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 σ̂2 of the host in the initial system should be replaced by the corresponding transformed value μ̂2. 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, σ̂e, of the initial σ-system, can be found from the effective value μ̂e 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.