ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅱ)-CRYSTAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS RESTRICTED BY VARIOUS MATERIAL SYMMETRIES

摘要

The explicit representations for tensorial Fourier expansion of 3-D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3-D ODF make up just a single irreducible mth-order tensor, the coefficients in the mth term of the Fourier expansion of a 3-D CODF constitute generally so many as 2m + 1 irreducible mth-order tensors.Therefore, the restricted forms of tensorial Fourier expansions of 3-D CODFs imposed by various micro- and macro- scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3-D CODFs contain remarkably reduced numbers of mth-order irreducible tensors than the number 2m + 1 . These results are based on the restricted forms of irreducible tensors imposed by various point-group symmetries, which are also thoroughly investigated in the present part in both 2- and 3-D spaces.