Orientation distribution functions for microstructures of heterogeneous materials (I) - Directional distribution functions and irreducible tensors

In this two-part paper, a thorough investigation is made on Fourier expansi ons with irreducible tensorial coefficients for orientation distribution fu nctions (ODFs) and crystal orientation distribution functions (CODFs), whic h are scalar functions defined on the unit sphere and the rotation group, r espectively. Recently it has been becoming clearer and clearer that concept s of ODF and CODF play a dominant role in various micromechanically-based a pproaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmon ics and these spherical harmonics can further be expressed in terms of irre ducible tensors. The fundamental importance of such irreducible tensorial c oefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the mat erial constitutions and defects. In Part (I), the investigation about the i rreducible tensorial Fourier expansions of ODFs defined on the N-dimensiona l (N-D) unit sphere is carried out. Attention is particularly paid to const ructing simple expressions for 2- and 3-.D irreducible tensors of any order s in accordance with the convenience of arriving at their restricted forms imposed by various point-group ( the synonym of subgroup of the full orthog onal group) symmetries. In the continued work -Part (U), the explicit expre ssion for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point-group symmetries are derived.