Modeling crystallographic texture evolution with finite elements over neo-Eulerian orientation spaces

A novel methodology is presented for the modeling of crystallographic texture based on the application of finite elements to represent and compute the orientation distribution function (ODF) over explicit discretizations of orientation space. Various orientation spaces are examined for this purpose. The neo-Eulerian axis angle spaces of Frank are preferred over the conventional Euler angle spaces for their superior properties. Properties of the neo-Eulerian spaces required by the modeling are derived. These include the reduction of the spaces under crystal symmetries to fundamental regions, the consequent boundary symmetry relationships, and the various Riemannian metrical properties of the spaces. The structure of crystal flow generated under the uniaxial extension of FCC crystals is examined over the cubic fundamental region of Rodrigues' space. Stabilized finite element schemes for the ODF conservation equation, developed previously for the texturing of planar polycrystals, are extended to the three-dimensional texturing considered. Properties of the schemes are illustrated by application to the texturing of FCC polycrystals over Rodrigues' space.