Determination of Hashin–Shtrikman bounds on the isotropic effective elastic moduli of polycrystals of any symmetry

• An improved method to calculate Hashin–Shtrikman bounds for polycrystal aggregates elasticity is given.
• The algorithm is illustrated graphically to improve comprehension of the underlying theory and numerical methods.
• The algorithm, using modern numerical environments, results in compact code.
• The new method works for crystals of any symmetry class.
• Results for low symmetry crystals, that could not previously be analyzed, are reported.

Although methods to determine optimal Hashin–Shtrikman bounds for polycrystals of cubic to monoclinic symmetry have been described, the calculation of bounds for triclinic crystals has not previously been possible. The recent determination of elastic moduli of common minerals with low symmetry provides motivation to extend the Hashin–Shtrikman formulation to lower symmetry. Here, Hashin–Shtrikman moduli, valid for crystals of any symmetry, are calculated as a function of the properties of a reference isotropic material. Defining the difference between moduli of the crystal and the moduli of the reference isotropic material as the residual tensor, the optimal lower (and upper) bounding moduli are found by a search along the boundary of positive (or negative) definite regimes of the residual elasticity tensor. The new numerical approach reproduces earlier results for higher symmetry crystals and successfully provides optimal bounds for triclinic crystals that have previously not been subject to analysis. The algorithm is sufficiently compact that implementation is relatively easy within any modern computational environment. Hashin–Shtrikman bounds for triclinic minerals in the plagioclase solid solution series are reported. These bounds are significantly narrower than extremal Voigt–Reuss bounds. The Hill averages moduli lie within the Hashin–Shtrikman bounds.