Applications of the Clifford torus texture representation to disorientations in single and multi-phase materials

In this paper we apply the concept of the Clifford torus and the derived square torus maps to the study of disorientations in microstructures. First, we interpret the Clifford torus in terms of the more commonly used orientation representations (Rodrigues-Frank vectors, 3D stereographic vectors, and homochoric vectors) and show representations of the torus in those spaces. This leads to a simple graphical interpretation of the generation and meaning of the square torus maps. Then we apply this approach to the study of disorientations in polycrystalline materials (CSL boundaries in grain boundary engineered Nickel) as well as intervariant boundaries in martensitic and bainitic steels. We show that pre-computed theoretical square torus maps can be used to determine population fractions of specific boundaries. • We propose the representation of disorientation data sets by means of Clifford square torus maps. • The square torus map approach allows for the determination of CSL boundary types in experimental data. • Martensite and bainite intervariant boundaries can be characterized using the square torus map approach. • Square torus maps can be used to determine population fractions of specific boundary types.