Equivalence of Kröner and weighted Voigt-Reuss models for x-ray stress determination

Abstract

The determination of stress in polycrystalline aggregates by diffraction-based methods relies on the proper choice of grain interaction model that links the observed strain to the elastic stress state in the ensemble. It is shown that for single-phase, polycrystalline samples composed of crystals with cubic symmetry, x-ray elastic constants (XEC) calculated under the Kröner model are equal to those from a weighted combination of Reuss and Voigt XEC, where the weighting factor is only a function of the single crystal elastic tensor coefficients. This weighting factor, xKr, generally scales with elastic anisotropy factor, A, with a value close to the Neerfeld limit for elastically isotropic materials (A 1). Materials that possess large values of A, and correspondingly small xKr, exhibit a greater deviation between the Neerfeld and Kröner limit XEC. A dimensionless parameter, Q, based on a different combination of elastic coefficients than A, demonstrates a monotonic trend with respect to xKr and may serve as a better metric for describing the elastic response of a polycrystalline ensemble as interrogated by x-ray diffraction. For crystals possessing lower symmetry, a similar analysis reveals that Kröner XEC are not a unique combination of Voigt and Reuss limits. In the case of hexagonal crystal symmetry, x Kr for a particular material varies as a function of the orientation parameter of the crystal, indicating that the degree of elastic anisotropy of the constituent crystals may impact the determination of stress depending on the choice of x-ray reflection. © 2013 AIP Publishing LLC.