Mechanical elastic constants and diffraction stress factors of macroscopically elastically anisotropic polycrystals: the effect of grain-shape (morphological) texture

Abstract

Abstract The effect of the grain-shape ('morphological') texture of a polycrystal on the mechanical elastic constants and diffraction (X-ray) stress factors is investigated. To this end, the Eshelby–Kröner grain interaction model originally devised for polycrystals consisting of spherical grains is extended to ellipsoidal grain morphology. Results obtained for the mechanical elastic constants show that a polycrystal consisting of ellipsoidal grains with their principal axes aligned along common directions (i.e. when an ideal grain-shape texture occurs) is macroscopically elastically anisotropic. Also the diffraction (X-ray) stress factors are affected by the grain-shape texture; they reflect the macroscopic elastic anisotropy by resulting in nonlinear so-called sin2 ψ plots. In general, a grain-shape texture can have a moderate effect on the mechanical elastic constants and a pronounced effect on the diffraction elastic constants, depending on the crystal symmetry and single-crystal elastic anisotropy. Acknowledgement The authors thank Dr Sylvain Freour (Max Planck Institute for Metals Research, Stuttgart, Germany) for a helpful discussion. Notes Usually the arithmetic and not the geometric average is used. The geometric average of n numbers is defined as the nth root of the product of the n numbers. Thus, as an example, for Young's modulus, the geometric average E GA of the modulus E R according to the Reuss model and the modulus E V according to the Voigt model is defined as (E R E V)1/2. A body is said to be mechnically elastically transversely isotropic if the mechanical elastic constants exhibit rotational symmetry with respect to a particular symmetry axis. Of course, a real polycrystal cannot consist of ellispoidal grains (only). The ellipsoidal shape is an idealized shape, which is adopted in order to represent grains with (average) aspect ratios different from one, whereas a spherical shape is adopted for grains with an (average) aspect ratio of one. Taking a non-ideal grain-shape texture into consideration would require the definition of an additional reference frame P (attached to the principal shape axes of a grain), the definition of Euler angles α′, β′ and γ′ (describing the orientation of (the shape of) a grain in the specimen frame of reference) and the definition of a morphological ODF. EquationEquation (5), now being a threefold integral, would transform to a sixfold integral over both the Euler angles α, β and γ and the Euler angles α′, β′ and γ′. This case is not considered in this paper; see the discussion in § 3.1.1. Note that p S is the grain interaction tensor ϒ defined by Welzel and Mittemeijer (Citation2003) (cf. Equationequation (20) in the cited paper). Here and in the following, the term 'grain (crystallite)' is used in order to refer to all crystallites with the same crystallographic orientation in the specimen.