Bounds and self-consistent estimates for elastic constants of polycrystals composed of orthorhombics or crystals with higher symmetries

Methods for computing Hashin-Shtrikman bounds and related self-consistent estimates of elastic constants for polycrystals composed of crystals having orthorhombic symmetry have been known for about three decades. However, these methods are underutilized, perhaps because of some perceived difficulties with implementing the necessary computational procedures. Several simplifications of these techniques are introduced, thereby reducing the overall computational burden, as well as the complications inherent in mapping out the Hashin-Shtrikman bounding curves. The self-consistent estimates of the effective elastic constants are very robust, involving a quickly converging iteration procedure. Once these self-consistent values are known, they may then be used to speed up the computations of the Hashin-Shtrikman bounds themselves. It is shown furthermore that the resulting orthorhombic polycrystal code can be used as well to compute both bounds and self-consistent estimates for polycrystals of higher-symmetry tetragonal, hexagonal, and cubic (but not trigonal) materials. The self-consistent results found this way are shown to be the same as those obtained using the earlier methods, specifically those methods designed specially for each individual symmetry type. But the Hashin-Shtrikman bounds found using the orthorhombic code are either the same or (more typically) tighter than those found previously for these special cases (i.e., tetragonal, hexagonal, and cubic). The improvement in the Hashin-Shtrikman bounds is presumably due to the additional degrees of freedom introduced into the available search space.