Elastic mechanical grain interactions in polycrystalline materials; analysis by diffraction-line broadening

Abstract

Experimental investigations have revealed that the Neerfeld–Hill and Eshelby–Kröner models, for grain interactions in massive, bulk (in particular, macroscopically isotropic) polycrystals, and a recently proposed effective grain-interaction model for macroscopically anisotropic polycrystals, as thin films, provide good estimates for the macroscopic (mechanical and) X-ray elastic constants and stress factors of such polycrystalline aggregates. These models can also be used to calculate the strain variation among the diffracting crystallites, i.e. the diffraction-line broadening induced by elastic grain interactions can thus be predicted. This work provides an assessment of diffraction-line broadening induced by elastic loading of polycrystalline specimens according to the various grain-interaction models. It is shown that the variety of environment, and thus the heterogeneity of the stress–strain states experienced by each of the individual grains exhibiting the same crystallographic orientation in a real polycrystal, cannot be accounted for by traditional grain-interaction models, where all grains of the same crystallographic orientation in the specimen frame of reference are considered to experience the same stress–strain state. A significant degree of broadening which is induced by the heterogeneity of the environments of the individual crystallites is calculated on the basis of a finite element algorithm. The obtained results have vast implication for diffraction-line broadening analysis and modelling of the elastic behaviour of massive polycrystals.

Keywords:

• elastic grain interaction
• single-crystal elastic anisotropy
• diffraction-line broadening
• elastic loading
• grain-interaction models
• quasi-isotropic materials
• stress–strain variation

Notes

Generally, the integral breadth of the only strain broadened profile is proportional to the square root of the strain variance [17, 27–29]. If the strain broadening is small as compared to the instrumental broadening, then, if a Gaussian-shape function is adopted for the instrumental and the only strain-broadened profiles, upon convolution, it follows that the additional broadening in the measured diffraction line profile as compared to the instrumental profile roughly scales with the strain variance, which, in turn, scales with for isotropic grain interaction [22]. Note, however, that if a Lorentzian (Cauchy) shape function is adopted for the instrumental and the only strain-broadened profiles, then the additional line broadening in the  profile as compared to the  profile would scale with the square root of the strain variance, and thus with . For a rigorous discussion on (also line-profile shape of) micro-(lattice-)strain broadening, see Ref. [30].

Roe convention is adapted for the grain-interaction model calculations, Section 4; Bunge convention has been applied to calculate grain orientation in Appendix B.