Conduction in 2-D and 3-D dimensional spherically-symmetric anisotropic-coating inclusion composites

Abstract

Based on the variational three-point correlation results for the conductivity (thermal, electrical,...) of the multi-coated sphere assemblages, limiting procedures have been developed to construct the explicit expressions of the macroscopic conductivity and the respective microscopic gradient and flux fields for spherically-symmetric inclusion composites with anisotropic coating in d dimensions (d=2,3) under both the imposed macroscopic gradient or flux fields. Numerical examples are provided, verifying the convergence of the results for the macroscopic conductivity obtained from the minimum energy upper and lower bound approaches, and also that derived from the differential one. Explicit formal and numerical expressions of the microscopic conduction fields and illustration pictures of the microscopic flux lines within and around the typical anisotropic coating inclusion are also given. When the volume proportion of the outermost spherical shell increases to be the predominant one, one obtains the respective exact results for the important specific case of dilute solution of the anisotropic-coating inclusions suspended in the major matrix phase.