**5. Plane-strain compressed fcc metallic single crystals**

#### **5.1. Channel-die compressed {110}<001> aluminum single crystal**

The annealing texture of single-phase crystals of Al-0.05% Si of the Goss orientation {110}<001> deformed in channel-die compression was studied by Ferry et al. [37]. In the channel-die compression, the compression and extension directions were <110> and <001> directions, respectively. Their experimental results showed that, even after deformation to a true strain of 3.0 which is equivalent to a compressive reduction of 95%, the original orienta‐ tion was maintained as shown in Figure 25a. Figure 25b shows one (110) pole figure typical of a deformed crystal after annealing at 300 °C for 4 h. The comparison of Figures 25a and 25b suggests that the annealing texture is essentially the same as the deformation texture. 12 expected from SERM.

Running Title

They also reported that even after 90% reduction and annealing for up to 235 h, the orienta‐ tion was the same as that of the as-deformed crystal. For deformed specimens electropolish‐ ed and annealed for various temperatures between 250 and 350 °C, no texture change took place before and after annealing, although grains which had different orientations were sometimes found to grow from the crystal surface after very long annealing treatments. For samples deformed over the true strain range of 0.5 to 3.0 in their work, annealing at a given temperature resulted in similar microstructural evolution. They called the phenomenon dis‐ continuous subgrain growth during recovery. They stated that crystals of an orientation which was stable during deformation were generally resistant to Rex. This statement cannot be justified in light of single crystal examples in Sections 5.2 to 5.4. 15 The annealing texture of single-phase crystals of Al-0.05% Si of the Goss orientation {110}<001> 16 deformed in channel-die compression was studied by Ferry et al. [37]. In the channel-die 17 compression, the compression and extension directions were <110> and <001> directions, 18 respectively. Their experimental results showed that, even after deformation to a true strain of 3.0 19 which is equivalent to a compressive reduction of 95%, the original orientation was maintained as 20 shown in Figure 25a. Figure 25b shows one (110) pole figure typical of a deformed crystal after 21 annealing at 300˚C for 4 h. The comparison of Figures 25a and 25b suggests that the annealing 22 texture is essentially the same as the deformation texture. They also reported that even after 90% 23 reduction and annealing for up to 235 h, the orientation was the same as that of the as-deformed 24 crystal. For deformed specimens electropolished and annealed for various temperatures between 250 25 and 350˚C, no texture change took place before and after annealing, although grains which had 26 different orientations were sometimes found to grow from the crystal surface after very long 27 annealing treatments. For samples deformed over the true strain range of 0.5 to 3.0 in their work, 28 annealing at a given temperature resulted in similar microstructural evolution. They called the

29 phenomenon discontinuous subgrain growth during recovery. They stated that crystals of an

1 and the decrease indicates the texture change during subsequent grain growth, that is, AGG. A similar

3 Cho et al. [36] measured the drawing and Rex textures of 25 and 30 m diameter Au wires of over 4 99.99% in purity, which had dopants such as Ca and Be that total less than 50 ppm by weight. The Au 5 wires were made by drawing through a series of diamond dies to an effective strain of 11.4.

Figure 24 shows the grain size and the volume fraction of the <111> and <100> grains as a function of annealing time at 300 and 400°C. These values are based on EBSD measurements. The aspect ratio of grain shape was in the range of 1.5 - 2, which is little influenced by annealing time and temperature [36]. The grain growth occurs in whole area of the wire and is more rapid at 400°C than at 300°C as expected for thermally activated motion of grain boundaries. The volume fraction of the <111> grains decreases and that of the <100> grains increases with annealing time when Rex takes place, as

2 phenomenon is observed in drawn Ag wire during annealing (Figure 20).

14 5.1 Channel-die compressed {110}<001> aluminum single crystal

17

the positions at the midthickness of the top crystal, the bicrystal boundary, and the midthick‐ ness of the bottom crystal, respectively. The deformation textures of the two bicrystals, (123)[4 1-2]/(123)[-4-1 2] and (123)[4 1-2]/(-1-2-3)[4 1-2], channel-die compressed by 90%, are repro‐ duced in Figure 26. The initial orientation of the component crystals is also indicated in these pole figures. The annealing textures are shown in Figure 27. As Bricharski *et al*. pointed out, the

tion relationship with the deformation textures (compare Figures 26 and 27). Lee and Jeong [41] dicussed the Rex textures based on SERM. The slip systems activated during deformation and their activities (shear strains on the slip systems) must be known. Figure 28 shows the ori‐ entation change of crystal {123}<412> during the plane strain compression. Comparing the cal‐ culated results with the measured values in Figure 28, the measured orientation change during deformation seems to be best simulated by the full constraints strain rate sensitivity model. Figure 29 shows the calculated shear strain increments on active slip systems of the (123)[4 1-2] crystal as a function of true thickness strain, when subjected to the plane strain compression. The experimental deformation texture is well described by (0.1534 0.5101 0.8463)[0.8111 0.4242 -0.4027], or (135)[2 1-1], which is calculated based on the full constraints strain rate sensitivity model with *m* = 0.01. The reason why the measured deformation texture is simulated at the re‐ duction slightly lower than experimental reduction may be localized deformation like shear band formation occurring in real deformation. The localized deformation might not be reflect‐ ed in X-ray measurements. The scattered experimental Rex textures may be related to the non‐ uniform deformation. Now that the shear strains on active slip systems are known, we are in position to calculate AMSD. For a true thickness strain of 2.3, or 90% reduction, the *γ* values (Eq. 8) of the (111)[1 0-1], (111)[0 1-1], (1-1 1)[110], (1-1-1)[110], (1-1 1)[011], and (1-1-1)[101] slip systems calculated using the data in Figure 29 are proportional to 2091, 776, 1424, 2938, 76, and 139, respectively. The contributions of the (1-1 1)[011] and (1-1-1)[101] slip systems are negligi‐ ble compared with others. Therefore, the (111)[1 0-1], (111)[0 1-1], (1-1 1) [110], and (1-1-1)[110] systems are considered in calculating AMSD. It is noted that all the slip directions are chosen so that they can be at acute angle with RD, [0.8111 0.4242 -0.4027]. AMSD is calculated as follows:

209[10 1] 776[0 1 1] 1424 0.577[110] 2938 0.577[110] [4608 3293 2867] / /[0.7259 0.5187 0.4516]unit vector - - - - (12)

where the factor 0.577 originates from the fact that the slip systems of (1-1 1)[110] and (-111) [110] share the same slip direction [110] (Eq. 7). Two other principal stress directions are obtained as explained in Figure 6. Possible candidates for the direction equivalent to **S** in Figure 6 are the [011], [101], and [1-1 0] directions, which are not used in calculation of AMSD among six possible Burgers vector directions. The [011], [101], and [1-1 0] directions are at 87.3, 78.8

directions equivalent to **B** and **C** in Figure 6 are calculated to be [-0.0345 0.6833 0.7294] and [0.6869 -0.5139 0.5139] unit vectors, respectively. In summary, OA, OB, and OC in Figure 30, which are equivalent to **A**, **B**, and **C**, are to be parallel to the <100> directions in the Rexed grain. If the [0.7259 0.5187 -0.4516], [0.6869 -0.5139 0.5139] and [-0.0345 0.6833 0.7294] unit vectors are set to be parallel to [100], [010] and [001] directions after Rex (Figure 30), compo‐ nents of the unit vectors are direction cosines relating the deformed and Rexed crystal coordinate axes. Therefore, ND, [0.1534 0.5101 0.8463], and RD, [0.8111 0.4242 -0.4027], in the

, respectively, with AMSD. The [011] direction is closest to 90° (Figure 30). The

and 81.6o

<111> rotational orienta‐

Recrystallization Textures of Metals and Alloys

http://dx.doi.org/10.5772/54123

23

Rex textures of the fully annealed bicrystal specimens do not have 40o

**Figure 25.** pole figures for 95% channel-die compressed Al single crystal (a) before and (b) after annealing at 300 ˚C for 4 h. (Contour levels: 2, 5, 11, 20, 35, 70 x random) [37].

The result was discussed based on SERM [38]. The (110)[001] orientation is calculated by the full constraints Taylor-Bishop-Hill model to be stable when subjected to plane strain compression. The active slip systems for the (110)[001] crystal are calculated to be (111)[0-1 1], (111)[-101], (-1-1 1)[011], and (-1-1 1)[101], whose activities are the same. It is noted that all the slip direc‐ tions are chosen so that they can be at acute angle with the maximum strain direction [001]. AMSD is [0-1 1] + [-101] + [011] + [101] = [004]//[001], which is MYMD because [*S*44-2(*S*11-*S*12)] < 0 from compliances of Al [39]. When AMSD in the deformed state is parallel to MYMD in Rexed grains, the deformation texture remains unchanged after Rex (1st priority in Section 2).

#### **5.2. Aluminum crystals of {123}<412> orientations**

Blicharski et al. [40] studied the microstructural and texture changes during recovery and Rex in high purity Al bicrystals with S orientations, e.g. (123)[4 1-2]/(123)[-4-1 2] and (123)[4 1-2]/ (-1-2-3)[4 1-2], which had been channel-die compressed by 90 to 97.5% reduction in thickness. The geometry of deformation for these bicrystals was such that the bicrystal boundary, which separates the top and bottom crystals at the midthickness of the specimen, lies parallel to the plane of compression, i.e. {123} and the <412> directions are aligned with the channel, and the die constrains deformation in the <121> directions. The annealing of the deformed bicrystals was conducted for 5 min in a fused quartz tube furnace with He + 5%H2 atmosphere. The tex‐ tures of the fully Rexed specimens were examined by determining the {111} and {200} pole fig‐ ures from sectioned planes at 1/4, 1/2 and 3/4 specimen thickness. This roughly corresponds to the positions at the midthickness of the top crystal, the bicrystal boundary, and the midthick‐ ness of the bottom crystal, respectively. The deformation textures of the two bicrystals, (123)[4 1-2]/(123)[-4-1 2] and (123)[4 1-2]/(-1-2-3)[4 1-2], channel-die compressed by 90%, are repro‐ duced in Figure 26. The initial orientation of the component crystals is also indicated in these pole figures. The annealing textures are shown in Figure 27. As Bricharski *et al*. pointed out, the Rex textures of the fully annealed bicrystal specimens do not have 40o <111> rotational orienta‐ tion relationship with the deformation textures (compare Figures 26 and 27). Lee and Jeong [41] dicussed the Rex textures based on SERM. The slip systems activated during deformation and their activities (shear strains on the slip systems) must be known. Figure 28 shows the ori‐ entation change of crystal {123}<412> during the plane strain compression. Comparing the cal‐ culated results with the measured values in Figure 28, the measured orientation change during deformation seems to be best simulated by the full constraints strain rate sensitivity model. Figure 29 shows the calculated shear strain increments on active slip systems of the (123)[4 1-2] crystal as a function of true thickness strain, when subjected to the plane strain compression. The experimental deformation texture is well described by (0.1534 0.5101 0.8463)[0.8111 0.4242 -0.4027], or (135)[2 1-1], which is calculated based on the full constraints strain rate sensitivity model with *m* = 0.01. The reason why the measured deformation texture is simulated at the re‐ duction slightly lower than experimental reduction may be localized deformation like shear band formation occurring in real deformation. The localized deformation might not be reflect‐ ed in X-ray measurements. The scattered experimental Rex textures may be related to the non‐ uniform deformation. Now that the shear strains on active slip systems are known, we are in position to calculate AMSD. For a true thickness strain of 2.3, or 90% reduction, the *γ* values (Eq. 8) of the (111)[1 0-1], (111)[0 1-1], (1-1 1)[110], (1-1-1)[110], (1-1 1)[011], and (1-1-1)[101] slip systems calculated using the data in Figure 29 are proportional to 2091, 776, 1424, 2938, 76, and 139, respectively. The contributions of the (1-1 1)[011] and (1-1-1)[101] slip systems are negligi‐ ble compared with others. Therefore, the (111)[1 0-1], (111)[0 1-1], (1-1 1) [110], and (1-1-1)[110] systems are considered in calculating AMSD. It is noted that all the slip directions are chosen so that they can be at acute angle with RD, [0.8111 0.4242 -0.4027]. AMSD is calculated as follows:

They also reported that even after 90% reduction and annealing for up to 235 h, the orienta‐ tion was the same as that of the as-deformed crystal. For deformed specimens electropolish‐ ed and annealed for various temperatures between 250 and 350 °C, no texture change took place before and after annealing, although grains which had different orientations were sometimes found to grow from the crystal surface after very long annealing treatments. For samples deformed over the true strain range of 0.5 to 3.0 in their work, annealing at a given temperature resulted in similar microstructural evolution. They called the phenomenon dis‐ continuous subgrain growth during recovery. They stated that crystals of an orientation which was stable during deformation were generally resistant to Rex. This statement cannot

**Figure 25.** pole figures for 95% channel-die compressed Al single crystal (a) before and (b) after annealing at 300 ˚C

The result was discussed based on SERM [38]. The (110)[001] orientation is calculated by the full constraints Taylor-Bishop-Hill model to be stable when subjected to plane strain compression. The active slip systems for the (110)[001] crystal are calculated to be (111)[0-1 1], (111)[-101], (-1-1 1)[011], and (-1-1 1)[101], whose activities are the same. It is noted that all the slip direc‐ tions are chosen so that they can be at acute angle with the maximum strain direction [001]. AMSD is [0-1 1] + [-101] + [011] + [101] = [004]//[001], which is MYMD because [*S*44-2(*S*11-*S*12)] < 0 from compliances of Al [39]. When AMSD in the deformed state is parallel to MYMD in Rexed

grains, the deformation texture remains unchanged after Rex (1st priority in Section 2).

Blicharski et al. [40] studied the microstructural and texture changes during recovery and Rex in high purity Al bicrystals with S orientations, e.g. (123)[4 1-2]/(123)[-4-1 2] and (123)[4 1-2]/ (-1-2-3)[4 1-2], which had been channel-die compressed by 90 to 97.5% reduction in thickness. The geometry of deformation for these bicrystals was such that the bicrystal boundary, which separates the top and bottom crystals at the midthickness of the specimen, lies parallel to the plane of compression, i.e. {123} and the <412> directions are aligned with the channel, and the die constrains deformation in the <121> directions. The annealing of the deformed bicrystals was conducted for 5 min in a fused quartz tube furnace with He + 5%H2 atmosphere. The tex‐ tures of the fully Rexed specimens were examined by determining the {111} and {200} pole fig‐ ures from sectioned planes at 1/4, 1/2 and 3/4 specimen thickness. This roughly corresponds to

The annealing texture of single-phase crystals of Al-0.05% Si of the Goss orientation {110}<001> deformed in channel-die compression was studied by Ferry et al. [37]. In the channel-die compression, the compression and extension directions were <110> and <001> directions, respectively. Their experimental results showed that, even after deformation to a true strain of 3.0 which is equivalent to a compressive reduction of 95%, the original orientation was maintained as shown in Figure 25a. Figure 25b shows one (110) pole figure typical of a deformed crystal after annealing at 300˚C for 4 h. The comparison of Figures 25a and 25b suggests that the annealing texture is essentially the same as the deformation texture. They also reported that even after 90% reduction and annealing for up to 235 h, the orientation was the same as that of the as-deformed crystal. For deformed specimens electropolished and annealed for various temperatures between 250 and 350˚C, no texture change took place before and after annealing, although grains which had different orientations were sometimes found to grow from the crystal surface after very long annealing treatments. For samples deformed over the true strain range of 0.5 to 3.0 in their work, annealing at a given temperature resulted in similar microstructural evolution. They called the phenomenon discontinuous subgrain growth during recovery. They stated that crystals of an

<sup>13</sup>5. Plane-strain compressed fcc metallic single crystals

1 and the decrease indicates the texture change during subsequent grain growth, that is, AGG. A similar

3 Cho et al. [36] measured the drawing and Rex textures of 25 and 30 m diameter Au wires of over 4 99.99% in purity, which had dopants such as Ca and Be that total less than 50 ppm by weight. The Au 5 wires were made by drawing through a series of diamond dies to an effective strain of 11.4.

Figure 24 shows the grain size and the volume fraction of the <111> and <100> grains as a function of annealing time at 300 and 400°C. These values are based on EBSD measurements. The aspect ratio of grain shape was in the range of 1.5 - 2, which is little influenced by annealing time and temperature [36]. The grain growth occurs in whole area of the wire and is more rapid at 400°C than at 300°C as expected for thermally activated motion of grain boundaries. The volume fraction of the <111> grains decreases and that of the <100> grains increases with annealing time when Rex takes place, as

2 phenomenon is observed in drawn Ag wire during annealing (Figure 20).

17

be justified in light of single crystal examples in Sections 5.2 to 5.4.

30 orientation which was stable during

36 a b

for 4 h. (Contour levels: 2, 5, 11, 20, 35, 70 x random) [37].

**5.2. Aluminum crystals of {123}<412> orientations**

14 5.1 Channel-die compressed {110}<001> aluminum single crystal

Running Title

12 expected from SERM.

22 Recent Developments in the Study of Recrystallization

#### 209[10 1] 776[0 1 1] 1424 0.577[110] 2938 0.577[110] [4608 3293 2867] / /[0.7259 0.5187 0.4516]unit vector - - - - (12)

where the factor 0.577 originates from the fact that the slip systems of (1-1 1)[110] and (-111) [110] share the same slip direction [110] (Eq. 7). Two other principal stress directions are obtained as explained in Figure 6. Possible candidates for the direction equivalent to **S** in Figure 6 are the [011], [101], and [1-1 0] directions, which are not used in calculation of AMSD among six possible Burgers vector directions. The [011], [101], and [1-1 0] directions are at 87.3, 78.8 and 81.6o , respectively, with AMSD. The [011] direction is closest to 90° (Figure 30). The directions equivalent to **B** and **C** in Figure 6 are calculated to be [-0.0345 0.6833 0.7294] and [0.6869 -0.5139 0.5139] unit vectors, respectively. In summary, OA, OB, and OC in Figure 30, which are equivalent to **A**, **B**, and **C**, are to be parallel to the <100> directions in the Rexed grain. If the [0.7259 0.5187 -0.4516], [0.6869 -0.5139 0.5139] and [-0.0345 0.6833 0.7294] unit vectors are set to be parallel to [100], [010] and [001] directions after Rex (Figure 30), compo‐ nents of the unit vectors are direction cosines relating the deformed and Rexed crystal coordinate axes. Therefore, ND, [0.1534 0.5101 0.8463], and RD, [0.8111 0.4242 -0.4027], in the

deformed crystal coordinate system can be transformed to the expressions in the Rexed crystal coordinate system using the following calculations (refer to Eq. 9):

$$
\begin{bmatrix} 0.7259 & 0.5187 & -0.4516 & 0.1534 & -0.0062 \\ 0.6869 & -0.5139 & 0.5139 & 0.5101 & -\begin{bmatrix} -0.0062 \\ 0.2781 \\ -0.0345 & 0.6833 & 0.7294 \end{bmatrix} \begin{bmatrix} 0.5101 \\ 0.8463 \end{bmatrix} & \tag{13}
$$

$$
\begin{bmatrix} 0.7259 & 0.5187 & -0.4516 \\ 0.6869 & -0.5139 & 0.5139 \\ -0.0345 & 0.6833 & 0.7294 \end{bmatrix} \begin{bmatrix} 0.8111 \\ 0.4242 \\ -0.4027 \end{bmatrix} = \begin{bmatrix} 0.9907 \\ 0.1322 \\ -0.0319 \end{bmatrix} \tag{14}
$$

**Figure 26.** pole figures for 90% channel-die compressed Al crystals of {123}<412> orientations [40]. A/B indicates bi‐ crystal composed of A and B crystals. ● ~{135}<211>; ■ ~{011}<522>. Running Title 1 Figure 26. (111) pole figures for 90% channel-die compressed Al crystals of {123}<412> orientations

2 [40]. A/B indicates bicrystal composed of A and B crystals. ~{135}<211>; ~{011}<522>.

19

]241)[123( ]214)[123( ]214)[321( ]241)[321(

**Figure 29.** Calculated shear strain rate with respect to thickness reduction of 0.01, dγ(i)/dε33, on active slip systems of

rd ]010//[0.5139] 0.5139- 6869.0[

**Figure 30.** Orientation relations in deformed and Rexed states. Subscripts d and r indicate deformed state and Rexed

The calculated result means that the (0.1534 0.5101 0.8463)[0.8111 0.4242 -0.4027] crystal, which is obtained by the channel die compression by 90% reduction, transforms to the Rex texture (-0.0062 0.2781 0.9606)[0.9907 0.1322 -0.0319]. Similarly, crystals deformed by channel die compression from (123)[-4-1 2] and (-1-2-3)[4 1-2] orientations transform to (-0.0062 0.2781 0.9606)[-0.9907 -0.1322 0.0319] and (0.0062 -0.2781 -0.9606)[0.9907 0.1322 -0.0319], respectively, after Rex. The results are plotted in Figure 27 superimposed on the experimental data. It can be seen that the calculated Rex textures are in good agreement with the measured data.

]011[

**S**

87.3

**B**

O

rd ]100//[0.4516]- 0.5187 7259.0[

**A**

rd - ]001//[0.7294] 0.6833 0345.0[

**Figure 28.** Orientation rotations of {123}<412> crystals during plain strain compression by 90% [41].

calculated orientation rotation

Recrystallization Textures of Metals and Alloys

http://dx.doi.org/10.5772/54123

25

measured orientation of 90% compressed crystal

(123)[4 1-2] crystal as a function of true thickness strain, ε33, [41].

**C**

state, respectively.

18 Figure 27. (111) pole figures of 90 and 95% channel-die compressed Al bicrystals after annealing at 19 125 and 185 °C for 5 min [40]. ■ SERM-calculated Rex orientations: ■1 (-0.0062 0.2781 20 0.9606)[0.9907 0.1322 -0.0319]; ■2 (-0.0062 0.2781 0.9606)[-0.9907 -0.1322 0.0319]; ■3 (0.0062 - 21 0.2781 -0.9606)[0.9907 0.1322 -0.0319] [41]. **Figure 27.** pole figures of 90 and 95% channel-die compressed Al bicrystals after annealing at 125 and 185 °C for 5 min [40]. ■ SERM-calculated Rex orientations: ■1 (-0.0062 0.2781 0.9606)[0.9907 0.1322 -0.0319]; ■2 (-0.0062 0.2781 0.9606)[-0.9907 -0.1322 0.0319]; ■3 (0.0062 -0.2781 -0.9606)[0.9907 0.1322 -0.0319] [41].

 ]241)[123( ]214)[123( ]214)[321( ]241)[321( measured orientation of 90% compressed crystal calculated orientation rotation

Figure 28. Orientation rotations of {123}<412> crystals during plain strain compression by 90% [41]. calculated based on the full constraints strain rate sensitivity model with *m* = 0.01. The reason why the measured deformation texture is simulated at the reduction slightly lower than experimental reduction may be localized deformation like shear band formation occurring in real deformation. The localized deformation might not be reflected in X-ray measurements. The scattered experimental Rex textures may be related to the nonuniform deformation. Now that the shear strains on active slip systems are known, we are in position to calculate AMSD. For a true thickness strain of 2.3, or 90%

**Figure 28.** Orientation rotations of {123}<412> crystals during plain strain compression by 90% [41].

deformed crystal coordinate system can be transformed to the expressions in the Rexed crystal

(13)

(14)

19

0.7259 0.5187 0.4516 0.1534 0.0062 0.6869 0.5139 0.5139 0.5101 0.2781 0.0345 0.6833 0.7294 0.8463 0.9606

<sup>é</sup> - - ùé ù é ù <sup>ê</sup> úê ú ê ú - <sup>ê</sup>

ê-ë ûë û ë û

0.7259 0.5187 0.4516 0.8111 0.9907 0.6869 0.5139 0.5139 0.4242 0.1322 0.0345 0.6833 0.7294 0.4027 0.0319

<sup>é</sup> - ùé ù é ù <sup>ê</sup> úê ú ê ú - <sup>ê</sup>

ê- - - <sup>ë</sup> ûë û ë û

]241)[123( ]214)[123( ]241)[123( ]241)[321( contour 35 15, 5, 1, :level

**Figure 26.** pole figures for 90% channel-die compressed Al crystals of {123}<412> orientations [40]. A/B indicates bi‐

90% 95%

]241)[123( ■1 ]214)[123( ■2 ]241)[123( ■1 ]214)[123( ■2 125 125

90% 95%

1 Figure 26. (111) pole figures for 90% channel-die compressed Al crystals of {123}<412> orientations 2 [40]. A/B indicates bicrystal composed of A and B crystals. ~{135}<211>; ~{011}<522>.

18 Figure 27. (111) pole figures of 90 and 95% channel-die compressed Al bicrystals after annealing at 19 125 and 185 °C for 5 min [40]. ■ SERM-calculated Rex orientations: ■1 (-0.0062 0.2781 20 0.9606)[0.9907 0.1322 -0.0319]; ■2 (-0.0062 0.2781 0.9606)[-0.9907 -0.1322 0.0319]; ■3 (0.0062 -

0.2781 0.9606)[-0.9907 -0.1322 0.0319]; ■3 (0.0062 -0.2781 -0.9606)[0.9907 0.1322 -0.0319] [41].

contour 8 4, 3, 2, 1, 0.5, :level

**Figure 27.** pole figures of 90 and 95% channel-die compressed Al bicrystals after annealing at 125 and 185 °C for 5 min [40]. ■ SERM-calculated Rex orientations: ■1 (-0.0062 0.2781 0.9606)[0.9907 0.1322 -0.0319]; ■2 (-0.0062

]241)[123( ■1 ]241)[321( ■3 ]241)[123( ■1 ]241)[321( ■3 185 185

Figure 28. Orientation rotations of {123}<412> crystals during plain strain compression by 90% [41]. calculated based on the full constraints strain rate sensitivity model with *m* = 0.01. The reason why the measured deformation texture is simulated at the reduction slightly lower than experimental reduction may be localized deformation like shear band formation occurring in real deformation. The localized deformation might not be reflected in X-ray measurements. The scattered experimental Rex textures may be related to the nonuniform deformation. Now that the shear strains on active slip systems are known, we are in position to calculate AMSD. For a true thickness strain of 2.3, or 90%

 ]241)[123( ]214)[123( ]214)[321( ]241)[321( measured orientation of 90% compressed crystal calculated orientation rotation

coordinate system using the following calculations (refer to Eq. 9):

24 Recent Developments in the Study of Recrystallization

crystal composed of A and B crystals. ● ~{135}<211>; ■ ~{011}<522>.

21 0.2781 -0.9606)[0.9907 0.1322 -0.0319] [41].

Running Title

**Figure 29.** Calculated shear strain rate with respect to thickness reduction of 0.01, dγ(i)/dε33, on active slip systems of (123)[4 1-2] crystal as a function of true thickness strain, ε33, [41].

**Figure 30.** Orientation relations in deformed and Rexed states. Subscripts d and r indicate deformed state and Rexed state, respectively.

The calculated result means that the (0.1534 0.5101 0.8463)[0.8111 0.4242 -0.4027] crystal, which is obtained by the channel die compression by 90% reduction, transforms to the Rex texture (-0.0062 0.2781 0.9606)[0.9907 0.1322 -0.0319]. Similarly, crystals deformed by channel die compression from (123)[-4-1 2] and (-1-2-3)[4 1-2] orientations transform to (-0.0062 0.2781 0.9606)[-0.9907 -0.1322 0.0319] and (0.0062 -0.2781 -0.9606)[0.9907 0.1322 -0.0319], respectively, after Rex. The results are plotted in Figure 27 superimposed on the experimental data. It can be seen that the calculated Rex textures are in good agreement with the measured data.

#### **5.3. Aluminum crystal of {112}<111> obtained by channel-die compression of (001)[110] crystal**

Butler et al. [42] obtained a {112}<111> Al crystal by channel-die compression of the (001)[110] single crystal. The (001)[110]orientation is unstable with respect to plane strain compression, to form the (112)[1 1-1] and (112)[-1-1 1] orientations as shown in Figure 31a. The Rex texture produced after annealing at 200 °C was a rotated cube texture (Figure 31b). Lee [43] analyzed the result based on SERM. Figure 32 shows shear strains/extension strain on slip systems of 1 to 6 as a function of rotation angle about TD [-110] of the (001)[110] fcc crystal obtained from the Taylor-Bishop-Hill theory. The contribution of the slip systems to the deformation is approximated to be proportional to the area under the shear strains *γ*(*i*) on slip systems *i* / extension strain - rotation angle *θ* curve in Figure 32. The area ratio becomes

$$\int\_0^{35} \gamma^{(1)} d\theta : \int\_0^{10} \gamma^{(3)} d\theta : \int\_{10}^{35} \gamma^{(5)} d\theta = 30 : 3 : 20.6\tag{15}$$

direction that is equivalent to **B** in Figure 6 is calculated to be [2-4 1] by the vector product of AMSD [3 1-2] and [112]. Thus, the [3 1-2], [2-4 1], and [112] directions become parallel to <100>

Figure 33. Vector sum of slip directions ① [1 0-1], ③ [0-1-1], and ⑤ [110] assuming that their

For the contribution of the former three slip systems to the crystal deformation, AMSD is obtained by the vector sum of the [1 0-1], [0-1-1], and [110] directions whose contributions are assumed to be proportional to the area ratio obtained earlier (30 : 3 : 20.6). The vector sum is shown in Figure 33. The resultant direction passes through point E, which divides line BC by a ratio of 1 to 2. Thus, AMSD // AE // [3 1-2]. Another high stress direction equivalent to **S** in Figure 6 is BD, or [-110] which is not used in calculation of AMSD among possible Burgers vectors. The [-110] direction is not normal to AMSD. The direction that is at the smallest possible angle with the [-110] direction and normal to AMSD must be on a plane made of AMSD and the [-110] direction. The plane normal is obtained to be the [112] direction by the vector product of AMSD and the [-110] direction, which is equivalent to **C** in Figure 6. The direction that is equivalent to **B** in Figure 6 is calculated to be [2-4 1] by the vector product of AMSD [3 1-2] and [112]. Thus, the [3 1-2], [2-4 1], and [112] directions

**Figure 33.** Vector sum of slip directions ① [1 0-1], ③ [0-1-1], and ⑤ [110] assuming that their activities are propor‐

Figure 33. Vector sum of slip directions ① [1 0-1], ③ [0-1-1], and ⑤ [110] assuming that their

For the contribution of the former three slip systems to the crystal deformation, AMSD is obtained by the vector sum of the [1 0-1], [0-1-1], and [110] directions whose contributions are assumed to be proportional to the area ratio obtained earlier (30 : 3 : 20.6). The vector sum is shown in Figure 33. The resultant direction passes through point E, which divides line BC by a ratio of 1 to 2. Thus, AMSD // AE // [3 1-2]. Another high stress direction equivalent to **S** in Figure 6 is BD, or [-110] which is not used in calculation of AMSD among possible Burgers vectors. The [-110] direction is not normal to AMSD. The direction that is at the smallest possible angle with the [-110] direction and normal to AMSD must be on a plane made of AMSD and the [-110] direction. The plane normal is obtained to be the [112] direction by the vector product of AMSD and the [-110] direction, which is equivalent to **C** in Figure 6. The direction that is equivalent to **B** in Figure 6 is calculated to be [2-4 1] by the vector product of AMSD [3 1-2] and [112]. Thus, the [3 1-2], [2-4 1], and [112] directions

Book Title

Figure 31. (a) (111) pole figure of Al single crystal with initial orientation (001)[110] after 70% reduction by channel-die compression; (b) (111) pole figure of measured Rex texture (contours), (100)[0-4 1], and (100)[041] [42]. (001)[√6 -1 0] and (001)[-√6 -1 0] are calculated by SERM [43].

Figure 31. (a) (111) pole figure of Al single crystal with initial orientation (001)[110] after 70% reduction by channel-die compression; (b) (111) pole figure of measured Rex texture (contours), (100)[0-4 1], and (100)[041] [42]. (001)[√6 -1 0] and (001)[-√6 -1 0] are calculated by SERM [43].

Book Title

Figure 32. Shear strains on slip systems of ① to ⑥ as a function of rotation angle about TD [-110] of

Figure 32. Shear strains on slip systems of ① to ⑥ as a function of rotation angle about TD [-110] of

**Figure 32.** Shear strains on slip systems of 1 to 6 as a function of rotation angle about TD [-110] of (001)[110] crystal

④

④

If the directions [3 1-2], [2-4 1], and [112], whose unit vectors are [3/√14 1/√14 -2/√14], [2/√21 -4/ √21 1/√21], and [1/√6 1/√6 2/√6], respectively, are set to be parallel to [100], [010] and [001] directions in the Rexed crystal, components of the unit vectors are direction cosines relating the deformed and Rexed crystal coordinate axes (Eq. 9). Therefore, ND, [112], and RD, [1 1-1], in the deformed crystal coordinate system can be transformed to the expressions in the Rexed

Therefore, the (112)[11-1] deformation texture transforms to the (001)[ √6-1 0] Rex texture. Simi‐ larly, from the (111)[0 1-1], (-1-1 1)[-1 0-1], and (-1 1-1) [110] slip systems, another AMSD AF, or

 / 14 1 / 14 −2 / 14 / 21 −4 / 21 1 / 21 / 6 1 / 6 2 / 6

 <sup>−</sup><sup>1</sup> / / <sup>6</sup> −1 

BE : EC = 1 : 2 FD : FC = 1 : 2 plane AEF // (112) ∠EAG = ∠FAG = 22o

BE : EC = 1 : 2 FD : FC = 1 : 2 plane AEF // (112) ∠EAG = ∠FAG = 22°

 (111)[1 0-1] (111)[0 1-1] (-1-1 1)[0-1-1] (-1-1 1)[-1 0-1] (1-1-1)[110] (-1 1-1)[110]

Recrystallization Textures of Metals and Alloys

http://dx.doi.org/10.5772/54123

 (111)[1 0-1] (111)[0 1-1] (-1-1 1)[0-1-1] (-1-1 1)[-1 0-1] (1-1-1)[110] (-1 1-1)[110]

crystal coordinate system using the following calculation:

activities are proportional to 30:3:20.6 (Eq. 15).

activities are proportional to 30:3:20.6 (Eq. 15).

become parallel to <100> in the Rexed grains.

become parallel to <100> in the Rexed grains.

and

in the Rexed grains.

tional to 30:3:20.6 (Eq. 15).

[43].

of (001)[110] crystal [43].

of (001)[110] crystal [43].

 / 14 1 / 14 −2 / 14 / 21 −4 / 21 1 / 21 / 6 1 / 6 2 / 6

All the slips may not occur on the related slip systems uniformly in a large single crystal. Some regions of the crystal may be deformed by 1, 3, and 5 slip systems, while some other regions by 2, 4, and 6 slip systems.

**Figure 31.** (a) (111) pole figure of Al single crystal with initial orientation (001)[110] after 70% reduction by channeldie compression; (b) (111) pole figure of measured Rex texture (contours), (100)[0-4 1], and (100)[041] [42]. (001)[√6 -1 0] and (001)[-√6 -1 0] are calculated by SERM [43].

For the contribution of the former three slip systems to the crystal deformation, AMSD is obtained by the vector sum of the [1 0-1], [0-1-1], and [110] directions whose contributions are assumed to be proportional to the area ratio obtained earlier (30 : 3 : 20.6). The vector sum is shown in Figure 33. The resultant direction passes through point E, which divides line BC by a ratio of 1 to 2. Thus, AMSD // AE // [3 1-2]. Another high stress direction equivalent to **S** in Figure 6 is BD, or [-110] which is not used in calculation of AMSD among possible Burgers vectors. The [-110] direction is not normal to AMSD. The direction that is at the smallest possible angle with the [-110] direction and normal to AMSD must be on a plane made of AMSD and the [-110] direction. The plane normal is obtained to be the [112] direction by the vector product of AMSD and the [-110] direction, which is equivalent to **C** in Figure 6. The

(100)[0-4 1], and (100)[041] [42]. (001)[√6 -1 0] and (001)[-√6 -1 0] are calculated by SERM [43].

Book Title

Book Title

**5.3. Aluminum crystal of {112}<111> obtained by channel-die compression of (001)[110]**

extension strain - rotation angle *θ* curve in Figure 32. The area ratio becomes

 10 35 (1) (3) (5) 0 10

a b

 q

q

by 2, 4, and 6 slip systems.


Butler et al. [42] obtained a {112}<111> Al crystal by channel-die compression of the (001)[110] single crystal. The (001)[110]orientation is unstable with respect to plane strain compression, to form the (112)[1 1-1] and (112)[-1-1 1] orientations as shown in Figure 31a. The Rex texture produced after annealing at 200 °C was a rotated cube texture (Figure 31b). Lee [43] analyzed the result based on SERM. Figure 32 shows shear strains/extension strain on slip systems of 1 to 6 as a function of rotation angle about TD [-110] of the (001)[110] fcc crystal obtained from the Taylor-Bishop-Hill theory. The contribution of the slip systems to the deformation is approximated to be proportional to the area under the shear strains *γ*(*i*) on slip systems *i* /

> q

All the slips may not occur on the related slip systems uniformly in a large single crystal. Some regions of the crystal may be deformed by 1, 3, and 5 slip systems, while some other regions

> ]110)[001( ]111)[112( ]111)[112( ]016)[001(

]140)[100( ]041)[100(

**Figure 31.** (a) (111) pole figure of Al single crystal with initial orientation (001)[110] after 70% reduction by channeldie compression; (b) (111) pole figure of measured Rex texture (contours), (100)[0-4 1], and (100)[041] [42]. (001)[√6

For the contribution of the former three slip systems to the crystal deformation, AMSD is obtained by the vector sum of the [1 0-1], [0-1-1], and [110] directions whose contributions are assumed to be proportional to the area ratio obtained earlier (30 : 3 : 20.6). The vector sum is shown in Figure 33. The resultant direction passes through point E, which divides line BC by a ratio of 1 to 2. Thus, AMSD // AE // [3 1-2]. Another high stress direction equivalent to **S** in Figure 6 is BD, or [-110] which is not used in calculation of AMSD among possible Burgers vectors. The [-110] direction is not normal to AMSD. The direction that is at the smallest possible angle with the [-110] direction and normal to AMSD must be on a plane made of AMSD and the [-110] direction. The plane normal is obtained to be the [112] direction by the vector product of AMSD and the [-110] direction, which is equivalent to **C** in Figure 6. The


*ddd* : : 30 : 3 : 20.6 òòò (15)

**crystal**

Recent Developments in the Study of Recrystallization

Figure 32. Shear strains on slip systems of ① to ⑥ as a function of rotation angle about TD [-110] of of (001)[110] crystal [43]. **Figure 32.** Shear strains on slip systems of 1 to 6 as a function of rotation angle about TD [-110] of (001)[110] crystal [43]. 

Figure 32. Shear strains on slip systems of ① to ⑥ as a function of rotation angle about TD [-110] of

the vector sum of the [1 0-1], [0-1-1], and [110] directions whose contributions are assumed to be proportional to the area ratio obtained earlier (30 : 3 : 20.6). The vector sum is shown in Figure 33. The resultant direction passes through point E, which divides line BC by a ratio of 1 to 2. Thus, AMSD // AE // [3 1-2]. Another high stress direction equivalent to **S** in Figure 6 is BD, or [-110] Figure 33. Vector sum of slip directions ① [1 0-1], ③ [0-1-1], and ⑤ [110] assuming that their activities are proportional to 30:3:20.6 (Eq. 15). For the contribution of the former three slip systems to the crystal deformation, AMSD is obtained by **Figure 33.** Vector sum of slip directions ① [1 0-1], ③ [0-1-1], and ⑤ [110] assuming that their activities are propor‐ tional to 30:3:20.6 (Eq. 15).

which is not used in calculation of AMSD among possible Burgers vectors. The [-110] direction is

the vector sum of the [1 0-1], [0-1-1], and [110] directions whose contributions are assumed to be

direction that is equivalent to **B** in Figure 6 is calculated to be [2-4 1] by the vector product of AMSD [3 1-2] and [112]. Thus, the [3 1-2], [2-4 1], and [112] directions become parallel to <100> in the Rexed grains. not normal to AMSD. The direction that is at the smallest possible angle with the [-110] direction and normal to AMSD must be on a plane made of AMSD and the [-110] direction. The plane normal is obtained to be the [112] direction by the vector product of AMSD and the [-110] direction, which is equivalent to **C** in Figure 6. The direction that is equivalent to **B** in Figure 6 is calculated to be [2-4 1] by the vector product of AMSD [3 1-2] and [112]. Thus, the [3 1-2], [2-4 1], and [112] directions proportional to the area ratio obtained earlier (30 : 3 : 20.6). The vector sum is shown in Figure 33. The resultant direction passes through point E, which divides line BC by a ratio of 1 to 2. Thus, AMSD // AE // [3 1-2]. Another high stress direction equivalent to **S** in Figure 6 is BD, or [-110] which is not used in calculation of AMSD among possible Burgers vectors. The [-110] direction is not normal to AMSD. The direction that is at the smallest possible angle with the [-110] direction and

If the directions [3 1-2], [2-4 1], and [112], whose unit vectors are [3/√14 1/√14 -2/√14], [2/√21 -4/ √21 1/√21], and [1/√6 1/√6 2/√6], respectively, are set to be parallel to [100], [010] and [001] directions in the Rexed crystal, components of the unit vectors are direction cosines relating the deformed and Rexed crystal coordinate axes (Eq. 9). Therefore, ND, [112], and RD, [1 1-1], in the deformed crystal coordinate system can be transformed to the expressions in the Rexed crystal coordinate system using the following calculation: become parallel to <100> in the Rexed grains. normal to AMSD must be on a plane made of AMSD and the [-110] direction. The plane normal is obtained to be the [112] direction by the vector product of AMSD and the [-110] direction, which is equivalent to **C** in Figure 6. The direction that is equivalent to **B** in Figure 6 is calculated to be [2-4 1] by the vector product of AMSD [3 1-2] and [112]. Thus, the [3 1-2], [2-4 1], and [112] directions become parallel to <100> in the Rexed grains.

$$
\begin{bmatrix} 3/\sqrt{14} & 1/\sqrt{14} & -2/\sqrt{14} \\ 2/\sqrt{21} & -4/\sqrt{21} & 1/\sqrt{21} \\ 1/\sqrt{6} & 1/\sqrt{6} & 2/\sqrt{6} \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix} 3/\sqrt{14} & 1/\sqrt{14} & -2/\sqrt{14} \\ 2/\sqrt{21} & -4/\sqrt{21} & 1/\sqrt{21} \\ 1/\sqrt{6} & 1/\sqrt{6} & 2/\sqrt{6} \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \begin{bmatrix} \sqrt{6} \\ -1 \\ 0 \end{bmatrix}
$$

Therefore, the (112)[11-1] deformation texture transforms to the (001)[ √6-1 0] Rex texture. Simi‐ larly, from the (111)[0 1-1], (-1-1 1)[-1 0-1], and (-1 1-1) [110] slip systems, another AMSD AF, or the [1 3-2] direction, can be obtained. In this case, the (112)[1 1-1] deformation texture trans‐ forms into the (001)[-√6 -1 0] Rex texture. The {001}<√6 1 0> orientation has a rotational relation with the {001}<100> orientation through 22° about the plane normal. The calculated Rex texture is superimposed on the measured data in Figure 31b. The calculated results are in relatively good agreement with the measured data. It is noted that Figure 32 does not represent the correct strain path during deformation. Therefore, there is a room to improve the calculated Rex tex‐ ture. The Rex texture is at variance with the {001}<100> Rex texture in polycrystalline Al and Cu.

( <sup>−</sup>0.745 <sup>−</sup>0.401 0.532 0.069 0.747 0.660 −0.663 0.529 −0.529

> 1 2 3

> 7

a

7

10

0.015

1 2 3

( ) *d* / *d<sup>i</sup>*

0.010

0.005

a

10

6 ●(112)[-1-1 1]; ○ (112)[11-1]; □(001)[100].

11

0.015

( ) *d* / *d<sup>i</sup>*

0.010

0.005

RD, TD, and ND. The changes in *dγ*

(001)[100] components with their density ratio being 2:1 [45].

RD, TD, and ND. The changes in *dγ*

a b

0.0 0.5 1.0 1.5 2.0 <sup>11</sup>

(111)[01 1]

(111)[1 1 0] (11 1)[1 1 0]

(111)[1 01]

0.0 0.5 1.0 1.5 2.0 <sup>11</sup>

(111)[1 01]

(111)[1 1 0] (11 1)[1 1 0]

11

6 ●(112)[-1-1 1]; ○ (112)[11-1]; □(001)[100].

showing orientation rotation from {123}<634> to {112}<111> [45].

) =( 0.049 3.543 −1.192

a b c

**Figure 34.** pole figures for (123)[-6-3 4] Cu single crystal after rolling by (a) 95%, (b) 99.5%, and (c) 99.5% and subse‐ quent annealing at 538 K for 100 s [44]. ▲(123)[-6-3 4]; ∆(321)[-436]; ●(112)[-1-1 1]; ○ (112)[11-1]; □(001)[100].

4 Figure 34. (111) pole figures for (123)[-6-3 4] Cu single crystal after rolling by (a) 95%, (b) 99.5%, 5 and (c) 99.5% and subsequent annealing at 538 K for 100 s [44]. ▲(123)[-6-3 4]; ∆(321)[-436];

Figure 35. (a) Shear strain rates *dγ/dε*11 with *dε*11=0.01 vs. *ε*<sup>11</sup> 8 on active slip systems (C, J, M, B) and

9 (b) (111) pole figure showing orientation rotation from {123}<634> to {112}<111> [45].

**Figure 35.** (a) Shear strain rates dγ/dε11 with dε11=0.01 vs. ε<sup>11</sup> on active slip systems (C, J, M, B) and (b) (111) pole figure

Figure 35. (a) Shear strain rates *dγ/dε*11 with *dε*11=0.01 vs. *ε*<sup>11</sup> 8 on active slip systems (C, J, M, B) and

b

▲ (123)[634] △ (321)[436] ● (112)[1 1 1]

○ (112)[11 1]

b

○ (112)[11 1]

▲ (123)[634] △ (321)[436] ● (112)[1 1 1]

4 Figure 34. (111) pole figures for (123)[-6-3 4] Cu single crystal after rolling by (a) 95%, (b) 99.5%, 5 and (c) 99.5% and subsequent annealing at 538 K for 100 s [44]. ▲(123)[-6-3 4]; ∆(321)[-436];

24 Book Title

Figure 36. (a) Orientation relationship between deformed (d) and Rexed (r 11 ) states and (b) (111) pole 12 figures of ○ (0 3-1)[100] and □ (001)[100] orientations. Contours were calculated assuming Gaussian 13 scattering (10°) of (0 3-1)[100] and (001)[100] components with their density ratio being 2:1 [45]. on active slip systems *i* as a function of *ε*11 14 for the (123)[-6-3 4] crystal, which was calculated by the *dε*13 and *dε*23 15 relaxed strain rate sensitive model (m = 0.02) with the subscripts 1, 2, and 3 indicating

(i)/*dε*11 depending on *ε*<sup>11</sup> 16 indicate that the (123)[-6-3 4] orientation is unstable with respect to the plane strain compression. A part of the (123)[-6-3 4] crystal, particularly the surface layers where *dε*23 18 is negligible due to friction between rolls and sheet, seems to rotate to the {112}<111> orientation. A part of the {112}<111> crystal further rotates to (321)[- 436] with increasing reduction. This is why (321)[-436] has lower density than (123)[-6-3 4] along with weak {112}<111>. The orientation rotation is shown in Figure 35b. Since important components in the deformation texture are (123)[-6-3 4], (321)[-436], and {112}<111>, their Rex

Figure 36. (a) Orientation relationship between deformed (d) and Rexed (r 11 ) states and (b) (111) pole 12 figures of ○ (0 3-1)[100] and □ (001)[100] orientations. Contours were calculated assuming Gaussian 13 scattering (10°) of (0 3-1)[100] and (001)[100] components with their density ratio being 2:1 [45]. on active slip systems *i* as a function of *ε*11 14 for the (123)[-6-3 4] crystal, which was calculated by the *dε*13 and *dε*23 15 relaxed strain rate sensitive model (m = 0.02) with the subscripts 1, 2, and 3 indicating

**Figure 36.** a) Orientation relationship between deformed (d) and Rexed (r) states and (b) (111) pole figures of ○ (0 3-1) [100] and □ (001)[100] orientations. Contours were calculated assuming Gaussian scattering (10°) of (0 3-1)[100] and

(i)/*dε*11 depending on *ε*<sup>11</sup> 16 indicate that the (123)[-6-3 4] orientation is unstable with respect to the plane strain compression. A part of the (123)[-6-3 4] crystal, particularly the surface layers where *dε*23 18 is negligible due to friction between rolls and sheet, seems to rotate to the {112}<111> orientation. A part of the {112}<111> crystal further rotates to (321)[- 436] with increasing reduction. This is why (321)[-436] has lower density than (123)[-6-3 4] along with weak {112}<111>. The orientation rotation is shown in Figure 35b. Since important components in the deformation texture are (123)[-6-3 4], (321)[-436], and {112}<111>, their Rex

(i)/*dε*<sup>11</sup> 23 on the active

(i)/*dε*<sup>11</sup> 24 on C, J, M, and B are 0.014, 0.01, 0.007, and 0.003, respectively, at zero strain. Therefore, AMSD is 0.014 [-101] + 0.010.577 [-1-1 0] + 0.007 0.577 [-1-1 0] + 0.003 [0-1 1] = [-0.02381 -0.01281 0.017], where the factor 0.577 originated from the fact that the duplex slip systems of (1-1 1)[-1-10] and (-111)[-1-10] share the same slip direction (Figure 4). The [-0.02381 -0.01281 0.017] direction, or the [-0.745 -

(i)/*dε*<sup>11</sup> 23 on the active

(i)/*dε*<sup>11</sup> 24 on C, J, M, and B are 0.014, 0.01, 0.007, and 0.003, respectively, at zero strain. Therefore, AMSD is 0.014 [-101] + 0.010.577 [-1-1 0] + 0.007 0.577 [-1-1 0] + 0.003 [0-1 1] = [-0.02381 -0.01281 0.017], where the factor 0.577 originated from the fact that the duplex slip systems of (1-1 1)[-1-10] and (-111)[-1-10] share the same slip direction (Figure 4). The [-0.02381 -0.01281 0.017] direction, or the [-0.745 -

textures are calculated using SERM. If the (123)[-6-3 4] orientation is stable, *dγ*

textures are calculated using SERM. If the (123)[-6-3 4] orientation is stable, *dγ*

slip systems do not vary with strain. It follows from Fig. 35a that *dγ*

slip systems do not vary with strain. It follows from Fig. 35a that *dγ*

a b

9 (b) (111) pole figure showing orientation rotation from {123}<634> to {112}<111> [45].

(111)[01 1]

24 Book Title

) and ( <sup>−</sup>0.745 <sup>−</sup>0.401 0.532 0.069 0.747 0.660 −0.663 0.529 −0.529

)( −6 −3 4

) =( −

Recrystallization Textures of Metals and Alloys

7.801 0.017 −0.275 ) 29

http://dx.doi.org/10.5772/54123

## **5.4. Copper crystal of (123)[-6-3 4] in orientation rolled by 99.5% reduction in thickness**

Kamijo et al. [44] rolled a (123)[-6-3 4] Cu single crystal reversibly by 99.5% under oil lubrica‐ tion. The (123)[-6-3 4] orientation was relatively well preserved up to 95%, even though the orientation spread occurred as shown in Figure 34a. However, the crystal rotation proceeded with increasing reduction. A new (321)[-4 3 6] component, which is symmetrically oriented to the initial (123)[-6-3 4] with respect to TD, developed after 99.5% rolling as shown in Figure 34b. It is noted that other two equivalent components are not observed. The rolled specimens were annealed at 538 K for 100 s to obtain Rex textures. In the Rex textures of the crystals rolled less than 90%, any fairly developed texture could not be observed, except for the retained rolling texture component. They could observe a cube texture with large scatter in the 95% rolled crystal and the fairly well developed cube orientation in the 99.5% rolled crystal after Rex as shown in Figure 34c. They concluded that the development of cube texture in the single crystal of the (123)[-6-3 4] orientation was mainly attributed to the preferential nucleation from the (001)[100] deformation structure. The cube deformation structure was proposed to form due to the inhomogeneity of deformation. Lee and Shin [45] explained the textures in Figure 34 based on SERM. Figure 35 shows *dγ*(i)/*dε*11, with *dε*11 = 0.01, on active slip systems *i* as a function of *ε*11 for the (123)[-6-3 4] crystal, which was calculated by the *dε*13 and *dε*<sup>23</sup> relaxed strain rate sensitive model (m = 0.02) with the subscripts 1, 2, and 3 indicating RD, TD, and ND. The changes in *dγ*(i)/*dε*11 depending on *ε*11 indicate that the (123)[-6-3 4] orientation is unstable with respect to the plane strain compression. A part of the (123)[-6-3 4] crystal, particularly the surface layers where *dε*<sup>23</sup> is negligible due to friction between rolls and sheet, seems to rotate to the {112}<111> orientation. A part of the {112}<111> crystal further rotates to (321)[-436] with increasing reduction. This is why (321)[-436] has lower density than (123)[-6-3 4] along with weak {112}<111>. The orientation rotation is shown in Figure 35b. Since important components in the deformation texture are (123)[-6-3 4], (321)[-436], and {112}<111>, their Rex textures are calculated using SERM. If the (123)[-6-3 4] orientation is stable, *dγ*(i)/*dε*11 on the active slip systems do not vary with strain. It follows from Fig. 35a that *dγ*(i)/*dε*<sup>11</sup> on C, J, M, and B are 0.014, 0.01, 0.007, and 0.003, respectively, at zero strain. Therefore, AMSD is 0.014 [-101] + 0.01×0.577 [-1-1 0] + 0.007 ×0.577 [-1-1 0] + 0.003 [0-1 1] = [-0.02381 -0.01281 0.017], where the factor 0.577 originates from the fact that the duplex slip systems of (1-1 1)[-1-10] and (-111) [-1-10] share the same slip direction (Figure 4). The [-0.02381 -0.01281 0.017] direction, or the [-0.745 -0.401 0.532] unit vector, will be parallel to one of the <100> directions, MYMDs of Cu, after Rex. Orientation relationship between the matrix and Rexed state is shown in Figure 36a, which is obtained as explained in Figure 6. The Rex orientation of the (123)[-6-3 4] matrix is calculated as follows:

Recrystallization Textures of Metals and Alloys http://dx.doi.org/10.5772/54123 29

$$
\begin{pmatrix} -0.745 & -0.401 & 0.532 \\ 0.069 & 0.747 & 0.660 \\ -0.663 & 0.529 & -0.529 \end{pmatrix} \begin{vmatrix} 1 \\ 2 \\ 3 \end{vmatrix} = \begin{pmatrix} 0.049 \\ 3.543 \\ -1.192 \end{pmatrix} \text{ and } \begin{pmatrix} -0.745 & -0.401 & 0.532 \\ 0.069 & 0.747 & 0.660 \\ -0.663 & 0.529 & -0.529 \end{pmatrix} \begin{vmatrix} -6 \\ -3 \\ 4 \end{vmatrix} = \begin{pmatrix} 7.801 \\ -0.017 \\ -0.275 \end{pmatrix}
$$

the [1 3-2] direction, can be obtained. In this case, the (112)[1 1-1] deformation texture trans‐ forms into the (001)[-√6 -1 0] Rex texture. The {001}<√6 1 0> orientation has a rotational relation with the {001}<100> orientation through 22° about the plane normal. The calculated Rex texture is superimposed on the measured data in Figure 31b. The calculated results are in relatively good agreement with the measured data. It is noted that Figure 32 does not represent the correct strain path during deformation. Therefore, there is a room to improve the calculated Rex tex‐ ture. The Rex texture is at variance with the {001}<100> Rex texture in polycrystalline Al and Cu.

28 Recent Developments in the Study of Recrystallization

**5.4. Copper crystal of (123)[-6-3 4] in orientation rolled by 99.5% reduction in thickness**

is calculated as follows:

Kamijo et al. [44] rolled a (123)[-6-3 4] Cu single crystal reversibly by 99.5% under oil lubrica‐ tion. The (123)[-6-3 4] orientation was relatively well preserved up to 95%, even though the orientation spread occurred as shown in Figure 34a. However, the crystal rotation proceeded with increasing reduction. A new (321)[-4 3 6] component, which is symmetrically oriented to the initial (123)[-6-3 4] with respect to TD, developed after 99.5% rolling as shown in Figure 34b. It is noted that other two equivalent components are not observed. The rolled specimens were annealed at 538 K for 100 s to obtain Rex textures. In the Rex textures of the crystals rolled less than 90%, any fairly developed texture could not be observed, except for the retained rolling texture component. They could observe a cube texture with large scatter in the 95% rolled crystal and the fairly well developed cube orientation in the 99.5% rolled crystal after Rex as shown in Figure 34c. They concluded that the development of cube texture in the single crystal of the (123)[-6-3 4] orientation was mainly attributed to the preferential nucleation from the (001)[100] deformation structure. The cube deformation structure was proposed to form due to the inhomogeneity of deformation. Lee and Shin [45] explained the textures in Figure 34 based on SERM. Figure 35 shows *dγ*(i)/*dε*11, with *dε*11 = 0.01, on active slip systems *i* as a function of *ε*11 for the (123)[-6-3 4] crystal, which was calculated by the *dε*13 and *dε*<sup>23</sup> relaxed strain rate sensitive model (m = 0.02) with the subscripts 1, 2, and 3 indicating RD, TD, and ND. The changes in *dγ*(i)/*dε*11 depending on *ε*11 indicate that the (123)[-6-3 4] orientation is unstable with respect to the plane strain compression. A part of the (123)[-6-3 4] crystal, particularly the surface layers where *dε*<sup>23</sup> is negligible due to friction between rolls and sheet, seems to rotate to the {112}<111> orientation. A part of the {112}<111> crystal further rotates to (321)[-436] with increasing reduction. This is why (321)[-436] has lower density than (123)[-6-3 4] along with weak {112}<111>. The orientation rotation is shown in Figure 35b. Since important components in the deformation texture are (123)[-6-3 4], (321)[-436], and {112}<111>, their Rex textures are calculated using SERM. If the (123)[-6-3 4] orientation is stable, *dγ*(i)/*dε*11 on the active slip systems do not vary with strain. It follows from Fig. 35a that *dγ*(i)/*dε*<sup>11</sup> on C, J, M, and B are 0.014, 0.01, 0.007, and 0.003, respectively, at zero strain. Therefore, AMSD is 0.014 [-101] + 0.01×0.577 [-1-1 0] + 0.007 ×0.577 [-1-1 0] + 0.003 [0-1 1] = [-0.02381 -0.01281 0.017], where the factor 0.577 originates from the fact that the duplex slip systems of (1-1 1)[-1-10] and (-111) [-1-10] share the same slip direction (Figure 4). The [-0.02381 -0.01281 0.017] direction, or the [-0.745 -0.401 0.532] unit vector, will be parallel to one of the <100> directions, MYMDs of Cu, after Rex. Orientation relationship between the matrix and Rexed state is shown in Figure 36a, which is obtained as explained in Figure 6. The Rex orientation of the (123)[-6-3 4] matrix

**Figure 34.** pole figures for (123)[-6-3 4] Cu single crystal after rolling by (a) 95%, (b) 99.5%, and (c) 99.5% and subse‐ quent annealing at 538 K for 100 s [44]. ▲(123)[-6-3 4]; ∆(321)[-436]; ●(112)[-1-1 1]; ○ (112)[11-1]; □(001)[100]. 2 3 24 Book Title

4 Figure 34. (111) pole figures for (123)[-6-3 4] Cu single crystal after rolling by (a) 95%, (b) 99.5%, 5 and (c) 99.5% and subsequent annealing at 538 K for 100 s [44]. ▲(123)[-6-3 4]; ∆(321)[-436];

1

1 2 3

7

7

10

10

9 (b) (111) pole figure showing orientation rotation from {123}<634> to {112}<111> [45]. **Figure 35.** (a) Shear strain rates dγ/dε11 with dε11=0.01 vs. ε<sup>11</sup> on active slip systems (C, J, M, B) and (b) (111) pole figure showing orientation rotation from {123}<634> to {112}<111> [45]. b a 0.005 0.0 0.5 1.0 1.5 2.0 <sup>11</sup> (111)[01 1]

Figure 35. (a) Shear strain rates *dγ/dε*11 with *dε*11=0.01 vs. *ε*<sup>11</sup> 8 on active slip systems (C, J, M, B) and

○ (112)[11 1]

Figure 35. (a) Shear strain rates *dγ/dε*11 with *dε*11=0.01 vs. *ε*<sup>11</sup> 8 on active slip systems (C, J, M, B) and

9 (b) (111) pole figure showing orientation rotation from {123}<634> to {112}<111> [45].

RD, TD, and ND. The changes in *dγ* (i)/*dε*11 depending on *ε*<sup>11</sup> 16 indicate that the (123)[-6-3 4] orientation 17 is unstable with respect to the plane strain compression. A part of the (123)[-6-3 4] crystal, particularly the surface layers where *dε*23 18 is negligible due to friction between rolls and sheet, seems 19 to rotate to the {112}<111> orientation. A part of the {112}<111> crystal further rotates to (321)[- 12 figures of ○ (0 3-1)[100] and □ (001)[100] orientations. Contours were calculated assuming Gaussian 13 scattering (10°) of (0 3-1)[100] and (001)[100] components with their density ratio being 2:1 [45]. on active slip systems *i* as a function of *ε*11 14 for the (123)[-6-3 4] crystal, which was calculated by the **Figure 36.** a) Orientation relationship between deformed (d) and Rexed (r) states and (b) (111) pole figures of ○ (0 3-1) [100] and □ (001)[100] orientations. Contours were calculated assuming Gaussian scattering (10°) of (0 3-1)[100] and (001)[100] components with their density ratio being 2:1 [45].

Figure 36. (a) Orientation relationship between deformed (d) and Rexed (r 11 ) states and (b) (111) pole

*dε*13 and *dε*23 15 relaxed strain rate sensitive model (m = 0.02) with the subscripts 1, 2, and 3 indicating

(i)/*dε*11 depending on *ε*<sup>11</sup> 16 indicate that the (123)[-6-3 4] orientation is unstable with respect to the plane strain compression. A part of the (123)[-6-3 4] crystal, particularly the surface layers where *dε*23 18 is negligible due to friction between rolls and sheet, seems to rotate to the {112}<111> orientation. A part of the {112}<111> crystal further rotates to (321)[- 436] with increasing reduction. This is why (321)[-436] has lower density than (123)[-6-3 4] along with weak {112}<111>. The orientation rotation is shown in Figure 35b. Since important components in the deformation texture are (123)[-6-3 4], (321)[-436], and {112}<111>, their Rex

(i)/*dε*<sup>11</sup> 23 on the active

(i)/*dε*<sup>11</sup> 24 on C, J, M, and B are 0.014, 0.01, 0.007, and 0.003, respectively, at zero strain. Therefore, AMSD is 0.014 [-101] + 0.010.577 [-1-1 0] + 0.007 0.577 [-1-1 0] + 0.003 [0-1 1] = [-0.02381 -0.01281 0.017], where the factor 0.577 originated from the fact that the duplex slip systems of (1-1 1)[-1-10] and (-111)[-1-10] share the same slip direction (Figure 4). The [-0.02381 -0.01281 0.017] direction, or the [-0.745 -

*dε*13 and *dε*23 15 relaxed strain rate sensitive model (m = 0.02) with the subscripts 1, 2, and 3 indicating

20 436] with increasing reduction. This is why (321)[-436] has lower density than (123)[-6-3 4] along 21 with weak {112}<111>. The orientation rotation is shown in Figure 35b. Since important 22 components in the deformation texture are (123)[-6-3 4], (321)[-436], and {112}<111>, their Rex

(i)/*dε*<sup>11</sup> 23 on the active

(i)/*dε*<sup>11</sup> 24 on C, J, M, and B are 0.014, 0.01, 0.007, and 0.003, respectively, at zero strain. Therefore, AMSD is 0.014 [-101] + 0.010.577 [-1-1 0] + 0.007 0.577 [-1-1 0] + 0.003 [0-1 1] = [-0.02381 -0.01281 0.017], where the factor 0.577 originated from the fact that the duplex slip systems of (1-1 1)[-1-10] and (-111)[-1-10] share the same slip direction (Figure 4). The [-0.02381 -0.01281 0.017] direction, or the [-0.745 -

textures are calculated using SERM. If the (123)[-6-3 4] orientation is stable, *dγ*

textures are calculated using SERM. If the (123)[-6-3 4] orientation is stable, *dγ*

slip systems do not vary with strain. It follows from Fig. 35a that *dγ*

slip systems do not vary with strain. It follows from Fig. 35a that *dγ*

RD, TD, and ND. The changes in *dγ*

The calculated Rex orientation is (0.049 3.543–1.192)[7.801-0.017-0.275] ≈ (0 3-1)[100]. Similarly the (321)[-436] crystal is calculated to have slip systems of (111)[-101], (111)[-110], (-1-1 1)[011], and (-1 1-1)[011], on which the shear strain rates at *dε*11=0 are 0.014, 0.003, 0.01, and 0.007, respectively. The (321)[-436] is calculated to transform to the (-0.049 3.543-1.192)[7.801 0.017 -0.275] ≈ the (0 3-1)[100] Rex texture. This result is understandable from the fact that the (0 3-1) [100] orientation is symmetrical with respect to TD as shown in Figure 36b and the deformation orientations, (123)[-6-3 4] and (321)[-436], are also symmetrical with respect to TD as shown in Figure 35b. The {112}<111> rolling orientation to the {001}<100> Rex orientation transformation is discussed based on SERM in Section 6.1.

cube component in Rex textures were observed at about 3% Mg (Figure 37). This implies that the Cu component is responsible for the cube component. However, these cannot prove that the Cu texture is responsible for the cube texture because deformation components with the

Changes in orientation densities of 95% rolled Cu during annealing at 400 to 500 °C (Figure 38), 95% rolled AA8011 Al alloy during annealing at 350 °C (Figure 39a), and 95% rolled Fe-50%Ni alloy during annealing at 600℃ (Figure 39b), and 95% rolled Cu after heating to 150 to 300℃ at a rate of 2.5 K/s followed by quenching showing that the Cu component disappears most rapidly when the cube orientation started to increase [52]. These results imply that the Cu component is responsible for the cube Rex texture. Rex is likely to occur first in high strain energy regions. It is known that the energy stored in highly deformed crystals is proportional

of specimen, respectively). The Taylor factor is calculated to be 2.45 for the cube oriented fcc crystal using the full constraints model, 3.64 for the Cu oriented fcc crystal using the ε<sup>13</sup> relaxed constraints rate sensitive model, 3.24 for the S oriented fcc crystal using the *ε*13 and *ε*23 relaxed constraints rate sensitive model, and 2.45 for the brass orientation using the ε12 and ε23 relaxed constraints rate sensitive model. In the rate sensitive model calculation, the rate sensitivity index was 0.01 and each strain step in rolling was 0.025. The measured stored energies for 99.99% Al crystals channel-die compressed by a strain of 1.5 showed that the Cu oriented region had higher energies than the S oriented region [53]. The Taylor factors and the measured stored energies indicate that the driving force for Rex is higher in the Cu oriented grains than in the S oriented grains. Therefore, the Cu component in the deformation texture is more

**Rolling reduction Brass Copper Goss S Cube**

73% 2.8 3.0 0.9 1.1 1.1 90% 0.7 5.7 0.1 0.7 1.3

73% 1.8 1.5 1.3 1.4 2.1 90% 0.2 0.8 0.2 0.4 20.0

Rolling texture 58% 3.6 2.6 1.1 1.4 0.6

Rex texture 58% 2.1 1.4 1.0 1.3 1.2

The copper to cube texture transition was first explained by SERM [4], and elaborated later [54]. The orientations of the (112)[1 1-1] and (123)[6 3-4] Cu single crystals remain stable in the center layer for all degree of rolling [55]. The Cu orientation (112)[1 1-1] is calculated to be stable by the *ε*13 relaxed constraint model [56,57]. For the (112)[1 1-1] crystal, the active slip systems are calculated by the RC model to be (-1 1-1)[110], (1-1-1)[110], (111)[1 0-1], and (111) [0 1-1], on which shear strain rates are the same regardless of reduction ratio. Almost the same

*/dεij* with *γ* and *εij* being shear strains on slip systems *k* and strains

Recrystallization Textures of Metals and Alloys

http://dx.doi.org/10.5772/54123

31

highest density are not always linked with highest Rex components [47].

responsible for the cube Rex texture than the S component.

**Table 3.** Texture component strength of high purity OFE copper [46]

to the Taylor factor (Σ*dγ*(*k*)

According to the discussion in Section 6.1, if the cube oriented regions are generated during rolling, they are likely to survive and act as nuclei and grow at the expense of neighboring {112}<111> region during annealing because the region tend to transform to the {001}<100> orientation to reduce energy. The grown-up cube grains will grow at the expense of grains having other orientations such as the {123}<634> orientation, resulting in the {001}<100> texture after Rex, even though the Cu orientation is a minor component in the deformation texture. Meanwhile, the main S component in the deformation texture can form its own Rex texture, the near (0 3-1)[100] orientation. In this case, the Rex texture may be approximated by main (001)[100] and minor (0 3-1)[001] components. Figure 36b shows the texture calculated assuming Gaussian scattering (half angle=10°) of these components with the intensity ratio of (001)[100]: (0 3-1)[001] = 2 : 1. It is interesting to note that the cube peaks diffuse rightward under the influence of the minor (0 3-1)[100] component in agreement with experimental result in Figure 34c.
