Effective conductivity tensor of ordered and disordered composite media: exact relations and numerical simulations

Generic three-dimensional (3D) exact relations were found recently (Phys. Rev. B (2002) 184416) between macroscopic or bulk effective moduli of composite systems with related microstructures which are, in general, different. As an example of possible application of these relations, a new numerical approach is proposed for simulations of composite systems with oblate inclusions: The initially anisotropic shape of the inclusions can be transformed to spherical, but the local conductivity tensor of the host in the initial system should be replaced by the corresponding transformed value . We simulate large 3D networks of circuit elements in this new μ-system using relaxation, network-reduction, and other methods. The effective value of the conductivity, , of the initial σ-system, can be found from the effective value of the transformed μ-system, using our exact relations. We propose to apply this approach for simulations of the phase transition in the high-Tc superconducting granular ceramics.