ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ)-DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS

摘要

In this two-part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micrornechanically-based approaches 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 harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N-dimensional (N-D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2- and 3-D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point-group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work - Part (Ⅱ), the explicit expression 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 syrnmetries are derived.