Analysis of the effective conductivity of composite materials in the entire range of volume fractions of inclusions up to the percolation threshold

The analytical interpolation formulas for the effective electrical conductivity of fiber-reinforced and particulate composites for any volume fractions of inclusions are derived in the present paper. The Garnett formulas are used for a limiting case of small volume fractions of inclusions. And the formulas based on the Shklovskii–De Gennes model are adopted for a limiting case of large volume fractions of inclusions approaching the percolation threshold. The derivation is based on application of the method of asymptotically equivalent functions. This approach presents a natural generalization of the two-point Padé approximants to the case, when in one of the limits the interpolated function cannot be represented in the form of power series. The application of this interpolation technique made it possible to derive the formulas for the effective conductivity of fiber-reinforced and particulate composites with very high volume fractions of inclusions, up to the percolation threshold. The numerical results are compared with the known asymptotic expressions, and also with the existing exact expressions in some limiting cases, for example, in the case of statistically equivalent arrangement of the constituent materials. Comparison with the experimental data confirms the satisfactory accuracy of the obtained analytical results.