Universal relations for three-dimensional thermal, electric and magnetic properties

A periodic fibre-reinforced two phase material is considered here. The material properties of the constituents are anisotropic and may be specified by either the thermal, electrical conductivity, dielectric or magnetic permeability tenser. The orientation of each of the characteristic directions of the material is not parallel to the axis of geometric symmetry. The asymptotic homogenization method is used to study this doubly periodic composite. The interface between fibres and matrix must conform with the periodicity, but otherwise is arbitrary and may be disjoint in the periodic cell. A connection is found among the solutions of the three local problems that appear in the implementation of the method. This relation is basic for the elementary derivation of three universal relations for this three dimensional scalar problem. Several examples show how to relate the structure of the material tensor considered in this paper to that provided by the material data.