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

Abstract

While the well-known Voigt and Reuss (VR) bounds, and the Voigt–Reuss–Hill (VRH) elastic constant estimators for random polycrystals are all straightforwardly calculated once the elastic constants of anisotropic crystals are known, the Hashin–Shtrikman (HS) bounds and related self-consistent (SC) estimators for the same constants are, by comparison, more difficult to compute. Recent work has shown how to simplify (to some extent) these harder to compute HS bounds and SC estimators. An overview and analysis of a subsampling of these results is presented here with the main point being to show whether or not this extra work (i.e., in calculating both the HS bounds and the SC estimates) does provide added value since, in particular, the VRH estimators often do not fall within the HS bounds, while the SC estimators (for good reasons) have always been found to do so. The quantitative differences between the SC and the VRH estimators in the eight cases considered are often quite small however, being on the order of ±1%. These quantitative results hold true even though these polycrystal Voigt–Reuss–Hill estimators more typically (but not always) fall outside the Hashin–Shtrikman bounds, while the self-consistent estimators always fall inside (or on the boundaries of) these same bounds.