Effective Conductivity of a Composite in a Primitive Tetragonal Lattice of Highly Conducting Spheres in Resistive Thermal Contact With the Isolating Matrix

Abstract

A state-of-the-art study and a physical and numerical 3D finite element study of anisotropic conduction through composites filled with isometric inclusions of different conductivity were performed by modeling the longitudinal conduction across a tetragonal lattice of spheres in imperfect contact with the surrounding matrix. In dimensionless variables, the effective conductivity E is expressible as a function of a geometrical parameter B, reflecting the relative thickness of the gap between spheres, the Kapitza resistance C of the contact inclusion/matrix, and the relative resistivity D of the filler. The computation of some 600 E values at some 25 levels of the factors B, C, and D allows one to find some features, such as the leading role of the factor whose value is the highest of three, the low effect of the interactions between factors, the imperfect equivalence of the effects of the three factors, the slow and linear E dependence on the second and third greatest factor, and finally, the asymptotically exact linear relationship between E and the logarithmated sum of factors, with a slope depending only slightly on the relative magnitudes of factors.