Bounds on the conductivity of a random array of cylinders

Abstract

We consider the problem of determining rigorous third-order and fourth-order bounds on the effective conductivity $\sigma_e$ of a composite material composed of aligned, infinitely long, equisized, rigid, circular cylinders of conductivity $\sigma_2$ randomly distributed throughout a matrix of conductivity $\sigma_1$. 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 $\phi_2$ (cylinder volume fraction) values in the range $0 \leqslant \phi_2 \leqslant 0.65$. The physical significance of the parameter $\zeta_2$ for general microstructures is briefly discussed. For a wide range of $\phi_2$ and $\alpha = \sigma_2/\sigma_1,$ 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 $\sigma_e$ are found to be sharp enough to yield good estimates of $\sigma_e$ for a wide range of $\phi_2$, even when the phase conductivities differ by as much as two orders of magnitude. When the cylinders are perfectly conducting ($\alpha = \infty$), moreover, the fourth-order lower bound on $\sigma_e$ provides an excellent estimate of this quantity for the entire volume-fraction range studied here, i.e. up to a volume fraction of 65%.