REFERENCE ODF CALCULATIONS FOR THE TORSION OF FCC RATE SENSITIVE POLYCRYSTALS

Analytic calculations are presented for the deformation induced ODF in a sheared FCC polycrystal. The uniform strain (Taylor) hypothesis and the theory of rate dependent slip are used. The fibre considered contains three interesting ideal orientations, and periodic ODF variations are observed. The present predictions, which can be readily and rapidly obtained, are useful for testing computer codes for the simulation of large deformation textures. positive divergences lead to ,ODF intensity de- creases. Here, rate sensitivity is taken into account, and a whole orientation fibre is analyzed.


INTRODUCTION
The development of deformation textures is usually simulated by using incremental procedures in which large numbers of individual grains and strain steps are treated by computer. The incremental nature of such simulations, however, leads to limitations in precision as a result of the appreciable number of steps required to attain large applied strains. Moreover, the orientation distribution function (ODF) is not easily deduced from these results for discrete orientations.
In the present study, the possibility is considered of predicting ODF's for the torsion of polycrystals by non-incremental analytical or semi-analytical means. Although limited to special situations, such results are of interest for the precise testing of computer-based incremental simulations. Moreover, the details of texture changes such as texture transitions and periodicity can be observed, which are not evident from the usual computations.
To the authors' knowledge, only one such analytical approach can be found in the literature, and it concerns a very special case. For a rate insensitive polycrystal to which the uniform strain (Taylor) model is applied, Wierzbanowski and C16ment t showed in 1984 that the ODF increases exponentially for (isolated) stable orientations when the divergence of the orientation velocity is negative. Shnilarly, positive divergences lead to ,ODF intensity decreases. Here, rate sensitivity is taken into account, and a whole orientation fibre is analyzed.

PRELIMINARY DEFINITIONS
A polycrystal consisting of face centered cubic (FCC) crystals deforming by means of {111}<110> slip is considered and submitted to simple shear. The direction of macroscopic shearing is denoted by the index 1, the normal to the shearing plane by 2, and the direction perpendicular to 1 in this plane by 3, see Of course, analytical or semi-analytical solutions can only be obtained in sufficiently simple situations. An example is the case where the set of orientations considered is a trajectory in orientation space; under these conditions, any orientation belonging to the fibre remains on it when deformation proceeds, and a single parameter is sufficient to specify the orientation change. In simple shear, two such cases can immediately be found: orientations with either a <100> or a <110> direction along the 3 axis. Each of these two series of orientations is specified by a single parameter defining the position about the 3 axis, and they are self-symmetric with respect to the synmetry plane of simple shear.
The first fibre was referred to as 'cube-on-face' by Gilormini eta/.5, and will not be considered any further in the present paper. The second fibre, 'cube-on-edge', is studied in more detail below, with emphasis put on the periodicity of the ODF. It is det'med by the angle between the 1 axis and the <100> direction in the 1-2 plane, see Figure

ANALYTICAL CALCULATION FOR THE CUBE-ON-EDGE FIBRE
In the reference (1,2,3) axes, the non-zero components of the prescribed strain rate are D12=D2--/2, and those of the associated stress state are It is shown below that values of S and z can be found such that the induced slips are responsible for D, and that such stress states fulfil the prescribed zero traction condition. (It has already been proved that the solution to this problem is unique, T6th et al. 4.) The resolved shear stresses induced by the above stress state on the 12 slip systems of the FCC structure can now be calculated; the corresponding slip rates are deduced from equation (1) and they are combined to give the resultant strain rate. Comparison with the prescribed strain rate yields the relations that must be satisfied by rtd S" [(a +d +2c +4b) sin2ql-2/2 (a -d) cos2q)] cos2a + Calculation of the ODF intensity variations re.quires knowledge of the divergence of orientation flow. More precisely, the continuity equation for the ODF intensity f derivexi by C16ment and Coulomb6, C16ment?, and Arminjon g is needed, which is Here a given orientation is followed as its moves through the 3-dimensional Euler space deemed by the following angles: a rotation q about the 3-axis (axis 1 rotates to 1') followed by a rotation 4> about the 1'-axis (axis 3 moves to 3') and a f'mal rotation about the 3'-axis. (The latter rotation is absent in the case of the cube-on-edge fibre, for which 4)--45.) g denotes the triplet of Euler angles.
There are three terms in the expression for the divergence the first one represents the variation of 0 when p varies, and it can be obtained from equations (2) Figure.2 Rotation rate (a) and divergence (b) along the cube-on-edge fibre.
ning two terms det'me the variations of the other components of the orientation rate when the two last angles vary. They can be deduced by a first-order analysis around the orientation fibre of interest. The calculations are quite lengthy but not difficult (see Gilormini  The results obtained for various m values are shown in figure 2b, from which it can be seen that the divergence is zero for C, near AI* and, because of symmetry, near A2*.
Due to the 180-period of the (q) function, any orientation located on the fibre rotates clockwise 180 about the 3-axis for an applied shear equal to (7) r d It can be seen from figure 3 that r increases sharply when tn approaches zero; the r=2g value for m=l was reported previously by Canova eta/. 3  that the ODF is also 1--periodic along this fibre for an initially centro-symmetric texture. (The period is 21" when the initial texture is not centro-symmetric.) This periodicity could not be determined with the previous combersome computer simulations; it also leads to an interpretation of the oscillatory axial stresses found by Harren eta/. 9 (using a more complicated model including elasticity): the period shown in figure 18 of their paper is about the 1" value deduced from the present simple analysis, viz. 6.9 for m=0.5, and 12.5 for m---0.2. This suggests that the cube-on-edge fibre is responsible for most of the axial stress, since the other orientations are not 1--periodic (see Gilormini

ODF CALCULATIONS
Along the cube-on-edge fibre, equation (5) can be integrated easily and gives div (9) f (q,T) fo () exp where fo is the initial ODF (for T=0) and is the initial location of the orientation which travels to the angle after a shear of t. By introducing the following one-variable functions that can be deduced from (4) and (6) and tabulated in the 0-180 range for any m value, a(x)= f: dqp and f'(x) = -div, dp equation (9) (10) shows that an initial ODF def'med by f* remains remarkably unchanged along the fibre of interest as deformation proceeds. Examples of the ODF's predicted along the cube-on-edge fibre by the present method are shown in figure 4 for various m values and for an initially uniform f=l ODF. Note that, due to the periodicity, the ODF for an applied shear of r-T is obtained from the results at T by symmetry with respect to --90 .The peaks near the ideal orientations (defined by dashed lines) appear clearly, and the locations of the tilted peaks considered in T6th et al. 4, for instance, can be studied precisely. Whatever the m value, the following developments take place in terms of the period r: (i) From 0 to 0.32r: three peaks appear, which are shifted first in the sense opposite to the applied shear and then in the shear direction, with intensities such that AI*<C<A2* reaching a maximum when the shift is zero. (ii) At about 0.32 a texture transition occurs; a peak develops rapidly near C, which is shifted opposite to the shear direction.
(iii) From 0.32r to 0.50r: smaller peaks appear near At* and A2* with the intensity of greater than that of A2*; the three peaks reach a maximum when the shift, which decreases, is zero.
(iv) At 0.50r: the ODF is symmetric with respect to the maximum at p--90 , the peak near is shifted in the sense of the applied shear (and that near A2* in the opposite sense).
Of course, the intensifies of the peaks vary with m, and they can be extremely high for low m values, but the peaks are very narrow under these conditions; moreover, it should be recalled that the peak intensifies grow exponentially in the rate insensitive case. Predicted ODFs for an initially isotropic texture, m--0.333 (a), 0.2 (b), 0.125 (c).