A NEARLY EQUAL DISTANT GRID OF ORIENTATIONS FOR QUANTITATIVE TEXTURE ANALYSIS

The construction of a nearly equal distant grid of orientations is presented. Based on that grid texture components can be determined automatically. The obtained set of components can be interpreted as an ensemble of single orientations representing the texture. The grid may be used for the simulation of texture modifying processes. As examples textures of experimental deformed rock salt and aluminium are compared with results of numerical simulations of deformation (Taylor model).


INTRODUCTION
Solid state processes in polycrystalline materials are frequently associated with changes of texture caused by changing crystallite orientations. The textures of the initial and final material state can be calculated from diffraction pole figures (PF), e.g. by the component method (Truszkowski et al., 1973;Liicke et al., 1986;Helming and Eschner, 1990). This method can be applied for each crystal symmetry and also for multi-phase materials (Helming, 1993;Helming et al., 1994a). Furthermore texture estimates are possible even in those cases, where the quantity and quality of the measured PFs are limited (Helming et al., 1994b). A component is described by its preferred orientation, the halfwidth and the intensity/volume fraction (Matthies et al., 1987-90). Component orientations and halfwidths can be calculated by applying a non-linear iterative algorithm. For this first estimates are necessary which can be obtained interactively from the graphical representation of the diffraction data.
In this paper the construction of a nearly equal distant grid (NED grid) of orientations is explained (Helming, 1995). The NED grid is characterised by only one parameter, its resolution A, which is the maximum distance between two nearest neighbour orientations. For NED orientations and a resolution A, a set of texture components can be defined which completely fill the orientation space. The component intensities are determined using a linear fit algorithm. This procedure (NED grid and linear intensity fit) works automatically, i.e. independent of the user.
The set of components based on a NED grid corresponds to a discretisation of the texture. The component intensities may be interpreted as volumes of pseudo crystallites with NED orientations. This texture information can directly be used for numeric simulations of texture modifying processes, assuming that the simulation model does not consider interactions between the pseudo crystallites, like Taylor model (Taylor, 1938) or relaxed constraints (Van Houtte, 1988 (Bunge, 1982;Varshalovitc et al., 1975;Matthies et al., 1987-90: Helming, 1995  .toc fie sln---n sln-cos-+ n sln-cos--+ n x ncsi sln (3) A distance has to satisfy three demands: O(g, g2) C3(g2, g), o(g, g) O, c3(g, g2) < ((gl, g3) + cO(g2, g3), which are fulfilled by o3c. If two different orientations are rotated simultaneously by g, the distance t3 between them remains unchanged: o3(g, g2) (ggl, gg2). It is based on an arbitrary direction r marked rA if it is fixed in KA or rB if it is fixed in KB. rA has the same coordinates in KA as rB in KB. The trrrst (fight) operator rotates KA into K' through on rA. The second rotation turns K' into Ka by O around the node direction k. 0 is the angle (or distance) between rA (Ok, tpg) and rB (o, cosO cos 0, cos t + cos( tp^) sinO,, sinO.
After both operations rA is parallel to rB and KA is identical with Ka. The orientation distance follows from (2) cos cos cos- (8) The rotation axis n lies in a plane with (r + ra) and (r x ra) 2sin--cosn sin2---(rA + r)+ cos2---(rA x rB), in which fixes the direction of n. If r Z the angle is identical with tr (tpl + q)2) / 2. The angle tr was the basic quantity for the introduction of a nearly distortion free ODF presentation (Bunge, 1988;Helming et al., 1988). The orientation distance (cf. (8)) equally depends on the angle and direction distance O. Therefore the three-dimensional G-space can be understood as a combination of a two-dimensional direction space and a one-dimensional angle space.
where Kla, equals KA (cf. (5)). For 0 (Z1 not parallel to Z2). we have To construct a NED grid of orientations where the distances to nearest neighbours and its numbers are nearly constant for each orientation, one firstly needs a nearly equal-distant grid of directions. The equal area projection of that kind of direction grid (ref. to Ka) is shown in figure la. Its construction is based on a cube (Helming. 1995) with the edges parallel to the coordinate axes of Ka. Cube and pole sphere are concentric.
On the cube face for x const the directions shown are given by cartesian coordinates and-'<i,j <z.
where : 90/A must be an integer, even number. A is the maximum angle between nearest neighbour directions. It should be a divisor of 45. In the same manner the directions for the remaining five cube faces can be built. The special arrangement of directions in the cube grid allows two-dimensional interpolations.
Two steps are necessary to create a NED grid of orientations. At first each direction in figure l a should be identified as a ZA-axis (ref. to Ka) Secondly orientations with the same ZA-axis but different 0" (cf. (10)) are distinguished by different 0"-sections (cf. figure l a). The partition of 0" (cf. (7)) is 0-0, A, 2A 360-A with A of the direction grid. Without any symmetry the total number T of single orientations gt (0 since the number of 0--values for a ZA-axis is 47 and the number of directions on the cube is 6"t'2 + 2 (six faces with "t "2 points and two remaining edges). The NED grid is completely determined by the angle A called resolution. The resolution A characterises the maximum distance between nearest neighbour gt. For A 5 (: 18) only 140112 orientations are necessary unlike a 5-partition based on the Eulerian angles tpl, 4, tp2 with (360/5)2(180/5 + 1) 181476 orientations.
Since KB and not KA is the reference system (figure l a), an existing crystal symmetry can be used to reduce T in a simple manner. In the case of cubic symmetry one gets T 5838 (/t 5), because only the 24th part of a cube surface (presented in figure lb) has to be considered. If a Zg-axis is parallel to an n-fold symmetry axis of the crystallite the range of tr is reduced to 0 < cr _< 360/n,.

TEXTURE COMPONENTS
The texture of a crystalline phase in a polycrystalline material is quantitatively given by the orientation density function (ODF):  (Wassermann, 1939). A component is described by a model function f(g), which is locally restricted in the G-space. It has a maximum at a preferred orientation g and decreases with increasing orientation distance a (cf. (2)). The intensity I describes the volume fraction of all crystallites belonging to the component c. The quantity F gives the volume fraction of the crystallites, which are randomly oriented in the sample. It may be understood as the intensity of the only global component used in the model, which is given by f (g) 1 for each g G. The Gaussian model function (Matthies et al., 1987-90) on a PC monitor. This procedure is explained in more detail in (Helming and Eschner, 1990;Helming, 1993). Another possibility to define ge and b is based on the NED grid described above. The preferred orientations g are given by the T single orientations gt of the NED grid. The values b for all components should be equal to the resolution A. Only the intensities I must be calculated applying the least squares fit (20). This only requires a linear algorithm (I-fit) that can be performed automatically.
If all intensities F are set to 1/T an isotropic ODF with f(g) 1 is expected. In figure 2a the corresponding (111), (200) and (220) PFs are shown, prbduced by 218 components for A 15 (cubic crystal symmetry). Considering that the construction is based only on geometrical principles, the PFs show the expected isotropy. To improve the ODF the intensities F were refined fitting PFs with P,(y) 1. The recalculated PFs are shown in figure 2b. The visible deviations are caused by the geometric assumptions of the NED grid and the behaviour of the distribution (18).
To describe an arbitrary ODF with NED grid components the I-fit has to be performed with the corresponding PFs. To reduce a loss of texture information caused by the component model, the resolution A should be as small as possible (but not smaller than the resolution of the measured pole figures). This is explained in figure 3 where the texture of rock salt (natural state, PFs measured with neutrons, Scheffziik 1995) was approximated by means of three NED grids with A 45 , 15 , 11.25 (T 9. 218, 515 respectively).
If the I-fit results in some components with negative intensities, the solution should be considered as a first estimate of an iterative algorithm. Negative intensities may be caused e.g. by errors of measurement. The corresponding components should be deleted before the next iteration step, i.e. before repeating the I-fit. In figure  (111), (200) and (220) PFs of a casted A1 composite (Mg < 1,5%, measured with Xrays) are shown (Fels, 1996). Because of the large grain size (--100 /tm) they are affected by a poor statistics. The component description is based on two NED grids with A 15 and 11.25. As shown in figure 5 the number Tm of remaining positive components decreases with increasing iteration step m down to a limit of T4 60 and T5 106, respectively. The recalculated PFs of the final solution (m 5, A 11.95) are presented in figure 4b. The same material has been extruded (rectangular crosssection). Using NED components with A 15 (T 218) the I-fit yields 37 components after five steps. Figure 6 shows the measured and recalculated PFs

TEXTURE SIMULATIONS
The modelling of processes in polycrystals is often based on handling large numbers of independent single grain orientations. Their choice has to be suitable to represent the texture of the materials initial state. For A ---) 0 the NED components correspond to a discretisation of the ODF, i.e. the component intensities may be interpreted as volumes of pseudo crystallites with NED orientations. The NED crystallites generated by means of the automatic component method from PFs of natural rock salt (figure 3) and casted A1 (figure 4) have been taken as initial states for the simulation of  (220) PFs of a cast A1 composite (Mg < 1,5%, measured with X-rays) from Fels (1969). Because of the large grain size (= 100/tm) they are affected by a poor statistics.
b) The texture description by 106 components is based on a NED grid with A 11,25 and an iterative reduction (I-fit, five steps) of the component number (cf. tab. 1). deformation. Compression and extrusion (round cross-section) was simulated using the FC Taylor model (PC program by van Houtte, 1988). Z, was chosen parallel to the compression/extrusion-axis. The initial state PFs of rock salt show a (200)1 Z, -fibre. those of Lasted A1 a cube texture. In the case of rock salt the glide systems 100 < 110> and {110}<110> with equal critical resolved shear stresses and a total strain of 20% were used as input parameters. In additiopn to a (220)11 ZA fibre induced by the deformation a (200)_1_ ZA fibre is visible in the PFs (cf. figure 7a). The (200)1 ZA fibre is a remainder of the initial texture. The result corresponds to the PFs of the experimental deformed sample (20% strain; Scheffziik, 1995) shown in figure 7b. In the case of the A1 sample the simulation was carried out with glide systems 111 }<110> and a strain of > 99%. As expected for extrusion a (111),(100)llZa double fibre texture is obtained. The axial symmetry is disturbed by some spherical components. They should be meta-stable orientations occurring at the transition from the initial cube texture to the final double fibre texture. The corresponding PFs (figure 8a) are compared with the incompletely measured PFs of the experimentally deformed sample (80% strain; Fels, 1996) (80% deformation;Fels, 1996). The initial texture is shown in figure 4a.

1.25" -
of the extensive interactive search for preferred orientations. On the other hand the disadvantage of a large number of (initial) NED components must be accepted. For some applications like Taylor modelling the number of NED components should be as large as possible (A 0).
The used PC-version allows to calculate with a maximum of 1000 components. This is sufficient for the presented examples (A 10 , cubic crystal symmetry). For smaller resolutions or lower crystal symmetry improvements of soft-and hardware are required.
Based on the NED grid, graphical presentations of ODF for arbitrary directions of projections (not limited to ZA axis) Can be imagined. These presentations will reflect the orientation distance adequately like o-sections.