Stress determination through diffraction: establishing the link between Kröner and Voigt/Reuss limits

Abstract

The quantification of stress in polycrystalline materials 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 aggregate. X-ray elastic constants (XEC) relate the strain as measured using X-rays to the state of stress in a quasi-isotropic ensemble of grains. However, the corresponding interaction models (e.g., Voigt and Reuss limits) often possess unlikely assumptions as to mechanical response of the individual grains. The Kröner limit, which employs a self-consistent scheme based on the Eshelby inclusion method, is based on a more physical representation of isotropic grain interaction. For polycrystalline aggregates composed of crystals with cubic symmetry, Kröner limit XEC are equal to those calculated from a linear combination of Reuss and Voigt XEC, where the weighting fraction, xKr, is solely a function of the single-crystal elastic constants and scales with the material's elastic anisotropy. This weighting fraction can also be experimentally determined using a linear, least-squares regression of diffraction data from multiple reflections. Data on metallic thin films reveals that this optimal experimental weighting fraction, x*, can vary significantly from xKr, as well as that of the Neerfeld limit (x = 0.5).