Clarification of the Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries

Abstract

Bounds on the effective elastic moduli of randomly oriented aggregates of hexagonal, trigonal, and tetragonal crystals are derived using the variational principles of Hashin and Shtrikman. The bounds are considerably narrower than the widely used Voigt and Reuss bounds. The Voigt-Reuss-Hill average lies within the Hashin-Shtrikman bounds in nearly all cases. Previous bounds of Peselnick and Meister are shown to be special cases of the present results.

Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

“…The idea is to take the known Voigt and Reuss averages of the elastic system stiffnesses or compliances, and then make direct use of this information by computing either the arithmetic or geometric mean of these two limiting values. These formulas have been found to be very effective for fitting real data in a wide variety of circumstances [44][45][46]. Clearly the same basic idea can be applied to any pairs of bounds for scalars, such as the Hashin-Shtrikman bounds; or, for complex constants, a similar idea based on finding the center-of-mass of a bounded region in the complex plane could be pursued.…”

“…These bounds are of Hashin-Shtrikman type, but were first obtained for hexagonal symmetry by Peselnick and Meister [52] with some corrections supplied later by Watt and Peselnick [46].…”

“…The precise values of the parameters G ± and K ± (being shear and bulk moduli of the HS isotropic comparison material) were given algorithmically by Watt and Peselnick [46].…”

Geomechanical analysis with rigorous error estimates for a double-porosity reservoir model

“…Hashin-Shtrikman bounds on K * for random polycrystals whose grains have hexagonal symmetry [22,23] show in fact that the K R value lies outside the bounds in many situations [21]. The Voigt [4] average for bulk modulus of hexagonal systems is well-known to be…”

“…It has been shown elsewhere [21,24] that the Peselnick-Meister-Watt [22,23] bounds for bulk modulus of a random polycrystal composed of hexagonal (or transversely isotropic) grains are given by…”