DETERMINATION OF THE COMPLETE ODF OF CUBIC SYSTEM MATERIALS BY THE MAXIMUM ENTROPY METHOD

A maximum entropy method (MEM) suitable for determining the complete ODF of cubic system materials is developed. This method treats the cubic system material texture data just as if they were if tegragonal besides introducing the relations among W lmn which characterize the cubic system. As experimental evidence this method has been applied to determine the complete ODF of a cold-rolled deep-drawing steel sheet sample and the result shows that the texture of the sample, as is supposed, consists of a {111} fiber and an incomplete (110) fiber only while the negative regions and the weak ghosts presented in the incomplete ODF of the same sample are disappeared here.


INTRODUCTION
The maximum entropy method (MEM) provides the most probable solution of problems with insufficient information, so it ought to be suitable for the complete ODF determination. This method was at first successfully applied to the determination of the inverse pole figure of hexagonal system material in 1987 and its good reliability was proved by means of model simulation as well (Wang et al. 1987). Then the complete ODF determination of hexagonal system material by the MEM was also achieved (Wang et al. 1989). Here the determination of the complete ODF of cubic system material by the MEM is made, which is basically the same as for the tetragonal system material besides introducing the relations among W Iron which characterize the cubic system.

PRINCIPLE
Suppose there are three sample symmetry planes each perpendicular to one of the sample coordinate axes, three crystal symmetry planes each perpendicular to one of the crystallographic axes and a 4-fold symmetry axis coincides with the C axis in addition, then the dimension of the asymmetric sub-orientation space adopted is p, 0, q=0-90 . Divide this space into J orientation elements (sin 0AOApAtp) and let V be the sample volume fraction oriented within 55 56 F. WANG ET AL. element j, then the relation between to(0j, pj, tp) and V takes the form the average orientation density V 32o(0, p, q) sin 0jAOApA (1) and J g 1.
(2a) i=l Express the incomplete ODF series of the sample as for the tetragonal system, truncate the series at L max and let R be the total number of the non-zero W lmn, however owing to the sample being really cubic there are some relations (Roe 1966) existing among W lmn which may be used to derive all the R values from those known W lmn. Now W lmn holds (Wang et  was determined by H. Hu (1986). There the incomplete ODF series was truncated at L max 16 with R 152 while only 58 of them were independent and known. The sub-orientation space was divided into J 5832 orientation elements with Ap A0 Aq 5. Comparing Figure 2 with Figure 1 it is found that their configurations are like each other but the former appears more dense with peak level over 12 and no negative region while the later contains massive negative regions with valley level near -1.8 and the peak level is about 8 only. Moreover, the texture of the sample corresponding to the complete ODF, as is supposed, is just composed of a {111} fiber and an incomplete (110) fiber while the weak peaks which appeared merely in the incomplete ODF are not consistent with rational texture components. Hence it may be considered that the complete ODF derived by the MEM is rather reliable.
Incidently the authors have been advised to test the MEM by directly measuring the orientations of crystallites of a sample one by one (Morris 1989). Hence, the crystallite orientations of an electrical silicon steel sheet containing 308 crystallites have been measured accurately by a modified etch pit method. Surely this was not enough to examine the complete ODF determination with such a few crystallite orientations (Wagner 1986), but it could be verified (Wang et al. 1990) that the inverse pole figure determination by the MEM was excellent.

CONCLUSIONS
The MEM developed for determining the complete ODF of cubic system materials is suitable for use and its reliability is confirmed by experiment test.