Reduced Statistical Representation of Crystallographic Textures Based on Symmetry-Invariant Clustering of Lattice Orientations

As proven in numerous experimental and theoretical studies, physical and mechanical properties of materials are determined by their internal structure. In the particular case of polycrystalline metals and alloys, an important role is given to the orientation distributions of crystalline lattices, or, in other words, crystallographic textures. Physically reasonable models of texture formation are highly demanded in modern Material Science and Engineering since they can provide an efficient tool for designing polycrystalline products with improved operational characteristics. Models of interest can be obtained on the basis of statistical formulations of multilevel approaches and crystal elasto–visco–plasticity theories (in particular, Taylor–Bishop–Hill models and their various modifications are appropriate here). In such a framework, a representative volume element of a polycrystal is numerically implemented as a finite aggregate of crystallites (grains or subgrains) with a homogenized response at the macro-scale. Quantitative texture analysis of this aggregate requires estimating statistically stable features of the orientation distribution. The present paper introduces a clustering-based approach for executing this task with regard to preferred orientations. The proposed procedure operates with a weighted sample of orientations representing the aggregate and divides it into clusters, i.e., disjoint subsets of close elements. The closeness criterion is supposed to be defined with the help of a special pseudometric distance, which takes rotational symmetry of the crystalline lattice into account. A specific illustrative example is provided for better understanding the developed procedure. The texture in the clustered aggregate can be described reductively in terms of effective characteristics of distinguished clusters. Several possible reduced-form representations are considered and investigated from the viewpoint of aggregating elastic properties in application to some numerically simulated textures.