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Analysis of deformation textures of asymmetrically rolled aluminum sheets
Acta mater. 49 (2001) 2583–2595 www.elsevier.com/locate/actamat
ANALYSIS OF DEFORMATION TEXTURES OF ASYMMETRICALLY ROLLED ALUMINUM SHEETS K. -H. KIM1‡ and D. N. LEE2† 1
Sheet Products and Process Research Group, Technical Research Laboratories,Pohang Iron and Steel Co., Ltd., Pohang, Kyungbuk 790-785, South Korea and 2School of Materials Science and Engineering, Seoul National University, Seoul 151–742, South Korea ( Received 11 October 2000; received in revised form 15 January 2001; accepted 17 January 2001 )
Abstract—Asymmetric rolling, in which the circumferential velocities of working rolls are different, imposes shear deformation and in turn shear deformation textures to sheets through the thickness. A component of ND//具111典 in the shear deformation textures can improve the plastic strain ratios of aluminum sheets. In order to understand the evolution of ND//具111典, the strain histories and distributions in the sheets and the texture evolution during the asymmetric rolling have been measured and calculated. The shear deformation texture can vary with the ratio of shear to normal strain increments. As the ratio increases from zero to infinity, the texture moves from the plane strain compression texture (b fiber) to the ideal shear deformation texture consisting of {001}具110典, {111}具110典, and {111}具112典. The ratio increases with rolling reduction per pass in asymmetric rolling. However, it is practically difficult to develop a rolling reduction per pass high enough to obtain the ideal shear deformation texture. Imposing the positive and negative shear deformations on the sheet by reversal of the shearing direction can give rise to the ideal shear deformation texture. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Asymmetric/symmetric rolling; Aluminum; Shear deformation; Texture
1. INTRODUCTION
Aluminum alloy sheets are considered to be one of the high potential substitutes for steel sheets when considering the weight reduction of automobiles. However, aluminum alloy sheets have drawbacks such as higher prices and inferior deep drawability. The inferior deep drawability is mainly due to lower plastic strain ratios or Lankford values [1–3]. The plastic strain ratios of aluminum alloy sheets are low, however, due to the texture developed in conventional rolling and thermal processes [4]. The texture of aluminum alloy sheets cold rolled by the conventional rolling processes is characterized by the b fiber texture, which changes to the cube texture after annealing. However, the shear texture characterized by the {001}具110典, {111}具110典, and {111}具112典 orientations forms in the surface layer of aluminum sheets rolled at high frictions between sheet and rolls and at characteristic deformation geometries [5, 6]. Recently, the present authors [7, 8] have developed a process called asymmetric rolling which gives rise
† To whom all correspondence should be addressed. Tel.: +0082-2-880-7093; Fax: +0082-2-884-1413. E-mail address: [email protected] (D. N. Lee) ‡ E-mail address: [email protected] (K. -H. Kim)
to the shear deformation uniformly through the sheet thickness, which in turn form the shear texture. The {111}具110典 and {111}具112典 orientation components in the shear texture can enhance the plastic strain ratio. The uniform shear texture through the thickness means a substantial increase in the plastic strain ratio. Indeed, an average plastic strain ratio of 1.5 or higher could be obtained in asymmetrically rolled aluminum and aluminum alloys [9]. In this paper, the asymmetric rolling of aluminum sheets was analyzed and their deformation textures were measured and analyzed.
2. EXPERIMENTAL METHOD
Aluminum sheets of 2 mm in thickness used in the present study were prepared in the following way. A 99.99% aluminum billet of 75 mm in thickness was cold forged to 40 mm in thickness and annealed at 450°C for 10 min. The 40 mm thick specimen was cold rolled to 2 mm in thickness under lubrication. This deformation is equivalent to a thickness reduction of 95%. The 2 mm thick sheet had a typical plane strain rolling texture characterized by the b fiber texture. A rolling mill had working rolls with different diameters rotating at the same rate, resulting in different circumferential velocities. The ratio of the upper
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 0 3 6 - 2
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Fig. 1. Measure (111) pole figures of aluminum sheets asymmetrically rolled by 50% at roll ratios of (a) 1.25, (b) 1.5, and (c) 2.
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aluminum was assumed to consist of 978 randomly oriented crystallites in each element. The deformation gradient of each element is obtained from the FEM results. The deformation gradient history can yield the velocity gradient that is the sum of the deformation rate tensor and spin tensor. The velocity gradient times incremental time becomes the strain increment. The strain increment history of each element was used to calculate crystallographic orientations based on the full constraints Taylor theory [11] and the Renouward–Wintenberger theory [12]. The orientations were expressed as Gaussian peaks with a scattering angle of 10° and superimposed to obtain distributions in pole figures [13]. 4. RESULTS AND DISCUSSION
Fig. 2. Deformed mesh of aluminum sheets asymmetrically rolled by 50% at roll ratio of (a) 1.25, (b) 1.5, and (c) 2.
roll radius to the lower roll radius, the roll radius ratio, was varied from 1.25 to 2 with the lower roll of 63 mm in radius to obtain the optimum roll radius ratio. In order to investigate the effect of asymmetric rolling reduction on the texture development, the 2 mm thick specimens were asymmetrically rolled by three different schedules with a roll radius ratio of 1.5 at room temperature. In schedule I, the specimens were rolled by 80% through 16 passes, with a rolling depth of 0.1 mm per pass. In schedule II, the specimens were rolled by 80% through six passes, with a rolling depth of 0.25 mm per pass. In schedule III, the specimens were rolled by 80% through 2 passes, with a rolling depth of 1.0 mm in the first pass and 0.6 mm in the second pass. The textures of specimens were measured with an X-ray texture goniometer in the back reflection mode with Fe filtered Co–Ka radiation. The (111), (200), and (220) partial pole figures were measured and used to calculate the complete pole figures by the WIMV program [10]. 3. CALCULATION OF DEFORMATION TEXTURES
The deformation was analyzed by the elasto-plastic FEM. In the analysis, the friction coefficient between rolls and material was assumed to be 0.4, because sticking friction occurred between the rolls and material and the flow curve of aluminum was approximated by s = 179⑀0.22 [5]. The undeformed
Figure 1 shows the measured (111) pole figures of the upper surface, the center, and the lower surface layers of aluminum sheet asymmetrically rolled by 50% at roll radius ratios of 1.25, 1.5, and 2.0. As the angle between a measured major orientation and the center of (111) pole figure approaches zero, the texture approaches the ideal shear deformation texture. For a roll radius ratio of 2.0, the angle deviated from the shear texture is about 15° through the thickness. For a roll radius ratio of 1.25, the textures of the upper and lower layers are close to the shear texture, while the texture of the center layer is close to the plane strain deformation texture. For a roll radius ratio of 1.5, the texture of sheet is close to the relatively uniform shear deformation texture through the thickness. The result indicates that the roll radius ratio of 1.5 gives the best shear texture among the three roll radius ratios. To understand this unexpected result, the deformation was analyzed by the elasto-plastic FEM. Figure 2 shows the deformed meshes obtained by FEM analysis. In order to see the shear strain distributions in the asymmetrically deformed sheets, the shear deformation rates D13 in the upper surface, center, and lower surface layers were calculated. The calculated results are shown in Fig. 3. The total shear strains are related to the areas under curves in Fig. 3. In the upper surface layer, the smallest areas are obtained at a roll radius ratio of 2.0 and the similar areas at roll radii of 1.25 and 1.5. In the center layer, the smallest area was obtained at a roll ratio of 1.25 and the similar areas at roll radii of 1.5 and 2.0. In the lower surface layer, the largest area was obtained at a roll radius ratio of 1.5 and the smallest area at a roll radius of 1.25. The unexpected low D13 value at a radius ratio of 2.0 is due to a decrease in the roll pressure with increasing roll radius ratio. The sign of D13 changes at the neutral point. In Fig. 3(a) where the D13 curves of the upper surface layer are plotted, the positive D13 means that the circumferential velocity of the upper roll is faster than the moving velocity of sheet. Therefore, the neutral point is located at the exit side of the positive D13
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Fig. 3. Calculated deformation rate D13 in (a) upper surface, (b) center, and (c) lower surface layers of aluminum sheets asymmetrically rolled by 50% at roll radius ratios of 1.25, 1.5, and 2. (d) The relative position of neutral point (distance or time from entrance to neutral point N distance or time from entrance to exit L) as a function of roll radius ratio.
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS
Fig. 4. Calculated (111) pole figures of aluminum sheets asymmetrically rolled by 50% at roll radius ratio of (a) 1.25, (b) 1.5, and (c) 2.
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of upper surface moves toward the exit while the neutral point of lower surface moves toward the entrance, as the roll radius ratio increases. The volume elements between the neutral points are subjected to the opposite tangential forces, so the roll pressure in the middle region decreases. As the roll radius ratio increases or the difference between the circumferential velocities of rolls increases, the region between the two neutral points is widen and the roll pressure decreases [14]. Therefore, the optimum roll radius ratio can exist for the best shear results. In the present case, the optimum roll radius ratio is 1.5. Taking a random distribution of orientations, and imposing the deformations of Fig. 3, texture evolution is predicted through simulation (Fig. 4). The textures in Fig. 4 all deviate from the ideal shear deformation textures and are qualitatively comparable with the measured textures in Fig. 1. To better understand the shear texture formation, stable shear deformation textures were calculated as a function of a value. The strain state of sheets under rolling deformation can be approximated by a two-dimensional strain state of compressive strain along the normal direction plus simple shear strain along the rolling direction. The resulting strain can be expressed as
冤
e11 0 e13
Fig. 5. Rotation rates for fcc crystals undergoing plane strain compression at (a) positive and (b) negative a values.
[eij] =
0 0 0
冥
= L dt
(1)
0 0 e33
where eij is displacement gradients with suffixes 1, 2, and 3 indicating the rolling, transverse, and normal directions, L is the velocity gradient, and dt is time increment. Equation (1) is valid at incremental strains. In simple shear, e31 is zero and the shear to normal strain ratio a becomes a=
Fig. 6. Stable orientations rotated from Dillamore orientations about transverse direction as a function of a for fcc crystal.
curve. In Fig. 3(c) where the D13 curves of the lower surface layer are shown, the positive D13 means the sheet moves faster than the circumferential velocity of the lower roll. Therefore, the neutral point is located at the entrance side of the positive D13 curve. Figure 3(d) shows the relative position of neutral point (the distance or time from the entrance to the neutral point divided by the distance or time from the entrance to the exit) as a function of roll radius ratio. It can be seen from the figure that the neutral point
L13 e13 = . L11 e11
(2)
In order to obtain stable orientations whose transverse direction is parallel to [11¯ 0] as a function of a, their rotation rates were calculated. The calculated results are shown in Fig. 5, in which a = ±⬁ cannot be achieved in rolling, so calculated by setting e11 = e33 = e31 = 0, e13 = ±0.1. The rotation is defined as positive when the rotation rate is positive, and vice versa. Therefore, an orientation is stable when the slope of the curve is negative at a rotation rate of zero. For a positive slope at a rotation rate of zero, the orientation is metastable. When a = 0, in which no shear strain exists, the Dillamore orientation {4 4 11}具11 11 8典 and the Goss orientation {110}具001典 are stable as is well known. As the absolute value of a increases, the stable orientation moves to a shear deformation texture and finally to the ideal shear deformation texture consisting of {001}具110典 and {111}具112典 at a = ±⬁.
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS
Figure 6 shows the stable orientation plotted against the a value. As the absolute value of a increases, the orientation approaches the {001}具110典 and {111}具112典 orientations as already pointed out, except in the range of 1.8⬍|a|⬍2.1, in which the {111}具112典 orientation does not appear. The following hypothetical calculation shows it is true. Figure 7(a) shows the texture calculated assuming that an aluminum sheet is plane strain compressed up to ⑀=2.0 at a=0, which is characterized by the ideal plane strain deformation texture. If the compressed sheet is deformed further up to ⑀=0.5 at a=2.0 to have accumulated strain ⑀=2.5, the texture shown in Fig. 7(b) is obtained, which is not symmetric about transverse direction. However, when the sheet having the texture in Fig. 7(b) is deformed up to ⑀=0.1 at a=⫺2.0 to have accumulated strain ⑀=2.6, the sheet is calculated to have the texture in Fig. 7(c), which is near symmetric about the transverse direction and approximated by (001)[1¯ 10] and does not contain the {111}具112典 orientation. It is noted that the symmetry is achieved by the opposite shear. The effect of the opposite shear on the symmetric texture formation will be discussed in the next section. Figure 8 shows experimental textures of the upper surface layers of aluminum sheets asymmetrically rolled by the schedules I, II, and III. It can be seen that the initial texture rotates as the rolling reduction increases. However, for the schedule I, in which the rolling depth is 0.1 mm per pass, the end texture is slightly away from the plane strain deformation texture, even after 80% reduction. As the rolling depth per pass increases, the deformation texture approaches increasingly to the ideal shear deformation texture. The asymmetric rolling gives rise to rotation about the transverse axis. The rotation is expected to increase with increasing a value. Therefore, the textures of asymmetrically rolled sheets may be calculated using strains defined by the a values. Indeed, the textures of 80% rolled aluminum sheets in Fig. 8 could be approximated by the textures calculated at
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⑀11=1.6 (80%) using appropriate a values and the initial {225}具554典 orientation, which approximated the texture of plane strain rolled aluminum sheets. The calculated ideal orientations are shown as a function of ⑀11 in Fig. 9(a–c), which were calculated using a=1.1, a=1.3, and a=10, respectively. The solid square symbols indicate the orientations at 80% reduction. The texture in Fig. 9(c) is very close to the ideal shear deformation texture. Figure 10 shows the orientations rotated from two components of the Dillamore orientation, the (4 4 11)[11 11 8] and (4 4 11)[11118] orientations, calculated as a function of a with d⑀11=0.01 at ⑀11=2. At a small value of a, the rotation about the transverse direction is small. As the a value increases, the rotation rate increases and rapidly approaches the ideal shear deformation texture. The two Dillamore components rotate in the same direction and both the {001}具110典 and {111}具112典 orientations appear at high a values, except in the range of 1.8⬍|a|⬍2.1, as already pointed out in Fig. 6. The calculated results are in qualitative agreement with the experimental results in Fig. 9. It was pointed out that the ideal shear deformation texture could not be obtained by unidirectional asymmetric rolling. It has been known that the ideal shear deformation texture can be obtained in the surface layer of the usual symmetric rolling [5, 15]. In symmetric rolling, the surface layer undergoes the positive shear from the entrance to the neutral point and the negative shear from the neutral point to the exit. Therefore, it is necessary to investigate the effect of the opposite shear strains on the texture development. In order to understand the differences in deformation behavior in symmetric and asymmetric plane strain rolling of aluminum, the finite element analysis was made at a roll radius ratio of 1.5 with a lower roll diameter of 126 mm, an initial sheet thickness of 2 mm, and a rolling reduction of 50%. The friction coefficient was set to be 0.4. In the symmetric rolling, the upper and lower roll diameters are all 126 mm
Fig. 7. (111) pole figures of aluminum sheet calculated assuming that it is (a) deformed up to ⑀=2.0 at a=0, (b) additionally deformed by ⑀=0.5 at a=2.0 to have accumulated strain, ⑀=2.5, and (c) subsequently deformed by ⑀=0.1 at a=⫺2.0 to have accumulated strain, ⑀=2.6.
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Fig. 8. Experimental (111) pole figures of upper surface layers of aluminum sheets asymmetrically rolled by schedules (a) I, (b) II, and (c) III.
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS
Fig. 9. Experimental (111) pole figures of aluminum sheets 80% asymmetrically rolled by schedules (a) I, (b) II, and (c) III. Curves in (a), (b), and (c) indicate orientations rotated from initial orientation of {225}具554典 as a function of ⑀11, calculated using a=1.1, a=1.3, and a=10, respectively. 䊏, Orientations at 80% reduction (⑀11=1.6).
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and other conditions are the same as in the asymmetric rolling. Figure 11 shows meshes of the 50% symmetrically and asymmetrically rolled sheets and distortion of mesh in the deformation zone of the asymmetric rolling. The symmetrically rolled sheet shows severe shear deformations in the surface layers and the severity decreases with approaching the center in which no shear deformation takes place, whereas the asymmetrically rolled sheet shows shear deformation taking place through the thickness. The shear strains in symmetric rolling are known to be influenced by the friction between rolls and material and the deformation zone geometry characterized by the L/h value with L and h being the projected contact length and the average thickness [16–23]. In the asymmetric rolling, the circumferential speed of the upper roll is higher than that of the lower roll, which in turn gives rise to the highest speed in the upper surface layer at the exit and the lowest speed in the lower surface layer at the entrance, resulting in the distortion in Fig. 11(c). Figure 12(a) shows calculated shear stress distributions in the symmetric and asymmetric roll deformation zones. The parameter s in the figure is defined by s = 2d/t with d and t being a distance from the center layer and the sheet thickness, respectively, that is, s = 0 at the center, s = 1 at the upper surface, and s = ⫺1 at the lower surface. In the symmetric rolling the shear stress is highest at the surface and rapidly decreases, as the center is approached. The shear stress direction changes at the neutral point. In the asymmetric rolling, most layers are under the positive shear stresses. It is noted that even the center layer is under the positive shear stress throughout deformation and the negative s layers are under the negative shear stress only in the initial stage. This is related to the distortion in Fig. 11(c). Figure 12(b) shows the calculated shear components D13 of the deformation rate tensor D along the thickness direction during symmetric and asymmetric rolling. In the symmetric rolling, the shear strain increment is the largest at the surface layer and zero at the center layer and the direction of the shear strain increment changes at the neutral point. However, during the asymmetric rolling, all the layers undergo positive shear deformation except in the initial stage of the lower surface layer. Figure 12(c) shows the a values through the thickness during rolling. The a distribution is similar to the shear strain distribution (b), even though the difference in a value between the surface and center layers is higher than in the shear strain. Figure 13 shows the textures calculated, as explained in Section 3, along the thickness direction of symmetrically and asymmetrically rolled aluminum sheets. In the symmetric rolling, the pole figures in most layers (s = 0, s = 0.1, s = 0.7, and s = 0.9) except the middle layer (s=0.5) are almost symmetric about the transverse direction. However, in the asym-
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Fig. 10. Orientations rotated from Dillamore orientations during plane strain compression of fcc crystal up to ⑀11=2 at positive a (first two rows) and negative a (last two rows) values.
Fig. 11. Deformed meshes of (a) symmetrically and (b) asymmetrically rolled aluminum sheets and (c) distortion of mesh in deformation zone of asymmetric rolling. Reduction and friction coefficient are 50% and 0.4, respectively.
metric rolling, all layers do not show the symmetry as in the symmetric rolling as already discussed in the previous section. The symmetric pole figures are natural, when the shear strain is small compared with the normal strain (small a values). It can be seen that the symmetric shear deformation texture or the ideal shear deformation texture is related to the reversion of shear direction. The texture of the lower surface layer (s = ⫺0.9) is closer to the ideal shear deformation texture than others in the asymmetrically rolled sheet is attributed to the fact it undergoes a relatively large negative shear strain in the entrance zone. The texture inhomogeneity along the thickness in symmetric rolling has been studied by many workers [16–18, 23, 24], even though the symmetry problem has not been dealt with. Figure 14 shows the measured textures of the upper surface layers of aluminum sheet to see an effect of the reversal of shear direction in rolling. The aluminum sheet having b fiber texture [Fig. 14(a)] was asymmetrically rolled by 35% through 6 passes to form the texture in Fig. 14(b). Figure 14(a and b) have a rotational relation of 5° about the transverse direction. The sheet of Fig. 14(b) was further asymmetrically rolled by 23% in the same shear direction to have the texture shown in Fig. 14(c), which is
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Fig. 12. Calculated distributions of (a) shear stress s13, (b) shear strain increment D13, and (c) a value in deformation zone of symmetric (left) and asymmetric rolling (right).
rotated through 3–4° from the texture in Fig. 14(b) and has a higher intensity than Fig. 14(b). The specimen in Fig. 14(b) was asymmetrically rolled by 23% in the reverse direction to form the texture shown in Fig. 14(d), whose center peak just crossed the transverse line and was close to the ideal shear deformation texture. It should be emphasized that the reverse shear deformation could bring about the ideal shear deformation texture, as already seen in the calculation in Fig. 7. This is the reason why unidirectional asymmetric rolling does not give rise to the ideal shear deformation texture, whereas symmetric rolling does develop the ideal shear deformation texture in the surface layers under high friction between rolls and sheet.
increasing ratio of the shear to normal strain. For given roll radius ratio and total rolling reduction, the shear to normal strain ratio increases with increasing reduction per path. The relative position neutral point of the upper surface moves toward the exit while that of the lower surface moves toward the entrance, as the roll radius ratio increases. The ideal shear deformation texture cannot be obtained by unidirectional asymmetric rolling, but can be obtained by reversing the asymmetric rolling direction one after another. The surface layers of symmetrically rolled sheets readily have the ideal shear deformation texture, because they are subjected to the positive shear from the entrance to the neutral point and to the negative shear from the neutral point to the exit.
5. CONCLUSIONS
Asymmetric rolling, in which the upper and lower roll radii are different, imposes shear deformation and in turn shear deformation textures on sheets through the thickness. The shear deformation texture approaches the ideal shear deformation texture with
Acknowledgements—Financial support from the National Research Laboratory for Texture Control, Seoul National University is gratefully acknowledged.
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Fig. 13. Calculated (111) pole figures along thickness direction of (a) symmetrically and (b) asymmetrically rolled aluminum sheets.
REFERENCES
Fig. 14. (111) pole figures of upper surface layers of aluminum sheets. (a) b fiber texture of starting sheet, (b) texture of sheet asymmetrically rolled by 35% through six passes, (c) texture of sheet obtained by asymmetric rolling of sheet in (b) by 23%, and (d) texture of sheet obtained by asymmetric rolling of sheet in (b) by 23% in reverse direction.
1. Hosford, W. F. and Caddell, R. M., in Metal Forming. Prentice Hall, Englewood Cliffs, NJ, 1983, p. 274. 2. Lee, D. N., J. Mater. Sci. Lett., 1984, 3, 677. 3. Lee, D. N., in Strength of Metals and Materials, ed. H. J. McQueen, J. P. Balio, J. I. Dickson, J. J. Jonas, M. G. Akben and D. N. Lee. Pergamon, Oxford, 1985, p. 971. 4. Lequeu, Ph., Gilormini, P., Montheillet, F., Bacroix, B. and Jonas, J. J., Acta Metall., 1987, 35, 1159. 5. Choi, C. -H., Kwon, J. -W., Oh, K. H. and Lee, D. N., Acta mater., 1997, 42, 5119. 6. Kamijo, T., Kakota, S. and Inagaki, H., Acta Metall., 1993, 41, 1713. 7. Choi, C.-H., Kim, K.-H. and Lee, D. N., Mater. Sci. Forum, 1998, 273-275, 391. 8. Choi, C. -H., Kim, K. -H., Jeong, S. -Y. and Lee, D. N., J. Korean Inst. Met. Mater., 1997, 35, 429. 9. Choi, C. -H., Kim, K. -H. and Lee, D. N., in Synthesis/Processing of Lightweight Metallic Materials II, ed. C. M. Ward-Close, F. H. Froes, D. J. Chelleman and S. S. Cho. The Minerals, Metals and Materials Society, Warrendale, PA, USA, 1997, p. 37. 10. Matthies, S., Phys. Stat. Sol., 1980, 101, 111. 11. Taylor, G. L., J. Inst. Metals, 1938, 63, 307. 12. Renouard, M. and Wintenberger, M., C. R. Acad. Sci. Paris Serie B, 1981, 292, 385. 13. Bunge, H. -J., in Texture Analysis in Materials Science. Butterworths, London, 1982, p. 180. 14. Hwang, Y. -M. and Tzou, G. -Y., J. Mater. Engng Performance, 1995, 4, 265. 15. Lee, S. H. and Lee, D. N., Mater. Sci. Engng, 1998, A249, 84.
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS 16. Asbeck, H. O. and Mecking, H., Mater. Sci. Engng, 1978, 34, 111. 17. Truszkowski, W., Krol, J. and Major, B., Metall. Trans., 1980, 11A, 749. 18. Truszkowski, W., Krol, W. and Major, B., Metall. Trans., 1982, 13A, 665. 19. Bauer, R. E., Mecking, H. and Lu¨ cke, K., Mater. Sci. Engng, 1977, 68, 31. 20. Sakai, T., Saito, Y. and Kato, K., Trans. ISIJ, 1987, 27, 520.
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21. Sakai, T., Saito, Y., Matsuo, M. and Kawasaki, K., ISIJ Int., 1991, 31, 86. 22. Um, K. -K., Jeong, H. -T., An, J. -K., Lee, D. N., Kim, G. and Kwon, O., ISIJ Int., 2000, 40, 58. 23. Choi, C. -H. and Lee, D. N., Met. Mater. Trans., 1997, 28A, 2217. 24. Huh, M. -Y., Cho, Y. S., Kim, J. S. and Engler, O., Z. Metallkd., 1999, 90(2), 124.
ANALYSIS OF DEFORMATION TEXTURES OF ASYMMETRICALLY ROLLED ALUMINUM SHEETS K. -H. KIM1‡ and D. N. LEE2† 1
Sheet Products and Process Research Group, Technical Research Laboratories,Pohang Iron and Steel Co., Ltd., Pohang, Kyungbuk 790-785, South Korea and 2School of Materials Science and Engineering, Seoul National University, Seoul 151–742, South Korea ( Received 11 October 2000; received in revised form 15 January 2001; accepted 17 January 2001 )
Abstract—Asymmetric rolling, in which the circumferential velocities of working rolls are different, imposes shear deformation and in turn shear deformation textures to sheets through the thickness. A component of ND//具111典 in the shear deformation textures can improve the plastic strain ratios of aluminum sheets. In order to understand the evolution of ND//具111典, the strain histories and distributions in the sheets and the texture evolution during the asymmetric rolling have been measured and calculated. The shear deformation texture can vary with the ratio of shear to normal strain increments. As the ratio increases from zero to infinity, the texture moves from the plane strain compression texture (b fiber) to the ideal shear deformation texture consisting of {001}具110典, {111}具110典, and {111}具112典. The ratio increases with rolling reduction per pass in asymmetric rolling. However, it is practically difficult to develop a rolling reduction per pass high enough to obtain the ideal shear deformation texture. Imposing the positive and negative shear deformations on the sheet by reversal of the shearing direction can give rise to the ideal shear deformation texture. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Asymmetric/symmetric rolling; Aluminum; Shear deformation; Texture
1. INTRODUCTION
Aluminum alloy sheets are considered to be one of the high potential substitutes for steel sheets when considering the weight reduction of automobiles. However, aluminum alloy sheets have drawbacks such as higher prices and inferior deep drawability. The inferior deep drawability is mainly due to lower plastic strain ratios or Lankford values [1–3]. The plastic strain ratios of aluminum alloy sheets are low, however, due to the texture developed in conventional rolling and thermal processes [4]. The texture of aluminum alloy sheets cold rolled by the conventional rolling processes is characterized by the b fiber texture, which changes to the cube texture after annealing. However, the shear texture characterized by the {001}具110典, {111}具110典, and {111}具112典 orientations forms in the surface layer of aluminum sheets rolled at high frictions between sheet and rolls and at characteristic deformation geometries [5, 6]. Recently, the present authors [7, 8] have developed a process called asymmetric rolling which gives rise
† To whom all correspondence should be addressed. Tel.: +0082-2-880-7093; Fax: +0082-2-884-1413. E-mail address: [email protected] (D. N. Lee) ‡ E-mail address: [email protected] (K. -H. Kim)
to the shear deformation uniformly through the sheet thickness, which in turn form the shear texture. The {111}具110典 and {111}具112典 orientation components in the shear texture can enhance the plastic strain ratio. The uniform shear texture through the thickness means a substantial increase in the plastic strain ratio. Indeed, an average plastic strain ratio of 1.5 or higher could be obtained in asymmetrically rolled aluminum and aluminum alloys [9]. In this paper, the asymmetric rolling of aluminum sheets was analyzed and their deformation textures were measured and analyzed.
2. EXPERIMENTAL METHOD
Aluminum sheets of 2 mm in thickness used in the present study were prepared in the following way. A 99.99% aluminum billet of 75 mm in thickness was cold forged to 40 mm in thickness and annealed at 450°C for 10 min. The 40 mm thick specimen was cold rolled to 2 mm in thickness under lubrication. This deformation is equivalent to a thickness reduction of 95%. The 2 mm thick sheet had a typical plane strain rolling texture characterized by the b fiber texture. A rolling mill had working rolls with different diameters rotating at the same rate, resulting in different circumferential velocities. The ratio of the upper
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 0 3 6 - 2
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Fig. 1. Measure (111) pole figures of aluminum sheets asymmetrically rolled by 50% at roll ratios of (a) 1.25, (b) 1.5, and (c) 2.
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aluminum was assumed to consist of 978 randomly oriented crystallites in each element. The deformation gradient of each element is obtained from the FEM results. The deformation gradient history can yield the velocity gradient that is the sum of the deformation rate tensor and spin tensor. The velocity gradient times incremental time becomes the strain increment. The strain increment history of each element was used to calculate crystallographic orientations based on the full constraints Taylor theory [11] and the Renouward–Wintenberger theory [12]. The orientations were expressed as Gaussian peaks with a scattering angle of 10° and superimposed to obtain distributions in pole figures [13]. 4. RESULTS AND DISCUSSION
Fig. 2. Deformed mesh of aluminum sheets asymmetrically rolled by 50% at roll ratio of (a) 1.25, (b) 1.5, and (c) 2.
roll radius to the lower roll radius, the roll radius ratio, was varied from 1.25 to 2 with the lower roll of 63 mm in radius to obtain the optimum roll radius ratio. In order to investigate the effect of asymmetric rolling reduction on the texture development, the 2 mm thick specimens were asymmetrically rolled by three different schedules with a roll radius ratio of 1.5 at room temperature. In schedule I, the specimens were rolled by 80% through 16 passes, with a rolling depth of 0.1 mm per pass. In schedule II, the specimens were rolled by 80% through six passes, with a rolling depth of 0.25 mm per pass. In schedule III, the specimens were rolled by 80% through 2 passes, with a rolling depth of 1.0 mm in the first pass and 0.6 mm in the second pass. The textures of specimens were measured with an X-ray texture goniometer in the back reflection mode with Fe filtered Co–Ka radiation. The (111), (200), and (220) partial pole figures were measured and used to calculate the complete pole figures by the WIMV program [10]. 3. CALCULATION OF DEFORMATION TEXTURES
The deformation was analyzed by the elasto-plastic FEM. In the analysis, the friction coefficient between rolls and material was assumed to be 0.4, because sticking friction occurred between the rolls and material and the flow curve of aluminum was approximated by s = 179⑀0.22 [5]. The undeformed
Figure 1 shows the measured (111) pole figures of the upper surface, the center, and the lower surface layers of aluminum sheet asymmetrically rolled by 50% at roll radius ratios of 1.25, 1.5, and 2.0. As the angle between a measured major orientation and the center of (111) pole figure approaches zero, the texture approaches the ideal shear deformation texture. For a roll radius ratio of 2.0, the angle deviated from the shear texture is about 15° through the thickness. For a roll radius ratio of 1.25, the textures of the upper and lower layers are close to the shear texture, while the texture of the center layer is close to the plane strain deformation texture. For a roll radius ratio of 1.5, the texture of sheet is close to the relatively uniform shear deformation texture through the thickness. The result indicates that the roll radius ratio of 1.5 gives the best shear texture among the three roll radius ratios. To understand this unexpected result, the deformation was analyzed by the elasto-plastic FEM. Figure 2 shows the deformed meshes obtained by FEM analysis. In order to see the shear strain distributions in the asymmetrically deformed sheets, the shear deformation rates D13 in the upper surface, center, and lower surface layers were calculated. The calculated results are shown in Fig. 3. The total shear strains are related to the areas under curves in Fig. 3. In the upper surface layer, the smallest areas are obtained at a roll radius ratio of 2.0 and the similar areas at roll radii of 1.25 and 1.5. In the center layer, the smallest area was obtained at a roll ratio of 1.25 and the similar areas at roll radii of 1.5 and 2.0. In the lower surface layer, the largest area was obtained at a roll radius ratio of 1.5 and the smallest area at a roll radius of 1.25. The unexpected low D13 value at a radius ratio of 2.0 is due to a decrease in the roll pressure with increasing roll radius ratio. The sign of D13 changes at the neutral point. In Fig. 3(a) where the D13 curves of the upper surface layer are plotted, the positive D13 means that the circumferential velocity of the upper roll is faster than the moving velocity of sheet. Therefore, the neutral point is located at the exit side of the positive D13
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Fig. 3. Calculated deformation rate D13 in (a) upper surface, (b) center, and (c) lower surface layers of aluminum sheets asymmetrically rolled by 50% at roll radius ratios of 1.25, 1.5, and 2. (d) The relative position of neutral point (distance or time from entrance to neutral point N distance or time from entrance to exit L) as a function of roll radius ratio.
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS
Fig. 4. Calculated (111) pole figures of aluminum sheets asymmetrically rolled by 50% at roll radius ratio of (a) 1.25, (b) 1.5, and (c) 2.
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of upper surface moves toward the exit while the neutral point of lower surface moves toward the entrance, as the roll radius ratio increases. The volume elements between the neutral points are subjected to the opposite tangential forces, so the roll pressure in the middle region decreases. As the roll radius ratio increases or the difference between the circumferential velocities of rolls increases, the region between the two neutral points is widen and the roll pressure decreases [14]. Therefore, the optimum roll radius ratio can exist for the best shear results. In the present case, the optimum roll radius ratio is 1.5. Taking a random distribution of orientations, and imposing the deformations of Fig. 3, texture evolution is predicted through simulation (Fig. 4). The textures in Fig. 4 all deviate from the ideal shear deformation textures and are qualitatively comparable with the measured textures in Fig. 1. To better understand the shear texture formation, stable shear deformation textures were calculated as a function of a value. The strain state of sheets under rolling deformation can be approximated by a two-dimensional strain state of compressive strain along the normal direction plus simple shear strain along the rolling direction. The resulting strain can be expressed as
冤
e11 0 e13
Fig. 5. Rotation rates for fcc crystals undergoing plane strain compression at (a) positive and (b) negative a values.
[eij] =
0 0 0
冥
= L dt
(1)
0 0 e33
where eij is displacement gradients with suffixes 1, 2, and 3 indicating the rolling, transverse, and normal directions, L is the velocity gradient, and dt is time increment. Equation (1) is valid at incremental strains. In simple shear, e31 is zero and the shear to normal strain ratio a becomes a=
Fig. 6. Stable orientations rotated from Dillamore orientations about transverse direction as a function of a for fcc crystal.
curve. In Fig. 3(c) where the D13 curves of the lower surface layer are shown, the positive D13 means the sheet moves faster than the circumferential velocity of the lower roll. Therefore, the neutral point is located at the entrance side of the positive D13 curve. Figure 3(d) shows the relative position of neutral point (the distance or time from the entrance to the neutral point divided by the distance or time from the entrance to the exit) as a function of roll radius ratio. It can be seen from the figure that the neutral point
L13 e13 = . L11 e11
(2)
In order to obtain stable orientations whose transverse direction is parallel to [11¯ 0] as a function of a, their rotation rates were calculated. The calculated results are shown in Fig. 5, in which a = ±⬁ cannot be achieved in rolling, so calculated by setting e11 = e33 = e31 = 0, e13 = ±0.1. The rotation is defined as positive when the rotation rate is positive, and vice versa. Therefore, an orientation is stable when the slope of the curve is negative at a rotation rate of zero. For a positive slope at a rotation rate of zero, the orientation is metastable. When a = 0, in which no shear strain exists, the Dillamore orientation {4 4 11}具11 11 8典 and the Goss orientation {110}具001典 are stable as is well known. As the absolute value of a increases, the stable orientation moves to a shear deformation texture and finally to the ideal shear deformation texture consisting of {001}具110典 and {111}具112典 at a = ±⬁.
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS
Figure 6 shows the stable orientation plotted against the a value. As the absolute value of a increases, the orientation approaches the {001}具110典 and {111}具112典 orientations as already pointed out, except in the range of 1.8⬍|a|⬍2.1, in which the {111}具112典 orientation does not appear. The following hypothetical calculation shows it is true. Figure 7(a) shows the texture calculated assuming that an aluminum sheet is plane strain compressed up to ⑀=2.0 at a=0, which is characterized by the ideal plane strain deformation texture. If the compressed sheet is deformed further up to ⑀=0.5 at a=2.0 to have accumulated strain ⑀=2.5, the texture shown in Fig. 7(b) is obtained, which is not symmetric about transverse direction. However, when the sheet having the texture in Fig. 7(b) is deformed up to ⑀=0.1 at a=⫺2.0 to have accumulated strain ⑀=2.6, the sheet is calculated to have the texture in Fig. 7(c), which is near symmetric about the transverse direction and approximated by (001)[1¯ 10] and does not contain the {111}具112典 orientation. It is noted that the symmetry is achieved by the opposite shear. The effect of the opposite shear on the symmetric texture formation will be discussed in the next section. Figure 8 shows experimental textures of the upper surface layers of aluminum sheets asymmetrically rolled by the schedules I, II, and III. It can be seen that the initial texture rotates as the rolling reduction increases. However, for the schedule I, in which the rolling depth is 0.1 mm per pass, the end texture is slightly away from the plane strain deformation texture, even after 80% reduction. As the rolling depth per pass increases, the deformation texture approaches increasingly to the ideal shear deformation texture. The asymmetric rolling gives rise to rotation about the transverse axis. The rotation is expected to increase with increasing a value. Therefore, the textures of asymmetrically rolled sheets may be calculated using strains defined by the a values. Indeed, the textures of 80% rolled aluminum sheets in Fig. 8 could be approximated by the textures calculated at
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⑀11=1.6 (80%) using appropriate a values and the initial {225}具554典 orientation, which approximated the texture of plane strain rolled aluminum sheets. The calculated ideal orientations are shown as a function of ⑀11 in Fig. 9(a–c), which were calculated using a=1.1, a=1.3, and a=10, respectively. The solid square symbols indicate the orientations at 80% reduction. The texture in Fig. 9(c) is very close to the ideal shear deformation texture. Figure 10 shows the orientations rotated from two components of the Dillamore orientation, the (4 4 11)[11 11 8] and (4 4 11)[11118] orientations, calculated as a function of a with d⑀11=0.01 at ⑀11=2. At a small value of a, the rotation about the transverse direction is small. As the a value increases, the rotation rate increases and rapidly approaches the ideal shear deformation texture. The two Dillamore components rotate in the same direction and both the {001}具110典 and {111}具112典 orientations appear at high a values, except in the range of 1.8⬍|a|⬍2.1, as already pointed out in Fig. 6. The calculated results are in qualitative agreement with the experimental results in Fig. 9. It was pointed out that the ideal shear deformation texture could not be obtained by unidirectional asymmetric rolling. It has been known that the ideal shear deformation texture can be obtained in the surface layer of the usual symmetric rolling [5, 15]. In symmetric rolling, the surface layer undergoes the positive shear from the entrance to the neutral point and the negative shear from the neutral point to the exit. Therefore, it is necessary to investigate the effect of the opposite shear strains on the texture development. In order to understand the differences in deformation behavior in symmetric and asymmetric plane strain rolling of aluminum, the finite element analysis was made at a roll radius ratio of 1.5 with a lower roll diameter of 126 mm, an initial sheet thickness of 2 mm, and a rolling reduction of 50%. The friction coefficient was set to be 0.4. In the symmetric rolling, the upper and lower roll diameters are all 126 mm
Fig. 7. (111) pole figures of aluminum sheet calculated assuming that it is (a) deformed up to ⑀=2.0 at a=0, (b) additionally deformed by ⑀=0.5 at a=2.0 to have accumulated strain, ⑀=2.5, and (c) subsequently deformed by ⑀=0.1 at a=⫺2.0 to have accumulated strain, ⑀=2.6.
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Fig. 8. Experimental (111) pole figures of upper surface layers of aluminum sheets asymmetrically rolled by schedules (a) I, (b) II, and (c) III.
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS
Fig. 9. Experimental (111) pole figures of aluminum sheets 80% asymmetrically rolled by schedules (a) I, (b) II, and (c) III. Curves in (a), (b), and (c) indicate orientations rotated from initial orientation of {225}具554典 as a function of ⑀11, calculated using a=1.1, a=1.3, and a=10, respectively. 䊏, Orientations at 80% reduction (⑀11=1.6).
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and other conditions are the same as in the asymmetric rolling. Figure 11 shows meshes of the 50% symmetrically and asymmetrically rolled sheets and distortion of mesh in the deformation zone of the asymmetric rolling. The symmetrically rolled sheet shows severe shear deformations in the surface layers and the severity decreases with approaching the center in which no shear deformation takes place, whereas the asymmetrically rolled sheet shows shear deformation taking place through the thickness. The shear strains in symmetric rolling are known to be influenced by the friction between rolls and material and the deformation zone geometry characterized by the L/h value with L and h being the projected contact length and the average thickness [16–23]. In the asymmetric rolling, the circumferential speed of the upper roll is higher than that of the lower roll, which in turn gives rise to the highest speed in the upper surface layer at the exit and the lowest speed in the lower surface layer at the entrance, resulting in the distortion in Fig. 11(c). Figure 12(a) shows calculated shear stress distributions in the symmetric and asymmetric roll deformation zones. The parameter s in the figure is defined by s = 2d/t with d and t being a distance from the center layer and the sheet thickness, respectively, that is, s = 0 at the center, s = 1 at the upper surface, and s = ⫺1 at the lower surface. In the symmetric rolling the shear stress is highest at the surface and rapidly decreases, as the center is approached. The shear stress direction changes at the neutral point. In the asymmetric rolling, most layers are under the positive shear stresses. It is noted that even the center layer is under the positive shear stress throughout deformation and the negative s layers are under the negative shear stress only in the initial stage. This is related to the distortion in Fig. 11(c). Figure 12(b) shows the calculated shear components D13 of the deformation rate tensor D along the thickness direction during symmetric and asymmetric rolling. In the symmetric rolling, the shear strain increment is the largest at the surface layer and zero at the center layer and the direction of the shear strain increment changes at the neutral point. However, during the asymmetric rolling, all the layers undergo positive shear deformation except in the initial stage of the lower surface layer. Figure 12(c) shows the a values through the thickness during rolling. The a distribution is similar to the shear strain distribution (b), even though the difference in a value between the surface and center layers is higher than in the shear strain. Figure 13 shows the textures calculated, as explained in Section 3, along the thickness direction of symmetrically and asymmetrically rolled aluminum sheets. In the symmetric rolling, the pole figures in most layers (s = 0, s = 0.1, s = 0.7, and s = 0.9) except the middle layer (s=0.5) are almost symmetric about the transverse direction. However, in the asym-
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Fig. 10. Orientations rotated from Dillamore orientations during plane strain compression of fcc crystal up to ⑀11=2 at positive a (first two rows) and negative a (last two rows) values.
Fig. 11. Deformed meshes of (a) symmetrically and (b) asymmetrically rolled aluminum sheets and (c) distortion of mesh in deformation zone of asymmetric rolling. Reduction and friction coefficient are 50% and 0.4, respectively.
metric rolling, all layers do not show the symmetry as in the symmetric rolling as already discussed in the previous section. The symmetric pole figures are natural, when the shear strain is small compared with the normal strain (small a values). It can be seen that the symmetric shear deformation texture or the ideal shear deformation texture is related to the reversion of shear direction. The texture of the lower surface layer (s = ⫺0.9) is closer to the ideal shear deformation texture than others in the asymmetrically rolled sheet is attributed to the fact it undergoes a relatively large negative shear strain in the entrance zone. The texture inhomogeneity along the thickness in symmetric rolling has been studied by many workers [16–18, 23, 24], even though the symmetry problem has not been dealt with. Figure 14 shows the measured textures of the upper surface layers of aluminum sheet to see an effect of the reversal of shear direction in rolling. The aluminum sheet having b fiber texture [Fig. 14(a)] was asymmetrically rolled by 35% through 6 passes to form the texture in Fig. 14(b). Figure 14(a and b) have a rotational relation of 5° about the transverse direction. The sheet of Fig. 14(b) was further asymmetrically rolled by 23% in the same shear direction to have the texture shown in Fig. 14(c), which is
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS
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Fig. 12. Calculated distributions of (a) shear stress s13, (b) shear strain increment D13, and (c) a value in deformation zone of symmetric (left) and asymmetric rolling (right).
rotated through 3–4° from the texture in Fig. 14(b) and has a higher intensity than Fig. 14(b). The specimen in Fig. 14(b) was asymmetrically rolled by 23% in the reverse direction to form the texture shown in Fig. 14(d), whose center peak just crossed the transverse line and was close to the ideal shear deformation texture. It should be emphasized that the reverse shear deformation could bring about the ideal shear deformation texture, as already seen in the calculation in Fig. 7. This is the reason why unidirectional asymmetric rolling does not give rise to the ideal shear deformation texture, whereas symmetric rolling does develop the ideal shear deformation texture in the surface layers under high friction between rolls and sheet.
increasing ratio of the shear to normal strain. For given roll radius ratio and total rolling reduction, the shear to normal strain ratio increases with increasing reduction per path. The relative position neutral point of the upper surface moves toward the exit while that of the lower surface moves toward the entrance, as the roll radius ratio increases. The ideal shear deformation texture cannot be obtained by unidirectional asymmetric rolling, but can be obtained by reversing the asymmetric rolling direction one after another. The surface layers of symmetrically rolled sheets readily have the ideal shear deformation texture, because they are subjected to the positive shear from the entrance to the neutral point and to the negative shear from the neutral point to the exit.
5. CONCLUSIONS
Asymmetric rolling, in which the upper and lower roll radii are different, imposes shear deformation and in turn shear deformation textures on sheets through the thickness. The shear deformation texture approaches the ideal shear deformation texture with
Acknowledgements—Financial support from the National Research Laboratory for Texture Control, Seoul National University is gratefully acknowledged.
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Fig. 13. Calculated (111) pole figures along thickness direction of (a) symmetrically and (b) asymmetrically rolled aluminum sheets.
REFERENCES
Fig. 14. (111) pole figures of upper surface layers of aluminum sheets. (a) b fiber texture of starting sheet, (b) texture of sheet asymmetrically rolled by 35% through six passes, (c) texture of sheet obtained by asymmetric rolling of sheet in (b) by 23%, and (d) texture of sheet obtained by asymmetric rolling of sheet in (b) by 23% in reverse direction.
1. Hosford, W. F. and Caddell, R. M., in Metal Forming. Prentice Hall, Englewood Cliffs, NJ, 1983, p. 274. 2. Lee, D. N., J. Mater. Sci. Lett., 1984, 3, 677. 3. Lee, D. N., in Strength of Metals and Materials, ed. H. J. McQueen, J. P. Balio, J. I. Dickson, J. J. Jonas, M. G. Akben and D. N. Lee. Pergamon, Oxford, 1985, p. 971. 4. Lequeu, Ph., Gilormini, P., Montheillet, F., Bacroix, B. and Jonas, J. J., Acta Metall., 1987, 35, 1159. 5. Choi, C. -H., Kwon, J. -W., Oh, K. H. and Lee, D. N., Acta mater., 1997, 42, 5119. 6. Kamijo, T., Kakota, S. and Inagaki, H., Acta Metall., 1993, 41, 1713. 7. Choi, C.-H., Kim, K.-H. and Lee, D. N., Mater. Sci. Forum, 1998, 273-275, 391. 8. Choi, C. -H., Kim, K. -H., Jeong, S. -Y. and Lee, D. N., J. Korean Inst. Met. Mater., 1997, 35, 429. 9. Choi, C. -H., Kim, K. -H. and Lee, D. N., in Synthesis/Processing of Lightweight Metallic Materials II, ed. C. M. Ward-Close, F. H. Froes, D. J. Chelleman and S. S. Cho. The Minerals, Metals and Materials Society, Warrendale, PA, USA, 1997, p. 37. 10. Matthies, S., Phys. Stat. Sol., 1980, 101, 111. 11. Taylor, G. L., J. Inst. Metals, 1938, 63, 307. 12. Renouard, M. and Wintenberger, M., C. R. Acad. Sci. Paris Serie B, 1981, 292, 385. 13. Bunge, H. -J., in Texture Analysis in Materials Science. Butterworths, London, 1982, p. 180. 14. Hwang, Y. -M. and Tzou, G. -Y., J. Mater. Engng Performance, 1995, 4, 265. 15. Lee, S. H. and Lee, D. N., Mater. Sci. Engng, 1998, A249, 84.
KIM and LEE: DEFORMATION TEXTURES OF ALUMINUM SHEETS 16. Asbeck, H. O. and Mecking, H., Mater. Sci. Engng, 1978, 34, 111. 17. Truszkowski, W., Krol, J. and Major, B., Metall. Trans., 1980, 11A, 749. 18. Truszkowski, W., Krol, W. and Major, B., Metall. Trans., 1982, 13A, 665. 19. Bauer, R. E., Mecking, H. and Lu¨ cke, K., Mater. Sci. Engng, 1977, 68, 31. 20. Sakai, T., Saito, Y. and Kato, K., Trans. ISIJ, 1987, 27, 520.
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21. Sakai, T., Saito, Y., Matsuo, M. and Kawasaki, K., ISIJ Int., 1991, 31, 86. 22. Um, K. -K., Jeong, H. -T., An, J. -K., Lee, D. N., Kim, G. and Kwon, O., ISIJ Int., 2000, 40, 58. 23. Choi, C. -H. and Lee, D. N., Met. Mater. Trans., 1997, 28A, 2217. 24. Huh, M. -Y., Cho, Y. S., Kim, J. S. and Engler, O., Z. Metallkd., 1999, 90(2), 124.