscopic dc and ac response in such systems.