Elastic constants of a polycrystal with transversely isotropic grains, and the influence of precipitates

Abstract

The elastic moduli of an isotropic polycrystal containing aligned precipitates in its constituent grains, and of a polycrystal with hexagonal crystallites, are investigated theoretically. For the latter the moduli are determined by both Hill's and Walpole's self-consistent schemes for spherical grains and grain-shape effects, respectively, and by Hashin and Shtrikman's bounds. It is found that when the grains are somewhat elongated or deflated along the c- axis , elongated grains will give rise to a superior polycrystal than the deflated ones for most hcp crystals (e.g. Cd, Mg, Zn, α-Co). The self-consistent estimates generally lie inside the H-S bounds, but if the grains become too elongated or too flat, the estimates may lie outside the bounds. The elastic moduli of the precipitate-containing polycrystal are determined by the combination of Mori and Tanaka's theory (assuming the single crystal matrix to be elastically isotropic) and the self-consistent relations derived for hexagonal crystals. As the shape of precipitates varies from thin discs to spheres, and all the way to needles, the elastic constants of the polycrystals are found to always lie on or within the Hashin-Shtrikman bounds. With hard precipitates, discs provide the most effective reinforcement but, unlike for an ordinary two-phase composite, it never attains the upper bounds. When discs and needles become totally rigid or empty, the moduli are also given in terms of the disc and needle-density parameters.