Bounds on the Conductivity of a Random Array of Cylinders

We consider the problem of determining rigorous third-order and fourth-order bounds on the effective conductivity σ_e of a composite material composed of aligned, infinitely long, equisized, rigid, circular cylinders of conductivity σ_2 randomly distributed throughout a matrix of conductivity σ_1. Both bounds involve the microstructural parameter zeta_2 which is an integral that depends upon S_3, the three-point probability function of the composite (G. W. Milton, J. Mech. Phys. Solids 30, 177-191 (1982)). The key multidimensional integral zeta_2 is greatly simplified by expanding the orientation-dependent terms of its integrand in Chebyshev polynomials and using the orthogonality properties of this basis set. The resulting simplified expression is computed for an equilibrium distribution of rigid cylinders at selected φ_2 (cylinder volume fraction) values in the range 0 <=slant φ_2 <=slant 0.65. The physical significance of the parameter zeta_2 for general microstructures is briefly discussed. For a wide range of φ_2 and α = σ_2/σ_1, the third-order bounds significantly improve upon second-order bounds which only incorporate volume fraction information; the fourth-order bounds, in turn, are always more restrictive than the third-order bounds. The fourth-order bounds on σ_e are found to be sharp enough to yield good estimates of σ_e for a wide range of φ_2, even when the phase conductivities differ by as much as two orders of magnitude. When the cylinders are perfectly conducting (α = ∞), moreover, the fourth-order lower bound on σ_e provides an excellent estimate of this quantity for the entire volume-fraction range studied here, i.e. up to a volume fraction of 65%.