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Importance of the α → β transformation in the variant selection mechanisms of thermomechanically processed titanium alloys
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Scripta Materialia 59 (2008) 570–573 www.elsevier.com/locate/scriptamat
Importance of the a ? b transformation in the variant selection mechanisms of thermomechanically processed titanium alloys C. Cayron* CEA, LITEN, DTH, Minatec, 38054 Grenoble, France Received 19 March 2008; revised 6 May 2008; accepted 7 May 2008 Available online 21 May 2008
Selection of a variants is often reported in thermomechanically processed titanium alloys. Many workers have noticed that a variants are frequently situated at the grain boundary between b grains that share a common {1 1 0}b plane, but up to now such coincidence on the b grains could not be fully clarified. By careful examination of electron backscatter diffraction data published in literature, it has been established that the adjacent b grains are actually b variants inherited from the a ? b transition. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Titanium; Variant selection; Operator; EBSD; Phase transition
Titanium and its alloys have high strength-to-weight ratios, excellent mechanical properties and corrosion resistance, and hence are widely used in the aerospace, energy, chemical and health industries. Their high strength is due to a complex microstructure that results from a first-order transition during cooling: parent body-centered cubic (bcc) b phase is transformed into daughter hexagonal close-packed (hcp) a phase. The temperature of the b ? b + a transition is about 880 °C for pure titanium and 10–20 °C lower for its alloys. A parent b crystal and a daughter a crystal respect the Burgers orientation relationship (OR) [1]: [0 0 1]a// 1 1 1]b. Due to the symmetries of [1 1 0]b and [1 0 0]a//[ the parent and daughter phases, there are 12 equivalent ORs that correspond to equivalently orientated daughter a crystals called variants. These a variants are represented in Figure 1a. Commercialized titanium alloys are in general produced by a sequence of melting, casting, forging at temperatures higher than 950 °C in the b domain, and thermomechanical processing at lower temperatures between 600 and 900 °C in the b + a domain. The final microstructures (i.e. respective content of the phases, grain sizes and morphologies) are various and strongly depend on the process parameters (i.e. temperatures, cooling rates and amount of deformation during the hot working steps). In general the microstructures
* Tel.: +33 4 38 78 93 29; fax: +33 4 38 78 58 91; e-mail: [email protected]
are constituted of globular primary ap grains, colonies of secondary as grains and, in some alloys, of some retained b between these grains. The ap grains are formed during the first cooling after forging, and they grow and become globular during the thermomechanical process. The as grains are formed from b in the b + a domain during the final cooling after the thermomechanical processing. The as grains can nucleate at grain boundaries in the form of parallel colonies or inside the b part with basketweave morphologies [2]. The final textures are in general very strong and X-ray analyses have proved that this is in large part due to some variant selection during the thermomechanical processing [3–5]. The electron backscatter diffraction (EBSD) technique has developed rapidly and is increasingly being used to map locally the orientations of the grains in order to better understand the variant selection mechanisms. Many scientists have noticed that the a colonies are often situated at the boundary between two retained b grains that share a common {1 1 0}b plane [6–9], and that the a colonies are in Burgers OR with at least one of the two b grains, with their c axis perpendicular to the common {1 1 0}b plane. Variant selection is also supposed to be at the origin of the macrozones, i.e. millimetre ghost zones inherited from the forging process, that induce premature fatigue crack initiation and important scatter in the fatigue life [10]. Each individual macrozone shows a strong texture with the c axis of the ap grains tending to be aligned. The alignment directions vary from one macrozone to
1359-6462/$ - see front matter Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2008.05.013
C. Cayron / Scripta Materialia 59 (2008) 570–573
Figure 1. Representation in three dimensions of the variants (in blue) in OR with their parent crystal (in red) (a) for the b ? a transition and (b) for a ? b transition. In (a) the 12 a variants are grouped by pairs of twin sisters that share their c axis. The variants in the pairs are linked by a rotation of 10.5° about their common c axis. In (b) the six b P variants are grouped by entangled pairs of 3 variants that share a b h1 1 1i axis. The variants in the pairs are linked by a rotation of 60° about their common h1 1 1ib axis. The representations are created by the software GenOVa [17]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
another without any relation with the rolling directions [10], or with a preference to be in the axial direction or in the three directions located in the radial plane of the forged billet [11–13]. The global b textures in the macrozones were supposed to result from plasticity and dynamic recrystallization at high temperatures [11]. The presence of a grains that have parallel c axis was shown to result from a local anisotropy imposed by the adjacent b grains with common {1 1 0}b planes [12]. It should be noted that without this special b/b interfacial condition, the probability that a variants share their c axis is very low. Indeed, calculations based on minimization of elastic energy in isotropic conditions proved that such pairs—called here twin sister variants and shown in Figure 1a—are not energetically favoured [14]. Moreover, the absence of the a twin sister variant was later confirmed by EBSD analysis on titanium that had been formed by casting or by powder metallurgy [15]. One can therefore legitimately ask: why in thermomechanically processed Ti alloys do so many b grains share a common {1 1 0}b plane, and why does this facilitate the formation of a grains in Burgers OR? Many authors believe that the high incidence of common {1 1 0}b planes is due to the deformation imposed by the hot working. Moreover, Bhattacharyya et al. recently showed that two special misorientations occur frequently between these adjacent b grains: a rotation of 10.5° around a h1 1 0ib direction and a rotation of 60° around a h1 1 1ib [9]. They explained these findings in terms of low sigma coincidence site lattices. Plasticity and dynamic recrystallization at high temperatures might also be supposed to play a role in the formation of the special beta grain boundaries, but the precise mechanisms have not yet been discovered. The present paper gives an alternative explanation for the high incidence of specially misoriented adjacent b grains: the adjacent b grains that share a common {1 1 0}b plane and that are at the origin of the selection of the a variants are in fact b variants previously created by the a ? b transition. This result is new and surprising because it might be supposed that the nucleation of new b grains in Burgers OR with a a matrix is
571
thermodynamically less favourable than the grain growth of residual b. Many researchers have therefore assumed that, after the b ? a transition at the first forging step, the further a ? b transitions during the thermomechanical process lead to the same b orientations as those already formed during the previous transformations [4,8,11–13,16], which can be written: b0 ? {ai}i2[1,12] ? b0. It will be shown here that in fact other b variants are created and that the general b0 ? {ai}i2[1,12] ? {bij}i2[1,12],j2[1,6] must be considered, although variant selection probably occurs also for the a ? b transition. In order to establish this process, all the EBSD data published in the literature will be interpreted by assuming that adjacent b grains are b variants. We have used our software GenOVa [17] to generate the b variants inherited from the a ? b transition. For a global understanding, demonstrations and clear definitions of the vocabulary that will follow, see Ref. [18]. There are six b variants represented in three dimensions in Figure 1b and by pole figures in Figure 2. The b variants are linked by six operators reported in Table 1. Operator 0 is the operator identity, operators 1 and 2 are complementary polar operators, and the other ones are ambivalent operators. Of course, all the operators contain a rotation about the h1 1 0ib direction parallel to the c axis of the parent a crystal; however, since there are many equivalent rotations due to the symmetries of the b phase, the rotation with the minimum angle representing operator 5 is a 60° rotation about a h1 1 1ib direction. Operators 4 and 5 are the two operators experimentally found by Bhattacharyya et al. [9]. Operator 5 is aProtation of 60° about a h 1 1 1i axis; it links the pairs of 3 variants represented in Figure 1b. GenOVa also calculates the composition table between the operators (not reported here)
Figure 2. Pole figures of the b variants (in blue) in OR with their parent a crystal (in red) with c axis of the a crystal orientated (a) toward the reader and (b) horizontally. The projected directions are h0 0 1ia and h1 1 0ib. The pole figures are created by the software GenOVa [17]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
Table 1. List of the six operators that link the b variants Operator number
Rotation with the minimum angle (disorientation)
0 1 2 3 4 5
0° [0 0 1]b 60° [0 1 1]b 60° [0 1 1]b 49.47° [0 1 1]b 10.52° [0 1 1]b 60° [1 1 1]b
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C. Cayron / Scripta Materialia 59 (2008) 570–573
and the possibility for many b variants to be inherited from distinct parent a crystals. GenOVa found that the three b variants linked by operators 0, 1 and 2 can be inherited by two distinct parent a crystals that share their c axis. GenOVa also permits the pole figures to be orientated in manual or automatic mode. In order to interpret the experimental data published in the literature, we have manually oriented the theoretical pole figures of b variants with GenOVa until we obtained a good superposition between these figures (displayed on the computer screen) and the experimental ones (drawn on a tracing paper). The angular differences between the experimental and theoretical pole figures were carefully measured by reproducing the poles on a Wulff net (2° step). For the experimental pole figures containing dispersion, the average orientations were reproduced. For each comparison, we have deduced the higher angle of discrepancy, i.e. the maximum value of all the angular differences between the experimental and theoretical poles. First, we have validated our approach by comparing our simulations with experimental data obtained by in situ EBSD experiments on a commercially pure titanium alloy heated above 900 °C [19]. As shown in Figure 3, the comparison is very conclusive: the maximum angular difference is 2°. We have also simulated all the published experimental pole figures involving the adjacent b grains sharing a {1 1 0}b plane. As shown in Figures 4 and 5, our hypothesis of b variants agrees with all the experimental data with an angular accuracy better than 10°, as shown in Figures 4 and 5. Moreover, we notice that operators 3 and 5 are always found, except for the macrozones where the b variants are linked by operators 1 and 2. We will try to find the reasons of this difference in our subsequent studies. In conclusion, the adjacent b grains that share a common {1 1 0}b plane are b variants inherited from the inverse a ? b transition. The local selection of a variants can be simply explained. Let us consider the elaboration process as a sequence of three transitions: (1) b ? a, (2) a ? b and (3) b ? a transitions, with transition (1) occurring during cooling after the first b forging and cycles of transitions (2) and (3) occurring during the thermomechanical processing. The a daughter grains produced by (1) become the parent grains of the b
Figure 3. Comparison between (a) experimental h 1 1 0ib pole figures of b variants obtained by in situ EBSD experiments at 882°C (superposition of Fig. 1c and d of Ref. [19]), and (b) simulations with two b variants linked by operator 5. The common h110ib are marked by circles in (a). One of them corresponds to the c axis of the parent a crystal marked by the larger red disk in (b). The angular a–b discrepancy is 2°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
Figure 4. Comparison between experimental h1 1 0ib pole figures published in literature (first column) and simulated ones (second column). The common h1 1 0ib = h0 0 1ia are marked by circles in the experimental pole figures and by a red disk in the simulations. The experiments come from (a) Figure 7a and b of Ref. [7] simulated in (b) with two variants linked by operator 3; (c) Figure 7b and c of Ref. [7] simulated in (d) with operator 3; (e) Figure 8d of Ref. [9] simulated in (f) with operator 5; (g) Figure 6a of Ref. [8] simulated in (h) with operator 5; (i) Figure 5b of Ref. [6] simulated in (j) with operator 3. The angular discrepancies are 10° for a–b, 2° for c–d, 4° for e–f, 4° for g–h, and 4° for i–j. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
C. Cayron / Scripta Materialia 59 (2008) 570–573
Figure 5. Comparison between (a) an experimental h1 0 0ib pole figure reconstructed from the a grains in a macrozone (from Fig. 7a of Ref. [11]) and (b) its simulation with three variants linked by operators 1 and 2. The larger red disk close to the center in (b) represents the c axis of the parent a grain. There are more spots on the experimental pole figure because of the angular spreading that splits the directions close to the basal circle into two opposite directions. The angular discrepancy is 10°. (For interpretation of the references to colour in this figure legend, the readers is referred to the web version of this article.).
variants produced by (2). The a variants produced by (3) can be in Burgers OR simultaneously with two adjacent b grains if those b grains were the variants produced by (2) and if the a variants adopt the same orientation as the parent a grains of that transition (2). This can be noted for the total sequence of transitions: b0 ? {ai}i2[1, 12] ? {bij}i2[1,12],j2[1,6] ? {ai}i2[1,12]. Moreover, although this study is not statistical, we found that operators 3 and 5 occur more frequently in the literature than the other operators, and thus, selection of b variants for transition (2) seems to be also an important mechanism that should be studied statistically. We believe that the global textures of the forged titanium alloys in fact result from a mechanical texture of b produced by the first forging step and from the subsequent cycles of b ? a and a ? b transitions during the thermomechanical treatments. Since the grains in the final material obey nearly exactly the Burgers OR, we think that the important deformations induced by the thermomechanical treatments are mainly accommodated by variant selection mechanisms during the b ? a and a ? b transitions according to mechanisms similar to those implied in memory alloys. The different b grains reconstructed in the macrozones [11], i.e. the parent grains of transition (3), can also be interpreted as the daughter grains of transition (2), and the misorientations between them as operators. We can also infer that the parent a grains of transition (2) in the macrozones [11] were gigantic (larger than 5 mm) and the billet was probably monocrystalline or contained only few b grains during its first
573
forging in the b domain. Cycles of parent grain reconstructions with b ? a and a ? b transitions will be performed with the software ARPGE [18] on some experimental EBSD maps in order to confirm this point. According to our model, the textures are formed principally by the cycles of b ? a and a ? b transformations and not directly by the deformations imposed by the thermomechanical treatments. We suppose that these deformations mainly induce variant selections. In the alternative hypothesis of plasticity and continuous dynamical recrystallization [11], deformations are essential to explain the textures. Therefore, we can propose a method of distinguishing between the two hypotheses: after the forging step, two processes can be applied to the material: (A) the normal thermomechanical treatments, and (B) only the thermal treatments of A without deformation. According to the continuous dynamic recrystallization hypothesis, treatment B should not produce any texture. With our hypothesis, treatment B should produce a texture that should be weaker than for treatment A, and more precisely texture A should be a subtexture of texture B. [1] W.G. Burgers, Physica 1 (1934) 561. [2] E. Aeby-Gautier, F. Bruneseaux, J. Da Costa Teixeira, B. Appolaire, G. Geandier, D. Denis, JOM 59 (2007) 54. [3] Z.S. Zhu, J.L. Gu, R.Y. Liu, N.P. Chen, M.G. Yan, Mater. Sci. Eng. A 280 (2000) 199. [4] N. Gey, M. Humbert, Acta Mater. 50 (2002) 277. [5] P. Ari-Gur, S.L. Semiatin, Mater. Sci. Eng. A 257 (1998) 118. [6] N. Gey, M. Humbert, J. Mater. Sci. 38 (2003) 1289. [7] D. Bhattacharyya, G.B. Viswanathan, Robb Denkenberger, D. Furrer, H.L. Fraser, Acta Mater. 51 (2003) 4679. [8] N. Stanford, P.S. Bate, Acta Mater. 52 (2004) 5215. [9] D. Bhattacharyya, G.B. Viswanathan, H.L. Fraser, Acta Mater. 55 (2007) 6765. [10] K. Le Biavant, S. Pommier, C. Prioul, Fatigue Fract Eng. Mater. Struct. 25 (2002) 527. [11] L. Germain, N. Gey, M. Humbert, P. Bocher, M. Jahazi, Acta Mater. 53 (2005) 3535. [12] M. Humbert, L. Germain, N. Gey, P. Bocher, M. Jahazi, Mater. Sci. Eng. A 430 (2006) 157. [13] I. Lonardelli, N. Gey, H.R. Wenk, M. Humbert, S.C. Vogel, L. Lutterotti, Acta Mater. 55 (2007) 5718. [14] P. Barberis, F. Montheillet, C. Chauvy, Solid State Phen. 105 (2005) 133. [15] C. Cayron, J. Appl. Crystallogr. 40 (2007) 1183. [16] N. Gey, M. Humbert, M. Moustahfid, Scripta Mater. 42 (2000) 525. [17] C. Cayron, J. Appl. Crystallogr. 40 (2007) 1179. [18] C. Cayron, Acta Crystallogr. A 62 (2006) 21. [19] G.G.E. Seward, S. Celotto, D.J. Prior, J. Wheeler, R.C. Pond, Acta Mater. 52 (2004) 821.
Scripta Materialia 59 (2008) 570–573 www.elsevier.com/locate/scriptamat
Importance of the a ? b transformation in the variant selection mechanisms of thermomechanically processed titanium alloys C. Cayron* CEA, LITEN, DTH, Minatec, 38054 Grenoble, France Received 19 March 2008; revised 6 May 2008; accepted 7 May 2008 Available online 21 May 2008
Selection of a variants is often reported in thermomechanically processed titanium alloys. Many workers have noticed that a variants are frequently situated at the grain boundary between b grains that share a common {1 1 0}b plane, but up to now such coincidence on the b grains could not be fully clarified. By careful examination of electron backscatter diffraction data published in literature, it has been established that the adjacent b grains are actually b variants inherited from the a ? b transition. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Titanium; Variant selection; Operator; EBSD; Phase transition
Titanium and its alloys have high strength-to-weight ratios, excellent mechanical properties and corrosion resistance, and hence are widely used in the aerospace, energy, chemical and health industries. Their high strength is due to a complex microstructure that results from a first-order transition during cooling: parent body-centered cubic (bcc) b phase is transformed into daughter hexagonal close-packed (hcp) a phase. The temperature of the b ? b + a transition is about 880 °C for pure titanium and 10–20 °C lower for its alloys. A parent b crystal and a daughter a crystal respect the Burgers orientation relationship (OR) [1]: [0 0 1]a// 1 1 1]b. Due to the symmetries of [1 1 0]b and [1 0 0]a//[ the parent and daughter phases, there are 12 equivalent ORs that correspond to equivalently orientated daughter a crystals called variants. These a variants are represented in Figure 1a. Commercialized titanium alloys are in general produced by a sequence of melting, casting, forging at temperatures higher than 950 °C in the b domain, and thermomechanical processing at lower temperatures between 600 and 900 °C in the b + a domain. The final microstructures (i.e. respective content of the phases, grain sizes and morphologies) are various and strongly depend on the process parameters (i.e. temperatures, cooling rates and amount of deformation during the hot working steps). In general the microstructures
* Tel.: +33 4 38 78 93 29; fax: +33 4 38 78 58 91; e-mail: [email protected]
are constituted of globular primary ap grains, colonies of secondary as grains and, in some alloys, of some retained b between these grains. The ap grains are formed during the first cooling after forging, and they grow and become globular during the thermomechanical process. The as grains are formed from b in the b + a domain during the final cooling after the thermomechanical processing. The as grains can nucleate at grain boundaries in the form of parallel colonies or inside the b part with basketweave morphologies [2]. The final textures are in general very strong and X-ray analyses have proved that this is in large part due to some variant selection during the thermomechanical processing [3–5]. The electron backscatter diffraction (EBSD) technique has developed rapidly and is increasingly being used to map locally the orientations of the grains in order to better understand the variant selection mechanisms. Many scientists have noticed that the a colonies are often situated at the boundary between two retained b grains that share a common {1 1 0}b plane [6–9], and that the a colonies are in Burgers OR with at least one of the two b grains, with their c axis perpendicular to the common {1 1 0}b plane. Variant selection is also supposed to be at the origin of the macrozones, i.e. millimetre ghost zones inherited from the forging process, that induce premature fatigue crack initiation and important scatter in the fatigue life [10]. Each individual macrozone shows a strong texture with the c axis of the ap grains tending to be aligned. The alignment directions vary from one macrozone to
1359-6462/$ - see front matter Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2008.05.013
C. Cayron / Scripta Materialia 59 (2008) 570–573
Figure 1. Representation in three dimensions of the variants (in blue) in OR with their parent crystal (in red) (a) for the b ? a transition and (b) for a ? b transition. In (a) the 12 a variants are grouped by pairs of twin sisters that share their c axis. The variants in the pairs are linked by a rotation of 10.5° about their common c axis. In (b) the six b P variants are grouped by entangled pairs of 3 variants that share a b h1 1 1i axis. The variants in the pairs are linked by a rotation of 60° about their common h1 1 1ib axis. The representations are created by the software GenOVa [17]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
another without any relation with the rolling directions [10], or with a preference to be in the axial direction or in the three directions located in the radial plane of the forged billet [11–13]. The global b textures in the macrozones were supposed to result from plasticity and dynamic recrystallization at high temperatures [11]. The presence of a grains that have parallel c axis was shown to result from a local anisotropy imposed by the adjacent b grains with common {1 1 0}b planes [12]. It should be noted that without this special b/b interfacial condition, the probability that a variants share their c axis is very low. Indeed, calculations based on minimization of elastic energy in isotropic conditions proved that such pairs—called here twin sister variants and shown in Figure 1a—are not energetically favoured [14]. Moreover, the absence of the a twin sister variant was later confirmed by EBSD analysis on titanium that had been formed by casting or by powder metallurgy [15]. One can therefore legitimately ask: why in thermomechanically processed Ti alloys do so many b grains share a common {1 1 0}b plane, and why does this facilitate the formation of a grains in Burgers OR? Many authors believe that the high incidence of common {1 1 0}b planes is due to the deformation imposed by the hot working. Moreover, Bhattacharyya et al. recently showed that two special misorientations occur frequently between these adjacent b grains: a rotation of 10.5° around a h1 1 0ib direction and a rotation of 60° around a h1 1 1ib [9]. They explained these findings in terms of low sigma coincidence site lattices. Plasticity and dynamic recrystallization at high temperatures might also be supposed to play a role in the formation of the special beta grain boundaries, but the precise mechanisms have not yet been discovered. The present paper gives an alternative explanation for the high incidence of specially misoriented adjacent b grains: the adjacent b grains that share a common {1 1 0}b plane and that are at the origin of the selection of the a variants are in fact b variants previously created by the a ? b transition. This result is new and surprising because it might be supposed that the nucleation of new b grains in Burgers OR with a a matrix is
571
thermodynamically less favourable than the grain growth of residual b. Many researchers have therefore assumed that, after the b ? a transition at the first forging step, the further a ? b transitions during the thermomechanical process lead to the same b orientations as those already formed during the previous transformations [4,8,11–13,16], which can be written: b0 ? {ai}i2[1,12] ? b0. It will be shown here that in fact other b variants are created and that the general b0 ? {ai}i2[1,12] ? {bij}i2[1,12],j2[1,6] must be considered, although variant selection probably occurs also for the a ? b transition. In order to establish this process, all the EBSD data published in the literature will be interpreted by assuming that adjacent b grains are b variants. We have used our software GenOVa [17] to generate the b variants inherited from the a ? b transition. For a global understanding, demonstrations and clear definitions of the vocabulary that will follow, see Ref. [18]. There are six b variants represented in three dimensions in Figure 1b and by pole figures in Figure 2. The b variants are linked by six operators reported in Table 1. Operator 0 is the operator identity, operators 1 and 2 are complementary polar operators, and the other ones are ambivalent operators. Of course, all the operators contain a rotation about the h1 1 0ib direction parallel to the c axis of the parent a crystal; however, since there are many equivalent rotations due to the symmetries of the b phase, the rotation with the minimum angle representing operator 5 is a 60° rotation about a h1 1 1ib direction. Operators 4 and 5 are the two operators experimentally found by Bhattacharyya et al. [9]. Operator 5 is aProtation of 60° about a h 1 1 1i axis; it links the pairs of 3 variants represented in Figure 1b. GenOVa also calculates the composition table between the operators (not reported here)
Figure 2. Pole figures of the b variants (in blue) in OR with their parent a crystal (in red) with c axis of the a crystal orientated (a) toward the reader and (b) horizontally. The projected directions are h0 0 1ia and h1 1 0ib. The pole figures are created by the software GenOVa [17]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
Table 1. List of the six operators that link the b variants Operator number
Rotation with the minimum angle (disorientation)
0 1 2 3 4 5
0° [0 0 1]b 60° [0 1 1]b 60° [0 1 1]b 49.47° [0 1 1]b 10.52° [0 1 1]b 60° [1 1 1]b
572
C. Cayron / Scripta Materialia 59 (2008) 570–573
and the possibility for many b variants to be inherited from distinct parent a crystals. GenOVa found that the three b variants linked by operators 0, 1 and 2 can be inherited by two distinct parent a crystals that share their c axis. GenOVa also permits the pole figures to be orientated in manual or automatic mode. In order to interpret the experimental data published in the literature, we have manually oriented the theoretical pole figures of b variants with GenOVa until we obtained a good superposition between these figures (displayed on the computer screen) and the experimental ones (drawn on a tracing paper). The angular differences between the experimental and theoretical pole figures were carefully measured by reproducing the poles on a Wulff net (2° step). For the experimental pole figures containing dispersion, the average orientations were reproduced. For each comparison, we have deduced the higher angle of discrepancy, i.e. the maximum value of all the angular differences between the experimental and theoretical poles. First, we have validated our approach by comparing our simulations with experimental data obtained by in situ EBSD experiments on a commercially pure titanium alloy heated above 900 °C [19]. As shown in Figure 3, the comparison is very conclusive: the maximum angular difference is 2°. We have also simulated all the published experimental pole figures involving the adjacent b grains sharing a {1 1 0}b plane. As shown in Figures 4 and 5, our hypothesis of b variants agrees with all the experimental data with an angular accuracy better than 10°, as shown in Figures 4 and 5. Moreover, we notice that operators 3 and 5 are always found, except for the macrozones where the b variants are linked by operators 1 and 2. We will try to find the reasons of this difference in our subsequent studies. In conclusion, the adjacent b grains that share a common {1 1 0}b plane are b variants inherited from the inverse a ? b transition. The local selection of a variants can be simply explained. Let us consider the elaboration process as a sequence of three transitions: (1) b ? a, (2) a ? b and (3) b ? a transitions, with transition (1) occurring during cooling after the first b forging and cycles of transitions (2) and (3) occurring during the thermomechanical processing. The a daughter grains produced by (1) become the parent grains of the b
Figure 3. Comparison between (a) experimental h 1 1 0ib pole figures of b variants obtained by in situ EBSD experiments at 882°C (superposition of Fig. 1c and d of Ref. [19]), and (b) simulations with two b variants linked by operator 5. The common h110ib are marked by circles in (a). One of them corresponds to the c axis of the parent a crystal marked by the larger red disk in (b). The angular a–b discrepancy is 2°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
Figure 4. Comparison between experimental h1 1 0ib pole figures published in literature (first column) and simulated ones (second column). The common h1 1 0ib = h0 0 1ia are marked by circles in the experimental pole figures and by a red disk in the simulations. The experiments come from (a) Figure 7a and b of Ref. [7] simulated in (b) with two variants linked by operator 3; (c) Figure 7b and c of Ref. [7] simulated in (d) with operator 3; (e) Figure 8d of Ref. [9] simulated in (f) with operator 5; (g) Figure 6a of Ref. [8] simulated in (h) with operator 5; (i) Figure 5b of Ref. [6] simulated in (j) with operator 3. The angular discrepancies are 10° for a–b, 2° for c–d, 4° for e–f, 4° for g–h, and 4° for i–j. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
C. Cayron / Scripta Materialia 59 (2008) 570–573
Figure 5. Comparison between (a) an experimental h1 0 0ib pole figure reconstructed from the a grains in a macrozone (from Fig. 7a of Ref. [11]) and (b) its simulation with three variants linked by operators 1 and 2. The larger red disk close to the center in (b) represents the c axis of the parent a grain. There are more spots on the experimental pole figure because of the angular spreading that splits the directions close to the basal circle into two opposite directions. The angular discrepancy is 10°. (For interpretation of the references to colour in this figure legend, the readers is referred to the web version of this article.).
variants produced by (2). The a variants produced by (3) can be in Burgers OR simultaneously with two adjacent b grains if those b grains were the variants produced by (2) and if the a variants adopt the same orientation as the parent a grains of that transition (2). This can be noted for the total sequence of transitions: b0 ? {ai}i2[1, 12] ? {bij}i2[1,12],j2[1,6] ? {ai}i2[1,12]. Moreover, although this study is not statistical, we found that operators 3 and 5 occur more frequently in the literature than the other operators, and thus, selection of b variants for transition (2) seems to be also an important mechanism that should be studied statistically. We believe that the global textures of the forged titanium alloys in fact result from a mechanical texture of b produced by the first forging step and from the subsequent cycles of b ? a and a ? b transitions during the thermomechanical treatments. Since the grains in the final material obey nearly exactly the Burgers OR, we think that the important deformations induced by the thermomechanical treatments are mainly accommodated by variant selection mechanisms during the b ? a and a ? b transitions according to mechanisms similar to those implied in memory alloys. The different b grains reconstructed in the macrozones [11], i.e. the parent grains of transition (3), can also be interpreted as the daughter grains of transition (2), and the misorientations between them as operators. We can also infer that the parent a grains of transition (2) in the macrozones [11] were gigantic (larger than 5 mm) and the billet was probably monocrystalline or contained only few b grains during its first
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forging in the b domain. Cycles of parent grain reconstructions with b ? a and a ? b transitions will be performed with the software ARPGE [18] on some experimental EBSD maps in order to confirm this point. According to our model, the textures are formed principally by the cycles of b ? a and a ? b transformations and not directly by the deformations imposed by the thermomechanical treatments. We suppose that these deformations mainly induce variant selections. In the alternative hypothesis of plasticity and continuous dynamical recrystallization [11], deformations are essential to explain the textures. Therefore, we can propose a method of distinguishing between the two hypotheses: after the forging step, two processes can be applied to the material: (A) the normal thermomechanical treatments, and (B) only the thermal treatments of A without deformation. According to the continuous dynamic recrystallization hypothesis, treatment B should not produce any texture. With our hypothesis, treatment B should produce a texture that should be weaker than for treatment A, and more precisely texture A should be a subtexture of texture B. [1] W.G. Burgers, Physica 1 (1934) 561. [2] E. Aeby-Gautier, F. Bruneseaux, J. Da Costa Teixeira, B. Appolaire, G. Geandier, D. Denis, JOM 59 (2007) 54. [3] Z.S. Zhu, J.L. Gu, R.Y. Liu, N.P. Chen, M.G. Yan, Mater. Sci. Eng. A 280 (2000) 199. [4] N. Gey, M. Humbert, Acta Mater. 50 (2002) 277. [5] P. Ari-Gur, S.L. Semiatin, Mater. Sci. Eng. A 257 (1998) 118. [6] N. Gey, M. Humbert, J. Mater. Sci. 38 (2003) 1289. [7] D. Bhattacharyya, G.B. Viswanathan, Robb Denkenberger, D. Furrer, H.L. Fraser, Acta Mater. 51 (2003) 4679. [8] N. Stanford, P.S. Bate, Acta Mater. 52 (2004) 5215. [9] D. Bhattacharyya, G.B. Viswanathan, H.L. Fraser, Acta Mater. 55 (2007) 6765. [10] K. Le Biavant, S. Pommier, C. Prioul, Fatigue Fract Eng. Mater. Struct. 25 (2002) 527. [11] L. Germain, N. Gey, M. Humbert, P. Bocher, M. Jahazi, Acta Mater. 53 (2005) 3535. [12] M. Humbert, L. Germain, N. Gey, P. Bocher, M. Jahazi, Mater. Sci. Eng. A 430 (2006) 157. [13] I. Lonardelli, N. Gey, H.R. Wenk, M. Humbert, S.C. Vogel, L. Lutterotti, Acta Mater. 55 (2007) 5718. [14] P. Barberis, F. Montheillet, C. Chauvy, Solid State Phen. 105 (2005) 133. [15] C. Cayron, J. Appl. Crystallogr. 40 (2007) 1183. [16] N. Gey, M. Humbert, M. Moustahfid, Scripta Mater. 42 (2000) 525. [17] C. Cayron, J. Appl. Crystallogr. 40 (2007) 1179. [18] C. Cayron, Acta Crystallogr. A 62 (2006) 21. [19] G.G.E. Seward, S. Celotto, D.J. Prior, J. Wheeler, R.C. Pond, Acta Mater. 52 (2004) 821.