Bounds on the effective transport and elastic properties of a random array of cylindrical fibers in a matrix

This paper studies the determination of rigorous upper and lower bounds on the effective transport and elastic moduli of a transversely isotropic fiber-reinforced composite derived by Silnutzer and by Milton. The third-order Silnutzer bounds on the transverse conductivity σ_e, the transverse bulk modulus k_e, and the axial shear modulus ne, depend upon the microstructure through a three-point correlation function of the medium. The fourth-order Milton bounds on σ_e and µ_e depend not only upon three-point information but upon the next level of information, i.e., a fourpoint correlation function. The aforementioned microstructure-sensitive bounds are computed, using methods and results of statistical mechanics, for the model of aligned, infinitely long, equisized, circular cylinders which are randomly distributed throughout a matrix, for fiber volume fractions up to 65 percent. For a wide range of volume fractions and phase property values, the Silnutzer bounds significantly improve upon corresponding second-order bounds due to Hill and to Hashin; the Milton bounds, moreover, are narrower than the third-order Silnutzer bounds. When the cylinders are perfectly conducting or perfectly rigid, it is shown that Milton’s lower bound on σ_e or µ_e provides an excellent estimate of these effective parameters for the wide range of volume fractions studied here. This conclusion is supported by computer-simulation results for σ_e and by experimental data for a graphite-plastic composite.