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Recrystallization texture of aluminum bicrystals with S orientations deformed by channel die compression
Materials Science and Engineering A269 (1999) 49 – 58 www.elsevier.com/locate/msea
Recrystallization texture of aluminum bicrystals with S orientations deformed by channel die compression Dong Nyung Lee a,*, Hyo-Tae Jeong b b
a School of Materials Science and Engineering, Seoul National Uni6ersity, Seoul 151 -742, South Korea Research Center for Thin Film Fabrication and Crystal Growing of Ad6anced Materials, Seoul National Uni6ersity, Seoul 151 -742, South Korea
Received 27 July 1998; received in revised form 7 March 1999; accepted 7 March 1999
Abstract The texture changes during recrystallization in high-purity aluminum bicrystals with S orientations, e.g. (123)[412( ]/(123)[4( 1( 2] and (123)[412( ]/(1( 2( 3( )[412( ], deformed by channel die compression were investigated by Blicharski et al. There was no 111 rotational relationship between the recrystallized and the deformed matrix phases. This result could be explained by the strain energy release maximization model, in which the direction of absolute maximum internal stress due to dislocations in the deformed matrix becomes parallel to the minimum Young’s modulus direction in recrystallized grains, whereby the energy release during recrystallization can be maximized. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Recrystallization texture; Aluminium bicrystals; Channel die compression
1. Introduction Blicharski, Liu and Hu [1] studied the microstructural and texture changes during recovery and recrystallization in high purity aluminum bicrystals with S orientations, e.g. (123)[412( ]/(123)[4( 1( 2] and (123)[412( ]/ (1( 2( 3( )[412( ], which had been compressed by the 90– 97.5% reduction in thickness using a channel die. 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 direction. The annealing of the deformed bicrystals was conducted for 5 min in a fused quartz tube furnace with He+5%H2 atmosphere. The textures of the fully recrystallized specimens were examined by determining the {111} and {200} pole figures from sectioned planes at one-quarter, one-half and three-quarters specimen thickness. This roughly corresponded to the positions * Corresponding author. Tel.: + 82-2-880-7085; fax: + 82-2-8859671. E-mail address: [email protected] (D. Nyung Lee)
at the midthickness of the top crystal, the bicrystal boundary and the midthickness of the bottom crystal, respectively. The recrystallization texture, together with the EBSP measurements of the partially recrystallized specimens, did not seem to be related to the deformation textures of the matrix crystals by specific orientation relationships, such as a [111] rotation of 40o. They concluded that both oriented nucleation or oriented growth alone appeared to be inadequate to explain the recrystallization textures. Recently, one of the present authors [2] advanced a model for the evolution of recrystallization texture, in which the direction of absolute maximum internal stress due to dislocations in the deformed or fabricated material becomes parallel to the minimum Young’s modulus direction in recrystallized grains, whereby the energy release during recrystallization can be maximized because material concerned does not change macroscopically its shape and volume during recrystallization, and so the recrystallization is a displacement controlled process. This strain energy release maximization model originates from the presumption that the stored energy due to dislocations is the dominant driving force for recrystallization. This model could explain recrystallization textures developed from deformation textures [3– 8] and electrodeposition textures [9,10].
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The purpose of this work is to discuss the recrystallization textures of the high purity aluminum bicrystals with S orientations obtained by Blicharski et al. [1], based on the strain energy release maximization model. 2. Experimental textures The deformation textures of the two bicrystals, (123)[412( ]/(123)[4( 1( 2] and (123)[412( ]/ (1( 2( 3( )[412( ], deformed 90% by channel die compression, are reproduced by the {111} pole figures in Fig. 1(a),(b), respectively. As shown in these pole figures, the texture of crystal (123)[412( ] and that of crystal (123)[4( 1( 2] or (1( 2( 3( )[412( ] are given by the left and right figures, which were determined at the midthickness position of the component crystals. The central pole figures represent the textures measured at the initial boundary of the bicrystals, hence the deformation textures of both of the component crystals are shown. The initial orientation of the component crystals is also indicated in these pole figures. The recrystallization textures of bicrystal (123)[412( ]/ (123)[4( 1( 2] were determined from specimens annealed at 125oC for 5 min, and those of bicrystal (123)[412( ]/ (1( 2( 3( )[412( ], from specimens annealed at 185oC for 5 min. The {111} pole figures of these fully annealed specimens are reproduced in Figs. 2 and 3.
As Bricharski et al. pointed out, the recrystallization textures of the fully annealed bicrystal specimens (Figs. 2 and 3) do not have 40o 111 rotational orientation relationship with the deformation textures of the deformed bicrystals (compare Figs. 2 and 3 with Fig. 1). The 40o 111 rotational orientation relationship has traditionally been associated with the oriented growth theory. Such a misorientation relationship between a growing recrystallized grain and the surrounding deformation matrix has been reported to have a high mobility [11]. The relationship is also claimed to be associated with the oriented nucleation theory [12]. According to the latter theory, the preferred nucleation of recrystallized cube grains from cube bands surrounded by 40o 111 boundaries (i.e. surrounded by the S deformation texture component) is probably due to a combined effect of a higher stored energy in S component than in other deformation texture components and a low grain boundary energy of the 40o 111 boundary. However, as already mentioned, there is no 111 orientation relationship between the deformation and recrystallization textures. Therefore, the experimental results cannot be explained by the oriented nucleation and growth theories. The results may be explained by the strain energy release maximization model.
Fig. 1. (111) pole figures showing the deformation textures of the bicrystals after 90% reduction in thickness by channel die compression. (a) Bicrystal (123)[412( ]/(123)[4( 1( 2], (b) bicrystal (123)[412( ]/(1( 2( 3( )[412( ] [1]. Orientations of crystal: , {135}211; , {011}522.
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Fig. 2. (111) pole figures showing the texture of the fully annealed specimens of bicrystal (123)[412( ]/(123)[4( 1( 2]. (a) After 90% reduction and annealing at 125°C for 5 min, (b) after 95% reduction and annealing at 125°C for 5 min [1].
Fig. 3. (111) pole figures showing the texture of the fully annealed specimens of bicrystal (123)[412( ]/(1( 2( 3( )[412( ]. (a) After 90% reduction and annealing at 185°C for 5 min, (b) after 95% reduction and annealing at 185°C for 5 min [1].
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3. Strain energy release maximization model One of the present authors [2] advanced a model for the evolution of recrystallization texture, in which the direction of absolute maximum internal stress due to dislocations in the deformed or fabricated materials becomes parallel to the minimum Young’s modulus direction in recrystallized grains, whereby the strain energy release during recrystallization can be maximized (Fig. 4). Since this concept is not well known, it is briefly explained in the following. Suppose that a single crystal with a single slip system is plain strain deformed, then the dislocation array in the deformed crystal may be depicted as shown in Fig. 5. The stress fields of the array of an infinite number of edge dislocations can be calculated by superposing the stress fields of isolated dislocations. Following Sutton and Ballufi [13], the stress fields of this array are given by s12 = − s0 sin X1(cosh X2 −cos X1 −X2sin X2)
(1)
s11 =s0[2sinhX2(coshX2 −cosX1) − X2(coshX2cosX1 −1)]
(2)
Fig. 6. The principle stress distribution around parallel edge dislocation, which is calculated based on the array of 100 parallel edge dislocations using the principles of superposition. The dislocation spacing is 10b, b being the Burgers vector, and G is the shear modulus.
s22 = − s0X2(coshX2cosX1 − 1)
(3)
where X1 = 2px1/D, X2 = 2px2/D, and s0 = −Gb/ [2D(1− n)(cosh X2 − cos X1)2] with G, b, n and D being the shear modulus, the Burgers vector, Poisson’s ratio and the dislocation spacing, respectively. As x2 9 , it is seen that s22 and s12 tend to zero exponentially, but Fig. 4. Matrix and recrystallized grains constitute the constant volume system, in which energy release can be maximized when the absolute maximum internal stress direction becomes parallel to the minimum elastic modulus direction of the recrystallized grain.
Fig. 5. Edge dislocation array in plane strain deformed crystal. No strain direction is parallel to the dislocation line.
s11
Gb sgn(x2) D(1− n)
(4)
where sgn(x2)= − 1 if x2 \ 0 and sgn(x2)= 1 if x2 B0. The absolute maximum stress, smax = (s11 +s22)/ 2+ [(s11 − s22)2/4+s 212]1/2 approaches s11 exponentially as x2 increases above D/2p. Therefore, the absolute maximum stress direction is parallel to the Burgers vector or the slip direction (Fig. 6). For multiple slip, the shear strains on active slip systems may not be equal. The dislocation density r on a slip system is expected to increase with increasing the related shear strain g, i.e. r8g a. Since the relation between g and r is not known, a is assumed to be unity. It follows that s11 : smax 8 1/D 8r8 g. Therefore, the contribution of each active slip system to the absolute maximum direction will be proportional to its shear strain. When a crystal rotates during deformation, the shear strain increments on slip systems can vary with deformation. In this case, the contribution of each slip
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system to the absolute maximum stress will be proportional to
&
K = dg do
(5)
where dg is the absolute magnitude of the shear strain increment on corresponding slip system and o is the strain of specimen. If a deformation texture is stable, the shear strain increments on the slip systems are independent of deformation, and the contribution of the slip systems to the absolute maximum stress will be proportional to the shear strain increments on them. It is noted that the absolute maximum stress is an internal stress. In order for this method to be used, it is necessary to know active slip systems and shear strains on them. They can be obtained from the deformation texture or from a model which can simulate the deformation texture. If a small volume in a uniaxially stressed body whose ends are fixed is replaced by the same volume of stress free body, the strain energy of the system including the substituted region will be reduced. The released strain energy is represented by the area OAB in Fig. 7. The released energy will vary with Young’s modulus of the substituted body. The released energy will be maximized, when Young’s modulus of the substituted body is a minimum. In view of the fact that recrystallization is a displacement controlled process, and the absolute maximum internal stress due to the dislocation array is approximated by the uniaxial stress, the deformed or fabricated matrix and the recrystallized grain may be approximately equivalent to the stressed body and the substituted body in Fig. 7, respectively. In this case, we can state that the strain energy release can be maximized when the direction of absolute maximum internal stress due to dislocations in the deformed or fabricated mate-
Fig. 7. A stress free body (a) is elongated and both its ends are fixed (b). The strain energy of the body is represented by the area OAC. When a small portion of the stresses body is replaced by a stress free material, the strain energy of the system is reduced to the area OBC and the energy release is represented by the area OAB.
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rial becomes parallel to the minimum Young’s modulus direction in recrystallized grains. In the real situation, the stress field is triaxial. However, this simple concept can be a starting point. Another orientation relation between the matrix and recrystallized grain must be known to obtain the recrystallization texture. This relation will be discussed in the next section.
4. Discussion In order to use the strain energy release maximization model, the slip systems activated during deformation and the shear strains on them must be known. The slip systems and their shear strains can be obtained in the process of simulation of deformation. Fig. 8 shows the orientation change of crystal (123)[412( ] during the plane strain compression calculated using the full constraints [14,15], o31 relaxed constraints [16,17] and o31, o23 relaxed constraints [16,17] models, and the strain rate sensitivity models with m=0.01[18,19]. Comparing the calculated results with the measured values in Fig. 1, the measured orientation change during deformation seems to be best simulated by the full constraints strain rate sensitivity model. A strain rate sensitivity model, that which Blicharski et al. used, gives rise to the orientation change which is less sensitive to deformation. However, the crystal rotation is not the same as the measured. Therefore, the full constraints strain rate sensitivity model was used to simulate the deformation texture. Fig. 9 shows the calculated orientation changes of crystal (123)[412( ], (123)[4( 1( 2], (123)[412( ] and (1( 2( 3( )[412( ] when they are subjected to the plane strain compression. Fig. 10 shows the calculated shear strain increments on acting slip systems of crystal (123)[412( ] as a function of true thickness strain when subjected to the plane strain compression. The changes in shear strain increments on active slip systems are sensitive to strain up to a thickness strain of about 0.5 and not very sensitive afterwards. The experimental deformation texture is better described by (0.1534 0.5101 0.8463) [0.8111 0.4242 0.4027] or (135)[211( ], which is calculated based on the full constraints strain rate sensitivity model with m= 0.01, than (011)[522( ], which Blicharski et al.[20] described as shown in Fig. 1. 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 may not be reflected in X-ray measurements. The scattered experimental recrystallization 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 the recrystallization texture.
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Fig. 8. Changes of orientation of crystal (123)[412( ] during plane strain compression calculated using the Taylor – Bishop – Hill full constraints (a), o31 relaxed constraints (b), and o31, o23 relaxed constraints (c) models; and strain rate sensitivity full constraints (d), o31 relaxed constraints (e), and o31, o23 relaxed constraints (f) models. , Orientation of initial crystal (123)[412( ]; , measured orientation of 90% reduced crystal (135)[211( ]; , calculated orientation of crystal reduced by 90%.
For a true thickness strain of 2.3 or 90% reduction, the ratio of the K values in Eq. (5) of the slip systems, (111)[101( ], (111)[011( ], (1( 11( )[110], (11( 1( )[110],(1( 11( )[011] and (11( 1( )[101], which are calculated using the data in Fig. 10, are 2091, 776, 1424, 2938, 76 and 139, respectively. The contributions of the slip system, (1( 11( )[011] and (11( 1( )[101], are negligible compared with other active slip systems. Therefore, the slip systems, (111)[101( ], (111)[011( ], (1( 11( )[110] and (11( 1( )[110] are considered in calculating the absolute maximum internal stress direction. It is noted that all the slip directions are designated so that they can be at acute angle one another and the absolute maximum internal stress direction calculated in Eq. (6) can be at an acute angle with the major extension direction or the rolling direction, [0.8111 0.4242 0.4027]. The absolute maximum stress direction is calculated as follows:
tallization, according to the strain energy release maximization model. Another orientation relation between the matrix and the recrystallized grain is obtained in the following. If
2091[101( ]+ 776[011( ] +1424 ×0.577[110] +2938 × 0.577[110]= [4608 3293 2867]
(6)
where the factor 0.577 originated from the fact that the duplex slip systems of (11( 1)[110] and (1( 11)[110] are equivalent to a single slip system with slip direction [110] (see Appendix A). The [4608 3293 2867] direction or the [0.7259 0.5187 0.4516] direction will become parallel to one of the 100 directions, the minimum Young’s modulus direction of aluminum, after recrys-
Fig. 9. Changes of orientations of crystals {123}412 when subjected to the plane strain deformation, which were calculated using the full constraints strain rate sensitivity model with m= 0.01. , Orientations of initial crystal {123}412; , measured orientations of 90% reduced crystal (135)211; , calculated orientation of crystal reduced by 90%.
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Fig. 10. The calculated shear strain increments at a thickness reduction of 0.01 on acting slip systems of the crystal (123)[412] as a function of strain.
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but less than, the right angle. The direction which is at the smallest possible angle with the [011] and normal to the maximum internal stress direction at the same time must be on a plane made of the [011] direction and the maximum internal stress direction (OB in Fig. 11). The direction OB must be normal to both the plane normal (OC in Fig. 11) and the maximum internal stress direction (OA in Fig. 11). The direction OC is obtained by the vector product of the directions OA and [011]. The direction OB is obtained by the vector product of the directions OC and OA. The directions OC and OB are calculated to be the [0.6869 0.5139 0.5139] and [0.0345 0.6833 0.7294] directions, respectively. In summary, the [0.7259 0.5187 0.4516], [0.6869 0.5139 0.5139] and [0.0345 0.6833 0.7294] directions, which are normal to each other, will become parallel to the 100 directions in the recrystallized 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 recrystallization, components of the unit vectors are direction cosines relating the deformed and recrystallized crystal coordinate axes. Therefore, the rolling plane normal direction, (0.1534 0.5101 0.8463), and the rolling direction, [0.8111 0.4242 0.4027], in the deformed crystal coordinate system can be transformed to the expressions in the recrystallized crystal coordinate system using the following calculations:
Á 0.7259 0.5187 −0.4516ÂÁ0.1534Â Ã 0.6869 − 0.5139 0.5139 ÃÃ0.5101Ã Ä − 0.0345 0.6833 0.7294 ÅÄ0.8463Å
Fig. 11. The orientation relations in the deformed and recrystallized states. Subscripts d and r indicate deformed and recrystallized states, respectively.
the absolute maximum internal stress direction in the deformed state becomes parallel to one of the 100 directions in the recrystallized state, another 100 direction in the recrystallized state, which is normal to the previous 100, will tend to be parallel to a direction that is at the smallest possible angle with another high stress direction in the deformed state. Possible candidates of the high stress direction must be three Burgers vector directions, the [011], [101] and [11( 0] directions, which have not been used in calculation of the maximum internal stress direction among six possible Burgers vector directions. The [011], [101] and [11( 0] directions are at 87.28, 78.82 and 81.57o, respectively, with the absolute maximum internal stress direction, [0.7259 0.5187 0.4516]. The [011] direction is closest to,
Á − 0.0062Â = Ã 0.2781 Ã Ä 0.9606 Å
(7)
Á 0.7259 0.5187 −0.4516ÂÁ 0.8111 Â Ã 0.6869 − 0.5139 0.5139 ÃÃ 0.4242 Ã Ä − 0.0345 0.6833 0.7294 ÅÄ − 0.4027Å Á 0.9907 Â = Ã 0.1322 Ã Ä − 0.0319Å
(8)
The calculated result means that crystal (0.1534 0.5101 0.8463)[0.8111 0.4242 0.4027], which is obtained by the channel die compression of 90% reduction, transforms to the recrystallization 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( )[412( ] orientations transform to (0.0062 0.2781 0.9606)[0.9907 0.1322 0.0319] and
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Fig. 12. The (111) pole figures showing the textures of the fully annealed specimens of bicrystal (123)[412( ]/(123)[4( 1( 2] (a) after 90% reduction and annealing at 125°C for 5 min, and (b) after 95% reduction and annealing at 125°C for 5 min [1]. , (0.0062 0.2781 0.9606)[0.9907 0.1322 0.0319].
Fig. 13. The (111) pole figures showing the textures of the fully annealed specimens of bicrystal (123)[412( ]/(1( 2( 3( )[412( ] (a) after 90% reduction and annealing at 185°C for 5 min, and (b) after 95% reduction and annealing at 185°C for 5 min [1]. , (0.0062 0.2781 0.9606)[0.9907 0.1322 0.0319].
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Advanced Materials. The authors wish to thank Sang Heon Lee for his calculation of textures based on the rate sensitivity models.
Appendix A Fig. 14 shows two slip planes, S1 and S2, intersecting along the common slip direction, the X3 axis; the X2 axis bisects the angle between the poles of these planes. Suppose that the direction of loading lies within the quadrant drawn between S1 and S2, and the displacement Dx3 at any point P with coordinates (x1,x2,x3) is considered. If shear strains g1 and g2 occur on the slip systems, then Dx3 = g1PN1 + g2PN2 but PN1 = OP sin(a −b) and PN2 = OP sin(a +b), so Dx3 = (g1 + g2)OP sin a cos b+ (g2 − g1)OP cos a sin b (A.1)
Fig. 14. Schematic diagram of slip planes S1 and S2 that share a common slip direction.
(0.0062 0.2781 0.9606) [0.9907 0.1322 0.0319], respectively, after recrystallization. The results are plotted in Figs. 12 and 13, superimposed on the experimental data. The calculated values are not plotted in the textures of boundaries, because the boundary textures reflect the textures in the top and bottom layers. It can be seen that the calculated recrystallization textures are in good agreement with the measured data.
5. Conclusions 1. The channel die compression by 90 – 95% of aluminum bicrystal with (123)[412( ]/(123)[4( 1( 2] or (123)[412( ]/ (1( 2( 3( )[412( ] gave rise to a deformation texture similar to the original crystal orientations. This result could be well simulated using the full constraints strain rate sensitivity model. 2. Recrystallization textures calculated based on the strain energy release maximization model are in good agreement with the measured data.
Acknowledgements This study has been supported by Research Center for Thin Film Fabrication and Crystal Growing of
Since a\b and (g1 + g2)\ (g2 − g1), the second term of the right-hand side is negligible compared with the first term. It follows that Dx3 $ (g1 + g2)x2 sin a Therefore, the displacement Dx3 is linear with the x2 coordinate, and the deformation is equivalent to single slip in the x3 direction on the (g1S1 + g2S2) plane. The apparent shear strain ga is ga = Dx3/x2 $ (g1 + g2)sin a
(A.2)
When g1 = g2 = g, it follows from Eq. (A.1) that ga = 2g sin a
(A.3)
For a duplex slip of (1( 11( )[110] and (11( 1( )[110], sin a= 0.577.
References [1] M. Blicharski, J. Liu, H.sun Hu, Acta Mater. 43 (1995) 3125. [2] D.N. Lee, Scr. Metall. Mater. 32 (1995) 1689. [3] Y.B. Park, D.N. Lee, G. Gottstein, Mater. Sci. Technol. 13 (1997) 289. [4] Y.B. Park, D.N. Lee, G. Gottstein, Acta Mater. 46 (1998) 337. [5] D.N. Lee, Textures Microstruct. 26 – 27 (1996) 361. [6] S.H. Hong, H.-T. Jeong, C.-H. Choi, D.N. Lee, Mater. Sci. Eng. A229 (1997) 174. [7] C.-H. Choi, D.N. Lee, Metall. Mater. Trans. 28A (1997) 2217. [8] S.H. Lee, D.N. Lee, Mater. Sci. Eng. A249 (1998) 84. [9] D.N. Lee, S. Kang, J. Yang, Plating Surface Finishing 82 (3) (1995) 76. [10] D.N. Lee, Met. Mater. 2 (1996) 121. [11] B. Liebman, K. Lu¨cke, G. Masing, Z. Metall. 47 (1956) 57. [12] E. Nes, H.E. Vatne, Z. Metall. 87 (1996) 448. [13] A.P. Sutton, R.W. Ballufi, Interfaces in Crystalline Materials, Clarendon Press, Oxford, 1996, pp. 115 – 120.
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[14] G.I. Taylo, r, J. Inst. Met. 62 (1938) 307. [15] J.F.W. Bishop, R. Hill, Philos. Mag. 42 (1951) 414. [16] H. Honneff, H. Mecking, Proceedings of ICOTOM5, Achen, Springer, Berlin, Vol. 1, 1978, p. 265. [17] P. Van Houtte, Proceedings of ICOTOM6, Tokyo, Vol. 1, 1981, p. 428.
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[18] G.R. Canova, C. Fressengeas, A. Molinari, U.F. Kocks, Acta Metall. Mater. 36 (1961) 1988. [19] L.S. Toth, P. Gilormini, J.J. Jonas, Acta Metall. Mater. 36 (1988) 3077. [20] M. Blicharski, R. Becker, H.sun Hu, Acta Metall. Mater. 41 (1993) 2007.
Recrystallization texture of aluminum bicrystals with S orientations deformed by channel die compression Dong Nyung Lee a,*, Hyo-Tae Jeong b b
a School of Materials Science and Engineering, Seoul National Uni6ersity, Seoul 151 -742, South Korea Research Center for Thin Film Fabrication and Crystal Growing of Ad6anced Materials, Seoul National Uni6ersity, Seoul 151 -742, South Korea
Received 27 July 1998; received in revised form 7 March 1999; accepted 7 March 1999
Abstract The texture changes during recrystallization in high-purity aluminum bicrystals with S orientations, e.g. (123)[412( ]/(123)[4( 1( 2] and (123)[412( ]/(1( 2( 3( )[412( ], deformed by channel die compression were investigated by Blicharski et al. There was no 111 rotational relationship between the recrystallized and the deformed matrix phases. This result could be explained by the strain energy release maximization model, in which the direction of absolute maximum internal stress due to dislocations in the deformed matrix becomes parallel to the minimum Young’s modulus direction in recrystallized grains, whereby the energy release during recrystallization can be maximized. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Recrystallization texture; Aluminium bicrystals; Channel die compression
1. Introduction Blicharski, Liu and Hu [1] studied the microstructural and texture changes during recovery and recrystallization in high purity aluminum bicrystals with S orientations, e.g. (123)[412( ]/(123)[4( 1( 2] and (123)[412( ]/ (1( 2( 3( )[412( ], which had been compressed by the 90– 97.5% reduction in thickness using a channel die. 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 direction. The annealing of the deformed bicrystals was conducted for 5 min in a fused quartz tube furnace with He+5%H2 atmosphere. The textures of the fully recrystallized specimens were examined by determining the {111} and {200} pole figures from sectioned planes at one-quarter, one-half and three-quarters specimen thickness. This roughly corresponded to the positions * Corresponding author. Tel.: + 82-2-880-7085; fax: + 82-2-8859671. E-mail address: [email protected] (D. Nyung Lee)
at the midthickness of the top crystal, the bicrystal boundary and the midthickness of the bottom crystal, respectively. The recrystallization texture, together with the EBSP measurements of the partially recrystallized specimens, did not seem to be related to the deformation textures of the matrix crystals by specific orientation relationships, such as a [111] rotation of 40o. They concluded that both oriented nucleation or oriented growth alone appeared to be inadequate to explain the recrystallization textures. Recently, one of the present authors [2] advanced a model for the evolution of recrystallization texture, in which the direction of absolute maximum internal stress due to dislocations in the deformed or fabricated material becomes parallel to the minimum Young’s modulus direction in recrystallized grains, whereby the energy release during recrystallization can be maximized because material concerned does not change macroscopically its shape and volume during recrystallization, and so the recrystallization is a displacement controlled process. This strain energy release maximization model originates from the presumption that the stored energy due to dislocations is the dominant driving force for recrystallization. This model could explain recrystallization textures developed from deformation textures [3– 8] and electrodeposition textures [9,10].
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The purpose of this work is to discuss the recrystallization textures of the high purity aluminum bicrystals with S orientations obtained by Blicharski et al. [1], based on the strain energy release maximization model. 2. Experimental textures The deformation textures of the two bicrystals, (123)[412( ]/(123)[4( 1( 2] and (123)[412( ]/ (1( 2( 3( )[412( ], deformed 90% by channel die compression, are reproduced by the {111} pole figures in Fig. 1(a),(b), respectively. As shown in these pole figures, the texture of crystal (123)[412( ] and that of crystal (123)[4( 1( 2] or (1( 2( 3( )[412( ] are given by the left and right figures, which were determined at the midthickness position of the component crystals. The central pole figures represent the textures measured at the initial boundary of the bicrystals, hence the deformation textures of both of the component crystals are shown. The initial orientation of the component crystals is also indicated in these pole figures. The recrystallization textures of bicrystal (123)[412( ]/ (123)[4( 1( 2] were determined from specimens annealed at 125oC for 5 min, and those of bicrystal (123)[412( ]/ (1( 2( 3( )[412( ], from specimens annealed at 185oC for 5 min. The {111} pole figures of these fully annealed specimens are reproduced in Figs. 2 and 3.
As Bricharski et al. pointed out, the recrystallization textures of the fully annealed bicrystal specimens (Figs. 2 and 3) do not have 40o 111 rotational orientation relationship with the deformation textures of the deformed bicrystals (compare Figs. 2 and 3 with Fig. 1). The 40o 111 rotational orientation relationship has traditionally been associated with the oriented growth theory. Such a misorientation relationship between a growing recrystallized grain and the surrounding deformation matrix has been reported to have a high mobility [11]. The relationship is also claimed to be associated with the oriented nucleation theory [12]. According to the latter theory, the preferred nucleation of recrystallized cube grains from cube bands surrounded by 40o 111 boundaries (i.e. surrounded by the S deformation texture component) is probably due to a combined effect of a higher stored energy in S component than in other deformation texture components and a low grain boundary energy of the 40o 111 boundary. However, as already mentioned, there is no 111 orientation relationship between the deformation and recrystallization textures. Therefore, the experimental results cannot be explained by the oriented nucleation and growth theories. The results may be explained by the strain energy release maximization model.
Fig. 1. (111) pole figures showing the deformation textures of the bicrystals after 90% reduction in thickness by channel die compression. (a) Bicrystal (123)[412( ]/(123)[4( 1( 2], (b) bicrystal (123)[412( ]/(1( 2( 3( )[412( ] [1]. Orientations of crystal: , {135}211; , {011}522.
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Fig. 2. (111) pole figures showing the texture of the fully annealed specimens of bicrystal (123)[412( ]/(123)[4( 1( 2]. (a) After 90% reduction and annealing at 125°C for 5 min, (b) after 95% reduction and annealing at 125°C for 5 min [1].
Fig. 3. (111) pole figures showing the texture of the fully annealed specimens of bicrystal (123)[412( ]/(1( 2( 3( )[412( ]. (a) After 90% reduction and annealing at 185°C for 5 min, (b) after 95% reduction and annealing at 185°C for 5 min [1].
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3. Strain energy release maximization model One of the present authors [2] advanced a model for the evolution of recrystallization texture, in which the direction of absolute maximum internal stress due to dislocations in the deformed or fabricated materials becomes parallel to the minimum Young’s modulus direction in recrystallized grains, whereby the strain energy release during recrystallization can be maximized (Fig. 4). Since this concept is not well known, it is briefly explained in the following. Suppose that a single crystal with a single slip system is plain strain deformed, then the dislocation array in the deformed crystal may be depicted as shown in Fig. 5. The stress fields of the array of an infinite number of edge dislocations can be calculated by superposing the stress fields of isolated dislocations. Following Sutton and Ballufi [13], the stress fields of this array are given by s12 = − s0 sin X1(cosh X2 −cos X1 −X2sin X2)
(1)
s11 =s0[2sinhX2(coshX2 −cosX1) − X2(coshX2cosX1 −1)]
(2)
Fig. 6. The principle stress distribution around parallel edge dislocation, which is calculated based on the array of 100 parallel edge dislocations using the principles of superposition. The dislocation spacing is 10b, b being the Burgers vector, and G is the shear modulus.
s22 = − s0X2(coshX2cosX1 − 1)
(3)
where X1 = 2px1/D, X2 = 2px2/D, and s0 = −Gb/ [2D(1− n)(cosh X2 − cos X1)2] with G, b, n and D being the shear modulus, the Burgers vector, Poisson’s ratio and the dislocation spacing, respectively. As x2 9 , it is seen that s22 and s12 tend to zero exponentially, but Fig. 4. Matrix and recrystallized grains constitute the constant volume system, in which energy release can be maximized when the absolute maximum internal stress direction becomes parallel to the minimum elastic modulus direction of the recrystallized grain.
Fig. 5. Edge dislocation array in plane strain deformed crystal. No strain direction is parallel to the dislocation line.
s11
Gb sgn(x2) D(1− n)
(4)
where sgn(x2)= − 1 if x2 \ 0 and sgn(x2)= 1 if x2 B0. The absolute maximum stress, smax = (s11 +s22)/ 2+ [(s11 − s22)2/4+s 212]1/2 approaches s11 exponentially as x2 increases above D/2p. Therefore, the absolute maximum stress direction is parallel to the Burgers vector or the slip direction (Fig. 6). For multiple slip, the shear strains on active slip systems may not be equal. The dislocation density r on a slip system is expected to increase with increasing the related shear strain g, i.e. r8g a. Since the relation between g and r is not known, a is assumed to be unity. It follows that s11 : smax 8 1/D 8r8 g. Therefore, the contribution of each active slip system to the absolute maximum direction will be proportional to its shear strain. When a crystal rotates during deformation, the shear strain increments on slip systems can vary with deformation. In this case, the contribution of each slip
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system to the absolute maximum stress will be proportional to
&
K = dg do
(5)
where dg is the absolute magnitude of the shear strain increment on corresponding slip system and o is the strain of specimen. If a deformation texture is stable, the shear strain increments on the slip systems are independent of deformation, and the contribution of the slip systems to the absolute maximum stress will be proportional to the shear strain increments on them. It is noted that the absolute maximum stress is an internal stress. In order for this method to be used, it is necessary to know active slip systems and shear strains on them. They can be obtained from the deformation texture or from a model which can simulate the deformation texture. If a small volume in a uniaxially stressed body whose ends are fixed is replaced by the same volume of stress free body, the strain energy of the system including the substituted region will be reduced. The released strain energy is represented by the area OAB in Fig. 7. The released energy will vary with Young’s modulus of the substituted body. The released energy will be maximized, when Young’s modulus of the substituted body is a minimum. In view of the fact that recrystallization is a displacement controlled process, and the absolute maximum internal stress due to the dislocation array is approximated by the uniaxial stress, the deformed or fabricated matrix and the recrystallized grain may be approximately equivalent to the stressed body and the substituted body in Fig. 7, respectively. In this case, we can state that the strain energy release can be maximized when the direction of absolute maximum internal stress due to dislocations in the deformed or fabricated mate-
Fig. 7. A stress free body (a) is elongated and both its ends are fixed (b). The strain energy of the body is represented by the area OAC. When a small portion of the stresses body is replaced by a stress free material, the strain energy of the system is reduced to the area OBC and the energy release is represented by the area OAB.
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rial becomes parallel to the minimum Young’s modulus direction in recrystallized grains. In the real situation, the stress field is triaxial. However, this simple concept can be a starting point. Another orientation relation between the matrix and recrystallized grain must be known to obtain the recrystallization texture. This relation will be discussed in the next section.
4. Discussion In order to use the strain energy release maximization model, the slip systems activated during deformation and the shear strains on them must be known. The slip systems and their shear strains can be obtained in the process of simulation of deformation. Fig. 8 shows the orientation change of crystal (123)[412( ] during the plane strain compression calculated using the full constraints [14,15], o31 relaxed constraints [16,17] and o31, o23 relaxed constraints [16,17] models, and the strain rate sensitivity models with m=0.01[18,19]. Comparing the calculated results with the measured values in Fig. 1, the measured orientation change during deformation seems to be best simulated by the full constraints strain rate sensitivity model. A strain rate sensitivity model, that which Blicharski et al. used, gives rise to the orientation change which is less sensitive to deformation. However, the crystal rotation is not the same as the measured. Therefore, the full constraints strain rate sensitivity model was used to simulate the deformation texture. Fig. 9 shows the calculated orientation changes of crystal (123)[412( ], (123)[4( 1( 2], (123)[412( ] and (1( 2( 3( )[412( ] when they are subjected to the plane strain compression. Fig. 10 shows the calculated shear strain increments on acting slip systems of crystal (123)[412( ] as a function of true thickness strain when subjected to the plane strain compression. The changes in shear strain increments on active slip systems are sensitive to strain up to a thickness strain of about 0.5 and not very sensitive afterwards. The experimental deformation texture is better described by (0.1534 0.5101 0.8463) [0.8111 0.4242 0.4027] or (135)[211( ], which is calculated based on the full constraints strain rate sensitivity model with m= 0.01, than (011)[522( ], which Blicharski et al.[20] described as shown in Fig. 1. 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 may not be reflected in X-ray measurements. The scattered experimental recrystallization 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 the recrystallization texture.
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Fig. 8. Changes of orientation of crystal (123)[412( ] during plane strain compression calculated using the Taylor – Bishop – Hill full constraints (a), o31 relaxed constraints (b), and o31, o23 relaxed constraints (c) models; and strain rate sensitivity full constraints (d), o31 relaxed constraints (e), and o31, o23 relaxed constraints (f) models. , Orientation of initial crystal (123)[412( ]; , measured orientation of 90% reduced crystal (135)[211( ]; , calculated orientation of crystal reduced by 90%.
For a true thickness strain of 2.3 or 90% reduction, the ratio of the K values in Eq. (5) of the slip systems, (111)[101( ], (111)[011( ], (1( 11( )[110], (11( 1( )[110],(1( 11( )[011] and (11( 1( )[101], which are calculated using the data in Fig. 10, are 2091, 776, 1424, 2938, 76 and 139, respectively. The contributions of the slip system, (1( 11( )[011] and (11( 1( )[101], are negligible compared with other active slip systems. Therefore, the slip systems, (111)[101( ], (111)[011( ], (1( 11( )[110] and (11( 1( )[110] are considered in calculating the absolute maximum internal stress direction. It is noted that all the slip directions are designated so that they can be at acute angle one another and the absolute maximum internal stress direction calculated in Eq. (6) can be at an acute angle with the major extension direction or the rolling direction, [0.8111 0.4242 0.4027]. The absolute maximum stress direction is calculated as follows:
tallization, according to the strain energy release maximization model. Another orientation relation between the matrix and the recrystallized grain is obtained in the following. If
2091[101( ]+ 776[011( ] +1424 ×0.577[110] +2938 × 0.577[110]= [4608 3293 2867]
(6)
where the factor 0.577 originated from the fact that the duplex slip systems of (11( 1)[110] and (1( 11)[110] are equivalent to a single slip system with slip direction [110] (see Appendix A). The [4608 3293 2867] direction or the [0.7259 0.5187 0.4516] direction will become parallel to one of the 100 directions, the minimum Young’s modulus direction of aluminum, after recrys-
Fig. 9. Changes of orientations of crystals {123}412 when subjected to the plane strain deformation, which were calculated using the full constraints strain rate sensitivity model with m= 0.01. , Orientations of initial crystal {123}412; , measured orientations of 90% reduced crystal (135)211; , calculated orientation of crystal reduced by 90%.
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Fig. 10. The calculated shear strain increments at a thickness reduction of 0.01 on acting slip systems of the crystal (123)[412] as a function of strain.
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but less than, the right angle. The direction which is at the smallest possible angle with the [011] and normal to the maximum internal stress direction at the same time must be on a plane made of the [011] direction and the maximum internal stress direction (OB in Fig. 11). The direction OB must be normal to both the plane normal (OC in Fig. 11) and the maximum internal stress direction (OA in Fig. 11). The direction OC is obtained by the vector product of the directions OA and [011]. The direction OB is obtained by the vector product of the directions OC and OA. The directions OC and OB are calculated to be the [0.6869 0.5139 0.5139] and [0.0345 0.6833 0.7294] directions, respectively. In summary, the [0.7259 0.5187 0.4516], [0.6869 0.5139 0.5139] and [0.0345 0.6833 0.7294] directions, which are normal to each other, will become parallel to the 100 directions in the recrystallized 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 recrystallization, components of the unit vectors are direction cosines relating the deformed and recrystallized crystal coordinate axes. Therefore, the rolling plane normal direction, (0.1534 0.5101 0.8463), and the rolling direction, [0.8111 0.4242 0.4027], in the deformed crystal coordinate system can be transformed to the expressions in the recrystallized crystal coordinate system using the following calculations:
Á 0.7259 0.5187 −0.4516ÂÁ0.1534Â Ã 0.6869 − 0.5139 0.5139 ÃÃ0.5101Ã Ä − 0.0345 0.6833 0.7294 ÅÄ0.8463Å
Fig. 11. The orientation relations in the deformed and recrystallized states. Subscripts d and r indicate deformed and recrystallized states, respectively.
the absolute maximum internal stress direction in the deformed state becomes parallel to one of the 100 directions in the recrystallized state, another 100 direction in the recrystallized state, which is normal to the previous 100, will tend to be parallel to a direction that is at the smallest possible angle with another high stress direction in the deformed state. Possible candidates of the high stress direction must be three Burgers vector directions, the [011], [101] and [11( 0] directions, which have not been used in calculation of the maximum internal stress direction among six possible Burgers vector directions. The [011], [101] and [11( 0] directions are at 87.28, 78.82 and 81.57o, respectively, with the absolute maximum internal stress direction, [0.7259 0.5187 0.4516]. The [011] direction is closest to,
Á − 0.0062Â = Ã 0.2781 Ã Ä 0.9606 Å
(7)
Á 0.7259 0.5187 −0.4516ÂÁ 0.8111 Â Ã 0.6869 − 0.5139 0.5139 ÃÃ 0.4242 Ã Ä − 0.0345 0.6833 0.7294 ÅÄ − 0.4027Å Á 0.9907 Â = Ã 0.1322 Ã Ä − 0.0319Å
(8)
The calculated result means that crystal (0.1534 0.5101 0.8463)[0.8111 0.4242 0.4027], which is obtained by the channel die compression of 90% reduction, transforms to the recrystallization 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( )[412( ] orientations transform to (0.0062 0.2781 0.9606)[0.9907 0.1322 0.0319] and
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Fig. 12. The (111) pole figures showing the textures of the fully annealed specimens of bicrystal (123)[412( ]/(123)[4( 1( 2] (a) after 90% reduction and annealing at 125°C for 5 min, and (b) after 95% reduction and annealing at 125°C for 5 min [1]. , (0.0062 0.2781 0.9606)[0.9907 0.1322 0.0319].
Fig. 13. The (111) pole figures showing the textures of the fully annealed specimens of bicrystal (123)[412( ]/(1( 2( 3( )[412( ] (a) after 90% reduction and annealing at 185°C for 5 min, and (b) after 95% reduction and annealing at 185°C for 5 min [1]. , (0.0062 0.2781 0.9606)[0.9907 0.1322 0.0319].
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Advanced Materials. The authors wish to thank Sang Heon Lee for his calculation of textures based on the rate sensitivity models.
Appendix A Fig. 14 shows two slip planes, S1 and S2, intersecting along the common slip direction, the X3 axis; the X2 axis bisects the angle between the poles of these planes. Suppose that the direction of loading lies within the quadrant drawn between S1 and S2, and the displacement Dx3 at any point P with coordinates (x1,x2,x3) is considered. If shear strains g1 and g2 occur on the slip systems, then Dx3 = g1PN1 + g2PN2 but PN1 = OP sin(a −b) and PN2 = OP sin(a +b), so Dx3 = (g1 + g2)OP sin a cos b+ (g2 − g1)OP cos a sin b (A.1)
Fig. 14. Schematic diagram of slip planes S1 and S2 that share a common slip direction.
(0.0062 0.2781 0.9606) [0.9907 0.1322 0.0319], respectively, after recrystallization. The results are plotted in Figs. 12 and 13, superimposed on the experimental data. The calculated values are not plotted in the textures of boundaries, because the boundary textures reflect the textures in the top and bottom layers. It can be seen that the calculated recrystallization textures are in good agreement with the measured data.
5. Conclusions 1. The channel die compression by 90 – 95% of aluminum bicrystal with (123)[412( ]/(123)[4( 1( 2] or (123)[412( ]/ (1( 2( 3( )[412( ] gave rise to a deformation texture similar to the original crystal orientations. This result could be well simulated using the full constraints strain rate sensitivity model. 2. Recrystallization textures calculated based on the strain energy release maximization model are in good agreement with the measured data.
Acknowledgements This study has been supported by Research Center for Thin Film Fabrication and Crystal Growing of
Since a\b and (g1 + g2)\ (g2 − g1), the second term of the right-hand side is negligible compared with the first term. It follows that Dx3 $ (g1 + g2)x2 sin a Therefore, the displacement Dx3 is linear with the x2 coordinate, and the deformation is equivalent to single slip in the x3 direction on the (g1S1 + g2S2) plane. The apparent shear strain ga is ga = Dx3/x2 $ (g1 + g2)sin a
(A.2)
When g1 = g2 = g, it follows from Eq. (A.1) that ga = 2g sin a
(A.3)
For a duplex slip of (1( 11( )[110] and (11( 1( )[110], sin a= 0.577.
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