Bounds on the effective thermal conductivity of a dispersion of fully penetrable spheres

Abstract

We evaluate upper and lower bounds on the effective thermal conductivity K e of a model of two-phase composite materials in which one of the phases consists of spherical inclusions (or voids) of conductivity K 2 and volume fraction φ 2, dispersed randomly throughout a matrix phase of conductivity K 1 and volume fraction φ 1. Our evaluations compare third-order bounds of Beran and of Brown, which utilize the three-point matrix probability function of the model, with bounds of De Vera and Strieder (which apply only to the aforementioned “fully penetrable sphere” model) and of Hashin and Shtrikman. The comparisons are made over extended ranges of values of both φ 2 and α = K 2 K 1 and reveal that the best bounds among those we have tested (generally those of Beran) are sharp enough to give quantitatively useful estimates of K e for 0.1 ≤ K 2 K 1 ≤ 10 over a wide range of φ 2 values. They are sharp at high φ 2 values (i.e., φ 2 = 0.9) and very sharp at low φ 2 values (e.g. φ 2 = 0.1) where they remain useful for K 2 K 1 ≈ 100 . They are less sharp at intermediate values (e.g. φ 2 = 0.5). As is well known, such results immediately translate into equivalent results for the electrical conductivity, dielectric constant, or magnetic permeability of composites.