Generalization of the Concept of Sample Symmetry- Fuzzy Symmetry, Symmetroids, Similarity

The crystal orientation distribution function of polycrystalline materials, i.e. the texture,may exhibit internal symmetries due to symmetries of the production steps, or more generally,to the whole materials history. The “sharpness” of such symmetries can be quantifiedin terms of various symmetry parameters. If the symmetries of subsequent productionprocesses are different, e.g. of sheet rolling and deep drawing, then these symmetries maystill be recognized in the final texture. In the same way also similarities of textures andproperties of different materials can be quantified. Symmetry parameters have manypractical applications. Examples of that are the determination of rolling direction corrections,determination of the “plastic spin”, estimation of coarse-grained materials, orfinding the “correct” (symmetry adapted) axis system in a material.