Bounds and self-consistent estimates of the elastic constants of polycrystals

The Hashin–Shtrikman bounds on the elastic constants have been previously calculated for polycrystalline materials with crystallites having general elastic symmetry (triclinic crystallite symmetry). However, the calculation of tighter bounds and the self-consistent estimates of these elastic constants has remained unsolved. In this paper, a general theoretical expression for the self-consistent elastic constants is formulated. An iterative method is used to solve the expression for the self-consistent estimates. Each iteration of the solution gives the next tighter set of bounds including the well-known Voigt–Reuss and Hashin–Shtrikman bounds. Thus, all of the bounds on the elastic constants and the self-consistent estimates for any crystallite symmetry are obtained in a single, computationally efficient procedure. The bounds and self-consistent elastic constants are reported for several geophysical materials having crystallites of monoclinic and triclinic symmetries.