Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries

Abstract

Peselnick, Meister, and Watt have developed rigorous methods for bounding elastic constants of random polycrystals based on the Hashin-Shtrikman variational principles. In particular, a fairly complex set of equations that amounts to an algorithm has been presented previously for finding the bounds on effective elastic moduli for polycrystals having hexagonal, trigonal, and tetragonal symmetries. The more analytical approach developed here, although based on the same ideas, results in a new set of compact formulas for all the cases considered. Once these formulas have been established, it is then straightforward to perform what could be considered an analytic continuation of the formulas (into the region of parameter space between the bounds) that can subsequently be used to provide self-consistent estimates for the elastic constants in all cases. These self-consistent estimates are easily shown (essentially by construction) to lie within the bounds for all the choices of crystal symmetry considered. Estimates obtained this way are quite comparable to those found by the Gubernatis and Krumhansl CPA (coherent potential approximation), but do not require any computations of scattering coefficients. * berryman1@llnl.gov

Combining analysis of random elastic polycrystals with poroelasticity for granular composites having orthotropic porous grains and fluid-filled pores

“…Some recent work [13,14] has shown how to simplify these harder to compute estimators to some extent. The present discussion gives an overview with the main new point being to show how this extra work does provide added value, since -in particularthe well-known Voigt-Reuss-Hill estimators often do not fall within the Hashin-Shtrikman bounds, but the self-consistent estimators (for good reasons [15]) have always been found to do so.…”

“…Some recent work of the author [13,14] has addressed some of these issues, including the orthorhombic case that will be the main emphasis 2 here.…”

“…Definitions of another useful shear factor G v ef f depend on the specific crystal symmetry under consideration (see Refs. [13,14] for specifics).…”

“…This work will be only briefly elaborated. In particular, a useful "shooting method of optimization" (which makes use of the computed values of the selfconsistent estimates in order to speed up the search for HS bounds) was developed previously to streamline the algorithm used to produce the Hashin-Shtrikman bounds [11,13,14,22].…”

Evaluating bounds and estimators for constants of random polycrystals composed of orthotropic elastic materials

“…Some recent work [6,7] has shown how to simplify these harder to compute estimators to some extent. The present discussion gives an overview with the main point being to show that this extra work does provide added value, since -in particular -the well-known Voigt-Reuss-Hill estimators often do not fall within the Hashin-Shtrikman bounds, but the self-consistent estimators (for good reasons [8]) have always been found to do so.…”