Self-Consistent Calculation of the X-Ray Elastic Constants of Polycrystalline Materials for Arbitrary Crystal Symmetry

Abstract

For the calculation of load or residual stresses from measured strain data by means of x-rays or neutrons, the x-ray elastic constants are required. Usually they are calculated from the corresponding single crystal data assuming some models such as Voigt(strain compatibility), Reuss(stress equilibrium), Hill(arithmetic average of Voigt and Reuss) or Kroner. All these models, however, make assumptions of the coupling of the grains. Historically, the Voigt and Reuss limits have been investigated first assuming constant strain respectively constant stress in the grains. Hill proved that the average elastic constants for a macroscopically isotropic aggregate should fall within the limits imposed by the Reuss and Voigt models Although the Kroner model is often recommend for the calculation of the x-ray elastic constants, it assumes the polycrystalline material to behave as a model or perfect disorder. In this paper, the finite element method is used to guarantee both, stress equilibrium and strain compatibility across the grain boundaries. The polycrystalline material is assumed to behave as a three-dimensional mixture of macroscopic isotropic and fully orthotropic grains under elastic loading where the relative ratio of both types of grains can be varied between 0 and 100%. The macroscopically isotropic bulk values are calculated by an iteration procedure which was first proposed by Kneer. The method is applicable to materials with a random texture by calculating the hkl dependent quasiisotropic x-ray elastic constants as well as to specimens with a given texture using the information of the Orientation Distribution Function (ODF) by calculating the corresponding hkl- and orientation dependent stress factors. All crystal structures from cubic to triclinic are supported. The method is demonstrated on monoclinic SiO 2 with a random texture.