Simulation for Texture Formation of Both Face-Centered-Cubic Metals and Body-Centered-Cubic Ones Based on Rotational Symmetry among Principal Axes

Based on the rotational symmetry of the principal axes of X [100], Y [010] and Z [001], in fcc metal 24 possible combinations of the five slips on {111} planes on <110> direction while in bcc metal 72 possible combinations of the five slips on {110} planes on <111> direction by intersection of two kinds of {110} planes from the three ones composed of {110}, {101} and {011} are respectively chosen both based on Taylor’s formidable restriction rule of the five slips. In fcc metal, orientation at onset (minimum) of Taylor factor M value, i.e. the minimum total slip amount, shows the cube {100}<001> and the M value gradually increases by way of {100}<001>→ {100}<016>→ {100}<013>→ {100}<012>→ {100}<023> → {100}<0,9,11> with decrease of φ1 or does {100}<001>→ {016}<100>→{013}<100> →{0,6,13}<100> with increase of φ2, most of which were experimentally reported as indiscrete recrystallized orientations with lowest dislocation density named the cluster composed of cube and cube-family in fcc metal. In bcc metal, crystal rotation is carried out by only one solution among the 72 by the minimum total slip amount at every strain and simulates properly lengthy of accumulated researcher’s experimental results such as the three stable orientations of bcc metal in rolling {112}<110>, {11 11 8}<44 11> and {100}<011>.


Introduction
The principal axes of X [100], Y [010] and Z [001] are perpendicular to each other as the three orbits of , and by a rotational symmetry of mathematical group theory in such way that component X is not related to Y or Z one another whichever [1].There is a conservation quantity in the symmetry [2]. For example, a small ball at the top of Mexican hat may drop down on the bottom everywhere equally and symmetrically within 360°around the top due to existence of potential energy of the ball as a conservation quantity. Once the hat inclined, the ball would lose both the quantity and symmetry. As Taylor proved, crystal rotates so that slips occur associating themselves with the minimum total slip amount [3]. The minimum total slip amount in crystal by Taylor corresponds to both the conservation quantity in the rotational symmetry of cubic crystal and even Taylor factor M value itself of the material. These approaches will be useful for both face centered cubic (fcc) metal and body centered cubic (bcc) metal.

Fcc Metal
Here, dX i and dY i mean pair glides with common glide direction on neighbor planes in fcc metal and a, b, c, d are synthetic glide of them respectively. Figure 1 shows eight shears in four pairs of <110> direction on {111} plane around principal axis Z [001] in fcc metal. Similarly there are another eight shears in four pairs on both {111} planes around X [100] and Y [010] axes, which consequently satisfy the twelve glide systems of fcc metal equally. Relation among the eight shears dX 1 , dY 1 , dX 2 , dY 2 , dX 3 , dY 3 , dX 4 , dY 4 (1) for Z [001] group from geometrical relation in Figure 1.

and both of strain
There are another similar relations for X [100] and Y [010] groups.
The theory calculates sum of total shears by (2) each in these three groups of Z [001], X [100] and Y [010] and chooses minimum of them based on Taylor's theory [3] as the minimum total slip amount Γ and resultantly selects glide group which achieves maximum work satisfying yield surface of material.
Taylor factor M value has been also obtained as total shears, i.e., the minimum total slip amount Γ as shown in (2). In (2), it shall be remarked that if angle between plane whereon dX and one whereon dY is θ, there exists, on the other hand, also supplementary angle 180 °-θ owing to equivalent reverse slip on opposite side. Therefore if we ask average synthetic shear for both, their cosine terms cancel each other in average of the cosine theorem and finally there leaves only dX 2 + dY 2 term in the average irrespective of angle in fcc, bcc or NaCl whatever.
It may be important to understand that the M value depends solely both on crystal structure (fcc, bcc, NaCl) and deformation style, and doesn't depend on a degree of deformation. Thus, distribution (isosurface map) of Taylor factor M value in 3D ODF coordinates can be drawn in rolling.

Bcc Metal
There are three kinds of 110 planes having a common glide axis 111 in bcc crystal such as those illustrated on Figure 2 in case around Z [001] axis.    Calculation method is exemplified as follows.
Firstly, an intersection of two kinds of 110 planes from the three ones composed of 110 , 101 and 011 as illustrated on Figure 2 is chosen. As example, an intersection between 101 and 011 in Figure 2 is chosen and shears on 101 group planes on four 111 directions in case around Z [001] axis are named by turns dX 1 , dX 2 , dX 3 and dX 4 and those on 011 group planes are done dY 1 , dY 2 , dY 3 and dY 4 correspondently as shown in Figure 3. These are geometrically correlated to strain components dε 11 , dε 22 , dε 33 , dε 12 , dε 13 and dε 23 and crystal rotation dφ 1 , dφ 2 and dφ 3 around the principal axes X, Y and Z noted as 1, 2 and 3 respectively as demonstrated in (3) and accordingly a crystal rotation and texture formation can be calculated for the bcc crystal during any of compression, elongation, rolling and others by putting three of the eight shears to zero.
Second the eight slips, however, must be reduced to five shears according to Taylor's formidable combination (restriction) rule of the five slips as follows [3]. The geometrical condition rule for a given strain cannot be satisfied if the five shears are chosen so that two are taken from one group, i.e. one slip plane, and the remaining three are chosen one from each of the three remaining groups, in other words, all of which must be chosen so that two shears occur on each of two planes, one on the third and none on the fourth. This rule was properly applied to the present model as exemplified in Figure 3. By the Taylor In this case of the example between 101 and 011 from the three kinds of intersections around Z [001] axis in Figure 2, there are relations as demonstrated in (3) among the shears dX 1 , dX 2 , dX 3 and dX 4 on 101 group planes on four 111 directions as well as dY 1 , dY 2 , dY 3 and dY 4 on 011 group planes, and strain components, crystal rotations. Similar equations exist also in each case around X [100] and Y [010] cases. Further, applied to (3), a calculation is carried out for crystal rotation in the possible case (○) of dX 1 =dY 1 =0, dX 2 =0 on Figure 3 according to the Taylor's restriction rule of the five slips and its solution is introduced as in (4). The above example is only one of total 72 possible combinations of the five slips by the three kinds of intersections of two 110 planes on 111 directions in bcc metal and each of the 72 has respectively similar equations as (4). Third an actual crystal rotation by (4) proceeds only for the case of which the total slip amount Γ (gamma) defined by (2) is minimum among the 72 cases. Calculation of (2) is performed by inserting both equation (3) and (4)

Fcc Metal
According to the model in 2.1, Taylor factor M value is calculated as Γ itself, i.e. the minimum total slip amount in (2).

Bcc Metal
According to the model, Figure 6 demonstrates how final stable rolling orientation in bcc 11 11 8 4 4 11 is derived from initial orientation 1 1 2 1 1 1 with increase of strain. At each strain the model selects one solution among the 72 cases composed of 24 ones each belonging to any one of X, Y and Z group. In this Figure, it may be noted i) how the model gains the orientation with strain by way of selecting one of the partitions , and independently in the rotational symmetrical system and ii) how it holds a continuous value of Γ (gamma), the minimum total slip amount by way of (2) through the three X, Y and Z group, so smoothly as not to give such a fatal discontinuity in the value of Γ (gamma) at each strain as to lose the symmetry in the system. This is in reasonable accord with an expectation that amount of the external work by applied force to material shall be continuously changed with strain. As shown in Figure 6, due to symmetry [1], crystal selects one of X [100], Y [010] and Z [001] groups (mode) in each strain, holding a continuous value of Γ (gamma), the minimum total slip amount by way of (2) which is a conservation quantity [2] in the three X, Y and Z symmetry group and even Taylor factor M value itself of the material. In the column of "mode" in Figure 6, it is cleared how the model selects one solution among the 72 cases composed of 24 ones each belonging to any one of X, Y and Z group at each strain. Figure 7 illustrates by the model dynamically how orientations in ODFs map at rolling ratio of (a)6%, (b)38%, (c)62% and (d)95% are rigorously accumulated from initial random ones with strain in rolling by the present model and consequently the three stable orientations of bcc metal in rolling [4]- [9] such as 112 110 , 11 11 8 4 4 11 and 100 011 are attained as drawn on Figure 7(d).

Fcc Metal
As shown in Figure 4 and Figure 5 the isosurface of distribution of Taylor factor M value in the ODF coordinates of fcc metal under cold rolling is characterized by existence of the cube structure at onset (minimum) of the isosurface. The cube structure gradually develops and forms a skeleton of distribution of the M value in cold rolling of fcc metal. This cube M value may be plausibly related with the cube structure in recrystallized fcc metal. Figure 5(1) shows notation A, B and C of representative {100}<001> spots at the onset of Taylor factor M = 1.81 in fcc metal and Figure 5(2) shows details of the M value by 2D perspective of it through Φ axis. From those figures, Figure 8 is plotted for orientation change from the cube  {100}<001>  where  orientation  changes  gradually  {100}<001>→{100}<016>→{100}<013>→{100}<012>→{  100}<023>→ {100}<0, 9, 11> with decrease of φ 1 at spot "B" in Figure 5(1).     Figure 8 and Figure 9. Recently with development of observation with EBSD, the cube-family orientation grains are known to grow in clusters around the cube grain [12][13][14]. On the other hand, similar hypothesis [19] that "the minimum possible" Taylor factor M value of fcc metal in cold rolling at cube orientation with lowest dislocation density, shall encourage recovering from cold state more easily, could be linked with the present study. The present study may plausibly explain why the cube and the cube-family coexist in fcc recrystallized metal by means of low Taylor factor in addition to a fact that the {100}<0kl> grains in hot band inevitably rotate toward the cube by cold rolling [12]- [15].

Bcc Metal
According to the model, Figure 6 illustrates how final stable rolling orientation in bcc 11 11 8 [4 4 11] is derived from initial orientation 1 1 2 [1 1 1] with increase of strain sequentially selecting one among the 72 cases composed of 24 ones each belonging to any one of X, Y and Z group. By the model, it has been already shown that stable 112 [1 10] of bcc metal is derived from initial orientation 111 [1 10] in similar way [21], but at that time other stable orientation had not been ascertained yet.
As in Figure 6, Γ (gamma), i.e. the minimum total slip amount, by way of (2) in the three X, Y and Z group, is not constant and changes but so gradually as not to give such a fatal discontinuity in the value of Γ (gamma) as to lose the symmetry in the system. As generally known, even in the stable system composed of extreme symmetry, it can lose the symmetry immediately and transiently when exposed to external forces or other physical energy. Even in this case, however, as it is still the orthodoxy accepted by the majority, if a break of symmetry is so small where the symmetry recovers immediately and sustain still continuously the original symmetric state that the symmetry may still give birth to forceful analytical means to the phenomenon [22]- [24]. Besides example of Figure 6, it is supposed that such phenomena by the model may happen through the whole orientation as shown in Figure 7.
Similar model on fcc metal as described in model 2.1 requires 24 cases in total for one solution where two combinations of five slips are accepted by the Taylor's formidable restriction rule of the five slips similarly as in Figure 3 on 111 planes on every four directions of 110 for each group of X, Y and Z principal axes [25]- [27]. A ratio of the 24(=2×4×3) of fcc metal to the 72 of bcc metal which has three intersections of 110 planes, implies a ratio of one 111 slip plane on direction 110 of fcc metal to at least three slip planes 110 , 112 and 123 on direction of 111 in the pencil glide theory of bcc metal [4] [28] [29].

Conclusion
The principal axes of X [100], Y [010] and Z [001] are perpendicular to each other as the three orbits of ± , ± and ± by a rotational symmetry of mathematical group theory in such way that component X is not related to Y or Z one another whichever. There is a conservation quantity in the symmetry. As Taylor proved, crystal rotates so that slips occur associating themselves with the minimum total slip amount. The minimum total slip amount in crystal by Taylor corresponds to both the conservation quantity in the rotational symmetry of cubic crystal and even Taylor factor M value itself of the material. It will be demonstrated in this study that these approaches are useful for both face centered cubic (fcc) metal and body centered cubic (bcc) metal.
1. In fcc metal, distributions in 3D ODF coordinates for Taylor factor M value, i.e. the minimum total slip amount under cold rolling was calculated based on Taylor