Effective conductivity of periodic media with cuboid inclusions

Abstract

This paper presents a numerical solution for the effective conductivity of a periodic binary medium with cuboid inclusions located on an octahedral lattice. The problem is defined by five dimensionless geometric parameters and one dimensionless conductivity contrast parameter. The effective conductivity is determined by considering the flow through the “elementary flow domain” (EFD), which is an octant of the unitary domain of the periodic media. We derive practical bounds of interest for the six-dimensional parameter space of the EFD and numerically compute solutions at regular intervals throughout the entire bounded parameter space. A continuous solution of the effective conductivity within the limits of the simulated parameter space is then obtained via interpolation of the numerical results. Comparison to effective conductivities derived for random heterogeneous media demonstrate similarities and differences in the behavior of the effective conductivity in regular periodic (low entropy) vs. random (high entropy) media. The results define the low entropy bounds of effective conductivity in natural media, which is neither completely random nor completely periodic, over a large range of structural geometries. For aniso-probable inclusion spacing, the absolute bounds of K eff for isotropic inclusions are the Wiener bounds, not the Hashin-Shtrikman bounds. For isotropic inclusion and isoprobable conditions well below the percolation threshold, the results are in agreement with the self-consistent approach. For anisotropic cuboid inclusions, or at relatively close spacing in at least one direction ( p > 0.2) (aniso-probable conditions), the effective conductivity of the periodic media is significantly different from that found in anisotropic random binary or Gaussian media.