Orientation distribution functions for microstructures of heterogeneous materials (II) - Crystal distribution functions and irreducible tensors restricted by various material symmetries

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 restricte d forms of tensorial Fourier expansions of 3-D CODFs imposed by various mic ro- and macro-scopic symmetries are further established, and it is shown th at in most cases of symmetry the restricted forms of tensorial Fourier expa nsions of 3- D CODFs contain remarkably reduced numbers of mth-order irredu cible tensors than the number 2m + 1. These results are based on the restri cted 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.