On the conductivity of fiber reinforced materials

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Abstract
A two‐phase material in which the phase boundaries are cylindrical surfaces is considered. A technique exists for finding upper and lower bounds on the effective thermal conductivity (or electrical conductivity, permittivity, or magnetic permeability) of the composite in the direction perpendicular to the generators of the phase boundaries in terms of two different three point correlation functions. It is shown how a phase interchange theorem can be introduced into these bounds enabling us to express them in terms of a single geometrical constant of phase geometry. We determine what range of values of this factor is realizable for real phase geometries, and we show that the bounds thus obtained span exactly all realizable effective conductivities for such composites. Finally, we show that the bounds as expressed here enable us to use a knowledge of the effective conductivity of a composite for one ratio of constituent conductivities to narrow the bounds for some other ratio.