Texture Representation

In the figure below, part (a) shows a three-dimensional section of a representative microstructure.  The study of microstructure is only useful as it relates to observable properties though, and a variety of the effective tensor properties of materials are governed primarily by the crystallographic texture, or the distribution of crystal orientations in a material.  As a result, the analysis of the microstructure in (a) may often be simplified by neglecting the spatial information and considering the microstructure from the standpoint of texture alone.  The remaining information is nevertheless considerable, and requires the development of updated and more powerful means to represent and analyze orientation information.

Historically, Euler angles are used for the analysis and presentation of texture information.  Euler angles describe an orientation by three sequential rotations about fixed axes, as is depicted in part (b) of the figure below.  One conspicuous feature of Euler angles are their asymmetry, since they more naturally describe rotations about the x and z axes than about the y axis.  The asymmetry has a number of repercussions, among which is a singularity in the definition of certain rotations and in functions of Euler angles as well.

Other descriptions of orientations used by the materials science community rely on the description of a rotation by an angle of rotation about a variable axis, as shown in part (c) of the figure below, and do not exhibit the asymmetry that is inherent to Euler angles.  Although there are other benefits to using these representations, they are generally not used as frequently as Euler angles.

One reason for the continued preference for Euler angles is that the only method to analytically represent a texture has historically been as a linear combination of functions of Euler angles.  Our research group has recently developed an alterative method that represents a texture as a linear combination of functions of the axis and angle of rotation. This is enabling the refinement of our ability to interpret and analyze texture information, and continues to be an area of active research.

The progressive abstraction of microstructural information.  (a) Three-dimensional reconstruction of the microstructure of a commercial austenitic stainless steel.  (b) Axis-angle representation of the texture of a crystalline material, completely characterizing the crystallographic orientations present.  (c) Stereographic projection indicating the orientation of  crystallographic planes of a material with a copper texture.

Published Articles:

The relationship of the hyperspherical harmonics to SO(3), SO(4) and orientation distribution functions
Mason JK; ACTA CRYSTALLOGRAPHICA SECTION A 65: x-x Part 4 JUL 2009 (PDF)

Hyperspherical harmonics for the representation of crystallographic texture
Mason JK, Schuh CA; ACTA MATERIALIA 56 (20): 6141-6155 DEC 2008 (PDF)