Computer simulation results for bounds on the effective conductivity of composite media

This paper studies the determination of third- and fourth-order bounds on the effective conductivity σ_e of a composite material composed of aligned, infinitely long, identical, partially penetrable, circular cylinders of conductivity σ₂ randomly distributed throughout a matrix of conductivity σ₁. Both bounds involve the microstructural parameter ζ₂ which is a multifold integral that depends upon S₃, the three-point probability function of the composite. This key integral ζ₂ is computed (for the possible range of cylinder volume fraction φ₂) using a Monte Carlo simulation technique for the penetrable-concentric-shell model in which cylinders are distributed with an arbitrary degree of impenetrability λ, 0≤λ≤1. Results for the limiting cases λ=0 ("fully penetrable" or randomly centered cylinders) and λ=1 ("totally impenetrable" cylinders) compare very favorably with theoretical predictions made by Torquato and Beasley [Int. J. Eng. Sci. 24, 415 (1986)] and by Torquato and Lado [Proc. R. Soc. London Ser. A 417, 59 (1988)], respectively. Results are also reported for intermediate values of λ: cases which heretofore have not been examined. For a wide range of α=σ₂/σ₁ (conductivity ratio) and φ₂, the third-order bounds on σ_e significantly improve upon second-order bounds which just depend upon φ₂. The fourth-order bounds are, in turn, narrower than the third-order bounds. Moreover, when the cylinders are highly conducting (α≫1), the fourth-order lower bound provides an excellent estimate of the effective conductivity for a wide range of volume fractions.