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γ component. This is, however, not an unknown component since it was also present in some T R I P steels discussed in previous w o r k . U
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CRYSTALLOGRAPHIC ORIENTATION RELATIONSHIPS The data that were calculated as described above were first used to check which neighbouring grain fitted best with one of the predicted K S , N W and Pitsch transformation products of the retained austenite grains. As is discussed e l s e w h e r e " in m o r e detail, it was found that almost 2 out of 3 retained austenite grains had at least one neighbouring B C C grain that was within a 5° misorientation of an ideal K S , N W or Pitsch transformation product. From this result, the question arises why not all the retained austenite grains had a close K S , N W or Pitsch relationship with at least one of its neighbours. Firstly, it was d e m o n s t r a t e d that the average confidence index of the retained austenite grains that did not have a neighbour with a close correlation with an ideal K S , N W or Pitsch transformation product was markedly lower. Secondly, it should also be mentioned that an E B S D measurement only provides information of one 2 D section of a 3D microstructure. which might cause certain close-to K S , N W or Pitsch neighbours to remain undetected. 13
After considering which neighbouring BCC grains fitted best with one of the predicted K S . N W and Pitsch transformation products of the retained austenite, these data are investigated in more detail. For those retained austenite grains that had a neighbour transformation product with the misorientation angle below 5°, a frequency distribution showing which orientation relationship provided the best prediction of the real crystallographic orientation of the best fitting BCC grain was made. This distribution is displayed in Figure 4 and it is shown that the Pitsch orientation relationship is clearly the least dominant orientation relationship. This feature is in good accordance with all literature data, which always suggest a d o m i n a n c e of the KS or N W orientation relationship. In the present study, for both materials, the KS orientation relationship appears to be the more dominant one. This is in particular the case for T R I P 2 , where in nearly two out of three cases the KS relationship is observed. These findings should, however, always be stated with the necessary precautions. Several retained austenite grains might display a good correspondence with the ideal K S . N W or Pitsch orientation relationship with several of their B C C neighbours, whilst the conclusions m a d e above were only based on considering the best fitting neighbouring B C C grain. In order to illustrate this, one region (Figure 5) is selected from the detailed O I M measurements that were carried out on the TRIP2 material. Figure 5 consists of five retained austenite grains. At first, the crystallographic orientations of these grains were examined for possible correlations. It was found that the m i n i m u m misorientation angle between grain A and
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grain Β w a s b e l o w 1°. Such an orientation difference can easily be attributed to the fitting of the diffraction pattern by the O I M software. Consequently, during intercritical annealing, grain A and grain Β formed one intercritical austenite grain. The same conclusion follows when the orientations of grain D and grain Ε are compared. The misorientation between grain C and grain Β did not reveal a specific correlation, so there must have been a regular high angle grain boundary in between them during intercritical annealing. The misorientation between grain C and grain D consists, however, of the following axis/angle pair: [0.58 0.57 -0.58]59.9°. i.e. a nearly perfect twin correlation. The presence of a twin boundary in the austenite is not surprising since it was reported in literature that twin boundaries are typical for an annealed austenitic microstructure and are understood to b e generated by multiple twinning during the growth of austenitic grains during transformation. 7
Figure 4. Histogram showing for both materials which the orientation relationship was between the retained austenite grain and the best fitting measured BCC grain. Only those cases where the orientation difference was less than 5° were included.
Figure 5. Microstructural detail with several retained austenite grains and their B C C neighbours. The retained austenite grains are in red. B C C grains are in white. All grain boundaries ( ω > 5°) are in black.
T h e analysis of the orientations of the retained austenite was followed by a study of the orientation relationships between these grains and their B C C neighbours. Grain 1 and grain 10 are polygonal ferrite grains. O f particular interest were the orientation relationships of the grains that are surrounded by what initially was one retained austenite grain. The software indicated that grain 3 displays a Nishiyama-Wassermann orientation relationship with a misfit of 3.6° with grain A and a misfit of 2.9° with grain B . respectively. Grain 7 also shows a N W orientation relationship with grain D and grain E, with an orientation difference of 2.6° and 3.0°, respectively. The orientation relationship between grain C and grain 7 is of the KS type and has a misorientation of 2.8°. For the sake of completeness, it should also be mentioned that grain A and grain 2 are 1.5° away from an ideal K S relationship: grain Β and grain 8 show a misfit of 1.1° with a perfect K S relationship and grain D and grain 6 are 3.4° from an ideal Pitsch relationship. A s is s h o w n by these results, different orientation relationships with only a small deviation from the ideal orientation relationship were found between the austenitic grains and their B C C neighbours. Therefore, it can be deduced from these data that the performed analysis
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of the active orientation relationship when searching for the dominant orientation relationship should be treated with the necessary precautions. Indeed, macroscopic conclusions on which crystallographic orientation relationship prevails for a complete material should be made with relative care because for solid-solid transformations, in case of no pre-strain nor an external loading, the microscopic "environment" might play an important role in determining which orientation relationship governs the transformation of one particular grain. In the present case, possible influences from the microstructural environment of the grain might be the interaction of the grain with its neighbours, the possible presence of micro-stresses or residual stresses and the accommodation of the transformation strain. These factors are also mentioned in literature for other solid-solid transformations, such as martensitic transformations' ' ' , but in order to clarify their influence in the present case, further measurements of the internal stresses, as was for by neutron diffraction measurements are necessary. example done by Muransky el al. 4
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CONCLUSION E B S D measurements were performed in order to study the microstructural properties of phosphorus added T R I P steels. The brass component appeared to be the most dominant component in the retained austenite texture, with an intensity gradually decreasing along the alpha and beta fibre. The intensity of the texture of both B C C microconstituents was similar and the bainite texture contained transformation components. Most austenite grains were found to display a good correspondence with the ideal K S . N W or Pitsch orientation relationship with different B C C neighbours. The KS relationship appeared to be dominant. However, it was argued that this macroscopic conclusion should be made with care due to the possible influence of the local environment of the transforming grain which might have an impact in "the selection" of a specific orientation relationship. REFERENCES ' O. Matsumura, Y. Sakuma and H. Takechi. ISU Int., 2 7 , 570 (1987). M. De Meyer. Transformations and mechanical properties of cold rolled and intercritically annealed CMnAISi T R I P aided steels, 2 0 0 1 , P h D thesis Ghent University. L. Barbé. Physical Metallurgy of P-alloyed T R I P Steels, 2 0 0 5 , PhD thesis Ghent University, ISBN 90 8578-037-3. 2 0 0 5 . L. Barbé. Κ. Verbeken and E. Wettinck. ISU Int.. 4 6 , 1249 (2006). Y. He. S. Godet and J.J. Jonas. Acta Mat., 5 3 , 1179 (2005). S. Zaefferer. J. Ohlert and W. Bleck, Ada Mai.. 5 2 2675 (2004). G. Brückner. J. Pospiech, I. Siedl and G. Gottstein, Scripta Mat., 4 4 2635 (2001). Β. Verlinden, Ph. Bocher. Ε. Girault and E. Aernoudt, Scripta Mal., 4 5 909 (2001). M. De Meyer, L. Kestens and B.C. De C o o m a n , Mat. Sei. and Techn., 17 1353 (2001). B. Hutchinson. L. Ryde, E. Lindh and K. Tagashira. Mat. Sc. and Eng. A, A 2 5 7 9 (1998). V. Andrade-Carozzo and P. Jacques. Mai Sc. Forum. 5 3 9 - 5 4 3 , 4 6 4 9 (2007). J. Huang. W.J. Poole and M . Militzer. Met. Mat. Trans., 3 5 A , 3363 (2004). K. Verbeken. L. Barbe and D. Raabe, Met. Mat. Trans., submitted. K. Otsuka and X. Ren. Prog. Mai. Sc.. 5 0 . 511 (2005). Y. A y d o g d u . A. Aydogdu and O. A d i g u z e l . J . Mat. Proc. Tech.. 123, 498 (2002). O. Muransky, P. Lukas, J. Zrnik and P. Sittner, Physica B. 3 8 5 - 3 8 6 , 587 (2006). 2
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M I C R O S T R U C T U R A L A N D T E X T U R A L E V O L U T I O N S IN A C O L D R O L L E D HIGH MN TWIP STEEL Kim Verbeken
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*. Lieven B r a c k e ' , Leo Kestens'
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Department of Metallurgy and Materials Science Ghent University (UGent) Technologiepark 903 B-9052 Ghent, Belgium Tel. +32 9 264 57 83 Fax +32 9 264 58 33 [email protected] 2
Microstructure Physics Max-Planck-Institut fur Eisenforschung Max-Planck-Strasse 1 D - 4 0 2 3 7 Düsseldorf, Germany 3
Corus Research D e v e l o p m e n t and Technology PO Box 10000 N L - 1 9 7 0 CA IJmuiden, The Netherlands 4
Materials Science and Engineering Department Delft University of Technology Mekelweg 2 N L - 2 6 2 8 C D Delft, The Netherlands * Postdoctoral fellow with the Fund for Scientific Research-Flanders (Belgium) (F.W.O. Vlaanderen) ABSTRACT A fully austenitic T W I P steel w a s cold rolled. During cold rolling, the evolution of the crystallographic texture was monitored. The development of a brass type of texture was found, which is typical for low stacking fault energy (SFE) materials. Intensive electron microscopic observations, with T E M and S E M , revealed four active deformation mechanisms: micro twinning, dislocation slip, the formation of stacking faults and microscopic shear banding. The effect of macroscopic shear banding was expected to be minimal. Putting together all the results showed that both micro twinning and slip play a major role in the development of the observed brass texture. KEYWORDS Austenitic diffraction
steels,
deformation
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microscopy,
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INTRODUCTION Environmental issues push the automotive industry towards research and development of new alloys combining a high strength with an excellent formability. Consequently, the vehicle weight can be reduced and the emissions of green house gases can be reduced. The competition between the light metals industry and the steel industry has led over the past decades to the development of many new alloys. In the steel industry, the initial effort was mainly concentrated on improving the properties of low carbon steels with a B C C crystal structure. Typical e x a m p l e s are the ferrite-martensite dual phase (DP) steels and the multiphase steels which display the TRansformation Induced Plasticity (TRIP) effect during deformation. In order to meet the increasing d e m a n d s for higher ductility, austenitic steels were found to show a very interesting potential as a material for producing complex parts in forming operations. T h e mechanical behaviour and especially the strain-hardening, of these steels largely depend on the stacking fault energy ( S I E ) since this material parameter largely determines which deformation m e c h a n i s m will be a c t i v e ' ' . 1
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With increasing SFE, the active mechanism gradually changes from a"-martensite formation over ε-martensite formation to mechanical twinning, deformation by partial and by perfect dislocations. The formation mechanism of mechanical twinning and that of the hexagonal ε-martensite is similar. T h e only difference is that for the γ -> ε martensitic transformation a Shockley partial has to be generated on every second close packed plane, while in the case of mechanical twinning this has to be done on every close packed plane. Note, however, that the detailed activation mechanism for both p h e n o m e n a is still not fully solved. Both the formation of strain-induced ε-martensite and mechanical twins will result in an increase of the tensile strength and the uniform elongation, by the TRansformation (TRIP) and T W i n n i n g Induced Plasticity ( T W I P ) effect, respectively. The T W I P effect gives the best results regarding the uniform elongation and therefore, the critical value of the SFE for the transition from ε-martensite transformation to mechanical twinning is of particular i n t e r e s t . Although the exact value of the SFE is difficult to determine and depends on the chemical composition, performed on alloys with different compositions report an acceptable different studies agreement in the obtained transition value, which ranges from 9 to 18mJm' . The SFE is also k n o w n to have a strong influence on the development of the crystallographic texture during cold rolling. High SFE materials develop a strong ß-fiber texture with an increasing intensity from the brass component, over the S to the copper c o m p o n e n t ' ' ' '". Low SFE materials, which deform by twinning, develop a α-fiber texture with an increased intensity on the brass and G o s s components. This texture formation is often related to mechanical t w i n n i n g " . but it is also reported that mechanical twinning initiates macroscopic shear band formation, which results in a brass t e x t u r e . This texture has also been observed in low SFE materials before twinning occurred; this was attributed to microscopic shear b a n d s . In low SFE materials, the partial dislocations that form the stacking fault are widely separated. This restricts the possibilities for cross slip, since the required perfect screw character for a cross slipping dislocation can only be achieved by recombination of the two partials to a perfect dislocation. H e n c e , the slip is restricted to the plane in w h i c h the stacking fault originated, i.e. planar slip. Recently, the influence of the planarity of the slip systems and the localization of slip in fee metals with a low SFE has been modelled by Miriglia et a l . . They found that their model could only partially account for the experimental observations. 4
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The present work studies the development of the crystallographic texture during cold rolling of a low SFE Fe-Mn-C based T W I P alloy. Moreover, it is aimed to reveal the active
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deformation m e c h a n i s m s by performing a detailed investigation of the deformed microstructure and to use the results of this investigation to explain the texture evolution observed. EXPERIMENTAL PROCEDURE T h e material studied is a fully austenitic F e - 2 2 % M n alloy with considerable amounts of C and Ν to stabilize the austenite at room temperature. The SFE of the material was m e a s u r e d ' to be 15mJm" . The material was cast as an 80kg ingot in a laboratory vacuum induction furnace under a protective N i atmosphere of 1013hPa. After reheating at 1473K. the material was hot rolled o n a laboratory mill from 2 0 m m to 3 m m in four passes, keeping the finish rolling temperature higher than 1223K to allow for static recrystallization before water quenching. Samples of the hot rolled strip w e r e cold rolled to different strains varying between a cold rolling reduction of 5 and 4 0 % . X-ray diffraction texture measurements w e r e performed on a Siemens D5000 diffractometer. using MoK,, radiation ( λ = 0 . 0 7 0 9 2 6 n m ) at 50kV and 5 0 m A . by measuring 4 pole figures and calculating the Orientation Distribution Functions ( O D F ) . The { 1 0 0 ' . {110}. {211} and {310} incomplete pole figures w e r e measured by the Schultz reflection method using an Euler cradle goniometer. The orientation distribution functions ( O D F ) were calculated with the M T M - F H M software developed by Van H o u t t e . A Zeiss D S M 962 S E M operated at 15kV was used for microstructural investigation. Electron BackScatter Diffraction ( E B S D ) measurements were carried out on the plane parallel to the rolling and normal directions. Scans with step sizes of 0.1 to 0.5μηι were carried out on a FEI X L 3 0 E S E M with a LaB -filament, equipped with a T S L - O I M " ' E B S D attachment. The sample preparation for X R D and O I M observations consisted in mechanical polishing with 1pm diamond paste, followed by mechanical polishing with a Struers O P S solution. N o etching was applied. T h e Kernel Average Misorientation ( K A M ) was used to illustrate the energetic homogeneity of the cold rolled structure. The K A M is a measure for the local misorientation and is determined as follows. For each point measured in the E B S D scan, the average misorientation of that point and its neighbours is calculated using T S L - O I M " software, with the proviso that misorientations exceeding some tolerance value are excluded. This is necessary to exclude the effect of large misorientations, e.g. as a consequence of the presence of a grain boundary. ?
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Figure 1. S E M images of (a) after cold rolling, ε = 0.22. The arrows indicate possible microtwins. (b) after cold rolling ε - 0.51 : onset of macroscopic shear banding. N o etching was applied.
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MICROSTRUCTURAL EVOLUTION Figure 1(a) shows the microstructure after cold rolling to moderate strains (ε = 0.22). As shown on the picture, deformation lines are locally visible, which might be attributed to twinning, but the possibility that these lines are slip lines cannot be excluded as will be shown later. At higher strains (ε - 0.51). cf. Figure 1(b). shear bands became visible. The observed angle with respect to the rolling direction was between 20° and 4 5 ° , which is in fair agreement with the angle of 35° which is reported for cold rolled low SFE materials ''' . Higher strains could not be reached with the laboratory rolling mill used. In order to verify whether the deformation lines observed were t w i n s or slip lines, four similarly looking areas were isolated from t w o E B S D measurements. Figure 2(a) results from a measurement of a sample rolled to a strain of ε = 0 . 5 1 , whereas Figure 2(b) results from a sample rolled to a strain of ε = 0.22. In these deformed areas of these samples, misorientation profiles were taken in different d i r e c t i o n s . In area 1. misorientation profiles along the rolling (RD) and normal direction ( N D ) did not reveal the presence of twin boundaries. A point-to-origin misorientation of about 20° was built up along the N D and one of about 10° was built up along the RD. Because of the absence of twin misorientations, it can be concluded that defonnation in area 1 only occurred by slip. Similar observations could be made for area 2. 1
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Figure 2. Inverse pole figures o f 4 selected deformed areas. (1 to 3 : ε = 0 . 5 1 , 4: ε = 0.22). A misorientation profile along N D clearly showed the presence of a n u m b e r of twin boundaries in area 3. T h e twinned regions appeared to be about 1 μιη wide and correspond to bundles of microtwins. Individual microtwins. as can be observed by T E M , cannot be seen with the scanning electron microscope used because of a limited resolution. Similar observations were made for a misorientation profile along R D in area 4. The fact that both twinning and slip are observed in similarly looking areas leads to the conclusion that it is impossible to quantify the fraction of microtwins by looking at images that do not contain orientation information. A detailed T E M s t u d y was performed to reveal the deformation mechanisms that are active during cold rolling. The presence of microtwins, even a small strain of ε = 0.10, was clearly visible. T w i n s were localized in bands of narrowly spaced twin lamellae. The deformation of the material by slip in regions without microtwins was demonstrated as well. Microscopic shear bands, without features corresponding to overlapping stacking faults or micro 18
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twinning, have been observed in samples cold rolled to a strain of ε = 0.10. It was also found that the microstructure contained a high amount of regular stacking faults, as can be expected for a material with a S F E as low as the alloy studied. Despite the presence of different deformation m e c h a n i s m s , the deformed microstructure was energetically rather h o m o g e n e o u s as is illustrated by the calculating the Kernell Average Misorientation m a p . which is s h o w n e l s e w h e r e . 18
TEXTURAL EVOLUTION The texture evolution with increasing reduction is shown in Figure 3. The strain increases from ε = 0 in Figure 3(a) until ε = 0.51 in Figure 3(f). The hot rolled texture (Figure 3(a)) is weak with a m a x i m u m intensity of 2.0 times random for the cube component {001 )<100>. The deformation textures are relatively weak as well, with a m a x i m u m intensity of 5.0 times random for the sample with a rolling strain of ε = 0.51. With increasing reduction, the intensity-' of the cube component decreases and disappears for strains above ε — 0.22. In the early stages of cold rolling, the intensity of the copper component increases, remains constant up to strains of about ε = 0.35 and decreased afterwards. The main feature of the texture evolution is the development of an α-fiber texture. At every strain level, the intensity on the ß-fiber gradually decreases from the brass component over the S c o m p o n e n t to the copper component. The overall intensity on this fiber decreases with increasing strain, except for the brass component. This type of texture is typical for low SFE materials . An important feature is the appearance of the copper twin component ( C u T ) with increasing strain. This is a clear indication that mechanical twinning has an important influence on the texture, as will be discussed below. 7
Figure 3. Evolution of the texture during cold rolling, cp = 45°, φ = 65° and φ = 90° sections of the O D F (a) ε = 0. (b) ε = 0.05, (c) ε = 0 . 1 1 . (d) ε = 0.22. (e) ε = 0.36. (f) ε = 0.51 (Levels: 0.80, 1.00. 1.30. 1 . 6 0 . 2 . 0 0 . 2 . 5 0 . 3 . 2 0 , 4 . 0 0 . 5 . 0 0 ) 2
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T h e different deformation mechanisms that contribute to the development of the texture observed can be nicely illustrated with the crystallographic orientation data (Figure 4) obtained for the different areas shown in Figure 2. Area 1 and 2. w h i c h both deformed by slip, have a
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cr> stallographic orientation between Goss and brass (G/B) for area 1 and near the α -fiber ( N A F ) for area 2. It is clear from Figure 3(f) that these orientations are present in the final deformation texture. Therefore, it can be concluded that slip is important during the formation of the rolling texture of the T W I P alloy under consideration.
Figure 4. φ = 45°, 65° and 90° sections of the orientation distribution function, showing the orientations of the four areas of Figure 2. ( G / B : between Goss and brass. N A F : near a-fiber. Cu: copper, C u T : copper twin. B: brass. X: newly defined c o m p o n e n t ) 2
For area 3. the O D F displays two c o m p o n e n t s : the copper (Cu) and the copper twin (CuT). Taking into account the corresponding misorientation profile and the microstructural morphology observed, this can be considered as a proof that the appearance of the copper twin in the O D F s of Figure 3 really is the consequence of the twinning of the copper orientation and that it does not result from another deformation mechanism. For area 4, the O D F also s h o w s two texture components: the first o n e . not described in the literature, which will be referred to as component X . situated between the cube and the copper orientation on the (p2=45° section and with Miller indices close to { 1 1 4 j < 1 7 2 > and the well-known brass component. A s illustrated in Figure 3(d). the component X is clearly present in the O D F s obtained by X R D . This demonstrates the statistical relevance of the observation of area 4. It was found by applying a simple crystal rotation program that the brass component is indeed one of the possible twin products of the parent X c o m p o n e n t . It was also shown that one of the other possible twin variants of the parent X component is the copper orientation. This could explain the increase of the copper intensity at the beginning of the cold rolling, but this could, however, not be demonstrated in the particular case of area 4. 18
DISCUSSION The results of the microstructural and textural evaluation clearly indicate that several factors play an important role in the plastic behaviour of the F e - M n alloy during cold rolling.
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The development of a brass type texture is typical for low SFE alloys, but there exists some controversy about the deformation mechanism controlling the development of this type of . Studying these features is not only interesting from a scientific point of texture ' " ' ' ' view but it is also important with respect to the industrial processing of this material. Furthermore, it is essential to interpret the recrystallization behaviour during annealing, which determines the final microstructure and the ultimate mechanical p r o p e r t i e s . T h e presence of the copper twin orientation nicely illustrates the importance of twinning in the alloy studied. It has been demonstrated (cf. Figure 4) that the copper twin originates from the twinning of the copper orientation. Hirsch et al.' argue that grains with a copper orientation are likely to twin to the copper twin orientation during cold rolling as a consequence of the high Taylor factor for this specific twinning. Subsequent shear banding causes a further rotation towards the Goss orientation, which is also assumed to be metastable and which further rotates to the brass orientation. This m e c h a n i s m requires a high amount of twins to have a clear impact on the macroscopic texture, a condition which is fulfilled in this kind of alloys. However, as can be seen from Figure 3, the brass component already develops in the early stages of deformation and twinning already occurs at strains lower than those normally expected for extensive twinning of the copper component. This suggests that the twinning of other crystallographic orientations might play a role in the development of the brass texture, which seems to be confirmed by the presence of the X c o m p o n e n t (cf. Figure 4). The present results are. however, inconclusive to prove unambiguously that the X component is the parent orientation and the brass component is the twin product and not vice versa. The authors believe that the progressive disappearance of the X c o m p o n e n t and the increased intensity on the brass orientation is an indication that the X component really is a possible parent orientation for the brass orientation. Moreover, the brass orientation is very unlikely to twin under a rolling l o a d , which leads to the conclusion that the X component is relevant in the development of the brass texture by twinning. El-Danaf et al? studied cold rolled α-brass and their results indicate that twinning alone could not explain the formation of a brass texture. They concluded that microscale shear banding is responsible for the formation of the observed brass texture. Our observations also indicate the occurrence of microscopic shear banding, but a reliable quantitative idea of the amount could not be given. Therefore, the relative importance of this mechanism could not be given, but our results clearly point out that slip also plays a role in the texture development. Weidner et a l . attribute the brass texture they observed, for strains higher than ε = 0.69, to the presence of macroscopic shear bands which are typically formed with an inclination of 35° with respect to the rolling direction. These shear bands are also strongly related to preceding mechanical twinning. In the material w e studied, the initiation of macroscopic shear band formation has been observed for ε = 0 . 5 1 , which is in agreement with observations of Hirsch et al. in low S F E materials. However, there is a clear development of a brass type texture; even before these shear bands appear. Therefore, it seems unlikely that the formation of macroscopic shear bands is an important mechanism for the creation of a brass texture. report The texture evolution of our alloy is very similar to what V e r c a m m e n et al. ' ' for their F e - 3 0 % M n - 3 % S i - 3 % A l T W I P alloy. They state that the brass texture in their alloy is a consequence of twinning and they did not mention the influence of microscopic shear bands. At high strains (ε > 1). these authors report the formation of a γ-fiber as a consequence of the alignment of the twinning planes with the rolling plane. It is possible that our alloys will develop a similar texture at higher strains, but due to the limited m a x i m u m rolling force of the laboratory mill used, this could not be verified. 9
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CONCLUSION The evolution of the microstructure during cold rolling of a low SFE austenitic Fe-Mn-C based alloy was studied in detail. T E M and E B S D were used to identify four main deformation mechanisms: micro twinning, slip of dislocations, stacking faults and microscopic shear banding. The effect of macroscopic shear banding, which was observed at large rolling reductions, w a s believed to be negligible because the brass texture already developed before the appearance of these shear bands and because their volume fraction was rather low. Despite the presence of these different deformation m e c h a n i s m s , the microstructure was energetically rather homogeneous.The evolution of the crystallographic texture was studied as well. A brass type of texture, typical for low SFE alloys, was found. It was shown that twinning certainly played a very important role in the development of this texture, but it was also illustrated that slip, through regular dislocation glide or possibly through microscopic shear bands, significantly contributes in the texture development. ACKNOWLEDGEMENTS The authors a c k n o w l e d g e the financial support by Arcelor-Mittal and also thank dr. ir. C. Scott (Arcelor Research. Maizières-lès-Metz, France), for his highly appreciated input. REFERENCES B.W. Oh. S.J. C h o . Y.G. Kim. Y.P. K i m . W . S . Kim and S.H. Hong, Materials Science and Engineering A, 1 9 7 , 147 (1995). S. Allain. J.-P. Château, O. Bouaziz. S. Migot and N. Guelton, Materials Science and Engineering A. 3 9 7 - 3 9 8 . 158 (2004). ' L. Rémy and A. Pineau, Materials Science and Engineering, 2 8 , 99 (1977). L.Bracke, G. Mertens, J. Penning, Bruno C. De C o o m a n . M. Liebeherr, N. Akdut, Metallurgical and Materials Transactions Α. 3 7 A , 307 (2006). G. Frommeyer. U. Brüx and P. N e u m a n n , ISU International, 4 3 , 438 (2003). S. Allain, J.-P. Château and O. Bouaziz. Steel Research, 7 3 , 299 (2002). R.E. Smallman and D. Green, Acta Melallurgica, 1 2 , 145, (1964). F.J. Humphreys and M. Hatherly, Recrystallization and Related Annealing Phenomena, 1995, Pergamon. Elsevier Science Ltd., Oxford, U K . E. El-Danaf. S.R. Kalidindi, R.D. Doherty and C. Necker, Acta Materialia, 4 8 , 2665 (2000). R.K. Ray. Acta Melallurgica et Materialia. 4 3 , 3861 (1995). " G. Wasserman, Zeitschrift für Metallkunde, 5 4 , 54 (1963). S. V e r c a m m e n , B. Blanpain, B.C. De C o o m a n and P. Wollants, Acta Mat, 5 2 , 2005 (2004). A. Weidner and P. Klimanek, Scripta Materialia, 3 8 , 851 (1998). M. Miriglia, P. D a w s o n and T. Leffers. Acta Materialia, 5 5 , 799 (2007). L. Bracke, PhD Thesis, Ghent University, Belgium, 2007. P. Van Houtte, The M T M - F H M Software System, version 2. J. Hirsch. J. Lucke and M. Hatterly, Acta Melallurgica, 3 6 , 2905 (1988). L. Bracke, Κ. Verbeken. L. Kestens and J. Penning, Ada Materialia, submitted (1). " A.K. Vasudevan, W . O . Fricke. R.C. Malcolm, R.J. Bucci, M.A. Przystupa and F. Barlat, Metallurgical Transactions A, 1 9 A . 731 (1988). O. Engler. Acta Materialia. 4 8 . 4 8 2 7 (2000). L. Bracke, Κ. Verbeken, L. Kestens and J. Penning, Acta Materialia. submitted (2). S. V e r c a m m e n , PhD thesis, KU Leuven, Belgium, 2004. 1
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S T O R E D E N E R G Y E V O L U T I O N IN A C O L D - R O L L E D IF S T E E L
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A. Wauthier - , B. Bacroix , T. C h a u v e a u , O. Castelnau . H. Réglé Arcelor Research SA Voie Romaine BP 30320, 57 283 Maizières-les Metz, France L P M T M - C N R S , Université Paris 13 99, Av. J.B. Clément, 93430 Villetaneuse, France
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ABSTRACT During the deformation of low carbon steel by cold-rolling, dislocations are created and stored in grains depending on local crystallographic orientation, deformation, and deformation gradient. Orientation dependent dislocation densities have been characterized by the broadening of the X-ray diffraction lines measured on t w o complementary synchrotron experiments. Different cold-rolling levels (from 3 0 % to 9 5 % thickness reduction) and material states (deformed and restored) have been considered. O n e experiment was dedicated to the study of some specific crystallographic orientations and the other one to get a mapping of stored energy in the Euler space. The advantage of the g a m m a fibre grains during the recrystallisation is quantified, with a stored energy two times higher than the alpha fibre grains. Results were confirmed by an independent analysis of E B S D data. INTRODUCTION Polycrystalline metals, when subjected to heat treatment after cold-deformation, undergo a recrystallisation transformation, where n e w grains form in the deformed microstructure. This induces in the material new crystallographic textures and thus new anisotropic physical and mechanical properties. It is therefore very important to understand the genesis of these textures during recrystallisation. The energy stored during deformation acts as a driving force for the nucleation step of the recrystallisation process and can be measured by high resolution diffraction. Several attempts to estimate the orientation dependent stored energy (Es) have been carried out on steels. On stainless steel, Shin et al. showed the influence of cold-rolling level on Es with an increase up to 8 0 % strain and a slight decrease between 80 and 9 0 % rolling. On low carbon steel, m e a s u r e m e n t s show that Esjiooi < E s j 2 i i | < Es
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EXPERIMENTAL PROCEDURE Material The hot band of a Ti-IF steel with an average initial grain size of 20μιηι has been deformed by cold-rolling with different thickness reductions in a laboratory rolling mill up to 9 3 % . M i c r o s t m c t u r e s have been analyzed essentially by S E M and E B S D techniques. The crystallographic texture of these samples h a s been measured in laboratory by X R D on an Inel diffractometer. Acquisition of high resolution diffraction data Diffraction in Bragg - Brentano geometry has been carried out on two synchrotron beamlines. B M 0 2 at the E S R F (Grenoble. France) and 16.3 at the S R S (Daresbury. U K ) . A large beam cross section ( I m t t f ) and a relatively large beam energy (20keV, which provides an attenuation length of - 5 0 μ η ι for iron) were selected in order to scan a large n u m b e r of grains simultaneously and thus to get statistically representative results. For each measured Θ-2Θ profile. 310 points were acquired, with 10 points describing the peak itself and 300 points describing the intensity decay in the peak tails, important for line profile analysis. The 2Θ scan amplitudes ranged between 0.9° for the less deformed specimen and 2.9° for the most deformed ones. First, a reference p o w d e r of C e 0 was analyzed in order to characterize the resolution of the setup used at B M 0 2 . and an almost constant value (0.018°) of the Full Width at Half m a x i m u m has been recorded in the investigated 2Θ range (figure 1 ). 2
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Figure 1. Values of integral breadth for different analyzed planes (so for different Bragg angle) for the reference powder C e 0 . the less and the most deformed samples ( 2 9 % rolling in dashed lines and 9 3 % rolling). Measures in the centre of the poles figures, corresponding to a m a x i m u m texture intensity for the specimen. 2
A first experiment was dedicated to analyze the six specific crystallographic orientations indicated in figure 2. All of these orientations have significant v o l u m e fractions (more than 5%) and can thus be easily measured by X R D (note that the two orientations D and Ε exhibit higher volume fraction after recrystallisation). For these orientations, diffracted intensity was measured along the diffracting vector (Θ-2Θ scans) on {110}, {200}, [ 2 2 0 } . {211}, {310} and {400} diffracting planes according to their corresponding theoretical positions on the pole figures. But it must be made clear that, since X R D only provides an average measurement along the fiber
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determined by the orientation of the diffraction vector, the corresponding data do not only account for those specific orientations but also for other minor texture components. The second experiment aimed at obtaining a real stored energy distribution in the Euler orientation space. For this, we systematically scanned several pole figures, namely ( 1 1 0 ) . (200) and ( 2 1 1 ) . With a regular grid (indicated in figure 2). the acquisition consisted in 43 peak measures for one fourth of a pole figure.
Figure 2. (a) Specific orientations studied during the first experiment, and (b) their representation in the Euler cut at φ = 4 5 ° . (c) Position of poles measured during the second experiment 2
dedicated to a mapping of the stored energy.
DATA ANALYSIS Analysis of specific crystallographic orientations Data analysis for specific crystallographic orientations A to Ε relies on the modified Warren Averbach ( m W A ) method introduced by U n g a r and B o r b e l y and often used in the literature. The Fourier transform A(n) of the diffraction line I(K) is given, for small values of the Fourier parameter L (which corresponds to a physical length in the crystal), by
\n\A(L)\ = \n\A'{L)\-~pL
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+ (){Κψ ).
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Here, Κ is the n o r m of the diffraction vector ( Κ - 2 sin θ I λ. with λ the wavelength), ρ is the ). The first dislocation density, and Re an outer cutoff radius for the dislocations (R ~ 1/ term A expresses the broadening due to the small size of the diffracting subgrains. and becomes important essentially for sizes smaller than - 1 0 0 nm. It can be identified from two multiple diffraction peaks (such as {hkl} and {2h 2k 21}) measured for the same diffraction vector K. The dimensionless coefficient C is the average contrast factor, which accounts for the anisotropy of the deformation field created by the dislocations. Here, it has been assumed an equal density of edge and screw dislocations uniformly distributed over all possible slip planes, and it has been verified that this assumption does not critically affect the results presented below. Eventually, it is pointed out that line profiles measured on our specimens exhibit a m u c h larger width than those observed on the reference powder of C e O j (figure 1). indicating that, for this first e
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estimation, the instrumental broadening can be reasonably neglected. Consequently, measured data were not corrected from the instrumental response. An example of result is presented in figure 3.
f i g u r e 3. Application of the m W A method o n a specimen cold-rolled to 6 1 % for orientation Β ( ί 1 1 2 j < l 10>). Here, equation (1) nicely matches the Fourier transform of { 1 1 0 ) . {21 ! j . [ 2 0 0 ) . { 3 1 0 ) . | 4 0 0 ) planes, up to 1 = 30 n m . Construction of a Stored Energy Distribution Function ( E s D F ) X R D measurements provide an average over the fiber Γ which contains all grains with a normal to {hkl\
plane parallel to the diffraction vector K . T h e stored energy
E
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s
extracted from a single diffraction peak thus reads
(2)
with g denoting the lattice orientation in the Euler space, and / ( g ) the orientation Distribution Function ( O D F ) . Building a EsDF thus aims at retrieving the distribution of £
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space from the measured data E . This can be done using standard methods for texture analysis. s
The critical step of this inversion is the division by the volume fraction
j _ / ( g )
the position of measurement in the pole figure, since any errors in the determination of this volume fraction leads to large uncertainties for E . s
Here, used w a s made of the O D F estimated
by X R D data obtained on a diffractometer dedicated for texture development, rather than data obtained during the present synchrotron experiment. For each line profile, the stored energy was
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estimated according to Stibitz's formula , which advantage is its simplicity, but which does not account for the anisotropy of dislocation strain field (more accurate analysis is in progress). From the measurement of line profiles for several {hkl} diffracting planes, the product E ( g ) / ( g ) has been obtained by means of the Arbitrarily Defined Cells method used in the s
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software package L a b o t e x ® . A n example of result is given in figure 4 for specimen 6 1 D ( 6 1 % cold-rolling). The O D F , the weighted distribution of stored energy, and the EsDF are shown.
Figure 4. E x a m p l e of different distribution plotted for one specimen, here after 6 1 % cold-rolling, in cf>2=45° Euler cut (obtained with L a b o t e x ® software) (a) Texture of the material. f(g) measured in laboratory' (finer acquisition grid than in synchrotron) - (b) Distribution of £,(g)/(g)
-(C)ESDF'
RESULTS Specific crystallographic orientations Values obtained for specific orientations in the first experiment are presented in figure 5. For a high rolling level ( > 7 6 % ) . the stored energy in C and F orientations seems dominant, with a value in F twice the one in A. This result is in agreement with previous studies ' . But at lower reductions (< 6 1 % ) , the stored energy seems to be more uniformly distributed. Slight m a x i m a are however observed in Β and C texture components. 8
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Figure 5. Stored energy calculated with the modified Warren Averbach method for different orientations and cold-rolling levels.
Stored energy distribution function Section at ψ = 45° of the EsDF for specimens submitted to 2 9 % . 6 1 % . 7 6 % . and 8 8 % cold rolling are presented in figure 6. The wavy aspect of those distributions is due to a too high resolution of the inversion method used by Labotex software. But, independent to this, it is found that, for all investigated specimens, the stored energy is m a x i m u m for the orientation φι = 15°. φ = 3 0 ° . ψ2 — 4 5 ° . except for 7 6 % cold-rolled sample w h e r e a m a x i m u m at φι — 9 0 ° . φ = 6 0 ° . φ = 45° has been obtained. 2
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Figure 6. EsDF calculated from IB measured in laboratory and F W H M in synchrotron.
DISCUSSION It is easy to see that figure 5 and 6 do not exactly match with each other. In order to compare these two different evaluations of stored energy in terms of major texture component.
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we have extracted from the calculated EsDF the stored energy values at the exact orientations A to F. Results are shown in figure 7. It is worth reminding that during this treatment we cannot access Es in a specific orientation, so that only normalized value E ( g ) / E o are s h o w n . E r e p r e s e n t s the total a m o u n t of e n e r g y stored in the m a t e r i a l , different for each cold-rolling level. Here, results clearly indicate a higher stored energy value in the g a m m a fiber (D. E. and F orientations) than in the alpha fiber. Moreover, contrarily to figure 5, no radical change is observed between 6 1 % and 7 6 % . cold rolling level, both distribution of stored energy presenting similar trends. The discrepancies between the t w o set of experiments can be explained as follow. During the first experiment, one m a k e s the implicit assumption that the texture is entirely described by a finite set of ideal orientations, since one attributes each measurement for a given diffracting vector Κ to a single crystal orientation. In doing so. we omit the fact that, if the O D F presents clear m a x i m a (figure 4). it contains m u c h m o r e than a few set of ideal orientations. Furthermore, using the m W A method, several diffraction peaks are used simultaneously, and one therefore mix up many different crystal orientations for the analysis. In that sense, the benefit of the m W A method, w h i c h accounts for the specific displacement field of dislocations, is probably lost since one has to mix up several crystal orientations that do not have much in c o m m o n . For these reasons, we believe that the second method, based on a reconstruction of the E s D F . is more reliable, even if it does not account yet the specific strain field nature created by dislocations. s
iracr
m o c r o
Figure 7: Stored energy for different orientations and cold-rolling levels, extracted from the EsDF shown in figure 6. To further evaluate independently the stored energy distribution in the same specimens, w e have also considered the I m a g e Quality (IQ) of E B S D images, which provides information about dislocation density present in specific orientations (Image Quality is some average m e a s u r e m e n t of Kikuchi line broadening). Note that the IQ does not allow to compare two different specimens quantitatively, since its value depends on the specimen surface quality. However, the I Q distribution in different orientations of a given specimen is relevant for the stored energy distribution. According to figure 8. at low strain ( 2 9 % ) . good quality indexes are obtained for all orientations except for C {111 } < 1 1 0 > . However, after 7 6 % cold rolling, low IQ
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values are obtained for C. D. E. and F orientations, indicating a higher dislocation density along the g a m m a fiber. These results are in good agreement with our X-ray analysis based on the E s D F reconstruction.
Figure 8. E B S D Image Quality profiles for the specific orientations measured in the transverse section on 2 9 % (a) and 7 6 % (b) cold-rolled materials. CONCLUSION The stored energy in a Ti-IF steel has been estimated by different approaches. This first analysis has shown that, for X-Ray based analysis, it is important to consider the whole O D F of the material. Reducing the texture to a small set of specific orientations does not provide coherent results here. However, the method based on a complete reconstruction of the stored energy distribution function provides results in agreement with those based on analysis of Kikuchi diagrams. The influence of the cold-rolling is found to be very important, increasing the stored energy values in the g a m m a fiber, specifically after 6 1 % thickness reduction. The two orientations | 5 5 4 } < 2 2 5 > and [ 2 2 3 j < 5 8 2 > that develop during recrystallisation exhibit a higher energy (at the cold rolled stage) than orientations belonging to the alpha fiber. REFERENCES E. Shin, B-S. Seong. S.H. Park. H-R. Kim. Physica B. 3 8 5 - 3 8 6 . 600-603 (2006). R.L. Every. M. Hatherly. Texture. 4 . 183(1974). A. Borbély, J.H. Driver. T. Ungar. Acta Mater.. 4 8 . 2 0 0 5 - 2 0 1 6 (2000). O. Castelnau. T. Ungar. A. Miroux, T. Chauveau. B. Bacroix, Mater. Sei. Forum. 3 2 1 - 3 2 4 . 720-725 (2000). T. Ungar. A. Borbely. Appl. Phys. Lett.. 6 9 . 2 1 , 3 1 7 3 - 3 1 7 5 . (1996). C R Stibitz. Phys. Rev.. 4 9 . 872 (1937). \(http://\vww. labosofl. com.pl)} N . Rajmohan. Y. H a y a k a w a . J.A. Szpunar. J.H. Root, Acta Mater.. 4 5 . 6. 2485-2494 (1997). A. Samet-Meziou. X.L. Etter. T. Baudin, R. Penelle, Mater. Sc. Forum. 5 5 8 - 5 5 9 , 323-328 (2007). 1
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A C O M P A R I S O N O F T E X T U R E M E A S U R E M E N T S VIA E B S D A N D X - R A Y Stuart I. Wright and Matthew M. Nowell E D A X - T S L . 392 Ε 12300 S. Suite H Draper. Utah 84020 U S A ABSTRACT After introduction of the Electron Backscatter Diffraction ( E B S D ) technique, several studies compared textures measured using manual E B S D measurements of individual grain orientations with those obtained using the conventional X-Ray diffraction pole figure technique to ascertain the n u m b e r of grains required to reach the same statistical reliability. Similar studies were performed after automation of the E B S D technique. Since those early studies, E B S D scanning speeds have increased dramatically. Thus, we revisit the question of statistical reliability of textures measured using E B S D relative to X-ray diffraction. Textures were measured in rolled stainless steel and threaded steel rods. As expected, the results from the two techniques were comparable. Approximately 10,000 grains were found to produce a very good sampling in these materials. In addition, the effect of various microstructural features on texture analysis using E B S D is discussed. INTRODUCTION T h e idea of measuring textures by E B S D was first introduced to the texture community at I C O T O M 8 . Studies done in a semi-automated m o d e showed that for statistical reliability an automated technique would be required . After automation of the technique, several studies sought to determine the n u m b e r orientation m e a s u r e m e n t s required to achieve statistical reliability . These works suggested numbers ranging from 200 to 3000 depending on various factors. The baseline for these determinations was generally results obtained by x-ray diffraction. Since these early studies, E B S D analysis speeds have increased from 0.25 measurements per s e c o n d to over 300. It is n o w practical to sample numbers of grains comparable to those sampled by x-ray . Thus, it is worth revisiting some of the issues regarding statistical reliability. 1
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EXPERIMENTAL DETAILS T h r e e samples were investigated: two low-carbon steel threaded rods ( 9 m m and 15mm in diameter) and a 431 stainless steel sheet sample. The details of the processing history of the rods were unknown. High resolution orientation maps showed that the rods were in a cold worked state as evidenced by the presence of many low angle misorientations. The sheet was cold-rolled and then recovery annealed. It exhibited a classic B C C cold-rolled texture for a BCC material . X-ray texture measurements were performed on a Scintag X-ray system using m o n o c h r o m a t e d Fe Κ - α radiation. (110), (200) and (211) pole figures were measured. The sample was oscillated to average over significant sample volumes, and a relatively long 5 sec. dwell time was used to optimize the data statistics. Data analysis was performed using popLA (preferred orientation package - Los A l a m o s ) utilizing the W I M V ' algorithm. 6 7
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E B S D measurements were performed using an E D A X - T S L O I M system on an FEI XL30 field emission gun scanning electron microscope (FEG S E M ) . The texture analysis was performed using the generalized spherical harmonic series expansion ( G S H E ) m e t h o d " . Details of the scans performed on each sample are reported in table I. For the threaded rods, the coarser scans were obtained by removing every other column and every other row
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within the scan grid essentially doubling the step size. O n the rolled sheet, actual scans at different step sizes were collected over different scan areas. In all cases, the grids are quite coarse so as to sample many grain orientations. In some cases, the step size exceeds the average grain size and thus the n u m b e r of points is roughly equal to the n u m b e r of grains sampled. Table I. E B S D scan details. (The n u m b e r of points listed is the n u m b e r of indexed points with confidence i n d e x values greater than 0.1 after confidence index standardization".) Step Step Grid Grid Sample Sample Grains Grains Points Points (Area in m m ) (Area in m m ) (pm) (pm) 2.5 150,060 134.025 20.0 1.783 1,780 9 m m Rod 9 m m Rod 439 37,471 36.515 40.0 437 5 (1 χ 1) (1 χ D 10.0 7,053 7.014 112 112 80.0 4 99.768 27.681, 32 1,499 1.510 15mm Rod 15mm Rod 361 24,803 15,892 64 360 8 (1.2 χ 1.1) (1.2 χ 1.1) 6,162 5,655 128 98 96 16 1.7 1.264.118 32.197 1.966 42.5 1,955 Rolled Sheet 3.4 315.183 27,725 Rolled Sheet 62.75 904 864 50.177 85 499 497 8.5 19.583 (1.8 χ 1.7) (1.8 χ 1.7) 7,885 7.219 127.5 213 211 21.25 12
RESULTS Figure 1 shows (110) pole figures measured using X-ray diffraction and E B S D for the two threaded rods. There is a strong qualitative agreement between the pole figures - all s h o w i n g moderate (110) near- fiber textures with similar in-plane symmetry. Figure 2 s h o w s (111) pole figures and the φ = 45° section of the O D F for the sheet material measured using X-ray diffraction and E B S D . Once again, the results show strong qualitative agreement between the textures measured by X-ray and E B S D . There is a notable increase in detail at the center of the X-ray pole figure relative to the E B S D pole figure. 2
Figure 1.(110) pole figures obtained by X-Ray and E B S D on 9 m m and 15mm threaded rods. The gray scale is given in multiples of r a n d o m distribution ( M R D ) .
Figure 2. (111) pole figures and φ = 45° O D F sections obtained by X-Ray and E B S D on steel sheet. Sample symmetry has been enforced in the O D F sections but not in the pole figures. 2
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Three approaches are used to compare X-ray results with E B S D results: the first based on comparing actual times-random values from the pole figure and orientation distributions, a second based on volume fractions of specific orientation c o m p o n e n t s , and a third based on satisfying sample symmetry. For the results shown the. the series rank used for the G S H E was / = 2 2 . A detailed examination of the calculation factors involved has been previously reported . 3
Distribution Intensities Figure 3 s h o w s the (110) pole figures for the 9 m m bar measured using X-ray diffraction as well as E B S D and a plot tracking the (110) peak intensity as a function of the number of grain orientations sampled. While the pole figures all show a general (110) fiber texture, even with 52,000 grains sampled, the X-ray and E B S D pole figures do not match exactly. Some of this could be due to the depth sampled - X-rays will sample volume to a certain depth depending on the absorption for the materials investigated (~8 - 20 p m in this case) whereas E B S D measures grain orientations strictly at the sample surface (within approximately 20nm). Differences in areal coverage also contribute to variances in the measured textures. Differences can also be attributed to the method of data reduction ( W I M V versus G S H E ) . As the rod samples exhibit a (110) liber texture, the (110) peak intensities are used to compare the X-Ray and E B S D textures. It appears that once 10.000 grains are sampled, the texture is well characterized.
Figure 3 - (110) pole figures for 9 m m rod determined for X-Ray and for decreasing numbers of E B S D measurements along with a plot racking the (110) peak intensities. Texture can be m o r e rigorously compared using O D F intensities. This has been done for the rolled sheet material and the results shown in figure 4. For this comparison, o r t h o t o p i c sample symmetry has been enforced. The differences are calculated using a sum of squared differences approach for the O D F at each point in a 5°x5°x5° discretization of Euler Space. Once again, it appears that the differences reach a stable m i n i m u m at about 10.000 grains.
Figure 4 - q>2 = 45°sections of O D F s from rolled sheet calculated using varying number of E B S D measurements as well as from an X-ray measurement.
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V o l u m e Fractions Textures can also be c o m p a r e d using volume fractions of specific texture components. For the rod material this is accomplished by determining the fraction of material having (110) planes aligned with the sample service within in a given tolerance - 10°. For the E B S D data, the volume fraction calculated directly from the raw data. The orientations satisfying the 10° tolerance are counted and the count divided by the total n u m b e r of orientations in the scan. For X-ray textures an approach approximating that outlined by C h o and R o l l e n ' was used. The first step is to create a set of discrete orientations weighted by the O D F . Then follow the approached outlined for the E B S D data except taking into account the weights. This procedure w a s used to quantify the (110) fiber volume fractions for the threaded rods and the conventional texture components for rolled B C C materials for the sheet material. It was found that good agreement was achieved at less than 5000 measurements; however, the volume fractions were typically higher for the E B S D measurements. Other a u t h o r s ' ' have noted that volume fractions differences can arise from differences in the calculation methods used. 4
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Sample Symmetry Quantifying the degree of sample symmetry in the texture results provides another approach to ascertaining the n u m b e r of measurements needed to achieve the same statistical reliability as in X-ray texture measurements. Sample symmetry in texture analysis is not a true symmetry in the sense of crystal symmetry but a statistical symmetry generally arising from symmetry in the processing of a material. For example, in rolled sheet, we expect to see orientations related to orthotropic symmetry with the same frequency due to the orthotropic symmetry of the rolling process. This symmetry is generally most obvious in pole figures where the horizontal and vertical axes of the pole figure will coincide with mirror planes in the sample. The departure from perfect symmetry in the pole figures can thus be used as a measure of the statistical reliability of the individual orientation measurements. One method to measure the degree of sample symmetry is to c o m p a r e the intensities at symmetric positions in the pole figure. This was done using a sum of squared differences approach for the threaded rods. The results showed that once again 10.000 grains must be sampled to reach the same statistical relevance as in the X-Ray data. For rolled sheet data a m o r e rigorous approach was used based on the C-coefficients themselves. If the material exhibits perfect sample symmetry then some of the C-coefficients should be zero. T h e sum of the coefficients appropriately normalized (per the texture i n d e x " ) then serves as a measure of departure from symmetry. Results from these calculations suggest 20.000 measurements are needed to reach the same level of sample symmetry as in the X-Ray measured textures. DISCUSSION Other factors besides simply the n u m b e r of measured orientations should be addressed as well. This includes assessing the general quality of the data and the impact of microstructure on scanning strategies for measuring texture. Data Quality It is also important to consider the pattern indexing success rates for the automated E B S D measurements. If patterns from certain orientations are repeatedly misindexed this can lead to biases in the measured textures. Various post-processing procedures have been devised to clean
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up E B S D scan d a t a . The general objective is to identify isolated non-indexed or mis-indexed points within a grain or at grain boundaries and to modify the orientations of these points to match those of the surrounding neighbors. These methods often fail w h e n using E B S D to measure texture as the scans tend to be so coarse as to prevent the grains from being reconstructed from the scan data. Nonetheless, in order to study clean up effects in cases were the step size is small e n o u g h to allow the grains to be reconstructed, a relatively small dataset (9100 points) containing 150 grains w a s measured on a low carbon steel sample. The average n u m b e r of points per grain was 60. At each scan point, the E B S D pattern was saved. The scans were re-run based on the saved patterns with random noise added. This was done until 2 0 % , 4 0 % , 6 0 % and 8 0 % of the pixels in the pattern contained r a n d o m intensities. After the completion of each rescan the data was processed to clean up the measurements. The percentage of points modified for each of the noise levels was 1.8% for the original data, and 1 1 % , 7 0 % , 7 8 % and 9 7 % for the scans at increasing noise levels. Due to the sparseness of the data, the textures were calculated using L = 16 and Φο = 5°. A s expected, as the scan data quality decreases the measured textures increasingly deviate from the real texture. While the clean up procedure was found to improve the texture results, a threshold is reached where the cleanup can produce a dramatically different texture. Surprisingly, this was not observed until the 8 0 % noise level was reached. Microstructure O n e could easily argue that the more m e a s u r e m e n t s collected the better. However, such assumptions should not be made without first considering the sample itself. Consider for example, the threaded rod sample. In this case, both the E B S D and X-Ray measurements were taken near the center of the rod. If the entire rod cross-section is sampled, then any inhomogeneity in the texture from the centerline of the rod out to the threads will be included in the analysis. In fact, the threaded rod does exhibit a texture gradient. The ability of E B S D to measure spatially specific orientations enables the texture gradient to be measured. Figure 5 shows results obtained by performing a set of combined stage and beam scans of the sample. This resulted in nearly 3 million individual orientation measurements. The step size between the m e a s u r e m e n t s w a s 4 microns. Each of the squares shown in the figure contains 40.000 individual orientation measurements. Besides long range gradients, other microstructural features must be considered when using E B S D to measure texture. Consider a material with a bimodal grain size distribution. E B S D can differentiate the between the large grain and the small grain textures; however, it is imperative to consider the size of the large grains when measuring the texture(s). Basing the area to be scanned on the average grain size instead of the average size of the large grains could lead to sample area insufficient to capture the 10,000 grains needed for a statistically reliable texture measurement. Such ideas should also be considered w h e n attempting to measure textures of individual phases in a multiphase material. For example, consider a material with three phases where the average grain size and volume fraction of each phase is approximately equal. In order to reach the proposed critical value of 10,000 grains for each phase, an area three times larger need be sampled than for a single phase material. It is m o r e likely that the individual phases have very different grain size distributions and volume fractions. It may be more efficient to design sampling schemes for each individual phase in order to achieve a statistically relevant sampling. It is also important to consider orientation relationships between phases. For example, consider a ferritic steel sample with finely dispersed retained austenite. Since the orientations of the retained
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austenite grains are likely related in a specific way to the ferrite colonies in which they reside, a statistically representative n u m b e r of ferrite colonies needs to be sampled in order to accurately measure the texture of the austenite despite its small grain size.
Figure 5 . ( 1 1 0 ) pole figures for a combined stage and beam E B S D scan of a threaded steel rod cross section (image quality map). E B S D can also be used to study correlations between specific microstructural features and orientation in a statistical m a n n e r using the tools of texture analysis. Consider a mineral sample containing olivine, quartz and enstatite. The enstatite grains are elongated perpendicular to the enstatite-quartz interface. Is there a relationship between the elongation direction and a particular crystallographic direction? In order to study this, the orientation of each measurement point is rotated about an axis normal to the elongation direction of the grain the measurement point belongs to. Once the data has been rotated the texture of the enstatite is calculated in the usual manner. Figure 6 shows an inverse pole figure from these calculations. The results show a weak preference for the grains to have [nwO] directions aligned with the elongation direction of the grains with a specific preference for [010]. It should be noted that it is assumed the elongation direction of the grains is in the plane. Three-dimensional analysis w o u l d need to be performed to find the true elongation direction.
Figure 6. E B S D m a p colored according to phase and an elongation direction inverse pole figure for the enstatite phase.
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CONCLUSION From the various comparisons in the materials studied here, it is concluded that 10.000 grains provides a texture characterization nearly equivalent to that obtained by X-ray diffraction. The observed differences appear to be more related to various analytical factors as opposed to a not sampling enough grains. W e emphasize that the orientations of 10,000 grains is proposed, not simply 10,000 orientation measurements. A 1 million point scan could be concentrated in only a few grains if performed with a small step size relative to the grain size. Thus, a-priori knowledge of the grain size is helpful w h e n setting up an E B S D scan. Coarse grained materials may require multiple scan areas to collect a representative set of orientation measurements. The ability of E B S D to obtain spatially specific orientation information gives it a specific advantage over X-ray in many cases. For example, E B S D can differentiate the textures specific to the large and small grains in materials with a bimodal grain distributions. E B S D is also well suited to characterizing texture gradients. While this can be achieved by X-Ray. it generally requires complex sample sectioning. Care must be exercised to ensure in all EBSD measurements that enough data is acquired over large enough areas to be representative. An inherent advantage to measuring textures using E B S D over X-ray diffraction, is that the odd parts of the O D F can be directly determined. For some textures, the contribution of the odd portion of the O D F can be significant". Another advantage of the E B S D technique is that volume fractions of specific texture components can be calculated directly from the raw data. The differences between textures measured using X-ray diffraction and EBSD reported are on the scale of those observed in comparison between X-ray results and other techniques such as neutron diffraction ' . What may be m o r e critical than the actual one-to-one comparison of intensities or volume fractions is the actual effect of the texture differences on the prediction of a materials property of i n t e r e s t ' . 20
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ACKNOWLEDGEMENTS The authors gratefully acknowledge the following for their contributions to this work. The X-Ray diffraction m e a s u r e m e n t s were performed by John Bingert of Los Alamos National Laboratory. The samples were prepared and the E B S D scans run by Nathan Wright and John Carpenter of E D A X - T S L . The mineral sample was provided by Karsten Kunze of ΕΤΗ Zürich and scanned by Rene de Kloe of E D A X - T S L . Helpful discussions with Bevis Hutchinson of the Swedish Institute for Metals Research and Austin Day are also acknowledged. REFERENCES Ό . J. Dingley, On Line Microtexture Determination Using Backscatter Kikuchi Diffraction in a Scanning Electron Microscope, Proceedings of the Eighth International Conference on Textures of Materials (ICOTOM 8), J. S. Kallend and G. Gottstein (Eds.). 189-94 (1988). S . I. Wright and B. L. A d a m s , An Evaluation of the Single Orientation Method for Texture Determination in Materials of Moderate Texture Strength, Textures and Microstructures, 12, 6576(1990). S . I. Wright. M. M. Nowell and J. F. Bingert, A Comparison of Textures Measured Using XRay and Electron Backscatter Diffraction, Metall. Mat. Trans. A, 38, 1845-55 (2007). S . I. Wright and B . L. A d a m s , Automatic-analysis of electron backscatter diffraction patterns. Metall. Trans. A, 23, 759-67 (1992). C . Detavernier and C. Lavoie, Texture in thin films. Mater. Sei. Forum. 495-497. 1333-42 (2005). 2
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"R. K. Ray. J. J. Jonas and R. E. Hook. Cold Rolling and annealing textures in low carbon and extra low carbon steels. International Materials Reviews. 3 9 , 129-72 (1993). A . D. Rollett and S. 1. Wright. Typical Textures in Metals, Texture and Anisotropy. U. F. Kocks, C. T o m é and H.-R. Wenk (Eds.), 178-239 (1998). 7
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J . S. Kallend, U. F. Kocks, A. D. Rollett and H.-R. Wenk, Operational Texture Analysis. Mat. Sei. Eng.. A 1 3 2 . 1-11 (1991). S . Matthies and G. W. Vinel, On the Reproduction of the Orientation Distribution Function of Texturized Samples from Reduced Pole Figures using the Conception of a Conditional Gauss Correction, phys. stat. sol. (h). 112. K l 11-K20 (1982). S . Matthies, H. R. Wenk and G. W. Vinel, Some Basic Concepts of Texture Analysis and Comparision of Three M e t h o d s to Calculate Orientation Distributions from Pole Figures, J. Appl. Cryst.. 2 1 , 285-304 (1988). " H . - ! . Bunge. Texture Analysis in Materials Science. Mathematical Methods. Butterworths, London (1982). D . P. Field. Recent advances in the application of orientation imaging, Ultramicroscopy, 6 7 , 19(1997). " M . M. Nowell and S. I. Wright. Orientation effects on indexing of electron backscatter diffraction patterns. Ultramicroscopy, 103, 41-58 (2005). .l.-H. C h o . A. D. Rollett and Κ. H. O h . Determination of a Mean Orientation in Electron Backscatter Diffraction M e a s u r e m e n t s . Metall Mat. Trans. A, 3 6 A , 3427-38A (2005). " U . F. Kocks. J. S. Kallend and A. C. Biondo. Accurate Representations of General Textures by 1 4 - 1 8 . 199-204 (1991). a Set of Weighted Grains. Textures and Microstructures. H . Jin and D. J. Lloyd, T h e reduction of planar anisotropy by texture modification through asymmetric rolling and annealing in A A 5 7 5 4 , Mater. Sei. Eng. A, 3 9 9 , 358-67 (2005). P . Sonnweber-Ribic. P. Gruber. G. Dehrn and Ε. Arzt, Texture transition in Cu thin films: Electron backscatter diffraction vs. X-ray diffraction. Acta Mat., 5 4 , 3863-70 (2006). S . 1. Wright, R a n d o m thoughts on non-random misorientation distributions, Materials Science and Technology. 22, 1287-96 (2006). R . Penelle. T. Baudin, P. Paillard and L. Mora, Characterization of Recrystallization Textures in Fe-3%Si Sheets by E B S P : C o m p a r i s o n with X-ray Diffraction, Textures and Microstructures, 597-610(1991). H . - R . Wenk. Standard Project for Pole-Figure Determination by Neutron Diffraction. J. Appl. Cryst.. 2 4 . 9 2 0 - 7 ( 1 9 9 1 ) . H . E. Vatne, A. Oscarsson, O. Engler. H. Weiland, A. Reeves. P. S. Bate, P. Van Houtte, X.-H. Zeng. C. Johnson, J. J. Fundenberger, et al.. Results from a R o u n d Robin Test on Textures Measured by X-Ray Diffraction. Proceedings of the Eleventh International Conference on Textures of Materials (ICOTOM-11), Z. Liang. L. Z u o and Y. Chu (Eds.), 191-6 (1996). K . Mehnert, H. S. Ubhi and A. P. Day, Comparison of texture data measured by E S B D and conventional x-ray diffraction.. Proceedings of the Twelfth International Conference on Textures of Materials. J. A. Szpunar (Ed.), 217-22 (1999). 9
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U . F. Kocks. S. I. Wright and A. J. Beaudoin, The Sensitivity of Yield Surface Predictions to the Details of a Texture, Eleventh International Conference on Textures of Materials, Z. Liang, L. Z u o and Y. Chu (Eds.). 763-8 ( 1996).
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T H E M E A S U R E M E N T A P P R O A C H O F X - R A Y R E S I D U A L S T R E S S IN FE-NI ALLOY WITH DIFFERENT TEXTURES
a
a
a
b
b
c
Q i u l i n W u * , X i a o j u n H u , Z u m i n g F u , H a i y a n Z h a o , B i n g Z h a n g . Yong W a n g , Wei 3
Fang , Limin Li
a
Beijing BeiYe functional m a t e r i a l s c o r p o r a t i o n , Beijing 1 0 0 1 9 2 , C h i n a ;
b
c
a
D e p a r t m e n t o f m e c h a n i c a l e n g i n e e r i n g , T s i n g h u a U n i v e r s i t y , Beijing 1 0 0 0 8 4 , C h i n a ; N a t i o n a l k e y l a b o r a t o r y for r e m a n u f a c t u r i n g . a c a d e m y o f a r m o r e d force e n g i n e e r i n g ,
Beijing 1 0 0 0 7 2 , C h i n a
ABSTRACT Fe-Ni a l l o y s h a v e e x t e n s i v e industrial a p p l i c a t i o n s d u e to their low coefficient thermal e x p a n s i o n and h i g h s t r e n g t h . T h e F e - N i alloy plate, sheet, rod, and foil h a v e different t e x t u r e s , and in g e n e r a l , the residual stress is not m e a s u r e d in t h e t e x t u r e d metal. In the p r e s e n t study, t w o m e a s u r e m e n t a p p r o a c h e s , X - r a y diffraction ( X R D ) t e c h n i q u e and flexural strain a n a l y s i s , o f X - r a y residual stress in F e - N i alloy with different t e x t u r e s w e r e d e v e l o p e d . T h e e x p e r i m e n t a l r e s u l t s s h o w e d that X R D t e c h n i q u e can be a p r o v i s i o n a l w a y t o m e a s u r e X - r a y residual stress in F e - N i alloy with different t e x t u r e s v a l i d a t e d with flexural strain a n a l y s i s . INTRODUCTION F e - N i a l l o y s h a v e e x t e n s i v e industrial a p p l i c a t i o n s d u e t o their low coefficient t h e r m a l e x p a n s i o n a n d high strength [ 1 , 2 ] . T h e r e a r e different t e x t u r e s in t h e plate, sheet, rod and foil o f F e - N i a l l o y s , k n o w n as w e a k t e x t u r e and strong t e x t u r e that affect their p r e s s i n g . G e n e r a l l y , X - r a y diffraction ( X R D ) h a s b e e n u s e d for m a n y y e a r s as a reliable tool t o d e t e r m i n e residual s t r e s s e s in c r y s t a l l i n e m a t e r i a l s in a n o n - d e s t r u c t i v e w a y . e s p e c i a l l y in m a t e r i a l s w i t h o u t very s t r o n g c r y s t a l l o g r a p h i c t e x t u r e s [ 3 ] . T h e t e x t u r e can b e d e s c r i b e d b y t h e o r i e n t a t i o n d i s t r i b u t i o n function ( O D F ) [ 4 - 7 ] . T h e O D F a n a l y s i s can b e e x p l o i t e d to reveal t h e c l e a r l y different t e x t u r e s in F e - N i alloy sheets, such as g o s s - t y p e t e x t u r e , b r a s s - t y p e t e x t u r e and c u b i c - t y p e t e x t u r e . In a d d i t i o n , c a s e s in w h i c h a w e a k or s t r o n g c r y s t a l l o g r a p h i c t e x t u r e can p r e s e n t t h e r e s i d u a l stress after cool rolling and a n n e a l i n g p r o c e s s e s . T h e r e f o r e , t h e r e is a q u e s t i o n h o w t o m e a s u r e t h e residual s t r e s s e s in F e - N i a l l o y s with different t e x t u r e s . In the p r e s e n t study, t w o m e a s u r e m e n t a p p r o a c h e s , X - r a y diffraction ( X R D ) t e c h n i q u e and flexural strain a n a l y s i s , o f X-rayresidual stress in Fe-Ni a l l o y with different t e x t u r e s w e r e d e v e l o p e d . T h e e x p e r i m e n t a l results s h o w e d that X R D t e c h n i q u e w i t h o u t m o d i f i c a t i o n can b e a p r o v i s i o n a l w a y to m e a s u r e X - r a y residual stress in Fe-Ni alloy w i t h different t e x t u r e s validated with flexural
strain a n a l y s i s .
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MATERIALS AND METHODS Three t y p e s o f F e - N i alloy m a t e r i a l s , labeled T y p e - A , T y p e - B a n d T y p e - C , with t h e s a m e F C C m i c r o s t r u c t u r e a n d different t e x t u r e s w e r e o b t a i n e d from B e i j i n g
BeiYe
F u n c t i o n a l M a t e r i a l s C o r p o r a t i o n a n d used in t h e e x p e r i m e n t s . T h e r e are t w o p r o c e s s states p r e s e n t in t h e s a m p l e s , t h e c o o l - r o l l i n g state a n d t h e a n n e a l i n g state. Firstly, the s a m p l e s w e r e p l a c e d in flexural d e f o r m a t i o n to o b t a i n t h e d e f o r m a t i o n strain a n d stress. Different g r a d e s o f flexural d e f o r m a t i o n o f t h e s a m p l e w e r e used t o o b t a i n
different
flexural s t r e s s e s o f t h e s a m e s a m p l e s , n a m e d as D e f o r m - 1 . D c f o r m - 2 a n d D e f o r m - 3 . The d e f o r m e d s a m p l e s w e r e t h e n h e l d for X R D a n a l y s i s . T h e reflection m e t h o d w a s used to m e a s u r e i n c o m p l e t e p o l e figure a n d c o m p u t e t h e true O D F w i t h S i m e n s X - r a y diffraction i n s t r u m e n t . An X - r a y diffraction s p e c t r u m w a s o b t a i n e d o p e r a t i n g at 4 0 k V and 4 0 m A w i t h C u K a p e a k o f 1.54Â a l o n g a solid state d e t e c t o r . T h e residual s t r e s s e s in the s a m p l e s w e r e m e a s u r e d u s i n g C h i n e s e H D - A S T - 3 5 0 A X R D i n s t r u m e n t at 2 2 kV a n d 6 m A with C r K a .
RESULTS AND DISCUSSIONS T h e c o m p l e t e O D F a n a l y s i s is u s e d to a n a l y z e t h e t e x t u r e o f different state with t h e s a m e F e - N i alloy, as s h o w n
in F i g u r e
1. T h e s t r o n g g o s s - t y p e t e x t u r e a n d
weak
b r a s s - t y p e t e x t u r e can b e c l e a r l y seen in F i g . 1(a) in t h e s a m p l e s o f T y p e - Α . the w e a k g o s s - t y p e t e x t u r e , w e a k b r a s s - t y p e t e x t u r e a n d w e a k c u b i c - t y p e t e x t u r e in t h e s a m p l e s o f T y p e - B a n d T y p e - C in F i g . 1(b). In e a c h t e x t u r e state, the s m a l l figures a r e listed w i t h the p o l a r a n g l e interval of 5° as s h o w n in F i g s . 1(a) a n d 1(b).
Figure I
X - r a y diffraction
the O D F o f l h e texture: (a) c o o l rolling slate, (b) annealing state.
( X R D ) s p e c t r a o f t h r e e t y p e s o f F e - N i alloy m a t e r i a l s h a v e b e e n
s h o w n in F i g u r e 2. With t h e O D F a n a l y s i s a n d the c r y s t a l l i n e p e a k in t h e X R D spectra, flexural stress c a n be c a l c u l a t e d a n d o b t a i n e d directly from flexural strain a n a l y s i s , as s h o w n in F i g u r e 3 .
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Figure 2 . X R D patterns of the three of Fe-Ni alloy materials: (a) Type-Α sample, (b) Type-B sample, and (c) Type-C sample. Materials Processing and Texture
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Figure 3
Flexural stresses obtained from flexural strain analysis and X R D technique: (a) T y p c - A s a m p l e , (b) T y p e - B s a m p l e , and (c) T y p e - C sample.
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In F i g u r e 3 , t h e flexural stress o b t a i n e d from flexural strain a n a l y s i s a n d c a l c u l a t e d from t h e O D F a n d X R D s p e c t r u m s i n c r e a s e w i t h t h e a g g r a v a t i n g flexural d e f o r m a t i o n . T h e r e is a s i m i l a r i n c r e a s e t r e n d o b t a i n e d from X R D t e c h n i q u e a n d flexural strain a n a l y s i s , as s h o w n in F i g s 2 a n d 3 . In g e n e r a l , t h e w e a k t e x t u r e can be m e a s u r e d with
XRD
t e c h n i q u e a n d t h e s t r o n g t e x t u r e m a y b e not. T h e flexural strain a n a l y s i s m a y be a p o s s i b l e a p p r o a c h t o o b t a i n t h e c h a r a c t e r i s t i c s o f w e a k or s t r o n g t e x t u r e in e n g i n e e r i n g application.
CONCLUSION In t h e p r e s e n t study, t w o m e a s u r e m e n t a p p r o a c h e s . X - r a y diffraction ( X R D ) t e c h n i q u e a n d flexural strain a n a l y s i s , o f X - r a y r e s i d u a l stress in F e - N i a l l o y w i t h
different
t e x t u r e s w e r e d e v e l o p e d . T h e e x p e r i m e n t a l r e s u l t s s h o w e d that X R D t e c h n i q u e can be a p r o v i s i o n a l w a y to m e a s u r e X - r a y residual stress in F e - N i alloy with different t e x t u r e s v a l i d a t e d w i t h flexural strain a n a l y s i s .
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INFLUENCE OF GRAIN ORIENTATION N O N - O R I E N T E D SILICON S T E E L S H E E T
ON M A G N E T I C
A G I N G B E H A V I O R OF A
L.Xu, W.Mao, P.Yang, H.Feng Department of Materials, State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Xue-Yuan Road 30, 100083 Beijing, China. (e-mail:lingfengxu@ 126.com)
ABSTRACT The influence of grain orientation on magnetic aging behavior of a non-oriented silicon steel sheet was investigated. It was observed that the relative core loss increment went up with increasing {111} texture, as well as with m o r e second-phase particles which precipitated dispersively during the aging treatment (200°C for 24h) and created a pinning force against domain wall m o v e m e n t during magnetization. It was deduced that the magnetic driving force depends on grain orientation, that in which grains with <100> parallel to the magnetic field, is the most favorable texture component for reducing the core loss increment after aging. The magnetostrictive effect of the non-oriented silicon steel sheet may promote precipitation of second-phase particles. However, the sheet texture was not strong enough to induce notable diffences in magnetostriction a m o n g the observed samples. KEY WORDS non-oriented silicon steel, magnetic aging, texture, magnetostriction . INTRODUCTION The temperature increases during operation of a silicon steel core because a portion of the electrical energy is dissipated as heat. This dissipated energy is k n o w n as core loss. Non-oriented silicon steels often undergo a slow change and degradation in magnetic properties during their service life. This degradation is mainly associated with an increase in core loss as the temperature of steel sheets increases. This p h e n o m e n o n is k n o w n as magnetic aging. The important factor causing magnetic aging in silicon steel sheets is usually the precipitation behavior of the supersaturated alloying elements in the matrix. The precipitated second-phase particles will hinder domain wall m o v e m e n t and cause an increase in hysteretic loss . The domain wall m o v e m e n t is orientation dependent, which is related to the magnetization anisotropy of silicon steels. Therefore the sheet texture may affect the magnetic aging characteristics in electrical steels. Moreover, non-oriented silicon steels are subjected to alternating and rotating magnetic fields, which lead to expansion and contraction, i.e. magnetostriction of the magnetic domains inside the steel s h e e t s . Therefore, because of the magnetostriction, undesired acoustic noise is sometimes produced and also varies with the sheet texture . The magnetostriction may even promote the precipitation of the second-phase particles mentioned above. However, the relationship between sheet texture and magnetic aging has not been satisfactorily investigated thus far. Therefore some new observations about the influence of grain orientation on the magnetic aging behavior of a non-oriented silicon steel are presented in this paper. 1
2
3
EXPERIMENTAL PROCEDURE Three samples 0 . 5 x 3 0 x 3 0 0 m m (in the N D x T D x R D ) were cut from strips of a commercial 3
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non-oriented electrical steel, of which the chemical composition is listed in Table 1. The critical temperature A ι was determined to be about 1 0 0 0 - 1 0 2 0 " C . Samples A and Β were annealed at 1050"C for 80 seconds and sample C at 1000°C for 40 seconds followed by air cooling. Then, the three samples were annealed at 200°C for 24h in v a c u u m to simulate the aging process. The magnetic properties along the rolling direction before and after the aging treatment were determined by a N I M - 2 0 0 0 E magnetic test device. Incomplete {110}, (200} and {112} pole figures of the samples were measured using a Siemens D 5 0 0 0 X-ray diffractormeter. after which the orientation distribution function ( O D F s ) were calculated. c
Table 1 Chemical composition of the non-oriented steel (weight percent)
c
Si
Mn
Ρ
S
Al
Ν
0.003
1.15
0.35
0.01
0.005
0.002
0.0012
The optical mierostructures of the three samples are s h o w n in F i g . l . The average grain sizes are 73pm, 7 1 p m and 3 5 p m respectively. The magnetic properties measured along the rolling direction before and after the aging treatment are listed in Table II. According to the O D F sections shown in Fig.2. the main texture component of the three samples is a { I I I } / / N D component, while a weak ) 100J//ND component appears as well. However, the density of the {111} texture in sample C is lower than those in samples A and B . and the {111} texture in sample Β appears to be the strongest (Fig.2).
Fig.l Optical mierostructures of the sheet samples
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Sample C Orientations in φ2=45" section Fig.2 (p2=45° O D F sections from samples Α , Β and C (levels: 1. 2. 4. 6) Table II Magnetic properties before and after the aging treatment and Samples A Core loss before aging. P, , W/kg 4.268 Core loss after aging. P i , , W / k g 5.051 Core loss increment. Δ Ρ 1 . 5 , W/kg 0.783 Relative core loss increment 18.3% Magnetic induction before aging, B50. Τ 1.72 1.72 Magnetic induction after aging. B . Τ Average grain size, p m 73 Average orientation factor in R D . m 0.848 Average magnetostrictive coefficient in R D , λ, χ 10"° -3.2308 5
5
f0
the average Β 4.307 5.271 0.964 22.4% 1.71 1.71 71 0.840 -3.6657
grain size C 4.819 4.869 0.05 1.0%
1.75 1.75 35 0.855 -2.7503
INFLUENCE OF SHEET M I C R O S T R U C T U R E ON THE MAGNETIC PROPERTIES It is k n o w n that < 100> and < 111 > are the easiest and the hardest magnetization directions in silicon steels respectively . T h e <100> lie in the {100j plane, but not in the [111} plane. Therefore, it is observed that the magnetic induction B of sample C is higher than those in samples A and Β due to its lower density of {111} texture components. T h e hysteresis loss of steel 4
5 0
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Influence of Grain O r i e n t a t i o n o n M a g n e t i c A g i n g of a N o n - O r i e n t e d Silicon S t e e l S h e e t
sheets increases with decreasing grain size w h e n the grain diameter is below ΙΟΟμητ. because the grain boundaries pin the domain wall movement. Therefore the P[ 5 core loss of sample C is higher than those of samples A and Β (Table II). though they have the same chemical composition. The Β jo magnetic induction of the samples is almost unchanged after the aging treatment because there is no significant change in sheet texture. However, there are obvious increases in core loss, namely 18.3%, 2 2 . 4 % and 1 % . after aging in samples Α. Β and C respectively. It is known that the solubility of carbon in ferrite is 0.00023%C at 400°C and less than 0.0000007%C b e l o w 2 0 0 ° C . Therefore, the carbide will precipitate in the form of fine particles during long term aging of the supersaturated ferrite around 200°C' . Samples A and Β were annealed over Aci at 1050°C. which resulted in the transformation from ferrite to austenite. where the MnS (concerning the S content in table I) in the matrix would fully dissolve . However, sample C was annealed at 1000°C, where the M n S in the matrix could not be fully dissolved. It is understandable that full dissolution of M n S in advance would induce dispersive precipitation of very fine MnS particles during the aging treatment more than the case of incomplete M n S dissolution''. The fine dispersive M n S particles would inhibit drastically the movement of magnetic domain walls and increase the core loss, which explains the higher core loss increments in samples A and Β than that in sample C (Table II). a
7
8
I N F L U E N C E OF G R A I N O R I E N T A T I O N S A N D T E X T U R E O N T H E M A G N E T I Z A T I O N PROCESS Magnetic d o m a i n s form in ferromagnetic materials due to spontaneous magnetization, and their magnetization vectors in electrical steel are almost all along the <100> direction. The magnetic domains in steel sheets consist of 90° auxiliary d o m a i n s and 180° stripe domains, of which the 180° stripe d o m a i n s are the major component. The magnetization process of electrical steels in a magnetic field consists of two basic procedures, i.e. the movement of magnetic domain walls and the rotation of the d o m a i n s . The main magnetization procedure is the wall m o v e m e n t of magnetic d o m a i n s , i.e. of the 180° stripe d o m a i n s under low magnetic field intensity, which is the c o m m o n case for examining the magnetic properties of electrical steel sheets. Accordingly, the force F driving the wall movement of 180° stripe domains can be calculated as : 4
10
l0
F=2uoM>H
(1)
where μο. M and H are vacuum permeability, magnetization vector of unit domain and vector of external magnetic field respectively. It is obvious that the driving force F is grain orientation dependent and also influenced by the angle α between the vectors M and H, i.e. between the <100> crystal directions and the direction of the external magnetic field. Therefore, equation 1 can be transformed into: F=2u<)MHcosa=2uoMHm
(2)
where m = c o s a is introduced as the orientation factor. The smallest angle between one of the three <100> directions and the vector of magnetic field refers correspondingly to the highest orientation factor m as well as the highest force driving the wall movement of 180° stripe domains. The driving force for each individual grain and the corresponding expended energy depend on grain orientation, in which the expended energy contributes to the hysteretic loss of the sheet during the magnetization process of polycrystalline non-oriented silicon steels under a given
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Influence of G r a i n O r i e n t a t i o n o n M a g n e t i c A g i n g of a N o n - O r i e n t e d Silicon S t e e l S h e e t
external magnetic field- Therefore, sheet texture will influence the core loss, and the corresponding effect could be expressed as the average orientation factor in calculated from the texture. Supposing that the magnetic field vector is along the sheet rolling direction, the calculation method for the average orientation factor fn is similar to that described in [11]. The orientation space is divided into 936 identical subspaces represented by orientation gi ( i = 1 . 2 , 3 936). and the orientation factor nij refers to orientation g;. of w h i c h the orientation density fj can be calculated individually by using the O D F data. The average orientation factor m of a polycrystalltne steel sheet can be obtained by summarizing all grain orientation contributions, m, from the 936 subspaces to the m value, where the orientation density f, is used as the weighting coefficient. The calculated m values of samples Α. Β and C are 0.848. 0.840 and 0.855 respectively (Table II), and then the magnetization driving force F can be calculated according equation (2), where the m is replaced by m . The driving force F \ of sample A is higher than that F of sample B . ( F > F ) . however lower than that of sample C ( F < F ) . D
A
B
A
C
I N T E R A C T I O N E F F E C T O F S E C O N D - P H A S E P A R T I C L E S A N D T E X T U R E ON T H E SHEET CORE LOSS As mentioned above, second-phase particles could precipitate dispersively from supersaturated ferrite during long time aging at 200°C, of which the pinning effect against domain wall m o v e m e n t will decrease the driving force for magnetization. The decrease of driving force can be expressed by an obstacle force R which is equal for samples A and Β because they experienced the s a m e annealing and aging treatments. Then, the pure driving force becomes F - R and F - R for samples A and B . where F - R > F - R is valid. The ratio between the two pure A
D
A
D
< 1. which also represents the driving force
driving forces is expressed as
ratio before the aging treatment with R = 0 . It is obvious that this ratio will decrease with increases in R. indicating that the pure driving force declines more rapidly and induces a higher hysteretic loss increment in sample Β than that in sample A during aging. The analysis discussed above shows that a favorable sheet texture can directly reduce the hysteretic loss and the hysteretic loss increment induced by aging. The amount of fine dispersive M n S particles and therefore the obstacle force R in sample A is higher than that in sample C because of the full dissolution annealing and corresponding aging precipitation, where R » R < is combined with F < F . The ratio of the two pure driving forces F —R can be expressed as — - « 1 , which indicates that the pure driving force in sample C is F -R much higher than those in samples A and B. This should be the reason why the hysteretic loss increment in sample C after aging was so limited. A
A
c
C
c
THE POSSIBLE INFLUENCE OF TEXTURE ON THE SHEET MAGNETOSTRICTION The magnetostriction of non-oriented silicon steels is also orientation d e p e n d e n t . Therefore, sheet texture affects the magnetostriction behavior and may induce core vibration and noise during service life. The magnetostrictive coefficient λ along a [uvw] direction, as a magnetization direction of a single domain can be calculated as '", 10
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Influence of Grain O r i e n t a t i o n o n M a g n e t i c A g i n g of a N o n - O r i e n t e d Silicon S t e e l S h e e t
V»i
=
^\ioo|(ai* +«2
+
a
t
— —) + 3 X
2
llllJ
( « f c x ; +a a; 2
2
+α;α )
(3)
where the α, (i=l .2.3) are the direction cosines of [uvw] with respect to the three <1 11> axes, λμοοι and λ|ΐιΐ| are the saturation magnetostrictive coefficients in the [100] and [111] directions, in which λ | ΐ ο ο ρ 2 0 . 7 χ 1 0 " and λ [ ΐ π ] = - 2 1 . 2 1 0 " are valid. The average value of the magnetostrictive coefficient along the rolling direction of the polycrystalline sheet are - 3 . 2 3 0 8 x 1 0 " . - 3 . 6 6 5 7 x 1 0 " ° and - 2 . 7 5 0 3 x 1 0 " " for samples Α . Β and C respectively (see Table II), which were calculated according to Zhu et al.'s m e t h o d " . The absolute value of the magnetostriction coefficient is too low to produce noticeable differences in the magnetostrictive effect a m o n g the samples. However, the limited magnetostriction could still supply activation energy to promote the aging precipitation of second-phase particles, which needs to be investigated in details. 6
χ
6
6
CONCLUSION The precipitation of M n S or other particles besides carbides may result in obvious magnetic aging and core loss increases. The aging effect could be reduced with decreasing {111} texture or increasing {100} texture. The effective domain wall m o v e m e n t is along the < 1 0 0 > directions during magnetization, which results in an orientation d e p e n d e n c e of the magnetic driving forces. Therefore a texture, in which the <100> directions of grains are parallel to the magnetic field, is the most favorable texture for reducing the core loss increment during aging. The magnetostrictive effect might p r o m o t e the precipitation of second-phase particles. However, the difference in magnetostriction a m o n g the samples of non-oriented silicon steel sheets studied w a s not significant becuase of the weak sheet textures. REFERENCES O . M . M i c h a l and J.A. Slane. The kinetics of Carbide Precipitation in silicon-aluminum steels, Metallurgical Transactions A (Physical Metallurgy and Materials Science), 1 7 A ( 8 ) : 1287-1294( 1986). K . M . M a r r a . E.A.Alvarenga, V.T.L.Buono. Magnetic aging anisotropy of a semi-processed non-oriented electrical steel. Materials Science and Engineering A, 390:423-426(2005). G.H.Shirkoohi. Anisotropic d e p e n d e n c e of magnetostriction in electrical steels under applied linear stress. Journal of m a g n e t i s m and magnetic materials, 157/158:516-518(1996). D . R . A s k e l a n d and P.P.Phule. T h e science and engineering of materials, Fourth edition. T h o m s o n learning 2004. 2
3
4
'J.T.Park and J.A.Szpunar. Texture d e v e l o p m e n t during grain growth in nonoriented electrical steels. ISIJ International, 4 5 ( 5 ) 7 4 3 - 7 4 9 ( 2 0 0 5 ) . J . C h i p m a n . T h e r m o d y n a m i c s and phase diagram of the Fe-C system. Metallurgical Transactions. 3(1):55-64(1972). S . K . R a y . O.N.Monanty. O n predicting the extent of magnetic aging in electrical steels, Journal o f Magnetism and Magnetic Materials, 78:255-262(1989). K . H u l k a . C.Vlad, A.Doniga. T h e role of niobium as microalloy in electrical sheet, Technical Report. C o m p a n h i a Brasileira de Metalurgia e Mineraçào(2002). G. Lyukosky. P.D.Southwick. Effect of thermomechanical history upon the microstructure and magnetic properties of nonoriented silicon steels. Metallurgical Transactions A,
6
7
8
9
17A(8):1267-1275(1985). 10
W . Z h o n g , Ferromagnetics. Beijing: Science press 1992.
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"G. Zhu, W. M a o , Y. Yu, Calculation of misorientation distribution between recrystallized grains and deformed matrix, Scripta Materialia, 42(1 ):37-41(l 999). A . J . M o s e s , A.Ntatsis, T.Kochmann, J.Schneider, Magnetostriction in non-oriented electrical steel: general trends. Journal of M a g n e t i s m and Magnetic Materials. 215-216: 669-672(2000). 12
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Effects of Magnetic Fields
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E U T E C T O I D P O I N T SHIFT A N D O R I E N T A T I O N R E L A T I O N S H I P S B E T W E E N FERRITE A N D C E M E N T I T E IN P E A R L I T E IN A HIGH M A G N E T I C F I E L D . Y. D. Zhang Key Laboratory
for Anisotropy
and Texture of Materials
(Ministry
of
Education),
Northeastern University, Shenyang 110004, China LETAM, C N R S - U M R 7078, University of Metz, Ile du Saulcy. 57045 Metz, France C. Esling L E T A M , C N R S - U M R 7078, University of Metz, Ile du Saulcy, 57045 Metz, France M. Calcagnotto L E T A M , C N R S - U M R 7078, University of Metz, Ile du Saulcy, 57045 Metz, France H. Klein, Department of Crystallography, University of Göttingen. D-37077 Göttingen, Germany X. Zhao Key Laboratory
for
Anisotropy
and
Texture of Materials
(Ministry
of
Education),
Texture of Materials (Ministry
of
Education),
Northeastern University, Shenyang 110004. China L. Zuo Key Laboratory
for Anisotropy
and
Northeastern University, Shenyang 110004. China ABSTRACT The effect of a 12 Tesla magnetic field on microstructure. texture and orientation relationship (OR) between pearlitic ferrite and cementite in a 0.81C-Fe (wt.%) near eutectoid plain carbon steel is investigated. Results show that the magnetic field shifts the eutectoid composition from 0.77C%wt. to 0.84C%wt. Further calculation showed that this field increases the eutectoid temperature by 2 9 ° C . The field applied slightly enhances the <001> fiber component in the transverse field direction. This effect is related to the magnetic dipolar interaction between Fe atoms in the transverse field direction. There are mainly four O R s between pearlitic ferrite and pearlitic cementite that are Isaichev (IS), near Bagaryatsky (Bag) and t w o near Pitsch-Petch (P-P-l and P-P-2) O R s in both the field treated and non-field treated samples. However, with field the P-P-2 OR becomes more frequent. The promotion of a magnetic field on the high magnetization phase, i.e., ferrite accounts for the frequent occurrence of P-P 2 OR. K E Y W O R D S : EPM-electromagnetic processing of materials; phase transformation; eutectoid point, orientation relationship (OR). INTRODUCTION Study on solid-to-solid state phase transformation of steels under a high magnetic field (more than 10 tesla) has well developed for more than one decade. Plenty of results have been obtained, mainly on the phase equilibrium between austenite and ferrite in a Fe-X system by thermodynamic calculation
1
2
3
7
as the influence of the applied magnetic field on the microstructure ' 9
orientations '
10
6
. Phase transformation behaviors in medium carbon steels " as well and crystallographic
of the product ferritic phase have been investigated experimentally. However.
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there is still no experimental report on the shift of the eutectoid point by the application of a magnetic
field
under equilibrium transformation
condition. Moreover, the
crystallographic
orientation relationship between lamellar ferrite and lamellar cementite under a magnetic field has not been reported. Therefore, experimental identification of the eutectoid point shift
and
clarification of the orientation relationships between lamellar ferrite and cementite through modified eutectoid transformation by a high magnetic field will enrich the understanding of the effect of magnetic field on solid state phase transformation. On this basis, a near eutectoid plain carbon steel (0.81 C wt.%) was selected and heat-treated without and with being exposed to a 12 Τ magnetic field. The influence of the magnetic field on eutectoid point shift in composition scale and temperature scale is experimentally examined and theoretically calculated. Its influence on orientation of ferrite and on the orientation relationships between eutectoid ferrite and cementite was experimentally examined and analyzed. EXPERIMENTAL The material used in this study was a 0.81 C-Fe (wt.%) near eutectoid plain carbon steel. It was austenitized at 840°C for 50 min and cooled at a rate of 2, 5, 10 and 23°C/min without and with a 12-Tesla high magnetic field. During the heat treatment, the specimens were placed in the central (zero magnetic force) area - non-field gradient area - to avoid the influence of the field gradient on carbon diffusion". The microstructure transformed was observed with an O L Y M P U S BX61 microscope equipped with the a n a l y S I S ™ software. The percentage of bulk ferrite areas appearing in the field-treated specimens and the lamellar spacing of pearlite in both field- and non-field-treated specimens was measured. The measurement was performed in 20 images for each specimen to achieve statistical reliability. Synchrotron radiation measurements were performed to measure the in-complete pole figures of ferrite of the samples cooled at 2°C/min without and with a 12-Tesla high magnetic field. The data were analyzed with MAUD and represented in inverse pole figures. Individual orientations of ferrite and cementite in pearlite colonies were manually measured through acquiring and indexing their electron back-scattering diffraction ( E B S D ) Kikuchi patterns 12
and represented in the form of Euler angles (φι, φ. φ ) in Bunge n o t a t i o n . More than 30 areas 2
were randomly selected to achieve a statistical reliability. The O R s between two adjacent phases, ferrite and cementite. were identified and represented in the form of Miller indices for easy visualization. Moreover, the habit planes of ferrite/cementite interfaces were determined by the 13
""indirect two-trace m e t h o d " developed by our g r o u p . RESULTS AND DISCUSSION Shift of Eutectoid Point under the Magnetic Field Figure 1 shows the microstructure of the specimens cooled at 2°C/min (a) without and (b) with a 12-T magnetic field. T h o u g h lamellar pearlite is the main component in both micrographs, the striking difference between the two is that, in the non-field treated specimen, we could observe a small amount of proeutectoid cementite (arrowed in the z o o m image in the top right c o m e r of Figure 1 (a)) characterizing the hypereutectoid microstructure whereas, in the
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E u t e c t o i d Point Shift a n d O r i e n t a t i o n R e l a t i o n s h i p s in Pearlite
specimen, we could see some bulk ferrite or proeutectoid ferrite between pearlite colonies (the white areas circled in Figure 1 (b)) and that component - rather than proeutectoid cementite - is typical of the hypoeutectoid microstructure. The crystal structure of the proeutectoid cementite and the proeutectoid ferrite observed with the optical microscope were further confinned by the indexing of their Kikuchi patterns by S E M E B S D technique.
Figure 1 Optical micrographs of specimens austenitized at 840°C for 42 min and cooled at 2°C/min without (a) and with (b) a 12-T magnetic field (The field direction is horizontal). The zoom image in the right hand corner of Fig. 1 (a) s h o w s the secondary cementite. as indicated by the arrow. The magnification is 1.5 times that of the main image. The circles in Fig. 1 (b) mark out the proeutectoid ferrite between pearlite colonies. The percentages and standard deviation of the average areas of the bulk ferrite measured in the field-treated specimens at various cooling rates are s h o w n in Table I. It should be noted that the deviation in the amount of bulk ferrite is relatively large. This is due to the fact that the size and amount of such ferrite is small. T h e measurement of the area percentage was made under high magnification and, as a consequence, the inhomogeneity of the local microstructures became pronounced. However, the average value obtained is statistically representative for the bulk sample. It can be seen that the amount of bulk ferrite decreases as the cooling rate increases. This 14
form of influence of the cooling rate on the proeutectoid transformation is n o r m a l . The presence of bulk ferrite in the
field-treated
specimens suggests that the magnetic field shifts the eutectoid
point in the Fe-C binary system beyond the carbon content of the material tested (0.81 C%wt.). Using the lever law. the average carbon contents of the new eutectoid point at various cooling rates were calculated and also displayed in Table I. A s the carbon content of the equilibrium eutectoid point is obtained under equilibrium transformation condition, i.e.. under extremely slow cooling condition, the variation of the eutectoid carbon content with the cooling rate is extrapolated to a zero cooling rate to obtain the equilibrium eutectoid carbon content. The extrapolated equilibrium eutectoid carbon content is 0.8287wt.%. As it is difficult to measure - in temperature scale - the shift of the eutectoid point under in situ equilibrium condition and under a high magnetic field, it m a y be useful to determine this shift through theoretical calculation. By using the well established statistical t h e r m o d y n a m i c model to calculate the chemical Gibbs free energy of the related phases
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and introducing the magnetic field influence as magnetic Gibbs free energy for each concerned 15
p h a s e , eutectoid carbon content and temperature calculated without and with a 12-T magnetic field is 0.779wt.% C; 725.71°C and 0.847wt.%; 754.68°C. The eutectoid carbon contents without and with the magnetic field - as calculated - appears to be very close to the value determined experimentally (0.8287wt.%). This agreement shows that the increase in eutectoid temperature calculated by this method is reliable.
Table I Average area percentages of the bulk ferrite in 0.81C-Fe specimens austenitized at 840°C for 42 min and cooled at various cooling rates with a 12-T magnetic field, and the calculated eutectoid carbon content by using lever law. Cooling rate (°C/min)
Area percentage of ferrite (%)
Eutectoid carbon content (wt.%)
2
2.153
0.827
5 10 23
1.905 1.519 1.277
0.825 0.822 0.820
Texture under the Magnetic Field Figure 2 shows the inverse pole figures of ferrite of the samples heat treated at slow cooling rate (2°C /min) without and with the 12T magnetic field and the corresponding sample coordinate system. It is seen that under the magnetic field there is slight enhancement of < 0 0 1 > fiber component in both the sample normal direction (ND) and the widthwise direction (TD), as seen in Figure 2 (b). This result is quite close to what w e found in a medium carbon plain steel heat 10
treated under a 12T magnetic field . Actually in the present case, both N D and T D are transverse field directions. It is k n o w n that each Fe atom carries a magnetic moment. U n d e r the applied magnetic field, these m o m e n t s tend to align along the field direction. Then, there exists the dipolar interaction between neighboring Fe atoms. They attract each other along the field direction but repel each other along the transverse field direction ( N D and T D in the present study). Correlatively. the distance between neighboring atoms tends to decrease along FD and increase along N D and T D to minimize the total energy of the system. For ferrite, the carbon atoms are located in the octahedral interstices. The interstices are flat in the < 0 0 1 > direction. The occupation of the carbon atom in this interstice exerts an expansion stress on its neighboring iron atoms along the <001> direction. This gives rise to the lattice distortion and creates distortion energy. If such a <001> direction of a grain were parallel to the transverse field direction ( N D or T D ) , the lattice distortion energy would be reduced through increasing the atomic spacing in such < 0 0 1 > direction by the magnetic field. Therefore, the nucleation and growth of the grains having such <001> parallel to the N D and T D is most energetically favored by the magnetic field. In this way, the <001> component is enhanced.
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Figure 2 Inverse pole figures of the samples austenitized at 840°C for 4 2 min and cooled at a rate of 2°C/min without (a) and with a 12-T magnetic field (b). and the corresponding sample coordinate system. Orientation Relationships between Ferrite and Cementite in Pearlite 16
In our previous w o r k , w e have derived four different ferrite/cementite orientation relationships (ORs) with the sample cooled at 2°C/min after full austenitization without the 16
magnetic field as shown in Table I I . It w a s found that all the four O R s possess a c o m m o n feature of close-packed
plane
parallelism
between
ferrite
and
cementite. Their
crystallographic
compatibility with habit planes exhibit variety of possible habit plane and excludes the existence 1 6
of the exact conventional Bagaryatsky and Pitsch-Petch O R s . When the magnetic field was applied, the four O R s also appear but each has a different occurrence frequency from the same OR in the non-field treated sample. It is found that the n u m b e r of P-P 2 O R is obviously increased
Materials Processing and Texture
·
385
E u t e c t o i d Point Shift a n d O r i e n t a t i o n R e l a t i o n s h i p s in Pearlite
under the 12 Τ magnetic field, i.e.. from 13.16% of the 38 pearlite colonies selected randomly in the non-field treated sample to 4 2 . 8 6 % of the 36 colonies in the field treated sample. Many of such oriented lamellar ferrite and cementite appear next to the proeutectoid ferrite as shown in Figure 3. From the above analysis, it is clear that this proeutectoid ferrite is produced by the shift of the eutectoid point due to the introduction of the magnetic field. It is k n o w n that the pearlitic transformation involves two structural changes. O n e refers to the transformation from fee austenite into bec ferrite and orthorhombic cementite and the other to the final formation of ferrite/cementite interface. The first may lead to a misfit at the austenite/ferrite and austenite/cementite interfaces due to their difference in crystal structure, resulting in the transformation strain energy at these interface boundaries. The second, the formation of ferrite/cementite interface, creates a new interfacial energy item that d e p e n d s on the atomic misfit on interface planes. These two energy terms are considered as the energy barriers to the pearlitic transformation. To minimize these barriers, it requires some specific O R s between the parent and the product phase and the coherent habit planes connecting the product phases to minimize transformation strain and
interfacial
atomic misfit. It concluded in our previous work that the four O R s well satisfy the " e d g e - t o - e d g e " matching condition at the austenite/ferrite and austenite/cementite interface that helps to reduce 16
the transformation s t r a i n . However, the different O R s are correlated to different
nucleation
conditions. The P-P I OR occurs w h e n the pearlitic ferrite and cementite nucleate simultaneously, whereas the P-P 2 O R appears w h e n pearlitic ferrite forms before pearlitic cementite. However, the IS OR could happen either when pearlitic ferrite nucleates first or the pearlitic cementite nucleates 16
first .
This derivation is quite coherent with the present observation that under a 12 Τ magnetic
field the occurrences of P-P 2 is increased. Many studies have proved that the equilibrium condition between phases with different induced magnetization can be change by the application 1
5
of a magnetic field " . The phase with high-induced magnetization will become more stable or easily to nucleate than that with low-induced magnetization. As at the pearlitic transformation temperature of the present study, pearlitic ferrite is ferromagnetic with high-induced magnetization under the 12 Τ magnetic field and cementite is paramagnetic with low-induced magnetization, ferrite would nucleate before cementite, especially w h e n there already exists proeutectoid ferrite. In this way, the O R s that favor the nucleation of ferrite before would become more frequent.
Table II Orientation relationships (ORs) between pearlitic ferrite and pearlitic cementite in the non-field treated s a m p l e
16
IS OR
Near Bag. O R (103),
11(011),
\010\
//[///],
Habit plane (001),
- 3°from 11(211),.
(103),
11(011),.
P-P 2
P-P 1 (103),.
11(101),
(103\ll(l01),
[010],.//[Ill],
[010],.//[131],..
[3ll] //[lll]
Habit plane:
Habit planes
Habit planes (001), ~3J°from(2l5),
(101),
.11(112),.
(001), (103),.
.11(115),.
c
r
11(101),, Unknown
(101) 8.7°from(215), c
386
Materials Processing and Texture
E u t e c t o i d Point Shift a n d O r i e n t a t i o n Relationships in Pearlite
Table III Occurrence of different O R s with and with a magnetic field treatment Field intensity Near Bag O R P-P 1 O R P-P-2 O R IS O R 13.16% OT 18.42% 18.42% 50% 12T
CONCLUSIONS The study of the phase transformation of the 0.81C-Fe (wt.%) near eutectoid plain carbon steel under a 12 Τ magnetic field s h o w s that: (1) the eutectoid point of Fe-C system is shifted from 0.779wt.% C; 725.71 °C to 0.847wt.%; 754.68°C by the 12 Τ magnetic field : (2) the field applied slightly enhances the < 0 0 1 > fiber texture in the transverse field direction due to the dipolar interaction of the magnetic m o m e n t s carried by the Fe atoms. (3) the magnetic field is in favor of the occurrence of P-P 2 O R due to the promotion of nucleation of pearlitic ferrite. ACKNOWLEDGEMENTS A c k n o w l e d g e m e n t s to the National Science Fund for Distinguished Young Scholars (No. 50325102), N S F of China (No. 50234020 and 50571024). and PRA M X 0 4 - 0 2 project. REFERENCES: Ύ . D. Zhang, C. S. He, X. Z h a o . L. Z u o and C. Esling. N e w phase equilibrium in Fe-C binarysystem under a magnetic field. Solid State Phenomena. 2
105, 187-94 (2005).
M . C. Gao, T. A. Bennett. A. D. Rollett and D. E. Laughlin. The effects of applied magnetic fields
on the α/γ phase boundary in the F e - S i system. J. Phys. D: Appl. Phys.. 3 9 , 2890-96 (2006).
Materials Processing and Texture
387
Eutectoid Point Shift a n d Orientation Relationships in Pearlite
3
Y. D. Z h a n g . C. S. H e . X. Zhao, C. Esling and L. Z u o , A n e w approach for rapid annealing of
medium carbon steels. Adv. Eng. Mater.. 6. 310-13 (2004). J
Y. D. Zhang, C. S. He, X. Z h a o . L. Z u o . C. Esling and J. C. He, N e w microstructural features
occurring during transformation from austenite to ferrite under kinetic influence of magnetic field in a medium carbon steel,,/. Magn. Magn. Mater. 5
2 8 4 , 287-93 (2004).
Y. D. Z h a n g . C. S. He. X. Zhao and L. Zuo, T h e r m o d y n a m i c and kinetic Characteristics of High
Temperature Cooling Phase Transformation under High Magnetic Field,./. Magn. Magn.
Mater..
294, 267-72 (2005). °G. M.
Ludtka,
R.
A.
Jaramillo,
R.
A.
Kisner,
D.
M.
Nicholson,
J.
B.
Wilgen.
G.
Mackiewicz-Ludtka and P. N. Kalu. In situ evidence of enhanced transformation kinetics in a medium carbon steel due to a high magnetic field, Scripta Mater, 7
M . Shimotomai. K. Maruta, K. Mine and
5 1 , 171-74 (2004).
M. Matsui, Formation of aligned
two-phase
microstructures by applying a magnetic field during the austenite to ferrite transformation in steels. Acta Mater, 8
5 1 , 2921-32 (2003).
Y. D. Z h a n g . C. Esling. J. Muller, C. S. He, X. Z h a o and L. Z u o . Magnetic-field-induced grain
elongation under a high magnetic field in medium carbon steel in its austenitic decomposition, Appl. Phys. Lett.. 8 7 . 212504 (2005). Y. D. Z h a n g . G. Vincent. N . D e w o b r o t o , L. Germain. X. Z h a o , L. Z u o and C. Esling, The effect of thermal processing in a magnetic field on grain boundary characters of ferrite in a medium steel, J. Mater. Sei., 4 0 . 905-08 (2005). '" Y. D. Z h a n g . C. Esling. J. S. Lecomte. C. S. He, X. Z h a o and L. Z u o , Grain boundary characteristics and texture formation in a m e d i u m carbon steel during its austenitic decomposition in a high magnetic field. Acta Mater.
5 3 . 5213-21 (2005).
" S . Nakamichi. S. Tsurekawa, Y. M o r i z o n o . T. Watanabe, M. Nishida and A. Chiba. Diffusion of carbon and titanium in g a m m a - i r o n in a magnetic field and a magnetic field gradient,./.
Mater.
.SV/.. 40. 3191-98 (2005). l2
13
B u n g e HJ. Texture Analysis in Materials Science. Cuvillier Verlag: Goettingen; 1993. p4. Y. D. Zhang, C. Esling. X. Z h a o and L. Zuo, Indirect two-trace method to determine a faceted
low energy interface between two crystallographically correlated crystals, J. Appl.
Cryst..
40.
436-40 (2007). I 4
Y . D. Z h a n g . C. Esling. M . L. G o n g . G. Vincent. X. Z h a o , L. Zuo, Microstructural features
induced by a high magnetic field in a hypereutectoid steel during austenitic decomposition,
Scripta
Mater.. 54. 1897-1900 (2006). ' Ύ . D. Z h a n g . C. Esling, M. Calcagnotto, M. L Gong, X. Z h a o and L. Z u o , Shift of the eutectoid point in the Fe-C binary system by a high magnetic field. J. Phys. D: Appl. Phys., 40, 6501-06 (2007). " Ύ . D. Zhang. C. Esling. M. Calcagnotto. X. Z h a o and L. Z u o , N e w Insights into Crystallographic Correlations between Ferrite and Cementite in Lamellar Eutectoid Structures, obtained by S E M - F E G / E B S D and Indirect T w o - T r a c e Method, J. Appl. Cryst., 40, 849-56 (2007).
388
·
Materials P r o c e s s i n g a n d T e x t u r e
CHARACTERISTICS
OF RECRYSTALLIZATION
STEEL SHEET A N N E A L E D
TEXTURE
WITH A MAGNETIC
FIELD
OF COLD-ROLLED IN T H E
IF
TRANSVERSE
DIRECTION
1
1
2
2
Yan W u , Changshu H e ' , Yudong Z h a n g ' , Xiang Z h a o ' , Liang Z u o ' , Claude Esling * 1
Key
Laboratory
for
Anisotropy
and
Texture
of
Materials
(Ministry
of
Education),
N o r t h e a s t e r n University, S h e n y a n g 110004, L i a o n i n g P r o v i n c e , R R . C h i n a 2
L E T A M , C N R S - U M R 7 0 7 8 , University of Metz, Ile du Saulcy, 5 7 0 4 5 Metz, France * e-mail: [email protected]
ABSTRACT T h e effect o f m a g n e t i c field a n n e a l i n g on recrystallization texture at the early stage of recrystallization of a cold-rolled interstitial-free (IF) steel sheet w a s investigated by the S E M - E B S D analysis. T h e m a g n e t i c field a n n e a l i n g w a s p e r f o r m e d at 6 5 0 ° C for different t i m e s (0, 10 a n d 3 0 m i n ) with a 12-tesla m a g n e t i c field to obtain a partially recrystallized m i c r o s t r u c t u r e in s p e c i m e n s . D u r i n g a n n e a l i n g the t r a n s v e r s e direction ( T D ) of specimen was set to b e parallel to the m a g n e t i c field. It w a s found that the {] 11 }<112> texture c o m p o n e n t w a s favored by the applied high m a g n e t i c field in the early stage of nucleation and growth. INTRODUCTION A s an important external field, it h a s been e x p e r i m e n t a l l y evidenced that magnetic field can affect the evolution o f microstructure and recrystallization texture in plastically deformed 8
metallic m a t e r i a l s ' " , especially in ferromagnetic m a t e r i a l s ' " , a m o n g w h i c h the study of m a g n e t i c field a n n e a l i n g on recrystallization and t e x t u r e evolution in silicon steel sheet and 5
6
IF d e e p - d r a w i n g steel sheet has b e e n an object o f great interest. A s reported e l s e w h e r e ' , the magnetic
field a n n e a l i n g e n h a n c e d
the d e v e l o p m e n t
of recrystallization
a-fiber
texture
c o m p o n e n t s in IF steel sheet, w h e n it w a s annealed at 6 5 0 ° C for 2 5 m i n , with the magnetic field ( M D ) parallel to the rolling direction ( R D ) . H o w e v e r , in m o s t o f the p r e v i o u s studies, the field w a s applied in the rolling direction o f the s p e c i m e n s and only limited w o r k
7
was
d o n e with t h e m a g n e t i c field parallel to the transverse direction o f the s p e c i m e n . So in the present paper, w e particularly studied the effect of a m a g n e t i c field on the recrystallization texture o f a cold-rolled IF steel sheet at t h e initial stage of recrystallization with the magnetic field in the transverse direction. EXPERIMENTAL T h e material used w a s a 7 5 % cold-rolled IF steel sheet o f I m m t h i c k n e s s , having the chemical c o m p o s i t i o n o f (wt % ) : 0 . 0 0 2 3 C , 0.056Ti, 0.014Si, 0 . 1 6 M n , 0 . 0 1 1 R 0 . 0 0 6 4 S , 0.052AI, 0 . 0 0 1 8 N . S p e c i m e n s with the d i m e n s i o n s of 2 0 m m x l O m m x 1mm w e r e respectively subjected to isothermal a n n e a l i n g at 6 5 0 ° C for different t i m e s (0, 10 and 3 0 m i n ) in a furnace installed in a c r y o c o o l e r - c o o l e d s u p e r c o n d u c t i n g m a g n e t at a h e a t i n g rate of 5°C /min, with and w i t h o u t a 12-tesla m a g n e t i c field, and then cooled in the furnace. D u r i n g the magnetic field a n n e a l i n g , the m a g n e t i c field w a s kept constant in the w h o l e heating, holding and cooling p r o c e s s e s . T h e s p e c i m e n s w e r e placed at the c e n t e r o f the applied field with their
389
C h a r a c t e r i s t i c s of Recrystallization T e x t u r e of C o l d - R o l l e d IF S h e e t S t e e l
transverse direction parallel to the m a g n e t i c field direction. Microstructures o f longitudinal s e c t i o n s o f the s p e c i m e n s w e r e e x a m i n e d by orientation i m a g i n g m i c r o s c o p y ( O I M ) . In the O I M m a p s , the recrystallized grains are highlighted (dark grains). T h e scan was carried out o v e r the area of 1191 ><878 m e a s u r i n g points with a 0.07μιη and 0.21 μηπ step size, respectively, a c c o r d i n g to the d e g r e e o f recrystallization. T h e c o r r e s p o n d i n g O D F s w e r e calculated using the C h a n n e l 5 software. RESULTS AND DISCUSSION Figure 1 s h o w s the O D F φ = 4 5 ° s e c t i o n s o f all recrystallized grains in t h e s p e c i m e n s 2
annealed with different h o l d i n g t i m e s . A s Figure 1 (a) and (d) show, in the case of h o l d i n g for Omin. the orientation of the recrystallized nuclei d e t e r m i n e d
in the non-field
annealed
specimen is mainly {111 } < u v w > . while it is a little different for the field annealed s p e c i m e n , i.e. the recrystallized nuclei mainly have a strong { I I 1 } < 1 I 2 > c o m p o n e n t . In the case o f holding for lOmin. the recrystallized grains in the non-field a n n e a l e d s p e c i m e n exhibit a strong γ liber texture with intensity p e a k s b e t w e e n {111}<123> and {111}<110> (Figure 1 (b)), while for the field a n n e a l e d s p e c i m e n , the orientation o f the recrystallized g r a i n s is mainly of a strong { l l l } < 1 1 2 > and a w e a k
{ l l l } < 1 2 3 > c o m p o n e n t near
{I11}
(Figure 1 (e)). W h e n the a n n e a l i n g t i m e w a s further e x t e n d e d to 3 0 m i n , t h e orientation distributions o f the recrystallized g r a i n s in the field and non-field annealed s p e c i m e n s a r e similar with the intensity peak at {111 } < 1 1 2 > (Figure 1 (c), (0). T h e a b o v e results obviously indicate that the high m a g n e t i c field a n n e a l i n g favoured the d e v e l o p m e n t o f
{I1I}<1I2>
texture c o m p o n e n t at the initial stage o f recrystallization, w h e n t h e m a g n e t i c field
was
applied in the t r a n s v e r s e direction.
Figure I O D F φ = 4 5 ° sections for the recrystallized grains of the s a m p l e s a n n e a l e d at 6 5 0 ° C 2
for différent time with and without m a g n e t i c Held (a) 0 T * Omin
(b) Ο Τ χ I Omin
(c)0T><30min
x
(d) 1 2 T 0 m i n
(e) 12Tx 10min
(f) 1 2 T x 3 0 m i n We further investigated the orientation features a m o n g the large, m e d i u m and small recrystallized grains in the field and non-tield annealed s p e c i m e n s . Figure 2 and Figure 3 show the orientation (Band C o n t r a s t + Grain b o u n d a r y ) m a p s o f the s p e c i m e n s a n n e a l e d for 10 and 3 0 m i n , respectively. T h e dark colored grains in t h e m a p s are the recrystallized g r a i n s with different sizes. T h e c o r r e s p o n d i n g O D F φ = 4 5 ° sections of those recrystallized grains 2
are s h o w n in F i g u r e 4 .
390
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Materials Processing and Texture
E u t e c t o i d Shift a n d O r i e n t a t i o n R e l a t i o n s h i p s B e t w e e n Ferrite a n d C e m e n t i t e
F i g u r e 2 Recrystallized g r a i n s o f different sizes in t h e s p e c i m e n s annealed at 650°C for 10min with and w i t h o u t t h e m a g n e t i c field (a) 0 Ι"χ 1 Omin, large g r a i n s ; (b) 12Ί'χ 1 Omin, large g r a i n s : (c) O T x l O m i n , m e d i u m g r a i n s ; (d) 1 2 l ' x l 0 m i n . m e d i u m g r a i n s ; (e) O T x l O m i n , small g r a i n s : (f) 1 2 T x l 0 m i n , small grains.
Materials Processing and Texture
·
391
E u t e c t o i d Shift a n d O r i e n t a t i o n R e l a t i o n s h i p s B e t w e e n Ferrite a n d C e m e n t i t e
F i g u r e 3 Recrystallized g r a i n s o f different sizes in the s p e c i m e n s a n n e a l e d at 650°C for 3()min with and w i t h o u t the m a g n e t i c field (a) 0 T x 3 0 m i n , large g r a i n s ;
(b) 1 2 T x 3 0 m i n . large g r a i n s ;
x
(c) 0 T 3 0 m i n , m e d i u m g r a i n s ; (d) 1 2 T x 3 0 m i n , m e d i u m g r a i n s ; (e) 0 I'x30min, small g r a i n s ;
392
·
Materials Processing and Texture
(0 1 2 T x 3 0 m i n . small g r a i n s .
E u t e c t o i d Shift a n d Orientation R e l a t i o n s h i p s B e t w e e n Ferrite a n d C e m e n t i t e
Figure 4 O D F φ = 4 5 ° sections o f recrystallized grains o f different sizes in s p e c i m e n s 2
annealed at 650°C for 10 min and 3 0 m i n (a) O T x l O m i n : (b) 1 2 T x l 0 m i n ; (c) 0 T x 3 0 m i n ; (d) 12Tx 3 0 m i n It can be seen that, in the case o f holding for 10min w i t h o u t the m a g n e t i c field (Figure 4 (a)), the { 1 1 1 } < 1 1 0 > , { l l l } < 1 2 3 > and {111}<112> orientations are respectively associated with the large, m e d i u m and small recrystallized g r a i n s . H o w e v e r , it is obviously different w h e n t h e high m a g n e t i c field w a s applied. After h o l d i n g for 10min in the high m a g n e t i c field (Figure 4 (b)). the large, m e d i u m and small recrystallized grains all possess a much stronger {111}<112> orientation, and part of the large recrystallized g r a i n s s h o w a texture c o m p o n e n t near { l l l } < 1 2 3 > . W h e n the holding t i m e is 3 0 m i n (Fig. 4 (c). (d)), all the recrystallized grains with different sizes in the field and non-field annealed s p e c i m e n s exhibit a γ-fiber texture with the intensity peak at {111 }
speaking,
the
stored
energy
in
the d e f o r m e d
regions
of
{111}
orientation is higher than that in the regions of {111} < 110> and {111} < 123> orientations in 12
the cold rolled IF steel s h e e t . C o n s e q u e n t l y , nucleation o c c u r r i n g in the oriented matrix is c o n s i d e r e d to be energetically favored. M o r e o v e r ,
Jlll}<112> 13
in the i r o n ,
Materials Processing and Texture
the
393
E u t e c t o i d Shift a n d O r i e n t a t i o n Relationships B e t w e e n Ferrite a n d C e m e n t i t e
{111 }<110> oriented crystals is related to the {111 }<112> oriented matrix by a 30° rotation a r o u n d their c o m m o n
To
understand the origin of this effect, it is natural to e x a m i n e the effect o f the m a g n e t i c field on the energy o f the n e w grains formed d u r i n g recrystallization. T w o kinds of energy are involved here, crystal anisotropy e n e r g y and magnetoelastic e n e r g y . Generally, the m a g n e t o c r y s t a l l i n e anisotropy energy Gk (J-m" ), is the energy necessary 3
to deflect the m a g n e t i c m o m e n t in a single crystal from the easy to t h e hard direction. T h e easy and hard directions arise from the interaction o f the spin m a g n e t i c m o m e n t with the crystal lattice. In cubic crystals, like iron and steel, the m a g n e t o c r y s t a l l i n e anisotropy energy is given by a series e x p a n s i o n in t e r m s o f the a n g l e s b e t w e e n the direction o f magnetization and the c u b e a x e s : G'k = /M ( α ι α 2 + α 2 α + α 3 α ι ) + ^ 2 α ι α 2 α + K ( a i a W a + a a . | ) + 2
2
2
2
2
2
2
2
3
2
2
3
2
3
2
2
3
2
2
3
2
...
(1)
w h e r e , K\, K and K3 are the m a g n e t o c r y s t a l l i n e anisotropy constants, and αι. ci2, a are 2
3
the directional cosines b e t w e e n m a g n e t i z a t i o n M and three crystallographic a x e s . Usually, it is sufficient to represent the anisotropy energy in an arbitrary direction by j u s t t h e first t w o terms in the series e x p a n s i o n , i.e. with t h e first- and second order anisotropy c o n s t a n t s ΚI and K2. respectively. Then, the results o f calculation are G
2
3
Gk<no>= K]/4, ( a i = a 2 = l / V 2 Ί α = 0 ) ; 3
6\< ΐ2>=Α',/4+/Ϊ2/54, ( a , = a = l / V 6 , a = 2/Vë 1
2
Gk<
l l u
= Κ\β+Κ ΙΠ.
3
);
(ai=a =a =l/V3).
2
2
4
3
3
For iron. K\ (~ 10 ) > 0 , Λ ^ ( ~ 1 0 ) < 0 . So the s e q u e n c e of m a g n e t o c r y s t a l l i n e anisotropy energy of grains with different orientations is as follows: <
<
Gk
compared
with
the
(2) <112>,
<110>
h a s the h i g h e r
magnetocrystalline
anisotropy energy, thus, the nuclei with < 1 1 0 > orientation parallel to t h e m a g n e t i c
field
direction h a v e lower driving force for nucleation and g r o w t h than that of the nuclei with <112> orientations parallel to the m a g n e t i c field direction. However, on the other hand, w h e n a crystal is m a g n e t i z e d , it u n d e r g o e s an elastic deformation b e c a u s e of m a g n e t o s t r i c t i o n . If the crystal is part o f a polycrystalline a g g r e g a t e , and therefore constrained by its n e i g h b o r s , magnetoelastic (strain) energy will be p r o d u c e d , and the a m o u n t of energy so stored will d e p e n d on the magnetostriction coefficient λ. B a s i n g
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·
Materials Processing and Texture
E u t e c t o i d Shift a n d Orientation R e l a t i o n s h i p s B e t w e e n Ferrite a n d C e m e n t i t e
1
1
on the e x p e r i m e n t a l results o f T a t a u m m o t o and O k a m o t o . B h a n d a r y and Cullity " calculated the magnetostriction coefficient λ of different crystalline directions in crystalline iron. T h e results calculated for 6 5 0 ° C are as follows: |Λ| -
|Λ|
(3)
O b v i o u s l y , c o m p a r e d with the < l 10>, < 1 1 2 > h a s a h i g h e r magnetostriction coefficient value. S o at t h e a n n e a l i n g t e m p e r a t u r e applied in this study, crystals oriented with < 1 1 2 > parallel to the m a g n e t i c field direction w o u l d have larger strain energy and it is difficult for t h e m to grow c o m p a r e d with t h o s e h a v i n g
F i g u r e 5 S c h e m a t i c illustration of the relationship b e t w e e n {111 }
field
a l o n g the transverse direction, the
different effects of m a g n e t o c r y s t a l l i n e anisotropy and magnetostriction anisotropy on the <110>
and
<112>
orientation
accounts
for
the
selection
of the
{1U}<112>
texture
c o m p o n e n t at the initial stage o f the recrystallization. In t h e present study, magnetostriction a n i s o t r o p y m a y play a key role. SUMMARY A 12-T high m a g n e t i c field w a s applied during the a n n e a l i n g o f a cold-rolled IF steel sheet at 6 5 0 ° C with t h e m a g n e t i c field direction parallel to the t r a n s v e r s e direction. It w a s found that the m a g n e t i c field a n n e a l i n g favoured the first n u c l e a t i o n and subsequent growth o f {111 }<112> t e x t u r e c o m p o n e n t at the initial stage of recrystallization. ACKNOWLEDGEMENT T h i s w o r k w a s supported by t h e National Natural Science Foundation of C h i n a (Grant No.50501006),
the National
Science
Found
for
Distinguished
Young
Scholars
(Grant
N o . 5 0 3 2 5 1 0 2 ) , and the "111"' Project (Grant N o . B 0 7 0 1 5 ) .
Materials Processing and Texture
·
395
E u t e c t o i d Shift a n d Orientation Relationships B e t w e e n Ferrite a n d C e m e n t i t e
The a u t h o r s w o u l d like to a p p r e c i a t e t h e support o f the H i g h M a g n e t i c Field L a b o r a t o r y of Northeastern
University for p r o v i d i n g the facilities. S h a n g h a i B a o Steel is gratefully
appreciated for the provision o f studied m a t e r i a l s for this w o r k .
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R. S m o l u c h o w s k i , and R. W. Turner. Influence o f M a g n e t i c Field on Recrystallization. ./. Appl. Phys.,
2 0 , 745-6 (1949),
2
H. O . M a r t i k a i n e n , V. K. L i n d r o o s , O b s e r v a t i o n s on the effect o f m a g n e t i c field on the
1
T. W a t a n a b e . Y. S u z u k i . S. Tani. and H . O i k a w a . T h e effects o f m a g n e t i c a n n e a l i n g on
recrystallization in ferrite. ScandJ.
Metall,
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recrystallization and grain b o u n d a r y c h a r a c t e r distribution ( G B C D ) in iron-cobalt alloy. Philos. 4
Mag. Lett., 6 2 . 9 - 1 7 ( 1 9 9 0 ) .
N . M a s a h a s h i . M . M a t s u o , K. W a t a n a b e . D e v e l o p m e n t o f preferred orientation in a n n e a l i n g o f F e - 3 . 2 5 % S i in a high m a g n e t i c field. J. Mater. Res., 13. 4 5 7 - 4 6 1 ( 1 9 9 8 ) .
>
C.
S. He, Y. D. Z h a n g , X. Z h a o , L. Z u o . J.C. He, K. W a t a n a b e , T. Z h a n g , and G. N i s h i j i m a ,
Effects o f a High M a g n e t i c Field on M i c r o s t r u c t u r e a n d T e x t u r e Evolution in a Cold-rolled Interstitial-free (IF) Steel Sheet d u r i n g A n n e a l i n g , Adv. Eng. Mater., 5 ( 8 ) , 5 7 9 - 8 3 ( 2 0 0 3 ) . 6
C . S. He, Y. D. Z h a n g , X. Z h a o , L. Z u o , and C . Esling. C h a r a c t e r i s t i c s o f Recrystallization Texture Evolution in High M a g n e t i c Field for Interstitial-free (IF) Steel Sheet, Mater. Forum.
7
Sei.
4 9 5 - 4 9 7 , 4 6 5 - 7 0 (2005).
Y. Wu, C . S. H e . X. Zhao, L. Z u o , and T. W a t a n a b e , Effects o f high m a g n e t i c field and field direction on recrystallization and recrystallization texture in cold-rolled IF steel sheet. Mater. Sei. Forum.
s
558-559, 401-406(2007).
Z h o u S h i - c h u n , Pei Wei, S h a Yu-hui, et al. Effect o f m a g n e t i c a n n e a l i n g on recrystallization texture and m i c r o s t r u c t u r e o f n o n - o r i e n t e d silicon steel. Journal (Natutal
9
Science),
of Northeastern
A. D. S h e i k h - A l i , D. A . M o l o d o v , a n d H. G a r m e s t a n i , M a g n e t i c a l l y d e v e l o p m e n t in zinc alloy sheet. Scripta
1 0
University
2 8 (8), 1131-1135(2007). induced
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Mater., 4 6 , 8 5 7 - 6 2 ( 2 0 0 2 ) .
A. D. S h e i k h - A l i . D. A. M o l o d o v . and H . G a r m e s t a n i , Migration and R e o r i e n t a t i o n o f Grain B o u n d a r i e s in Zn Bicrystals d u r i n g A n n e a l i n g in a High M a g n e t i c Field,
Scripta
Mater., 4 8 , 4 8 3 - 8 ( 2 0 0 3 ) . " D. A. M o l o d o v . and A . D. S h e i k h - A l i . Effect o f M a g n e t i c Field on Texture E v o l u t i o n in T i t a n i u m . Acta Mater.. 5 2 ( 1 4 ) , 4 3 7 7 - 8 3 ( 2 0 0 4 ) . 1 2
N . R a j m o h a n , Y. H a y a k a w a , J. A . Szpunar, and J . H . Root, N e u t r o n diffraction m e t h o d for stored energy m e a s u r e m e n t in interstitial free steel. Acta Mater., 4 5 , 2 4 8 5 - 2 4 9 4 ( 1 9 9 7 ) .
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W. B . H u t c h i n s o n , Recrystallization t e x t u r e s in iron resulting from n u c l e a t i o n at grain b o u n d a r i e s , / ) eta Metall.
1 4
V. S. Bhandary,
3 7 , 1047-1056(1989).
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recrystallized in a m a g n e t i c field. Trans. Met. Soc,
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Materials Processing and Texture
AIME.
properties o f Iron
224, 1194-1200(1962)
wire
C R Y S T A L L O G R A P H I C F E A T U R E S D U R I N G M A R T E N S I T I C T R A N S F O R M A T I O N IN NiMn-Ga F E R R O M A G N E T I C SHAPE MEMORY A L L O Y S D. Y. C o n g
1
2
1
2
2
1
1
, Y. D. Z h a n g ' , C. E s l i n g ' , Y. D. W a n g , X. Z h a o , L. Z u o
1
'Key Laboratory for Anisotropy and Texture of Materials ( M O E ) , Northeastern
University.
Shenyang 110004, China 2
L E T A M , C N R S - U M R 7078, University of Metz, Ile du Saulcy, 57045 Metz. France e-mail: [email protected]
ABSTRACT The martensitic transformation crystallography in t w o Ni 3Mn2sGa22 (at. %) ferromagnetic 5
shape memory alloys ( F S M A s ) was investigated by means of misorientation calculation and pole figure analysis based on the orientation of the martensitic lamellae obtained from electron backscattered diffraction ( E B S D ) measurements. In the alloy that w a s first annealed at 1073K for 4h, and then cooled to 473K at ~4K7min and held for 30min. followed by cooling to room temperature at ~10K/min, there are only two kinds of differently orientated martensitic lamellae with a misorientation angle o f - 8 2 ° distributed alternatively in each initial austenite grain. There is a compound twinning orientation relationship between the t w o lamellae. The prevalent orientation relationship between austenite and martensite is Kurdjumov-Sachs (K-S) relationship with ( 1 1 1 ) / / ( 1 0 1 ) , [1-1 0 ] / / [ 1 1-1 ] - In the alloy that w a s annealed at 1173K for 4h followed A
M
a
m
by furnace cooling, nanoscale twins inside the martensitic lamellae were observed and the orientation relationships both between the nanotwins within one lamella and between the nanotwins in two neighboring lamellae were determined. The results presented in this paper will enrich the crystallographic data of the F S M A s and offer useful information for the development of novel F S M A s with optimal performances. INTRODUCTION N i - M n - G a ferromagnetic shape memory alloys ( F S M A s ) are a n e w class of active materials 1
8
that have attracted great attention in the past few years " due to their giant magnetic shape memory effect ( M S M E ) under external magnetic fields. These F S M A s integrate the advantages of both conventional shape memory alloys driven by thermal fields and magnetostrictive 5
6
materials actuated by magnetic fields, showing large output strain and short response t i m e ' . Therefore these F S M A s can be easily actuated and controlled, having promising prospect in the applications of actuators and sensors. In spite of extensive investigations on various aspects of these
alloys, there still
remain
many
fundamental
issues
unresolved.
Up
to now.
the
crystallographic features during martensitic transformation is still unclear, and the orientation relationship between austenite and martensite has not been widely reported. Insight into these aspects is of both fundamental significance and practical interest. In this paper, we report the crystallographic features occurring during martensitic transformation in two Ni5jMn25Ga22 F S M A s subjected to different heat treatments.
397
C r y s t a l l o g r a p h i c F e a t u r e s During M a r t e n s i t i c T r a n s f o r m a t i o n in N i - M n - G a Alloys
EXPERIMENTAL Two Ni5jMri25Ga22 (at. %) alloys were prepared by repeated melting of the constituent elements in an arc furnace. Subsequently, one alloy (denoted as alloy I) was first annealed at 1073K for 4h. and then cooled to 4 7 3 K (above martenstic transformation temperature) at ~4K/min and held for 30min. followed by cooling to room temperature at ~10K7min. The other alloy (denoted as alloy II) was annealed at 1173K for 4h followed by furnace cooling. The orientation information of the martensitic lamellae was obtained by electron diffraction
(EBSD)
m e a s u r e m e n t s . The
sample
for the
EBSD
backscattered
measurements
was
firstly
mechanically polished and then electrolytically polished. The orientation analysis was performed with a Jeol J S M 6 5 0 0 F scanning electron microscope equipped with Channel
5 software ( H K L
Technology. Denmark). The twinning relationship between martensitic lamellae was determined 9
by misorientation calculation and the orientation relationship between austenite and martensite w a s determined from the information of the orientation of the martensitic lamellae '
1
.
RESULTS A N D D I S C U S S I O N The orientation map of alloy I is shown in Fig. 1 ( a ) , from which one can clearly see the selfa c c o m m o d a t e d martensitic microstructure. Typical fine lamellar structure is located in certain colonies that represent the initial austenite grains. In the orientation m a p illustrated in Fig. 1 ( a ) , the lamellae are colored according to their orientations. From the color pattern presented in each colony, one can see that there are only two martensitic lamellae distributed alternately in each initial austenite grain. T h e misorientation angle distribution for the martensitic lamellae is demonstrated in Fig. 1 (b). It can be seen that the misorientation angle between the martensitic lamellae is in the range of 78-84°. with a m e a n value of 82°. The variation of the misorientation angle by several degrees can be attributed to the inhomogeneous local stress in the sample. The distribution
of the rotation
axes in the crystal coordinate
system, corresponding
to the
misorientation angles 78-84°, is illustrated in Fig. 1 (c). It is evident that the rotation axes are around <110>M (hereafter the subscripts M and A in the indices of the crystallographic planes and directions denote that the planes and directions are expressed in the tetragonal crystal coordinate
system of martensite
4
and
the cubic
crystal coordinate
system
of austenite ,
respectively). Therefore, the misorientation between the neighboring martensitic lamellae can be represented as - 8 2 ° around <1 10>M axis. It should be noted that the misorientation mentioned above is the misorientation with the m i n i m u m misorientation angle, which is conventionally used to describe the misorientation between two crystals. However, in order to clarify the specific orientation relationship, or twinning relationship, all the misorientations between the neighboring lamellae should be calculated. Table 1 lists all the misorientations between the two martensitic lamellae A and Β shown in Fig. 1 ( a ) . It is seen that there are eight sets of equivalent misorientations, each set being defined by a misorientation angle ω, and the eight corresponding rotation axes, equivalent through the rotation point symmetry group. A m o n g the eight sets of equivalent misorientations, there are two sets of nearly 180° rotations around two families of equivalent axes. The indices in the tetragonal structure
398
·
4
with c / a = 1.7066 for the two 180° rotation axes are <47.49 45.29
Materials Processing and Texture
C r y s t a l l o g r a p h i c F e a t u r e s During Martensitic T r a n s f o r m a t i o n in Ni-Mn-Ga Alloys
44.2 1>m a n d <53.51 53.21 3 8 . 4 4 > , which are 1.63° a w a y from < 1 1 1 > M
M
and 2.19° away from
Fig. 1. Orientation m a p (a), misorientation angle distribution (b) and distribution of the rotation axes in the crystal coordinate system, corresponding to the misorientation angles 78-84° (c) for the martensitic lamellae in alloy I. The macroscopic sample coordinate system XoYoZo for the pole figure construction in Fig. 2 is illustrated in (a). < 3 3 2 > . respectively. The two axes [ U 1 ] m and [ 3 3 - 2 ] M
The planes normal to < 1 1 1 >
Μ
and < 3 3 2 >
M
M
are almost perpendicular to each other.
are close to {113}m (with 0.8° deviation) and { 1 1 2 }
M
(with 0.8° deviation), respectively. According to the classical definition of t w i n n i n g " , if there are two 180° rotations around two rational axes and the planes normal to the rotation axes are also rational, the two crystals have a c o m p o u n d twin relationship, with one rotation axis being the twinning direction % and the other being the normal of the twinning plane K | . Hence there are two possibilities: (i) K i = [ 1 1 2 } m , T|i=<11-1>m and (ii) K i = { 1 1 3 } m . t | i = < 3 3 - 2 > m . However, the second possibility is excluded by pole figure analysis, as shown below.
Materials P r o c e s s i n g a n d T e x t u r e
·
399
C r y s t a l l o g r a p h i c F e a t u r e s During M a r t e n s i t i c T r a n s f o r m a t i o n in N i - M n - G a Alloys
Table I. The misorientation angles and rotation axes between the two lamellae A and Β in the frame of Fig. 1 (a). For each misorientation a n g i e ß ) , there is a family of 8 equivalent rotations with 8 equivalent axes d which transform according to the tetragonal rotation symmetry group. Coordinates of rotation axes, d Misorientation angles,ω{°) 82.03 98.00 115.31 115.71 123.29 126.14 178.21 179.76
d,
d
d
69.01 70.50 1.85 77.48 0.24 84.63 53.51 47.49
72.37 70.90 77.66 1.84 85.74 0.24 53.21 45.29
2
3
0.33 2.07 62.98 63.19 51.46 53.27 65.61 75.45
Fig. 2 •{112)M and j 1 1 3 ) M pole figures obtained from the twin lamellae enclosed in the frame of Fig. 1 (a). T h e sample coordinate XoYoZo is shown in Fig. 1 (a). The square in each pole figure shows the overlapping poles of the t w o lamellae in the sample coordinate system. The traces of the twin interface and the c o m m o n planes are also illustrated. The ! 11 2 ) M and 1113 j vi pole figures obtained from the twin lamellae enclosed in the frame of Fig. 1 (a) are s h o w n in Fig. 2. It is s h o w n that one of the { 1 1 2 )
Μ
planes (whose normal is
< 3 3 2 > M ) is c o m m o n between the twin-related lamellae. Similarly, one c o m m o n ( 1 1 3 } Μ plane
400
Materials Processing and Texture
C r y s t a l l o g r a p h i c F e a t u r e s During M a r t e n s i t i c T r a n s f o r m a t i o n in N i - M n - G a Alloys
(whose normal is <11 1>M) can be clearly seen. These c o m m o n planes correspond to the twinning plane and the plane normal to the twinning direction, respectively. A s seen from the traces of the twin interface and the c o m m o n planes, the trace of the twin interface and the trace of the common {112}
M
planes are parallel to each other, suggesting that the twinning plane is { 1 1 2 } M
In contrast, the trace of the c o m m o n { 1 1 3 }
M
planes is not parallel to the trace of the twin
interface, which excludes the possibility that the twinning plane is {113 } . M
In c o m p o u n d twins, all the four twinning elements Κ ι , K 2 (the conjugate twinning plane), ηι and η
2
(the conjugate twinning direction) should be r a t i o n a l " . According to the method of n
analyzing twinning elements used by Kishida el a l . the normal of K
2
must be close to ηι for 13
the twinning shear to be realistically small and have low and rational i n d i c e s . The most reasonable choice of K is then found to be { 1 1 - 2 } , the normal of which is 10.7° away from η , . 2
M
The plane of shear Ρ containing ηι and the normals of K | and K . can thus be determined to be 2
{1-10}M, and η , the intersection of K 2
2
and P , is < 1 1 1 > . The magnitude of shear, s. calculated M
as s=2cot0 w h e r e θ is the acute angle between Ki and K . is 0.379. T h u s the twinning elements 2
in the observed twin lamellae are finally summarized as Ki={ 1 1 2 } M , K-I= {1 1-2}M, 1 I
= <
1 1 -1>M-
η = < 1 1 1 > , P = { l - 1 0 } - i =0.379. 2
Μ
M
During martensitic transformation, there generally exists a specific orientation relationship (OR) between austenite and martensite. If w e assume this OR to be expressed by the matrix T, it is possible to calculate the orientation of the austenite grain from the measured orientations of 9
the martensitic lamellae within the initial austenite grain ' '". If the orientations of austenite grain calculated from the different martensitic lamellae inherited from the same austenite grain have at 14
least one orientation in c o m m o n , then the assumed O R is c o r r e c t . Using this method and 14
considering the minimization of martensitic transformation strain e n e r g y , the prevalent OR between austenite and martensite is determined to be Kurdjumov-Sachs (K-S) relationship with (111)A//(101)M,[1-10] //[11-1]M. a
The S E M image of alloy II is shown in Fig. 3 (a). Nanoscale twins inside the lamellae can be clearly seen. The emergence of the nanoscale twins may be due to the stress accommodation in the sample. It can also be seen from Fig. 3 (a) that the nanoscale twins in each lamellae are paired together and every two twins in each pair have quite different thickness. One is thicker than the other. The schematic illustration of the configuration
of the nanotwins in two
neighboring lamellae is illustrated in Fig. 3 (b). in which 1 and 2 denote the thicker and thinner nanotwins in one lamella, respectively, while 3 and 4 denotes those in the other lamella. By m e a n s of misorientation calculation, the orientation relationships between the adjacent nanotwins are determined. It is shown that the internal nanotwins (1 and 2. 3 and 4) inside each lamella have a perfect twinning relationship, while the thicker nanotwins in the two lamellae (1 and 3) have a twinning relationship with several degrees deviation. The thicker nanotwin in one lamella and the thinner nanotwin in the other lamella (1 and 4. 2 and 3) have a degraded twinning relationship with enlarged imperfectness. In contrast, no twinning relationship exists between the thinner nanotwins in the t w o lamellae (2 and 4). The in-depth study on the formation mechanism and crystallographic features of the nanoscale twins is in progress.
Materials Processing and Texture
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Crystallographic F e a t u r e s During M a r t e n s i t i c T r a n s f o r m a t i o n in N i - M n - G a Alloys
Fig. 3. (a) S E M image of alloy II; (b) Schematic illustration of the configuration of nanotwins in alloy II. CONCLUSIONS (1) There are only two kinds of differently orientated martensitic lamellae with a misorientation of ~82°around < 1 1 0 > axis distributed alternatively in each initial austenite grain in alloy I. M
(2) There is a c o m p o u n d twinning orientation relationship between the neighboring lamellae in alloy I. The twinning elements are K i = i l l 2 } , K = { 1 1 - 2 } - ηι=<11 - l > M , M
2
M
r|2
= <
lU> , M
P= Î1 -10 j M . i - 0 . 3 7 9 . (3) The prevalent orientation relationship between austenite and martensite in alloy I is Kurdjumov-Sachs (K-S) relationship with ( 1 1 1 ) / 7 ( 1 0 1 ) . [1-10] //[ 1 1 - 1 ] . A
M
A
M
(4) Nanoscale twins inside t h e lamellae are observed in alloy II. T h e orientation relationships both between the nanotwins within one lamella and b e t w e e n the nanotwins
in t w o
neighboring lamellae are determined. ACKNOWLEDGEMENT The authors are grateful to the National Natural Science Foundation of China (Grant No. 50325102. 50531020 and 50528102) and the Ministry' of Education of China with the N C E T - 0 4 0282. Financial support from the 111 Project (B07015) and the P R A project ( M X 0 4 - 0 2 ) is greatly acknowledged. O n e of the authors (D. Y. C o n g ) is grateful to the financial support from the P h . D . fund of Northeastern University. China and from Region Lorraine. France. REFERENCES 'K. Ullakko. J. K. H u a n g , C. Kantner, R. C. O ' H a n d l e y . and V. V. Kokorin. Large magneticfield-induced strains in N i M n G a single crystals, Appl. Phys. Lett.. 69. 1966-68 (1996). 2
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G R A I N B O U N D A R Y M I S O R I E N T A T I O N A N D C S L IN M A G N E T I C A L L Y A N N E A L E D Fe-0.75%Si C.M.B. Bacaltchuk, G.A. Castello-Branco Centra Federal de Educaçào Tecnologica-CSF-RJ. Mechanical Engineering Department. Avenida Maracanà 299 Rio de Janeiro. RJ, 2 0 2 7 1 - 1 1 0 . Brazil. H. Garmestani Georgia Institute of Technology. Materials Science and Engineering Department. 771 Ferst Drive, N . W . Atlanta, Georgia, 3 0 3 3 2 - 0 2 4 5 , U S A .
ABSTRACT The purpose of this work is to investigate h o w a 17Tesla magnetic field applied during primary annealing at 800°C for 3 , 15 and 30minutes affects grain boundary misorientation and CSL of G N O Fe-0.75%Si steel samples. Magnetic annealing seems to affect Goss-oriented grains by increasing the percentage of high-energy boundary grains and Σ5 type grains.
INTRODUCTION Grain boundaries and inter-phase boundaries are important elements of the microstructure of most engineering metallic and ceramic materials. Studies have revealed that grain boundaries are not structureless but have a wide variety of atomic arrangements, which generate structuredependent properties and therefore are known to affect mechanical, physical and chemical properties . To completely describe a grain boundary five macroscopic parameters are needed, namely, three terms for the orientation relationship between two adjacent grains, such as the three Euler angles, ψ, θ, φ, and two terms for the spatial orientation of the grain boundary plane normal η with respect to one of the adjacent crystals. The m i n i m u m rotation angle required to bring two lattices into coincidence is called misorientation or disorientation angle. Misorientations are similar to orientations, but instead of bringing the crystal lattice into coincidence with the sample axes, a misorientation refers instead to bringing the crystal lattice of one grain into coincidence with another grain. Grain boundary misorientation angle has been associated to high or low grain boundary mobility . In 1953, Cotrell' noted that the mobility of grain boundaries in sub-grains with a low angle of misorientation was very much lower than the mobility for a high-angle grain boundary. As a result of this mobility difference, only sub-grains that are highly misoriented, typically by more than 15° with respect to at least part of their surroundings, can g r o w quickly and become recrystallizing grains. According to D u n n ' s work using F e - 3 % S i , the grain boundary energy increases as a function of grain boundary misorientation angle. One of the theories involving the growth of Goss grains in Fe-Si " * is based on the assumption that boundaries with misorientation angle between 20° and 45° have high energy and are therefore more mobile than boundaries with misorientation angle lower than 20° and higher than 45°. The categorization of grain boundary misorientations as angle/axis pairs is often supplemented by further classification according to the coincidence site lattice (CSL) 1
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model, especially for cubic materials'". The existence of a C S L indicates that the pattern formed by lattice points of both crystal lattices is periodic with the periodicity of the C S L " . The parameter Σ is described as the reciprocal density of coinciding sites or in other words, Σ 5 . for example, is a relationship equivalent to a coincidence of 1 in 5 lattice sites. The population of grain boundaries is divided into 3 categories: low-angle boundaries, coincidence grain boundary (CSL) and others or random boundaries. The low angle-boundaries are the boundaries with misorientation lower than 10°. Coincident boundaries associated with low Zs, generally lower than Σ 2 9 ( Σ 3 - 2 9 ) . are of interest from a processing point of v i e w ' - ' and the random boundaries or high-angle boundaries, which have misorientation angle larger than 15° and have no special relationship with their neighboring grain boundaries. A qualitative explanation of the behavior of special boundaries is based on the differences in impurity segregation at different types of boundaries. Because coincident-site special boundaries are "good-fit" boundaries, they should accommodate fewer solute a t o m s than other "bad-fif' or random boundaries, reducing the solute drag effect o n the grain boundaries. According to some a u t h o r s " C S L boundaries ate responsible for the growth of Goss grains in Fe-Si steels. 2
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Iron silicon or electrical steel is a soft magnetic material extensively used for electrical applications due to its high permeability and low coercivity. Grain non-oriented ( G N O ) iron silicon has its main technological application in rotating electrical machinery and the final microstructure of this material has a great effect on its magnetic behavior. The term "magnetic annealing" is used regularly to indicate the application of a magnetic field while a material is being heat-treated and it has been noticed that magnetic annealing affects microstructure evolution in ferrous and non-ferrous alloys "" . 2
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EXPERIMENTAL PROCEDURE G N O iron silicon samples with 0 . 7 5 % silicon content were used in this investigation. The samples were hot and cold rolled at A C E S I T A - Brazil and table I presents the chemical composition of the samples. C o u p o n specimens 5 m m w i d e . 8 m m long and 0.5 m m thick were sampled from the "as received" cold rolled sheet with their longitudinal axis parallel to the rolling direction of the sheet. Table I- Chemical composition %C Material %Si 0.003 G N O Fe-Si 0.75
%Mn 0.50
%S 0.002
%N 0.003
%A1 0.002
The magnetic annealing treatments were performed at 800°C. 17T, for 3 different annealing times ( 3 . 15 and 30 minutes) and were carried out in a cylindrical furnace inserted into the 195mm bore of a 20Tesla resistive magnet. An alumina sample holder, containing the samples, was placed inside the furnace at the center of the magnetic field with the samples positioned with their rolling direction parallel to the direction of the field. Table 11 s h o w s the nomenclature of the samples according to their processing. In order to evaluate the effect of the magnetic field, annealing without field was conducted in a second set of specimens, with the same temperature, times and a u n o s p h e r e used during magnetic annealing. Both annealing with and without magnetic field were performed at the National High Magnetic Field Laboratory ( N H M F L ) in Tallahassee - Florida. To minimize oxidation during the heat treatments, an inert atmosphere composed of 9 5 % a r g o n - 5 % hydrogen was used in both m o d e s of annealing.
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Table II- N o m e n c l a t u r e of the samples Sample Annealing Processing 03 Without field for 3 minutes 015 Without field for 15 minutes 030 Without field for 30 minutes
Sample M3 M15 M30
Annealing Processing With field for 3 minutes With field for 15 minutes With field for 30 minutes
All O I M m e a s u r e m e n t s were m a d e using an E S E M Model Geol E 3 , operating at 20kV. The electron gun source was L a B 6 with a resolution of about 4 nm. The vapor pressure in the E S E M chamber w a s maintained during the measurements at about 0.2T (~ 267 bar). RESULTS The percentage of grains with misorientation angle lower than 20°. between 20° and 45° and higher than 45° for all the measured grains and particularly for the Goss-oriented grains of each annealed samples is shown in table III. For the samples annealed without field, small changes were found as the annealing time increased. The frequency of low-misoriented boundaries (less than 20°) decreased with annealing time while the frequency of high-energy boundaries increased from 4 8 . 7 % to 51.2%. The same trend was observed for the Goss-oriented grains but the percentage of low-misoriented boundaries for these grains was much higher than the percentage for the whole set of measured grains (all grains) at all 3 annealing times. After magnetic annealing, differently from the ordinary annealing, the frequency of low-misoriented boundaries increased with annealing time from 1 3 . 3 % after 3 minutes to 2 0 . 5 % after 30 minutes of annealing. The middle-misoriented boundaries (20° - 45°) did not change much with annealing time and the fraction of high misoriented boundaries decreased. The fraction of lowmisoriented boundaries for the Goss-oriented grains in the magnetically annealed samples also increased with annealing time and it was more pronounced than the increasing observed for the other measured grains. The frequency of middle-misoriented boundaries in the Goss grains, in contrast to the general set, decreased with time from 5 9 . 4 % to 5 1 . 8 % . Middle-misoriented boundaries or high energy boundaries were the most frequent type of boundaries a m o n g all six samples for the whole set of grains as well as for the Goss grains regardless the application of magnetic field during annealing. Table III- Grain boundary misorientation data for the entire set of measured grains (all grains) and for the Goss-oriented after annealing without and with magnetic field Grain misorientation (all grains) - % Grain misorientation (Goss grains) - % Sample 20° - 45° >45° <20° 20° - 45° >45° <20° 03 015 O30 M3 M15 M30
21.6 19.2 18.6 13.3 20.4
48.7 49.6 51.2 52.1 48.2
20.5
51.1
29.7 31.2 30.2 34.6 31.4 28.4
38.9 34.3 34.5 18.4 30.0 33.8
44.2 47.7 50.9 59.4 53.9 51.8
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Table IV presents the distribution of grain boundary structure for the samples annealed without and with magnetic field. Table IV- Distribution of grain boundary structure of samples annealed without and with magnetic field Low-angle (%) Sample CSL R a n d o m (%) (Σ = 3 - 29) (%)
03 015 030 M3 M15 M30
14.7 13.8 11.7 6.8 11.6 12.8
74.2 74.8 75.9 81.0 77.2 77.0
11.1 11.4 12.4 12.2 11.2 10.2
After annealing without magnetic field, the percentage of special boundaries (CSL) increased very slightly with annealing time and the percentage of low-angle boundaries showed a small decrease. Magnetic annealing, seems not to affect appreciably the occurrence of special boundaries and only a small decrease, from 12.2% to 10.2% w a s observed with annealing time. The percentage of low-angle boundaries, in contrast, increased significantly with annealing time from 6.8%. after 3 minutes, to 12.8%. after 30 minutes of annealing. A m o n g all the samples, the magnetically annealed specimen M 8 3 . annealed for 3 minutes, was the one that showed fraction of low-angle boundaries closer to the fraction for a r a n d o m polycrystal. Table V shows the type and frequency of the Goss-grain boundaries for the samples annealed without and with magnetic field. The frequency of low-angle boundaries was quite high, 2 9 . 8 % . after annealing without field for 3 minutes and it decreased as the annealing time increased reaching 2 1 . 1 % after annealing for 30 minutes. After 3 minutes of magnetic annealing the amount of low-angle boundaries was very low, if compared to the sample annealed outside magnetic field and as the annealing time increased, the percentage of these types of grains increased reaching 2 0 . 8 % after 30 minutes of annealing. Table V- Distribution of Goss-grain boundary structure of samples annealed without and with magnetic field Low-angle (%) Sample CSL R a n d o m (%) (Σ = 3 - 2 9 ) ( % )
03 015 O30 M3 M15 M30
29.8 25.4 21.1 9.7 26.7 20.8
8.2 11.1 12.6 12.2 9.3 10.8
62.0 63.5 66.3 78.1 64.0 68.4
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1 5 2
1 7
1 8 1 9
are According to the C S L t h e o r y " " - " , the C S L boundaries namely Σ 5 ° . Σ 7 . Σ 9 responsible for the growth of Goss grains. Hutchinson et a l . ' have found a strong dependence of Goss texture on the matrix texture in relation to C S L relationship. Lin et a l . " suggests that Σ 3 9 boundaries collectively are responsible for the abnormal grain growth of Goss grains. H a r a s e pointed out that the mechanism of the Goss secondary recrystallization texture evolution in 20
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silicon steel with single stage cold-rolling is due to Σ 9 C S L boundaries. Besides the fraction of low-angle. C S L (Σ3-29) and high-angle random boundaries for Goss-oriented grains, presented in tables 4 and 5, the n u m b e r fraction of Σ 5 , Σ 7 , Σ 9 and Σ 3 - 9 boundaries were also evaluated and are being s h o w n in figure 4. According to this figure, the most r e m a r k a b l e effect of magnetic annealing w a s observed after 3 minutes o f annealing when the amount of Σ5 boundaries found in the magnetically annealed sample w a s more than twice as m u c h as the fraction in the sample annealed without field. After 1 5 minutes, magnetic field seems to influence the number fraction of Σ7 and 1 9 boundaries, which exceeded the fraction obtained for the ordinarily annealed sample. After 30 minutes, magnetic field showed to be an advantage over ordinary annealing only for the Σ 7 type of boundaries. Lee and S z p u n a r investigating a Fe-3%Si observed that G o s s grains having higher a m o u n t of Σ5 CSL boundaries before secondary' recrystallization grew faster than other grains. 28
Figure 4 - Variation of C S L boundaries in Goss-grains for samples annealed without and with magnetic field. DISCUSSION After annealing with and without field, the most remarkable difference in the results of grain boundary formation w a s found for the samples annealed for 3 minutes. The presence of fewer sub-grains, denoted by a low percentage o f grains with misorientation angle lower than 20°. in sample M 3 can be an indication of nucleation retardation. The high-energy grains do not seem to be affected neither by the field nor by annealing time, since the percentage of grains within these angles of disorientation did not vary significantly a m o n g the samples annealed by
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both modes. For the Goss-oriented grains, on the other hand, magnetic field increased the percentage of high-energy boundary grains, mainly after 3 minutes of annealing. According to Hayakawa and S z p u n a r the high percentage of high-energy boundaries found in Goss-oriented grains after primary recrystallization of high-permeability and conventional electrical steels matrices is the cause for the strong Goss texture developed after secondary annealing of these materials. After annealing with and without field, the most remarkable difference in the results of grain boundary formation was found between the samples annealed for 3 minutes. Sample M 3 showed a much smaller percentage of low-angle boundaries, a higher percentage of random boundaries and a slightly higher percentage of CSL boundaries (Σ3 - 29) than sample 0 3 . Magnetic field tended to increase the percentage of low-angle boundaries and to decrease the percentage of C S L boundaries (Σ3 - 29) as the annealing time increased, while the opposite trend was observed in the samples annealed without field. After 3 minutes of annealing, no significant difference was observed in the percentage of the three distinct groups of boundaries. 8 9
In the case of Goss grains, it was found a large difference in the percentage of low-angle boundaries between the samples annealed for 3 minutes. In both annealing modes, the variation with annealing time in the percentage of low, high and special boundaries (table V) followed, basically, the same trend as the other grains. The major differences between the Goss grains and the general set of grains were the percentages of low-angle boundaries, that were m u c h higher in the Goss grains, and the percentage of random boundaries that were lower than the percentage found for the general set of grains. The analysis of CSL boundaries has shown that for relatively short annealing times. 3 minutes, recrystallization inside magnetic field can generate a higher n u m b e r of Goss grains with special boundaries. RLFERENCF.S 1
T. Watanabe. " T h e potential for grain boundary design in materials development". Materials Forum, 1 1 . pp.284-303 (1988). "R.D. Doherty. "Recrystallization and texture". Progress in Materials Science, 4 2 , 39-58 (1997). Ά.ΙΙ. Cotrell: Progress in Materials Science. 4, 255 (1953). C . G . Dunn. F.C. Daniels and M.J. Bolton: Journal of Metallurgy. 1 8 8 , 1245 (1950). Y . H a y a k a w a and J.A. Szpunar. " A new model of Goss texture development during secondary recrystallization of electrical steel". Acta Materialia, 4 5 , 4 7 1 3 - 2 0 (1997). Y . H a y a k a w a M. Muraki and J.A. Szpunar, "The changes of grain boundary character distribution during the secondary recrystallization of electrical steel ", Acta Materialia, 4 6 , 106373 (1998). N . Rajmohan. J.A. Szpunar and Y. Hayakawa, "Importance of fractions of highly mobile boundaries in abnormal growth of Goss grains", Materials Science and Engineering A, 2 5 9 , 8-16 (1999). Y . H a y a k a w a . J.A. Szpunar. G. Palumbo and P. Lin, " T h e role of grain boundary character distribution in Goss texture development in electrical steels", Journal of M a g n e t i s m and Magnetic Materials. 160, 143-46 (1996). Y. H a y a k a w a and J.A. Szpunar. " T h e role of grain boundary character distribution in secondary recrystallization of electrical steels", Acta Materialia. 4 5 , 1285-95 (1997). 4
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V. Randle and O. Engler: Introduction to Texture Analysis: Macrotexture, microtexture and Orienattion M a p p i n g . Gordon and Breach Science Publishers. 2000. " H . Grimmer, W. Bollmann and D.H. Warrington. "Coincidence-site lattices and complete pattern-shift in cubic crystals", Acta Crystallographic. A 3 0 . 197-207 ( 1974). V . Randle. N . Hansen and D.J. Jensen, " T h e deformation behaviour of grain boundary regions in polycrystalline a l u m i n i u m " . Philosophical M a g a z i n e . A 7 3 . pp. 265-82 (1996). A . P . Sutton and R . W . Balluffi. "Overview no. 61 On geometric criteria for low interfacial energy". Acta Metallurgica. 3 5 . 2177-2201 (1987). L . S . Shvindlerman and B.B. Straumal. "Regions of existence of special and non-special grain boundaries". Acta Metallurgica, 3 3 . 1735-49 (1985). P . Gangli and J.A. Szpunar. "The Role of Σ5 Coincidence Boundaries in the Growth Selection of Fe-Si'". Journal of Materials Processing Technology. 4 7 , 167-84 (1994). 12
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'"G. Abbruzzese. S. Fortunat! and A. C a m p o p i a n o . " O n the Parameters Controlling the Kinetics of Secondary Recrystallization in Grain Oriented Silicon Iron", Materials Science Forum. 9 4 - 9 6 . 405-12"( 1992)* l7
J . Harase, "Effect O f Cross Rolling Reduction on the Texture Evolution by Grain Growth in F e - 3 % Si Alloy". Materials Science Forum. 94-96. 419-24 (1992). Y . Yoshitomi. Y. U s h i g a m i . J. Harase. T. N a k a y a m a , H. Matsui and N. Nakahashi. "Influence Growth of Primary Recrystallized Grains on Secondary Recrystallization Texture in Fe-3%Si Alloy". Materials Science Forum. 113-115. 715-20 (1993). , 8
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Y . Ushigami, S. N a k a m u r a . S. Takebayashi and S. Suzuki. "Effect of Grain BoundaryCharacter on Selective Growth of Goss Grain in Fe-3%Si Alloy ". Materials Science Forum. 408-412. 973-78 (2002). N . C . Pease. D . W . Jones, M.H.L. Wise and W . B . Hutchinson: Metallurgical Science, 15. 203 (1981). ~ ' W . B . Hutchinson and H. H o m m a . "Misorientation dependence of growth rate in secondary- recrystallization in silicon iron", Proceedings of the Third International Conference of Grain G r o w t h . 3 8 7 ( 1 9 9 8 ) . P . Lin. G. P a l u m b o . J. Harase and K.T. Aust. '"Coincidence Site Lattice (CSL) grain boundaries and Goss texture development in F e - 3 % Si alloy". Acta Materialia. 44, pp.4677-83 (1996). N . M a s a h a s h i . M . Matsuo and K. W a t a n a b e . " D e v e l o p m e n t of Preferred Orientation in Annealing of Fe-3.25%Si in a High Magnetic Field". Journal of Journal of Materials Research. 1 3 , 4 5 7 - 4 6 1 (1998). T . Watanabe. Y. Suzuki. S. Tanii and H. O i k a w a . ""The Effects of Magnetic Annealing on Recrystallization and Grain-Boundary Character Distribution ( G B C D ) in Iron-Cobalt Alloy Polycrystals", Philosophical Magazine Letters. 62. 9-17 (1990). Y . Xu, H. Ohtsuka. K. Itoh and H. W a d a . "Effects of High Magnetic Field on Recrystallization and Coarsening Behavior in Fe-Si Steels". Journal of the Magnetics Society of Japan. 2 4 . 6 5 1 - 6 5 6 ( 2 0 0 0 ) . J . Harase. R. Shimizu and D.J. Dingley. "Texture evolution in the presence of precipitates in F e - 3 % Si alloy". Acta Metallurgica, 3 9 . 763-70 ( 1991). J . Harase. R. Shimizu, Y. Yoshitomi. Y. Ushigami and N. Nakahashi: Proc. Materials W e e k ' 9 2 . Chicago, Illinois. (1992). K . T . Lee and J.A. Szpunar. The Role of Special Boundaries during the Grain Growth in F - 3 % Si". Canadian Metallurgical Quarterly, 3 4 , 257-63 (1995). 2I)
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T H E E F F E C T O F U L T R A - H I G H M A G N E T I C F I E L D S ON G R A I N G R O W T H A N D R E C R Y S T A L L I Z A T I O N IN M E T A L S A.D. Sheikh-Ali* and H. Garmestani** *Department of Physics and Engineering, Kazakh-British Technical University 59 Tole bi Street, 0 5 0 0 0 0 . Almaty, Kazakhstan Present address: Dayton T. Brown. Bohemia, N Y 11716. U S A e-mail: askarfliS.yahoo.com " G e o r g i a Institute of Technology, Materials Science and Engineering. 771 Ferst Drive. N . W . , Atlanta, G A 3 0 3 3 2 - 0 2 4 5 . U S A ABSTRACT Cold rolled Zn-1.1%A1. Ti and Cu were annealed in a direct-current magnetic field with strength of 15.4 and 25.5 M A / m . For comparison the samples from the same metals were annealed out of the field. The textures in Zn-1.1%A1 and Ti were characterized by the two 0002 components tilted at ±15-20° about the transverse and at ±30° about the rolling direction, respectively. Treatment in the field results in a significant difference between texture peak components when the samples of Ti are tilted at +30 or -30° to the field direction about the rolling direction and samples of Zn-1.1%A1 are tilted at + 1 9 or -19° about the transverse direction. The microhardness measurements in Ti and Cu reveal no effect of the field at the initial stages of recrystallization. A small difference in microhardness after annealing with and without field has been observed at the end of recrystallization and at the beginning of grain growth. Higher microhardness has been found after treatment in the field. The observed effects are ascribed to the difference in magnetic free energy of different grains due to the difference in their orientations or dislocation densities. INTRODUCTION The exposure of crystalline solids to magnetic fields at high temperatures can affect their structures due to different effects such as the increase in dislocation mobility [1]. stimulation of recrystallization [2, 3] or retardation of recrystallization and grain growth [4-7]. suppression of abnormal grain growth in nanocrystalline materials [8]. Such effects depending on the nature of a material and its original structure can be interpreted in terms of magnetostriction [2]. magnetic ordering [6], magnetoplastic effect [9] and the driving forces exerted by the difference in magnetic free energy across grain boundaries [10]. Existence of alternative explanations of same effects demonstrates that we are still far from their understanding. The interpretation of such effects for ferromagnetic materials seems to be the most difficult due to simultaneous presence of few factors related to magnetic field. Therefore it looks very promising to study the behavior of nonferromagnetic materials bearing in mind that some effects may exist in both ferromagnetic and non-ferromagnetic materials. The increased availability of high-magnetic fields with the strength of 10 Τ or higher gives a lot of opportunities to explore the effects of thermomagnetic treatments on structure of non-magnetic materials. There are two mechanisms which can be responsible for the magnetically related effects during grain growth and recrystallization in dia- and paramagnetic materials: depinning of dislocations from paramagnetic centers [9] and boundary migration induced by the differences in magnetic free energy [10]. In this paper, we review our results on the influence of magnetic field on sùticture of nonmagnetic metals interpreted in terms of the difference in magnetic free energy across grain boundaries. 7
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M A G N E T I C A L L Y I N D U C E D F O R C E A C T I N G ON B O U N D A R Y If the volume density of the magnetic free energy ω in a crystal induced by a unifonn magnetic field is independent on crystal shape and size (the condition for this is% « 1 ) then the magnetic force acting on the boundary of two crystals that have different magnetic susceptibilities is given by Mullins [10]: 2
ρ =
ω
ι
- ω
2
U H =^y- (χι-χ )
(i)
2
where χ, and χ , are the susceptibilities of crystal 1 and 2. respectively. Η is the field strength. T E X T U R E M O D I F I C A T I O N BY S E L E C T I V E G R A I N G R O W T H For the first time the effect of ultra-high magnetic field on evolution of texture was demonstrated by Sheikh-Ali et al. in recrystallized Zn-1.1%A1 alloy [11]. Specimens of the rolled alloy ( 9 9 % reduction) were annealed at 390°C in the field of 32 Τ for 55 min. The specimens were oriented differently with respect to the field. The rolling direction of one set of specimens were parallel to the field, the other specimens were tilted at +19° and -19° to the field about their transverse direction (Fig. 1). Originally rolled texture is characterized by two 0002 components tilted s o m e 15-20° about the transverse direction (Fig. 2a). Annealing of these specimens with no field slightly changes intensities of the texture peaks preserving the type of the pole figure (Fig. 2b). In contrast, annealing in the field m a k e s drastic changes in texture. In the case of specimen oriented parallel to the field there is a unification of t w o peaks into one although the positions the most intense subcomponents correspond to the positions of the original components (Fig. 2c). Magnetic annealing of specimens tilted at +19° to the field completely removes the component Β and increases the intensity of component A (Fig. 2d). In the case of specimens tilted at -19° the component A completely disappears and the intensity of component Β increases significantly (Fig. 2e). The observed change in type of texture can be understood as a result of selective grain growth induced by additional driving force created by the anisotropy of the magnetic susceptibility of Zn. In the case of zinc, Eq. (1) is transformed to
2
2
p = M ^H (cos 6|-COS 6 ) 2
0
(2)
2
where Θ, and θ are the angles between the direction of magnetic field and the hexagonal (or c or <0001> axis) in both neighboring grains. Δ χ is the difference of susceptibilities parallel and perpendicular to the hexagonal axis. T h e force ρ is directed towards the grain with smaller value of θ and does not depend on the sign of the magnetic field. The difference in the magnetic free energy between different grains can be estimated using the diamagnetic susceptibilities of Zn crystals measured by MacClure and Marcus [12]. Conversion of these data from Gauss unit system gives for volume susceptibility in SI units as χ^= - 1 . 6 9 5 - 1 0 " and χ±= — 1.294-10" . In this case the orientation of c-axis of the second component with respect to the field direction ranges between 50° and 60° and the magnetic force according to Eq. (2) varies from 0.4 to 0.7 kJ/m3. 2
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RD f H
a
H
b
RD
RD
t
H
c
Figure 1. Orientations of specimens with respect to the field during annealing, (a) Rolling direction is parallel to the field, (b) and (c) rolling direction is tilted from the direction of the field at +19 and -19°. respectively.
Figure 2. 0002 pole figures of Zn-1.1%A1 sheet specimens before (a) and after annealing (b-e). (b) Annealing without field: (c) oriented parallel to the field: (d) tilted at +19° to the field about the T D : (e) tilted at -19° to the field about the T D .
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This force can be compared with usual capillary driving force for grain growth determined as
where σ is grain boundary energy. R - m e a n grain radius. A s s u m i n g grain boundary energy of typically 0.3 J/m" and using Rs0.5 m m the capillary force a m o u n t s to p s l . 2 k J / m . Thus, a comparison of the respective driving forces reveals that the magnetic force is at least of the order of the capillary forces and able to make a strong influence on grain growth increasing the growth rate of those grains which <0001> axis is perpendicular to the field. Similar results were obtained in commercially pure Ti annealed in the field of 19.4 Τ [13]. The original 0002 texture after annealing of rolled ( 8 2 % ) sheet at 750°C is shown in Fig. 3 (a.b). The annealing without the field does not change the texture. The annealing in the field of specimens tilted by +30° and -30° to the field direction around the rolling direction results in a noticeable difference between 0002 components which have the s a m e original intensity (Fig. 3 c, d). 3
0
The experiments with bicrystals of Z n can be considered as an additional proof of selective grain growth induced by annealing of magnetically anisotropic materials in magnetic
Figure 3. Magnetic annealing of commercially pure titanium at 750°C. 0002 pole figure (a) and pole figure profiles (b.c.d) in N D - T D plane after regular annealing (a, b) and annealing in the field with T D tilted at +30° (c) and -30° (d) to the field direction.
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a
b
Figure 4. Grain boundary displacement in Z n bicrystals during annealing for 100 h 7
without the field (a) and 5 min in the field of 1.35· 10 A / m (b) at T=663 K.
fields [14]. Optical micrographs of bicrystals subjected to magnetic annealing and annealing without field are shown in Fig. 4. Annealing during 100 h does not change the position of the whole boundary. Only capillary boundary migration of the boundary ends has been observed (Fig. 4a). In contrast, during magnetic annealing for 5 min boundaries migrate towards grain A for a distance of almost 6 m m (Fig. 4b). R E C R Y S T A L L I Z A T I O N IN M A G N E T I C F I E L D The experiments were performed on samples of cold rolled polycrystalline Cu (99.95%) and Ti (99.999%). C o p p e r was used in as-received cold-rolled condition. Titanium w a s cold rolled up to 81 % reduction. The samples were annealed with and without the field for 15 min at a constant temperature in the range from 373 to 873 K. The m a x i m u m working field of 32 Τ has been applied. The rolling direction of the samples was oriented along the field. Both annealing out of the field and in the field were completed by water quenching of the samples. The microhardness of the samples was measured across the thickness of the sheet using a Vickers indenter with 100 g loads. Fig. 5 shows that with the rise of the annealing temperature the hardness of specimens decreases and then stagnates. The specimens after thermomagnetic treatment show a higher hardness value at the temperatures corresponding to the end of hardness descent and the beginning of its stagnation. The obtained results demonstrate that annealing both in the field and without field m a k e s a decrease in microhardness which looks typical for the process of recrystallization [15] The differences between magnetically and regularly annealed specimens begin w h e n recrystallization is essentially complete and grain growth is starting. For all investigated metals representing diamagnetics and paramagnetics the magnetic annealing preserves a slightly higher microhardness in comparison with regular annealing. The observed increase in microhardness can be understood as a result of the enhanced dislocation density left after thermomagnetic treatment. Magnetic susceptibilities of cold worked pure metals such as Cu and Ti are found to be higher than that after annealing [16-18]. Therefore, grains with residual dislocations left after cold work have higher paramagnetic or lower diamagnetic susceptibilities than dislocation-free grains. The magnetic free energy of grains with dislocations should be
Materials Processing and Texture
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Effects of U l t r a - H i g h M a g n e t i c Fields o n G r a i n G r o w t h a n d Recrystallization in M e t a l s
lower than the energy of " n e w " recrystallized grains. The inhibition force exerted by the difference in the magnetic free energy of neighboring grains due to different dislocation density can be calculated using Equation 1 where χ, and fa are magnetic susceptibilities of neighboring grains with different density of dislocations. According to Lowance and Constant [17] the magnetic susceptibility of Cu in preferred units before and after cold work is - 1 . 2 2 - 1 0 " and - 0 . 8 8 Ί 0 " , respectively. Conversion of these data to volume susceptibility gives - 1 . 0 9 - 1 0 ° and -0.79-10" . In the case of the field strength of 2.55-10 A / m and difference in volume susceptibilities of neighboring grains Δ χ = χ ι - χ 2 = 3 . 0 5 - 1 0 " and p s l . 2 5 k J / m . T h e driving force for primary recrystallization is strain energy of stored dislocations. For the migrating boundary moving under a driving pressure arising from the dislocation density difference the driving force will be 6
6
5
7
6
p =ApGb
3
2
(4)
r
where Δρ is the difference in dislocation density across the boundary, G is the shear m o d u l u s and b is the Burgers vector. For A p ~ 1 0 m" , G ~ 1 0 k J / m , b ~ 1 0 " ' m , p = 1 0 k J / m . The essential difference between the magnetic inhibition force and driving force for primary recrystallization allows the latter to be almost completed. The growth rate during recrystallization w a s found to decrease significantly with annealing time [19]. The main reason for the change in growth rate seems to be the variations in the driving force of primary recrystallization [20]. The latter decreases due to microstructural inhomogeneity and recovery which occurs during primary recrystallization. T h u s the process of recrystallization in the field should stop when its driving force reaches the magnetic inhibition force. 15
2
8
3
2
9
2
4
r
418
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Materials Processing and Texture
3
Effects of U l t r a - H i g h M a g n e t i c Fields o n Grain G r o w t h a n d Recrystallization in M e t a l s
SUMMARY Annealing of non-magnetic metals in high magnetic fields can affect the processes of grain growth and recrystallization. It is demonstrated that selective grain growth and textural changes occurred in zinc and titanium are exerted by the difference in magnetic free energy due to anisotropy of magnetic susceptibility. Magnetic driving force has the same order of magnitude as the capillary driving force at grain sizes of - 0 . 1 mm and higher. The difference in microhardness has been established for the final stage of primary recrystallization and beginning of grain growth following recrystallization for the samples annealed with and without the field. It is suggested that this difference is attributed to different density of dislocations in recrystallized and non-recrystallized grains. The grains with higher dislocation density have lower magnetic free energy and therefore more stable in the field. ACKNOWLEDGEMENTS The research was sponsored by the US Army Grants D A A D 19-99-1-0311 and 19-01-10742. The help of Dr. Bruce Brandt, Dr. Scott T. Hannahs and Mr. Bobby J. Pullum (DC Field Facilities of N H M F L ) is gratefully acknowledged. REFERENCES 'V.l. Alshits, E.V. Darinskaya. I.V. Gektina, and F.F. Lavrent'ev: Sov. Phys. Crystallogr. 3 5 . 597(1990). T . Chen, and W.E. Stutius: J. A p p . Phys. 4 5 , 4622 (1974). D . A . M o l o d o v , S. Bhaumik, X . Molodova, and G. Gottsein: Scripta Mater. 5 4 , 2161 (2006). Y u . N . Markov, and R.A. Adamesku: Fizika Metallov i Metallovedenie 3 2 , 800 (1971). H . O . M a r t i k a i n e n , and V.K. Lindroos: Scand. J. Metall. 1 0 , 3 (1981). °T. Watanabe. Y. Suzuki, S. Tani, and H. Oikawa: Phil. M a g . Lett. 6 2 , 9 (1990). Y . Xu, H. Ohtsuka, K. Itoh, and H.Wada: J. Magnetic Soc. Japan 2 4 , 6 5 1 . (2000). K . Harada, S. Tsurekawa, T. W a t a n a b e , and G. P a l u m b o : Scripta Mater. 4 9 , 367 (2003). M . I . Molotskii: Sov. Phys. Solid State 3 3 , 1760 (1991). W . W . Mullins: Acta Metall. 4 , 421(1956). " A . D . Sheikh-Ali, D A . M o l o d o v , and H. Garmestani: Scripta Mater.. 4 6 , 857 (2002). J . W. M a c C l u r e , and J. A. Marcus: Phys. Rev., 84, 787 (1951). D . A . M o l o d o v , and A . D . Sheikh-Ali: Acta Mater. Vol. 5 2 , 4 3 7 7 (2004). A . D . Sheikh-Ali, D.A. Molodov. and H. Garmestani: Appl. Phys. Lett. 18, 3005 (2003). " W . D . Callister: Materials Science and Engineering. An Introduction. (John Wiley & Sons, 6 edition 2 0 0 5 . p. 183). F . Bitter: Phys. Rev. 3 6 , 978 (1930). F . E . L o w a n c e , and F.W. Constant: Phys. Rev. 3 8 , 1547 (1931). S . V . Vonsovskii, V.A. Pavlov, A.I. Deryagin, and K.V. Vlasov. in: Phase Transformations and Structure of Metals and Alloys. Collection of Papers, edited by V.D. Sadovskii / Acad. SSSR, U S S R , (1982), p. 7 1 . B . B . Rath, R.J. Ledrerich, C F . Yolton, and F.H. Froes: Metall. Trans. 1 0 A , 1013 (1979). F . J . H u m p h r e y s and M. Hatherly: Recrystallization and Related Annealing Phenomena (Pergamon, 1995, p. 196). 2
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A NEW APPROACH TO TEXTURING AND GRAIN BOUNDARY ENGINEERING BY M A G N E T I C F I E L D A P P L I C A T I O N Tadao Watanabe Formerly T o h o k u
University, Sendai, Japan, N o w Visiting Professor,
Key Laboratory of
Anisotropy and Texture of Materials, Northeastern University, Shenyang, China. Sadahiro T s u r e k a w a Faculty o f Engineering, K u m a m o t o University, K u m a m o t o . 860-8555, Japan. Yudong Z h a n g . Xiang Zhao, Liang Z u o Key Laboratory of Anisotropy and Texture of Materials, Northeastern University. Shenyang, 110004. China. Claude Esling : L E T A M , C N R S - 7 0 7 8 , University o f Metz. Metz. 57045, France. ABSTRACT Recent achievements of the texturing and grain boundary engineering (GBE) by magnetic field application have been introduced. The versatility and high potential of magnetic field application have been effectively utilized as non-contacting processing for the control of texture and grain boundary microstructures to endow polycrystalline materials with high performance and excellent bulk properties. It has been confirmed by experiments that various types of magnetic processing can be applied to ferrous and nonferrous materials. Grain boundary-related metallurgical phenomena and bulk propertie s in polycrystalline materials have been proved to be well controlled by magnetic field application. INTRODUCTION The control of microstructures is the primary issue of materials design and development of high performance structural and functional materials, based on the discipline of "Materials , 2
Science and Engineering" established in the last century ' . Grain and interphase boundaries are the most important microstructural c o m p o n e n t s generally existing in all polycrystalline materials. It seems m o r e appropriate to use the term "the grain boundary microstructure" in discussing structure-dependent bulk properties o f polycrystalline materials. Since almost all properties o f crystallite strongly depend on their crystal structures and orientations, the orientation distribution for a huge n u m b e r of crystallites, i.e. texture is one o f key factors controlling bulk properties o f polycrystalline materials
The control of bulk properties by texturing has been extensively
applied to development of structural and functional engineering materials. On the other hand, in the last t w o decades, the control of the grain boundary microstructure defined by the grain boundary character distribution ( G B C D ) and other boundary-related parameters has been increasingly applied to development of high performance materials by the concept of Grain Boundary Engineering
4 _ 6
. Particularly in the past 10 years there has been great strides in the
grain boundary engineering
7 I
° . N o w w e have c o m e to the stage in which we should effectively
and precisely control the texture and grain boundary microstructure so as to produce high performance polycrystalline materials by more sophisticated processing, as much as we can.
421
N e w A p p r o a c h t o T e x t u r i n g a n d Grain B o u n d a r y E n g i n e e r i n g by M a g n e t i c Field A p p l i c a t i o n
BASES OF MAGNETIC TEXTURING AND GRAIN B O U N D A R Y ENGINEERING The application of external fields during material fabrication processing is known to be useful and powerful for introducing an o p t i m u m microstructure to endow a polycrystalline material 1
with high performance and desirable bulk p r o p e r t i e s " ' ' . A m o n g applicable external fields, a magnetic
field
is unique as non-contacting
field
having the feasibility
of handling,
the
preciseness and flexibility o f field control. The application of a magnetic field was attempted for the control of microstructure and bulk properties of engineering materials like magnetic materials, at the beginning of the last century " . We know that the effect of a magnetic field can be increased rapidly, as expected from the second power dependence of the magnetic free energy U on the
field
strength H. U= - 1 / 2 μ χ Η
2
0
(μ„ the magnetic permeability in vacuum, χ . the
susceptibility). Until recently, it w a s difficult to perform experimental study on the effect of a strong magnetic field on metallurgical p h e n o m e n a . For this purpose, we need two experimental tools: one is a magnetic which can produce a magnetic field strong enough to study the effect on metallurgical p h e n o m e n a associated with bulk properties o f polycrystalline materials. N o w a d a y s a high magnetic field up to 10T (Tesla) is available with a helium-free superconducting magnet at university laboratory. Figure 1(a) and
1(b) show the whole view and schematic of a
helium-free superconducting magnetic field heat treatment system used by the authors at Tohoku University. The m a x i m u m field strength is 7T and the m a x i m u m heating temperature is 1773K in vacuum or inert gas. The most important feature of this heating system is that the programmed magnetic processing can be started within a few minutes, by inserting a specimen into the field zone
keeping
programmed
test
magnetic
field
and
temperature.
F E - S E M - E B S D / O I M (Orientation Imaging Microscope) technique
14
The
computer-assisted
has enabled us to perform
quantitative analyses of local textures and grain boundary microstructures for polycrystalline specimens with grain size ranging from ordinary size of l-ΙΟΟμιη to submicron up to 30nm in nanocrystalline m a t e r i a l s
Fig.l
Overall v i e w
(a)
151
".
and
schematic
illustration
(b)
o f a liquid He-free superconducting
high
magnetic field heating system u s e d at T o h o k u University. The m a x i m u m field strength is 7T and
the
m a x i m u m heating temperature 1773K.
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N e w A p p r o a c h t o T e x t u r i n g a n d G r a i n B o u n d a r y E n g i n e e r i n g b y M a g n e t i c Field Application
T E X T U R I N G BY M A G N E T I C F I E L D A P P L I C A T I O N It is expected that the application of a magnetic field during the formation of a new solid crystalline phase and its subsequent growth results in the evolution of a large variety of mierostructures. depending o n processing condition and the detail of applied magnetic field. This is primarily due to magnetocrystalline anisotropy associated with intrinsic structural anisotropy of crystalline phase and shape effect. It would be possible to produce different mierostructures and textures by using a well-controlled magnetic field in order to e n d o w a polycrystalline material with desirable bulk properties and high performance.
Theoretical Prediction of Relationship between Texture and G B C D Theoretical predictions of the grain boundary character distribution ( G B C D ) have been 1 7
performed by several g r o u p s "
2 0
for differently textured polycrystalline materials and compared 21
23
with the experimental data reported on Fe-6.5mass%Si alloy r i b b o n s " , showing a good agreement with experimental observations. Moreover, the grain boundary character distribution (GBCD) for polycrystals with a mixture of different types of textures, such as <100> + < 1 1 1 > 17
has been performed by Garbacz and G r a b s k i . '
20
The physical significance of the occurrence of a
high frequency o f low-energy boundaries is such that severe brittleness caused by intergranular fracture
can be effectively
controlled
by increasing the frequency
of low-Σ
coincidence
boundaries higher than 4 5 % , reversely by decreasing the fraction of weak random high-energy boundaries less than 5 5 % . This can be intuitively understood from the threshold value of the fraction
of
strong
boundaries
for
percolation-controlled
intergranular
fracture
of
a
two-dimensional polycrystal with the grain boundary microstructure composed of hexagonal 24
network of grain b o u n d a r i e s . Development of high performance polycrystalline materials by the grain boundary engineering has been discussed on the basis of the relation between texture 3
and the grain boundary microstructure" .
Magnetic Annealing in Thin Ribbons. Probably the papers by Watanabe et al. on the texture and the grain boundary character 2
23
distribution ( G B C D ) in Fe-6.5Mass%Si r i b b o n s ' " a r e the first that reported a close relationship between the type and sharpness of" texture and G B C D . and explained the reason for high ductility of "intrinsically brittle" polycrystalline Fe-6.5mass%Si ribbons. As-rapidly solidified ribbons and slightly annealed one showed a random grain orientation distribution (random texture). Of great interest is that when annealed at 1363K or 1473K for Ihr, a very sharp {100} or {110} texture occurred respectively. T h e most important feature is that the frequency of low-Σ coincidence boundaries with specific Σ values which are considered as low-energy boundaries, occurred preferentially: low-Σ coincidence boundaries with Σ 1 . 5, 13, 25 occurred in descending order in the {100} textured ribbon, while Σ 1 , 3, 9, 11. 17. 1 9 i n t h e {110 J textured ribbon. These Σ values are explained by the possible coincidence orientation relationships for <100> or <110> rotation on the basis of the theory of coincidence-site-lattice (CSL). There is direct evidence that 1
low'-Σ coincidence boundaries have low-energy so that they can occur preferentially* . As seen
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N e w A p p r o a c h t o T e x t u r i n g a n d G r a i n B o u n d a r y Engineering b y M a g n e t i c Field A p p l i c a t i o n
from Fig.2. the relative grain boundary energy determined for the {110}textured Fe-6 . Smass^oS i ribbon showed that Σ 3 / < 1 1 0 > and Σ 9 < 1 1 0 > coincidence boundaries have low-energies, being 0.77-0.81 or 0.82-0.87 o f t h a t o f r a n d o m boundary, respectively. Only a little difference o f the relative boundary energy due to the boundary inclination effect was observed: smaller than 5 % for Σ3. and 6 % for Σ 9 coincidence boundaries. This gives us a useful information to estimate a possible c h a n g e o f the grain boundary energy depending on the boundary type and boundary inclination. Next let us look at recent finding that the application o f a magnetic field during annealing c a n produce a quite different-Upe o f texture compared with those obtained by nonmagnetic/ordinary annealing for the same material. We have recently found that magnetic annealing at
1373K 27
produced a sharp < 1 1 0 > type-texture which is normally formed by ordinary annealing at 1 4 7 3 K . Thus it was revealed that the magnetic annealing of F e - 6 . 5 m a s s % S i ribbon can p r o d u c e the same type o f texture even at lower annealing temperature. This finding suggests the possibility o f development o f a new technique for texturing by magnetic field application to introduce a specific type o f texture with variable sharpness and grain boundary microstructure. This may be attributed to the retardation o f surface diffusion by magnetic field application which has been recently found from the m e a s u r e m e n t o f the dihedral angle o f grain boundary in magnetically 28
annealed α-iron p o l y e r y s t a l s . It is reasonable to consider that surface diffusion
plays an
important role in grain g r o w t h in thin ribbon specimen so that it can directly affect the grain growth and texturing in Fe-6.5mass%Si alloy ribbons.
Fig.2
Relative grain boundary energies as a
function o f the misorientation angle for [110J 26
textured Fe-6.5mass%Si r i b b o n .
Fig.3.
The evolution o f {110J t e x t u r e i n
Fe7gSiyBi3 alloy after magnetic crystallization 2<
in a magnetic field up to 6T at 853K for 1.3ks *.
Magnetic Crystallization from the A m o r p h o u s Phase When a n e w phase is formed in the matrix with different phase or structure, the formation o f interphase boundary between the n e w phase and the matrix inevitably occurs by spending some extra energy associated with interphase boundaries, or grain boundaries. There are several cases
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Materials Processing and Texture
N e w A p p r o a c h t o T e x t u r i n g a n d G r a i n B o u n d a r y E n g i n e e r i n g b y M a g n e t i c Field Application
of structural combinations between a new phase and the matrix. Here we discuss the effect o f magnetic field application on a solid/solid phase transformation, particularly crystallization from the amorphous state. The application of a magnetic field can affect the formation of a new phase from the a m o r p h o u s phase, as expected from a standard t h e r m o d y n a m i c s of nucleation process. The Gibbs free energy change AGr during the transformation under a magnetic field is given by, 2
AG ι = 4jrr γ 5
4/3πΓ
where r is the radius of a new phase, y AG
W
3
s
( Δ&\ + AG,„ )
(1)
interphase boundary energy. AG, volume energy change.
the magnetic energy change given by AG =4!6nr'( w
μ - μ ι ) / / " , μ,and p .the susceptibility of 2
2
a new phase and the matrix, respectively. Η is magnetic field strength. It is likely that the critical size for the stable nucleus r ' c a n be smaller by magnetic field application because of additional contribution of the magnetic volume energy A G t o the Gibbs free energy of the whole system m
AGr, in comparison with non-magnetic crystallization. M o r e o v e r the magnetic
crystalline
anisotropy m a y affect the nucleation of a n e w phase by directing the easy magnetization direction of a n e w phase to the magnetic field direction. This means that those crystallites which have a specific orientation parallel to the field direction may occur preferentially and then further grow in the later stage of crystallization during magnetic annealing. Thus, the magnetic crystallization can bring about the evolution o f nanocrystalline material with a specific type of a sharp texture. This was the motivation and expectation of o u r work ribbons of
Fe
7
8
Si9Bu
2<>
3 0
. We used amorphous
alloy rapidly solidified from the melt by single-roller melt spinning at roller
speed of 31.4m/s in vacuum. The thickness and the width of ribbon were 0.75 mm and 15pm, respectively. A m o r p h o u s ribbons w e r e annealed at temperatures ranging from 653K to 853K. for 1.8ks in v a c u u m and in a magnetic field up to 6T using the specially designed superconducting magnet heating system mentioned before. F i g u r e 3 shows X-ray diffraction ( X R D ) profiles for the specimens crystallized at 853K in magnetic field with different strengths applied in the direction parallel to the ribbon surface. An X R D profile for as-solidified a m o r p h o u s ribbon is s h o w n at the bottom of the figure. It is evident that a magnetic field of 6T intensified the [110] peak associated with the α-Fe ( Si ) phase, indicating the evolution of [110] texture. The evolution of [110] sharp texture by the application of a magnetic field parallel t o the ribbon surface has been explained by the magnetocrystalline anisotropy and preferential g r o w t h of (110) grains w h o s e [001] axis is parallel to the ribbon surface
I 9
. It should be also noted that the shape of grains is more or less square or rectangular.
As expected, it has been found that the application of a magnetic field can enhance the formation and subsequent growth of specifically orientated nuclei, resulting in the evolution of a sharply [110] textured α - F e grain structure with very fine grain size ranging from about 400-800nm. The
magnetically
crystallized
characteristic of nanocrystalline
Fe7sSioBi_i ribbons soft
magnetic
showed
material,
31
different
magnetic
properties*''
depending on the condition
of
crystallization from the a m o r p h o u s phase.
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N e w A p p r o a c h t o T e x t u r i n g a n d G r a i n B o u n d a r y E n g i n e e r i n g b y M a g n e t i c Field A p p l i c a t i o n
Magnetic Solidification from the Melt The effect of a magnetic field on texturing via solidification processing, i.e. liquid/solid transformation, was studied o n the Bi-Pb binary eutectic alloy system
3 2
. In comparison with the
magnetic crystallization from the a m o r p h o u s phase.i.e. solid/solid transformation, the magnetic solidification has a large variety of scientific and engineering interest and worth studying. In view of this, we studied the magnetic solidification in pure Bi and binary Bi-Pb hyper-eutectic alloys with Pb content from 20 to 40at.% in which the primary Bi phase solidifies from the melt and finally the eutectic alloy ε phase ( the eutectic composition is 56.3at%Bi-43.7at.%Pb) forms with the primary bismuth phase at the eutectic temperature of 398K.. Accordingly we can expect some difference in processing time during the magnetic solidification depending on the alloy composition. M o r e o v e r we k n o w that Bi is a special element which does not show a shrinkage but an expansion during solidification, unlike ordinary metals and alloys, and has diamagnetic properties with the susceptibility of level o f - l x l 0 " 6 emu. at RT. Our magnetic solidification experiments were carried out by using a specially designed DC coil-winding magnet which could generate a magnetic field up to almost 2T and a quartz capsule in which the specimen was kept during magnetic solidification.
T h e grain orientation distribution
for differently
solidified
specimens of the alloys was analyzed by the O I M technique. A magnetic field was applied in the direction parallel to observed specimen surface. Solidification experiments were carried out by repeatedly cycling between without and with a magnetic field for the same specimen.
We
observed the different features of solidification curves between pure Bi and the Bi-Pb alloys. In the former the solidification occurred at a constant melting temperature, while in the latter it occurred at continuously descending melting temperature as expected from the phase diagram. The melting temperatures seem little affected by the application of a magnetic field of 2.15T for pure Bi and the Bi-Pb alloys studied. F i g u r e s 4 and 5 show the results of O I M analyses of grain orientation distributions for the specimens of B i - 3 5 a t . % P b , and Bi-40at.%Pb alloys. The inverse pole figures given on the upper column show the result of the solidification with a magnetic field of 2.15T and the lower column for the solidification without magnetic field. The magnetic field application produced a sharp texture around [001 ] which is the c axis of bismuth crystal and parallel to the field direction, while more random texture
was observed
by solidification
without magnetic
field.
It is
interesting that a much sharper texture with a spread of about 10 deg. around the c axis was observed in the B i - 4 0 a t . % P b alloy than the Bi-35at.%Pb alloy. It is k n o w n that the c axis is the easy magnetization direction of diamagnetic Bi crystal. Here it should be mentioned that recent study of vapor deposition of bismuth under a strong magnetic field of 5T has revealed that bismuth crystals tend to align along their a- or b-axis parallel to the magnetic
field
with
33
increasing the field s t r e n g t h . This is the case of magnetic phase üansformation via gas/solid phase transformation. Thus the effect of a magnetic field can occur generally o n different types of phase transformation. Particular interest in recent work is the combined effect of magnetic and electric fields associated with the Lorentz force on solidification structure in M g alloy."
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Materials Processing a n d Texture
1
N e w A p p r o a c h t o T e x t u r i n g a n d G r a i n B o u n d a r y E n g i n e e r i n g by M a g n e t i c Field Application
Fig.4
Fig.5
Pole figures for B i - 3 5 a t . % P b alloy
Pole figures for Bi-40at.% alloy
specimens solidified with a magnetic field
specimens solidified with a magnetic field
of 2.15T (top) and without field (bottom).
o f 2.15T (top) and without field (bottom).
GRAIN B O U N D A R Y ENGINEERING BY MAGNETIC FIELD APPLICATION It is only recently that the importance and possibility o f the grain boundary engineering by magnetic field application have b e e n increasingly recognized 12. 35. 36 and a number of new attempts have been m a d e in the last ten years, as briefly introduced as follows.
The Control o f Grain Boundary Microstructure Probably the first reported work on the effect of magnetic annealing o n the evolution of grain boundary microstructure and G B C D s w a s by Watanabe et al. in 1990 o n iron-9at.% cobalt alloy 7
performed by applying the S E M - E C P technique'' . The retardation o f recrystallization w a s found to occur during magnetic annealing in DC magnetic fields up to 5 k O e at 1073K for 18ks in cold rolled sheet o f the alloy. Similar retardation was reported by Martikainen and L i n d r o o s
38
o n the
magnetic annealing o f hot-forged and cold-rolled iron at 973K. and 9 9 8 K under a magnetic field o f 1.5T.
F i g u r e 6 shows one o f the results of G B C D s obtained; the frequency o f occurrence o f
low-angle (Σ1) boundary and l o w - Σ coincidence boundaries as a function of magnetic
field
strength for the magnetically annealed iron-9at.%cobalt alloy polycrystals. It is evident that the frequency
o f low-angle boundaries increases with magnetic field strength up to 7.8%, being 3.4
times larger than the prediction for a r a n d o m polycrystal. The frequency o f low-Σ coincidence boundaries ranged from 7 % to 1 3 % . The frequency o f high-angle random boundaries tends to increase as recrystallization proceeds, resulting in grain growth. Another interesting finding obtained w a s that the frequency o f r a n d o m boundaries showed the highest frequency (89.3%) for the specimen magnetically annealed under a magnetic field of 5 k O e at 1173K ( paramagnetic temperature) above the Curie temperature T = 1 1 3 0 K for iron-9at.%cobalt alloy. L
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427
N e w A p p r o a c h t o T e x t u r i n g a n d Grain B o u n d a r y Engineering b y M a g n e t i c Field A p p l i c a t i o n
Fig.6
Effect o f magnetic annealing on the
Fig.7
frequency o f low-angle boundaries and low-Σ coincidence boundaries in Fe-9at.%Co a l l o y . '
Grain size distributions o f for
Fe-50at.%Co ribbons annealed without 7
(top) and with (bottom) magnetic field 1.5T.
3
Figure 7 shows the results on F e - 5 0 % C o alloy ribbon that ordinary / non-magnetic annealing produced a very heterogeneous grain structure whose grain size distribution ranges from ΙΟμηι to 300μηι. due to the occurrence o f abnormal grain growth, see Fig.7 (a), while the magnetic annealing under 1.5T produced a more h o m o g e n e o u s grain structure with grain size ranging from ΙΟμιη to 70μηι without abnormal grain growth, as seen in Fig.7(b). Thus it is evident that the magnetic field application can control abnormal grain growth and produce a h o m o g e n e o u s and uniform grain structure. This finding is of great engineering importance to an improvement in the thermal stability of nanocrystalline materials. It is well k n o w n that grain growth during annealing can be drastically affected by grain boundary segregation. However it w a s found quite recently that grain boundary segregation at random high energy boundaries can be controlled by magnetic 40
field application in iron-tin a l l o y . Enhancement of Thermal Stability o f Nanocrystalline Materials. Nanocrystalline materials s h o w many interesting properties which cannot be replaced by 41
ordinary polycrystalline m a t e r i a l s " " , but there still remain some problems to be solved: o n e o f them is a lack o f thermal stability o f their microstructures and properties. T h e materials consist of a large volume fraction o f the grain-or interphase-boundary component reaching about 5 0 % for the grain size o f a few nm. They often show severe microstructural heterogeneity due to the occurrence o f abnormal grain growth during processing. Such microstructural
heterogeneity
drastically degrades special properties and performance of nanocrystalline materials. It is very urgent to solve this problem and to establish a new fabrication method o f nanocrystalline
428
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Materials Processing a n d Texture
N e w A p p r o a c h t o T e x t u r i n g a n d G r a i n B o u n d a r y E n g i n e e r i n g b y M a g n e t i c Field Application
materials. Here we demonstrate that magnetic annealing can solve this long-pending problem by producing m o r e thermally stable and h o m o g e n e o u s microstructure than that of ordinarily 45
processed, showing recent findings on nanocrystalline nickel produced by e l e c t r o d e p o s i t i o n . Figures 8 and 9 show t w o sets of micrographs obtained from observations of grain growth and microstructural changes during non-magnetic and magnetic annealing at 693K. ( > the Curie temperature T
c
= 627K
) without and with a DC magnetic
nanocrystalline nickel w h o s e initial grain size was 3 0 n m .
46 4 7
field
of 15kOe ( 1.5T ) in
Abnormal grain growth occurred
during ordinary annealing without a magnetic field, as seen from Fig.8 (c),(d). On the other hand, grain growth occurred rapidly in the very early stage of magnetic annealing, but in later stage of annealing after
10h it occurred homogeneously, as seen from Fig.9(c),(d)). M o r e recently
Hibbard et al. have revealed that the origin of abnormal grain growth occurring at late stage of annealing in electrodeposited nanocrystalline nickel is ascribed to the presence of a wetting sulfur-rich second phase at the abnormal growth interface
4 8
. Accordingly the observed abnormal
grain g r o w t h in the material is likely ascribed to the presence of sulfur-rich second phase probably
enhanced
by grain
boundary
segregation
of sulfur
which
shows a very
low
4
solid-solubility ( order o f 10" ) and a strong propensity to grain boundary segregation in nickel.
Fig.8
S E M micrographs of heterogeneous grain
structure due to abnormal grain g r o w t h in n a n o crystalline Ni during non-magnetic a n n e a l i n g .
47
Fig.9
S E M micrographs of homogeneous
grain structure by magnetic annealing at 693K in a magnetic field o f 6 T for different t i m e s .
47
Effect of Magnetic field on Carbon Diffusion in Steel. The diffusion of carbon which plays the most important role in phase transformation and microstructural modification in iron-base alloys and steels. To the authors' knowledge, until presently there has been no available literature on the effect of a magnetic field o n the diffusion of carbon in iron-based alloys and steels, although the
finding
o f aligned
ferrite/pearlite
two-phase mierostructures by magnetic field application during α/γ or γ / α phase transformation has been drawing an increasing interest of many researchers involving in microstructural control
Materials Processing and Texture
•
429
N e w A p p r o a c h t o T e x t u r i n g a n d Grain B o u n d a r y E n g i n e e r i n g b y M a g n e t i c Field A p p l i c a t i o n
and development of advanced steels. The roles of carbon diffusion
in α/γ or γ / α
phase
transformation is the most important issue of microstructural control in iron-base alloys and steels. We applied a decarburization technique to obtain reliable information on the carbon diffusion caused by the reaction of carbon with titanium for explosively joined specimens of hypoeutectoid steel ( 0 . 0 9 m a s s % C ) and commercially pure titanium sheets.
Decarburization
annealing was carried out at temperatures ranging from 873 Κ to 1323K under a magnetic field up to 6T or under magnetic field gradients from 30 to 45T/m in a vacuum. Further details of this 4
work should refer to the original paper by Nakamichi et a l ' ' . F i g u r e 10 shows the temperature dependence of the diffusion coefficient of carbon in γ-iron in a magnetic field of 6T and in a magnetic field gradient of 4 5 T / m . compared with data for non-magnetic diffusion annealing. It was found that the effect of a magnetic field o n the diffusion of carbon is quite different depending o n whether the specimen was kept in a uniform magnetic field or a magnetic field gradient; a retardation of the diffusion of carbon was observed in a uniform magnetic field, while an acceleration of carbon diffusion was observed in a magnetic gradient. The diffusion coefficient of carbon w a s found to increase by about four times with increasing magnetic field gradient. In contrast to the diffusion of carbon, a magnetic field was found to affect little influence on the diffusion of titanium. The effect of magnetism o n 3
diffusion is expected to extend by magnetic field application beyond the Curie temperature ".
Fig. 10
t e m p e r a t u r e dependence of
diffusion coefficients of carbon in γ-iron in a 4 5 T / m field gradient and constant field of O T .
49
Fig.l 1
Aligned q ferrite (white ) and pearlite
dark) two-phase microstructure observed in carbon steel magnetically transformed by cooling from γ-phase in magnetic field of 1 4 T .
36
Microstructure Control by Magnetic Phase Transformation Here we discuss the possibility o f microstructural control by magnetic field application during diffusion-controlled phase transformation. We take some e x a m p l e s from recent w o r k s on the effect of magnetic field application o n α/γ or γ / α phase transformation which is k n o w n as diffusion-controlled processes occurring in iron-base alloys and steels. The Gibbs free energy change o f the γ / α transformation in pure iron is given by
430
·
Materials Processing and Texture
N e w A p p r o a c h t o T e x t u r i n g a n d Grain B o u n d a r y E n g i n e e r i n g b y M a g n e t i c Field Application
AG'
T, H) = AGY^'( TM ) - « Gv,': ( Τ, Η )-l / 2 χ , Η
where Τ and Η are temperature, magnetic field susceptibility o f γ - i r o n . and
experimentally
α /γ transformation steels
j
(2)
strength, respectively, χ-, is the magnetic
On the basis of similar considerations, several workers have predicted
evidenced temperature
composition and field strength 5 3
2
that
by
application
is raised by
v
of
a
strong
magnetic
field,
the
1-3° per one Tesla depending on the alloy-
and that the solubility o f carbon in ferrite phase increases in
. A s a result of the effect of magnetic field application, the evolution of unique
microstructures was found to form w h i c h is c o m p o s e d of fen-ite grains and pearlite ( mixture of ferrite and cementite ) structure elongated and aligned in the direction parallel to the magnetic 53
56
field direction in plain carbon s t e e l s ' . F i g u r e 11(a) shows the microstructure produced bycooling at l r / K / m i n
in a magnetic field of 14T after heating at 1153K. The bright area
corresponds to the ferrite phase and the dark area pearlite colonies. A mechanism of the evolution of elongated pearlite microstructure has been proposed by Z h a n g et a l
5 6
o n the basis of
preferential nucleation o f ferrite grains at triple junctions which w a s experimentally evidenced by in-situ observations
5 7
and their elongation and binding due to magnetic dipole interaction in the
direction of a magnetic field, as schematically s h o w n in F i g . l l ( b ) . Hao et al.reported that the degree of elongation s h o w s the m a x i m u m temperature Τ ·=1043Κ of i r o n .
at a temperature corresponding to the
Curie
58
ς
N o w let us consider the origin of aligned elongated ferrites and pearlite microstructures produced my magnetic transformation, o n the basis of our recent observations on the orientation relationship between ferrite and cementite in p e a r l i t e .
59
Careful orientation analyses have revealed
that there exist four possible orientation relationships (ORs) between ferrite and cementite phases: (103) //(011), ( 1 0 3 ) / / ( 0 1 1 ) . (103) //(101 ) . (I03) //(101 ) . Surprisingly all the four O R s have a c
C
F
c
F
c
F
c o m m o n feature o f close-packed plane parallelism between ferrite and cementite. Moreover, when a magnetic field of 12T was applied, it was found that the number of the two specific O R s ( P-P2. IS ) a m o n g the four O R s increased. It is very likely that O R s between a product phase and the matrix meets specific "Plane M a t c h i n g " condition which is defined by matching of close-packed planes between two grains or phases, originally proposed by Pumphrey
6 0
and recently properly-
termed by Purdy " E d g e - t o - E d g e Plane M a t c h i n g " in interface-controlled phase transformation
6 1
.
It is reasonable to consider the plane-matching condition for the nucleation and growth of the product phase surrounded by newly formed interphase boundaries, particularly under a magnetic field. It is very likely that the flux of carbons, not single carbon atoms, may be directed by some distortion of the lattice induced by magnetic field application. Therefore the "Plane-Matching" condition may explain the observed O R s and also the selective growth and elongation of the primary ferrite and pearlite phases aligned in the field direction, likely controlled by the local 2
equilibrium of carbon at moving α/γ interphase boundaries* . A m e c h a n i s m o f α/γ
transformation
associated with plane-matching α/γ interphase boundaries, proposed by Watanabe et a l ,
57
may help
our understanding of the origin o f aligned ferrite / pearlite by magnetic phase transformation.
Materials Processing and Texture
·
431
N e w A p p r o a c h t o Texturing a n d Grain B o u n d a r y Engineering by M a g n e t i c Field Application
CONCLUSION We have discussed recent achievements of the texturing and grain boundary engineering by magnetic field application, showing experimental evidences for the versatility and potential of magnetic field application, particularly in connection with grain boundary-related p h e n o m e n a and phase transformation for ferrous materials and nonferrous materials. We have also s h o w n that several unsolved problems remained have been solved by magnetic field application.
ACKNOWLEDGEMENTS The authors are grateful to their coworkers who were deeply involved in our long-term project of the texturing and grain boundary engineering by magnetic field application almost in the last two decades. We also greatly appreciate financial supports provided by our governments and private research foundations. T.W. appreciates Visiting professorship by Northeastern University.
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EFFECTS OF MAGNETIC
FIELD DIRECTION
ON TEXTURE
EVOLUTION
IN A
C O L D - R O L L E D IF S T E E L S H E E T D U R I N G H I G H M A G N E T I C F I E L D A N N E A L I N G
Yan Wu, Changshu He, Xiang Zhao. Liang Z u o Key
Laboratory
for
Anisotropy
and
Texture
of
Materials
(Ministry
of
Education).
N o r t h e a s t e r n University, S h e n y a n g 110004, L i a o n i n g P r o v i n c e , P.R. C h i n a ABSTRACT S h e e t s o f cold rolled ( 7 6 % ) IF steel w e r e annealed at 7 5 0 ° C for 3 0 m i n under a 12-tesla m a g n e t i c field. During the m a g n e t i c field a n n e a l i n g , they w e r e placed at t h e center of the applied
field
respectively, being oriented differently
with respect to the m a g n e t i c
field
direction. It is s h o w n that altering the s p e c i m e n orientation to the m a g n e t i c field direction during a n n e a l i n g d o e s not c h a n g e t h e final annealing t e x t u r e s . T h e a v e r a g e intensity o f γ-fiber texture o f field a n n e a l e d s p e c i m e n s is higher than the non-field a n n e a l e d specimen, especially at t h e orientation angle o f 4 5 ° , w h e r e the result is m o r e p r o n o u n c e d . T h e intensity of {111} texture c o m p o n e n t s presents a similar periodic variation with respect to the specimen orientation to the m a g n e t i c field. A s the angle increases, their intensity first weakened, then strengthened to a m a x i m u m v a l u e at 4 5 ° , and then w e a k e n e d again as the rotation angle increased. INTRODUCTION The application of a high m a g n e t i c field to the texture d e v e l o p m e n t in metals and alloys 1
has been the object of great a t t e n t i o n "
12
in the area o f e l e c t r o m a g n e t i c processing. In 1 9 4 9 Ί
1
S m o l u c h o w s k i et. al. studied the effect of m a g n e t i c field a n n e a l i n g on the recrystallization texture d e v e l o p m e n t in iron-cobalt alloy. T h e y found that the influence o f a m a g n e t i c field 2
a p p e a r e d to be a strongly e n h a n c e o f the (001)f 110] recrystallization texture. He ct. al. ""also o b s e r v e d the similar influence o f a m a g n e t i c field on the recrystallization process and texture evolution p h e n o m e n a in a c o l d - r o l l e d IF steel. H o w e v e r , in the p r e v i o u s research, m o s t of the researchers considered only the a n n e a l i n g t e m p e r a t u r e and p r o l o n g e d annealing time, w h i l e 4
only limited r e s e a r c h e r s '
8
2
" " ' c o n s i d e r e d t h e orientation b e t w e e n t h e s a m p l e and m a g n e t i c 4
6
field. M o l o d o v and c o w o r k e r s " h a v e found that d e p e n d i n g on the specimen orientation to the applied field, the texture c o m p o n e n t s can be strengthened, r e m a i n u n c h a n g e d , or disappear entirely in zinc alloy and t i t a n i u m sheet after high m a g n e t i c a n n e a l i n g . T h e w o r k of Li et a l . " s h o w e d that the effects o f a high m a g n e t i c field on the recrystallized texture d e v e l o p m e n t in pure Ni varied as a function of the angle b e t w e e n the s a m p l e ' s normal and field direction. In a p r e v i o u s p a p e r
12
we reported preliminary results o f e x p e r i m e n t s with 7 6 % cold rolled
IF steel sheet a n n e a l e d in a high m a g n e t i c field, with the s p e c i m e n s ' rolling direction (RD) having different a n g l e s to the m a g n e t i c field direction ( M D ) . w h i l e k e e p i n g the rolling planes parallel to the M D . T h e results s h o w e d that the evolution o f the
{111}<112> texture
c o m p o n e n t is a little m o r e d e p e n d e n t on the orientation angle o f the s p e c i m e n with respect to the field direction, as c o m p a r e d with the {111} < 1 1 0 > texture c o m p o n e n t . At t h e orientation a n g l e of 30° and 9 0 ° , the intensity o f the {111 }<112> texture c o m p o n e n t had a peak value. M o r e o v e r , an a n a l y s i s of the results indicates that m a g n e t i c a n n e a l i n g d o e s not dramatically
435
Effects of M a g n e t i c Field Direction o n T e x t u r e Evolution in a C o l d - R o l l e d IF S t e e l S h e e t
c h a n g e the final a n n e a l i n g textures, w h i c h are only a little lower in intensity than that of the non-field a n n e a l e d s p e c i m e n . T h i s paper reports further investigations into t h e m a g n e t i c a l l y affected recrystallization t e x t u r e evolution o f 7 6 % cold-rolled IF steel sheet by v a r y i n g the a n g l e s b e t w e e n m a g n e t i c field direction and the s a m p l e ' s rolling p l a n e s . MATERIALS AND METHODS T h e material used in this study w a s a 7 6 % cold-rolled IF steel sheet o f 1mm t h i c k n e s s , with a c h e m i c a l c o m p o s i t i o n o f (wt % ) : C: 0 . 0 0 2 3 . Ti: 0.056. Si: 0 . 0 1 4 . M n : 0.16. P: 0 . 0 1 1 , S: 0.0064. Al: 0 . 0 5 2 . N : 0 . 0 0 1 8 . S h e e t s with the d i m e n s i o n s o f 1 6 m m x l 2 m m > < l m m
were
inserted into the variable angle t r o u g h of a n o n m a g n e t i c stainless steel m o u l d , a l l o w i n g the samples to be oriented differently with respect to t h e m a g n e t i c field direction. The t r a n s v e r s e direction ( T D ) of all s p e c i m e n s w a s kept p e r p e n d i c u l a r to t h e m a g n e t i c field, w h i l e t h e rolling direction ( R D ) w a s tilted to different a n g l e s (ζ) in relation to the m a g n e t i c field direction ( M D ) . as shown in F i g u r e 1. H e r e ζο=0° represents the orientation w h e r e the R D is parallel to the M D and ζ9(ΐ=90° represents the orientation w h e r e the R D is n o r m a l t o the M D . T h e s p e c i m e n s w e r e then subjected to isothermal a n n e a l i n g at 7 5 0 ° C for 3 0 m i n in a furnace integrated within a c r y o cooler-cooled s u p e r c o n d u c t i n g m a g n e t at a heating rate o f 5 ° C / m i n . and then c o o l e d in the furnace. D u r i n g m a g n e t i c a n n e a l i n g , the s p e c i m e n s w e r e placed at t h e center of the m a g n e t i c field for the duration of t h e heating, isothermal h o l d i n g and cooling processes. For c o m p a r i s o n , non-field a n n e a l e d s p e c i m e n w a s also processed in the s a m e furnace u n d e r the s a m e heat t r e a t m e n t c o n d i t i o n s . T h e global t e x t u r e s w e r e obtained at the 1/4 layer o f s p e c i m e n t h i c k n e s s by m e a s u r i n g three i n c o m p l e t e { 1 1 0 } . {200} and {211} pole m e t h o d " with C o K method
14
a
figures
using the Schulz
back-reflection
radiation. T h e c o r r e s p o n d i n g O D F s w e r e calculated with the t w o - s t e p
and the results are presented in t e r m s o f constant φ = 4 5 ° section ( R o e ' s notation).
Figure 1 Orientation of s p e c i m e n s d u r i n g
Figure 2 O D F constant φ = 4 5 ° section of
the a n n e a l i n g with respect to the m a g n e t i c
the as-rolled s p e c i m e n
field direction ( M D ) RESULTS AND DISCUSSION F i g u r e 2 s h o w s the φ = 4 5 " O D F section o f the as-rolled s p e c i m e n . T h e cold rolled texture consists o f t w o c o m p o n e n t s , i.e. a stronger α-fiber texture ( < 1 1 0 > / / R D ) and a w e a k e r γ-fiber texture ( < 1 1 1 > / / N D ) , w h i c h arc traditionally found in m a n y cold-rolled bec materials. Figure 3 c o m p a r e s the c o n s t a n t φ = 4 5 ° O D F section o f s p e c i m e n s a n n e a l e d at 7 5 0 ° C with and w i t h o u t a m a g n e t i c field. A s e x p e c t e d , only γ-fiber c o m p o n e n t s remain after full a n n e a l i n g .
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Effects of M a g n e t i c Field Direction o n T e x t u r e Evolution in a C o l d - R o l l e d IF S t e e l S h e e t
T h e orientation distributions o f t h e fully recrystallized s a m p l e s are very similar for both the non-field annealed s p e c i m e n and the field annealed s p e c i m e n s despite the different orientation angles to t h e m a g n e t i c field.
Figure 3 C o n s t a n t φ = 4 5 ° O D F sections for 7 6 % cold-rolled IF steel s p e c i m e n s annealed with no m a g n e t i c field (a) and a 12T m a g n e t i c field with varying angles o f ζ (b-h) (levels: 2 4 6...) Figure 4 s h o w s the variation in orientation intensity a l o n g the γ-oriented and α-oriented lines for s p e c i m e n s a n n e a l e d with and w i t h o u t a m a g n e t i c field. It is o b v i o u s that all the annealed s p e c i m e n s display final recrystallization textures consisting o f a relatively strong {111}<112> ( ψ = 0 ° or 60°), {111}<123> ( ψ = 1 0 ° or 70°) and w e a k e r {111}<110> texture ( ψ = 3 0 ° or 90°) c o m p o n e n t s , w h i c h are the c o m m o n texture c o m p o n e n t s found in fully recrystallized IF steel. After a n n e a l i n g at 7 5 0 ° C for 3 0 m i n , recrystallization is complete, resulting in α-oriented line intensities and shapes with only small deviations present within the group of annealed specimens.
Figure 4 Variation in orientation intensity a l o n g the γ-orienled and α-oriented lines for s p e c i m e n s annealed with and w i t h o u t m a g n e t i c field T h e a b o v e results imply that altering the s p e c i m e n orientation to the magnetic field direction during a n n e a l i n g does not ultimately c h a n g e the m e c h a n i s m s o f formation and
Materials Processing and Texture
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437
Effects of M a g n e t i c Field Direction o n T e x t u r e Evolution in a C o l d - R o l l e d IF S t e e l S h e e t
d e v e l o p m e n t o f recrystallized t e x t u r e s in cold rolled IF steel sheet in the present c a s e . A s Figure 4 s h o w s , c o m p a r e d with the non-field a n n e a l i n g s p e c i m e n , the m a g n e t i c field annealed s p e c i m e n s present a h i g h e r a v e r a g e intensity o f γ-fiber textures, especially at an orientation angle o f 4 5 ° , w h e r e the result is m o r e p r o n o u n c e d . This is different from the results of p r e v i o u s studies o n t e x t u r e e v o l u t i o n d u r i n g a n n e a l i n g of IF steel u n d e r a high m a g n e t i c 2
field ""' '". For the m a g n e t i c a l l y a n n e a l e d s p e c i m e n s , the γ-oriented lines are sorted into three o b v i o u s g r o u p s with nearly identical line s h a p e s and intensities. T h e first g r o u p has the highest value of intensity and c o n t a i n s s p e c i m e n s w i t h orientations o f 4 5 ° a n d 6 0 ° . T h e s e c o n d g r o u p contains intermediate value of intensity and includes s p e c i m e n s with angles of 0°, 3 0 ° , 7 5 ° . T h e final g r o u p includes the s p e c i m e n s with orientations o f 15° and 9 0 ° , and had the lowest intensity. Figure 5 s h o w s t h e variation in orientation intensity o f m a i n {111} t e x t u r e c o m p o n e n t s with respect to the orientation a n g l e ζ in the field-annealed s p e c i m e n s . It is o b v i o u s that the intensity of all {111} t e x t u r e c o m p o n e n t s present a similar periodic variation with respect to the s p e c i m e n orientation to the m a g n e t i c field. As the a n g l e increases, t h e intensity
first
decreased, followed by an increase to a m a x i m u m value at 4 5 ° , and then further d e c r e a s e d in intensity as ζ i n c r e m e n t e d .
Figure 5 Variation in orientation intensity of {111} texture c o m p o n e n t s with respect to the s p e c i m e n orientation a n g l e (ζ) So far. the effect o f high m a g n e t i c fields and their influence o n the recrystallization process 13
17
and texture evolution is not well u n d e r s t o o d . A s pointed e l s e w h e r e " , a possible m e c h a n i s m s m i g h t be related to m a g n e t i c a l l y o r d e r e d state and d o m a i n wall r e a r r a n g e m e n t induced by the m a g n e t i c field, w h i c h arc c o n s i d e r e d as t w o main factors affecting a t o m i c diffusion and grain boundary migration. 8
A c c o r d i n g to Tse and D u g g a n ' , the d e f o r m e d γ orientations display higher stored e n e r g y and a fairly sharp orientation gradient; thus it is r e a s o n a b l e to c o n s i d e r that d u r i n g m a g n e t i c a n n e a l i n g the nuclei with r a n d o m orientations are further restrained and the nuclei with {111} orientations are m o r e favored. B e c a u s e of this a d v a n t a g e in nucleation rate and size, the {111} oriented nuclei preferentially form and g r o w further d u r i n g t h e recrystallization stage, w h i c h s u b s e q u e n t l y leads to a relatively strong γ-fiber texture. In the present e x p e r i m e n t , the results s h o w high m a g n e t i c field can h a v e different effects on the recrystallizedYtexture w h e n the s a m p l e is placed at v a r i o u s angles to the m a g n e t i c field. T h e s e differences m a y be attributed to b o u n d a r y migration induced by the m a g n e t i c
438
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Materials Processing and Texture
field
Effects of M a g n e t i c Field Direction o n T e x t u r e Evolution in a C o l d - R o l l e d IF S t e e l S h e e t
w h e n the angle b e t w e e n the s a m p l e and m a g n e t i c field direction is adjusted. However, it is a very c o m p l i c a t e d p r o c e s s that includes m a g n e t i c field effects as well as the evolution of recrystallization t e x t u r e s . Further detailed research on this process will be d o n e in the next step. SUMMARY A l t e r i n g the s p e c i m e n orientation to the m a g n e t i c field direction during annealing does not c h a n g e the m e c h a n i s m s o f formation and d e v e l o p m e n t of recrystallization texture in cold rolled IF steel sheet in the present c a s e , and the a n n e a l e d s p e c i m e n s have the s a m e
final
annealing textures. T h e intensity of γ-fiber textures of
field-annealed
s p e c i m e n s is higher than the non-field
annealed s p e c i m e n , especially at t h e orientation a n g l e of 4 5 ° . the result is m o r e p r o m i n e n t . T h e intensity of main {111} texture c o m p o n e n t s in
field-annealed
s p e c i m e n s presents a
similar periodic variation with respect to t h e s p e c i m e n orientation to the magnetic field. With an increase o f orientation angle, the {111J texture w a s o b s e r v e d to first w e a k e n , then increase to a m a x i m u m v a l u e at 4 5 ° , followed by a s u b s e q u e n t reduction in intensity as the angle continued to increase. T h e results o f this study d e m o n s t r a t e that the t e x t u r e in f e r r o m a g n e t i c IF steel sheet can be effectively
changed when annealed
in a high m a g n e t i c
field
by altering the specimen
orientation angles to the field direction. T h i s m a y p r o v i d e a new m e t h o d to control the d e v e l o p m e n t of crystallographic texture in magnetically anisotropy materials. ACKNOWLEDGEMENT T h i s w o r k w a s supported by the National Science Found for Distinguished
Young
Scholars ( G r a n t NO. 5 0 3 2 5 1 0 2 ) , the key project of N a t i o n a l N a t u r a l Science Foundation of C h i n a ( G r a n t NO. 5 0 2 3 4 0 2 0 ) . and t h e National Youth Science Foundation of C h i n a (Grant N0.50501006). T h e a u t h o r s w o u l d like to e x p r e s s appreciation to the C e n t e r for Materials A n a l y s i s and Testing and the H i g h M a g n e t i c Field Laboratory of N o r t h e a s t e r n University for providing the facilities. T h e provision o f studied materials for this w o r k by S h a n g h a i B a o Steel is gratefully acknowledged. REFERENCES 1
R. S m o l u c h o w s k i . and R. W. Turner, Influence of M a g n e t i c Field on Recrystallization, J. Appl. Phys.,
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A. D. S h e i k h - A l i , D. A . M o l o d o v , and H. G a r m e s t a n i . M i g r a t i o n and Reorientation o f Grain B o u n d a r i e s in Z n Bicrystals d u r i n g A n n e a l i n g in a High M a g n e t i c Field, Scripta
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483-8 (2003). D. A. M o l o d o v , P. J. K o n i j n e n b e r g , N . B o z z o l o , and A. D . S h e i k h - A l i , M a g n e t i c a l l y Affected
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C. M. B. B a c a l t c h u k . G. A . C a s t e l l o - B r a n o , B . Gault, a n d H. G a r m e s t a n i , Effect of high magnetic field during p r i m a r y a n n e a l i n g on texture of silicon steel, Mater.
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Materials Processing and Texture
Hexagonal Metals
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E L E C T R O C H E M I C A L B E H A V I O R O F ( 0 0 1 ) , ( 100) A N D ( 110) Ti S I N G L E CRYSTALS UNDER SIMULATED BODY FLUID CONDITION
2
2
M . A z z i ' , S. F a g h i h i , M . T a b r i z i a n , J. A . S z p u n a r '
'Department of Materials Engineering, McGill University, Montréal. Qc. Canada. 2
D e p a r t m e n t of Biomedical Engineering, McGill University. Montréal. Q c . Canada
ABSTRACT In t h i s p a p e r , t h e e l e c t r o c h e m i c a l b e h a v i o r o f Ti s i n g l e c r y s t a l s a n d C P Ti u n d e r s i m u l a t e d b o d y fluid c o n d i t i o n w a s i n v e s t i g a t e d . ( 0 0 1 ) , ( 1 0 0 ) a n d ( 1 1 0 ) s i n g l e c r y s t a l s w e r e u s e d for t h i s i n v e s t i g a t i o n . E l e c t r o c h e m i c a l I m p e d a n c e S p e c t r o s c o p y ( E I S ) w a s u s e d to c h a r a c t e r i z e t h e e l e c t r o c h e m i c a l b e h a v i o r o f t h e i n t e r f a c e b e t w e e n t h e s u b s t r a t e s a n d t h e e l e c t r o l y t e . P o t e n t i o d y n a m i c p o l a r i z a t i o n w a s a l s o p e r f o r m e d t o c h a r a c t e r i z e the c o r r o s i o n r e s i s t a n c e o f t h e d i f f e r e n t T i s u r f a c e s . T h e E I S s p e c t r a w e r e p r e s e n t e d in B o d e p l o t a n d i n t e r p r e t e d in t e r m s o f a p p r o p r i a t e e l e c t r i c a l c i r c u i t s . It w a s found that t h e r e s i s t a n c e t o c h a r g e t r a n s f e r , R e t , o f t h e ( 0 0 1 ) s i n g l e c r y s t a l w a s 5.1 Μ Ω χ η ι t o 3.1 M Q x m
2
a n d 3.3 Μ Ω χ η ι
2
2
Rc, o f C p T i w a s 1.1 Μ Ω χ η ι .
2
compared
for t h e ( 1 0 0 ) a n d ( 1 1 0 ) s i n g l e c r y s t a l s r e s p e c t i v e l y . T h e It s h o u l d b e m e n t i o n e d t h a t t h e r e s i s t a n c e to c h a r g e
t r a n s f e r is i n v e r s e l y p r o p o r t i o n a l t o t h e c o r r o s i o n r a t e . T h e P o t e n t i o d y n a m i c p o l a r i z a t i o n curve h a v e also s h o w n that the corrosion rate a n d passive current of (001) single crystal w e r e l o w e r t h a n t h o s e o f ( 1 0 0 ) a n d ( 1 1 0 ) s i n g l e c r y s t a l s . T h e s e r e s u l t s h a v e s h o w n that t h e ( 0 0 1 ) b a s a l p l a n e w h i c h is t h e m o s t d e n s e l y p a c k e d p l a n e h a d h i g h e r c o r r o s i o n r e s i s t a n c e c o m p a r e d to ( 1 0 0 ) a n d ( 1 1 0 ) p l a n e s w h i c h h a v e l o w e r p l a n a r a t o m i c d e n s i t i e s . A l s o , T h e C p Ti w a s f o u n d t o h a v e t h e least c o r r o s i o n r e s i s t a n c e , m o s t p r o b a b l y d u e to t h e p r e s e n c e o f g r a i n b o u n d a r i e s w h i c h are a r e a s o f d i s o r d e r in t h e m a t e r i a l .
1-INTRODUCTION T i t a n i u m a n d its a l l o y s h a v e b e e n w i d e l y u s e d for b i o m e d i c a l a p p l i c a t i o n s d u e to their excellent corrosion
resistance
and
biocompatibility.
These excellent
corrosion
properties are d u e to the very thin, passive oxide layer w h i c h protects the surface
from
c o r r o s i o n [ 1 ] . In a d d i t i o n , t h e l o w e r stiffness o f t i t a n i u m c o m p a r e d to s t a i n l e s s steel and c o b a l t - c h r o m i u m a l l o y s , m a k e s t h e m a n ideal c h o i c e for o r t h o p e d i c a p p l i c a t i o n w h e r e the i m p l a n t s s h o u l d f o r m a r i g i d b o d y w i t h t h e b o n e s w h i c h h a v e a l o w m o d u l u s o f elasticity. It h a s b e e n r e p o r t e d t h a t t h e u s e o f t i t a n i u m r e d u c e s t h e s e v e r i t y o f s t r e s s - s h i e l d i n g a n d cortical osteoporosis associated
w i t h t h e u s e o f m o r e r i g i d m a t e r i a l s for total
hip
a r t h r o p l a s t y c o m p o n e n t s a n d fixation d e v i c e s [ 2 ] . Metallic materials used as implants must have high corrosion resistance; the release of m e t a l l i c i o n s into t h e s u r r o u n d i n g t i s s u e m a y n e g a t i v e l y affect t h e r e s p o n s e o f t h e host body to the biomaterial [3]. T h e c o m m e r c i a l l y p u r e t i t a n i u m C P Ti is a p o l y c r y s t a l l i n e m a t e r i a l t h a t is p r o d u c e d in v a r i o u s f o r m s ( b a r s , r o d s , s h e e t s . . . ) . It is v e r y i m p o r t a n t to u n d e r s t a n d t h e r o l e o f grain
443
Electrochemical Behavior of (001 ), (100) a n d (110) Ti Single Crystals
o r i e n t a t i o n s a n d g r a i n b o u n d a r i e s in t h e c o r r o s i o n p r o c e s s o f C P Ti in c o n t a c t
with
biological solutions. The a i m o f t h i s w o r k is t o i n v e s t i g a t e t h e c o r r o s i o n r e s i s t a n c e o f d i f f e r e n t Ti s i n g l e crystals
with
different
orientations
and
to c o m p a r e
the results
with
the
corrosion
resistance of C P Ti.
2- E X P E R I M E N T A L 2-1 M a t e r i a l s Ti s i n g l e c r y s t a l s w i t h ( 0 0 1 ) , ( 1 0 0 ) a n d ( 1 1 0 ) c r y s t a l l o g r a p h i c o r i e n t a t i o n s w e r e u s e d for t h i s i n v e s t i g a t i o n . T h e s i n g l e c r y s t a l s w e r e s u p p l i e d in f o r m o f d i s c s w i t h 10 m m d i a m e t e r a n d 2 m m t h i c k n e s s . C P Ti d i s c s w i t h t h e s a m e d i a m e t e r w e r e a l s o u s e d to c o m p a r e t h e r e s u l t s . R i n g e r ' s s o l u t i o n o f p H 6.6 w a s u s e d a s e l e c t r o l y t e to s i m u l a t e t h e b o d y fluid. Its c o m p o s i t i o n is g i v e n in T a b l e I. Tablel. T h e composition of R i n g e r ' s solution Compound
NaCl
KCL
CaCl
C o m p o s i t i o n (g/1)
9.0
0.4
0.17
2
NaHCOj 2.1
2 - 2 C o r r o s i o n cell T h e c o r r o s i o n cell c o n s i s t e d o f a T e f l o n c o n t a i n e r w i t h a c i r c u l a r o p e n i n g in t h e m i d d l e a s s h o w n in F i g . 1. T h e s a m p l e w a s p r e s s e d a g a i n s t t h e r u b b e r O - r i n g u s i n g a back-plate to prevent leakage. A copper plate w a s inserted b e t w e e n the sample and the b a c k - p l a t e t o p r o v i d e e l e c t r i c a l c o n n e c t i o n to t h e s a m p l e . W h e n t h e s a m p l e w a s p r e s s e d a g a i n s t the O - r i n g , o n l y a r o u n d s u r f a c e w i t h 6 m m o f t h e s p e c i m e n w a s e x p o s e d to t h e electrolyte. A Standard Calomel Electrode (SCE) w a s used as reference electrode and a g r a p h i t e rod o f 5 m m w a s u s e d a s a c o u n t e r e l e c t r o d e . T h e r e f e r e n c e e l e c t r o d e
and
c o u n t e r e l e c t r o d e w e r e fixed in t h e c o v e r o f t h e cell a s s h o w n in F i g . 1. T h e w o r k i n g e l e c t r o d e ( t h e s a m p l e ) , t h e r e f e r e n c e a n d t h e c o u n t e r e l e c t r o d e s w e r e c o n n e c t e d to a n A u t o l a b P G S T A T 3 0 2 p o t e n t i o s t a t e q u i p p e d w i t h a F r e q u e n c y R e s p o n s e A n a l y s e r for E I S measurements.
444
·
Materials P r o c e s s i n g a n d T e x t u r e
E l e c t r o c h e m i c a l Behavior of (001), (100) a n d (110) Ti S i n g l e Crystals
F i g . 1 E l e c t r o c h e m i c a l cell
2-3 C o r r o s i o n test T h e c o r r o s i o n e x p e r i m e n t s c o n s i s t e d o f e l e c t r o d e s t a b i l i z a t i o n in t h e e l e c t r o l y t e for 1 h o u r , d u r i n g w h i c h t h e O p e n C i r c u i t P o t e n t i a l ( O C P ) w a s m o n i t o r e d . N e x t , t h e Electrochemical I m p e d a n c e Spectroscopy ( E I S ) w a s performed over a frequency range of 1 0 H z - 10" H z , at t h e O C P . w i t h a n A C a m p l i t u d e o f + / - 1 0 m V . F i n a l l y , a p o t e n t i o d y n a m i c p o l a r i z a t i o n m e a s u r e m e n t w a s m a d e s t a r t i n g from 1 0 0 m V b e l o w t h e O C P in t h e c a t h o d i c z o n e , t o a p o t e n t i a l o f 2 0 0 0 m V a b o v e t h e O C P . T h e p o t e n t i a l scan rate w a s 1 m V / s e c . T h e c o r r o s i o n c u r r e n t w a s f o u n d u s i n g t h e Tafel i n t e r p o l a t i o n technique. 5
2
EIS technique allows us to investigate the electrochemical reactions, the dielectric p r o p e r t i e s a n d t h e b e h a v i o r o f p a s s i v e s u r f a c e in c o r r o s i v e m e d i u m . Briefly, this t e c h n i q u e c o n s i s t s o n s u p e r i m p o s i n g different l o w a m p l i t u d e s i n e w a v e s o f v o l t a g e , w i t h different f r e q u e n c i e s , b e t w e e n t h e w o r k i n g e l e c t r o d e a n d t h e r e f e r e n c e e l e c t r o d e . F r o m this p e r t u r b a t i o n , a l o w a m p l i t u d e s i n e w a v e o f c u r r e n t is m e a s u r e d b e t w e e n t h e w o r k i n g e l e c t r o d e a n d t h e c o u n t e r e l e c t r o d e . A t e a c h f r e q u e n c y , t h e i m p e d a n c e is g i v e n b y Z(w)
= 4rr I(w)
E
e
X
P
(
J
W
= , " ' \ I„exp(jwt-
= T - e x p ( / * ) = Z „ e x p ( ^ ) = Z c o s ( f l + Z„ sin(fl I 0
0
W h e r e E is t h e v o l t a g e a m p l i t u d e i m p o s e d , in V I is t h e c u r r e n t a m p l i t u d e m e a s u r e d , in A 0
0
Z is t h e i m p e d a n c e m a g n i t u d e 0
Materials Processing a n d Texture
·
445
E l e c t r o c h e m i c a l B e h a v i o r of ( 0 0 1 ) , (100) a n d ( 1 1 0 ) Ti S i n g l e Crystals
w is t h e f r e q u e n c y , in r d / s e c t is t h e t i m e , in s e c o n d s φ is t h e p h a s e shift b e t w e e n t h e i m p o s e d v o l t a g e s i g n a l a n d t h e m e a s u r e d c u r r e n t signal T h e e x p r e s s i o n o f t h e i m p e d a n c e Z ( w ) is c o m p o s e d o f a real a n d i m a g i n a r y p a r t s . If t h e real part is p l o t t e d o n t h e x - a x i s a n d t h e i m a g i n a r y part is p l o t t e d o n t h e y - a x i s o f a c h a r t , w e g e t t h e " N y q u i s t " p l o t o f t h e i m p e d a n c e . A n o t h e r p o p u l a r p r e s e n t a t i o n m e t h o d is t h e B o d e p l o t ; t h e i m p e d a n c e is p l o t t e d w i t h t h e l o g f r e q u e n c y o n t h e x - a x i s a n d t h e a b s o l u t e v a l u e s o f t h e i m p e d a n c e \7.\ = Z
0
a n d t h e p h a s e shift φ o n t h e y - a x i s . In t h i s p a p e r . B o d e
plot w a s u s e d to p r e s e n t t h e i m p e d a n c e s p e c t r a .
3- R E S U L T S A N D D I S C U S S I O N In
this
paragraph,
the
OCP
measurements
during
the
first
hour
electrode
s t a b i l i z a t i o n , t h e E I S s p e c t r a a n d t h e p o l a r i z a t i o n c u r v e s o f t h e Ti ( 0 0 1 ) , Ti ( 1 0 0 ) , Ti ( 1 1 0 ) a n d C P Ti a r e p r e s e n t e d . 3-1 O C P m e a s u r e m e n t s b e f o r e E I S T h e O C P m e a s u r e m e n t s o f t h e d i f f e r e n t s a m p l e s d u r i n g t h e first h o u r e l e c t r o d e s t a b i l i z a t i o n a r e p r e s e n t e d in F i g . 2. C P Ti h a d t h e l o w e s t O C P c o m p a r e d to t h e Ti s i n g l e c r y s t a l s . Ti ( 1 0 0 ) h a d t h e h i g h e s t O C P , f o l l o w e d by Ti ( 0 0 1 ) a n d t h e n b y Ti ( 1 1 0 ) . T h e O C P d o e s n ' t p r e d i c t t h e c o r r o s i o n r e s i s t a n c e o f a s u r f a c e , it j u s t s h o w s t h e c o r r o s i o n potential of the surface.
F i g . 2 O p e n C i r c u i t p o t e n t i a l m e a s u r e m e n t s d u r i n g t h e first h o u r i m m e r s i o n o f Ti ( 0 0 1 ) , I i ( 1 0 0 ) . T i ( 1 1 0 ) a n d C P Ti
446
Materials Processing and Texture
Electrochemical Behavior of (001), (100) a n d (110) Ti Single Crystals
3-2 HIS results The EIS spectra of Ti ( 0 0 1 ) , Ti ( 1 0 0 ) , Ti ( 1 1 0 ) and CP Ti are presented in Bode diagram in Fig. 3. The left y-axis is the absolute value of the impedance Log(7. ). and the right y-axis is the phase shift φ. It is clearly seen that the impedance of the CP Ti is lower than the impedances of the single crystals. The spectra of the single crystals were very close, the difference was only noticeable on the Arg(Z) curve at low frequencies; the Ti (001) sample showed slightly higher phase shift. The impedance spectra were interpreted in terms of appropriate Equivalent Electrical Circuits. Randle circuit, shown as an inset to Fig. 4 , was used for the simulation. It consists of the following elements: Rs, which corresponds to the solution resistance of the test electrolyte between the working electrode and the reference electrode. C is the capacitance, which represents the charge buildup at the interface between the sample surface and the electrolyte, and R^ is the charge transfer resistance; Ret is inversely proportional to the corrosion rate. As the electrochemical reactions at the interface are complicated, EIS spectra modeling requires the use of complex circuit clement such as the constant phase element, Q, which is often used to replace the capacitor C. Its impedance is expressed as 0
• n
y„(M 2
Where Y is the parameter of Q (Fsec "'cm" ). w is the angular frequency and η is the empirical exponent of Q that always lies between 0.5 and 1 [ 4 ] , If n=l, Q can be related to pure capacitance, if n
Fig. 3 EIS spectra of Ti ( 0 0 1 ) , Ti ( 1 0 0 ) , Ti ( 1 1 0 ) and CP Ti presented in Bode plot
Materials P r o c e s s i n g a n d Texture
·
447
E l e c t r o c h e m i c a l Behavior of (001), (100) a n d (110) Ti S i n g l e Crystals
F i g . 4 s h o w s t h e HIS s p e c t r u m a n d t h e fitting c u r v e o f T i ( 0 0 1 ) s a m p l e . Il s h o w s a l s o t h e Randle circuit
u s e d for s i m u l a t i o n .
Rc
t
was found
2
t o b e 5.1
ΜΩ.αη , η was
0.96
indicating thai the constant p h a s e clement, Q, could be considered as a capacitor with a very s m a l l d e v i a t i o n . T h e c a p a c i t a n c e w a s f o u n d t o b e 0 . 1 6 x l 0 "
4
2
F/cm .
Fig. 4 E I S s p e c t r u m a n d fitting c u r v e o f t h e T i ( 0 0 1 ) . R a n d l e c i r c u i t u s e d for s i m u l a t i o n T h e d a t a o b t a i n e d from t h e s i m u l a t i o n o f all E I S s p e c t r a a r e p r e s e n t e d in T a b l e II. Ti ( 0 0 1 ) h a d t h e h i g h e s t r e s i s t a n c e t o c h a r g e t r a n s f e r FCc, a n d t h e o t h e r s i n g l e c r y s t a l s h a d very c l o s e R t- T h i s d i f f e r e n c e c o u l d b e e x p l a i n e d b y t h e d i f f e r e n c e in p l a n a r a t o m i c c
densities
the different
planes
had. (001)
is t h e m o s t
densely
packed
plane
in
the
hexagonal closed p a c k e d ( H C P ) structure, followed by (110) a n d then by (100). T h e s e r e s u l t s s h o w e d that it is m o r e d i f f i c u l t t o d i s s o l v e T i i o n s f r o m a p l a n e that h a s a h i g h e r a t o m i c d e n s i t y t h a n a p l a n e w i t h l o w e r a t o m i c d e n s i t y . T h e C P Ti h a d v e r y low
R^,
c o m p a r e d to t h e s i n g l e c r y s t a l s ; t h i s c o u l d b e the r e s u l t o f t h e g r a i n b o u n d a r i e s t h a t a r e p r e s e n t in t h e p o l y c r y s t a l l i n e C P T i . T a b l e 11 T h e d a t a o b t a i n e d f r o m t h e E I S s p e c t r a s i m u l a t i o n Sample Ti ( 0 0 1 )
Rs (Ω)
Q (Yo) (F/cm )
Q(n)
41
0.161x10""*
2
Rct 2
(n.cm ) 0.96
5.1χ10
ό
Ti ( 1 0 0 )
38
0.180xl0"
4
0.96
3.1x10"
Ti(110)
39
0.185xlO"
4
0.96
3.3x10
C P Ti
36
0.304x10"
4
0.9
448
Materials Processing a n d Texture
6
1.1x10°
Electrochemical Behavior of (001), (100) a n d (110) Ti Single Crystals
3-3 P o t e n t i o d y n a m i c test T h e p o t e n t i o d y n a m i c p o l a r i z a t i o n c u r v e s o f Ti ( 0 0 1 ) . Ti ( 1 0 0 ) . Ti ( 1 1 0 ) a n d C P Ti are p r e s e n t e d in F i g . 5. The c o r r o s i o n r a t e s o f t h e d i f f e r e n t s a m p l e s w e r e f o u n d u s i n g Tafel s l o p e m e t h o d a n d a r e p r e s e n t e d in T a b l e III. C P Ti h a d t h e h i g h e s t c o r r o s i o n rate a n d T i ( 0 0 1 ) h a d t h e l o w e s t o n e . T h e c o r r o s i o n r a t e o f Ti ( 1 0 0 ) a n d Ti ( 1 1 0 ) w e r e very c l o s e . T h e s e r e s u l t s c o n f i r m e d t h e r e s u l t s o b t a i n e d in t h e E I S t e c h n i q u e . T h e p a s s i v e c u r r e n t s w e r e a l s o p r e s e n t e d in T a b l e III. In t h e p a s s i v e r e g i o n , t h e d i f f e r e n c e b e t w e e n C P Ti a n d t h e s i n g l e c r y s t a l s b e c a m e s m a l l e r ; t h i s is p o s s i b l y d u e t o t h e fact t h e p a s s i v e layer b e c a m e s l i g h t l y t h i c k e r in t h i s r e g i o n a n d t h e r e f o r e t h e g r a i n b o u n d a r i e s h a d less effect o n t h e d i s s o l u t i o n r a t e .
F i g . 5 P o t e n t i o d y n a m i c c u r v e s o f Ti ( 0 0 1 ) , Ti ( 1 0 0 ) , Ti ( 1 1 0 ) a n d C P Ti in R i n g e r ' s solution T a b l e III T h e c o r r o s i o n r a t e s a n d t h e p a s s i v e c u r r e n t s o b t a i n e d from t h e p o t e n t i o d y n a m i c tests Sample
C o r r o s i o n rate 2
Passive current 2
(A/cm )
(A.cm )
Ti(001)
8.91xl0
v
Ti(100)
1.02xl0"
8
Ti(110)
1.01x10
s
C P Ti
6.02xl0"
s
5.20x10"
6
5.25xl0"
6
5.89x10° 6.95xl0"
6
Materials P r o c e s s i n g a n d Texture
·
449
E l e c t r o c h e m i c a l B e h a v i o r of ( 0 0 1 ) , (100) a n d ( 1 1 0 ) Ti S i n g l e C r y s t a l s
4- S U M M A R Y 1- ( 0 0 1 ) p l a n e s in Ti w e r e f o u n d t o h a v e t h e h i g h e r c o r r o s i o n r e s i s t a n c e . Its r e s i s t a n c e to c h a r g e t r a n s f e r , R d , w a s 5.2 Μ Ω χ η ι
2
c o m p a r e d t o a p p r o x i m a t e l y 3.2 M Q . c m
both (100) and (110) planes. T h e corrosion rate on (001) w a s 8.91xl0" 8
9
2
for
A/cm
2
2
c o m p a r e d t o a p p r o x i m a t e l y l x l O " A / c m for b o t h ( 1 0 0 ) a n d ( 1 1 0 ) p l a n e s . T h i s is d u e to t h e fact t h a t ( 0 0 1 ) is t h e m o s t d e n s e l y p a c k e d p l a n e in t h e H C P c r y s t a l s t r u c t u r e . 2 - C o m m e r c i a l p u r i t y ( C P ) Ti w a s f o u n d t o h a v e l o w e r c o r r o s i o n r e s i s t a n c e c o m p a r e d t o t h e s i n g l e c r y s t a l s . T h i s c o u l d b e d u e to t h e p r e s e n c e o f g r a i n b o u n d a r i e s w h i c h a r e a r e a s o f h i g h d i s o r d e r in t h e m a t e r i a l .
REFERENCES [1] E . E . S t a n s b u r y , R . A . B u c h a n a n , A S M I n t e r n a t i o n a l ( 2 0 0 0 ) [2] M . B a r a t z , A . W a t s o b , J. I m b r i g l i a . O r t h o p a e d i c S u r g e r y ; T h e e s s e n t i a l s , T h i e m e [3] A. S h a h r y a r i , S. O m a n o v i c , J. A . S z p u n a r , M a t e r i a l s S c i e n c e & E n g i n e e r i n g
C,
A r t i c l e in p r e s s [4] Z . B o u - S a l e h , A . S h a h r y a r i , S. O m a n o v i c , T h i n S o l i d F i l m 5 1 5 ( 2 0 0 7 ) 4 7 2 7 - 4 7 3 7 [5] H . G . K i m . S.H. A h n . J . G . K i m , S.J. P a r k , K . R . L e e , T h i n S o l i d F i l m 4 7 5 ( 2 0 0 5 ) 2 9 1 297
450
·
Materials Processing and Texture
I D E N T I F I C A T I O N A N D R A T I O N A L I Z A T I O N O F S E C O N D A R Y T W I N V A R I A N T S IN A MAGNESIUM ALLOY M. R. Barnett and P. Cizek A R C Centre of Excellence for Design in Light Metals, C M F I , Deakin University, Pigdons Road. Waurn P o n d s . Victoria 3217, Australia ABSTRACT The present work c o m b i n e s electron backscatter diffraction, transmission electron microscopy and S c h m i d analysis to investigate secondary twinning in the magnesium alloy M g 3 A l - l Z n . Inspection of the misorientations between the parent matrix and {101 1} — {1012} doubly twinned volumes reveals that there are four possible variants. One of these variants characterized by 38°< 1210 > misorientation with the matrix is favoured much more than the others. This variant involves activation of the secondary twinning systems that are quite inconsistent with the Schmid-type behaviour. For the secondary twin to grow significantly it must take on a shape enforced by the primary twin, however, this is not optimal for strain compatibility. It is argued that the 3 8 ° < 1 2 1 0 > variant occurs most frequently because it provides the closest match between the primary and secondary twinning planes, thus minimizing the compatibility strain. This conjecture is confirmed by the simulations of twin activity in ellipsoidal grains performed using the visco-plastic self-consistent crystal plasticity model. INTRODUCTION The mechanical behaviour of magnesium single crystals is k n o w n to be sensitive to the nature of the mechanical twins that form during plastic deformation
1
Extension along the hep
c-axis favours the formation of " e x t e n s i o n " twins on the {1012} plane and the corresponding flow stress r e m a i n s low and displays little work hardening '. Compression along the c-axis favours "contraction" twinning m o d e s due to the polarity of twinning. Although different modes ,
have been observed by a n u m b e r of authors (e.g. " ) , it is quite c o m m o n for contraction twinning to occur on the {101 1} plane *". In m a n y cases this is followed by "re-twinning'* or " s e c o n d a r y " twinning o n the {10 1 2} plane. The {101 1} -{10 1 2} double twinning rotates the basal plane into a position where it is m u c h m o r e favourably oriented for dislocation glide, which can lead in single crystals to softening, rapid unloading, high strains and internal ductile failure " \ It has been suggested that a similar p h e n o m e n o n to this might also determine ductile failure in 6
8
polycrystals " . Indeed, it has been observed in several polycrystalline Mg-based alloys that ductile fracture w a s preceded by the formation of voids that frequently appeared to be initiated by {10 I 1} - {10 1 2} double twins
One of the present authors has developed a constitutive model
assuming "soft" double twins that is able to predict the onset of the observed macroscopic work softening, w h i c h eventually leads to strain localization and failure
The experimental and
modelling results have thus indicated that the formation of {10T1} -{10 1 2} double twins might cause early shear failure of Mg-based alloys due to the combined effects of strain softening of the continuum and localised void formation. The present investigation is concerned with secondary {101 2} twinning occurring inside primary
{1011} twins in polycrystalline M g - 3 A l - l Z n ( A Z 3 1 ) , the most c o m m o n
wrought
451
Identification a n d Rationalization of S e c o n d a r y T w i n V a r i a n t s in a M e g n e s i u m Alloy
m a g n e s i u m alloy. Electron backscatter diffraction ( E B S D ) studies performed on samples of rolled and extruded material
111
have frequently detected a 3 8 ° < 1 2 1 0 > double twin/matrix
misorientation relation. There are six {1012} planes and the c o m m o n l y reported 3 8 " < 1210 > reorientation holds for secondary twinning on only one of these planes, ft is not clear if this reorientation is physically favoured m o r e than the other reorientations or if studies to date have simply overlooked the possibility of other reorientations. The Schmid factor has proven to be useful in rationalizing the observed twinning m o d e s in a n u m b e r of materials ' and the present paper follows this line of investigation as a first attempt at e x a m i n i n g the problem. In order to establish a statistically meaningful data set, the present work uses about 30 E B S D m a p s of material tested in tension in both wrought and as-cast states. Selected specimens were also examined using transmission electron microscopy ( T E M ) . First, the double twin variants were identified and the Schmid factors were subsequently calculated and summarized pictorially. This was followed by the presentation of the data and a comparison of the findings with the predictions m a d e using the Schmid analysis. Finally, some speculation is offered to explain w h y some of the results seem to disagree with the Schmid predictions. 1
BACKGROUND Double Twin Variants In the following we examine the reorientations that a c c o m p a n y primary {10 1 1) twinning followed by secondary {1012) twinning. The objective is to identify the misorientation relations between the original matrix and the doubly twinned volume. T h e reorientations produced by the six secondary {101 2} twinning systems are illustrated in Fig. 1 on a pole figure (equal area projection), labelled A - F . The final orientations fall in three pairs. The m e m b e r s of each pair
(a)
(b)
Figure 1. (a) Reorientation of the basal poles during double twinning (the open circles refer to the original orientation of the basal pole subjected to compression, the closed circle refers to the reoriented basal pole after primary {101 1} twinning and the triangles correspond to the final basal poles after secondary {10 1 2} twinning); (b) {10 1 2} twin plane poles after primary { 1 0 Ï 1) twinning. The orientations produced by each twinning variant are tracked with the letters A to F.
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correspond to positive and negative rotations, in relation to the primary twin, of 86" around a single < 1210 > axis. There are three of these axes, h e n c e the three groups of final orientations. Furthermore, it turns out that the resulting misorientations b e t w e e n the initial matrix and the doubly twinned orientation are equivalent for twinning o n the planes C and E. T h e same is true for twin planes Β and F. T h e r e are therefore four misorientation relations to be considered and these are given in Table 1 and identified by the n u m b e r s 1-4. T h e most c o m m o n l y observed type discussed a b o v e corresponds to type 1. Table I. {10 1 1}-{101 2} double twinning variants identified in the present study. The capital letters refer to the secondary twinning planes indicated in Figure 1. Twinning variants Misorientation relation
2(A)
3 (B and F)
4 (C and E)
30.1°<1210>
66.5° < 5 9 4 3 >
69.9° < 2 4 2 1 >
1 (D) 37.5°<1210>
Schmid Factor Calculations T h e Schmid factors for {101 1} twinning under uniaxial tension were calculated and the t w o most heavily stressed systems (in the sense that permits c-axis contraction) were subjected to further analysis. For the orientations produced by these primary twin systems, the Schmid factors, m, for secondary {10 1 2} twinning w e r e calculated. T h e secondary twin variants corresponding to the t w o systems with the highest Schmid factors were recorded. The favoured variants were identified in terms of the corresponding misorientation relations given in Table I. The outcomes of these calculations are given in inverse pole figure space in Fig. 2, which shows the broadest extent of the orientations where type 1/2 variants and type 3/4 variants are predicted. These groupings are m a d e to simplify the analysis and because of a similarity of behaviour. It turns out that this approach is also convenient for comparing with experimental results. Variants 3 and 4
Variants 1 and 2
[0001]
[Ϊ2Ϊ0]
[0001]
[T2T0]
f [îoio]
[To 10]
Figure 2. Orientations of the stress axis where type 1 or 2 twins and type 3 or 4 twins are expected (non-shaded areas). This is based on a consideration of the two most heavily stressed primary and secondary twin systems. T h e dark regions are "forbidden" and the hatched regions are "not predicted" . 12
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EXPERIMENTS The material used in the present study is commercial-grade magnesium alloy AZ31 ( M g 3 % A l - l % Z n - 0 . 2 % M n ) obtained in three different forms: " a s - c a s t " , " e x t r u d e d " and " r o l l e d " (Fig. 3). The as-cast material, purchased in the form of a cast billet, possessed a random texture and had an average linear intercept grain size of - 3 5 0 p m . T h e extruded material, purchased in the form of a 75 m m diameter bar, had an average linear intercept grain size of 8 μιη and a strong axisymmetric texture, in w h i c h the majority of grains had their c-axes approximately perpendicular to the extrusion axis. The rolled sheet was received with a thickness of 2.5 m m and had an average linear intercept grain size of 7 μηι. The rolled sheet texture possessed a strong basal fibre component with the grain c-axes approximately perpendicular to the rolling plane. For both the as-cast and rolled material, machined samples were annealed for 2 h at 350 "C and aircooled prior to deformation. Machined samples from the extruded material w e r e tested in the " a s - e x t r u d e d " condition. Room-temperature tensile testing was conducted for the as-cast and rolled material along the RD (rolling) direction and for the extruded material parallel to the E D (extrusion) direction (see Fig. 3) using a range of strains up to failure. E B S D was employed to examine the crystallographic reorientations caused by deformation twinning and T E M was used to investigate the fine structure of twins in selected specimens.
Figure 3. Basal pole figures of: (a) as-received as-cast billet (intensity levels 0.5, 1, 2 , . . . , m a x = 6.2 times random): (b) as-received extruded bar (intensity levels 1, 2, 4 , . . . , max = 7.2 times random); (c) as-received rolled sheet (intensity levels 1, 2, 4 , . . . , m a x = 10.1 times random). RESULTS Observed Double T w i n Variants E B S D m a p s produced from the experimental materials revealed presence of all the four double twin variants identified above (Table I). Examples of these variants obtained by the E B S D technique are presented elsewhere
Figure 4 shows an example of the {101 1}-{I01 2 J d o u b l e
twin in the rolled material analyzed using the T E M technique. The corresponding habit plane w a s determined to be approximately parallel to a {3034J plane. The above figure shows that this particular double twin w a s dominated by the type 1 secondary twin variant and contained two small segments of type 2 variant. This would suggest that after their fonnation within the {101 1} primary twin interior, the nuclei of type 1 secondary twin variant grew much faster compared to those of the other variant and ultimately c o n s u m e d the majority of the primary twin volume. Overall, the n u m b e r of primary {10 1 lj twins and {10 1 1}-{10 1 2}double twins detected per grain varied between one and four. In two-thirds of the cases, one twin was seen per grain. Three or four twins were seen in 1 5 % of the grains examined. It should be also noted that, in
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Figure 4. T E M analysis of a {101 1}-{10 1 2}double twin, e m b e d d e d in the matrix M, largely composed of the type 1 secondary twin variant Ti and containing t w o segments of the type 2 variant T : (a) Dark-field image obtained using (0001 ) i reflection (the circles indicate the 2
T
locations w h e r e selected area diffraction ( S A D ) patterns were taken): (b) S A D pattern from T | / M interface showing the 38°< 1210 > type 1 reorientation; (c) S A D pattern from T / M interface 2
showing the 30"< 1210 > type 2 reorientation. The zone axes in (b,c) are close to [1210]. quite a n u m b e r of cases, boundaries with a misorientation m i d w a y between twin variants 1 and 2 or 3 and 4 w e r e observed. The most probable source of this discrepancy is the presence of additional deformation induced lattice rotations. In some cases this was evident as an orientation gradient of several degrees. Such gradients introduce a degree of uncertainty into the present analysis. In order to acknowledge this uncertainty, the four secondary twin variants are grouped into t w o categories in much of the following analysis: type 1/2 and type 3/4. Cases of the type 1/2 variant w e r e m o r e c o m m o n and in some situations these variants consumed very high fractions of the primary twin volume (see Fig. 4). Schmid Factors and the Observed Primary Twin Variants The Schmid factors corresponding to the primary {101 1} twins were determined for all cases (130) examined using E B S D w h e n the primary twinning system could be unambiguously identified. The sign of the Schmid factors was assigned so that a negative value refers to the case where the sense of the resolved shear stress corresponds to that expected to cause contraction along the c-axis. Approximately 6 0 % of the primary {101 1} twins formed o n systems with a negative Schmid factor > 0 . 3 . A small n u m b e r of isolated instances were observed of twins forming on systems " f o r b i d d e n " by the sense of the stress. In these cases, a nearby source of a departure from the imposed stress could be detected, such as an irregular grain boundary shape. About 5 5 % of the primary {101 1} twins formed on either the most or the second most heavily
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stressed system. For eases where only one {10 1 1} twin w a s observed in the grain, this fraction increased to - 6 5 % . It is clear that in these cases the Schmid factor is a reasonable predictor of the probability of primary twin formation. For present purposes, we are interested in the behaviour of secondary twinning and this is examined below for a reduced data set of primary twins that formed on either the most or the second most heavily stressed primary twin system. Schmid Factors and the Observed Secondary Twin Variants The observed secondary twinning variants grouped into either the type 1/2 class or the type 3/4 class are plotted in Fig. 5 in inverse pole figure space of the parent orientation. This manner of representation permits ready comparison with predictions (see Fig. 2). It also avoids the need to determine the primary twinning reorientation, which in many cases is not available experimentally. It can be seen in Fig. 5 that there is a significant disagreement between experiment and the Schmid factor predictions for the occurrence of the type 1/2 variant. For the type 1/2 variant, > 8 0 % of the occurrences are on systems other than one of the two most heavily stressed systems (as determined by the S c h m i d factor). This is in contrast to the occurrence of the primary {101 1{ twins considered above, w h e r e , for grains with one twin, - 3 5 % of the time the twin formed on a system other than one of the two most heavily stressed systems.
Figure 5. Matrix orientations corresponding to the twin variants observed in the present study. The data shown are for secondary twinning occurring within the primary twins that formed on the most or second most heavily stressed system. The shaded regions correspond to the hatched regions in Figure 2 . U
It thus appears that the type 1/2 secondary twin variant occurs m o r e readily in tension than one would expect from a consideration of the Schmid factor. In order to determine if there was a preference for the type 1 over the type 2 variant, the measured misorientation angles for twins in this category were examined. T h e histogram presented in Fig. 6 shows a clear skewing towards the type 1 variant. W e can thus conclude that the occurrence of the type 1 variant appears to be a n o m a l o u s in terms of a basic Schmid factor analysis. Furthermore, the data obtained using the wrought and as-cast (and h o m o g e n i z e d ) material were combined. The main t w o differences between these categories of samples are: the grain size (coarser for as-cast material) and the texture (sharper for wrought material). H o w e v e r , no differences in frequencies of secondary twin variants were detected between these t w o sample sets.
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100
24
26
28
30
32
34
36
38
40
42
44
46
48
Misorientation angle Ο
Figure 6. Distribution of misorientation angles for double twins identified as being of the type 1 or type 2 class. There is a clear preference for type 1 double twin variant. DISCUSSION Schmid Factors and Primary Extension Twins The non-Schmid behaviour of secondary (10 1 2J twins contrasts with the formation of primary {101 1} twins, the latter of which agrees reasonably well with the Schmid predictions. This m a y reflect a m o r e general difference between the t w o twinning m o d e s . In order to examine this possibility, the occurrence of primary {10 1 2} twinning in a sample of extruded AZ31 was inspected. Extruded material was strained in tension perpendicular to the extrusion direction to a strain of - 0 . 0 4 . The test direction w a s chosen to ensure a wide range of resolved shear stresses on the twinning planes. In all, 46 twins were examined and these were restricted to cases where one twin dominated the grain. T h e {1012} primary twins in extruded material were found to form o n planes with both a low Schmid factor magnitude and a low Schmid factor rank. The lesser stressed (third or less most heavily stressed) primary {10 1 2} twinning systems account for - 5 5 % of all the cases examined. This is greater than the value obtained for the singly forming primary
{101 1}
twinning systems ( 3 5 % ) , but still falls below the high ( > 8 0 % ) value obtained for the secondary 1/2 variant {101 2} systems. So, although it does appear that primary {1012) twinning occurs more readily on the lesser stressed planes than {101 1) twinning, the occurrence of the secondary {1012} twinning on lowly stressed planes occurs even more frequently. This difference is well illustrated in Fig. 7. In the cases shown for secondary twinning (which correspond to situations where the tensile stress is near to lying in the basal plane), the formation of the type 1 variant involves activation of twin planes stressed considerably less than the most heavily stressed plane. It can thus be concluded that the predominance of the above variant is a non-Schmid p h e n o m e n o n that appears to be unique to secondary {101 2) twinning.
Possible Double Twin Variant Selection Mechanism T h e present results showed that there was a clear preference for the formation of type 1 double twin variant. This preference seems to relate more to an enhancement of growth of the above secondary twin variant rather than to an enhancement of its nucleation. It appears that once nucleated this variant displays a high rate of lateral growth (see Fig. 4) and it was largely
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observed that the type 1 variant c o n s u m e d significant lengths of primary twin while no cases of obvious copious nucleation were found.
Figure 7. Ratio between the Schmid factor of the active secondary twinning system and that of the most heavily stressed system for: (a) T h e observed primary {1012} twins; (b) The type 1 variant for tension within 20° of the basal plane of the parent (i.e. the rightmost region of the inverse pole figure). The formation of the type 1 variant for these orientations requires activation of twinning on planes that are significantly less stressed than the most heavily stressed system. One reason why the type 1 variant might display an enhanced rate of growth has to do with the tendency of twins to minimize the compatibility strains. This principle explains the c o m m o n lenticular shape of deformation twins with the short axis normal to the twinning plane and is described mathematically in the Eshelby inclusion problem In the case of lateral growth of secondary {10 1 2} twins, the final shape is enforced by the shape of the primary {10 1 1} twin, which leads to an increase in the compatibility strain. The greater the departure of the secondary twinning plane from the habit plane of the primary twin, the greater the additional compatibility strain and, thus, the higher the expected impediment to secondary twin growth. The angles between the primary and secondary twin planes for the four twinning variants are shown in Table 11. It can be seen that the t w o planes are significantly closer to coincidence for the type 1 variant than for any of the others. Thus one might expect that the stress for propagating the secondary twin throughout the primary twinned volume would be least for the type 1 variant. This may account for its over-representation observed in the present work. In order to assess the a b o v e suggestion, some calculations were performed using the visco-plastic self-consistent ( V P S C ) crystal plasticity model, which has been well described in . The model regards each grain as an ellipsoidal inclusion e m b e d d e d in a the literature h o m o g e n e o u s effective m e d i u m . The externally applied (macroscopic) boundary conditions are fulfilled in a self-consistent manner by plastic deformations at the grain level. Ten similarly l 3 1 4
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Table II. Angles between the primary {101 1) twin plane and the secondary {1012} twin planes for each double twin variant. T w i n n i n g variant
1
2
3
4
Misorientation angle
18.8°
74.9°
87.6°
49.9°
oriented primary {101 1} twins were first constructed as e m b e d d e d ellipsoidal grains in a random texture. Strictly speaking these should be e m b e d d e d in the matrix of their respective parents but the present purpose is served well enough by e m p l o y i n g a r a n d o m environment. The initial crystallographic orientations of the " t w i n s " were grouped closely around that expected to form in an ideally oriented parent where all the Euler angles are zero and where tension was applied along the χ direction. The corresponding basal poles are s h o w n in Figure 8a. This scenario is quite c o m p a r a b l e to Fig. 1. T h e orientations of the ellipsoid axes were chosen such that the two equivalent long axes lay approximately in the relevant {10 1 1} twinning plane. Three ratios of the ellipsoid short axes to the t w o long axes w e r e considered: 1:1, 1:10 and 1:20. The aggregate w a s subsequently deformed in tension to a simulated strain of 0.05 assuming that the active deformation m o d e s are basal, prismatic and second order pyramidal slip and extension {1012} twinning. Values of the respective constitutive parameters in the Voce hardening laws w e r e equivalent to those employed by A g n e w at al. for the slip modes. For extension twinning, the critical resolved shear stress ( C R S S ) value was lowered to 15 M P a t o ensure twin activity. The reorientations produced during these simulations are shown in Figures 8b-8d. In all simulations, large reorientations occurred due to the activation of twinning. Furthermore, it is evident that with changing the aspect ratio there is a change in the active twin variant. A s the ellipsoid aspect ratio is increased, the type 1 variant is favoured. This is in agreement with the view advanced above, that this variant minimizes most the mismatch in preferred habit of the primary and secondary twins. Ζ
(a)
(b)
(c)
(d)
Figure 8. Basal pole reorientation produced during the simulated tensile deformation using the V P S C m o d e l : (a) Starting cluster of ten primary twin ( e m b e d d e d grain) orientations; (b) Partial reorientation to form the variant 3/4 secondary twins for the ellipsoid axis ratio of 1:1 (sphere); (c) Partial reorientation to form both the variant 3/4 and 1/2 secondary twins for the ellipsoid axis ration of 1:10; (d) Full reorientation to form the variant 1 secondary twins for the ellipsoid axis ration of 1:20. Tensile axis is labelled x.
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CONCLUSIONS - Inspection of the misorientations between the parent matrix and
{10 I 1}-{I0 1 2} doubly
twinned volumes in AZ31 m a g n e s i u m alloy deformed in tension reveals that there are four possible misorientation variants that are strongly orientation dependent. - The occurrence of secondary {1012} twinning within primary {101 1} twins favours the formation of misorientation variant characterized by 38° < 1210 > . This involves the activation of secondary twinning systems inconsistent with the simple Schmid-type behaviour. - One possible cause of the non-Schmid behaviour of secondary twinning is that for the secondary twin to grow it must take on a shape enforced by the primary twin. This is non-optimal for strain compatibility. The 38°< 1210 > variant provides the closest match between the primary and secondary twinning planes, thus minimizing the compatibility strain. This is also confirmed by V P S C simulations of twin activity in ellipsoidal grains. REFERENCES 1. E.W. Kelley and J.W.F. Hosford, Plane-Strain Compression of Magnesium and Magnesium Alloy Crystals, Trans. Melau. Soc. AIME, 2 4 2 , 5-13 (1968). 2. B.C. Wonsiewicz and W.A. Backofen, Plasticity of M a g n e s i u m Crystals, Trans. Metall. Soc. AIME, 2 3 9 , 1422-1431 (1967). 3. W . H . Harrt and R.E. Reed-Hill, Internal Deformation and Fracture of Second-Order {101}{102} Twins in M a g n e s i u m , Trans. Metall. Soc. AIME, 2 4 2 , 1127-1133 (1968). 4. C S . Roberts, The Deformation of M a g n e s i u m , in M a g n e s i u m and Its Alloys, John Wiley & Sons, N e w York, 8 1 - 1 0 7 ( 1 9 6 0 ) . 5. H. Yoshinaga, T. Obara, and S. M o r o z u m i , Twinning Deformation in Magnesium Compressed along the C-Axis, Mater. Sei. Eng., 1 2 , 255-264 (1973). 6. S.L. Couling, J.F. Pashak, and L. Sturkey, Unique Deformation and A g e i n g Characteristic of Certain M a g n e s i u m - B a s e Alloys, Trans. ASM, 5 1 , 94-107 (1959). 7. M.R. Bamett, M.D. N a v e , and C.J. Bettles, Deformation Mierostructures and Textures of Some Cold Rolled M g Alloys, Mater. Sei. Eng. A, 3 8 6 , 205-211 (2004). 8. M.D. Nave and M.R. Barnett, Mierostructures and Textures of Pure Magnesium Deformed in Plane-Strain Compression, Scripta Mater., 5 1 , 881-885 (2004). 9. M.R. Bamett, Twinning and the Ductility of Magnesium Alloys Part II: " C o n t r a c t i o n " Twins, Mater. Sei. Eng. A, 4 6 4 , 8-16 (2007). 10. L. Jiang, J.J. Jonas, A.A. Luo, A.K. Sachdev, and S. Godet, Twinning-Induced Softening in Polycrystalline A M 3 0 M g Alloy at Moderate Temperatures, Scripta Mater., 5 4 , 771-775 (2006). U . S . Godet, L. Jiang, A.A. Luo, and J.J. Jonas, Use of Schmid Factors to Select Extension Twin Variants in Extruded M a g n e s i u m Alloy T u b e s , Scripta Mater., 5 5 , 1055-1058 (2006). 12. M.R. Bamett, Z. Keshavarz, A.G. Beer, and X. Ma, N o n - S c h m i d Behaviour during Secondary Twinning in a Polycrystalline M a g n e s i u m Alloy, Acta Mater., 5 6 , 5-15 (2008). 13. R.A. Lebensohn and C.N. T o m e , A Study of the Stress State Associated with Twin Nucleation and Propagation in Anisotropic Materials, Philos. Mag. A, 6 7 , 187-206 (1993). 14. C.N. T o m e and G.R. Canova, Self-Consistent Modelling of Heterogeneous Plasticity, in Texture and Anisotropy: Preferred Orientations in Polycrystals and their Effect on Materials Properties, U.F. Kocks et al. eds., C a m b r i d g e University Press, 4 6 7 - 5 1 0 (1998). 15. S.R. A g n e w , C.N. T o m e , D.W. Brown, T.M. Holden, and S . C Vogel, Study of Slip Mechanisms in a Magnesium Alloy by Neutron Diffraction and Modelling, Scripta Mater., 4 8 , 1003-1008 (2003).
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T E X T U R E S IN H C P T I T A N I U M A N D Z I R C O N I U M : I N F L U E N C E O F T W I N N I N G
Nathalie Bozzolo and Francis Wagner L E T A M (Laboratoire d'Etude des Textures et Application aux Matériaux). U M R C N R S 7078. Université Paul Verlaine - Metz, Ile du Saulcy, F-57045 Metz cedex, France [email protected] ABSTRACT This paper is based on the comparison of the cold-rolling and subsequent annealing behavior of two hep metals (cp-Ti and Zr702). A special attention is paid to the consequences of deformation twinning on texture and microstructure evolutions. The Ti sheet develops an important amount of twins in the first stages of deformation, whereas the Zr702 sheet deforms by slip only. Twinning is a very efficient grain fragmentation mechanism. It generates specific texture c o m p o n e n t s at m e d i u m strains which smear out after 8 0 % thickness reduction. Recrystallization m e c h a n i s m s and kinetics are sensitive to the deformation substructure types and therefore to the occurrence of twinning. However the texture evolves quite few during recrystallization after 8 0 % cold-rolling in both materials. Twinning orientation relationships are promoted in the misorientation distributions of the deformed mierostructures but also slightly show up in the grain boundary populations of recrystallized materials. INTRODUCTION Titanium and Zirconium are both hexagonal-close-packed (hep) metals at ambient temperature. The alpha (hep) phase is predominant in many industrial alloys used in high performance fields such as aeronautics or nuclear industry. The crystallographic texture directly controls anisotropic mechanical properties " , but also controls other properties like corrosion resistance " which is dependent on the crystallographic orientation of the crystal surface, or stress corrosion cracking resistance which may be sensitive to both crystallographic texture and grain boundary character distribution. Deformation twinning is well-known to play an important role for plastic deformation in hep metals because of the limited n u m b e r of active slip s y s t e m s ' " . As a general rule, twinning generates m a n y different orientations and decreases the texture s t r e n g t h . Allowing twinning is often necessary to correctly simulate texture evolution during deformation of hep m e t a l s ' . Twinning therefore indirectly influences anisotropic properties because it influences the texture, and may contribute to crack but it has also a direct effect on crystal plasticity p r o p e r t i e s ' initiation m e c h a n i s m s . In the present paper, the consequences of deformation twinning on static annealing will also be discussed. The texture and microstructure evolutions during cold-rolling and annealing will be described for commercially pure titanium (cp-Ti), which undergoes twinning during coldrolling, and for low alloyed zirconium (Zr702), which deforms by slip only. A special attention will be paid to the influence of deformation twinning on the texture evolution during cold-rolling and subsequent annealing, on deformation substructures development and recrystallization m e c h a n i s m s , and on misorientation distributions. 1
5
4
8
9
10
412
4
0 1 6 - 1 8
19
MATERIALS AND EXPERIMENTAL PROCEDURES The crystalline structure of the investigated materials (cp-Ti and Zr702) is hexagonalclose-packed. Their chemical compositions are given in Table 1. T h e as-received sheets were
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T e x t u r e s in H C P T i t a n i u m a n d Z i r c o n i u m : Influence of T w i n n i n g
3 m m thick and in a recrystallized state, with equiaxed microstructures (mean grain sizes : 3 0 p m and 7μηι for cp-Ti and Zr702, respectively). Sheets were cold-rolled to 8 0 % thickness reduction and subsequently annealed under inert atmosphere. Table 1. Chemical composition of the studied materials (weight ppm) ΛΙ
C
Cr
52
cp-Ti 25
Zr702
54
206
Fe
Η
237
3
802
<3
Hf
Ν
Ni
41 127
55
60
Ο
Si
1062
<100
1308
21
Sn
Ti
Zr
balance 2769
175
balance
Microstructure and texture were analyzed at different stages of the deformation and annealing processes by X - r a y diffraction ( X R D ) texture goniometry and electron back scattered diffraction (EBSD) coupled to a field emission gun scanning electron microscope ( F E G - S E M ) . The harmonic method was applied to compute orientation distribution functions ( O D F s ) from either X R D pole figures or E B S D individual orientations. Crystal orientation is described by means of Euler angles ( φ ι . Φ. φ ) , as defined by B u n g e . The crystal coordinate system used here is <X=[10-10], Y = [ - 1 2 - 1 0 ] , Z = [ 0 0 0 1 ] j . It is worth noticing that another convention for crystal coordinate system is sometimes used in the literature, with X axis aligned with in stead of Y, which leads to a 30° difference in the φ values. Only O D F sections at (pi=0° in the Euler space are shown here because they contain the main features of the investigated textures. The misorientation distributions were obtained from E B S D data and plotted as density m a p s in the fundamental zone of the Rodriguès-Frank (RF) space. The RF space basis vectors R i , R and R are aligned with the crystallographic directions [2-1-10], [01-10] and [0001], respectively. 20
2
2
2
}
MICROSTRUCTURE EVOLUTION DURING COLD-ROLLING Fig.l reveals a clear difference in the microstructure evolution of cp-Ti and Zr702 sheets during cold rolling. A large amount of twins is observed in some grains of the titanium sheet after 3 0 % cold rolling. Additional cold-rolling leads to the fragmentation of those lamellae as a result of slip activity and multiple twinning. After 8 0 % thickness reduction, the microstructure is fragmented d o w n to the sub-micrometer scale. Very poor E B S D pattern indexing rate is obtained, except in some grains which have not twinned but stretched out along the rolling direction by slip activity. In Z r 7 0 2 , almost no twin is observed, deformation occurs by slip only. Right from the beginning, the strain is heterogeneously distributed in the material. S o m e grains being obviously m u c h harder than others keep an equiaxed shape after 5 0 % thickness reduction. Hard grains are oriented with [0001] axis close to the sheet normal (ND), which is consistent with the results reported by . Additional cold-rolling leads to the stretching of the hard grains and to the development of lamellar substructures at the micrometer and sub-micrometer scales. This difference in the behavior of the two materials is consistent with the fact that twinning is more easily activated when increasing grain size and decreasing impurity c o n t e n t ' . 2 I
14
22
2 3
I N F L U E N C E O F T W I N N I N G ON G R A I N F R A G M E N T A T I O N T w i n n i n g is a very efficient m e c h a n i s m for grain fragmentation. The twins observed after 3 0 % cold-rolling are subdivided into smaller and smaller features with increasing strain. The exact twin orientation relationships are lost due to additional slip activity in and around the twin lamellae and to the accumulation of dislocations in the twin boundaries. This m e c h a n i s m results in very small fragments (around 200 n m in size) which are highly misoriented to each o t h e r . The occurrence or non occurrence of twinning leads to very distinct microstructure types which can be better seen on the band contrast m a p s of Fig.2. Starting from a larger grain size compared to the Zr702 material, the cp-Ti sheet develops a finer deformed microstructure due to twinning. 24
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T e x t u r e s in H C P Titanium a n d Zirconium: Influence of Twinning
Figure 1. Microstructure evolution during cold-rolling of Zr702 and cp-Ti (Scale bars = 10 p m )
Materials P r o c e s s i n g a n d Texture
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Figure 2. Zr702 and cp-Ti mierostructures after 8 0 % cold-rolling ( E B S D band contrast m a p s in the transverse plane) INFLUENCE OF TWINNING ON TEXTURE EVOLUTION DURING C O L D - R O L L I N G The global texture evolutions a c c o m p a n y i n g cold-rolling are shown in Fig.3, together with the initial texture of the as-received materials. Initial textures, characterized by an O D F m a x i m u m around (0° 35° 15°-20°) for cp-Ti. and (0° 30° 30°) for Zr702 are typical of recrystallized materials having undergone moderate grain g r o w t h " . 25
27
Figure 3. Texture evolution during cold-rolling of Zr702 and cp-Ti
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In cp-Ti, after 8 0 % cold-rolling, the O D F m a x i m u m is at (0 40 0). with a considerable spread around it. At intermediate strains ( 3 0 % and 5 0 % thickness reduction), secondary components develop at (0 0 30) and (0 90 0) in the (Pi=0° O D F section. T h e first one disappears again at larger strains while the second one remains until 8 0 % thickness r e d u c t i o n . In Zr702, the texture also w e a k e n s during cold-rolling. The O D F m a x i m u m progressively shifts from (0 35 30) to (0 30 0) which b e c o m e s a very distinct texture feature at 8 0 % " ' ° . After 8 0 % thickness reduction, both materials develop a texture with basal poles tilted by +/- 20 to 40° from N D towards T D and <10-10> aligned with R D . This texture is well known for h e p metals with c/a < 1.633 submitted to c o l d - r o l l i n g ; twinning induces secondary components only. 424 2 8
2,
31
T h e additional c o m p o n e n t s developed in cp-Ti are indeed due to twinning, as it can be shown by analyzing the orientations of the twinned and untwinned areas in moderately deformed samples (Fig. 4). The untwinned grains are characterized by orientations close to (0 50 30) after 3 0 % ftiickness reduction. The twins have been identified, based on their misorientation to the matrix grains, as being either compressive or tensile twins. A higher occurrence was observed for compressive twins ({11-22}<11-2-3>). The {10-12}<10-1-1> tensile twins were about half as frequent and {11-21 ]<11-2-6> tensile twins were only rarely o b s e r v e d . The analysis of the E B S D m a p s showed that multiple twinning occurred also, with often tensile twins appearing in compressive ones. Multiple twinning was also mentioned by other authors working on a similar material . 24
28
Figure 4. Identification of the twinning related texture c o m p o n e n t s o n an E B S D m a p of the 3 0 % cold-rolled cp-Ti sheet: a) white = Φ < 1 5 ° . black = Φ > 7 5 ° . light grey = boundaries with misorientation > 5 ° . b) Untwinned areas in dark grey and related pole figures. INFLUENCE OF T W I N N I N G ON RECRYSTALLIZATION The difference in the 8 0 % cold-rolled microstructures of cp-Ti and Zr702 resulting from twinning activity discrepancy, leads to distinct behavior during recrystallization (Fig. 5). Recrystallization always starts in the most deformed areas, or finest fragment structures. In Zr702, new grains appear first along wavy lines (corresponding to the severely deformed areas, dark o n Fig. 2) and will progressively invade coarser and coarser substructured a r e a s . Full 27
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recrystallization is achieved after 4h at 500°C. In cp-Ti. the 80 v o l . % of the material which underwent severe fragmentation as a consequence of twinning, give rise to a high nucleation density and recrystallize very fast (40 min at 500°C), mainly by recover) of the highly misoriented deformation cells (continuous recrystallization). Regenerating. 1the other 20 v o l . % is much slower• (ends after 4h at 500°C) 500°C)
3
3 4
Figure 5. Early recrystallization stage in Zr702 and cp-Ti after 8 0 % cold-rolling
Figure 6. Recrystallization textures of Z r 7 0 2 and cp-Ti after 8 0 % cold-rolling (percentages are recrystallized volume fractions)
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Recrystallization weakens the global texture in both materials but the position of O D F maxima is maintained (Fig. 6. to be c o m p a r e d to the 8 0 % rolling textures of Fig. 3 ) . The texture change was quantified by integration of the positive difference between the full-recrystallization O D F and the deformation O D F . With the textures presented here, this calculation leads to 22vol.% and 26vol.% for cp-Ti and Zr702, respectively. The global texture evolution during recrystallization looks therefore very similar in both materials, despite different microstructure regeneration sequences and kinetics. 3 2
The twinning-related (0 90 0) texture component is still remaining after recrystallization completion in cp-Ti but will disappear very fast during subsequent grain growth (Fig. 7 T due to the small size of the corresponding grains. Fig. 8 shows partial textures computed from grainsize-based sub-populations of grains. The twin-related component indeed appears only in the partial texture of small grains. 6
5 c. Annealing for 30 min at600°C min at 600°C ( M G S = 5.1 μπι) (MGS = 8 . 3 p m ) Figure 7. Grain growth texture evolution in cp-Ti
a. recrystallized state (MGS = 3.3
μπι)
b. Annealing for
e. Annealing for d. Annealing for 100 min at 800°C 100 min at 700°C (MGS = 64.0 pm) ( M G S = 3 2 . 8 pm) (same color code as on Fig. 6)
Figure 8. Grain size based partial textures in a fully recrystallized cp-Ti sample (same color code as Fig. 6). a) Texture of the largest grains accounting for I5surf.%. b) Texture of the smallest grains also accounting for 15surf.%. a) large grains
b) small grains
INFLUENCE OF TWINNING ON MISORIENTATIONS Deformed materials Twinning strongly influences the misorientation populations in deformed
materials.
Using E B S D data, the amount of low and high angle misorientations has been calculated at different stages of cold-rolling and for both cp-Ti and Zr702 (Fig.9). It is worth mentioning that such an analysis is dependent on the E B S D measurement step-size and is biased by the decrease in indexing rate while increasing deformation. It nevertheless gives an interesting deformation structure-type fingerprint. Slip activity in Zr702 promotes the formation of orientation gradients and increases that way the amount of low angle misorientations. This is counterbalanced in cp-Ti by twinning-induced high misorientations.
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Figure 9. Relative frequencies of the low and high angle misorientations measured by E B S D in Zr702 and cp-Ti, as a function of the thickness reduction ratio Twinning induced misorientations can be investigated in m o r e detail by using for example Rodriguès-Frank space plots. The striking e x a m p l e of 3 0 % cold-rolled cp-Ti is presented on Fig. 10. Consistently with the E B S D map analysis, the highest peak corresponds to the compressive twin to matrix orientation relationship (circle). The peak related to (1-102)< 1 ΟΙ -1 > tensile twin (square) is about twice less intense, and a weak one is observed at the position of the second type of tensi le twi η ( 11 -21 )< 11 -2-6> (triangle).
Figure 10. Misorientations measured by E B S D in the 3 0 % cold-rolled cp-Ti sheet, plotted in the Rodriguès-Frank space (R,//[2-1-10], R / / [ 0 1 - 1 0 ] . R //[0001 ]) 2
3
Symbols : Circle: Compressive twin (11-22)<11-2-3>. Square: Tensile twin type 1 ( 1-102)<10-1-1>, Triangle: Tensile twin type 2 ( 11-21 )<11 -2-6>. Star: Multiple twinning induced misorientation
Another strong peak s h o w s up in this plot, corresponding to a misorientation by 41° around an axis close to < 5 - l - 4 3 > . The latter is induced by multiple twinning. This is the misorientation obtained at the intersection of a compressive twin and a tensile twin of type 1 in a given matrix grain. It is also the misorientation between the matrix and the tensile twin developed in a compressive one.
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Grain boundary populations in recrystallized cp-Ti After 8 0 % cold-rolling and subsequent recrystallization completion, the microstructures were equiaxed and without any
annealing twin. The m a x i m u m densities in the
misorientation
distributions (Fig. 11a) are of course much lower compared to the twinned previous case. The texture itself might completely control such a smeared distribution. In order to check for the existence of preferred misorientations, texture effect has to be evaluated (done here by means of the T - M D ) .
Figure 11. Misorientations measured by E B S D in a 8 0 % cold-rolled cp-Ti sheet after recrystallization completion, plotted in the Rodriguès-Frank space. ( R , / / [ 2 - l - 1 0 ] . R / / [ 0 1 - 1 0 ] , R //[0001]) Only grain boundaries with misorientation angles > 5° have been taken into account. 2
3
Symbols : Circle: Compressive twin (11-22)<11-2-3>. Square: Tensile twin type 1 (1-102)<10-1-1>, Triangle: Tensile twin type 2 (11-21)<11-2-6>. Star: Multiple twinning induced misorientation
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Fig. 11 b shows misorientations calculated between pairs of pixels randomly selected in the E B S D m a p . which leads to a texture-induced misorientation distribution ( T - M D ) slightly different from the one obtained from pairs of neighbor pixels (microstructure induced misorientation distribution M - M D , Fig. 1 l a ) . The T - M D would fit with the actual M - M D if the orientations would be randomly distributed in the microstructure. The difference observed between both distributions reveals that some misorientations are preferred in the grain boundary population. To better quantify this difference, the M - M D was divided by the T - M D (Fig. 1 l c ) . The values above 1 in this latter plot indicate which misorientations are favored in the grain boundary population: (i) misorientations with angle below 30° whatever the misorientation axis is. (ii) to a lower extend, misorientations of the compressive twin (circle) and tensile twins (triangle and square symbols), and (iii) other misorientations which might correspond to boundaries with particular properties (for e x a m p l e low energy). Twin orientation relationships still slightly show up. even after having been smeared out by further slip-deformation and after subsequent recrystallization. This might be the result of the continuous recrystallization of areas including fragments originally produced by twinning. CONCLUSION The deformation and recrystallization behavior of a Zr702 sheet, which deformed by slip only, was compared to the one of a cp-Ti sheet which underwent both twinning and slip during cold-rolling. A special attention has been paid to the twinning consequences all over the deformation and subsequent annealing processes. • •
After 8 0 % cold-rolling, the O D F maxima converge on the well-known "stable orientations" (0 30/40 0) for both alloys, regardless of twinning has been active or not. In the studied cp-Ti sheet (initial texture with [0001] at 35° from N D in the N D - T D plane), secondary texture c o m p o n e n t s with [0001] close to N D and perpendicular to N D are produced by twinning during cold-rolling.
•
Twinning is a very effective m e c h a n i s m for grain fragmentation during deformation. The twin lamellae formed at the beginning are deformed and fragmented by slip afterwards. The final cp-Ti microstructure is finer than in Zr702 despite a larger initial grain size.
•
Recrystallization m e c h a n i s m s and kinetics are closely related to the nature of the deformation substructures arid therefore depend on whether twinning occurred or not. Global texture evolution upon recrystallization nevertheless s h o w s similar features in both materials: O D F m a x i m a slightly decreased but kept in the same position. Twinning influences the texture evolution in the beginning of the grain growth stage because it indirectly contributes to a size-orientation correlation for the recrystallized grains. Misorientation distributions allow for identifying and quantifying twin types in deformed materials. Multiple twinning induces specific misorientations with an occurrence comparable to the one of the twin orientation relationships themselves. Even after cold rolling to 8 0 % thickness reduction and subsequent full recrystallization of cp-Ti. the twin orientation relationships still appear as slightly preferred misorientations in the grain boundary population.
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ACKNOWLEDGMENTS The authors would like to a c k n o w l e d g e their PhD students w h o have contributed to this work (N. Dewobroto (2004), F. Gerspach (2007). G. Sawina (thesis to be defended in 2008)). Dr Κ, Sztwiertnia, co-advisor of the later P h D work, and Pr. A . D . Rollett for many valuable discussions. REFERENCES 1 E.W. Kelly, W.F. Hosford, Deformation characteristics of textured magnesium. Trans. TMSAIME,242, 654-661 (1968) 2 D.S. M c D a r m a i d , P.G. Partridge, The effect of strain rate, temperature and texture on anisotropic deformation in Ti-6A1-4V, Journal of Materials Science. 2 1 , 1525-1532 (1986) 3 J.J. Fundenberger, M.J. Philippe, F. Wagner. C. Esling. Modelling and prediction of mechanical properties for materials with hexagonal symmetry (zinc, titanium.and zirconium alloys), Acta Materialia. 4 5 , 4 0 4 1 - 4 0 5 5 (1997) 4 A.K. Singh, R.A. Schwarzer, Texture and anisotropy of mechanical properties in titanium and its alloys, Zeitschrift: für Metallkunde, 9 1 , 702-716 (2000) 5 M. Schweinsberg, Α. Michaelis, J.W. Schultze, Growth of oriented anodic films on single grains of Zr: structure and epitaxy from anisotropy-micro-ellipsometry, Electrochimica Acta, 42, 3303-3310(1997) 6 H.G. Kim, T.H. Kim, Y.H. Jeong. Oxidation characteristics of basal (0002) plane and prism (11-20) plane in H C P Zr, Journal of Nuclear Materials. 3 0 6 . 44-53 (2002) 7 Y. Choi, E.J. Shin, H. Inoue, Study o n the effect of crystallographic texture on the corrosion behaviour of pilgered zirconium by neutron diffraction. Physica B, 3 8 5 - 3 8 6 , 529-531 (2006) 8 W. Qin, C. N a m , H.L. Li, J.A. Szpunar, Tetragonal phase stability in Z r 0 2 film formed on zirconium aloys and its effects on corrosion resistance, Acta Materialia, 5 5 , 1675-1701 (2007) 9 V.V. Likhanskii, L.V. Matweev, The development of the crack growth model in zirconium claddings in iodine environment, Nuclear Engineering and Design, 2 1 3 , 133-140 (2002) 10 P.G. Partridge, The crystallography and deformation m o d e s of hexagonal close-packed metals, Metallurgical Reviews, 1 1 8 , 169-193 (1957) 11 M.H. Y o o , J.K. Lee. Deformation twinning in hep metals and alloys. Philosophical Magazine A, 63, 9 8 7 - 1 0 0 0 ( 1 9 9 1 ) 12 C U . Nauer-Gerhardt, H.J. Bunge, Orientation distribution functions and twinning activation in Titanium, Proceedings of the 8th ICOTOM, edited by J.S. Kallend and G. Gottstein. The Metallurgical Society, 505-510 (1988) 13 R.A. Lebensohn, C.N. T o m é , A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals : application to zirconium alloys. Acta Metallurgica et Materialia, 4 1 , 2611-2624 (1993) 14 M.J. Philippe, M. Serghat, P. van Houtte, C. Esling, Modelling of texture evolution for materials of hexagonal symmetry - II. Application to zirconium and titanium a or near a alloys Acta Metallurgica et Materialia, 4 3 , 1619-1630 ( 1995) 15 G. Proust, C.N. T o m é , G.C. Kaschner, Modeling texture, twinning and hardening evolution during deformation of hexagonal materials, Ada Materialia, 5 5 , 2137-3148 (2007) 16 S. Nemat-Nasser, W.G. Q u o , J.Y. Cheng, Mechanical properties and deformation m e c h a n i s m s of a commercially pure titanium, Acta Materialia. 4 7 . 3705-3720 (1999)
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17 A.S. Salem. S.R. Kalidindi. R.D. Doherty. Strain hardening regimes and micorstructure evolution during large strain compression of high purity titanium, Scripta Materialia, 4 6 , 4 1 9 4 2 3 (2002) 18 A.A. Salem, S.R. Kalidindi, S.L Semiatin, Strain hardening due to deformation twinning in alpha-titanium : constitutive relations and crystal-plasticity modeling, Acta Materialia. 5 3 , 3495-3502(2005) 19 M.H. Y o o . Slip, twinning and fracture in hexagonal close-packed metals. Metallurgical Transactions Λ. 1 2 A . 4 0 9 - 4 1 7 ( 1981 ) 20 H.J. Bunge. Texture Analysis in Material Science. Butterworths, London (1982) 21 H. Francillerte. B. Bacroix. M. Gasperini. J. L. Béchade, Grain orientation effects in Zr702 polycrystalline samples deformed in channel die compression at r o o m temperature, Acta Materialia. 4 6 , 4 1 3 1 (1998) 22 H. Conrad, Effect of intersticial solutes on the strength and ductility of titanium. Progress in Materials Science, 2 6 , 123-403 (1981) 23 M.A. Meyers. O. Vöhringer, V.A. Lubarda, The onset of twinning in metals: a constitutive description. Acta Materialia. 4 9 . 4 0 2 5 - 4 0 3 9 (2001) 24 N. Bozzolo, N. D e w o b r o t o , H.-R. Wenk, F. Wagner, Microstructure and Microtexture of Cold-Rolled C o m m e r c i a l l y - P u r e Titanium. Journal of Materials Science, 4 2 , 2 4 0 5 - 2 4 1 6 (2007) 25 N. Bozzolo. N. D e w o b r o t o . T. Grosdidier, P. Barberis. F. Wagner, Grain growth texture evolution in Zirconium (Zr702) and Titanium (T40), Materials Science Forum, 4 6 7 - 4 7 0 , 441 446(2004) 26 N. Bozzolo. N . D e w o b r o t o , T. Grosdidier, F. Wagner, Texture evolution during grain growth in recrystallized commercially pure titanium, Materials Science and Engineering A, 3 9 7 . 346355(2005) 27 N. D e w o b r o t o . N. Bozzolo. P.Barberis and F. Wagner. On the m e c h a n i s m s governing the texture and microstructure evolution during static recrystallization and grain growth of low alloyed zirconium sheets (Zr702), International Journal of Materials Research (formerly Z. Metallkd), 9 7 , 826-833 (2006) 28 Y.B. Chun. S.H. Yu. S.L. Semiatin. S.K. H w a n g , Effect of deformation twinning on microstructure and texture evolution during cold rolling of cp-titanium, Materials Science Engineering A, 3 9 8 , 2 0 9 - 2 1 9 (2005)
and
29 N . D e w o b r o t o . Etude de l'évolution de texture lors de la recristallisation et de la croissance de grains d'alliages de titane et de zirconium. PhD thesis, University of Metz, France (2004) 30 H. Francillette. B. Bacroix. R.A. Lebensohn, J.L. Béchade, Final texture on cold rolled Zr702 alpha polycrystalline samples. Journal Physics IV. 1 1 , 4 - 8 3 (2001) 31 E. Tenckhoff. Deformation m e c h a n i s m s , texture and anisotropy in zirconium and zircaloy, A S T M special technical publication ( S T P 9 6 6 ) , Philadelphia (1988) 32 F. Wagner. N . Bozzolo. Ο. Van Landuyt, T. Grosdidier, Evolution of recrystallisation texture and microstructure in low alloyed titanium sheets, Acta Materialia. 5 0 , 1245-1259 (2002) 33 Y.B. Chun, S.L. Semiatin. S.K. H w a n g . M o n t e Carlo modeling of microstructure evolution during the static recrystallization of cold-rolled, commercially pure titanium. Acta Materialia. 5 4 . 3 6 7 3 - 3 6 8 9 (2006) 34 Y.B. C h u n . S.L. Semiatin. S.K. H w a n g . Role of deformation twinning in cold rolling and recrystallization of titanium. Proceedings of ICOTOM 14 (Leuven), 108-113 (2005)
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TEXTURE A N D M I C R O S T R U C T U R E EVOLUTION DURING A S Y M M E T R I C ROLLING OF MG ALLOYS 1
2
3
Jaehyung C h o , Hyung-Wuk Kim , Suk-Bong Kang Korea Institute of Materials Science, 66 Sangnam-dong, Changwon-city, Kyungnam, South Korea 641-010 2
3
Email: 'jhcho@ kims.re.kr, [email protected], [email protected] ABSTRACT Asymmetric rolling (ASR) conditions can be achieved either by changing rotational velocities of each roll or diameters of the rolls. Under asymmetric deformation conditions, materials were subjected to enhanced shear deformation, and high-angle boundaries developed with increasing strain, and finer grains were formed by following recrystallization processes. Conventional rolling process has a symmetric plane in the center of the work piece and the speeds of the top and bottom rolls are the same. ASR process has differential speeds for the rolls and it is necessary to consider the whole thickness of the sheet. To consider plastic deformation at the elevated temperatures, constitutive equation for AZ31 magnesium was implemented to finite element method. Effective deformation rates, shear strains, velocity and locations of neutral points were investigated for the various reduction ratios and roll speed ratios. As the ratio between top and bottom roll speeds increases, shear strain increases. The peak shear strains show a diagonal distribution located from the inlet contact position of the higher speed roll to the outlet contact position of the lower speed roll. Neutral positions moved in the opposite directions during asymmetric rolling. In the higher speed roll side, the neutral position proceeded to the outlet, while it retreated to the inlet in the slower speed roll side. Main texture components of magnesium alloys during conventional rolling process were basal poles and some shear components due to by friction. Shear deformation increased by asymmetric rolling and the basal poles were splited. Higher friction deepened the shear layer into the sheets. 1.
INTRODUCTION
Asymmetric rolling (ASR) has been considered as one of the useful processes to improve the formability of metal sheets. Shear deformation can be produced throughout the whole thickness during ASR. It has been known that there are two main advantages obtained by ASR over conventional rolling. Smaller working forces are used for ASR process than conventional rolling. In addition, mierostructures of finer grain size and shear textures favorable to mechanical properties are obtained [1. 2. 3. 4]. Shear deformation usually generates γ-fiber. { I l l ) in fee metals and increases r-value of the sheets. Conventional rolling process also can produce shear deformation via friction between rolls and workpiece [5, 6]. The shear effects, however, are limited to the surface region of the sheet. These days, light metals such as aluminum and magnesium alloys have been gaining more attentions in automobile applications. High strength and enhanced formability is the facing problems to be solved in the light metals. The formability of Al alloys is better than Mg alloys, but it still has lower r-values than steels. Many researchers have carried out various processes including severe deformation processes, i.e. torsion, equal channel angular process (ECAP) and ASR to enhance the strength and formability. Among these, ASR is a potential way to enable mass production of sheets commercially. Mg alloys have hexagonal close packed (hep) structure and their deformation modes to accommodate external loads are limited. ASR also has been reported to improve the microstructural features of Mg alloys. Compared with the mierostructures after conventional rolling, grain sizes after ASR are reduced and the intensities of basal poles are weakened [7]. The lowered intensities of basal poles can increase the formability.
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In this paper, we focused on shear deformation and texture evolution of Mg alloys during ASR. Two-dimensional Eulerian domain for ASR was modeled using triangle elements. To consider the differential velocities of the top and bottom rolls, the whole sheet without symmetric plane in the center of the plate was modeled. Viscoplastic constitutive equation for magnesium alloys, AZ31 was used and elastic strains were neglected. Effects of various ASR process parameters of differential velocity, friction, and reduction on crystallographic textures were investigated based on finite element method and self-consistent model for texturing. Calculated 0002 pole figures were presented across the outlet boundary after ASR modeling. The simulations presented here used the way that decouples the computations of the velocity gradients from the integration of the texture evolution equations. This decoupling approach for texture evolution has been used previously for thermo-mechanical modeling [8]. The procedure consists of three basic steps. The first step is to compute the velocity (or temperature) fields over the Eulerian domain of interest. Second, it is followed by a step to extract thermomechanical histories including velocity gradient and temperature along streamlines. Finally, integration of texture evolution equations was made along the streamlines using the thermomechanical data. In using this decoupled procedure we assume that the plastic anisotropy present in the yield surface due to crystallographic texture has little influence on the overall flow field. 2. CONSTITUTIVE EQUATION OF MAGNESIUM ALLOYS Several constitutive equations based on a phenomenological representation have been proposed [9, 10, 11]. They often contain constants that have no metallurgical significance. The shape of the stress-strain behaviors can be described using peak stress, peak strain and some other parameters and material constants. The peak stress and strain variables are related to the traditional metallurgical formula for constitutive equations which incorporate the power law with an activation energy for hot working. The peak and following lowered plateau in the flow curves reflect dynamic recrystallization (DRX) and recovery (DRV) during hot working, respectively. The phenomenological curve model was applied to Mg alloys in [ 12]. The stress-strain descriptions are expressed in terms of peak stress, peak strain, steady state stress and some additional parameters. (1)
where CT„ . σ , ε . Ci are steady state stress, peak stress, peak strain and characteristic parameter. ρ
respectively. C is a material constant which is assumed to be independent of temperature and strain rate. By introducing Zenner-Hollomon parameters, the peak stress, peak strain and steady state stress can be given. Nonlinear regression also determines both C] and C. (3)
(4)
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166928 V
{
σ„ = 2 . 7 2 x 1 0 " '
(5)
RT
Various uniaxial compression tests of AZ31 magnesium alloys raging from 523K to 673K with strain rates of 0.001s"' to Is" were used for model parameters. Fig. 1 shows predicted temperature profiles at 523K. Each curve consists of two regions displaying strain hardening and softening effects, respectively. 1
Figure 1. Predicted strain-stress curves at 523K for the various strain rates. The flow stress starts to increase with strain in the beginning of deformation during work hardening. After the peak stress, it gradually decreases and has a plateau because of softening by dynamic recrystallization and recovery. 3. M O D E L G E O M E T R Y AND B O U N D A R Y CONDITIONS Crystallographic texture is a major source of plastic anisotropy during metal forming. Many attentions are paid on texture and microstructure of light metals with lower formability. Rolled plates for metal forming are fabricated through various rolling conditions and the effects of the rolling conditions of reduction area, friction, temperature and roll speeds on the texture and microstructrue evolution have been investigated by experiments and modeling. In this paper, we mainly focused on texture evolution associated with modeling parameters of rolling conditions and microstructure evolution will be discussed yet.
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T e x t u r e a n d M i c r o s t r u c t u r e Evolution During A s y m m e t r i c Rolling of Mg Alloys
ASR introduces shear strain throughout the thickness direction. Fig. 2 shows a schematic rolling geometry and boundary conditions. Two-dimensional Eulerian domains are shown for 10 percent reduction area (RA) and 50 percent RA. Conventional rolling process has the same speeds of the top and bottom rolls and a symmetric plane is assmumed to be located in the center of the work piece. In this case, only half of the work piece can be modeled. ASR process has differential speeds for top and bottom rolls and it is necessary to consider the whole thickness of the sheet. The shape of Euler domains changes with parameter conditions. Here we tried to investigate the deformation features during ASR according to variations of rolling parameters, i.e. differential roll speed, friction coefficient, and reduction ratio. Effects of differential roll speeds on deformation and texture evolutions are stressed. 4.
MODELING TEXTURE EVOLUTION
Figure 2. A schematic diagram of the rolling process. (a) geometry and boundary conditions, (b) 10% RA. (c) 50% RA. Polycrystal plasticity models usually assume that the loading and displacement conditions at the boundary of the polycrystal are uniform, and the volume average of stress, stain and strain rate over all grains coincide with the overall stress, strain and strain rate at the boundary [13, 14]. There are some compromises to satisfy such stress and strain assumption. Taylor assumption of uniform strain throughout the polycrystal confirms strain compatibility and has been favored for modeling cubic materials which have multiple slip in the grains and mild anisotropy. Low symmetry materials, such as hep are characterized by a variety of active deformation mode, twinning activity and severe directional anisotropy in the single crystals. These crystals cannot accommodate certain deformation components because of lack of the necessary' deformation systems or because of high resolved shear stress (RSS). Self-consistent approach is a more precise method to model heterogeneous deformation of the materials. Self-consistent polycrystal models deduce the overall response of the aggregate from the known properties of the constituent grains and an assumption regarding the interaction of each grain with its environment. A polycrystalline aggregate can be represented by means of weighted orientations. The orientations represent grains and the weights are volume fractions. The solutions for the problem of a viscoplastic incompressible inclusion embedded in a viscoplastic incompressible effective medium being subject to external loading conditions can be used for construction of a polycrystal model. Each
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T e x t u r e a n d M i c r o s t r u c t u r e Evolution D u r i n g A s y m m e t r i c Rolling of M g Alloys
grain is treated as an ellipsoidal viscoplastic inclusion embedded in an effective viscoplastic medium. Both, inclusion and medium have fully anisotropic properties. The effective medium represents the average environment seen by each grain [ 1 5 , 16. 17]. Deformation is based on crystal plasticity mechanisms (slip and twinning systems) activated by a RSS. The viscoplastic constitutive behavior at local level (in a given grain) is described by means of the non-linear rate-sensitivity equation,
This equation can be linearized inside the domain of a grain (r), ε (χ) ιι
where M)' ] and k
= Μβσ (χ)
+ ε«'
ίι
)
(7)
are the viscoplastic compliance and the back-extrapolated term of grain (r).
respectively. The same relation is valid for the average strain rate and stress in grain (r),
and homogeneous equivalent medium (HEM) assumed with uniform properties has an analogous linear relation, Self-consistent equations for the homogeneous compliances, M
u
and back-extrapolated, £ ° are
given, {r)
u)
M, =(M :B ); lkl
1
1
Μ
f,', = { m " ' :ft"' +ε' )
(10)
for ellipsoids with same shape and orientations. If the grains have each a different shape, they have associated different Eshelby tensors, and the interaction tensors cannot be factored from the averages. In this case, the general self-consistent expressions should be used. M
l)kl
=(μ>":Β"):(β<Ύ;
(Η
E° = ( m ' " :b<" + ε ")-(Μ'"
1
:ß "):(β
Ι Γ ,
) " ' :{/>'"}
(11)
for general ellipsoids. 5. RESULT AND DISCUSSION
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Asymmetric conditions can be introduced by different velocities in the top and bottom parts of the sheet using differential velocitv or diameter rolls. To better understand the texture evolution during asymmetric rolling, effects of a variety of modeling parameters on textures were investigated.
Figure 3. Distributions of shear rates (el 3) according to various rolling speeds. Velocity ratios of the top to bottom rolls are given by (a) 1 : 1 . (b) 1:1.2, (c) 1:1.4, (d) 1:1.6, (e) 1:1.8, (t) 1:2. The rolling direction is specified with / and the plate normal direction is 3. When the sheet enters into the rolling gap. it experiences a compressive stress in the normal direction (ND). The sheet is extended to the rolling direction (RD). but is restricted along the transverse direction (TD) by friction [ 1 8 , 19]. In the center layer, plane strain compression condition can be assumed. In the surface, however, the geometrical changes within the rolling gap generate the shear strain component. In addition, friction between the roll and workpiece contributes shear deformation. The peak shear strains show a diagonal distribution located from the inlet contact position of the bottom roll side to the outlet contact position of the top roll. Bottom roll has higher velocities. In Fig. 3a, shear strain shows symmetric distribution. With the change of velocity ratios, most of deformation region has negative shear strain of ei.i.
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Figure. 4 Transition of the neutral points with various rolling speeds. The numbers in the figures are corresponding to various velocity ratios: 1(1:1), 2(1:1.2). 3(1:1.4). 4(1:1.6), 5(1:1.8). and 6(1:2).
Fig. 5 Texture evolution ( 0 0 0 2 PF) from initial homogeneous orientation (1000 single crystal aggregates)
A neutral point is the location where the sheet velocity equals to the roll velocity. In the symmetric rolling, neutral points in the both sides of the sheet are located at the same position along the rolling direction (Nr. 1 position in Fig. 4). Around the neutral point, the direction of the contact friction reverses. In the beginning of the deformation within the plastic zone, the left side from the neutral point, the velocity of the workpiece is lower than that of the rolls and the friction between the rolls and workpiece attract workpiece into the roll gap. To the right from the neutral point, the velocity of the workpiece is higher than that of the rolls, and thus the friction direction is reversed. The neutral points moved in the opposite directions during asymmetric rolling. In the higher speed roll side (bottom roll in Fig. 4). the neutral points proceeded to the outlet, while they retreated to the inlet in the slower speed roll side (top roll). Texture evolution across the outlet boundary (RA=10%) is presented in Fig. 5. The initial texture is uniform and is represented using 1000 discrete orientations. Main texture components of magnesium alloys during conventional rolling process are known as basal poles. Shear strain makes the basal pole
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deviated from the center of the 0002 pole figures. Seven streamlines are selected to compute texture evolution during ASR. Positions 1 and 7 aie located in the surface layer. These positions are influenced by friction more than any other locations. This results in the peak splitting in the basal pole figures. Inside the sheet, peak splitting is weakened and strong basal poles are found. Experimental textures and optical microstructures were measured after asymmetric rolling. The velocity ratio of the top and bottom rolls is 1.2 (83mm/s for the top roll, and 100mm/s for the bottom). The temperatures of the rolls and magnesium sheets were 200°C and 175°C. respectively. Intermediate heating of the sheets was carried out between each rolling pass at 175"C for 3minutes. The diameter of the rolls was 250mm. and the thickness of the initial sheets was 4.85mm. Fig. 6 displays the microstructures of sheets. Initial sheets have deformed grain structures through the thickness direction. After four passes of warm rolling (RA=50%), most grains were fragmented and there existed a small number of equi-axed grains. More reduction (RA=75%) resulted in more shear bands. The shear bands consisted of a large number of small grains. Using X R D , four incomplete pole figures of ( 10-10). (0002), (10-11) and (10-12) were measured and recalculated (0002) pole figures were presented in Fig.7. Two main features were found in the pole figures. First, major texture components are (0002)//ND fibers, and the basal intensities increased with RA. The bottom parts of the sheets have stronger intensities than the other parts. Second, maximum intensities of the basal poles were found 5 or 10 degree away from the center of the pole figures during asymmetric rolling.
Fig. 6 Optical microstructures for A Z 3 1 Β during warm rolling o f magnesium sheets, (a) initial sheets, (b) R A ^ 5 0 % . and (c) R A = 7 5 % .
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Fig. 7 ( 0 0 0 2 ) pole figures for A Z 3 IB during warm rolling o f magnesium sheets, (a) initial sheets, (b) R A = 5 0 % . and (c) RA=75%.
6.
CONCLUSION
Texture evolution during asymmetric rolling process was investigated using two-dimensional modeling and self-consistent polycrystal plasticity. Phenomenological stress-strain description was implemented to finite element to consider the material behaviors during hot working. Peak shear strains show a diagonal distribution located from the inlet contact position of the higher speed roll to the outlet contact position of the lower speed roll during ASR. Neutral positions moved in the opposite directions. In the higher speed roll side, the neutral position proceeded to the outlet, while it retreated to the inlet in the slower speed roll side. Shear deformation increased by asymmetric rolling and the basal poles were splited. Experimental textures and optical mierostructures are measured after asymmetric rolling. The number of shear bands and small grains increase with reduction area. Compared with the experimental textures, the predicted ones by crystal plasticity overestimated the effects of the shear deformation on rotations of the basal poles during asymmetric warm rolling. Temperature effects on slip and twining activities, and dynamic recrystallization need to be investigated yet. REFERENCES [1] K.H. Kim and D.N. Lee: Acta Mater. 49, p. 2583 (2001). [2] H. Jin and D.J. Lloyd: Materials Science and Engineering A399. p. 358 (2005). [3] S.B.Kang. B.K.Min. H.W.Kim, D.S.Wilkinson, J.Kang:Metall. Mater. Trans. 36A. p.3141 (2005)
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[4] Q. Cui. K. Ohori: Mater. Sei. Technol. 16. p. 1095 (2000) [5]W. Truszkowski. J. Krol and B. Major: Metall. Trans. 13A, p.665 (1982). [6] H. Jin and S. Saimoto: Scripta Mater. 46 p. 275 (2002). [7] S.H Kim. B.S You. C D . Yim and Y.M. Seo: Materials Letters 59. p.3876 (2005). [8] J.H. Cho. D.Boyce and P.R.Dawson: Mater. Sei. & Eng., A398, p. 146 (2005). [9j R. Kishore. T.K. Sinha. Metall. Mater. Trans.. 27A, pp. 3340-3343 (1996). [10] Cingara. H.J. McQueen. J. Mater. Process Technol., Vol. 36, pp. 3 1 - 4 2 (1992). [11] R. Ebrahimi. S.H. Zahiri. A. Najafizadeh, J. Mater. Process Technol.. 171 301-305 (2006). [12] Yujing Liu and Wei Shi. ICPNS2007. ZhengZhou. China, p. 101 (2007) [131 R- Hill. J. Mech. Phys. Solids. 15, 79-95 (1967). [14] U.F. Kocks. C.N. Tomé and H.-R. Wenk, 2000. "Texture and Anisotropy", Cambridge University Press (2nd edition). [15] R. Masson.. M. Bornert. P. Suquet and A. Zaoui, J. Mech. Phys. Solids 48, 1203-1227 (2000). 116] R.A. Lebensohn, C.N. Tomé and P.J. Maudlin, J. Mech. Phys. Solids 52, 249-278 (2004). [17] R.A. Lebensohn. C.N. Tomé and P. Ponte Castaneda, Materials Science Forum 495-497, 955-964 (2005). [18] O. Engler. M.-Y. Huh and C.N. Tome. Metall. Mater. Trans., 31A, p.2299 (2000). [19] W.F. Hosford and R.M. Caddell: Metal Forming: Mechanics and Metallurgy, Prentice-Hall. Englewood Cliffs. N J ( 1 9 9 3 ) .
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P R E D I C T I O N O F A N I S O T R O P I C P R O P E R T I E S IN M G A L L O Y S H E E T S U S I N G T H E CRYSTAL PLASTICITY FINITE E L E M E N T M E T H O D S.-H. Choi, J.K. Choi, H.W. Lee and D.H. K i m
Department of Materials Science and Metallurgical Engineering. Sunchon National University, Sunchon 5 4 0 - 7 4 2 . South Korea
ABSTRACT The effect
of crystallographic texture on the texture evolution and
macroscopic
anisotropic properties of polycrystalline M g alloy sheets w a s investigated using the crystal plasticity finite element method ( C P F E M ) . A predominant twin reorientation (PTR) scheme was successfully
implemented to capture grain reorientation due to deformation twinning. The
material behavior for the polycrystal model was described using the crystal plasticity theory', in which
each integration
point
in the element
is considered
to be discrete grains of a
polycrystalline M g alloy. T h e experimental anisotropic properties of a polycrystalline AZ31 ( M g - 3 % A I - l % Z n - 0 . 2 % M n ) M g alloy sheet were compared with the anisotropic properties predicted by the C P F E M .
INTRODUCTION M g alloys exhibit excellent strength-to-weight and stiffness-to-weight ratios. Therefore, wrought M g alloys have been used in electric and lightweight structural parts during the last few decades. However, low formability
using conventional
process conditions limits
broader
application of w r o u g h t M g alloys in consumer goods. It is well known that crystallographic texture plays an important role in formability of M g alloys. To understand texture evolution in M g alloys during plastic deformation, a n u m b e r of experimental and simulation studies have been conducted. The deformation behavior of M g alloys is complicated by the existence of various deformation m o d e s , such as basal slip, prismatic slip, pyramidal
. a tensile
twin is easily activated by c-axis tension. Many studies have investigated the effect of twin reorientation on the evolution of texture and macroscopic properties during plastic deformation [1]. A predominant twin reorientation (PTR) scheme [2] w a s successfully implemented in a visco-plastic self-consistent ( V P S C ) model to capture grain reorientation due to deformation twinning [3]. However, the V P S C model has limited applicability in the simulation of M g alloy structural behavior. T h e present study used the crystal plasticity finite element method ( C P F E M ) to simulate the texture evolution and macroscopic anisotropy of a polycrystalline M g alloy sheet
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P r e d i c t i o n of A n i s o t r o p i c P r o p e r t i e s in M g Alloy S h e e t s
during uniaxial tension and compression.
EXPERIMENTAL To measure macroscopic mechanical properties of a commercial AZ31 M g alloy sheet, tensile specimens were cut from the sheet at angles of 0 ° . 45° and 90° relative to the rolling 8
direction. The specimens were placed in a G L E E B L E ' 3 5 0 0 C thermo-mechanical simulator and heated by means of an inductive heating device at a rate of 5°C/sec. After holding the specimens 1
at 250 °C for 1 min. they were deformed at strain rate of 0.1 sec" . Neutron beam penetration was used to examine the macroscopic texture of the un-deformed specimens. For this measurement, cube-shaped samples ( l c m x l c m x l c m )
were prepared by stacking un-deformed
specimens.
Measurement of the complete pole figure was conducted using a four-circle diffractometer at the H A N A R O reactor at the Korea Atomic Energy Research Institute. The sample was mounted on a four-circle goniometer with the rolling direction of the sample parallel to the b e a m direction. A neutron beam from a Ge (113) m o n o c h r o m a t o r with a wavelength of 0.99 Â w a s used. The four — ( 0 0 0 2 ) . (10 1 0). (10 1 1)—complete pole figures were measured on a 5°x5° angular grid at the azimuth and polar distance over the entire pole hemisphere. T h e crystallographic orientation distribution function ( O D F ) w a s calculated from the four complete pole figures using the W I M V (William. Imhof. Matthies. and Vinel) method. The macrotexture of the as-received AZ31 M g alloy sheet is shown in Fig. 1.
Fig. 1. (0002) and ( 1 0 1 0 ) pole figures of the as-received A Z 3 1 M g alloy sheet.
THEORETICAL PROCEDURE Uniaxial tension and compression were simulated using the commercial finite element
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code. A B A Q U S . with the material model programmed according to continuum plasticity theory. A rate-dependent slip system constitutive relation w a s implemented into the user material subroutine UMAT. This model is fundamentally based on multiplicative decomposition of the deformation gradient into a plastic part characterized by shearing rates on active slip and twin systems and a part that a c c o u n t s for the rotation and elastic distortion of the crystal lattice [4],
i ' = ^ Y ® f f l "
(1)
The summation represents all of deformation m o d e . Ν ( = N + N ) . consisting of slip. N . and twin S
t
s
systems, N . Expressing the plastic work rate in the reference v o l u m e . t
τ D'= f f :P"
=£γ" "
t T
(2)
Τ
Here X is the symmetric Kirchhoff stress which is related to the Cauchy stress by Τ=\/σ, where J = D e t (F). The J a u m a n n rate of Kirchhof stress can be expressed as
«=/
where Κ is a fourth order tensor based on the anisotropic elastic m o d u l u s . C. D is the rate of deformation tensor (symmetric part o f the velocity gradient), and R ° is a tensor depending on the current slip plane normal and direction, the applied stress and the elastic modulus. The overall modulus and stress in each integration point are determined from the modulus and stress of each grain using a volume averaging method. For rate-dependent material, shear rates are given explicitly in terms of the resolved shear stress on the active slip/twin systems and the resistance of the active slip/twin systems to shear. For these simulations, this d e p e n d e n c e is given by
V'=Y'^7
r
sign f τ")
The evolution of the slip and twin hardening w a s evaluated by the following
(4)
microscopic
hardening law
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P r e d i c t i o n of A n i s o t r o p i c P r o p e r t i e s in M g Alloy S h e e t s
β
β
τ:=Σθ" \γ \
α,β-1
(Ν, + Ν,)
Q"" = a" h,^l - ^
(5)
where Q"" is a hardening matrix that is introduced to account for interaction between the slip ,<s
and twin systems. q' activity on system
accounts for the hardening rate of slip and twin system α due to slip
β . The fitting simulation was carried out by varying the
parameters (a, h„ and t
s a l
hardening
) until agreement between the predicted and the measured uniaxial
tension curve was achieved. In this study, three slip systems and one twin system
were
considered: basal ( { 0 0 0 2 } < 1 1 2 0 > ) , prismatic ({1 TOO} < 1 1 2 0 > ) , pyramidal
scheme requires tracking of the shear strain, y' , associated volume fraction,
V
f
1
= γ -*/S'
(S'
contributed by each twin system, t, and of the is the characteristic twin shear), within each
grain, g.
Table I .Microscopic hardening coefficients used in the C P F E M simulation.
Deformation m o d e
h,(MPa)
a
x"(MPa)
Basal < a > slip
70
1.1
32
80
Prismatic < a > slip
100
1.1
74
150
Pyramidal < c + a > slip
100
1.1
80
150
Tensile twin
50
1.1
64
120
The initial mesh ( 1 0 x 1 0 x 2 elements) of the polycrystal model is shown in Fig. 2. The initial length of the model region is given by l = 5 m m . w„=5mm, t „ = l m m . To impose initial element 0
orientations, 20 orientations were randomly selected from 1000 orientations and were distributed in each element. True strains for tension and compression were ε=0.182 and 0.223, respectively. The boundary condition was applied to the four planes comprising the 3-D mesh, as shown in Fig. 2.
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Fig. 2. Finite element mesh used in the present study: Description of the boundary.
For the uniaxial loading simulation, prescribed displacements in the χ direction were imposed on the 2-3-6-7 face. Because the tensile axis (TA) and compressive axis (CA) were parallel to the R D . 45° and T D . these specimens were denoted as follows: TA or C A / / R D ; TA or CA//45°; and. TA or C A / / T D .
(a)
(b) Fig. 3. Simulated textures after uniaxial compression:
(a) true strain=0.1 (CA//RD) (b) true strain=0.1 (CA//TD)
RESULTS AND DISCUSSION Fig. 3 shows the simulated (0002) pole figures for the specimens deformed by uniaxial compression. Triclinic sample symmetry was considered in the calculation of the orientation distribution function. Texture evolution for the C A / / R D sample is shown in Fig. 3(a). The C P F E M successfully
simulated an abrupt texture evolution during plastic deformation,
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in
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P r e d i c t i o n of A n i s o t r o p i c P r o p e r t i e s in M g Alloy S h e e t s
particular, rotation of the c-axis from the N D to the R D , Texture evolution of the C A / / T D sample is shown in Fig, 3(b). The C P F E M simulated rotation of the c-axis from the N D to T D . It should be noted that texture evolution during uniaxial compression was strongly dependent on the loading direction. The simulation results were shown to be in good agreement with the corresponding experimental results reported in the literature [5], Variation in the relative activity of each deformation m o d e with increasing true strain is shown in Fig. 4(a) and (b) for the C A / / R D and C A / / T D specimens, respectively. In the C A / / R D specimen, plastic deformation began with the activation of a tensile twin as the primary m o d e and a basal
slip as the secondary
m o d e . Subsequently, the tensile twin
decreased
continuously and pyramidal
(a)
(b)
Fig. 4 . Relative activities of the four deformation m o d e s during uniaxial compression: (a) C A / / R D
(b)
CA//TD
The uniaxial tension tests were interrupted by unloading and. with the aid of a micrometer, the plastic strains were measured relative to the R D ( D R D ) . T D (O >) and N D ( D D ) to determine the u
N
r-value (or plastic strain ratio): D H / D M J , The r-value anisotropy predicted by the C P F E M is shown in Fig. 5. The behavior of the r-value directionality for uniaxial tension w a s similar to the experimental data for all tensile directions relative to the R D . However, uniaxial compression tests were not been performed in the present study due to buckling. The r-value anisotropy for the C A / / T D specimen was simulated by the C P F E M . It was found that the r-value anisotropics
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for the different loading conditions were nearly identical, except for the absolute values of the rvalue. The experimental r-value for uniaxial compression tests w a s taken from the literature [5]. The r-value anisotropy for the C A / / T D specimen w a s simulated by the C P F E M . It was found that the r-value anisotropics for the different loading conditions were nearly identical, except for the absolute values of the r-value. These results indicate that asymmetric r-value as well as asymmetric yielding should be considered in the computational modeling for Mg alloy sheets.
Fig. 5. Comparison of experimental results and C P F E M predictions for an AZ31 M g alloy sheet.
REFERENCES 'S.R. A g n e w , M.H. Yoo and C.N. T o m e . Application of Texture Simulation to Understanding Mechanical Behavior of M g and Solid Solution Alloys Containing Li or Y. Acta mater,
49.
4 2 7 7 - 4 2 8 9 (2001). 2
C . N . T o m e , R.A. Lebenshon. U.F. K o c k s . A Model for Texture Development Dominated by Deformation Twinning: Application to Zirconium Alloys. Ada mater. 3 9 . 2667-2680 (1991).
"S.-H. C h o i . E.J. Shin and B.S. Seong. Simulation of Deformation Twins and Deformation Texture in an AZ31 M g Alloy under Uniaxial Compression. Ada 4
mater, 55. 4181-4192 (2007).
S . - H . Choi, Simulation of Stored Energy and Orientation Gradients in Cold-Rolled Interstitial Free Steels, Acta mater, 51. 1775-1788 (2003).
"A. Jain and S.R. Agnew, Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy A Z 3 1 Β sheet, Mater Sei and Eng A. 4 6 2 . 29-36 (2007).
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STRAIN HARDENING BEHAVIOUR OF AN INITIALLY TEXTURED TI6AL4V T I T A N I U M A L L O Y AS A F U N C T I O N O F S T R A I N R A T E A N D C O M P R E S S I O N DIRECTION Frederik C o g h e Royal Military A c a d e m y Brussels, Belgium
Luc Rabet Royal Military A c a d e m y Brussels, Belgium
Paul Van Houtte K.U. Leuven Leuven, Belgium
ABSTRACT This work shows some first results of a study aiming to determine the deformation mechanisms of a commercially available titanium alloy TÎ6A14V as a function of initial texture and strain rate. In particular this work will deal with the strain hardening behaviour of the material. Cylindrical specimens were fabricated from a textured extruded bar in mill annealed condition. One set of cylindrical samples (LD) had its axis aligned to the axis of the rod, a second set (TD) had its axis perpendicular to the rod axis and parallel to the majority of the caxes of the α-phase. Both sets were tested under uni-axial compression loading for a wide range of strain rates (0.001/s up to 1000/s) using a servo-hydraulical testing machine (lower strain rates) and a S H P B - s e t u p (higher strain rates). The LD-samples showed lower flow stresses than the T D - s a m p l e s for the same amount of strain, as could be expected from the lower C R S S available slip systems. T h e LD-samples also supported m u c h more strain before final fracture. The strain hardening of the LD-samples showed a clear tendency to increase significantly during deformation, this tendency being less pronounced for the T D - s a m p l e s . This was clearly illustrated by the concave-up true stress - true strain curves. Friction effects could be ruled out as being the source of this behaviour and after the origin of this hardening behaviour could be attributed to extensive twinning as illustrated by optical microscopy, but to be confirmed by T E M analysis. Texture measurements ( O I M and X R D ) before and after testing showed for both sets rotating of the c-axes of the α-phase towards the compression axis. INTRODUCTION Titanium and especially its TÎ6A14V alloy have always been of great interest for mechanical applications as they c o m b i n e a relative low density (compared to steel) with a high strength and moderate to high ductility. Due to the hexagonal crystal structure of the titanium a phase, the deformation behaviour of titanium and its alloys is not completely understood yet. Not only leads this hexagonal crystal structure to a marked anisotropy, it also leads to deformation twinning as a major deformation mechanism \ For commercially pure or α-titanium the twinning behaviour has well been established. For the Ti6A14V alloy the situation is less clear. Research typically focusses on the α-phase as this phase is present in large amounts compared to the ß-phase, and is the hardest phase as well. Several publications reported a lack of considerable twinning in the α - p h a s e of this alloy and only very few have mentioned an amount of deformation twinning, in most cases limited and under dynamic conditions . It has been reported that for single crystals the initiation of twinning in titanium-aluminum alloys is inhibited by increasing a l u m i n u m content ', meaning that for Ti6A14V no twinning was expected, which is clearly in contradiction with the actual results. 1
1 4
6
491
Strain H a r d e n i n g B e h a v i o u r of a n Initially T e x t u r e d T Î 6 A I 4 V T i t a n i u m Alloy
EXPERIMENTAL SETUP Material The TÎ6AI4V alloy bar used in this project is a commercially available, aerospace graded material. The extruded bars, with a diameter of 16mm. w e r e delivered in the mill-annealed condition. The material consisted mostly of an equi-axial. hexagonal a - p h a s e ( 9 7 % . m e a n grain size 1 Ιμιτι). A limited amount (3 w t % . mean grain size approximately l μηι)) of B C C ß-phase particles were finely dispersed at the grain boundaries of the α-phase grains. A majority of the caxes of the α - p h a s e were aligned to one another and perpendicular to the longitudinal axis of the original rod (transverse texture). The chemical composition of the material can be found in table I. Figure l illustrates the different orientations of the two sample sets and their corresponding texture. Table I: Chemical composition (atom absorption spectrophotometry) Al <wt%) V (wt%) Fe ( w t % ) Ü ( w t % ) 6.28 4.18 0.18 <0.2
Figure l : Orientation of samples and initial texture of the c-axes (OOOl direction) of the a - p h a s e . Mechanical testing The two sample sets were tested in uni-axial compression at room temperature for different strain rates. The strain rates that will be discussed in this work are 0.001/s. 1/s and 1.10 Vs. For the lower strain rates (0.001/s and 1/s). a servo-hydraulical testing machine ( M T S ) was used. The force was measured using a piezoelectric load sensor. Deformation w a s measured by measuring the crosshead displacement of the M T S - m a c h i n e . T h e deformation was corrected for the elastic deformation of the crosshead and the two anvils, by the use of calibration curves measured during anvil-on-anvil testing. For the higher strain rates ( 1.1 OVs). a Split Hopkinson Pressure Bars setup* ( S H P B ) w a s used. For more information on the S H P B setup, reference is made to \ Different specimen geometries and impactor lengths w e r e used in order to get a good c o m p r o m i s e between constant strain rate, useful plastic deformation and signal-to noise ratio. All compression testing was done following the guidelines o f . which includes a critérium for the barreling effect. None of the samples evaluated by this criterion showed unacceptable barreling. Friction was kept low by the use of teflontape for the M T S and a commercial lubricant for the S H P B .
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Microstructural characterisation T h e original bar material and some of the deformed samples were characterized using optical microscopy, E B S D - m e a s u r e m e n t s ( O I M ) and X R D texture measurements. The samples for optical microscopy were polished and etched during the last polishing step by adding 20vol% of H 2 0 2 to 8 0 v o l % of polishing solution. After this treatment, the samples were investigated using polarized light. The O I M - s a m p l e s were etched using a diluted Kroll's solution (97vol% H 2 0 , 2 v o l % H N 0 3 , 1 v o l % H F ) , prior to E B S D - m e a s u r e m e n t s in a F E G scanning microscope. Because of difficulties in obtaining good O I M - m e a s u r e m e n t s on deformed samples, the global texture evolution of the α - p h a s e in the deformed samples w a s measured using an X R D diffractometer (Siemens D 5 0 0 ) . As the majority of the material consisted of α - p h a s e grains and taking into consideration the relative softness of the small ß-phase particles compared to the α-phase, this work focuses only on the α - p h a s e . N o attempt has been made yet to have a closer look on the influence of the ß-phase particles on the deformation behaviour of this TÎ6A14V alloy. EXPERIMENTAL RESULTS AND DISCUSSION Mechanical behaviour Figure 2.a shows the true stress - true strain curves for the compression test on the longitudinal samples, for the three different strain rates (0.001/s. 1/s and 1.10 /s). For convenience compressive strains and stresses have been given positive signs. Due to the high ductility of the TÎ6AI4V alloy it w a s impossible to deform the samples up to fracture on the S H P B setup - this in spite of the very high susceptibility of Ti6A14V to adiabatic shear localization. The curves for a strain rate of 0.001/s and 1000/s clearly illustrate the dynamic behaviour of H C P materials, which is a combination of a BCC (increasing yield stress) and FCC (increasing strain hardening) material behaviour. It also s h o w s the excellent ductility of the TÎ6A14V alloy for this set of samples. It can be clearly seen that increasing the strain rate results into higher yield stresses. Only the plastic strain has been taken into account into figure 2.a. A s yield critérium, the 0.002 offset critérium was used instead of backwards extrapolation. In this way no hypothesis on the deformation behaviour, by the choice of a constitutive equation like Johnson-Cook or ZerilliArmstrong, has to be made. It has the disadvantage that the first part of the results of the S H P B setup s h o w s large fluctuations inherent for the S H P B setup, since it takes a certain time and hence strain (typically 0.05) to reach equilibrium state. The curve for a strain rate of 0.001 /s shows a concave upwards trend which is due to an increasing strain hardening rate (figure 2.b). This kind of behaviour is typically seen in materials which show twinning as one of the main active deformation systems, like commercially pure or a - t i t a n i u m " " " or m a g n e s i u m . This would m e a n that the Hall-Petch effect and/or the Basinski mechanism for deformation twinning, as mentioned by Salem et al., would be more important than the effect of texture softening due to twinning. For a strain rate of 1/s the curve initially is situated - as expected - above the curve for a strain rate of 0.001/s, which corresponds to the higher resistance of dislocation movement at higher strain rates (which is thermally equivalent to a lower temperature). For a strain of approximately 0.3 the 1/s curve crosses the 0.001/s curve due to the adiabatic heating of the sample and the following thermal softening. This is also illustrated by the decreasing strain hardening exponent in figure 2.b. Previous work has shown that the TÎ6AI4V alloy used in this work demonstrates adiabatic behaviour for a strain rate of as low as 0.1/s, due to its extremely 3
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low thermal conductivity which leads to failure by adiabatic shear localization. D u e to this thermal softening the alloy also s h o w s a higher failure strain (strain 0.66) than for a strain rate of 0.001/s. The Π6Α14 V alloy s h o w s a very high strain hardening rate for a strain rate of 1.10 /s (figure 2.b). and as in the case of the low strain rates, although much more pronounced, a concave upwards trend (figure 2.a). Again this suggests twinning being an important deformation mechanism. Microstructural investigations will have to confirm this. The increasing strain hardening is clearly visible in figure 2.b which shows an evolution of the strain hardening exponent from a value almost 0 to a value about 0.5. This is a possible explanation why publications o n the use of constitutive equations for the simulation of deformation behaviour of Ti6AI4V tend to have high values for the strain hardening exponent "'. In fact, a constitutive equation that uses a constant strain hardening exponent will be unable to reproduce the 0.001/s and 1.10 /s curves in figure 2.a. unless it uses a strain hardening exponent greater than one! 3
3
(a)
(b)
Figure 2: True stress - true strain curves as a function of strain rate (a) and evolution of the strain hardening exponent (b) for the longitudinal samples. Figure 3.a illustrates the true strain-true stress behaviour for the transverse samples. Again it can be seen that increasing the strain rate increases the yield strength and that for the higher strain rates the samples could not be deformed up to fracture. Compared to the longitudinal samples, the transverse samples show a remarkably higher yield strength and flow stress. This is due to the lower number of available slip systems. T h e alignment of the compression axis to the majority of the c-axes of the hexagonal α - p h a s e leads to the blocking of the basal J0001J < 1120 > . prismatic (t 0T0J < 1120 > and pyramidal {10T1J<1120> slip systems and to the activation of the 1 order {10T Ij < 1123 > and 2 order',1122] < 1123 > pyramidal slip systems and/or {10T2}, {l 122} or ( l O l l} twinning. These slip systems show a significantly higher C R S S " (Critical Resolved Shear Stress), which leads to a higher yield stress. The influence of the texture is also visible in the clearly reduced plasticity of the transverse specimens (true fracture strain of 0.20 and 0.18 for respectively a strain rate of 0.001/s and 1/s). As for the longitudinal samples a strain rate of 1/s produces enough thermal softening to make the 1/s curve cross the 0.001/s curve. D u e to the higher sensitivity to adiabatic shear localization for this orientation of the compression axis, this does not lead to a higher fracture strain, as for the longitudinal set of samples. st
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In figure 3.b it can be seen that in contrast to the strain hardening exponent curves of figure 3.a. all curves s h o w a m o n o t o n o u s decreasing trend. The peaks that can be seen at the end of both strain hardening exponent curves for a strain rate of 1.1 OVs might be due to the SHPB test, although these peaks have not been seen for other materials (steel, aluminum) tested under the s a m e conditions. The curves in figure 3.b have a rather constant value of approximately 0 . 1 .
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Figure 3: True stress - true strain curves as a function of strain rate (a) and evolution of the strain hardening exponent (b) for the transverse samples. Microstructural evolution Figure 4 shows on the left hand side the evolution of the orientation of the c-axes of the longitudinal samples for a strain rate of 0.001/s. for different a m o u n t s of plastic strain (compression axis perpendicular to the paper). In contrast with what has been published by Majorell et al. ' \ the c-axes tend to line up with the loading direction. T h e pole figures in figure 4 have been produced by X R D - m e a s u r e m e n t s and were confirmed by independent E B S D measurements. There is a clear evolution from a transverse texture to a basal texture. With increasing strain, the basal texture tends to become more diffuse which could partially be due to extensive twinning. On the right hand side of figure 4 optical microscopy reveals microstructural features, probably twins. T E M and E B S D measurements will have to confirm the twin relations between the parent grain and the twin. Even for the lower strains (ε = 0.058) twins can be seen. The amount of twins increases considerably at larger strains, which could be at the origin of the concave-upwards stress-strain curves previously shown, by a Hall- Petch effect or the Basinski mechanism. Furthermore it can be clearly seen that several grains d o n ' t seem to have twinned. A g a i n E B S D - w o r k will have to point out whether these grains were unfavorably oriented for twinning, or m a y b e were very favorably oriented to a c c o m m o d a t e strain by regular dislocation glide on the available slip systems. The transverse samples show a similar behaviour as the longitudinal samples in that the orientation of the c-axes is aligned with the compression axis (perpendicular to the paper), although the initial central peak tends to disappear and a randomization around the compression axis takes place as the strain increases. As for the longitudinal set, twins can be seen in the microsü'ucture, although in a m u c h lesser extent. This could be explained by an unfavourable orientation to provoke twinning, although the absence of soft slip systems for this orientation of the c-axes relative to the compression axis, should lead to the activation of twinning as an active deformation mechanism. Quantification of the twin fraction will have to clear this seemingly
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paradoxal result. The limited amount of twins could explain the small increase in strain hardening exponent in figure 3.b for a strain rate of 0.001/s, although more tests will be necessary to confirm this.
Figure 4: Orientation evolution of the c-axes ot the α - p h a s e ( X R D ) and evolution of the microstructure (optical microscopy. x500) for the longitudinal samples for a strain rate of 0.001/s.
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Figure 5: Orientation evolution of the c-axes of the α - p h a s e (XRD) and evolution of the microstructure (optical microscopy, x500) for the transverse samples for a strain rate of 0.001/s. CONCLUSION The evolution of the strain hardening as a function of initial texture and strain rate for different compression directions has been clearly illustrated. The macroscopic behaviour can, for the lowest strain rates, be explained by the microstructural and texture evolution of the TÎ6A14V alloy. Deformation twinning seems to play an important role as an alternative deformation system. There might be an important coupling between deformation twinning and dislocation glide, possibly due to a Hall-Petch effect or the Basinski mechanism. Further research is needed to identify the different active deformation mechanisms and to understand the competition between dislocation glide and twinning as a function of strain rate, initial texture and the compression direction.
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ACKNOWLEDGEMENTS This work was carried out in the frame of the "Inter University Attraction Poles programBelgian Science P o l i c y " under contract number I U A P P6/24. FOOTNOTES * Also known as Kolsky bars. Kolsky was in fact the first to c o m b i n e t w o Hopkinson bar s to measure dynamically mechanical properties of materials. REFERENCES M. Yoo, Slip, twinning, and fracture in hexagonal close-packed metals, Metall.
Trans.,
12A ,
409-417(1981). "N. Munroe, X. Tan, Orientation dependence of slip and twinning in H C P metals, Scripta
Mat.,
36. 1383-1386 (1997). 'J. Lecomte, Plastic deformation of a Ti-6%A1-4%V alloy at intermediate an high temperatures, Proc. ICOTOM 4
12, 712-717 (1999).
R . Picu, A. Majorell, Mechanical behaviour of Ti-6A1- 4V at high and moderate temperatures -
Part II: Constitutive modeling, Mat. Sei. Eng., A 2 3 6 , 306-316 (2002). Ύ . Xu, Y. Bai, M. Meyers, Deformation, phase transformation and recrystallization in the shear bands induced by high-strain rate loading in titanium, J. Mater. Sei. Technol,
22, (2006).
S. Liao, J. Duffy, Adiabatic shear bands in a Ti-6A1-4V titanium alloy, J. Mech. Phys.
Solids,
46, 2201-2231 (1998). J. Williams, R. Baggerly, N. Paton, Deformation behavior of H C P Ti-Al alloy single crystals, Metall. Trans.. 3 3 A , 837-850 (2002). "H. Kolsky, An investigation of the mechanical properties of materials at very high rates of loading, Proc. Phys. Soc. Umdon,
B 6 2 , 676 (1949).
B. Roebuck, J. Lord, R. Varma, M . Loveday, Measurement Good Practice Guide No. 3: Measuring Flow Stress in Hot Axisymmetric Compression Tests, Nat. Phys. Lab., (1996). "'S. Balasubramanian, L. Anand, Plasticity of initially textured hexagonal polycrystals at high homologous temperatures: application to titanium, Acta Mater., 50, 133-148 (2002). " S . Kailas, Y. Prasad, S. Biswas, Influence of initial texture on the microstructural instabilities during compression of commercial _-titanium at 25°C to 400°C, Metall.
Trans., 25A, 1425-1434
(1997). "A. Salem, S. Kalidindi, R. Doherty, S. Semiatin, Strain hardening due to deformation twinning in _-titanium: mechanisms, Metall.
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" D . Chichili, K. Ramesh, K. Hemker, The high-strain-rate response of alpha-titanium: experiments, deformation m e c h a n i s m s and modeling, Acta Mater., 46, 1025-1043 (1998). 14
P. Backx, Improved formability of magnesium and m a g n e s i u m alloy AZ31 by texture and
microstructure control, PhD thesis. University
of Ghent
(2006-2007).
" C o g h e F., Rabet L. and Kestens L., Deformation m e c h a n i s m s of a commercial titanium alloy TÎ6A14V as a function of strain rate and initial texture, J. de Phys. IV, 134, 845-850 (2006). ,6
H . Meyer, A modified Zerilli-Armstrong constitutive model describing the strength and
localizing behavior of TÏ-6A1- 4 V , Army Research
Laboratory,
A R L - C R - 0 5 7 8 (2006).
"J.J. Fundenberger, M . J. Philippe, F. Wagner and C. Esling, Modelling and prediction of mechanical properties for materials with hexagonal symmetry (zinc, titanium and zirconium alloys), Acta Mater., 4 5 , 4 0 4 1 - 4 0 5 5 (1997). " Ά . Majorell, S. Srivatsa, R. Picu, Mechanical behaviour of T1-6A1- 4V at high and moderate temperatures - Part I: Experimental results, Mat. Sei. Eng., A 2 3 6 , 297-305 (2002).
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T E X T U R E E V O L U T I O N D U R I N G T H E STATIC R E C R Y S T A L L I Z A T I O N OF T H R E E BINARY MG-Y ALLOYS R. Cottam, J. Robson, G. Lorimer *School of Materials, Materials Science Centre, The University of Manchester, Grosvenor Street, Manchester, Ml 7HS B. Davis Magnesium Elektron Ltd., PO Box 23, Rake Lane, Swinton, Manchester M 2 7 OD
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ABSTRACT The evolution of texture during the static recrystallization of three cold rolled binary Mg-Y alloys was characterised using electron backscattered diffraction (EBSD). It had been shown in previous studies that alloys that contain yttrium can develop more random texture, during thermomechanical processing and this can reduce or eliminate the yield asymmetry common to wrought magnesium alloy. The three alloys were initially hot rolled to break up the as cast structure then cold rolled using multiple passes of 1% reductions to avoid cracking and impart a high attainable level of stored energy. The as rolled texture was a typical basal texture with a small component outside the basal ring formed as a result of {10-11} - {10-12} double twinning. Isothermal annealing was carried out in 325°C at either 20 or 50 minutes to produce both partial and fully recrystallized structures. The texture produced after recrystallization was a weak basal texture. It was found that for all the alloys the orientations and boundary misorientation of the double twins was carried through to the recrystallized texture due to preferential recrystallization of the twinned regions which were extensive throughout the structure. INTRODUCTION Magnesium alloys offer the potential for significant weight savings in applications such as aerospace and automotive industries due to its low density of 1,6g/cm . The limited application of magnesium alloys can in part be attributed to their poor formability and hence the inability to manufacture wrought products. The poor formability has been attributed to the plastic anisotropy that occurs at a granular level [1], which during the deformation of a polycrystal produces incompatibility stresses which lead to early fracture [2], It has been shown that with a combination of a favourable texture and an ultra fine grain size the room temperature ductility can be significantly improved [3, 4], 3
Static recrystallization (SRX) offers the potential to refine the grain structure of magnesium alloys and potential produce new textures that can yield that desired combination of an ultra fine grain size and favourable texture. To date there has only been a small amount of work on the SRX of magnesium alloy [5-7], which leave a lot of scope for developing a better understanding of the nature of the process and the role of alloying elements. There has also been a small amount of work on the recrystallization of Titanium [8, 9] which has shown that the heterogeneity of stored energy due to the anisotropic deformation play a significant influence on the recrystallization kinetics which is probably also an important factor in the recrystallization kinetics of magnesium alloys.
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The field of SRX has been around for many years and has been mainly studied in copper, aluminium and steel [10-12]. The main current theory of static recrystallization texture formation is that the texture of the recrystallized grains c o m e s from the deformation structure and microstructural features such as shear band, transition bands, regions near grain boundaries and particles, which produce nuclei for recrystallization that are o f higher energy or have favourable orientation relationships with the surrounding regions, [11, 13, 14]. The microstructural development of magnesium alloys due to plastic deformation is complicated by the plastic anisotropy o f the material and the formation of deformation twins. Hence the recrystallization of magnesium alloys is anticipated to be complex which when coupled with the effect of solute should allow an understating of how SRX in magnesium alloys occurs from both a kinetic and texture point of view.
EXPERIMENTAL PROCEDURE Three binary magnesium yttrium alloys were prepared by magnesium electron at the Manchester Site with the following compositions: 0.5, 1.1 and 2 . 2 w t % Y other impurity elements were kept to below 0.01wt%. The alloys were cast in to 25mm thick square blocks. The blocks were cut to into 60x60mm blocks and then homogenised at 490°C for 8hrs. The samples were then hot rolled at 450°C to a true strain o f 0.9. Due to dynamic recrystallization the grain structure of all three alloys was uniform. The samples were then cold rolled using 1% reductions per pass to a true strain of 0.2.The billet was rotated between passes to reduce edge cracking. Edge cracking still occurred but the majority of the metal remained in tact. Squares of 15x 15mm were cut form the intact region of the plate and then pickled in a solution of 3 % sulphuric acid and water for 15mintures to remove 0.5mm of surface material. The pickled samples were then annealed at 325"C for either 20 or 50 minutes to achieve partial and full recrystallization respectively. The mierostructures after cold rolling were revealed by grinding to half the thickness of the sample and polishing down to a colloidal silica. The grain structure was revealed using a solution of acetic and picric acid in ethanol. Samples were prepared for electron backscattered diffraction (EBSD) by electro polishing in a solution of 3 0 % nitric acid in ethanol cooled to -30°C . using a voltage of 12v for 30 seconds. The initial textures of the cold rolled alloys and the partially and fully recrystallized were measured by E B S D using a Philips XL30 FEG SEM with a spot size of 4, working distance o f 20mm and an accelerating voltage of 20kV. Euler maps of the cold rolled and partially recrystallized structure were also conducted. RESULTS After cold rolling all three alloys had there mierostructures characterised by optical microscopy and E B S D . The optical micrographs Figure I show significant amounts of deformation twinning and a uniform grain size for all three alloys.
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a b c Figure 1 - optical micrographs of cold rolled Mg-Y alloys: a - 0 . 5 w t % Y: b - l . l w t % Y ; c 2.2vvt%Y The initial textures of the three alloys after cold rolling, Figure 2, all have a basal texture with similar intensities.
a b c Figure 2 - {0002} pole figures where the R D is to the right of the pole figure: a - 0.5wt% Y: b l.lwt%Y; c - 2 . 2 w t % Y Using the misorientation histogram in conjunction with a plot of the rotation axes for grains between 30-35° the type of deformation twin was determined. It should be noted that this analysis was performed for all three alloys but the results of only one alloy are shown here to save space. It is apparent that there is a slight peak in the misorientation histogram about 30-40° and reference to the rotation axes for this misorientation shows that there is a congregation of poles about the {-1210} orientation which is consistent with the double twinning [15]. After annealing for 20 minutes at 325°C the samples had partially recrystallized as shown the Euler map. Figure 4. What is interesting about this image is that there appears to be no twins left in the remaining large grains, which suggests that the twins preferential recrystallize. This suggestion is further supported by the presence of a peak in the misorientation distribution between 30-40° for the entire population and no peak for the large grain population. Figure 5, hence the recrystallized grains have the twin boundary.
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Figure 3 - Misorientation histogram and rotation axes inverse pole figure for 2.2wt%Y alloy after cold rolling The pole Figure of the two populations of grains is shown in Figure 6. The fine grains show a more random distribution of orientations where as the parent grains have the original basal texture. This suggests that grains that recrystallize first tend to not have the basal orientation. It should be noted that this analysis has been performed for all three alloys and the results were consistent with the example shown.
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Figure 4 - Euler map of 2.2vvt%Y after partial static recrystallization. The black boundaries are misorientations of greater then 15°. The lighter areas are the non-recrystallized regions.
a b Figure 5 - Misorientation histogram and rotation axis inverse pole figure tor the 2.2 \vt%Y alloy; a - misorientation histogram for entire data set: b - misorientation histogram for large grain population.
The pole figures for all three alloys after full recrystallization are shown in Figure 7. It is apparent that with an increase in the yttrium concentration the texture weakens and tends to split.
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a b Figure 6 - [0002} Pole figures for partially recrystallized 2.2wt%Y where the R D is to the right of the pole figure: a- recrystallized grains: b - parent grains. D u e to a limitation in the software intensity contours could no be used in the pole figures. The different colours represent different Euler angles.
a b c Figure 7 - ,0002) Pole figures for Mg-Y alloys after static recrystallisation, where the R D is to the right of the pole figure: a - 0.5wt%Y; b - I. I \vt%Y: 2.2wT%Y DISCUSSION O F R E S U L T S Looking at the change in the texture data alone from the cold rolled. Figure 1 to the fully recrystallized Figure 7 there is essentially a weakening of the texture as a result of recrystalization. The exception being the 2.2wt%Y alloys which exhibits a slight split in the basal texture about the N D . This weakening of the texture has been reported for the D R X of magnesium alloys [16, 17]. The reason for this weakening is not revealed in these publications. Figure 6 shows that texture for the population of the recrystallized grains and the parent grains. It is apparent that the texture of the recrystallized grains is far more dispersed. Given that there is a general consensus that the deformation microstructure is responsible for the texture of the recrystallized grains [11. 14] this suggests that grains that do not have the basal texture have higher stored energy and hence preferentially undergo recrystallization. These off basal orientations then increase the components of these orientations in the final texture and hence weaken the initial texture. This idea is supported by the observation of the compressive twin boundary persisting after recrystallization in the grain boundary misorientation histograms, Figures 3 and 5. The twin orientation is off basal (about 35° of the N D [15]) and given that the boundaries are still present the texture will also be present. The twins present in the initial structure are fine. Figure 1. so their contribution of the overall texture will be slight. However for the 2.2wf%Y alloys the orientation of the split is characteristic of the compression double twin [15] which suggests that this alloy has
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undergone significantly more twinning during the cold rolling. The volume fraction of twinning in all three of the cold rolled alloys would need to be determined to verify this statement. 1
Another interesting observation in relation to the twinning is that there are very few twins present in the partially recrystallized structure Figure 4. There are two potential reasons for why the twins preferentially recrystallized. The first is that the distance between the twin boundaries is very small so a potential nucleus for recrystallization will have a higher chance of having a high angled grain boundary as part of is configuration which will help to drive recrystallization which is similar to the argument used by Nave and Barnett [14]. The other is that twin boundaries are special boundaries (coincident boundaries) which are know to have greater mobility which would also promote recrystallization [18]. Essentially there has only been one other study of the static recrystallization of magnesium alloys which was conducted by Jain et al [6]. In this work it was shown that the basal texture persisted after recrystallisation when the recrystallization temperature was lower than the temperature which the particles dissolved. Whilst there was no particle present in the alloy of this study the result is consistent with this study. The problem is that in Jain's study when the recrystallisation temperature was about the solvus temperature for the particle a new texture was produced which is not consistent with the finding of this work. This suggests that the yttrium present in both alloys is not responsible for the change in texture. Instead it appears that the particle present in Jain's alloy is responsible for the change in final texture observed. CONCLUSIONS There is a strong indication that off basal orientations preferentially recrystallized and increases their component of the texture which ultimately weakens the recrystallized texture. Deformation twin preferentially recrystallized during the early stage of static recrystallization which is evident from the persistence of the twin boundary misorientation and rotation axes in the fully recrystallized structure and the lack of deformation twins in the partially recrystallized structure. REFERENCES 1. Agnew, S.R. and O. Duygulu, "Plastic Anisotropy and the Role of Non-Basal Slip in Magnesium Alloy A Z 3 1 B " International Journal of Plasticity,. 2 1 , 1 161-1193 (2005) 2. Agnew, S.R., D.W. Brown, and C.N. Tome, "Validating a polycrystal model for the elastoplastic response of magnesium alloy AZ3I using in situ neutron diffraction." Acta Materialia, 54, 4841-4852 (2006) 3. Koike, J., et al., The Activity of N o n B a s a l Slip Systems and Dynamic Recovery at Room Temperature in Fine-Grained A Z 3 1 B Magnesium Alloys." Acta Materialia, 51, 2055-2065 (2003) 4. Agnew, S.R., et al., "Enhanced ductility in strongly textured magnesium produced by equal channel angular processing." Scripta Materialia, 50, 377-381 (2004) 5. Jager, Α., et al., "Influence of annealing on the microstructure of commercial Mg alloy AZ31 after mechanical forming." Materials Science and Engineering A,. 432. 20-25 (2006) 6. Jain, J., W.J. Poole, and C.W. Sinclair. "A study on the static recrystallisation of cold rolled magnesium alloy A Z 8 0 . " In Magnesium Technology. 2006. 7. Senn, J.W. and S.R. Agnew. "Texture Randomization During Thermomechancial Processing of a Magnesium-Yttrium-Neodynium Alloy." in Magnesium technology in the global age. 2006. Montreal.
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T e x t u r e Evolution During t h e S t a t i c Recrystallization of T h r e e Binary M g - Y Alloys
8.
Chun, Y.B., S.L. Semiatin, and S.K. Hwang, "Monte Carlo modelling of microstructure evolution during the static recrystallization of cold rolled, commercial-purity titanium." Ada Materialia,. 54 3673-3689 (2006) 9. Bozzolo, N., et al„ "Texture evolution during grain growth in recrystallized commercial pure titanium." Materials Science and Engineering A, 3 9 7 346-355 (2005) 10. Weiping, Y.. R. Le Gall, and G. Saindrenan, "A study of the recrystallisation of an IF steel by kinetics models." Materials Science and Engineering A, 3 3 2 , 41-46 (2002) 11. Engler, O. and L. K., "Mechanisms of Recrystallization Texture Formation in Aluminium Alloys." Scripta Melallurgica et Materialia, 2 7 , 1527-1532 ( 1992) 12. McDonald. D.T., P. Bate, and W.B. Hutchinson, "Effect of strain path change on recrystallization in copper." Materials Science and Technology, 2 1 , 639-700 (2005) 13. Humphreys. F.J., "A new analysis of recovery, recrystallisation, and grain growth." Materials Science and Technology 15, 37-44 (1999) 14. Nave, M.. D. and M.R. Bamett, "Texture change near grain boundaries and triple points in cold-rolled interstitial-free steel." Materials Science and Engineering A, 3 9 6 , 244-253 (2004) 15. Nave. M„ D. and M.R. Bamett, "Mierostructures and Textures of Pure Magnesium Deformed in Plane Strain Compression." Scripta Materialia. 5 1 , 881-885 (2004) 16. Cottam, R., et al., "Dynamic recrystallisation of Mg and Mg-Y alloys: Crystallographic Texture Development." Materials Science and Engineering A, (2007) doi:10.1016/j.msea.2007.08.016.
17. Gehrmann, R. and G. Gottstein. "Texture and Microstructre Development During Plastic Deformation of Magnesium. In twelfth international conference on texture of materials." (1999) Montreal. 18. Humphreys. F.J. and M. Hatherly. "Recrystallization and Related Annealing Phenomena." (2004) Elsevier.
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G R A I N SIZE A N D O R I E N T A T I O N D I S T R I B U T I O N F U N C T I O N O F HIGH P U R I T Y (XTITANIUM Bradley S. Fromm, Brent L. A d a m s , Sadegh Ahmadi Department of Mechanical Engineering, Brigham Y o u n g University Provo, U T 84602, U S A Marko Knezevic Department of Materials Science and Engineering, Drexel University Philadelphia, PA 19104, U S A ABSTRACT A method to incorporate grain size effects into crystal plasticity is presented. The classical Hall-Petch equation inaccurately predicts the macroscopic yield strength for materials with non-equiaxed grains or materials that contain unequal grain size distributions. These deficiencies can be o v e r c o m e by incorporating both grain size and orientation characteristics into crystal plasticity theory. Homogenization relationships based on a viscoplastic Taylor-like approach are introduced along with a new function, the grain size and orientation distribution function ( G S O D F ) . Estimates of the G S O D F for high purity α-titanium are recovered through orientation imaging microscopy coupled with chord length measurements. A comparison between the new method and the traditional viscoplastic Taylor approach is made by evaluating yield surface plots. 1. I N T R O D U C T I O N During the early 1950's, E.O. Hall and N.J. Petch independently established what is k n o w n as the Hall-Petch relationship " . Through experimentation, they discovered that the macroscopic yield strength of a material is proportional to the inverse square root of the average grain size. This robust relationship has been documented for many materials and indicates that yield strength can be increased by simply reducing grain size. However, it does not hold true for materials with non-equiaxed grains or materials that contain unequal grain size distributions. The relationship has also been shown to breakdown for ultrafine grained materials . Further, it does not take into account the orientation of grains or the crystal anisotropy inherent in the microstructure. Similarly, crystal plasticity theory dates back to 1938 when G.I. Taylor postulated his uniform strain model in order to predict yield strength . This method calculates the stresses in individual grains of a material by resolving the strain rate for each grain in terms of slip rates on individual slip systems. The microscopic stress of each grain is then volume averaged to obtain an upper-bound estimate for the macroscopic yield strength. Although this method takes grain orientation into account and is valid for non-equiaxed and anisotropic materials, it does not distinguish the effect of local grain size on the local yield properties. The purpose of this paper is to extend crystal plasticity theory to incorporate both grain orientation and grain size effects into the model, thus overcoming current deficiencies in yield strength calculations. A new approach that incorporates a slip system-level Hall-Petch relationship into a rigid-viscoplastic model is described. This n e w methodology is then implemented for a high purity α-titanium material and a comparison is m a d e between the old and 1
2
3
4
5
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Grain S i z e a n d O r i e n t a t i o n Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
new methods to determine the extent to which grain size affects the mechanical strength of the material. 2. H A L L - P E T C H R E L A T I O N S H I P S 2.1. Macroscopic Hall-Petch equation Relation (1 ) is the well k n o w n Hall-Petch equation where σ
is the macroscopic yield
χ
strength. σ„ is the stress required to initiate dislocation m o v e m e n t (incorporating all strengthening effects except the grain size effect), Κ is the Hall-Petch slope, and D is the average grain size of the material
σ<=σ +-^
(1)
0
This empirical relationship has been established for n u m e r o u s metal alloys, including high purity α-titanium, which is studied in this paper. 3
Both < 7
0
and Κ are obtained through
2
mechanical testing. Values of 0.53 and 0.671 M N / m ' for the slope were found in the literature for α-titanium " . These large values of slope indicate that grain size effects are important when modeling the yield stress of titanium. 6
7
2.2. Microscale Hall-Petch correlation The
macroscopic
Hall-Petch
relationship
has
been
successfully 8
9
microscale by studying slip transmission across grain boundaries " .
extended
to
the
In this method, nano-
indentation is employed to determine the applied shear stress. r , necessary to force dislocations o
across a grain boundary according to r„ = r + - ^ L , 0
(2)
where r„ is the intrinsic frictional shear stress and D is the average grain size, as delineated by the distance between the indenter and the adjacent grain boundary. Further, it is postulated that k = Im
τ
is the equation for the slope where m represents the misorientation between the
slip systems on each side of the grain boundary, r
t
is the critical shear stress required to initiate
slip across the boundary, and r is the distance to the dislocation source in the neighboring grain. 2.3. New mesoscale Hall-Petch relationship The general nature of the Hall-Petch relationships suggests the possibility of extending them to the concept of a critical resolved shear stress in rate-insensitive plasticity, or to a reference shear stress in viscoplasticity theory. Although Relation (2) appears simple at first, implementation is problematic due to the grain boundary character dependence of both τ and r. Further, homogenization procedures used to connect the two relationships are not well understood. A new relationship, as shown in Equation (3), can be used to integrate grain size information into the viscoplasticity model:
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Grain S i z e a n d Orientation Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
· " · = γ"'
γ
(
+ *Ι1.
1
(3) 1
Here τ" " is the reference shear stress (or slip resistance) and k' '' is the Hall-Petch like slope, resolved on each slip system Λ of the model. It should be noted that D, which has been substituted in the place of D. is no longer the average grain size of the bulk material but the actual grain size for each individual grain within the material; this is allowed to vary within the crystal plasticity model. 3. M O D I F I E D T A Y L O R V I S C O P L A S T I C M O D E L F O R H I G H P U R I T Y Ot-TITANIUM 3.1. Power law equations The crystal plasticity model utilized in this research is the standard power-law viscoplasticity approach of Asaro and N e e d l e m a n as implemented by Kalidindi et a l " " . The power law, 10
relates strain rates έ
14
in terms of slip rates f" on individual slip systems s. with the geometry of
the slip system defined by the geometric slip t e n s o r / / „ " ' , according to: 1
A i
/»=i(è '»A;"+À/V'). 2
Here 6 a n d / P ' a r e defined as the unit slip and normal directions respectively. isotropic hardening, the slip rates can be expressed as: I M
Γ
By assuming
signif)
=γ
0
where r ' ° =<τ' μ)^
is the resolved shear stress associated with each slip system, r
resistance, and d
is the deviatoric component of the local Cauchy stress.
(
(5)
i
(6) ( > 1 K
i s the slip
By inserting
1
/ ' ' ' f r o m Equation ( 3 ) in place of r , / ' , the model is modified to allow grain size to vary for each individual grain in the Taylor model 3.2. Yield strength calculations The power law equation can be re-written in terms of the viscoplastic c o m p l i a n c e . Μ , as: 15
έ» = Μ σ[,.
(7)
φ
5
By applying the Taylor assumption wherein the local and macroscopic strain rates are equal . Equation (7) can be solved to estimate the local stress. Differing from the classical theory, a modified Taylor factor is introduced by the relationship:
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G r a i n S i z e a n d O r i e n t a t i o n Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
o r
°'u =™Λ
m =-^.
(8)
u
Here m is a new variable that contains the orientation dependence of the stress, like the well u
known Taylor Factor, but adjusted for grain size. frictional shear stress is replaced with f ,
A further difference is that the intrinsic
which is here defined to be the simple (arithmetic)
0
average value of the basal, prismatic, and pyramidal values of the reference shear stress for hexagonal slip systems.
T h u s the n e w factor, m , kl
includes not only grain size, but also the
particular ratios of slip system dependent reference shear stress relative to the average
T. 0
Volume averaging the local Taylor factor, and scaling it by the average slip resistance, recovers a new estimate of the macroscopic deviatoric stress: σύ
={">u)%>-
(9)
3.3. Grain size and orientation distribution function It is proposed that a new distribution function, called the grain size and orientation distribution function ( G S O D F ) . be defined so that the modified Taylor factor can be evaluated explicitly. This function is similar to the orientation distribution function ( O D F ) in that it contains volume fractions of grain orientation occurrences, but differs in that it also includes the grain size. It is expressed as f(g,D)dgdD
=~ ,
(10)
and is defined as the probability density of finding an occurrence of grain size D with an orientation g inside a single phase polycrystalline material sample. If the G S O D F is integrated over the full range of grain size, it returns the familiar O D F : j f(g,D)dD
= f(g).
(8)
On the other hand, if the G S O D F is integrated over the full range of possible lattice orientations (i.e., the fundamental zone. FZ). then the overall grain size distribution of the microstructure f(D)
is recovered: \\\f(g.D)dg=J(D)-
(9)
Moreover, the following normalization condition must hold: /;•„„.,
J JJJ/(«,D)flfe
(13)
3.4. New grain size dependent Taylor factor Next, a new macroscopic Taylor factor is defined by integrating the local grain size dependent Taylor factor with the G S O D F to yield:
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Grain S i z e a n d Orientation Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
Mm,
= j \\\f(g,D)m (g.D)dgdD
{m (g,D)) u
(14)
ii
0
1-7.
Whereas the original Taylor factor expresses the efficiency by which deformation is affected by the lattice orientation of crystallographic slip, this n e w grain size dependant Taylor factor expresses the same efficiency as a function of both lattice orientation and grain size. By inserting Relation (14) into Equation (9), the macroscopic yield strength can be evaluated as both a function of orientation and grain size. 3.5. Chord length distribution A second distribution, called the Chord Length Distribution function ( C L D F ) " 16
be introduced to recover the G S O D F experimentally.
17
, may
The C L D F is defined as the probability
that a random chord traversing a grain will sample a grain of orientation g with a chord length of D ±dDI
2 within invariant orientation measure dg and along an infinite line containing the unit 1
vector c " . ]
P
(g,D\c" )dgdD
(15)
The superscript s is used to indicate that we have chosen to resolve chord lengths in directions that correspond to the intersection of slip planes with the metallographic section plane.
The
1
direction of the chord, ë ' ' , for each slip system is obtained by taking the cross product between the section plane normal Ν and the slip plane normal
(s)
h. , )
c'° = /Vxn' .
(16)
Because the chords traverse a grain from one side to the other, they are closely related to the grain's size, D. Hereafter, we shall make no distinction between the term "chord length" and the term "grain size." Additionally, because multiple slip systems within each grain are sampled, a distribution of chord lengths will result for each grain. T h u s , even a single grain will present a range of grain sizes that can be utilized for the purpose of distinguishing grain size effects within the methodology of this paper. The reader will note the similarity between the G S O D F and the C L D F . In fact, they are essentially the same function but specific to each slip system for equiaxed grain structures. However, for microstructures with peculiar grain shapes, differences would be expected between the two functions. The normalization of the C L D F is obvious from its definition: } \\\p{g,D\c^)dgdD 0
= \.
(17)
17
For that which follows in this paper, w e shall not distinguish local grain size among the differing slip systems, but only the variation of grain size with grain orientation. In this case the G S O D F will be expressed in terms of the C L D F as an average over the total number of slip systems. ,9 : f(g,D)
=
^f.p(g.D\^).
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Grain S i z e a n d O r i e n t a t i o n Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
Consequently, the output of the G S O D F contains for any given orientation, g. a range of grain sizes, and this distribution is affected not only by the distribution of sizes of grains of a particular class, but also by the specific chord length distribution o f t h a t class. 4. E X P E R I M E N T A L M E T H O D S O N H I G H P U R I T Y Ot-TITANIUM 4 . 1 . u-titanium background information The high purity α-titanium material used in this study was supplied by the Alta G r o u p of Johnson Matthey Electronics. Inc. ( S p o k a n e . W A ) . The received plate was 9 9 . 9 9 9 8 % pure and measured 352 m m in diameter by 12 m m in thickness. The material was heat treated at 530° C for one hour and water quenched to produce a recrystallized grain structure. As detailed in previous research and s h o w n in the [0001] pole figure of Figure 1, the material exhibits a strong fiber texture with the c-axes of the grains distributed uniformly within 20-35° of the plate normal ( N D ) following heat treatment. 6 7
Figure 1 : Pole figure plots illustrating texture of heat treated α-titanium plate with c-axis of grains distributed around N D direction 4.2. Oblique sectioning scheme In order to recover the C L D F and G S O D F of the high purity α-tUanium in a statisticallyunbiased way. an oblique sectioning technique was employed. The sphere of directions in Figure 2(a) represents the surface normal directions for the 13 oblique section cuts along with their inverses. The points circled in black are found on the front of the sphere, while the points circled in gray are located on the back side of the sphere. Figure 2(b) is a rendering of the titanium sample with the section cuts r e m o v e d and shows the N D , T D . and R D directions of the sample. The samples w e r e electrical discharge machined from the heat treated titanium plate and carefully polished to remove deformation incurred from the sectioning process.
514
Materials Processing and Texture
G r a i n S i z e a n d Orientation Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
Figure 2: (a) Sphere of directions representing surface normal directions for oblique section cuts, (b) Rendering of titanium plate with 13 section cuts removed and N D , R D . and T D directions defined 4.3. O I M analysis Orientation imaging microscopy (OIM) was performed on a Philips X L - 3 0 S F E G scanning electron microscope for each of the thirteen samples in order to obtain the orientation and grain size statistics. A hexagonal grid with a 1 μηι step size yielded 208,247 scan points within each 300 p m χ 600 p m scan window. A total of 19.642 grains were resolved but since edge grains were excluded from the analysis, only 17.437 were used to ensure statistical reliability. 4.4. Chord length distribution plots A chord length distribution plot is presented in Figure 3 for three of the primary hexagonal slip systems of titanium. In this way the C L D F was used to calculate the average grain size for each of the 17.437 grains. However, the complete subset of primary slip systems totaling 18 w e r e sampled and then averaged to obtain the grain size values input into the Taylor model. T h e smallest grain sampled was 0.34 p m vs. the largest at 38.13 pm. The overall average grain size for the α-titanium material as calculated by the C L D F method was 7.72 ± 4.35 pm.
Figure 3: Chord length distribution plot for three of the eighteen slip system of α-titanium. (Note that only 10 grains are s h o w n and the spacing between chords is exaggerated to illustrate the concept). 4.5. True stress - true strain plots In order to calibrate and verify the accuracy of the Taylor model predictions, uniaxial compression testing was performed for titanium test samples electrical discharge machined from the plate along the R D . T D , and N D directions. T h e testing w a s performed at room temperature and at a constant strain rate of 10-2 s"'. During the test, Teflon sheets, high pressure grease, and regular lubrication were used to negate frictional effects. The r a w load and displacement data w a s corrected for machine compliance before the true stress - t r u e strain curves were calculated. A s plotted in Figure 4, the uniaxial compressive yield strength in the N D . R D . and T D directions w a s 352 M P a . 192 MPa, and 174 M P a respectively. A plane strain compression test in the N D direction resulted in a fourth value of 199 MPa.
Materials Processing and Texture
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Grain S i z e a n d O r i e n t a t i o n Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
Figure 4 : True stress - equivalent true strain curves from uniaxial and plane strain compression testing 5. C A L I B R A T I O N A N D E V A L U A T I O N O F M O D E L 5.1. Calculation of reference shear stresses Reference shear stresses for the three primary slip families (basal, prismatic, and pyramidal) of hexagonal close pack titanium are necessary to calibrate the crystal plasticity model presented in this paper. These values were determined by curve-fitting the predicted yield strengths in simple deformation m o d e s to the experimentally obtained curves s h o w n in Figure 4. The values obtained through this process were 200 MPa, 10 M P a . and 120 M P a respectively for basal, prismatic, and pyramidal. Additionally, values of 57.1 M P a , 5.4 M P a . and 44.4 M P a for the basal, prismatic, and pyramidal intrinsic frictional shear stresses were estimated based o n the work of C h u r c h m a n . Further, values of 0.013 M N / n r " , 0.397 M N / m ' , and 0.210 M N / m ' for the mesoscale Hall-Petch slopes were calculated by substituting the frictional shear stresses along with the average value of grain size into Relation (3). 19
2
3
2
3
2
5.2. Numerical methods Solving the Taylor viscoplastic equations for thousands of grains is computationally demanding. These multi-dimensional equations are known to converge poorly due to their stiff, non-linear nature. T h e situation becomes even more c o m p l e x when grain size effects are incorporated into the equations, as the calculations must be repeated for multiple grain sizes. A computationally efficient method w a s required to eliminate the need to repeatedly solve the equations for every combination of grain orientation and size. 5.3. Database approach
516
•
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G r a i n S i z e a n d Orientation Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
T h e database approach of Knezevic et al was utilized is this study, wherein the necessary variables are computed only once and then stored for later retrieval. As shown in Equation (19). the strain rate equation can be written in terms of a single angular variable, θ. w h e n expressed in the principle f r a m e . In this way. only the diagonal terms of the strain rate space need to be sampled, and the time necessary to probe the entire five dimensional strain rate space is conserved. A tessellation scheme of 10° was e m p l o y e d for orientation space and 3° degrees for θ intervals. Additionally, the grain size interval was incremented with a distance of 1 μηι between adjacent centroids 17
6. D I S C U S S I O N O F R E S U L T S Yield surface plots were chosen as a convenient way to visually compare the results found in this paper. Figure 5(a) represents a deviatoric shear stress subspace that can be extracted from the full five dimensional stress space and illustrated as a three dimensional object. The yield loci in Figure 5(b) represent yield surfaces in the κ -plane for the α-titanium material. A total of 1,500 yield points were sampled to create the plots.
Figure 5: (a) Deviatoric stress subspace plot for high purity α-titanium; (b) π-plane plot comparing the yield surface of the new model to the traditional Taylor and V o n - M i s e s models The dashed yield surface in Figure 5(b) represents the Von-Mises or isotropic case, the curve with data points composed of asterisks corresponds to the new grain size adjusted Taylor model, the curve with data points composed of circles represents the traditional Taylor
Materials Processing and Texture
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Grain S i z e a n d O r i e n t a t i o n Distribution F u n c t i o n of H i g h Purity α - T i t a n i u m
viscoplastic solution, and the triangles show the experimentally obtained yield points. It is observable that the traditional method, which only accounts for variations in texture, accurately predicts the yield surface in the σ'„ and σ'„ directions but substantially underestimates the yield surface in the a' , direction of the material. T h e new grain size differentiated model on the other r
hand slightly over-predicts the yield surface in the σ'
η
and σ'
22
direction but does a reasonable
j o b in predicting the anisotropic yield response of the material in the o~' direction. The empty n
regions between the two Taylor models of Figure 5 substantiate the idea that a material's grain size distribution contributes significantly to its yield strength. We would expect this effect to be even more pronounced for materials with a larger Hall-Petch slope, materials with large grain size variations, partially recrystallized textures, or materials with elongated grain structures. 7. C O N C L U S I O N S The goal of this work has been to introduce a n e w methodology whereby the grain size distribution can be introduced into crystal plasticity. A new distribution function, similar to the orientation distribution function, but adjusted for grain size, has been defined. This new grain size and orientation distribution function can be recovered through orientation imaging microscopy that simultaneously recovers the chord length distribution. The methodology has been demonstrated for a high purity sample of α-titanium. Experimental methods used to calibrate the new model were described and results were presented as yield surfaces in deviatoric stress space. The following conclusions and observations can be drawn from this study: • Grain size and its distribution have a significant impact on the yielding characteristics of α-titanium, in that both the size and shape of the yield surface were affected • •
The chord length distribution is an effective tool in recovering grain size statistics A mesoscale Hall-Petch relationship can be successfully incorporated into Taylor viscoplasticity
•
Introducing grain size as a variable in Taylor viscoplasticity m o r e accurately predicts the anisotropic yield loci of H C P α-titanium as compared to the traditional approach
ACKNOWLEDGEMENT This work w a s supported at Brigham Y o u n g University by a grant from the U S A r m y Research Office. Metallurgy Program, Dr. David Steep, Program Director. The authors gratefully acknowledge the assistance of Dr. Surya Kalidindi and his research group for providing the material and technical expertise in Taylor viscoplasticity. REFERENCES 'N.J. Petch, The Cleavage Strength of Polycrystals, Journal of the Iron and Steel Institute, 174, 25-28(1953). E . O . Hall. The Deformation and Ageing of Mild Steel: III Discussion of Results, Proceedings of the Physical Society of London, B 6 4 , 747-753 (1951). R . A . M a s u m u r a . P.M. Hazzledine, and C.S. Pande, Yield stress of fine grained materials. Acta Materialia. 4 6 . 4 5 2 7 - 4 5 3 4 (1998). J. Schiotz. K. Jacobsen. A m a x i m u m in the strength of nanocrystalline copper. Science. 3 0 1 , 1357-1359 (2003). G . I . Taylor, Plastic Strain in Metals. Journal of the Institute of Metals, 6 2 . 307-324 (1938).
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A . A . Salem, S.R. Kalidindi, and R.D. Doherty, Strain hardening regimes and microstructure evolution during large strain compression of high purity titanium. Scripta Materialia, 4 6 , 4 1 9 423 (2002). A . A . Salem, S.R. Kalidindi, R.D. Doherty, and S.L. Semiatin, Strain Hardening Due to Deformation Twinning in a-Ti: Mechanisms, Metallurgical and Materials Transactions A. 37A, 259-268 (2006). V . Bata and E. Pereloma, An alternate explanation of the Hall-Petch relation. Acta Materialia. 52, 657-665 (2004). W . A . Soer, K.E. Aifantis, and J.T.M De Hosson, Incipient plasticity during nanoindentation at grain boundaries in body-centered cubic metals, Acta Materialia. 5 3 , 4665-4676 (2005). R . J . Asaro, A. N e e d l e m a n , Texture development and strain hardening in rate dependent polycrystals. Acta Materialia. 3 3 , 923 (1985). " A . A . Salem, S.R. Kalidindi, and S.L. Semiatin, Strain hardening due to deformation twinning in α-titanium: Constitutive relations and crystal-plasticity modeling, A eta Materialia. 53, 34953502 (2005). S . R . Kalidindi, H.K. Duvvuru, Spectral methods for capturing crystallographic texture evolution during large plastic strains in metals. Acta Materialia. 5 3 . 3613-3623 (2005). X . P . Wu, S.R. Kalidindi, C. Necker, and A.A. Salem, Prediction of Crystallographic texture evolution and anisotropic stress-strain curves during large plastic strains in high purity atitanium using a Taylor-type crystal plasticity model, Acta Materialia. 5 5 , 423-432 (2007). M . K n e z e v i c , S.R.Kalidindi, D.Fullwood, Computationally efficient database and spectral interpolation for fully plastic Taylor-type crystal plasticity calculations of face-centered cubic polycrystals, International Journal of Plasticity, (accepted). J . W . Hutchinson, B o u n d s and self-consistent estimates for creep of polycrystalline materials. Proceedings of the Royal Society of London, 3 4 8 . 1 0 1 - 1 2 6 ( 1 9 7 6 ) . G . Matheron, R a n d o m Sets and Integral Geometry, Wiley & Sons, N e w York (1975). S . Torquato and B. Lu, Chord length distribution function for two-phase random media,
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Physical Review E.. 4 7 , 2950-2953 (1993). A . A . Salem, S.R. Kalidindi, R . D . Doherty, Strain hardening of titanium: role of deformation twinning, Acta Materialia, 5 1 , 4 2 2 5 - 4 2 3 7 (2003). A . T . C h u r c h m a n , The Slip M o d e s of Titanium and the Effect of Purity on their Occurrence during Tensile Deformation of Single Crystals, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 226, No. 1165, 216-226 (1954).
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TEXTURES IN T I T A N I U M ALLOYS - AN INDUSTRIAL DEFORMATION, TRANSFORMATION AND PROPERTIES. 1
2
David R u g g , David Furrer and Nigel Brewitt 'Rolls-Royce pic, ' R o l l s - R o y c e Corporation
PERSPECTIVE
ON
1
ABSTRACT The development of texture within titanium alloys can be critical for the development of optimal final component mechanical properties. Understanding of hexagonal close packed alpha phase texture formation can lead to predictive models to guide engineers in material and process design methods. A l p h a texture can lead to highly anisotropic behaviour, including directional strength, m o d u l u s and ultrasonic inspection capabilities. Preferred texture can result from deformation and/or transformation from the higher temperature beta phase. A summary of how alpha phase texture develops, the impact o n mechanical properties and implications for commercial application of titanium components will be provided. 1. I N T R O D U C T I O N Aero-engines are designed such that failure rates of safety critical structures are extremely remote, being generally less than one failure in 1 0 flights '. T o achieve this low level of risk, the designer must work with m i n i m u m rather than typical or m a x i m u m properties. Statistical m e t h o d s are applied to determine the m i n i m u m strength or life for a given confidence level. This m a y e n c o m p a s s tensile strength, fatigue strength, elongation to failure or creep strength and will depend on loading regimes that the component experiences. The design process is well-documented elsewhere" and only key issues of interest are highlighted here. Clearly texture has an important role to play in all of the above properties, yet understanding of process / texture property relationships is still at an early stage for Ti alloys. It is important to note that safety critical rotative c o m p o n e n t s are produced to a fixed method of manufacture with strict change control procedures. This is necessary because of the inherent and significant shortfalls of using specifications as the sole means of demonstrating fitness for purpose. Whilst material specifications give some confidence in basic mechanical properties they are unable to capture behaviour in more complex load regimes or reflect property drift within the specification limits. For this reason it is usual to define process control limits for a variety of measurable variables in the production process. Therefore, whilst texture is not measured directly its consistency is nominally captured by inference from simple mechanical tests. This is not an inherently robust technique as demonstrated by some of the examples cited later in this paper. It is the mandated duty of the metallurgist within an aero-engine c o m p a n y to ensure that changes to manufacturing route result in an equivalent integrity of component to that initially approved during the engine development and certification process. These changes m a y be either deliberate (change in supplier etc) or unexpected (manufacturing non conformance etc.). Process control limits have much tighter ' w i n d o w s ' than specifications and are therefore of more use in establishing minima and process drift . As such, the level of k n o w l e d g e and skill that the aerospace sector has, or has access to, needs to be sufficient to maximise commercial return whilst ensuring a high level of product reliability. T h e main c o m p o n e n t s in a typical aero gas turbine and associated load regimes are given in Figure 1. R
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Figure 1. Aero-gas turbine - loading and test regimes that are influenced by preferred texture. 2. D E F O R M A T I O N T E X T U R E S Qualitative relationships between processing route and deformation textures have been established for many years. Deformation textures in titanium alloys are particularly important since; (i) Marked elastic and plastic anisotropy can be developed which strongly influence subsequent component manufacture and behaviour in service. (ii) Texture ' m e m o r y ' is particularly pronounced resulting in the necessity to control most stages of the thermo-mechanical process route with great care. Potentially this extends all the way back to initial ingot cogging. Remarkably, a preferred alpha deformation texture, once developed, m a y be retained even through a beta heat treatment cycle as a result of strong variant selection during the transformation. In the aerospace industry, deformation textures are generally developed as a consequence of strain required to generate shape change and / or microstructural development. For s o m e products such as fan blades, deformation textures are deliberately manipulated in order to achieve optimised mechanical properties - typically impact or fatigue resistance. Quantitative relationships developed by end users are usually generated from extensive testing / release data which m a y then be modelled by empirical means. Such work is typically proprietary in nature and seldom reported in the open literature. In practice, the intensity of deformation textures m a y be significant:- in rolled plate product basal intensities m a y be as high as x30 and macroscopic elastic anisotropy may approach 10%. In forged products texture intensity is normally lower but m a y still be significant, particularly in relation to ' m a c r o z o n e ' formation. It is interesting to note that the < 1 1 0 > bcc texture developed during beta forging ( < 1 1 0 > transverse preferred texture) is consistent with the hexagonal transverse basal deformation texture as a result of the {110} beta I I {0001} alpha
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burgers orientation relationship . As yet, mechanistically sound quantitative models for the prediction of hexagonal transverse basal textures produced during alpha beta processing are in their infancy. Further development would represent an important step forward since it will require detailed knowledge of the role of the compliant beta phase and alpha beta interface behaviour as a function of temperature and strain rate. During early stages of alpha beta processing such modelling will not be trivial as a result of the concurrent microstructural evolution. O f particular significance is the break up and spheroidisation of the colony alpha platelets formed earlier during the beta to alpha transformation. Such colonies are typically coarse in practice as a result of the slow cooling rates arising from large section sizes at the time beta heat treatment is performed. T h e size of the micro-texture region comprised of equiaxed alpha with the same orientation is often still called a colony or an effective structural unit (ESU). as the length scale of this unit can control the mechanical behaviour - this will be discussed in more detail later in this paper. 3. T R A N S F O R M A T I O N T E X T U R E S Another important texture from an industrial standpoint is that formed during the beta to alpha transformation, when the alloy is cooling from above the beta transus. This transformation dictates the colony size of the alpha platelets, and, from the extent of beta boundary misorientation, the probability of adjacent colonies with the same alpha phase crystallographic orientation. This colony size largely dictates the mechanical properties in some very important load r e g i m e s . This influence m a y apply to material used in either the beta heat treated condition or in the alpha beta thenno-mechanically processed condition if the alpha / beta strain and directionality during processing h a s been inadequate to randomise the primary alpha present in addition to spheroidising it. Transformation textures from alpha beta heat treatment are generally not significant for near alpha or moderately beta stabilised alpha beta alloys. This is because in practice most alloys of this type have cooling rates that are insufficient to produce martensite o n cooling. As such, alpha coarsening occurs for very slow cooling rates such as recrystallisation anneals. For slow and moderate cooling rates (eg air cool) alpha platelet growth occurs nucleating on primary alpha present or from a m e c h a n i s m of interface instability and thus inheriting the primary alpha texture. This mechanism explains w h y the macroscopic c o m p o n e n t stiffness is not generally influenced b y alpha beta heat treatment during manufacture. For m o r e heavily stabilised alloys, the transformation texture is potentially more significant since under-cooling can result in transformation occurring at lower temperatures. Indeed, for m a n y such systems, large volume fractions of beta are retained to room temperature and transform on low temperature ageing. Kinetics associated with ageing will therefore not favour pronounced coarse alpha colony growth from existing primary alpha or grain boundary alpha. Strain history and dislocation network generation along with beta strain during transfonnation are potential m e c h a n i s m s that influence variant selection, which could lead to pronounced transformation textures in beta-processed alloys. In s u m m a r y , both elastic and inelastic deformation can be modified by alpha phase texture in titanium alloys. The practical impact of these changes will be discussed in sections 4 to 6. 3
4. E L A S T I C P R O P E R T I E S C h a n g e s in elastic properties have significant implications for the designer of high duty c o m p o n e n t s and the performance of products. For instance, initial geometric definition for a titanium alloy fan blade in an aerospace gas turbine requires only elastic modulus and density. From this, most of the key behaviours of the aerofoil in service can be predicted. For modulus and therefore texture related issues, these include:-
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(i)
Blade shape whilst operating - fan blades have a significant twist along their radial axis to optimise aerodynamic efficiency. Since centrifugal loads will result in the blade untwisting, the 'as manufactured' or static shape must have an over-twist to compensate. Since the fan blades are responsible for around 7 5 % of the engines thrust, accurate prediction and control of the running blade shape is essential in order to achieve required engine performance. As an example, blade tip stagger angle is controlled to within less than 1 degree and a typical total elastic untwist during running results in a leading edge tip m o v e m e n t in excess of 15mm.
(ii)
Resonant m o d e shape and resonant frequency - blades are subject to aerodynamic and mechanical forcing w h i c h cause them to resonate. The stresses and frequencies can be very high ( > 5 0 0 M P a , > 5 K H z ) . It is therefore imperative that the blade is ' t u n e d ' so that none of its resonant m o d e s coincide with engine forcing functions at cruise. A s an example, if on resonant condition, a high-pressure compressor blade from a Rolls-Royce Trent engine used on the Boeing 777 would accumulate 10° stress cycles on one flight from London to New York. Figure 2 illustrates an example of modal shape and frequency for a 'high order' m o d e for a typical compressor aerofoil.
Figure 2. High order resonant m o d e stress contour plot for a intermediate pressure compressor aerofoil. Absolute levels are arbitrary since they depend on excitation amplitude. Resonant frequency - 1 8 K H z , Scale - 1 : 1 .
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5. N O N - D E S T R U C T I V E E V A L U A T I O N A N D E B S D Clearly, direct measurement of texture on a production basis would provide a direct means of assessment and therefore potential control of the key manufacturing variables. Existing techniques leave m u c h to be desired in this regard; x-ray and E B S D are essentially destructive, slow and sample only small areas of material, macroscopic texture evaluation by ultrasonic methods such as compression w a v e velocity have limited spatial resolution. As such, new developments such as S R A S (Spatially Resolved Acoustic Spectroscopy) are an exciting innovation . By using lasers to m e a s u r e surface acoustic wave velocity, the modulus and therefore the texture can be inferred. This technique combines sub-millimetre spatial resolution with the ability to scan large areas 'in the field' rather than b e i n g laboratory based. Figure 3 illustrates textures measured by E B S D and corresponding S R A S images for a titanium alloy sample. The qualitative agreement is good, although it should be noted that for some hexagonal metals, such as zirconium, the Y o u n g ' s modulus displays a ' d i p ' between the c and a direction, meaning texture cannot be unambiguously determined by this technique. 6
Figure 3. Comparison of S R A S (left) and E B S D (right) for large grained IMI685, acquisition times were 2hours and 14 hours respectively. Specimen size 1mm χ 2 m m . Standard triangle applies to E B S D image only. S R A S is at an early stage of application and a standard m e a n s of representing orientations has yet to be decided. The technique is inherently different from E B S D in that velocity scans in two or m o r e directions are required to quantify crystallographic orientation. It should also be noted that the technique functions by measurement of elastic anisotropy. Therefore differentiation of some textural features is not possible - an e x a m p l e being prismatic intensity in alpha phase titanium.
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Sonic wave velocity differentials in hexagonal materials are also critical in determining inspection capability. Efforts have been made to relate texture in titanium alloys to ultrasonic capability . High ultrasonic noise restricts the resolution of the technique and therefore limits the size of defects that can be found for a given probability of detection. Understanding the relationship between the thermo-mechanical processing route of a semi-finished or finished component and 'sonic-ability' is difficult. One of the main difficulties associated with these studies is the temptation to relate sonic response to optical metallography. This approach is fundamentally flawed since the effective structural unit size of the material m a y span several ' g r a i n s ' due to a small misorientation distribution. Significant potential exists for learning a great deal about the structure and properties of metallic systems by understanding ultrasonic noise and its origins. Efforts to describe the effective unit size as it relates to ultrasonic inspection have resulted in a determination of a figure of merit for a given material . Figure of merit represents the size and distribution of textured regions (grains, colonies and/or macro-zones) within a material. 7
8
6. P L A S T I C P R O P E R T I E S O f all the issues related to texture in aerospace, few are as beguiling or complex as titanium alloy notch fatigue b e h a v i o u r . ' M a c r o - z o n e s ' within the material dictate the fatigue strength in loading regimes where there is a high ratio of mean to alternating strength. M a c r o zones are related to colonies of primary alpha that have similar orientation as a result of ingot to billet conversion processes or lamellar alpha that have similar orientation arising during the beta to alpha transformation. It is also probable that both macro-zone geometry and m e a n misorientation are critical in controlling properties. It is interesting to note that during typical 'cross rolling' of titanium alloy plate macro-zones exhibit basal poles with a very high degree of misorientation and therefore high elastic and plastic incompatibility . 9
10
Another truly fascinating aspect of texture and mechanical properties is that the 'effective structural unit size' of the material depends on the load regime. For low amplitude high cycle fatigue, the length scale of importance m a y be the primary alpha (microns in length). For cold dwell fatigue or notched fatigue with high mean stress the macro-zone size and misorientation may dictate properties'. These issues are of e n o r m o u s industrial significance - titanium cold dwell fatigue has caused numerous in-service failures of critical rotative components, spawning extensive research p r o g r a m m e s to understand the formation and behaviour of m a c r o z o n e s . 11
Macro-zone formation and the "effective structural unit s i z e " is also a factor influencing the properties of non-rotative titanium c o m p o n e n t s or c o m p o n e n t s that are subjected to sustained loads. In one instance, a large Ti 6A1 4V statically loaded, non-aerospace component exhibited in-service cracking and was subsequently investigated by Rolls-Royce pic. Fractography revealed quasi-cleavage facetting over most of the fracture surfaces as illustrated in Figure 4. These zones were also present sub-surface. Microcracking of this type was shown to arise on basal planes and grain to grain basal alignment within the zones differed less than 15°. These features are illustrated in Figure 4. Evans et i - - - showed that facet formation of this type is a result of stress redistribution within the alloy. This redistribution w a s considered to be a result of weaker regions, those with slip systems favourably aligned with the load direction, and stronger regions, those with basal planes aligned perpendicular with the direction of load. l 2
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Crystal plasticity modelling was advanced by Dunne et al w h o showed the concept of a "rogue grain combination". In this model, the degree of stress redistribution was directly related to the angle between an active slip system in one grain and a strong or hard adjacent grain i.e. with basal planes aligned perpendicular with the direction of loading. The model also showed a reliance on grain boundary orientation between t w o such grains. M a c r o z o n e s can be considered as a single hard orientated grain (an "effective structural unit") and the remaining grains between the zones contain the active slip systems which allow for off-loading of stress onto the zone. Whilst this model was developed to assist the aerospace sector with the issue of titanium cold dwell fatigue, the applicability to non-aerospace sectors is also clear.
Figure 4. (a) sub-surface cracking predominately within macrozones, (b) fracture surfaces illustrating faceting, (c) E B S D map of crack on basal planes within macro zone an (d) associated inverse pole figure.
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7. M O D E L L I N G - T H E W A Y F O R W A R D Modelling tools are b e i n g utilized to provide insight into the effects of manufacturing processing for specific component configurations. These tools aid engineers in m o r e rapidly designing and manufacturing c o m p o n e n t s with optimal performance with unique combinations of properties. The need for modelling and simulation of materials and processes is driving n u m e r o u s focussed efforts to further mature these technologies and is resulting in enhanced collaboration between the academic and the industrial engineering communities. Physics-based and phenomenological-based analytical tools are being developed to predict the evolution of microstructure and mechanical properties within materials as a result of thermo-mechanical processing. Modelling based on diffusion processes has been developed and validated on production manufacturing routes and has been s h o w n to accurately predict the growth of primary alpha during heat treatment cooling processes"'. This modelling approach has been successful in predicting the resultant primary alpha volume fraction and particle size as a result of sympathetic growth during cooling. This m e c h a n i s m of alpha grain structure evolution has significant implication with respect to perpetuating prior primary alpha texture within a component. Phase-field modelling is being extensively used to model the physics of a range of m e c h a n i s m s including beta grain growth and lamellar alpha precipitation within titanium. Recent efforts have shown excellent results for the prediction of alpha colony formation during cooling of beta grain structures' . The thermodynamic driving forces and mobility limitations within beta to alpha transfonnations result in changes in size of colonies and widths of alpha lamellae. Incorporation of interface elastic strain between the alpha and beta phase results in predictions of specific alpha phase orientation variants relative to an original beta grain orientation. This modelling approach can be utilized to assess microstructure evolution within full-scale c o m p o n e n t s via linking of predicted thermal histories and beta gTain structure at specific locations. The use of crystal-plasticity models linked to finite-element m o d e l s has recently led to the prediction of anisotropic flow behaviour of textured m a t e r i a l s . This has also allowed the prediction of texture evolution during large strain deformation processes, such as component forging. For this, the primary alpha texture is tracked-based on rigid-body rotation from finite element analysis, combined with rotation due to crystallographic slip from crystal plasticity analysis. This has been successful in accurately predicting the evolution of texture within a fullscale titanium forging ' ' . Figure 5 shows an example of a measured and predicted alpha phase texture in a full-scale Ti64 forging. T h e correlation of the modeled and measured texture is very good. Improvement in this prediction could result from incorporation of alpha variant selection rules to allow separation of the texture contribution from the primary and secondary alpha phases. The prediction for the alpha phase texture is c o m p o s e d of both the texture of the primary alpha that has evolved during deformation and the texture of the secondary alpha, which is a result of precipitation from the prior beta grains. In addition to being able to track the evolution of primary alpha texture, this modelling approach also allows for tracking of the deformation within the beta phase. A series of lamellar alpha crystallographic variant selection rules have been developed and are under investigation based on prior beta deformation structure. 18
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Figure 5. (a) Measured and (b) predicted pole figures for the alpha phase within a fullscale T i 6 4 forging. Prediction of mechanical properties within titanium alloys is of greater interest to component designers than prediction of texture and poses a significant challenge. Complete physics-based understanding of the m e c h a n i s m s relating microstructure (and chemistry) to mechanical properties of titanium alloys is elusive due to the complex nature of the microstructure. Models that explain the effect of specific elemental species, (such as Ni), on the creep behaviour of titanium has been developed, but general m o d e l s that govern the overall deformation processes within titanium are not available. 0
2
Statistics-based' and neural-network b a s e d ' m o d e l s have been developed that provide predictive capabilities for fatigue and tensile properties. These pragmatic models relate a combination of microstructural features, mechanical properties and/or processing histories to final component properties. Neural-network models can relate location-specific componentstrain (e.g. radial, axial, or tangential) chemistry, location-specific cooling rates and other processing history information to tensile strength. This modelling approach allows prediction of spatial distribution of tensile properties. The incorporation of component-strain captures the effects of deformation texture, which are k n o w n to have a significant effect on the mechanical properties. Figure 6 shows an example prediction of the tensile properties for a forged component as compared to measured values. It can be seen that the predictions match well with the m e a s u r e m e n t s and the effect of local deformation texture is also captured.
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Figure 6. Tensile-strength (MPa) for a full-scale Ti-6A1-4V forging. T h e contours represent predicted values, while the listed n u m b e r s are m e a s u r e d averages/ measurement variations for samples tested at the indicated locations. The incorporation of microstructure and mechanical property modeling, along with component and manufacturing process design, has a n u m b e r of benefits. It allows designers and manufacturing engineers to develop optimal designs and processes for new c o m p o n e n t s m o r e rapidly by: providing insight into component/process capabilities (and limitation) without shop trials providing process robustness assessment guiding engineers to locations of interest within c o m p o n e n t s for qualification testing supporting selection of critical processing parameters for control and to what range of parameter tolerance allowing early assessment of potential fixed process changes allowing assessment and screening of radically different processes to achieve unique component properties to exploit location-specific benefits of texture
8. C O N C L U S I O N S It is evident that the area of texture / process / property relationships in titanium alloys is of cntical importance to industry. The combination of improved models and cutting-edge experimental techniques offer the prospect of dramatic improvements encompassing all aspects of titanium processing, component design and in-service behavioural modelling.
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REFERENCES Rugg D, T i - 2 0 0 3 , Proc. 10th World Conference on Titanium Alloys, Hamburg, Germany, eds G Lutjering, J Albrecht, 2003, 2727-2736. " Yang X, Liu C R, J. Manufacturing Science and Engineering, 124, 2002, 745-753. Rugg D, Dixon M , D u n n e F Ρ E D, J. Strain Analysis for Engineering Design, vol42, No4, 2007, 269-279. Evans W J, M c E l h o n e M, R u g g D 'Designing for variability in fatigue performance' TÎ2007 Science and Technology: Proceedings of the 1 1 * World Conference on Titanium, July 2007, Kyoto, Japan. Lutjering G 'Influence of processing on microstructure and mechanical properties of (cc+ß) titanium a l l o y s ' . Materials Science and Engineering A 2 4 3 , 1998, 3 2 - 4 5 . Sharpies S D, Clark M, S o m e k h , M G, 'Spatially resolved acoustic sprectroscopy for fast noncontact imaging of material microstructure' Optics Express 2006, vol.14 No22 10435-10440. M.P. Blodgett and D. Eylon, "The Influence of Texture and Phase Distribution on Ultrasonic Attenuation in TÎ-6A1-4V", J. of Nondestructive Evaluation. Vol. 2 0 . N o . 1, 2 0 0 1 , pp. 1-16. R.B. T h o m p s o n , F.J. Margetan, Y.H.K. Han, A.J. Paxson and C E . Shamblen, "Relationship of Microstructure to Backscattered Ultrasonic Noise", Review of Progress in Quantitative Nondestructive Evaluation, Vol. 11, 1992, pp. 1685-1691. ' T a y l o r K, Bowen P, Rugg D, 'Effects of Microstructure on the H C F Behaviour of Ti 6A1 4V Alloys' T i - 2 0 0 3 . Proceedings of the 10 World Conference on Titanium held at CCH-Congress Centre, H a m b u r g , G e r m a n y , 13-18 July 2003, Eds G Lutjering, J Albrecht, 2737-2744, 17671774. Bantounas 1, Lindley Τ C, Rugg D, Dye D, 'Effect of microstructure on fatigue cracking in Ti 6A1 4 V , Acta Materialia. 55 (16):5655-5665, 2007 ' D u n n e , F P E , Walker, A, R u g g , D , "A systematic study of hep crystal orientation and morphology effects in polycrystal deformation and fatigue",Proc. R. Soc. Lond.. in press, 2007. Bache M R, Ά review of dwell-sensitive fatigue in titanium alloys: the role of microstructure, texture and operating conditions, Intl.Jnl.Fatigue, 2003, 25, 1079-1089 ' Bache M R, Evans W J, ' D w e l l sensitive fatigue response of titanium alloys for p o w e r plant applications, Jnl. Eng. For Gas Turbines and Power, 2003, 125,241-245 Evans W J, 'optimising mechanical properties in alpha beta titanium alloys' Mats. Sei Engg. 1998, A 2 4 3 , 89-96 Evans W J, Bache M R, 'dwell sensitive fatigue under biaxial loads in the near alpha titanium alloy I M I 6 8 5 , Fatigue, 1 6 , 4 4 3 - 4 5 2 . ""Semiatin, S.L. Knisley, P.N. Fagin, F. Zhang, and D.R. Barker, "Microstructure Evolution During Alpha-Beta Heat Treatment of Ti-6A1-4V," Metall, and Mater. Trans. A, 2003, vol. 34A, pp. 2377-2386. W a n g Y. Z, W a n g N, M a Q, Chen F. Z h a n g S L, Chen A and Chang Y A Predicting phase equilibrium, phase transformation, and microstructure evolution in titanium alloys' J O M vol.57, N o 9, Sept 2005, 1047-4838 G l a v i c i c M G, Goetz R L. Barker D R, Boyce D, Dawson Ρ R and Semiatin S L. "Integration of a Texture-Modeling Package into D E F O R M ™ " , A e r o m a t - 2 0 0 7 , Baltimore, Maryland, USA, June 25-28, 2007. '"Olavicic M G, Goetz R L, Barker D R, Shen G, Furrer D, Woodfield A, and Semiatin S L, " M o d e l i n g of Texture Evolution during Hot Forging of Alpha/Beta Titanium Alloys", to be published. "°Cohen F S, " H C F M e a n Stress Sensitivity in Titanium Alloys", A e r o m a t - 2 0 0 7 , Baltimore, Maryland. U S A , June 25-28, 2007. 1
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"'Furrer D and Thaler S, "Neural-Network M o d e l i n g , " A d v a n c e d Materials and Processes, November, 2 0 0 5 , pp. 42-46.
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ANNEALING
RELATED
MICROSTRUCTURAL
DEVELOPMENTS
IN A
TWO-
PHASE ZR-2.5 N B A L L O Y
3
a
3
b
V . D . H i w a r k a r , S.K. S a h o o , I. S a m a j d a r , K. N a r a s i m h a n " , * Κ. V . M a n i k r i s h n a , G . K . D e y \ D. Srivastav
b
b
R. T i w a r i , S. B a n a r j e e
b
a
Department of Metallurgical Engineering and Materials Science, IIT Bombay, M u m b a i - 4 0 0 0 7 6 , India
b
Materials Science Division, Bhabha Atomic Research Center, M u m b a i , India
ABSTRACT Deformed/pilgered
two-phase,
10-15%
cubic
second
phase
-
rest
being
primary
hexagonal phase, Zr-2.5 N b alloy w a s subjected to various annealing treatments treatments ranging
from
recovery to recrystallization
and grain growth.
-
Associated
microstructural developments were monitored through combinations of characterization techniques - bulk crystallographic texture & microtexture measurements and estimations o f lattice s t r a i n a n d r e s i d u a l s t r e s s . S i g n i f i c a n t t e x t u r e c h a n g e s w e r e a s s o c i a t e d o n l y w i t h grain g r o w t h of the primary p h a s e - a p r o c e s s facilitated by s e c o n d phase coarsening. F r o m n e a r l y c o n t i n u o u s p r e s e n c e at t h e p r i m a r y p h a s e g r a i n b o u n d a r i e s , l a t t e r s t a g e s o f g r a i n g r o w t h h a d s h o w n c o a r s e n e d s e c o n d p h a s e p r e s e n t o n l y at t h e t r i - j u n c t i o n s . T h i s p r o c e s s w a s a s s o c i a t e d w i t h s i g n i f i c a n t c h a n g e s in p h a s e - b o u n d a r y n a t u r e . A n effort w a s m a d e t o e x p l a i n s u c h c h a n g e s f r o m a n ' e x t e n d e d ' , i.e. e x t e n d e d t o p h a s e b o u n d a r i e s , C S L (coincident site'lattice) theory.
INTRODUCTION Z i r c o n i u m ( Z r ) a n d its a l l o y s a r e w i d e l y u s e d for S t r u c t u r a l a p p l i c a t i o n in n u c l e a r i n d u s t r y [ 1 - 4 ] . Z r - 2 . 5 N b is a t w o - p h a s e , h e x a g o n a l p h a s e p l u s c u b i c s e c o n d p h a s e , a l l o y w i t h t y p i c a l a p p l i c a t i o n s a s p r e s s u r e t u b e in p r e s s u r i z e d h e a v y w a t e r r e a c t o r s ( P H W R ) . The
tubes
are
fabricated
through
stages
of T M P
(thermo-mechanical
processing).
T y p i c a l l y h o t e x t r u s i o n is f o l l o w e d b y s t a g e s o f c o l d r e d u c t i o n s ( c o l d p i l g e r i n g a n d / o r tube
drawing)
and
annealing
to maintain
close
dimensional
tolerances
as well
as
microstructural control [1,3. 5].
533
Annealing R e l a t e d M i c r o s t r u c t u r a l D e v e l o p m e n t s in a T w o - P h a s e Zr-2.5 Nb Alloy
In t h e p r e s e n t s t u d y , p i l g e r e d Z r - 2 . 5 N b p r e s s u r e t u b e w a s s u b j e c t e d to s e r i e s o f annealing treatments - both recovery and recrystallization/grain growth. T h e objective w a s to c o m p r e h e n d o v e r a l l m i c r o s t r u c t u r a l c h a n g e s w i t h a n n e a l i n g .
EXPERIMENTAL
METHODS
S e m i - f a b r i c a t e d , after first p i l g e r i n g [ 6 ] , Z r - 2 . 5 N b a l l o y w a s s u b j e c t e d to a s e r i e s o f r e c o v e r y a n d r e c r y s t a l l i z a t i o n t r e a t m e n t s in a salt b a t h r e s p e c t i v e l y at 4 0 0 ° C
and
700°C. The microstructural changes were monitored through both bulk and microtextural measurements. T h e former w a s taken through X-ray O D F s from a Panalytical
MRD
s y s t e m , w h i l e Fei q u a n t a - 2 0 0 h v w i t h T S L - O I M p a c k a g e w a s u s e d for E B S D ( e l e c t r o n b a c k s c a t t e r e d d i f f r a c t i o n s t u d i e s . A d d i t i o n a l c h a r a c t e r i z a t i o n s , for e x t e n t o f r e c o v e r y & recrystallization,
were
brought
in t h r o u g h
micro-hardness
data
and
through
X-ray
e s t i m a t e d l a t t i c e s t r a i n [ 7 , 8] v a l u e s .
RESULTS & DISCUSSION F i g u r e 1 s h o w s t h e effects o f r e c o v e r y ( 4 0 0 ° C a n n e a l i n g ) a n d
recrystallization
( 7 0 0 ° C a n n e a l i n g ) o n h a r d n e s s a n d lattice s t r a i n ( a s e s t i m a t e d by X - r a y d i f f r a c t i o n , u s i n g s t a n d a r d p r o c e d u r e [7, 8 ] ) . T h e m i c r o s t r u c t u r a l d e v e l o p m e n t s d u r i n g r e c r y s t a l l i z a t i o n a r e s h o w t h r o u g h t h e E B S D i m a g e s o f f i g u r e 2. A s s h o w n in f i g u r e 2 , a n d s h o w n in further d e t a i l s in figure 3 , t h e i m p o r t a n t f e a t u r e o f m i c r o s t r u c t u r a l d e v e l o p m e n t is c o a r s e n i n g o f 2
n d
534
p h a s e a n d c o n c u r r e n t g r a i n g r o w t h for t h e p r i m a r y p h a s e .
·
Materials Processing a n d Texture
A n n e a l i n g R e l a t e d Microstructural D e v e l o p m e n t s in a T w o - P h a s e Z r - 2 . 5 N b Alloy
(a)
(b)
(c)
F i g u r e 2. E B S D i m a g e s o f Z r - 2 . 5 N b after 7 0 0 ° C a n n e a l i n g for (a) 3 0 s ( b ) 4 h (c) 1 4 d a y s . T h e hexagonal and the cubic phases are m a r k e d as grey and white respectively.
Materials Processing and Texture
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535
A n n e a l i n g R e l a t e d M i c r o s t r u c t u r a l D e v e l o p m e n t s in a T w o - P h a s e Z r - 2 . 5 N b Alloy
(a)
(b)
Figure 3 . E B S D e s t i m a t e d g r a i n s i z e s for ( a ) h e x a g o n a l p r i m a r y p h a s e a n d ( b ) c u b i c s e c o n d p h a s e after 7 0 0 ° C a n n e a l i n g for different t i m e s .
(a)
(b)
(c)
(d)
Figure 4. 3-d X - r a y O D F s o f the p r i m a r y h e x a g o n a l p h a s e ( a ) in a s - p i l g e r e d stale a n d after a n n e a l i n g at 7 0 0 ° C for ( b ) 3 0 s , ( c ) 4 h a n d ( d ) 14 d a y s .
(a)
(b)
(c)
Figure 5 . E B S D e s t i m a t e d p h a s e b o u n d a r y n a t u r e , b e t w e e n p r i m a r y h e x a g o n a l p h a s e a n d s e c o n d a r y c u b i c p h a s e after 7 0 0 ° C a n n e a l i n g for ( a ) 3 0 s , ( b ) 4 h a n d ( c ) 14 d a y s .
536
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A n n e a l i n g R e l a t e d Microstructural D e v e l o p m e n t s in a T w o - P h a s e Z r - 2 . 5 N b Alloy
F i g u r e 6. C o m p a r i s o n b e t w e e n g r o w i n g a n d n o n - g r o w i n g b o u n d a r i e s , (a) s c h e m a t i c s h o w i n g i d e n t i f i c a t i o n o f r e s p e c t i v e b o u n d a r i e s a n d ( b ) c o i n c i d e n c e for s u c h b o u n d a r i e s . In ( b ) a n g u l a r p r o x i m i t y t o c o i n c i d e n c e v s . p r o b a b i l i t y o f c o i n c i d e n c e h a s b e e n p l o t t e d for g r o w i n g a n d n o n - g r o w i n g b o u n d a r i e s .
A s s h o w n in figure 4 . c h a n g e s in b u l k c r y s t a l l o g r a p h i c t e x t u r e w a s o b s e r v e d only d u r i n g g r a i n g r o w t h . A l s o a s s o c i a t e d , a s in figure 5, w i t h g r a i n g r o w t h ( o f p r i m a r y p h a s e ) and coarsening (of 2
n d
p h a s e ) w e r e s i g n i f i c a n t c h a n g e s in p h a s e b o u n d a r y n a t u r e . T h e
E B S D scans could clearly distinguish between growing and non-growing boundaries
-
s e e figure 6 a . It w a s c o n s i d e r e d n e c e s s a r y t o b r i n g o u t p o s s i b l e r a t i o n a l e b e h i n d s u c h g r o w i n g a n d n o n - g r o w i n g b o u n d a r i e s . In an e a r l i e r s t u d y . C S L ( c o i n c i d e n t site lattice) m o d e l w a s e x t e n d e d to h e x a g o n a l s y s t e m [ 9 ] . In t h i s a p p r o a c h , a l s o n o w part o f s t a n d a r d TSL-EBSD
package,
hexagonal
unit
cells
are
'superimposed
on
each
other
after
a p p r o p r i a t e r o t a t i o n s . N u m b e r s o f c o i n c i d e n t lattice p o i n t s , w i t h i n a specified t o l e r a n c e , a r e t h e n e s t i m a t e d . In t h e p r e s e n t s t u d y , t h i s a p p r o a c h w a s further e x t e n d e d to c o v e r different u n i t c e l l s , e.g. h e x a g o n a l a n d c u b i c , a n d t h e n t o e s t i m a t e a n g u l a r p r o x i m i t y t o c o i n c i d e n c e v s . p r o b a b i l i t y o f c o i n c i d e n c e . A c o m b i n a t i o n w a s c o n s i d e r e d n e c e s s a r y , as c o i n c i d e n c e ( b e t w e e n h e x a g o n a l a n d c u b i c unit c e l l s ) at l o w e r a n g u l a r p r o x i m i t y is r a r e a n d t h e s o - c a l l e d p r o b a b i l i t y o f c o i n c i d e n c e (a statistical f u n c t i o n i d e n t i f y i n g n u m b e r s o f coincident
sites) d e p e n d s
on the angular tolerance. A s
in f i g u r e
6 b . the
growing
boundaries clearly have a higher degree of coincidence than the non-growing boundaries.
Materials Processing and Texture
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537
A n n e a l i n g R e l a t e d M i c r o s t r u c t u r a l D e v e l o p m e n t s in a T w o - P h a s e Z r - 2 . 5 N b Alloy
CONCLUSIONS: •
Significant changes bulk crystallographic texture w a s observed only during grain g r o w t h o f t h e p r i m a r y h e x a g o n a l p h a s e . G r a i n g r o w t h in t h e p r i m a r y p h a s e w a s c o n c u r r e n t w i t h c o a r s e n i n g o f s e c o n d a r y c u b i c p h a s e . T h e c o a r s e n i n g led to presence of 2
•
EBSD
could
n d
p h a s e o n l y in t h e t r i - j u n c t i o n s o f t h e p r i m a r y p h a s e g r a i n s . clearly
distinguish
between
growing
and
non-growing
phase
b o u n d a r i e s . E x t e n d i n g t h e C S L t h e o r y to p h a s e b o u n d a r i e s , it w a s p o s s i b l e t o b r i n g o u t that t h e s o - c a l l e d g r o w i n g b o u n d a r i e s h a d h i g h e r d e g r e e o f c o i n c i d e n c e than the non-growing boundaries.
ACKNOWLEDGEMENT Support from B R N S (Board of Research on Nuclear Science) and from D S T (Department of Science & Technology) are acknowledged.
REFERENCES: [1]
V e r l i n d e n B , S a m a j d a r I, D r i v e r J, D o h e r t y R. T h e r m o m e c h a n i c a l p r o c e s s i n g o f
metallic materials, p h e n o m e n a , Elsevier Scinece limited, U K , 2 0 0 7 [2]
Charles
OS.
Nuclear
reactor
materials.
In
Reading,
MA;
Addison-Wesley:
1967.p.l30 [31
Maussner
G,
Ortlieb
E.
Weidinger
H-G.
Materials
for
nuclear
reactor
core
a p p l i c a t i o n s . In: L o n d o n : B N E S ; 1 9 8 7 . p . 4 9 [4] S r i v a s t a v a D . G, D e y , G K , B a n e r j e e S. T e c h n i c a l R e p o r t . B h a b h a A t o m i c R e s e a r c h Centre, B o m b a y ; 1992. B A R C / 1 9 9 2 / 1 / 0 1 1 [5] S r i v a s t a v a D . G , D e y . G K , B a n e r j e e S. M t e a l l l M a t e r T r a n s 1 9 9 5 ; 2 6 A : 2 7 0 7 [6] K i r a n k u m a r M . S a m a j d a r
I, V e n k a t r a m a n i N , D e y G K . T e w a r i R, S r i v a s t a v a D ,
B a n a r j e e S. A c t a M a t e r i a l i a ; 2 0 0 3 : 5 1 ; p . 6 2 5 [7] C u l l i t y Β D , E l e m e n t s fo X - r a y diffraction
,In:2"
d
ed. R e a d i n g , M A :
Addision-
W e s l e y ; 1978 [8] S t i b i t z G R . P h y s . R e v . , 1 9 3 7 : 4 9 ; p . 8 6 2 [9] M a n i K r i s h n a Κ V , S a i n A . S a m a j d a r I, D e y G Κ, S r i v a s t a v a D . N e o g y S. T i w a r i R, B a n e r j e e S, A c t a M a t e r l i a , 2 0 0 6 : 5 4 ; p . 4 6 6 5
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T E X T U R E A S P E C T S O F D E L A Y E D H Y D R I D E C R A C K I N G IN P R O D U C T S FROM Zr-BASED ALLOYS Margarita Isaenkova and Yuriy Perlovich Physical Problems of Materials Science, M o s c o w Engineering Physics Institute, M o s c o w , Russian Federation ABSTRACT Delayed hydride cracking ( D H C ) in products from Zr-based alloys under tensile loading is considered as a texture-dependent p h e n o m e n o n . D H C development is anisotropic due to crystallographically regulated operation of plastic deformation m e c h a n i s m s within zones of stress concentration near m o v i n g cracks. As applied to both plain and notched samples from Z r - l % N b sheet, it w a s shown by X-ray study, that α-Zr crystallites under tensile loading change their initial orientation in different m a n n e r s by m e a n s of slip or twinning depending on the direction of this loading. Features of the plastic deformation zone at the tip of moving crack vary in accordance with operating m e c h a n i s m s . T h e revealed regularities of local reorientation are valid in the case of D H C in channel C A N D U tube from Z r - 2 . 5 % N b alloy as well. The orientation of δ-hydrides, observed near the fracture surface, testifies that they reprecipitate in α-Zr matrix both by its initial texture and after twinning. T h e proposed mechanism of D H C involves the twinning by {10.2} planes within the plastic deformation zone near the crack tip, formation of the distinct boundary between deformed and undeformed regions with the increased gradient of lattice distortion, the intense diffusion of hydrogen to this boundary, the preferential precipitation of stress-oriented hydrides at its favorably positioned sections, and the growth of hydrides both inside and outside the plastic deformation zone till the next step of the crack between boundaries, decorated by hydrides. The k n o w n anisotropy of D H C reflects variation of the capacity of the α - Z r matrix to deform by twinning depending on the direction of tensile loading. INTRODUCTION T h e p h e n o m e n o n of delayed hydride cracking ( D H C ) in products of Zr-alloys is a consequence of stress-oriented reprecipitation of hydrides under loading of the sample at the temperature of hydride formation . Whereas primary hydrides dissolve, new hydride platelets form perpendicular to the tensile axes. Though the study of D H C has already a many-years history, the question of the relative importance of different factors controlling this process, up to n o w w a s not answered unambiguously. T h e purpose of this paper is to consider the problem of D H C in Zr-alloy products from the standpoint of texture analysis. This approach has led us to the conclusion, that the texture controls the D H C development and proves to be the main factor, responsible for its anisotropy. An acceptable mechanism, explaining the D H C anisotropy, can be constructed only using m o d e l s of the texture formation theory and by interpretation o f experimental data in terms of the texture analysis. 1
2
Known data on habit planes of hydride precipitates in Zr a l l o y s are diverse and contradictory - for the δ-hydride planes of prismatic slip {10.0}, twinning planes {10.2}. {11.1} and {11.2}, planes of pyramidal slip {10.1}, planes of basal slip (00.1), planes {10.L} and, in particular, {10.7}were observed as habit planes. Intergranular and interphase boundaries often prove to be preferable sites for hydride formation as well. Both internal and external stresses are responsible for local variations in hydride growth. T h e evidences exist on the local plastic deformation of the matrix near hydride particles . Though the texture influence on the anisotropy of D H C is undeniable, its concrete m e c h a n i s m s w e r e not revealed up to n o w . 3
T h e new idea of the D H C mechanism consists in the different development of plastic deformation at the tip of the moving crack depending on its direction and in resulting variation of local conditions for reprecipitation of hydrides. T h i s idea has originated from X-
539
T e x t u r e A s p e c t s of D e l a y e d H y d r i d e C r a c k i n g in P r o d u c t s f r o m Z r - B a s e d Alloys
ray texture measurements of fracture surfaces, influenced by the w a v e of plastic deformation moving in front of the crack. We had carried out these m e a s u r e m e n t s as applied to the Zrl % N b sheet and to the hydrogen-charged Z r - 2 . 5 % N b channel t u b e . Since D H C intensifies with increase of the tensile stress and finally results in fracture o f the product, consideration of hydride reprecipitation near the tip of crack nucleus, being the typical stress concentrator, is the most expedient, because such regions of the matrix give the greatest input into D H C . T h e proposed mechanism of D H C deals with local reorientation of the crystalline lattice under loading at the tip of the m o v i n g crack and is based on concepts of texture formation theory 4
5
6
R E O R I E N T A T I O N O F α-Zr L A T T I C E W I T H I N P L A S T I C D E F O R M A T I O N Z O N E Plastic deformation of metal materials is always accompanied by rotations of the grain lattice to definite orientations, stable relative to this deformation. Both tension and compression have their o w n stable orientations, which, as a rule, are characterized by coincidence of the loading direction with a high-symmetric crystallographic axis. In the case of tensile loading these processes of grain reorientation and resulting texture c h a n g e s reproduce locally within the plastic deformation zone at the tip of a m o v i n g crack, where the stress concentration is sufficient for activation of deformation m e c h a n i s m s . 7
It w a s established as applied to rolled metal materials by the method of dividing g r i d s , that the form and the size of the plastic deformation zone at the notch tip at the m o m e n t of crack initiation d e p e n d s on the crack orientation relative to the direction of previous rolling: under RD-tension the plastic deformation zone near the notch, parallel to transverse direction (TD), is butterfly-shaped and has up to 2 m m in thickness, w h e r e a s under TD-tension this zone near the notch, parallel to R D , is flat and narrower by several times (Fig. 1). T h e fracture occurs, when the plasticity resource near the crack tip proves to be exhausted by local attainment o f the limiting deformation degree. It is a c c o m p a n i e d necessarily by local formation of the texture of tension with the axis perpendicular to the fracture surface. As for rolled Zr-based alloys, their anisotropic behavior under tensile loading is conditioned by different activated deformation w mechanisms.
" \
By X-ray study of these processes the investigation of the small zone at the crack tip w a s replaced by the analysis of texture changes in samples of t w o types from Z r - l % N b a l l o y : ( I ) plain samples, subjected to tensile testing along RD and I D up to different deformation degrees; (2) notched samples, fractured by passing of the crack along I D and R D . S a m p l e s (1) were assumed to be deformed h o m o g e n e o u s l y and provided correlation of observed texture c h a n g e s with the degree of tensile deformation. T h e surface of fractured samples (2). which w e r e cut out from the same sheet with plain samples (1). was formed by the w a v e of plastic deformation, passed through s a m p l e ' s cross-section. This w a v e caused local structure and texture c h a n g e s in the layer with a thickness equal to that of the plastic deformation z o n e . Fig. 1. Shapes of the plastic deformation zone at the crack tip in sheet samples by tensile loading along RD and I D .
4
Automated X-ray diffractometers D R O N - 3 and S I E M E N S D 5 0 0 0 (radiation Cu K„) were used by texture m e a s u r e m e n t s . For the study of fracture surfaces, the composite samples were assembled from macroscopically flat fragments of these surfaces with a total area of several square millimeters. Then Inverse Pole Figures (IPF), constructed by integral intensities of X-ray lines, reflect texture features of fracture surfaces, including the layer of a micro-relief, formed as a result of fracture. By m e a n s of X-ray layer-by-layer analysis, the
540
Materials Processing and Texture
T e x t u r e A s p e c t s of D e l a y e d H y d r i d e C r a c k i n g in P r o d u c t s f r o m Z r - B a s e d Alloys
distribution of deformation degree within the layer, adjacent to the fracture surface, was determined using the comparison with data, obtained for plain samples. Results of texture m e a s u r e m e n t s of samples of both types are presented in Fig. 2 as inverse pole figures (IPF) for cross-sections and fracture surfaces, perpendicular to the axis of tensile loading ( R D - and TD-seclions in cases of tension along RD and I'D, respectively). T h e initial textures of R D - and TD-sections of the studied sheet, typical for rolled metal materials with H C P crystalline lattice, differ sharply. IPF for RD-section (see Fig. 2-a) contains the single texture m a x i m u m with a peak between poles (10.0) and (11.0), testifying to the partial recrystallization of the initial sheet. IPF for TD-section also contains only one texture m a x i m u m , but it is severely scattered and shifted to the center of stereographic sector (Fig. 2-d). A s for textures of the fracture surfaces, it is seen in Fig. 2-c,f that to the first approximation they are similar and fall in the same type with the initial texture of RD-section, though being significantly more perfect. An essential grain reorientation near the fracture surface 0001 is especially evident in TD-sample, w h e r e it is connected with principle changes of the initial texture. The similarity of textures at both fracture surfaces confirms the above assertion, according to which grains in the sample under tensile deformation up to sufficiently high deformation degree eventually approach to some final stable orientation.
ιοίο
T h e following technique was used for determination of the 6001 deformation degree at fracture surfaces of RD- and I D - s a m p l e s . The tension texture of α-Zr is characterized by predominant alignment of crystallographic axes <10.0> along R D . Hence, as a result of tension the texture m a x i m u m in IPF for the section, perpendicular to the tension axis, would shift to pole (10.0). while its scattering would decrease. This prediction is supported MO by textures of stretched plain samples, Fig. 2. Texture changes in R D - (a, b, c) and T D measured in the corresponding sections (d,e,f) of the sheet Z r - l % N b alloy as a sections. Thus, in the IPF for the R D result of tensile testing of plain samples (b, e) and section the peak of texture maximum, fracture of notched samples (c, f): a - RD-section, initial state; d - TD-section, initial state; which initially was offset from pole (10.0) by 15°, due to RD-tension by e-TD-tension by 10%; b RD- tension by 2 2 % ; 2 2 % neared this pole by 7°. At the c - fracture surface f- fracture surface same time, an angular width of the ofRD-sample: of TD-sample. texture m a x i m u m at a half of its height decreased from 45° d o w n to 34° (Fig. 2-b). In the case of TD-tension of plain samples, al the very first deformation stage the texture m a x i m u m m o v e s abruptly to the external boundary of the stereographic sector and then, more slowly, m o v e s to pole (10.0). After tension of I D - s a m p l e only by 1 0 % the peak of texture m a x i m u m in the IPF approached pole (10.0) at an angular distance o f 5 ° a l r e a d y , while its half-width decreased from 64° down to 6
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T e x t u r e A s p e c t s of D e l a y e d H y d r i d e C r a c k i n g in P r o d u c t s f r o m Z r - B a s e d Alloys
30° (Fig. 2-c), testifying to the higher reorientation rate o f α-Zr grains, compared to the R D sample. On the basis of texture c h a n g e s in the extended plain samples the degrees of plastic deformation near the fracture surface of notched s a m p l e s w e r e determined. T h e ratio q = p(IO.O)/p(l 1.0) was used as a characteristic parameter of the IPF, increasing with the deformation degree. Using the curve q(s), constructed for extended plain samples, the deformation degrees near the fracture surface were estimated by the procedure, demonstrated in Fig. 3. T h e hatched strips show, h o w the sought deformation degrees ε w e r e found by values q, determined from the fracture surfaces. Obtained values characterize the average deformation degrees for the layer up to 20 p m in thickness, contributing to X-ray diffraction by IPF measurement. N e a r the fracture surface in the R D - s a m p l e the material experiences the tensile deformation by 15-19%, while in the T D sample - by 3 - 5 % . T h e real local deformation degrees for the layer of several microns in thickness, adjoining immediately to the fracture surface, are essentially higher, than the a b o v e indicated values. T h e diagram w a s calculated as applied to aZr for determination of the plastic deformation m e c h a n i s m s , activated under rolling depending on the orientation of basal axes relative to the loading scheme (Fig. 4 ) . When taking into account that, at the first approximation, rolling is Fig. 3. Estimation of the deformation the combination of mutually perpendicular degree at the fracture surface by use of the texture parameter q = p( 10.0)/p( 11.0): extension and compression, it b e c o m e s evident, that RD-tension realizes mainly prismatic slip, • - RD-tension; ο - TD-tension. whereas TD-tension causes at first intense twinning, resulting in the j u m p - l i k e shift of the initial texture m a x i m u m , and only then within the twinned area that slip b e c o m e s possible. Reorientation of grains at the tip of a m o v i n g crack follows from their regular tendency to attain an RD orientation, which would be stable relative to the predominant tensile deformation. In α-Zr stability of the final orientation of the tension axis is provided by mutually balanced operation of t w o prismatic slip systems. In the case of notched samples deformation begins first of all in the region, adjoining immediately to the notch tip. where due to stress concentration the maximal stress is attained. Only when the strain TD hardening of material within this region begins to Fig. 4. T h e plastic deformation increase, deformation extends gradually to the more m e c h a n i s m s , activated in α-Zr distant layers. T h e less the critical shear stress for the under rolling d e p e n d i n g on the active deformation m e c h a n i s m , the further from the orientation of basal a x e s relative notch tip extends the zone of localized plastic to the loading s c h e m e . deformation. The layer-by-layer distribution of 8
8
deformation degree near the notch tip in the R D sample depends on the strain hardening of α-Zr grains by prismatic slip. By tensile testing of the notched T D - s a m p l e the loading increases until activation of twinning within s o m e thin layer near the notch tip. In c o n s e q u e n c e of the resulting abrupt reorientation and the corresponding increase of the shear stress in prismatic slip systems, the twinned grains obtain
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at once the ability for subsequent deformation by m e a n s of prismatic slip. Maintenance of the prismatic slip requires less loading, than that achieved before, and this circumstance prevents extending of twinning for further layers. Only in the event that the strain hardening due to prismatic slip in the first layer e x c e e d s the difference between critical shear stresses for twinning and prismatic slip, will the following rise of loading reach the level necessary to activate twinning in the next layer. T h u s , extension of the plastic deformation zone from the notch tip inwards the sample takes place in different m a n n e r s in cases of R D - and T D - s a m p l e s : as the deformation degree near the notch tip increases, the plastic deformation zone in the R D - s a m p l e widens continuously, whereas in the T D - s a m p l e it widens discretely, remaining unaltered in the course of some loading stages. While the plastic deformation zone in the R D - s a m p l e does not have a distinct boundary, such a zone in the T D - s a m p l e is restricted by an abrupt border. Outside of this border the twinning was not activated Fig.5. A schematic sketch of and, result, its crossing is accompanied by the j u m p the plastic deformation zone at H k h g e of the local texture (Fig. 5). Therefore the f localized deformation in the RD-sample is the tip of the moving crack for i the cases of R D - (a) and T D thicker than in the T D - s a m p l e . (The denser the hatching tension (b). j pjg. 5 , the higher is the degree of local deformation e
a y e r
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by slip). FRACTURE SURFACE OF HYDROGEN CHARGED CANDU TUBE: TEXTURE AND PRECIPITATION OF HYDRIDES UNDER TENSILE TANGENTIAL LOADING The above-described process of α-Zr grain reorientation within zones of stress concentration under tensile loading takes place not only in sheets, but also in hydrogen charged t u b e s . As applied to channel tubes from Z r - 2 . 5 % N b alloy for C A N D U nuclear reactor the similar experimental evidences of twinning at the fracture surface w e r e obtained. Additionally, for the first time the texture o f hydride phase, localized at the fracture surface, w a s measured. T h e studied fracture surface of tube was obtained by testing a cantilever specimen at 250°C during 12 hours after its charging with hydrogen up to the content of 50 ppm. Beforehand the tube was annealed at 400°C / 24 hours for removal of residual stress. The sample w a s cut out of the tube perpendicular to its L-axis and the notch, coinciding initially with the tangential section (Tscction), was 3 9
inflicted on internal surface. Fig. 6. Diffraction spectra for C A N D U tube: a - distanced T-section: b - f r a c t u r e surface.
t e s t i n g
t h e
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the By l e
P experienced tension along 1 -axis. The
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fracture surface of this sample was 4x5 m m in size and consisted of t w o regions, showing, as a first approximation, brittle and viscous types of fracture. T h e region of brittle fracture comprised the majority of the fracture surface and w a s relatively flat and parallel to Tsection. The tube had a texture of the tangential type, being characterized by the alignment of basal normals in parallel with T-axis. Beside the fracture surface, the section, parallel to this surface and distanced from it by 20 m m , w a s examined also in order to c o m p a r e features of surface and inner layers. It w a s supposed that observed differences in structure and texture features of these layers are conditioned by the plastic deformation, a c c o m p a n y i n g the crack movement. The main experimental result of this w o r k consists of the direct X-ray observations of hydrides at the fracture surface of the sample that experienced D H C . In Fig. 6 the diffraction spectra, obtained by m e a s u r e m e n t s of sample 1, are presented: (a) - for the T-section. distanced from the fracture surface by 20 m m ; (b) - for the fracture surface. T h e diffraction spectrum of the fracture surface contains, besides the X-ray lines o f t t - p h a s e , two lines of δhydride. having the F C C crystalline lattice. - (111) and (200), w h e r e a s these lines are absent in the spectrum of the distanced T-scction. Hence, the content of hydride precipitates at the fracture surface is much higher, than in the inner T-section o f the same sample. This result gives a reason for the following interconnected assertions: (1) the fracture occurs in the plane of preferential arrangement of hydride platelets; (2) formation of new hydride precipitates with the stress-determined orientation proceeds near the fracture surface. I lydride particles at the fracture surface are responsible for the brittle type of fracture. 1 lowever, in this case the plastic deformation zone is present at the tip of the moving crack, as in the case of viscous fracture. It is seen from reorientation of α-Zr matrix at the fracture surface, parallel to T-scction (Fig. 7): the density of basal axes near T-dircction decreases (compare Fig. 7-a and 7c). w h e r e a s the density of axes <10.1> increases (compare Fig. 7-b and Ια). This change in orientation of the α-Zr matrix can be connected with twinning by planes {10.2} . 5
T h e presence of two X-ray lines of δ-hydridc in the diffraction spectrum of fracture surface testifies, that there are hydrides of two orientations
Fig. 7. P F ( 0 0 0 l ) ( a , c ) and PF{ 10.1} (b,d) f o r a - Z r o f C A N DU tube: a.b - distanced T-section; c,d - fracture surface. PF were constructed in units of measured intensity. Axis Τ is in the center of PF. Angular radius of PF - 70°.
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crystallographic planes {111} or {001} of hydride particles are parallel to the surface. A v o l u m e fraction of hydrides with the latter orientation is less, as it is evident from the essentially lower intensity of the X-ray line (20()) . fi
Two orientations of hydrides correspond to their different fractions.
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precipitating within (a) the α-matrix with the initial texture and (b) the plastic deformation zone, where this texture w a s changed as a result of twinning. T h e X-ray line ( l l l ) a is a reflection from fraction (a), whereas the X-ray line (200)g - from fraction (b). The following orientation relationships between δ-hydride and α-Zr are k n o w n : (111)δ ! Ι ( 0 0 0 1 ) and (11 l)s 1 1 ( 1 0 . 1 ) . According to these orientation relationships, hydrides of fraction (a) satisfy both the initial orientation of α-Zr matrix and its orientation after twinning. As for hydrides of fraction (b), the first orientation relationship can be valid for them, if they form within twinned α - Z r . 10
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PRECIPITATION OF STRESS-ORIENTED HYDRIDES: MAIN STAGES A s applied to C A N D U pressure tubes, characterized by the preferred alignment of basal axes along the T-direction, the proposed model of D H C corresponds to the following scheme: (1) T h e tensile T-loading of the C A N D U tube causes the local plastic deformation by {10.2} twinning within the zone of stress concentration near the tip of the crack or some phase precipitate. (2) T h e twinning and the a c c o m p a n y i n g reorientation of the α-matrix take place only within the zone where a critical shear stress in {10.2} planes is attained. Therefore, the twinned volume is surrounded by the distinct boundary, which coincides with an isoline of the critical shear stress for twinning and separates regions of sharply differing textures. (3) T h e new orientation of twinned crystallites of α - Z r associates with a significant increase of shear stresses in prismatic planes, so that their further plastic deformation is realized by means o f prismatic slip, resulting in some additional rotation of the crystalline lattice. (4) N e a r stress concentrators the substructure is characterized with the highest gradients of texture and, as a result, of lattice distortion. T h e latter is conditioned by the close neighbourhood of regions, where the plastic deformation was rather significant (inside the twinned volume) or absent (outside this volume). ( 5 ) T h e high gradient of lattice distortion under the tensile loading results in intensifying of the diffusion of hydrogen to the plastic deformation zone from the surrounding matrix, so that the final content of hydrides within this zone proves to be significantly increased as compared with neighbouring regions. (6) T h e boundary of the plastic deformation zone, separating regions with sharply different textures, is a preferable place for precipitation of hydrides, as any other high-angle intergranular boundary. Under conditions of tensile loading this boundary shows the preferential precipitation of hydrides only in the case of its appropriate orientation relative to the loading direction. Obviously, stress-oriented hydrides preferably precipitate at those segments of the boundary, which are perpendicular to the loading direction. (7) T h e habit planes of hydride platelets are determined by the most probable configuration of the plastic deformation zone under T-loading. ( 8 ) T h e discontinuous character of hydride images, taken from the T-section of C A N D U tubes, reflects the process of hydride formation by successive positions of the crack tip and the plastic deformation z o n e . Since the boundary of this zone is decorated by hydride precipitates, the j u m p - w i s e m o v e m e n t of the crack is seen from the series of hydride segments. (9) Processes of cracking and hydride reprecipitation are interconnected, that is each of both is influenced by other. Precipitation of hydrides and their growth from boundaries of the plastic deformation zone create a narrow passage for m o v e m e n t of the crack, which becomes brittle because of the increased hydrogen influx. As a result of hydride formation, the plastic deformation zone can transform into a polycrystalline hydride platelet, also containing remains of the α-matrix. T h e brittle crack goes through this platelet in a j u m p - w i s e manner and stops before the next part of the undeformed α-matrix, w h e r e the crack tip proves to be a stress concentrator, so that the above cycle repeats, including twinning, diffusion of hydrogen and formation of hydrides. Materials Processing and Texture
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CONCLUSION The proposed model of" D H C in products from Zr-based alloys under tensile loading explains the d e p e n d e n c e of this p h e n o m e n o n on the texture of the product and on the loading direction. T h e model is based on experimental data, obtained by X-ray study of fracture surfaces, resulting from passing of the crack with the a c c o m p a n y i n g plastic deformation zone through the body of the sample. T h e texture of rolled products is stable under tension along R D and unstable under tension along T D . In the case of rolled Zr-based alloys, tensile loading along R D activates prismatic slip at crack tips as well as at other stress concentrators and does not cause essential texture c h a n g e s in the plastic deformation z o n e . In contrast, tensile loading along T D at first activates {10.2} twinning, resulting in the abrupt local reorientation of the α-Zr crystalline lattice. Therefore in the case of TD-loading the plastic deformation z o n e at the crack tip sharply differs in its texture from the neighboring matrix and proves to be separated from this matrix by the distinct boundary. When the s a m e processes of local deformation d e v e l o p under similar conditions in the hydrogen charged C A N D U tubes from Z r - 2 . 5 % N b alloy, reprecipitation of hydrides by D H C is controlled by texture changes in the α-Zr matrix. Since basal axes in C A N D U tubes have the tangential orientation, the twinning in the plastic deformation zone under TD-loading is especially intense, so that the gradient of lattice distortion at the boundary of this zone is increased and therefore the diffusion of hydrogen to the boundary is intensified. Stressoriented hydrides precipitate preferentially at appropriately positioned sections of the boundary and grow both inside and outside the plastic deformation z o n e . T h e crack propagates in a j u m p - w i s e m a n n e r between the z o n e ' s boundaries, decorated by hydrides, and stops before the next part of the undeformed α-matrix. REFERENCES 'D.E. Douglas, Metallurgy of Zirconium (Intern. Atomic Energy Agency, Vienna), (1971). C . E . Ells -./. Nuct. Mater., 2 8 , 129 (1968). J . B . Bai, N. Ji, D. Gilbon et a l . - M e r . Mat. Trans. A, 25A, 1 1 9 9 - 1 2 0 8 ( 1 9 9 4 ) . Y u . Perlovich, M. Isaenkova and Goltzev V - JOURNAL DE PHYSIQUE IV, C o l l o q u e C 6 . supplement au Journal de Physique 111, 6. 335-342 (1996). Y u . Perlovich, M. Isaenkova, Y.S. Kim and S.S. Kim - Problems of Atomic Science and Technology, (1999) Ν 2; Series: " P h y s i c s of Irradiation D a m a g e and Irradiation Metal Science'" (77). 58-70 (in Russian). G . Wasserman and I. Greven, Texturen metallischer Werkstoffe (Springer Verlag, Berlin Gottingen - Heidelberg), (1962). V . Goltcev. A. Zelenskiy, O. Kudryavzev and Yu. Matvienko, T h e zones of localized plastic deformation in preliminary deformed sheet plastic materials. - In: Investigations of strength of materials and constructions of nuclear technique (Energoatomizdat, M o s c o w ) , pp.68-73 (1984) (in Russian). I . V . Matcegorin, A.A. Rusakov and A.I. Evstyukhin, Analysis of the texture formation mechanism in α-Zr by use of c o m p u t e r modelling. - In: Metallurgy and Metal Science of Pure Metals, Ν 14, (Atomizdat, M o s c o w ) , pp.39-52 (1980) (in Russian).
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Y . S . Kim, Yu. Perlovich. M. Isaenkova et al. - J. Nucl. Mater., 291 (3), 292-302 (2001). J . S . Bradbrook, G . W . Lorimer and N. Ridley - J. Nucl. Mater., 42, 142-160 (1972).
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E F F E C T O F T W I N N I N G ON T H E STRAIN H A R D E N I N G B E H A V I O R S O F T W O ALLOYS D E F O R M E D ALONG FOUR DIFFERENT STRAIN PATHS
Mg
L. Jiang and J.J. Jonas Department of Materials Engineering, McGill University. Montreal, Q C , H3A 2 B 2 , Canada ABSTRACT T h e strain hardening behaviors of two Mg-based (+A1. M n , Zn) alloys were investigated under conditions where twinning plays an important role. The results indicate that the types of twins that form (as well as their volume fractions) depend on the initial texture and the particular strain path employed. The different twinning behaviors are s h o w n to be responsible for the sharply contrasting strain hardening characteristics of the experimental flow curves. Contraction and double twinning generally generate net softening effects. W h e n such twins interact with extension twins, there is s o m e w h a t less softening. O n the other hand, the sole occurrence of extension twinning generally introduces a net hardening effect. In this case, the extent of hardening depends not only on the volume fraction of extension twins, but also on the surface area per unit volume of the twin boundaries and the distribution of the twins. INTRODUCTION Strain hardening is generally used to evaluate the flow localization resistance of a material. The former is intrinsically coupled with other aspects of plastic deformation, such as twinning, which plays an important role during the deformation of M g [1-6] and other metals [79], On the one hand, the twin boundaries can act as barriers to dislocation motion, as do grain boundaries, leading to an increase in the work hardening rate. In addition, they transform glissile dislocations into sessile dislocations within the twin interiors and hence contribute to strengthening in this way [7]. On the other hand, they a c c o m m o d a t e strain along the c-axis, which can give rise to a decrease in the work hardening rate. Furthermore, the lattice rotation introduced by twinning can enhance or reduce the rate of work hardening, depending on the type of twin formed [4-6]. In m a g n e s i u m alloys, two types of twins are frequently reported: ( 10-12}<10-1-1> extension and {10-11 } < 1 0 - l - 2 > contraction twins [1-6]. Extension twins are formed when there is an extension strain component parallel to the c-axis, while contraction twins are activated when there is a contraction strain component parallel to this axis. In addition to primary twinning, secondary twinning can take place within the reoriented primary twins. This is k n o w n as double twinning. Generally, contraction twins form first, after which {10-12} extension twins are propagated within the original contraction twins. {10-11} twinning and {10-11)-{10-12} twinning reorient the basal planes by 56° and 38°, respectively [1]. This means that basal planes originally unfavorably oriented for slip can be reoriented to more favorable orientations, which in turn leads to softening. {10-12} twinning reorients the basal planes by 86° [1]. Therefore, grains that are oriented unfavorably for slip still remain in unfavorable orientations after (10-12} twinning. It is well k n o w n that twinning has an effect on the rate of work hardening in M g alloys [1-6]. A better understanding of the effect of twinning on the strain hardening behaviour is therefore of considerable importance with regard to improving the accuracy of crystal plasticity
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calculations pertaining to intermediate temperatures, a range in which twinning plays such an influential role. EXPERIMENTAL METHOD The samples were taken from tubes extruded using porthole dies by T i m m i n c o Metals of Aurora, Ontario, Canada; they had nominal outer diameters of 7 0 m m and wall thicknesses of 4 m m . The chemical compositions of the t w o materials are presented in Table 1. The average grain sizes of the A M 30 and AZ31 materials were determined to be 3 9 p m and 107pm. Table 1. Chemical compositions of the A M 3 0 and A Z 3 lalloy tubes (in w t . % ) . Al Zn Mn Fe Ni Cu AM30 3.4 0.16 0.33 0.0026 0.006 0.008 AZ31 3.1 1.05 0.54 0.0035 0.007 0.008 The textures of the as-received A M 3 0 and AZ31 tubes are presented in Fig. 1. The initial grain orientations can be divided into two groups: one with their c-axes approximately parallel to the radial direction (called the R D or (10-10)<0001> c o m p o n e n t ) and the other with their c-axes approximately parallel to the tangential direction (called the T D or (11-20)<10-10> component). In the A M 3 0 , the two c o m p o n e n t s have similar volume fractions: 4 8 % (on average) for the T D and 3 9 % (on average) for the R D . In the A Z 3 1 , the intensity of the T D c o m p o n e n t is quite strong (65%) while that of the R D is relatively weak (15%) [10].
Figure 1. Textures of the as-extruded (a) A M 3 0 and (b) AZ31 tubes; ED=extrusion direction, RD=radial direction. Three different testing methods (uniaxial tension, uniaxial compression and ring hoop tension testing) were e m p l o y e d to study the influence of strain path. By taking advantage of the initial textures and the characteristics of twin reorientation, the present experimental arrangement provided a unique opportunity to investigate the combined effect of initial texture and strain path. The uniaxial tension and compression specimens were machined from the as-received tube walls with their axes aligned along the extrusion direction. Ring hoop tension testing ( R H T T ) specimens are narrow circumferential slices taken from as-received tubes. T w o initial strain rates. 0.001 and 0.1s"'. were used during testing. Since the twinning behavior w a s of particular interest in this study, most of the tests were conducted at ambient and moderate temperatures (at intervals of 50°C in the temperature range from 100 to 2 0 0 ° C ) and at a strain rate of 0.1/s. Further details of the experimental method can be found in ref. [10-12]. The effect of contraction and double twins on the subsequent compression behavior w a s examined by carrying out a tensile prestrain. Tensile samples were first prestrained to true strains
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of 0.15 at ambient temperature and a rate of 0.1/s. After 1 5 % extension, a volume fraction of approximately 2 4 % of the A M 3 0 and 5 1 % of the A Z 3 1 underwent contraction and double twinning [10]. Then, compression samples were machined from the gage sections of the prestrained tensile samples along the tension direction. These prestrained samples were compressed at a strain rate of 0.1s" and at ambient temperature as well as at 200°C to allow extension twins to be formed. The introduction of contraction twins during tensile prestraining and of extension twins during subsequent compression occurred because most of the basal plane normals were perpendicular to the t w o loading directions in the present material [10]. 1
RESULTS AND DISCUSSION When specimens of the present type are subjected to uniaxial tension, both the RD and T D grains experience transverse contraction and therefore undergo contraction and double twinning. By contrast, extension twins are formed in both texture components when a specimen undergoes uniaxial compression. In the case of R H T T . the T D component grains are extended circumferentially (i.e. along the c-axis) and therefore undergo extension twinning. By contrast, the R D c o m p o n e n t grains experience contraction along the c-axis (because of transverse contraction) and therefore undergo contraction and double twinning. Since the C R S S value for {10-12} extension twinning is much lower than that for {10-11} contraction twinning, during R H T T testing, {10-12} extension twinning will take place first in the T D component grains. The RD c o m p o n e n t grains will not experience contraction and double twinning until the local stress reaches the relatively high C R S S value for contraction twinning. M o r e detailed descriptions of the types of twins formed during straining along the different strain paths can be found in ref. [10]. Examples of the true stress-true strain curves corresponding to deformation along different strain paths are presented in Fig. 2. These were determined at ambient temperature and 200°C. The uniaxial tension curve exhibits the normal concave-down appearance, while the compression curve displays an unusual concave-upwards shape. This is caused by the occurrence of {10-12} extension twinning during compression [12]. The flow stresses in compression and R H T T were generally lower than those in tension at strains below 10%; however, they increased quickly on further deformation. {10-12} extension twinning dominated in the compression and R H T T tests. This indicates that twinning-induced hardening is much m o r e significant in the latter tests. The θ-ε curves pertaining to the tensile σ-ε curves of Fig. 2 are displayed in Fig. 3. The small increases early on are associated with yielding. After that, the strain hardening rates for both the AZ31 and A M 3 0 decreased with increasing strain as usual. Although the initial AZ31 strain hardening rate was lower than that of the A M 3 0 , it decreased m o r e slowly afterwards. However, at 200°C, the rate of decrease was similar to that of the A M 3 0 . During tensile straining, texture softening resulting from the formation of contraction and double twins reduced the η value [11]. As a result, at RT/O.ls" . the A Z 3 1 , which had a higher volume fraction of twins, displayed the lower strain hardening rate. On the other hand, the higher volume fraction of contraction and double twins also produced a higher proportion of twin boundaries; the latter acted as barriers to dislocation glide. Therefore, with increasing strain, hardening from these additional barriers was enhanced. This explains why, at room temperature, the AZ31 material 1
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had a lower rate of decrease in strain hardening rate than the A M 3 0 material, although the latter had a higher initial strain hardening rate (Fig. 3). Once slip replaced twinning as the dominant deformation mechanism, the differences in the rate of strain hardening and flow stress between the t w o materials disappeared.
Figure 2. Selected uniaxial tension, compression and R H T T flow curves. ' Ή " and " L " refer to tests carried out at strain rates of 0.1s"' and 0.001 s" : " M " and "Z" refer to the A M 3 0 and A Z 3 1 . 1
Some e x a m p l e s of strain hardening rate (θ-c) curves are presented in Fig. 4 for samples deformed in both uniaxial compression and R H T T (Fig. 2). U p to a strain of 0.06, the strain hardening behaviors were similar under all conditions, i.e. the strain hardening rate decreased with increasing strain. After 6 % deformation, the compression samples exhibited an increase in work hardening rate in both materials; here, the volume fractions of extension twins increased with strain at about the same rate. In the R H T T samples, the A Z 3 1 strain hardening rate remained almost constant instead of increasing. M e a n w h i l e , the A M 3 0 strain hardening rate actually decreased during R H T T . In the uniaxial compression tests, the strain hardening rates in both materials changed with strain at about the same rate. This can be linked directly to the similar evolutions of the volume fractions of the extension twins in the two alloys. Such an observation also indicates that the effect of extension twinning is m u c h stronger than those of grain size and Z n addition. Extension twinning also plays an important role in R H T T . However, the R H T T samples contain lower volume fractions of extension twins and the twin boundaries m o r e rapidly coalescence [10]. All these factors contribute to the R H T T strain hardening rates being lower than in the uniaxial compression samples. In addition, the strain hardening rates of the two materials differ
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more than in uniaxial differently in the R H T T component and a higher As a result, the A M 3 0 hardening rate.
compression. This is because the R D and T D components behave samples. There is a lower v o l u m e fraction of extension twins in the T D volume fraction of contraction and double t w i n s in the R D component. material (which has m o r e of the R D c o m p o n e n t ) has a lower strain
Figure 3. Strain hardening rate θ-ε curves for the AZ31 and A M 3 0 deformed in uniaxial tension at room temperature and a strain rate of 0.1s" . 1
Figure 4. Strain hardening rate θ-ε curves for the A Z 3 1 and A M 3 0 tested in compression and in R H T T . The samples were deformed at 200°C.
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The effect of extension twinning on the rate of strain hardening is evident w h e n the A Z 3 1 θ-ε curve determined at 200°C in uniaxial tensile testing (Fig. 3) is compared with that established in R H T T (Fig. 4). In the R H T T test, the strain hardening rate remained almost constant, while in uniaxial tensile testing, the rate decreased continuously with increasing strain. On the other hand, although extension twinning is an important deformation mechanism during uniaxial compression. R H T T and T-C testing at room temperature, the resulting strain hardening rates are not the same, as displayed in Fig. 5. After about 1 0 % deformation, the R H T T θ-ε curve was the only one that kept increasing slightly, while the uniaxial compression and T-C θ-ε curves increased at first but decreased later. After reaching a peak, the strain hardening rate during T-C testing decreased m o r e slowly than during uniaxial compression.
Figure 5. Strain hardening rate θ-ε curves for the AZ31 deformed along different strain paths at room temperature and a strain rate of 0.1 s" . Here, T-C testing refers to compression testing after a tensile prestrain of 1 5 % . 1
Strain hardening analy sis (Fig. 6) shows that the strain hardening rate has the highest rate of increase in the uniaxial compression samples, but it also drops the most quickly. At a strain of about -0.16. the θ value has essentially dropped to zero. By contrast, in the T-C and R H T T tests, the rates of increase and/or decrease in 0 are m u c h m o r e gradual; the θ values only drop to zero at about 0.19 and 0.20, respectively. The physical reasons for these differences are clearly of considerable importance and are taken up in what follows. T h e hardening effect of (10-12] extension twins involves three mechanisms: (1) HallPetch hardening (reduction in the effective slip length), (2) Basinski hardening (the trapping of glissile dislocations inside twins), and (3) texture hardening (crystal reorientation to harder orientations for slip). The first t w o m e c h a n i s m s are related to the surface area of twin boundaries per unit volume while the third is associated with the v o l u m e fraction of twins.
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Figure 6. Strain hardening rate θ-ε curves for the A Z 3 1 deformed along different strain paths at 200°C and a strain rate of 0.1s"'. In the present study, it was observed that the volume fraction of extension twins produced by deforming in uniaxial compression at 200°C/0.001s"' increases dramatically (from 4 9 % to 92%) during the strain interval from -0.08 to -0.15. Nevertheless, it is not accompanied by an increase in the strain hardening rate, see Fig. 6. This is because the increment in the volume fraction is due to twin coarsening. The surface area per unit volume of the twin boundaries does not increase, although the volume fraction increases. A s a result, the corresponding flow curves remain flat instead of exhibiting the concave upward shape associated with twinning, see Fig. 2. Such an observation indicates that texture hardening h a s less influence w h e n the surface area per unit v o l u m e of the twin boundaries decreases. More recently, by studying the strain hardening due to deformation twinning in α-Ti, Salem et al. [9] have pointed out that the Basinski rather than the crystal reorientation m e c h a n i s m is responsible for the higher hardness of the twinned regions (as opposed to the matrix). Their finding suggests that texture hardening may not be the only factor that contributes to the extra hardening w h e n extension twins form in M g alloys. Microstructural examination indicated that neither the R H T T nor the T-C samples contained m o r e than - 8 0 % volume fraction of extension twins [10. 13]. In the R H T T samples, the R D component grains do not develop extension twins; the T D component grains, which are more favorably oriented for extension twinning, gradually undergo extension twinning during the later stages of straining. In the T-C samples, extension twinning does not take place until the available basal slip dislocations have been used u p . In addition, the m o r e than 2 4 % of the volume fraction occupied by contraction and double twins cannot experience extension twinning. The contraction and double twin boundaries introduced during tensile prestraining interact with the extension twins and further enhance the rate of hardening. This indicates that if the occurrence of
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extension twinning and its development can be retarded and the volume fraction can be controlled, it should be possible to increase the ductility. CONCLUSIONS 1 In the previous study, it was found that contraction and double twins decrease the strain hardening coefficient (η-value) when a certain volume fraction ( 2 2 0 % in the present study) is reached. Nevertheless, w h e n the surface area per unit v o l u m e of the twin boundaries increases, the rate of decrease in the strain hardening rate is retarded. In addition, w h e n these twins interact with extension twins, an extra hardening component can be introduced. 2 _ Extension twinning has been s h o w n to introduce a hardening component beyond that due to dislocations. T h e extent of such hardening depends, not only on the volume fraction of extension twins, but also on the surface area per unit volume of the twin boundaries. It is greater when the twins are uniformly distributed. When the twins are wide or the twin boundaries coalesce, the amount of such extra hardening decreases, suggesting that the boundaries are contributing to the hardening. ACKNOWLEDGMENTS This research w a s sponsored by General Motors of Canada and the Natural Sciences and Engineering Research Council of Canada. Discussions with Professors Matthew Barnett and Sean A g n e w are gratefully acknowledged.
REFERENCES Wonsiewicz BC, Backofen WA., Trans. T M S - A I M E 2 3 9 , 1422-1431 (1967). J . Koike, Metallurgical and Materials Transactions A, 3 6 , 1689-1696 (2005). D . W . Brown, S.R. A g n e w , M . A . M . Bourke, T.M. Holden, S.C. Vogel and C.N. T o m é , Materials Science and Engineering A, 3 9 , 1-12 (2005). S . R . A g n e w , M.H. Yoo and C.N. T o m é , Acta Materialia, 4 9 , 4 2 7 7 - 4 2 8 9 (2001). M . R . Barnett, Materials Science and Engineering A, 4 6 4 , 1-7 (2007). "M.R. Barnett. Materials Science and Engineering A, 4 6 4 , 8-16 (2007). Z . S . Basinski, M . S . Szczerba, M. N i e w c z a s , J.D. Embury and S.J. Basinski, Revue de Metallurgie, 9 4 , 1037-1043 (1997). A . Rohatgi, K . S . V e c c h i o and G.T. Gray I I I , Metallurgical and Materials Transactions A, 3 2 , 135-143 (2001). Ά.A. Salem, S.R. Kalidindi, R.D. Doherty and S.L. Semiatin, Metallurgical and Materials Transactions A. 37, (2006). 2 5 9 - 2 6 8 . L . Jiang, J.J. Jonas, A.A. Luo, A.K. Sachdev and S. Godet, Acta Materialia, 5 5 , 3899-3910 (2007),. " L . Jiang. J.J. Jonas. A.A. Luo, A.K. Sachdev and S. Godet, Scripta Materialia, 5 4 , 771-775 (2006),. J i a n g L, Jonas JJ, Luo A A , Sachdev AK, Godet S, Mat. Sc. & Eng. A, 3 0 2 , 4 4 5 - 4 4 6 (2007). L . Jiang and J.J. Jonas, Effect of T w i n n i n g on the Flow Behavior during Strain Path Reversals in T w o M g (+A1, Zn, M n ) Alloys, Scripta Materialia, (2008), in press. 1
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EBSD STUDY OF ANNEALING ROLLED ZIRCONIUM a
b
b
L. J i a n g * , M.T. Pérez-Prado , Ο. A. R u a n o , Μ. Ε. K a s s n e r
1
a. Dept. of Aerospace and Mechanical Engineering, University of Southern California ( U S C ) . 3650 McClintock Ave., Los Angeles, C A 9 0 0 8 9 - 1 4 5 3 . U S A ' b. Dept. of Physical Metallurgy, National Center for Metallurgical Research ( C E N I M ) , CSIC Avda. de Gregorio del A m o , 8. 2 8 0 4 0 Madrid, Spain "•Corresponding author. Email: lingiianfgusc.edu K E Y W O R D S : Recrystallization, E B S D , texture, Zr, twins ABSTRACT The microstructure and texture evolution of mechanically twinned Zr ( 9 9 . 8 % purity) deformed by rolling was investigated during annealing by Electron Back-Scatter Diffraction ( E B S D ) . X-ray diffraction and optical microscopy. It w a s observed that recovery and recrystallization occur in pure Zr. It w a s suggested that mechanically induced twins in the asreceived materials may disappear by shrinking and leaving an untwinned grain or. alternatively, they m a y g r o w into a grain of a size comparable to the grain in w h i c h the twin resides in the general microstructure. N e w recrystallized grains preferentially nucleate at twin intersections and display t w o texture c o m p o n e n t s , (0001} < 1010 > and {0001} < 112 0 > . Both texture components remain present after annealing under the conditions investigated. INTRODUCTION T h e evolution of microstructure and texture during recrystallization is important in order to understand the mechanical behavior of Zr and Zr alloys. Limited data is available for Z r and Zr alloys, c o m p a r e d with the considerable amount of data on the recrystallization mechanism of cubic materials. Early work provided only limited information on the microstructure and texture during recovery and recrystallization of Zr and Zr alloys using optical microscopy, hardness measurements and X-ray diffraction The details of the restoration m e c h a n i s m s remained unclear. During the last decade. Electron Back-Scattered Diffraction ( E B S D ) , which enables the study on the evolution of grain orientations, has emerged to allow to a deeper understanding on the recrvstallization mechanisms. Recent studies on the recovery and recrystallization of Zr-702 , Zr-Hf and Zircaloy-4 revealed that recrystallization begins in highly deformed zones. Modelling by Van Landuyt et al. suggests that the grains which are easily mechanically twinned are then the ones offering the highest possibility for nucleation of new grains. According to T r e c o ', nuclei form preferentially at twin intersections or along twin bands during the recrystallization of relatively pure zirconium. Zhu et al. also reported the presence of recrystallized grains at twin intersections in the case of a Zr-2Hf alloy w h i c h was annealed at 530"C for 7 min and then annealed at 600"C for 4 min after 2 5 % plane strain compression. However, the microstructure and texture evolution m e c h a n i s m s during early stages of static annealing remain unclear. 1
4
ί β
7
8
In the present work, the evolution of microstructure. texture and grain orientations was investigated during recovery and recrystallization of pure, mechanically twinned Zr.
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E B S D S t u d y of A n n e a l i n g Rolled Z i r c o n i u m
EXPERIMENTAL The Zr ( 9 9 . 8 % purity) composition is listed in Table I. This metal w a s purchased from Haines & Maassen, Bonn, G e r m a n y , in the form of ( 5 . 2 x 1 4 0 x 2 0 0 ) m m rolled slabs. It was rolled, but no detailed information of its history is known. 3
Table I. Composition of the Zr material under study (ppm) Μη Hf Fe S 27 452 330 <550
Nd <50
Zr Rest
In order to investigate the static recrystallization of this pure Zr, the as-recieved material was annealed at 6 0 0 C for 10 min. and 4 0 min. The microstructure and texture were characterized through the thickness of the sample by a combination of several techniques. Electron backscatter diffraction ( E B S D ) was used to m e a s u r e the grain size, the distribution of grain boundary misorientations, and the microtexture. E B S D w a s performed in a scan area of 300 χ 210 p m , by a step of 80 nm and at 20 KV in a FEG SEM* located at the Max Planck Institute for Metals Research in Stuttgart. Germany, using computer s o f t w a r e " . Sample preparation for E B S D examination consisted of grinding on increasingly finer SiC papers of 600, 1200 and 1600 grit, followed by polishing with 6 p m and 1 p m diamond paste. Final surface finishing was performed with a suspension of 0.05 p m silica particles. Special care was taken in order to avoid the formation of twins during grinding and polishing. Here, special care includes using a low speed s a w for cutting samples, using a low rotation speed polishing wheel and using low pressure for polishing. T h e samples w e r e additionally etched with a solution of nitric acid (45%), distilled water ( 4 5 % ) and hydrofluoric acid ( 1 0 % ) for E B S D examination. The etched surface was rinsed with running water for 20 min in order to eliminate all the residues from the etching solution. The purpose of etching is to eliminate the superficial oxide layer which readily forms at the surface and thus does not allow to obtain good measurements. Optical microscopy (OM) was employed to investigate the microstructure of the initial rolled material. Texture was characterized by E B S D and X-ray diffraction. X-ray diffraction w a s performed in a Siemens D5000 diffractometer furnished with an open eulerian cradle at the National Center for Metallurgical Research ( C E N I M ) in Madrid, Spain. Sample preparation for X-ray diffraction is identical to that for E B S D , except that the etching step was not necessary. U
2
RESULTS Characterization of the initial state Fig. 1 illustrates the microstructure of the as-received material. A large n u m b e r of twins are observed in the grains resulting from the rolling. The average grain size, measured by the linear intercept method, is 13 p m if twin boundaries are counted and 31 p m without taking into account twin boundaries. T h e material possesses a weak rolling texture, with c-axes rotated about 30° from N D towards T D (Fig.2), which is represented by the texture component 3
9
0
{0001} < 1010 > ( { r o l l i n g plane}
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Figure 1. Microstructure of the as-received material (optical microscopy).
Figure 2. Macrotexture of the as-received material measured by X-ray diffraction. Evolution of microstructure and texture after annealing Separate "as received" specimens were annealed at 6 0 0 C during 10 minutes and 40 minutes. Figures 3 and 4 illustrate the microstructure resulting from both annealing treatments using E B S D orientation maps. After annealing at 600"C for 10 minutes, the microstructure shows evidence of the occurrence of both recovery and recrystallization. Very few twins remain present (the remnants of twins are m a r k e d by arrows A in Fig. 3a.) after annealing 600°C during 10 min. The thin lines present in s o m e grains appear to reveal the rearrangement of dislocations and the formation of subgrains (misorientation angle (0)<15°). Additionally, recrystallized grains, smaller in size than those present in the as-received material, are clearly visible (three recrystallized grains are marked by arrows Β in Fig. 3b.) These small recrystallized grains have P
t w o main orientations, i.e. {0001} < 1010 > and {0001} < 112 0 > . After annealing at 600°C for 40 minutes the material is almost fully recrystallized, as shown in Fig. 4. N o evidence of twinning is apparent after this heat treatment.
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Figure 3. E B S D m a p viewed along the N D (a) and along the R D (b) showing the microstructure after the as-received (rolled) material w a s annealed at 600°C for 10 minutes. Thick lines correspond to boundaries with misorienations higher than 15°. while the thinner lines correspond to boundaries with misorientations between 4° and 15° (boundaries less than 4° can not be reliably detected here).
Figure 4. E B S D m a p viewed along the N D (a) and along the R D (b) showing the microstructure of the as-received (rolled) Zr after annealing at 600°C for 40 minutes. Thick lines correspond to boundaries with misorienations higher than 15°.
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The grain size of the annealed samples w a s measured by the linear intercept method using only high angle grain boundaries (misorientation higher than 15"). The grain size corresponding to Zr annealed for 10 min. and 40 min. are both 17 p m , which is less than 31 μηι. the grain size for the as-received Zr, without counting twin boundaries. The grain size distributions are depicted in Fig. 5. It is clear that there are no small grains (<4 μηι) present in asreceived Zr. H o w e v e r , small grains (< 4 p m ) appear after annealing for 10 min. and 40 min. This suggests that some n e w grains are nucleated by static recrystallization. The fraction of high angle grain boundaries increases from 6 1 % (10 min.) to 9 2 % (40 min.), suggesting that, after the longer treatment, subgrain boundaries annihilate.
Figure 5. Grain size distribution of the as-received (rolled) material (a) and after annealing at 600°C for 10 nun. (b) and for 4 0 min. (c). Fig. 6 s h o w s the texture of the Zr after annealing at 600°C for 10 min. and 40 min. In both cases, both the texture component {0001} < 1010 > and {0001} < 1 1 2 0 > a r e observed.
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E B S D S t u d y of A n n e a l i n g Rolled Z i r c o n i u m
Figure 6. Direct pole figures corresponding to rolled Zr after annealing at 600°C for 10 minutes (a) and 40 minutes (b). DISCUSSION There is little evidence in the literature about the behavior of deformation twins resulting from the annealing of zirconium. Backx et al. " suggest that during annealing of the m a g n e s i u m alloy A Z 3 1 . twins can disappear by g r o w i n g and taking over an entire grain or shrinking and leaving completely untwinned grains. In this work, the twins that are left in the interior of the grains after annealing, marked by arrows A in Fig. 3, are of island-shape, which suggests that twins gradually shrink and ultimately disappear. This observation may not eliminate the possibility that, additionally, twins may grow and take over an entire grain at the very early stages of annealing, i.e., for annealing time smaller than 10 min. Static recrystallization appears to occur after annealing pure Zr at 600°C after j u s t 10 min (Fig.3). The average grain size (17 p m ) after annealing at 600"C for 40 min. does not change, compared with that (17 μηι) after annealing at 600°C for 10 min. Both are less than the average grain size. 31 μηι. of the as-received material without counting twins and only slightly larger than the grain size (13 μηι) with counting twin boundaries. This suggests that there m a y not be significant grain growth after annealing for 40 min. It is k n o w n that the texture of rolled Zr and Zr alloys consists of the alignment of 11
t h e < 1010 > directions along the rolling direction (RD) and that the c-axes are tilted 2 5 " to 40" from the normal direction ( N D ) toward the transverse direction ( T D ) along the N D - T D plane. This deformation texture is represented by the texture component. {0001} < 1010 > ({rolling 3 <
plane}
6 1 2 , 3
3 , 6
7
grains display two texture c o m p o n e n t s . {0001} < 1 0 1 0 > and{0001} < 1 1 2 0 > , while, D e w o b r o t o
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4
et a l . reported that in Zr-702 there is no preferential orientation for the n e w recrystallized grains during the annealing at 500°C. In the present work, small recrystallized grains that are apparent after annealing for 10 min., mainly display t w o texture c o m p o n e n t s , {0001} < 1010 > and {0001} < 1 1 2 0 > . With increasing annealing time (i.e. after annealing for 40 min.), both of those texture c o m p o n e n t s remain present. These observations are in good agreement with observations from Zhu et al.
6
that at higher temperature (650"Cj small grains still show the two texture
components, {0001} < 1010 > and {0001} < 1 1 2 0 > . Our results show that during primary recrystallization and the first stage of grain growth n e w texture component {0001} < 1 1 2 0 > occurs and the "rolling" texture c o m p o n e n t {0001} < 1010 > remains present.
Figure 7. Schematic showing of observed textures in this study, (a) Rolling texture represented by the texture component {0001} < 1010 > (b) Annealing texture of a rolled sheet at T=600°C for
10 and
40
min. represented
by
the two
texture
components,
{0001} < 1010 > and
{0001} < 1 1 2 0 > . CONCLUSION T h e evolution of microstructure and texture during annealing of mechanically twinned Zr resulting from rolling was investigated by E B S D , X-ray diffraction and optical microscopy.
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1. The twins present in the as-received material shrink and disappear during the first stages of annealing. However, the possibility that twins m a y g r o w and take over an entire grain at the very early stages of annealing can not be eliminated. 2 Grain size remains stable after annealing at 600°C for 40 min. 3 N e w recrystallized grains may preferentially occur at twin intersections and mainly display two texture c o m p o n e n t s , {0001} < 1010 > and{0001} < 1 1 2 0 > . With increasing the annealing time up to 40 min., those two texture c o m p o n e n t s remain present and macrotexture does not significantly change. FOOTNOTES ' LEO 1530-VP FEG S E M HKL Channel 5 software REFERENCES ' R . M . Treco, J. Metals., 10, 1304 (1956). R . K . McGeary and B. Lustman, Trans. A I M E . , 197, 284 (1953). J . H . Keeler, W.R. Hibbard and B.F. Decker, Trans. A I M E . , 197, 932 (1953). N . Dewobroto. N. Bozzolo, P.Barberis and F. Wagner, Mater. Sei. Forum., 467-470, 453-58 (2004). K . Y . Zhu, D. Chaubet, B . Bacroix and F. Brisset, Acta. Mater., 53, 5131-40 (2005). K . Y . Z h u . D. Chaubet, B. Bacroix and J.L. Bechade, Mater. Sei. Forum., 467-470, 537-42 (2004). D . Chaubet, B . Bacroix and J.L. Bechade, Mater. Sei. Forum., 408-412, 797-802 (2002). 0 . Van Landuyt, T. Grosdidier and F. Wagner, Proceedings of 1 2 I C O T O M , Edited by J. A. Szpunar, N R C Research Press, Montréal- Cananda, 895 (1999). E . Tenckhoff, Deformation M e c h a n i s m s , Textura, and Anisotropy in Zirconium and Zircaloy. A S T M Special Technical Publication ( S T P 966), (1988). P . R . Morris and A.J. Heckler, Trans. A I M E . , 245, 1877-81 (1969). " P . Backx, R. Petrov and L. Kestens, Mater. Sei. Forum., 550, 375-80 (2007). H . Hu and C. Cline, Trans. Metall. Soc. A I M E . , 242, 1013-23 (1968). M . Isaenkova and Y. Perlovich, Proceedings of 11" I C O T O M , Xian-China, edited by Z. Liang, International A c a d e m i c Publishers, Beijing, 4 7 2 - 7 7 (1996).
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Materials Processing and Texture
TEXTURE EVOLUTION DURING MULTI-PASS EQUAL CHANNEL ANGULAR EXTRUSION OF BERYLLIUM Saiyi Li School of Mechanical Engineering, South China University of Technology Guangzhou 510640, P. R. China David J. Alexander, Irene J. Beyerlein, Donald W. Brown Los A l a m o s National Laboratory Los A l a m o s , N M 8 7 5 4 5 , U S A
ABSTRACT Equal channel angular extrusion ( E C A E ) is applied to refine grains in beryllium (Be), using die angles of 90° and 120° via route Be- C o m p a r i s o n of the experimental textures measured by neutron diffraction and the ideal orientations predicted by crystal plasticity simulations suggests that the main deformation m e c h a n i s m s during E C A E of this material are (a)-type dislocation glide and tensile twinning. T h e effect of die angle is seen as the rotation of texture about the axis normal to the flow plane. For both cases the main texture components are located at orientations with the crystallographic shear plane parallel to the macroscopic shear plane or crystallographic shear direction parallel to the macroscopic shear direction. Since the textures after E C A E b e c o m e m u c h stronger that of the initial material, other processing routes should be sought to obtain weakly-textured Be materials that are ideal for potential capsule applications. INTRODUCTION Beryllium (Be) is the material of choice for fabrication of target capsules for the US National Ignition Facility (NIF) because of its combination of attractive neutronic, electronic, physical and mechanical properties [1]. Since the combination of x-ray heating and compression caused by material ablating from the outer part of the capsule during thermonuclear ignition can potentially cause fluctuations in pressure due to the crystals' anisotropic response, the beryllium must be equiaxed and isotropic (or with little to no texture) to allow uniform propagation of the shock front. It should also be comprised of crystals that are sufficiently small to give numerous grains through the capsule wall. Equal channel angular extrusion ( E C A E ) is one of the most promising techniques for grain refinement in bulk materials through severe plastic deformation [2]. Field et al. [3] have applied this technique to a p o w d e r metallurgy source Be alloy with 0.72wt% B e O . Their results demonstrate significant grain refinement and texture changes in the material after one pass of E C A E . T o eliminate the oxide content that is inevitably present in conventional powdermetallurgy materials, an ingot source produced by arc-melting was recently developed for Becapsule material, and in this alloy, significant grain refinement by E C A E was also demonstrated [1], The m a i n goal of the present study is to investigate the texture evolution in a powder metallurgy source Be during multi-pass of E C A E , with the emphasis on the effects of pass n u m b e r and die angle. The long-term goal of our investigations is to unravel the mechanisms of texture formation during E C A E , and thus to obtain Be material with not only fine grains but also weak textures.
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T e x t u r e Evolution During M u l t i - P a s s E q u a l C h a n n e l A n g u l a r Extrusion of Beryllium
EXPERIMENTAL The starting material used in this study was a commercial source S200F powder metallurgy Be designed for potential Be capsules. Billets with 5 m m diameter by 25 m m long were extracted from the Be ingot. To prevent release of Be particles into laboratory environment and to prevent oxidation, the billets were placed in cans made of commercial purity Ni (Ni-200). Ni plugs were electron-beam welded on the ends of the can. T h e assembly was 9.5 m m diameter by 45 m m long. T h e billet and tooling were well lubricated with a MoS2-containing grease. The assembled billets were processed at 4 2 5 ° C using an E C A E tooling (Fig. 1(a)) with a die angle (Φ) of 120°, for up to four passes via route Be. With this processing route, the billet was rotated by 90° about the long axis between successive passes [4]. To inspect the effect of Φ. another billet was processed for four passes via the s a m e route using a Φ- 90° tooling. As shown in Fig. 1(b), after E C A E the Ni cans remain intact even after four passes, with no sign of thinning or failure of the walls. The billets generally retained their original cross-sectional shape as in conventional ECAE without cans, which indicates that the cans and billets w e r e deformed in a similar manner. Fig. 2 shows the microstructures in the billets after one and four passes of E C A E using the 90° die. Fig. 2(a) indicates the original grain boundaries, with the grains slightly elongated due to the first pass, and Fig. 2(b) shows the grains further subdivided and distorted after four passes T h e bulk textures of the initial and E C A E processed billets w e r e measured on the timeof-flight H I P P O (High-Pressure Preferred Orientation) neutron diffractometer at the L o s A l a m o s Neutron Science Center ( L A N S C E ) . The samples for texture measurement (5 m m in diameter and 10 m m long) were sectioned from the center of the billets by electro-discharge machining. Thirty detector panels w e r e used for the measurements, providing 30 probed sample directions per sample orientation. By measuring four sample orientations (0°, 4 5 ° , 67.5° and 90° around the sample cylinder axis), a total of 120 sample directions w e r e probed. 98 of the 120 measured histograms were chosen for the texture analyses using the spherical harmonics representation of the orientation disnibution function [5] implemented in the Rietveld package G S A S [6], The m a x i m u m order of the spherical harmonics w a s increased sequentially from 6 up to 16. without imposing any sample symmetry.
Figure 1. (a) Schematic of the E C A E process and the correlation between the E C A E reference system ( 1 - 2 - 3 ) and simple shear reference system ( T - 2 " - 3 ) ; (b) Photo taken for the Ni-canned Be billet after four-pass E C A E with Φ = 120° and then cut into two halves along the flow plane.
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Figure 2. Optical micrographs taken from the B e billets after (a) o n e and (b) four passes o f E C A E using the Φ= 120° die. T o aid the characterization o f the present textures, rate-dependent single crystal plasticity simulations as explained in Ref. [7], were performed with the c/a ratio of Be. T h e simulations assumed simple shear deformation at the intersection plane of die die. Five slip modes were made available to the crystals in their plastic deformation, i.e. basal slip {0001}(1210), prismatic slip {10T0}(1210> (denoted as prism), first-order pyramidal (a) slip {1θΤΐ}<1210) (denoted as pyr(a)), first-order pyramidal (c+a) slip
{1θΤΐ}(1 123)
(denoted a s pyr(c+a)), and second-order
pyramidal (c+a) slip {1122}(T 123) (denoted as pyr2nd). In addition, compression {1122}(1 123)
Table I. Fibers with relatively stable orientations of interest predicted for E C A E o f Be (c/a 1.568) using an
hl„
0+ 0
0-90
0
(a)//l '-axis
{0ll0}<2110> ( 0 = 0 ° )
basal, prism.
{01Ï1K2H0}, (0=30°)
py(a)
s
{000I}<2H0>„ ((£=90°) 0
h3„
0
+ Θ + Θ
90 30
0-60
(c)//2'-axis
30-60 ( c ) at ±30° from the 3a x i s in the l "-plane
{000I[(2"H0>„ (p> = 0°)
basal, p y r ( a ) .
{0001ΚΠ00},, (φ, = 30°)
pyr(c+a)
(0lTlj(2110> (?* = 0°)
pyr(a>
e
(TT22!
h6
e
-40 + θ
90
0-60
( c ) at 140° from the 2"-a.\is in the 3-plane
a
(2ΤΤ3}(ΐΤθΤ) (
;i T02KI ToT>„ ( ^ = 30°)
See Ref. [7] for definition.
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(denoted as ctvv) and tension twinning {1012j(1011) (denoted as ttw), were also considered. Seven cases of numerical simulations were designed to promote (ideally) one deformation m o d e in each case by setting the reference shear stresses of that m o d e to be m u c h lower than those of the other m o d e s (see the values for C a s e s 1-6 and 7X in Ref. [7]). Using an angular resolution of 5° in the Euler angles, the results are found to be very similar to those of da = 1.633 (i.e. the ideal case in Ref. [7]). Table I summarizes the fibers (hi h 2 h3et h6») for the relatively stable orientations that are relevant to the textures o b s e r v e d for the present Be. Their locations o n the main pole figures are indicated in Fig. 3(e). Here, the convention of Bravais-Miller indices in the form of {hkil\(uvlw) is adopted, and it represents an orientation with the [hkil} plane parallel to the macroscopic shear plane ( M S P ) or 2'-plane and the (uvtw) direction parallel to the macroscopic shear direction ( M S D ) or 1'-direction. The subscript θ in the above representation denotes the rotation angle of the simple shear (or local) reference system 1 * - 2 ' - 3 from the fixed E C A E reference system 1 - 2 - 3 , and it is equal to half" of the die angle (see Fig. 1(a)). The definition of Euler angles m a y be found in Ref. [7]. ft
f t
β
RESULTS AND DISCUSSION The texture evolved substantially in E C A E from the as-received material, which had little texture (with a m a x i m u m orientation density of 1.2 χ random). The experimental textures after one to four passes of E C A E with the 120° die are shown in Figs. 3(a-d) by the (0002) and (1010) pole figures. As s h o w n , the texture evolves gradually with the increase of pass number, similar to that observed in E C A E of Cu and IF-steel using the s a m e processing route [8,9], As expected from the macroscopic simple shear deformation, the billet after the first pass depicts
Figure 3. (0002) and (1010) pole figures of the experimental textures in the Be billets after (a) one, (b) t w o . (c) three and (d) four passes of E C A E using the Φ= 120° die; (e) key pole figures showing the locations of the ideal fibers and orientations of interest.
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monoclinic sample symmetry, showing a symmetry center in the pole figures. Such symmetry is however lost for those processed after multiple passes, as a result of the 90°-rotation about the longitudinal axis of the billet between successive passes. The texture after the first pass (Fig. 3(a)) is featured with strong nig and h6g fibers, together with a partial h l « fiber from {0001}(2110), to {01 Tl}<2110)„. The m a x i m u m pole density is near (35°. 90°, 30°), which is in-between the {0001}<1 T00>
ff
orientation along the h 2
e
fiber and the {1 102}{1 101)„ orientation along the h6o fiber. T h e presence of these orientations indicates the activation of mainly basal slip and tensile twinning [7]. The textures become less fiber-like with an increase of pass number. The texture after two passes (Fig. 3(b)) is qualitatively similar to that after the first pass, but is stronger and features additional orientation concentrations near the j l 1221(1100)^ orientation along the h 3 fiber. The textures after three (Fig. 3(c)) and four (Fig. 3(d)) passes share similar features but the latter is slightly weaker. Both textures show further orientation rotations from the h 2 e a n d hoy fibers towards the h3« fiber, with the primary components near {1 1 2 2 ΐ ( 1 1 0 0 ) and {01 11}<2110) . The latter orientation is s
β
β
,
located at the intersection of the h l « a n d h 3 f i b e r s . A secondary component near {0 110 ,(2110) e
e
is also evident. A similar representation of the texture after four passes of E C A E using the Φ- 90° die is shown in Fig. 4. Comparison of Figs. 3(d) and 4 indicates that the four-pass textures obtained with two different die angles are similar in features, though the one for Φ - 90° is slightly weaker. An important result is that, the texture in the case of Φ= 90° die is CW-rotated by about 15° from that of the 120° die about the 3-axis. Such an effect of Φ on the texture formation was also observed in IF-steel processed after four passes of route Be using 90° and 120° dies [9] and discussed in Ref. [10]. According to [9], textures developed with t w o different die angles Φ] and Φι are expected to differ by a rotation about the 3-axis. and the rotation angle equals to half of the difference between their die angles, i.e. Αθ= (Φν-Φ\)Ι2. In the present case, A # s h o u l d be 15° between textures developed in Φ\ — 90° and Φι = 120° dies, which is in agreement with the measurement. An important observation common to these textures is that, as shown in Table I. the fibers for the main texture components have particular orientation relationships with respect to the macroscopic deformation axes, i.e. the simple shear axes or equivalently the E C A E axes after a rotation of Θ. Moreover, for some orientations along these fibers, the crystallographic
Figure 4. (0002) and (1010) pole figures of the experimental texture in the Be billet after four passes of E C A E using the Φ = 90° die.
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shear plane ( C S P ) and shear direction ( C S D ) of the associated active deformation m e c h a n i s m s (see also Table 1) are parallel to the M S P and the M P D , respectively. This tendency is also seen for the ideal orientations in cubic materials after E C A E [11]. It is interesting to note that for the case of Φ = 90°, similar textures after multiple passes of E C A E via route Be were also found in other hexagonal materials, such as commercial purity Zr [12] and M g alloys [13], processed via the same route using similar die set-ups. Lastly, believed to be the o p t i m u m route for grain refinement [4], it appears from though route 5 c the present results that the textures after E C A E deformation via this route b e c o m e stronger than that of the initial material and thus this route is not o p t i m u m for weakening the texture of the material. From this point of view, other processing routes should be sought to obtain fine-grained and weakly-textured Be capsule material. O n e possible candidate for this is the so-called route C (i.e. rotation of 180° about the billet long axis b e t w e e n successive passes). Previous investigations on cubic materials have revealed that this route leads to relatively weaker E C A E textures than other routes as a result of the strain reversal between consecutive passes [8]. w
a
s
CONCLUSIONS The texture evolution during E C A E of Be via route Βς using tooling with Φ= 120° and 90° is investigated. The results for the case of Φ= 120° indicate a gradual texture evolution with the increase of pass number. C o m p a r i s o n of the experimental textures and the ideal orientations predicted by crystal plasticity simulations suggests that after the first pass the billet features high orientation densities along the h2» and h6« fibers, indicating deformation by (a)-type dislocation glides and tensile twinning. Increasing pass n u m b e r leads to orientation rotations from the h2# and h o s fibers towards the h3# fiber in the textures. The effect of die angle is seen as the rotation of texture about the 3-axis, as expected from theory [9,10], For both die angles, the main texture c o m p o n e n t s are found at orientations featuring the C S P parallel to the M S P or the C S D parallel to the M S D . Overall, the textures after E C A E b e c o m e much stronger that of the initial material; other processing routes should be sought to weaken the textures. ACKNOWLEDGEMENTS This work was supported in part by a Program for N e w Century Excellent Talents in University ( N C E T - 0 6 - 0 7 4 1 ) and a Los A l a m o s Laboratory-Directed Research and Development project (20030216). The texture measurements have benefited from the use of L A N S C E , a user facility funded by the US Department of Energy under Contract W - 7 4 0 5 - E N G - 3 6 . REFERENCES 'D.J. Alexander, J.C. Cooley, D.J. T h o m a , and A.J. Nobile, Fusion Sei. Tech., 4 5 , 137 (2004). V . M . Segal. Mater. Sei. Eng. A, 197, 157 (1995). R . D . Field. K.T. Hartwig, C T . Necker, J.F. Bingert, and S.R. A g n e w , Metall. Mater. Trans., 3 3 A . 965 (2002). M . Furukawa, Y. Iwahashi, Z. Horita, M. N e m o t o , and T.G. Langdon. Mater. Sei. Eng. A, 2 5 7 . 328 (1998). H . J . Bunge, Texture Analysis in Materials Science, London, Butterworth (1982). R . B . Von Dreele,7. Appl. Cryst., 30, 517 (1997). S . Li, Acta Mater., 56, 1031 (2008). S . Li, I.J. Beyerlein, D.J. Alexander, and S.C. Vogel, Acta Mater, 5 3 , 2111 (2005). 2
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9
Pereloma, Acta Mater..
1 0
3 9 4 , 66 (2005).
,2
8 5 , 345 (2005). Liaw, Mater. Sei. Eng. A. 408, 72
S . Li, A.A. Gazder, I.J. Beyerlein. C.H.J. Davies, and E.V. (2007). S . Li, I J . Beyerlein, and M . A . M . Bourke, Mater. Sei. Eng. A. " S . Li, Acta Mater., 56, 1 0 1 8 ( 2 0 0 8 ) . S . H . Yu, Y . B . Chun, S.K. H w a n g , and D.H. Shin, Phil. Mag.. S . R . A g n e w , P. Mehrotra, T.M. Lillo, G.M. Stoica, and P.K. (2005). I3
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A M E C H A N I S M O F D E T E R M I N I N G T H E F O R M A B I L I T Y O F AZ31 N E A R R O O M TEMPERATURE Hualong L i , Emilie Hsu , Jerzy Szpunar McGill University, Department of Metals and Materials Engineering, 3610 University, Montreal, Quebec H 3 A 2 B 2 , Canada
ABSTRACT In our previous studies, it was shown that the basal slip and extension twinning were the primary deformation m o d e s of AZ31 alloy at room temperature. It was postulated that the maximum amount of cold reduction that a specimen with a given initial texture could undergo coincides with the exhaustion of twin deformation. In this paper, the formability of AZ31 with different initial textures is simulated under both uniaxial compression and tension. The textures considered are all fiber type with their fiber axes covering all possible crystallographic directions in the orientation space in 5° angular intervals. The basal fiber texture exhibits the worst formability, random texture exhibits improved formability, and (10-11) fiber texture exhibits the best formability. The simulation results are presented in the inverse pole figure space and they clearly demonstrate the relationship between the formability and the initial texture. This study ultimately generates a map of formability of AZ31 as a function of initial texture. INTRODUCTION In recent years, m a g n e s i u m alloys have attracted increasing interest due to their low density and high specific strength. They are being considered as potential replacements for heavier materials like steel and aluminum in various applications. However, m a g n e s i u m alloys do not have good room temperature formability [1] and forming at high temperature entails additional cost. There is considerable interest in improving room temperature formability of magnesium by microstructure and texture modifications. This requires identifying and understanding the role of different deformation m e c h a n i s m s active at room temperature. It is generally agreed that basal slip m o d e has the lowest C R S S and this slip system is active during room temperature deformation. Additional slip systems were also found possibly active at room temperature by other researchers [2, 3], In our previous studies [4], it was demonstrated that basal slip and extension twinning were the active deformation m o d e s of A Z 3 1 at room temperature. Based on this a computer model was established to predict the formability of AZ31 with a given initial texture. The forming limit of a specimen was determined by tracking exhaustion of twin deformation as measured by m a x i m u m twinning strain. The goal of this paper is to study, by using the previously developed computer model, the effect of initial texture on the formability of A Z 3 1 .
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TEXTURE SIMULATION PROCEDURE Computer simulation is based on the V P S C (Visco Plastic Self Consistent) model [5], In this model, the specimen is represented by a n u m b e r of grains. The orientations of these grains are defined in such a way that they reproduce the initial texture of the specimen. Each grain is treated as an inclusion e m b e d d e d in an effective medium. The effective m e d i u m represents the matrix that each grain interacts with. Thus, the grains do not interact with each other directly. Each grain, dependent on its o w n visco-plastic compliance or stiffness, experiences a strain or stress different from that of the m e d i u m . Deformation takes place by slip and twinning, activated when the resolved shear stress exceeds corresponding C R S S . The twinning is simulated in a manner similar to slip. It differs from the slip only in its directionality. The twinning is activated only if the resolved shear stress is positive (along the Burgers vector of the twin). The twinning process will also cause a rigid orientation flip of grains and it is controlled by the Predominant Twin Reorientation Scheme ( P T R ) proposed by T o m é et al [6], The main idea of PTR is that once the twin fraction reaches a certain level, one grain will be randomly chosen to flip its orientation completely.
Figure l. Initial fiber texture defined in inverse pole figure space, a) schematic representation of 153 fiber textures used in the simulations: b) (0001 ) fiber texture; c) ( 10-10) fiber texture and d) ( 11 -20) fiber texture.
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DEFINING INITIAL TEXTURE M g alloy AZ31 has a Hexagonal crystal structure. The m i n i m u m orientation space of Hexagonal crystal structure with the orthogonal sample symmetry is 0 ° < φ ι < 9 0 ° ; 0 ° < Φ < 9 0 ° and 0 ° < φ < 6 0 ° . where φ ι , Φ and φ are three Euler angles used to define an orientation. Without the orthogonal sample symmetry, the orientation space will be four times larger. If every orientation in this space is used a s an initial texture for the simulation, there would be a total of 19X19X13 = 4 6 9 3 different initial textures assuming that the Euler angles change in 5° interval. This requires 4693 simulations to generate a m a p of formability as a function initial texture. To simplify, the initial textures in this study were defined as fiber type. For the fiber texture, the crystallographic plane parallel to the sample surface is defined; the crystallographic direction in the sample surface is distributed evenly. Thus, the total n u m b e r of different initial textures is reduced to 153. The inverse pole figures in Figure l a show the space of fiber texture of hexagonal crystal system. The radial coordinate ranges from 0° to 90°; and angular coordinate ranges from 0° to 30°. Each point in the inverse pole figure indicates a crystallographic direction that is the normal direction of a fiber plane. Therefore, each point in the inverse pole figure represents a fiber texture. In the simulation required to generate the map of formability as a function of initial texture, the points were selected in every 5° interval in the inverse pole figure space as illustrated in Figure la. For each of those initial textures, it was assumed that the volume fraction of the ideal fiber was 2 0 % and the rest of grains represented the random texture component. There are three initial textures given in Figure lb-d. The m a x i m a in Figure l b , l c and Id are at [0001], [10-10] and [11-20] directions and they represent the normal directions to (0001), (10-10) and (11-20) planes and accordingly these three types of fiber textures. 2
2
DEFINING LOADING CONDITIONS Uniaxial compression and uniaxial in-plane tension loading conditions were applied in the simulations. In the simulation of uniaxial compression as shown in Figure 2a. the deformation was represented by 833 c o m p o n e n t up to the failure level, and σι 1 and 022 were assumed to be zero. The strain deformation at failure can be determined from the simulation using the formability model [4], The fiber texture was defined by its fiber axis parallel to the compression direction. In the simulation of uniaxial in-plane tension as shown in Figure 2b, the strain was applied to ει i up to the failure level, and o n and 022 were equal zero. The fiber texture was defined by the plane parallel to sheet surface. MEHTHODOLOGY OF DETERMINING FORMABILITY To a c c o m m o d a t e arbitrary shape changes of the specimen during deformation, a minimum of 5 independent slip and twinning systems are required. Since basal slip m o d e only provides two independent slip systems, the extension twinning mode, which has the second lowest C R S S at room temperature, must be activated. T h e existence of active basal slip and the extension twinning m o d e s at room temperature have been verified experimentally [5], The contributions of basal slip and extension twinning m o d e s to the total strain are dependent on the grain orientation and'the C R S S ratio, which can be calculated.
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A M e c h a n i s m of D e t e r m i n i n g t h e Formability of A Z 3 1 N e a r R o o m T e m p e r a t u r e
Figure 2. Loading conditions used in the simulations. A ) uniaxial compression: and b) uniaxial in-plane tension.
The m a x i m u m strain that can be generated by extension twinning is limited and is about 7 % [1] assuming that all grains are in the most favorable orientations for twinning deformation. For polycrystalline M g alloys with r a n d o m or moderate texture, the m a x i m u m strain that can be obtained as a result of twinning alone is about 3 . 5 % . W h e n the strain contributed by twinning reaches this value, the twinning activation in the specimen is saturated. Further deformation has to be supported solely by basal slip, which limits ductility and leads to fracture. According to this analysis, the m a x i m u m amount of strain in a specimen with a given initial texture can be determined by tracking exhaustion of twin deformation. Based on this idea the model to predict the formability was established [4], F O R M A B I L I T Y A S A F U N C T I O N O F INITIAL T E X T U R E Each fiber texture presented by a point o n the grid in Figure 1 a w a s taken as an initial texture to the simulation of formability under the compression and tension loading conditions. The simulation results are s h o w n in Figure 3 where Figure 3a represents the m a p of formability under compression loading condition and 3b represents the m a p under tension loading condition. The values of contour levels indicate the percentage of reduction in the case of compression and the engineer strain in case of tension. In Figure 3a, the m a x i m u m value represents (10-11) fiber texture that has the highest reduction strain of 4 4 % . Three corners of the inverse pole figure represent (0001). (10-10) and (11-20) fiber textures respectively; their forming limits under unixial compressions are 14%, 16%, 18%. The better formability of (10-11) fiber texture is generated because this texture is favorable for the activation of basal slip and thus requires less contribution from twinning. The simulation
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results in Figure 3a cannot be verified directly by experiments because the initial textures used in the simulation were artificially created. However, previous results [4] predicted that the random specimen could be deformed roughly to 3 0 % cold reduction while the basal specimen to 10% and off-basal specimen to 2 0 % reduction. The off-basal specimen had the basal pole parallel to the T D and therefore the texture was 90° rotated from basal texture. These failure strains predicted in simulation were in a good agreement with the experimental failure strains that were 2 0 - 2 5 % , 1 1 - 1 3 % , and 2 0 % respectively for the as-cast, basal and off-basal specimens. Figure 3 b s h o w s the simulated forming limits as a function of the initial texture w h e n unaxial inplane tension applied. Similarly as the texture presented in Figure 3a, (10-11) fiber texture also exhibits a improved formability. The simulated m a x i m u m engineering strain of (10-11) fiber
Figure 3. Simulated forming limits as functions of initial textures under a) uniaxial compression and b) uniaxial in-plane tension loading conditions. T h e values of contour levels represent reduction rate in a) and engineer strain in b)
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texture is 2 4 % . For (0001), (10-10) and (11-20) fiber textures that are marked at the three corners of the inverse pole figure, the forming limits are 14%, 2 0 % and 2 2 % respectively. The slightly increased formability of the (10-11) fiber texture results from favorable orientation of the basal slip system.
REFERENCE 1. 2. 3.
4.
R. W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials. Fourth edition, John Wiley & Sons, Inc. N e w York, 1995. A. Styczynski, C. Hartig, J. Bohlen, D. Letzig, "Cold Rolling Textures in AZ31 Wrought Magnesium A l l o y " Scripta Materialia 50 (2004), 9 4 3 . S. R. A g n e w . M. H. Yoo and C. N. T o m e , "Application of Texture Simulation to Understanding Mechanical Behavior of M g and Solid Solution Alloys Containing Li or Y", Acta Mater. 49 (2001). 4277. H. Li, E. Hsu. J. Szpunar, R. Verma and J. T. Carter, "Determination of Active Slip/Twinning M o d e s in AZ31 M g Alloy Near R o o m Temperature". JMEPEO, 16 (2007), 321-326_
5.
R. Α., L e b e n s o h n and. C. N . T o m e , Acta Metall.,
6.
C. N . T o m e , R. A. L e b e n s o h n and U. F. K o c k s , Acta.
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41 ( 1 9 9 3 ) , 2 6 1 1 . Metall.
Mater.,
39 (1991), 2667.
AN EBSD STUDY OF THE MISORIENTATIONS RELATED TO D Y N A M I C RECRYSTALLIZATION IN Mg A M 3 0 D E F O R M E D AT HIGH T E M P E R A T U R E S 1
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Etienne Martin , Stéphane Godet , Lan Jiang , Abdelbaset Elwazri , Pascal J. Jacques and John J. Jonas' 'Department of Materials Engineering, McGill University, Montreal, QC. H3A 2B2, Canada Service Matières et Matériaux, Université Libre de Bruxelles (U.L.B.), 50 Avenue F.D. Roosevelt, C P . 194/03, 1050 Brussels Université catholique de Louvain, Département des Sciences des Matériaux et des Procédés, IMAP. 2 Place Sainte Barbe, B-1348 Louvain-la-Neuve, Belgium. 2
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ABSTRACT The misorientation relationships that apply to the various types of nuclei that form during dynamic recrystallization (DRX) were investigated in a Mg-3.4%A1- 0.33%Mn alloy using electron backscatter diffraction (EBSD). Compression tests were carried out at 350°C and a strain rate of 0.001s"' on samples machined from extruded tubes. Two different types of DRX nuclei were observed at this temperature, i.e. those formed by bulging and by continuous dynamic recrystallization (CDRX). The misorientations with respect to the matrix that apply to the two types were determined and are expressed in Rodrigues-Frank space. These misorientations are seen to increase with strain. For small misorientations, the rotation axes have low inclinations with respect to the basal plane. These axes are evenly distributed around the c-axis without any particular direction being favored. Although visually distinct, the misorientations in the two types of nuclei both develop by means of the continuous mechanism. INTRODUCTION The occurrence of dynamic recrystallization in magnesium and magnesium alloys produces uniform fine-grained structures during deformation at elevated temperatures. This leads to significant enhancement in strength and ductility at room temperature, so studies of the evolution of fine grains are of importance for the production of improved magnesium alloys. Moreover, DRX may also lead to the randomization of initial textures, and so can be of practical interest with regard to the subsequent forming of hep materials. According to the literature, the mechanisms involved in DRX in magnesium are fairly complex and many types of nuclei have been identified to date. [1-3] Many researchers have studied the recrystallization textures observed in magnesium and magnesium alloys. Yi et al. [4] have reported that the O D F intensities are similar and high for the initial material and for material deformed into the DRX regime; nevertheless, there was a slight decrease in the intensity. Other investigators [5-7] have reported that most of the newly recrystallised grains have orientations similar to those of the matrix grains and thus serve mainly to weaken the texture. However, these researches were all carried out on different alloys and under different deformation conditions, without consideration being given to the type of nucleus produced. The present study was therefore undertaken with the aim of characterising the different DRX nuclei and linking their characteristics to the deformation parameters. The misorientations involved in the DRX pf AM30 deformed at a specific temperature and strain rate will be outlined in order to indicate the type of nuclei that are formed in this regime. EXPERIMENTAL This investigation was carried out on an experimental wrought magnesium alloy (AM30) developed at General Motors [8]. The tubes from which the samples were made were extruded using porthole dies and produced by Timminco Metals of Aurora, Ontario, Canada; they had nominal outer diameters of 70 mm and wall thicknesses of 4 mm. The samples were machined out of the tube walls
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with their axes aligned along the extrusion direction and the compression samples had heights of 4.5 mm and diameters of 3 mm so that the height/0 ratio was 1.5. The initial texture can be divided into two component orientations: one with its c-axis parallel to the radial direction and one of the prismatic plane normal's aligned with the extrusion direction (called the RD component or [1010}<0001>) and the other called the TD component ({11 -20}<10-10>). Compression tests were carried out at a fixed strain rate of 0.001s" and temperature of 350 °C. Each sample was heated to the deformation temperature and then held for five minutes prior to compression to the required strain. The orientations of the recrystallised and unrecrystallised grains were examined with the aid of a field emission type scanning electron microscope equipped with an EBSD detector. As the diameter of the smallest nucleus was considered to be a few microns [9]. 0.3 and 0.35 p m step sizes were employed, to provided a minimum ratio of 1:3 between the step size and the size of the smallest feature of interest. 1
RESULTS In order to be able to follow the appearance of nuclei, compression tests were conducted to the critical strain for DRX (0.05 in this case) with increments of 0.05 until 0.35 strain. Two different types of nuclei were identified, referred to here as the CDRX and bulge types. These results are consistent with those of earlier researchers. [2] CDRX Nucleation Mechanism An example of an RD-orientation grain that is undergoing DRX is presented in Figure 1 (a). The grain is subdivided by low and high angle boundaries. The relative misorientations associated with the boundaries crossed by the dashed and full lines, respectively, are illustrated in Figure 1 (b) and (c). Here the scans begin in the upper left corner and continue towards the lower right corner. Boundaries of 2" to 34° misorientation coexist in the same grain due to the continuous transformation of low angle into high angle boundaries. The higher misorientations are considered to pertain to the initial grain boundaries.
Figure 1 (a) OIM map of a dynamically recrystallizing microstructure at 0.1 strain deformed along the x-axis; relative misorientations of the boundaries crossed by (b) the dashed line and (c) the full line. Here tire white lines signify misorientations between 5° and 10°; the thin black lines between 10° and 15", and the thick black lines > 15". At low strains, the new grains are bounded by medium angles (15-30°) and can have quite large diameters (e.g. grain Y in Figure 1 (a)). As straining progresses, new low angle boundaries form inside the grains and are gradually transformed into high angle boundaries, reducing the sizes of the
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previously fonned DRX grains in this way. In Figure 1 (a), for example, DRX grain Y is bounded by two newly formed high angle boundaries that separate it from initial grain X (peaks 6 and 8 on Figure 1 (c)). Inside this grain, a new sub-boundary is being formed that will eventually divide it into two parts. The same phenomenon is visible in the upper part of grain X. As the strain is increased, substructures are continually being created within the previous ones, thus consuming some of the available dislocations and slowing down the increase in misorientation of the previously formed boundaries. These observations are consistent with those of Miura et al. [10, 11 ] as well as of Gourdet et al. [12], who have suggested that the misorientation increase during CDRX is related to the accumulation of dislocations in the subgrain boundaries. Here the strain dependence of the mean misorientation is shown in Figure 2.
Figure 2 Dependence of the average misorientation angle on strain Bulge Nucleation Mechanism Initial grain boundaries are privileged sites for the nucleation of CDRX. In Figure 1 (a), four DRX grains have formed at the boundary between initial grains X and Z. In this case, great care must be taken to distinguish between the bulge and C D R X types of nuclei. The misorientation between DRX grain 1 and initial grain X is 80°; this is similar to that between grains X and Z, indicating that it is a bulge nucleus formed by the migration of grain Ζ into grain X. By contrast, grains 2, 3 and 4 are of the C D R X type; they all have the same parent grain X. Apart from the fact that bulges can only form at grain boundaries, the two mechanisms have many common features. In Figure 3 (a) and (b). low angle boundaries of two different misorientations. 5° and 12° respectively, are created at the serrated boundaries. In Figure 3 (c), two new grains have resulted from the bulging of grain 1 (TD orientation) into grain 2 (RD orientation) while a third is being formed. This nucleation mechanism is also progressive with the strain, which is in good agreement with the observations of earlier researchers [2, 13].
Figure 3 O I M map of sample deformed to 0.05 strain along the x-axis and displaying the bulge type of nucleation. Here the white lines signify- misorientations between 5" and 10"; the thin black lines between 10° and 15°, and the thick black lines > 15°.
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The researches of Prior [14] and Wilkinson [15] indicate that the misorientation axis is reasonably well-defined when the misorientation angle is large. However as the misorientation angle is decreased, calculation of the misorientation axis leads to a larger amount of scatter. For this reason, measurements were only taken from boundaries with misorientations of 5° and more. DISCUSSION Misorientations Between Nuclei and their Parent Grains The misorientations between 865 individual DRX nuclei and their parent grains are presented in Figure 4 in the form of Rodrigues vectors. Here no account is taken of the individual nucleation mechanism. Figure 4 (a) to (c) correspond to low (5-15°) angle boundaries and (d) to (f) to high (1593°) angle ones. Each point corresponds to the end of a vector linked to the center of Rodrigues-Frank space.
Figure 4 Substructure (a) to (c) and nucleus (d) to (f) misorientations expressed with respect to the parent grain and displayed in Rodrigues-Frank space. 865 measurements are represented in the form of Rodrigues vectors. From Figure 4, it is fairly clear that no variant selection took place during DRX of the material of interest. This conclusion applies to both the low angle as well as the high angle boundaries. That is there are no specific misorientations or rotation axes that are favored. Figure 4 (b) and (e) show that the basal component of the Rodrigues vector is evenly distributed between the <10-10> and <-12-10> family of axes, again for both the low and high angle boundaries. This explains why DRX cannot create new texture components: the new grains are evenly rotated around the orientations of the initial components. In Figure 4, the dots are concentrated in the centers of the spaces and this is due to the low angles of misorientation involved. In Figure 4 (c) and (f), the vectors are projected onto the R]-R3 plane in order to show the inclinations of the Rodrigues vectors with respect to the basal plane (R3=0) and the c-axis ( R i = 0 ) . A small amount of clustering in the center of Rodrigues space is evident. When the symmetry elements associated with the hexagonal structure are applied to both crystals (including the inversion), the
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EBSD S t u d y of Misorientations Related t o D y n a m i c Recrystallization in Mg AM30
fundamental region of Rodrigues Frank space can be reduced to 1 /24' of its initial size. In this way, all the vectors can be represented in the first quadrant of the orthonormal frame; Ri, R 2 . Ri>0. Then, R3 can be plotted as a function of yJR' +R\ . so that the inclinations of the Rodrigues vectors can be displayed directly and the lengths of the vectors preserved. The 865 misorientations of Figure 4 processed in this way are displayed in Figure 5. The black quarter circles represent the measured 2
misorientation angles; the (R3) or Y axis corresponds to the c-axis and the ( -Jr* + R ) or X axis to the 2
basal plane. Note also that the two dashed black lines represent vectors that are inclined at 30" and 60" to the c-axis. They bound three groups of misorientations referred to below as A, Β and C.
Figure 5 Distribution of misorientations (axes and angles) between the DRX nuclei and the matrix presented in the form of Rodrigues vectors. There is clearly a concentration of Rodrigues vectors within 30° of the basal plane (region (A)), especially for misorientation angles within 5-10". As the inclination with respect to the basal plane increases (regions (B) and then ( Q ) , the Rodrigues vector concentration decreases. The main observations displayed in Figure 5 are summarized in Table 1. The Rodrigues vectors were first divided with respect to their lengths (misorientation angles) in increments of 5" (columns 2 to 7 of Table 1). The second column, for example, represents all the boundaries that have misorientation angles between 5 and 10°. In Figure 5, it corresponds to the points bounded by the 5" and 10° quarter circles. Then, they were divided according to the three inclination ranges described above. The second column reveals that 8 3 % of the Rodrigues vectors lie within 30° of the basal plane. It can therefore be concluded that preferred misorientations develop within the subgrains as they begin to form. These misorientations correspond to small rotations about axes lying close to or within the basal plane: the latter are associated with the pile-up of <1-210> dislocations. Table 1 Frequency distribution of the misorientations associated with the DRX nuclei. Deg.
5-10
11-15
16-20
21-25
26-30
31-93
R e g i o n (A)
83%
59%
. 60%
64%
61%
71%
R e g i o n (B)
11%
28%
32%
26%
23%
27%
R e g i o n (C)
6%
13%
8%
10%
16%
2%
The misorientation distribution changes suddenly when the misorientations exceed 10°, that is when the sub-boundaries begin to be transformed into high angle boundaries. Although most of the
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Rodrigues vectors remain within 30° of the basal plane, i.e. within region (A), the percentage decreases to 60%, which is still considerable. The region near the c-axis is clearly not favoured. At low misorientations. this can be due to the pile-up of <1-210> dislocations referred to above; on the other hand, misorientations above 30° are ruled out by hexagonal symmetry [16]. In the range from 11° to 30". the three orientation distributions (i.e. Α. Β and C) remain approximately constant. The frequency distribution of the misorientation angles associated with the nuclei is presented in Figure 6. For low angle boundaries, there is clearly a peak between 5" and 10", while the frequency is fairly evenly distributed between 10" and 35°. Although most of the Rodrigues vectors (83%) appear in Figure 5 and Table 1 at misorientations between 5° and 10", the actual peak is not related to anyspecific axis of misorientation since these are evenly distributed between the {-12-10! >d {10-10} axes, as can be seen from Figure 6 (b). Thus, from the start to the intermediate stages of nucleation. i.e. from 5°-35° of misorientation. it does not appear as if coincidence site lattices are involved in the two main high temperature mechanisms of DRX. ar
Figure 6 (a) Misorientation angle distribution of the DRX nuclei, and (b) Rodrigues vectors associated with the 5-10° misorientation nuclei (displayed in the reduced fundamental zone of Rodrigues-Frank space). Misorientations Associated with the Bulge and CDRX Types of Nuclei. In order to distinguish between the two types of nucleation, only nuclei formed at the boundaries of the T D and RD grains have been investigated. In this way. the IPF coloring enables the two types of nuclei to be identified. The misorientations described above apply to the two types of DRX nuclei discussed to date. i.e. those of the CDRX and bulge types. The present observations indicate that the Rodrigues vectors are randomly distributed about the c-axis. This can be seen from the fundamental zone projection displayed in Figure 7 (a). It is evident here that the distribution of Rodrigues vectors is fairly homogeneous in the zone bounded by the {10-10} and {-12-10} families of axes for the two types of nuclei. The three ranges of misorientation identified in Figure 5 are presented again in Figure 7 (b) and (c). but in this case the CDRX and bulge nuclei are shown separately. The small variations in the distributions of Figure 7 (b) and (c) can be explained by the limited number of bulge nuclei present in the microstructure. It is again clear from the two figures that the frequency decreases as the Rodrigues vectors move away from the basal plane and that few of the vectors are located near the c-axis.
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Figure 7 (a) Misorientation distributions of the CDRX (324 crosses) and bulge (244 black circles) nuclei in the reduced fundamental zone of Rodrigues-Frank space: misorientations between the DRX nuclei and the matrix for the (b) CDRX and (c) bulge types of nuclei.
Once again, the two types of nuclei display especially high Rodrigues vector intensities in the vicinity of the basal plane at the beginning of recrystallization, i.e. between 5"-10" in Figure 7 (b) and (c), or when the misorientations are just beginning to develop. It is of interest that bulges as large as 1 Oum in length were found without any sub-boundaries enclosing them. After a certain amount of growth, subboundaries eventually formed behind them. Thus sub-boundary formation and similar misorientation distributions seem to be involved in both types of nucleation. Instead, the two types of nuclei can be distinguished morphologically. CONCLUSIONS The crystallographic features of magnesium alloy A M 3 0 deformed in the high temperature regime were studied by means of EBSD techniques. Dynamic recrystallization was observed to occur and two different types of DRX nuclei were identified. The misorientations of these nuclei with respect to the matrix were measured and the results obtained can be summarized as follows: 1. There was a uniform distribution of the rotation axes around the c-axis. Such random behaviour during DRX helps to explain the loss of texture intensity during hot deformation. 2. The Rodrigues vector concentration increases as it approaches the basal plane. Most of the Rodrigues vectors were located within 30" of the basal plane, while c-axis rotations were almost absent during DRX. 3. A sharp change in the frequency distribution is observed when the low angle boundaries begin to take on a high angle character. For misorientation angles of 5" to 10", the Rodrigues vectors lie principally in or near the basal plane; these correspond to misorientation relationships attributable to the polygonisation of < 1 -210> dislocations. Beyond misorientations of 10°. the frequency distribution (with regard to inclination with respect to the basal plane) becomes more spread out and remains constant as the misorientation increases. 4. Although geometrically distinct, both bulge and CDRX nucleation involve the continuous transformation of sub-boundaries into high angle boundaries. Their rotation axes have similar distributions with regard to the c-axis and their operation leads to similar texture changes.
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ACKNOWLEDGMENTS This research was sponsored by the Natural Sciences and Engineering Research Council of Canada. Discussions with Phuong Vo are gratefully acknowledged. REFERENCES [1] M. R. Barnett, Journal of light metals, 1, 167-178, (2001). [2] A. Galiyev, R. Kaibyshev, G. Gottstein, Acta Materialia, 49, 1199-1207, (2001). [3] R. Kaibyshev, O. Sitdikov, in: T. R. McNelley (Ed.) Third international conference on recrystallization and related phenomena, ReX'96, p. 7, (1997). [4] S. B. Yi, S. Zaefferer, H.-G. Brokmeier, Materials Science and Engineering A, 424 no.1-2, 275281,(2006). [5] M. R. Barnett, Materials science forum 419-422, 503-508, (2003). [6] J. A. del Valle, M. T. Pérez-Prado, Ο. A. Ruano, Metallurgical and materials transactions A 36A, 1427-1438, (2005). [7] F. Zarandi. R. Verma, E. Essadiqi, S. Yue, in: The minerals, metals & materials society, Orlando, (2007). [8] A. A. Luo, in: The minerals, metals & materials society, 333-339, (2006). [9] A. G. Beer, The evolution of hot working stress and microstructure in Mg-3Al-lZn, in: School of Engineering and Technology, Doctor of Philosophy thesis, Deakin University, p. 219 (2004). [10] X. Yang. H. Miura. T. Sakai. Materials science forum. 426-432, 611-616 (2003). [11 ] X. Yang, H. Miura, T. Sakai, Materials transactions, 44, 197-193, (2003). [12] S. Gourdet, F. Montheillet, Materials Science and Engineering A, 283, 274-288, (2000). [13] O. Sitdikov, R. Kaibyshev, Materials transactions, 42, 1928-1937, (2001). [14] D. J. Prior, Journal of microscopy, 195, 217-215, (1999). [15] A. J. Wilkinson, Scripta Materialia, 4 4 , 2 3 7 9 - 2 3 8 5 , ( 2 0 0 1 ) . [16] A. Kumar, P. R. Dawson, Computer methods in applied mechanics and engineering, 153, 259302.(1998).
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T E X T U R E E V O L U T I O N IN B O R O N M O D I F I E D T I - 6 A L - 4 V A L L O Y 1
1
2
3
Shibayan R o y , Nilesh G u r a o , Satyam S u w a s ' , S. Tamirisakandala , R. Srinivasan and D.B. Miracle 4
'Laboratory for Texture and Related Studies, Department o f Materials Engineering, Indian Institute o f Science, Bangalore, I N D I A , d e p a r t m e n t o f M e c h a n i c a l Engineering, O h i o University, U S A Mechanical and Material Engineering Department, Wright State 4
University.
Dayton.
USA
A i r Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson A F B , U S A
ABSTRACT Owing to their high strength-to-weight ratio, excellent mechanical properties and corrosion resistance, titanium (Ti) and its alloys, especially ( α + β ) alloys like Ti-6A1-4V is the backbone materials for aerospace, energy, and chemical industries. Trace boron addition (~0.1 wt. % ) to the alloy Ti-6A1-4V produces a reduction in as-cast grain size by roughly an order of magnitude resulting in enhanced ductility, higher stiffness, strength and good fracture resistance. Boron addition could also affect the evolution of texture and microstructure in the material. The solidification mierostructures of Boron free as well as Boron containing Ti-6A1-4V are found to be almost h o m o g e n e o u s from periphery towards the center of as-cast ingot in terms of both a colony size and distribution. Boron addition substantially reduces α-colony size (~50-80 pm). A gradual change in α texture from periphery towards the center has been observed with orientations close to specific texture components suggesting the formation of texture zones. The mechanism of texture evolution can be visualized as a result of variant selection during solidification through ( α + β ) phase field. INTRODUCTION: Since the introduction of titanium and titanium alloys in the early 1950s, these materials have in a relatively shorter time b e c o m e backbone materials for the aerospace, energy, and chemical industries. They are very useful light materials that exhibit high specific strength and fracture toughness with a good corrosion resistance at temperatures up to 5 5 0 - 8 0 0 K . The basis for tailoring the microstructure and texture during heat treatment of titanium and titanium alloys centers on the β (body-centered cubic) to α (hexagonal close packed) transformation at ~ 8 8 5 ° C . The crystallography of the t w o phases is c o m m o n l y found to be related by the Burgers orientation relationship ( O R ) where close packed planes [0001 ]α = [110]β and close packed directions [1 l - 2 0 ] a = [111]β are p a r a l l e l ' . Seward et a l have made in situ observations (above 900°C) of the α—>β phase transformation in commercially pure titanium using E B S D and found that the strong c o m p o n e n t s in the β crystallographic texture can be correlated directly to the initial α texture via the Burger O R . T h e most widely used titanium alloy is the Ti-6A1-4V (α+β) alloy which is most c o m m o n l y used in the annealed condition . Microstructure and texture evolution in this alloy was studied with great details by many researchers. Kobryn et a l have found that metal-mold-cast Ti-6AI-^IV tends to have equiaxed prior β grain morphology, while direct-laser-fabricated T i - 6 A 1 - 4 V usually has a columnar morphology. T h e alpha-phase texture in direct-laser-fabricated T i - ÖAI^IV was found to be fairly weak (i.e. m a x i m u m IPF intensity of ~ 1.6 times random). In another study, Kalinyuk et a l showed that the microstructure of two electron beam melted TÎ-6A1-4V ingots at all locations are identical and characterized by coarse beta grains decorated by a layer of grain-boundary a . The texture of both the ingots are found to 1
2
3
4
1
5
6
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T e x t u r e Evolution in B o r o n M o d i f i e d T Î - 6 A I - 4 V Alloy
be completely random suggesting the possibility that the texture distribution not being affected by anisotropy in grain growth during solidification. In a v a c u u m arc melted Ti-6A1-4V ingot, Glavicic et a l have utilized the room temperature measured orientations of α phase variants from a n u m b e r of prior β grains to generate the prior β texture. The results of the analysis demonstrated that the solidification of the β phase produces columnar grains in the ingot along <100> preferred-growth directions, where the texture of the beta phase was inherited by the α phase in a manner consistent with the burgers relationship; in contrast to the equiaxed grains at the center of the ingot having r a n d o m α and β phase textures. 7
The primary problem in wrought processing of titanium is the large grain size (often in the millimeter range) that evolves during the solidification of the as-cast titanium ingot. Extensive hot working in the ß-phase field followed by recrystallization is an effective industrial practice for reducing the grain size of the wrought product. This process, popularly k n o w n as 'ingot b r e a k d o w n ' increases the production cost of the finished titanium products and restricts its extensive use to some extent. A n y development in terms of grain refinement in as-cast titanium and its alloys will certainly popularize its massive use. Recently, Zhu et a l have observed that small amount ( - 0 . 0 8 6 to 0.14 m a s s % ) of boron addition induces a significant refinement of ascast structure and improvement of mechanical properties like tensile ductility, strength and hardness for cast C P Titanium and Ti-0.5Si alloys. Similar observations are reported by other . Tamirisakandala et a l have shown a grain size reduction of as-cast Ti-6AM1V researchers by about an order of magnitude (from 1700 to 200 μηι) with an addition of only 0.1wt.% boron, while much weaker dependence is observed for boron additions from > 0 . 1 % to 1.0%. The concept of boron addition for grain refinement is, however, not a new thought and is quite . The grain refinement of cast Ti alloys via frequently being adopted for aluminium a l l o y s ' boron addition are hypothesized as the effect of constitutional super-cooling, caused by the boron rejection from the primary ß-Ti grains into the liquid ahead of the solidification front, which in turn increases the nucleation rate but prevents the nucleus to g r o w . Wherein Z h u et a l have reported that the tensile ductility of cast TW6A1-/4V alloy can not be improved with boron addition, a m o r e recent study by Sen et a l " have shown that with the refinement in the microstructure, the yield and ultimate tensile strengths increase whereas the fracture toughness and the threshold for fatigue crack propagation decreases. All these experimental findings leads to the increasing importance of the boron modified Ti-alloys. The texture evolution of as cast boron modified Ti-6A1-4V alloy is, however, still lacking, which is the prima facie of the present work. 8
9 1 0 1 1
9
2 1 3 1 4
9
8
EXPERIMENTAL Material: The materials used in this present study are the widely used Ti alloys, T i - 6 A 1 - 4 V , with and without the boron addition (referred to hereafter as Ti64 and T i 6 4 - B , respectively). The detailed composition in terms of weight per cent is given in T a b l e - 1 . Ingots from each of these compositions were melted at Flowserve Corporation, Dayton, O H in an induction skull melting chamber and cast into graphite m o l d s . The cast ingot dimensions were 72 m m diameter X 500 m m length. The boron was added in the form of elemental boron that completely dissolved in the liquid melt. All the ingots were subjected to a standard hot isostatic pressing (HIP) treatment at 900"C and 100 M P a for 2 hours. Since the HIP temperature is well b e l o w the ß-transus
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temperature (~1000°C). the microstructural features of as-cast and cast + H I P will be essentially identical. T h e ingots were radiographed after HIP and confirmed to be free from porosity. Table-1 : Chemical compositions (in wt. % ) of boron modified Ti alloys used in the study Alloy
Al
V
Ti64
6.6
4.1
TÎ64-B
6.0
4.0
Β
0.1
Ο
Η
C
Ν
Fe
Ti
0.18
0.008
0.02
0.01
0.23
Balance
0.15
0.005
0.02
0.01
0.13
Balance
Microstructural Characterization A s s h o w n in F i g . l , a thin strip of dimension 3 5 m m (length) X 7mm (width) X 5mm (thickness) was cut from the periphery to the center of the as-cast ingot. This thin strip was then divided into six equal pieces. The microstructures of the samples were characterized by scanning electron microscopy . The samples w e r e first metallographically polished up to 2500 grit sized silicon carbide paper and finally electro-polished using Struers® Lectropol-5 . The electrolyte contains 6 0 0 m l methanol. 360ml butoxyethanol and 60ml of perchloric acid. The samples were etched with a custom made etchant containing 5 % H F . 1C/oHNOj and rest water for 30 seconds. +
72 mm
Periphery
Fig. 1 : Experimental procedure showing (a) a bar is cut from the as-cast ingot and (b) the bar is divided into six equal pieces for texture measurement. This method is applied for both Ti64 and Ti64-B.
Textural Characterization Each of the pieces from the original thin strip was were subjected to Orientation Imaging Microscopy using a Field Emission G u n Scanning Electron Microscope* with Electron BackScattered Diffraction ( E B S D ) facility. All the samples were prepared in the same way as described above, ultrasonically cleaned at every step of polishing and degassed completely before using for the E B S D scan. F r o m each of the samples, an area of l m m X 4 m m was scanned
' QUANTA 200, FEI, The Netherlands * Struers A/S, Germany FEG-SEM, Sirion XL-40, FEI, The Netherlands :
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with a step size of 5 p m . The accelerating voltage used was 20 K V and m i n i m u m boundary misorientation was taken to be 2°. Titanium alloys, as reported in the literature, are very prone to oxidation in the normal condition so that sufficient care w a s taken before putting the samples for E B S D scan. Textural analysis (pole figures, inverse pole figures, IPF maps etc.) w a s done using O I M ™ Analysis Software . In the entire study, only alpha textures have been studied though these alloys contain substantial amount of beta phase ( 1 0 % ) . This is because beta phase is present between alpha laths and grain boundaries and hence one has to use smaller step size in E B S D scans to detect it. In order to get a quantitative estimate of alpha texture, larger area needs to be scanned that in turn makes impractical to detect beta phase using a smaller step size. Hence, the entire study is focused on the texture of the alpha phase that constitutes about 9 0 % of these alloys. Another important point to ponder over is the comparability between the sizes of the prior β grains and the area used for E B S D scans (4 square m m ) . As h a s been observed by in an earlier study of one of the authors, for Ti64, β grain size is ~ 1 . 7 m m . It may so appear that the area scanned in E B S D is insufficient to capture too many β grains. In order to increase statistical significance, therefore, a suitable area where at least 3-4 β grains meet is selected. This difficulty, however, is not critical for Ti64-B as boron addition reduces β grain size to ~ 0 . 2 m m , so that sufficient n u m b e r of β grains does interact with the E B S D scanning area. 8
RESULTS AND DISCUSSIONS:
Microstructural Characterization: Fig. 2 shows the microstructural m a p p i n g of the as-cast Ti64-B sample starting from the periphery towards the center of the ingot. The microstructure is almost h o m o g e n o u s and uniform throughout in terms of all kinds of microstructural length scales. This observation is quite consistent with the earlier works by different researchers ' and can be attributed to the lower solidification rate during the induction skull melting process which ensures the formation of equiaxed beta grains instead of c o l u m n a r grains ''. The microstructural refinement in Ti64 by boron addition is clearly visible from Fig. 2b and Fig. 2b inset. Hexagonal TiB crystals can be seen sitting at the grain boundary (see Fig. 2a inset) and forming a typical ' n e c k l a c e ' structure (see Fig. 2c inset), as has also been observed by earlier r e s e a r c h e r s ' ' °. Additional evidence can be deduced from the inverse pole figure m a p s generated from E B S D analysis, which supports both the facts that boron refines the grain size tremendously as well as induction skull melting produces an uniform macro as well as microstructure throughout the ingot. 1 0
1
8
5
TSL Crystallography. AMETEK Inc.. USA
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T e x t u r e Evolution in B o r o n M o d i f i e d T Î - 6 A I - 4 V Alloy
Fig. 2: Microstructural mapping of the as-cast TÎ64-B sample from the periphery' towards the Center (a-c); (a) B S E micrograph taken from C u t - 1 ; inset: hexagonal TiB crystals are frequently found along a grain or colony boundary, (b) SE micrograph taken from Cut-3; inset: micrograph taken at A lower magnification from the corresponding section of Ti64 sample shows much less microstructural refinement without the boron addition: (c) SE micrograph taken from Cut-3; inset: typical ' n e c k l a c e ' structure of TiB particles at the prior β grain boundary, (d) BSE micrograph taken from Cut-6; inset: TiB particles decorate the prior β grain boundary.
Texture Characterization: T h e texture data for the boron modified Ti64 is given in Fig. 3 (pole figures) vis-à-vis normal Ti64. T o estimate the gradient in the texture of the solidified ingot, bulk texture measurement using X R D is inefficient as it fails to give localized information. The only way to study texture gradient is by using E B S D ; however large areas need to be scanned in order to get statistically significant information. W h e n brought to the same intensity level, the pole figures for Ti64-B actually show less n u m b e r of contour lines than that of Ti64. Also, the number of intensity m a x i m a is relatively high for Ti64-B. This randomness in the solidification texture of the as-cast material can be attributed to the basic grain refinement m e c h a n i s m by boron addition in titanium alloys. As discussed earlier, boron addition increases β nucleation and suppresses its growth. In the process, it not only reduces the prior β grain size, but also increases the number of
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T e x t u r e Evolution in Boron Modified TÎ-6AI-4V Alloy
α colonies in a given sampling area as compared to Ti64. F r o m a single β orientation. 12 different α orientations of (0001 ( type can be generated consistent with the B u r g e r ' s orientation relationship' . Thus, any refinement in the prior β grain size will also refine the α colony size that can lead to randomization in the α phase texture, unless a variant selection process takes place during the transformation. T h e pole figures show that in general, the intensity levels are lower for the ΤΪ64-Β samples than Ti64 (see Fig. 3). This indicates that boron addition leads to a weaker texture. Also, there are no strong c o m p o n e n t s present in any of the samples. O n the other hand, the samples without boron shows higher intensity for certain texture components in different cuts. Boron addition thus not only refines the microstructure but also considerably weakens the texture and m a k e s it m o r e h o m o g e n e o u s . 0
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T e x t u r e Evolution in B o r o n M o d i f i e d T Î - 6 A I - 4 V Alloy
Fig. 3 : (0001) pole figures of α - p h a s e for as-cast Ti64-B vis-à-vis as-cast Ti64. Both types of pole figures are expressed in terms of similar intensity levels. T h e α phase texture that is investigated in the present study has its origin in the solidification texture of β phase w h i c h in t u m undergoes phase transformation to α phase. Thus, the solidification texture of α phase and the transformation texture during β to α transformation govern the final texture. The increase in nucleation rate of β due to rejection of boron leads to a m o r e r a n d o m beta texture. During transformation from beta to alpha phase, each beta orientation can be transformed to 12 equivalent α orientations with same probability. However, under certain conditions, only some of these alpha orientations are formed. This phenomenon is known as variant selection. In the present case, T i B needles are found to be present at the prior β grain boundary and hence it can act as an additional site for the nucleation of α from β during phase transformation. This further contributes to weakening the α texture upon transformation. Thus, the addition of boron has two fold effects on the texture evolution in Ti64; firstly, it affects the solidification texture of β phase and secondly it again alters the β to α transformation texture. CONCLUSION: The microstructural and micro texture evolution during solidification of E B M Ti64 and TÎ64-B ingot has been studied using E B S D . It is concluded that addition of boron refines the microstructure as well as modifies/weakens the texture of the solidified ingot. The uniform finer grain size along with a w e a k e r texture is expected to facilitate further thermomechanical processing of the ingot to finished products. ACKNOWLEDGEMENT: The authors cordially acknowledge the Asian Office of Aerospace Research and Development ( A O A R D ) for the financial support and Institute N a n o s c i e n c e Initiative (INI). Indian Institute of Science. Bangalore, India for providing the required research facilities to carry out the present investigation.
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REFERENCES: 1
A S M Metals H a n d b o o k V o l u m e 2 . "Properties and selection of nonferrous alloys and special purpose materials", A S M International, Tenth Edition " E. W. Collings. " T h e Physical Metallurgy of Titanium A l l o y s , " American Society for Metals. Metals Park. O H 4 4 0 7 3 G. Liitjering. and J. C. Williams. " T i t a n i u m " , Springer-Verlag Berlin Heidelberg 2003 G. G. E. Seward . S. Celotto, D. J. Prior. J. Wheeler, and R . C . Pond. "In situ S E M - E B S D observations of the hep to bcc phase transformation in commercially pure titanium." Acta Materialia, 5 2 . 8 2 1 - 8 3 2 (2004) P. A. Kobryn. S. L. Semiatin, "Microstructure and texture evolution during solidification processing o f T i - 6 A 1 - 4 V , " Journal of Materials Processing Technology, 1 3 5 . 3 3 0 - 3 3 9 (2003) A. N. Kalinyuk. N. P. Trigu, V. N . Z a m k o v . O . M. Ivasishin, P. E. M a r k o v s k y . R. V. Teliovich. and S. L. Semiatin, "Microstructure. texture, and mechanical properties of electron-beam melted Ti-6A1-4V," Materials Science and Engineering A. 3 4 6 , 178-188 (2003) M. G. Glavicic. P. A. K o b r y n . F. Spadafora. and S. L. Semiatin, "Texture evolution in v a c u u m arc remelted ingots of Ti-6A1-4V," Materials Science and Engineering A, 3 4 6 . 8-18 (2003) J, Z h u . A. K a m i y a . T. Y a m a d a , W. Shi, and K. N a g a n u m a , "Influence o f boron addition on microstructure and mechanical properties of dental cast titanium alloys." Materials Science and Engineering A. 3 3 9 , 53-62 (2003) '' S. Tamirisakandala. R. B . Bhat. J. S. Tiley. and D . B . Miracle, "Grain refinement o f cast titanium alloys via trace boron addition," Scripta Materialia, 5 3 , 1 4 2 1 - 1 4 2 6 ( 2 0 0 5 ) '" S. Tamirisakandala, R. B . Bhat. D, B. Miracle. S. Boddapati, R. Bordia. R. Vanover. and V. K. Vasudevan, "Effect of boron on the beta transus of T i - 6 A 1 ^ 4 V alloy," Scripta Materialia. 5 3 , 2 1 7 - 2 2 2 (2005) " I. Sen. S. Tamirisakandala. D. B . Miracle, and U. R a m a m u r t y . "Microstructural effects on the mechanical behavior of B-modified T i - 6 A 1 - 4 V alloys." Acta Materialia, 5 5 . 4 9 8 3 - 4 9 9 3 (2007) B. Hu, and H. Li, " G r a i n refinement of D1N226S alloy at lower titanium and boron addition levels," Journal of Materials Processing T e c h n o l o g y , 7 4 , 5 6 - 6 0 (1998) " H. Li. T. Sritharan, and H . P. S e o w , "Grain refinement in D I N 2 2 6 alloy at high titanium and boron inoculotion levels." Scripta Materialia, 3 5 ( 7 ) , 869-872 (1996) M . W a n g . S W a n g . Z. Liu, Z. Liu. T. Song, and X. Z u o , "Effect of B/Ti mass ratio on grain refining of low-titanium a l u m i n u m produced by electrolysis," Materials Science and Engineering .4.416.312-316(2006) J. P. K u a n g . R. A. Harding, and J. C a m p b e l l , "Microstructures and properties of investment castings of γ-titanium A l u m i n i d e . " Materials Science and Engineering A . 3 2 9 - 3 3 1 , 3 1 - 3 7 (2002) L. Z e n g , T. R. Bieler, "Effects of working, heat treatment, and aging on microstructural evolution and crystallographic texture of a . a", a " and β phases in T i - 6 A 1 — W w i r e , " Materials Science and Engineering A, 3 9 2 , 4 0 3 - 4 1 4 (2005) 1
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H E T E R O G E N E O U S D E F O R M A T I O N IN S I N G L E - P H A S E Z I R C A L O Y 2 1
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S. K. S a h o o , V. D. H i w a r k a r , 1 . Samajdar , P. P a n t , D. Srivastav , R. T e w a r i , G. K. D e y and S. Banerjee 2
'Department of Metallurgical Engineering and Materials Science, I IT Bombay, Mumbai-400076, India 2
B h a b h a Atomic Research Centre. Trombay, M u m b a i , India
ABSTRACT Deformed single-phase Zircaloy 2 shows a clear pattern of heterogeneous deformation. The deformed mierostructures could be generalized as grains/orientations which got fragmented and those w h i c h did not. The presence of deformation twinning, observed only at low strains, does not explain this phenomenon, which existed even at relatively higher (200-600°C) deformation temperatures where deformation twinning is absent. The source of this heterogeneous deformation appears to exist in the frame-work of differences in dislocations interactions at different crystallographic orientations. Though the exact source/origin of the heterogeneous deformation can still be debated, its effects on the deformed mierostructures were apparent in terms of clear bi-modal distributions in grain size, in in-grain misorientation developments and in the patterns of stored energy of cold work and residual stress. Key words: Heterogeneous Stress, Stored Energy.
Deformation,
Twinning,
Zirconium,
Grain Fragmentation,
Residual
INTRODUCTION In single phase metals and alloys, a large a m o u n t of literature exists on heterogeneous deformation [1-11]. Such studies are mostly concerned with differences in the extent of deformation, and corresponding differences in microstructural developments. Real heterogeneous deformation, where different grains/phases of the microstructure have remarkably different strains, is reported only in multi-phase alloys. Zircaloy 2 (chemical composition given in table 1), is used in nuclear p o w e r reactors as tubes and claddings [12], Typical microstructure involves equiaxed or elongated grains (based on prior processing) of hep structure, with submicron intermetallic precipitates. Such a Zircaloy 2 w a s subjected to near plane .strain deformation and uniaxial compression. The objective of the present study was to identify the extent of heterogeneous deformation in hep Zirconium. Table 1. Chemical composition, in weight % alloying elements, of Zircaloy 2.
Sn 1.54
Fe 0.15
Cr 0.12
Ni <0.05
0 0.12
Zr Balance
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H e t e r o g e n e o u s D e f o r m a t i o n in S i n g l e - P h a s e Z i r c a l o y 2
EXPERIMENTAL DETAILS Fully recrystallized single phase Zircaloy 2 was subjected to cold rolling in a laboratory rolling mill to 20 and 5 0 % reductions and w a s also subjected to uniaxial compression in a servohydraulic M T S to 2, 5, 7.5. 10, 13. 16, 20 and 4 2 % reductions. Samples from mid-thickness section (sectioned parallel to rolling direction and perpendicular to compression direction respectively) of each deformation conditions were electropolished using an electrolyte of 80:20 methyl alcohol and perchloric acid under 21V at -40°C and w e r e then subjected to E B S D (electron backscattered diffraction) measurements and also to X-ray diffraction ( X R D ) . The E B S D were performed using a T S L - O I M package on a Fei quanta-200 S E M (scanning electron microscope), while a Panalytical M R D system was used for X R D measurements. RESULTS Cold rolling of Zircaloy 2 Fig. 1 s h o w s the inverse pole figure (IPF) m a p s of the Zircaloy 2 before (Fig. l a ) and after (Fig. 1 b. c and d) the cold rolling. The initial equiaxed structure shows two types of grains after deformation - (i) grains which had undergone refinement in size - termed as deforming grains (below 3 microns) and (ii) larger grains - non-deforming grains (above 3 microns). Equiaxed type (ii) grains were observed, albeit to a lesser extent, even after 5 0 % reduction and at all crosssections - see fig. 1. Fig. 2 s h o w s the grain size distribution before and after cold rolling. As shown in fig. 3 non-deforming grains w e r e mostly near basal i.e. (011 5) < u v t w > . while deforming grains had a range of orientations (though developments of certain preferred orientations are visible). A s s h o w n in table 2. the non-deforming grains had insignificant grain average misorientation. while the smaller deforming grains had lower grain orientation spread, and the non-deforming grains were elastically harder.
100μπι Figure 1. Inverse pole figure m a p s of (a) initial microstructure. (b) & (c) normal and transverse direction microstructure after 2 0 % cold rolling and (d) microstructure after 5 0 % cold rolling. Rolling ( R D ) . normal direction (ND) and transverse ( T D ) directions are marked.
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ο υ CO
Grain Size (microns) [logscale] Figure 2 . Grain size distribution before and after rolling Ψ, (0.0°-90.0°) ι
>
Ψ Φ (0.0° - 90.0°) Constant Angle: φ, = 0
(a) (b) (c) (d) Figure 3 . Euler space plots of φι = 0 ° sections of (a) starting structure, (b) (a) after 2 0 % cold reduction, (c) non-deforming grains in (b) and (d) deforming grains in (b). Distinction between (c) & (d) was m a d e from grain size - (c) above 3 p m and (d) below 3 p m in size. Intensity levels were 0 . 5 . 1, 2 , 3 , 4 , 5 and 6 times random texture. Table 2 . T h e average Grain average misorientation ( G A M ) , the average grain orientation spread (GOS) and the orientation estimated average elastic moduli are provided. Different Properties
Undeformed Zircaloy 2
All grains
Deformed Zircalov 2 Non-deforming grains
Deforming grains
Average G A M 0.40 0.63 1.79 1.16 Average GOS 0.43 2.52 2.21 1.48 A v e r a g e Elastic M o d u l u s 98 GPa 6 7 GPa 73 GPa (These were estimated by providing a criterion for grain boundary (misorientation exceeding 15°) and then isolating the grains. From such 'isolated' grains G A M (misorientation between neighboring points of a grain) and G O S (misorientation between all measurement points of a grain and the grain average orientation) were estimated. Corresponding values for undeformed, deforming (below 3 micron), non-deforming (above 3 micron) and all (deforming + nondeforming) grains are listed. Based on known single-crystal elastic constants of hexagonal Zr, orientation information and appropriate strain tensor, average polycrystalline elastic modulus
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values were estimated using the Reuss (uniform stress assumption-inverse rule of mixtures) averaging scheme [13. 14].) Uniaxial compression of Zircaloy 2 The same recrystallized Zircaloy 2 samples, albeit at different orientation - see fig. 4, were taken for controlled cold uniaxial compression. Fig. 4 s h o w s IPF m a p s to bring out overall microstructural changes and IPF from bulk crystallographic texture at different percentage compressions. Note that only {1012} < 1011 > tensile twins were observed. Both frequency and relative length of the twin boundaries increased with deformation and then dropped. Significant presence of twins also coincided with substantial developments in crystallographic texture, while beyond 1 3 - 1 6 % deformation twin boundaries were relatively rare and changes in bulk texture were insignificant. The last set of results, fig. 5 and table 3 , summarizes observations on microstructural developments. T h e poles used in fig. 5 correspond to the suspected parent (0111), ( 0 2 2 1 ) . (1231), and product (0114) and (0115) orientations/fibers (fibers have s a m e crystallographic plane with all possible crystallographic directions). Suspected parent and product orientations/fibers were classified based on relative Taylor factor values (for prismatic,{0110} < 2110 > , slip). Suspected parents have relatively high value of Taylor factor compared to suspected products. Their orientations can be estimated from easy partitioning of the data in E B S D . As shown in fig. 5b and table 3 , they showed clearly different patterns of stored energy and residual stress developments. Stored energy of suspected product was slightly higher, but dropped drastically beyond 2 0 % deformation. Suspected product had tensile residual stresses in contrast with compressive residual stresses in suspected parent. Finally, the pattern of lattice strain, see fig. 5a, also provides an interesting picture. In the regime of significant twin presence, below 1 6 % deformation, lattice strain d e v e l o p m e n t s w e r e relatively small. A significant increase in lattice strain took place between 16-20% deformations, while a noticeable drop in lattice strain (indicative of softening) was observed above 2 0 % deformation.
DISCUSSIONS Cold rolling of Zircaloy 2 The cold rolled Zircaloy 2, clearly had t w o types of grains - deforming and non-deforming. This classification, though simplified in approach, is based on refinement in size. The classification can also be extended to changes in grain shape or aspect ratio and to developments in grain average misorientation, see figs. 1 and table 2. For example, near basal grains, ( 0 1 1 5 ) < u v t w > being the average orientation, did not get fragmented or changed their shape - they acted more like non-shearable particles. Difference in elastic stiffness appears to be the source of this ' e x t r e m e ' heterogeneous deformation. Uniaxial compression of Zircaloy 2 At relatively low strains{l012} < 1011 > twins were observed on orientations (<1010> & <21 10>). The suspected parent and product orientations had differences in stored energy of cold work, fig 5, and even stronger differences in residual stress, table 3. In other words, at the initial deformation stages (upto 1 3 - 1 6 % compression) twinning could create a clear pattern of heterogeneous deformation - especially in terms of stored energy and residual stress distribution.
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Figure 4. Microstructure and texture of Zircaloy 2 with progressive deformation by uniaxial compression. Texture, in inverse pole figure representation, was obtained from XRD; while inverse pole figure maps were obtained from EBSD. Intensity levels in bulk texture were 3.5, 4, 5, 6, 7 and 7.5 times random texture.
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Figure 5. Effects of deformation on X R D estimated values of (a) lattice strain, (b) stored energy. Respective poles, first 3 being suspected parent & the last two of suspected product, were taken for (b). Lattice strains were estimated from the X-ray peak broadening i.e. F W H M (full width half m a x i m u m ) values [15] and stored energv values were estimated using Stibitz formula [16, 17]. Table 3. X R D estimated values of residual stress (RS) with standard deviations (SD) for different percentage compressions. Respective poles of suspected parent & suspected product were taken for the residual stress estimation. For RS standard Sin"v|/ method [18] w a s used, while SD values were estimated using ψ splitting algorithm [19].
Suspected Parent
Percentage Compression
598
Suspected Product
(0221)
(01ÎI)
(01Ϊ4)
(1231)
RS
SD
RS
SD
RS
SD
RS
2.0
-42.4
14.1
46.2
7.6
45.5
19.2
-
5.0
-65.1
13.2
-64.1
22.0
-54.3
2.0
-
7.5
-92.1
20.0
-133.1
12.0
-64.7
1.4
113.2
(0ΙΪ5) SD
RS
SD
-
-
-
-
-
-
10.4
127.5
16.0
10.0
-134.6
32.0
-166.3
16.0
-79.6
15.0
143.3
7.0
146.9
6.0
13.0
-140
3.0
-277.1
16.0
-71.6
3.0
125.0
31.0
128.3
4.0
16.0
-1798
12.0
-277.5
12.0
-119.3
4.0
123.0
24.0
125.7
6.0
20.0
-206
20.0
-296.6
12.0
-134.7
8.0
60.0
14.0
64.5
14.0
42.0
-91
6.0
-45
22.0
-25
17.0
118.0
34.0
133.0
15.0
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H e t e r o g e n e o u s Deformation in S i n g l e - P h a s e Zircaloy 2
Such patterns of heterogeneity continued, albeit modified significantly - see fig 5, even beyond 16% compression, a deformation state where visible (i.e. visible through E B S D ) twinning was absent. At such reductions, mierostructures also revealed presence of deforming and nondeforming grains. T o summarize, a clear picture of orientation dependent heterogeneous deformation in single phase Zircaloy 2 emerges in the present study. The first part of the heterogeneous deformation was through orientation dependence of deformation twinning, while the second part is presently hypothesized to be a reflection of possible differences in dislocation interactions between different grains/orientations. On-going research experimental as well as modeling through dislocation d y n a m i c s , is aimed at testing the aforementioned hypothesis CONCLUSIONS •
•
The plane strain deformed mierostructures could easily be distinguished as deforming and non-deforming grains. Non-deforming grains were of larger size and with insignificant grain average misorientation. Even after 5 0 % reduction in thickness, they remained nearly equiaxed. U n d e r uniaxial compression & at relatively low strains (upto 13-16% compression), < 1 0 1 0 > to < 2 1 1 0 > oriented grains showed {1012} < 1011 > type of tensile twins <0001> grains were the apparent twinning product. Significant changes in bulk crystallographic texture were observed only during early deformation stages. Upto 13-16% compression, where deformation twinning had a significant presence, heterogeneous deformation was mostly in terms of differences in residual stress and stored energy distribution. Beyond 13-16% compression, in addition to differences in residual stress and stored energy values, clear presence of deforming and nondeforming grains (a distinction made from noticeable fragmentation - or reduction in grain sizes) was also observed.
ACKNOWLEDGEMENT Support from B R N S (Board of Research on Nuclear Science) and from D S T (Department of Science & Technology) are acknowledged.
REFERENCES 1.1.L. Dillamore, J.G. Roberts and A . C . Bush, Metal Sei., 13 . 7 3 , (1979). 2. J. Gil Sevillano, P. Van Houtte and E. Aernoudt, Prog. Mater. Sei. 2 5 , 379, (1981). 3. F. Delaire, J. L. Raphanel and C. Rey, Acta Mater., 4 8 , 1075. (2000). 4. D. Raabe, M . Sachtleber, Ζ. Zhao, F. Roters and S. Zaefferer, Acta mater. 4 9 , 3433, (2001). 5. T. R. Bieler and S.L. Semiatin, Int. J. Plasticity, 18, 1165, (2002). 6. L. Delannay, R.E. Loge, Y. Chastel and P. Van Houtte, Acta Mater.,59, 5127, (2002). 7. A. Tatschl and O. Kolednik, Mater. Sei. Eng. A 3 4 2 , 152, (2003). 8. S. R. Kalidindi, A. Bhattacharyya and R. D. Doherty, Proc. R. Soc. lond. A 460, 1935, (2004). 9. N . Z h a n g and W. Tong, Int. J. Plasticity 20, 5 2 3 , (2004).
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10. L. B. Zuev. V. I. Danilov, T. M. Poletika and S. A. Barannikova, Int. J. Plasticity 20, 1227, (2004). 11. M. Kiran Kumar, C. Vanitha, I. Samajdar, G. K. D e y , R. Tewari, D. Srivastava and S. Banerjee, Mater. Sei. Tech. 22(3), 3 3 1 , (2006). 12. C. Lemaignan and A.T. Motta, Materials Science & Technology, eds. R.W. Cahn, P. Hassen and E.J. K r a m m e r , v o l . 10B ( p a r t II), V C H . W e i n h e i m - G e r m a n y , 1,(1993). 13. T. Mura. Micromechanics of defects in solids, Matrinus Nikhoff, Dordrecht ( 1987), 4 2 1 . 14. R. Bauri, V. Pancholi, I. Samajdar and M.K. Surappa, Science and Technology of Advanced Materials, 6, 9 3 3 , (2005). 15. B. D. Cullity. ' E l e m e n t s of X-ray diffraction, 2nd edn, Addision-Wesley Publishing, 4 6 0 , (1978). 16. G. R. Stibitz, Phys. Rev., 4 9 , 862, (1937). 17. G. R. Stibitz, Phys. Rev., 8. 147. (1947). 18. P. Van Houtte and L. De B u y s e r . ^ c W Metall. Mater., 41(2), 3 2 3 , (1993). 19.1. C. N o y a n and J. B. Cohen, Residual Stress, Springer-Verlag, N e w York, (1987).
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T E X T U R E O F M A G N E S I U M A L L O Y S H E E T S H E A V I L Y R O L L E D BY HIGH S P E E D WARM ROLLING 1
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Tetsuo S a k a i , Satoshi Minamiguchi'", Hiroaki K o h , Hiroshi U t s u n o m i y a 'Division of Materials and Manufacturing Science, Graduate School of Engineering, Osaka University G r a d u a t e student Suita, Osaka, Japan a
ABSTRACT M a g n e s i u m alloy sheets had to be rolled at elevated temperatures to avoid cracking. The poor workability of m a g n e s i u m alloy is ascribed to its hep crystallography and insufficient activation of independent slip systems. Present authors have succeeded in one-pass large draught rolling of AZ31 and Z K 6 0 A magnesium alloy sheet below 200 °C by raising rolling speed above lOOOm/min. Heavy reduction larger than 50 % can be applied by one-pass high speed rolling even at room temperature. The improvement of workability at lower rolling temperature is due to adiabatic temperature rise of a material during rolling by plastic working. The texture of heavily rolled m a g n e s i u m alloy sheet is investigated in the present study. T h e texture of A Z 3 1 B sheet rolled 60 % at room temperature was <0001>//ND basal texture. At the rolling temperature above 100 °C, the peak of (0001) pole tilted 10-15 degrees along R D direction around T D axis to form a double peak texture. The texture varied through the thickness. At the surface, the (0001) peak tilted 10-15 deg along T D direction around RD axis. The direction of (0001) peak splitting rotated 90 degrees from the surface to the center of thickness. T h e texture of heavily rolled Z K 6 0 A sheets is almost similar to that of A Z 3 1 B sheets. Heavily rolled m a g n e s i u m alloy sheets have non-basal texture. The sheets having non-basal texture are expected to show better ductility than sheets with basal texture. INTRODUCTION Magnesium alloy sheets had to be rolled at elevated temperatures to avoid edge cracks. The need for processing at elevated temperatures requires additional cost of magnesium alloy sheet products and is a significant impediment to their introduction to automotive industries. The poor workability of m a g n e s i u m alloy is ascribed to its hep crystallography and insufficient activation of independent slip systems. They are mostly shaped by casting or thixo-moulding. Wrought m a g n e s i u m products are still limited, though d e m a n d for them increases rapidly. Thin sheets and strips are expected to be used in quantity for electronic applications and automotive parts in near future. M a g n e s i u m sheets are sometimes produced by rolling from cast slabs. In cold rolling, applicable reduction in thickness is less than 10% due to the poor ductility at room temperature. Therefore the thickness is mostly reduced in hot or warm rolling. In the process, multi-pass operation with small reduction accompanied by intermediate annealing is employed to suppress edge cracks or fracture of the material and to maintain the workability'. Rolls are often heated to minimize the temperature drop of sheets during rolling. Because of all these procedures, fabrication of m a g n e s i u m alloy sheets is less productive and m a g n e s i u m alloy sheets are more expensive than cast m a g n e s i u m products. If the productivity is improved, they should be used in quantity, especially in automotive industries.
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T e x t u r e of M a g n e s i u m Alloy S h e e t s H e a v i l y Rolled b y H i g h S p e e d W a r m Rolling
The authors have proposed high speed rolling for fabrication of m a g n e s i u m alloy sheets". During rolling of a sheet, heat is generated by plastic deformation and by friction between the material and rolls. M e a n w h i l e the heat transfers from the material to cold rolls. With increasing the roll speed, the duration where the material and the rolls are in contact b e c o m e s shorter. This results in effective temperature rise of the sheet during rolling. So fracture or cracks due to the low-temperature brittleness can be suppressed considerably by high speed rolling. Furthermore, higher strain rate deformation at lower temperature, i.e., high Zener-Hollomon parameter, is advantageous for grain refinement by dynamic recrystallization. T o improve plastic workability of magnesium alloys at lower temperature, especially at room temperature, grain refinement is a very promising method. Another important and promising method is controlling or changing the texture of sheets, because poor workability of m a g n e s i u m alloy is ascribed to its limited activity of slip systems and its sharp basal texture. T h e one pass large draught high speed rolling at relatively lower temperatures may provide new tools for manipulating the texture of m a g n e s i u m alloy sheets. In the present study, the rolling texture of magnesium alloy sheets rolled to large reduction in one pass at high speed up to 2000 m/min is investigated. EXPERIMENTAL A two-high laboratory high speed rolling mill with φ530 m m rolls schematically shown in Fig.l w a s used. The rolling speed can be varied from 200 m/min to 2600 m/min. Detailed specification can be found e l s e w h e r e , where the authors studied microstructure and texture evolution of steels and other metals systematically. Commercial 2.5 m m thick A Z 3 1 B ( M g - 3 % A l - l % Z n - 0 . 4 % M n ) annealed sheets were received. T h e initial microstructure was covered with equiaxed grains of which m e a n grain size was 12 p m . C o m m e r c i a l Z K 6 0 A ( M g - 5 . 6 % Z n - 0 . 5 % Z r ) sheets with the thickness of 2.5 m m sliced from an extruded bar were also received. Z K 6 0 A samples were solution treated at 500 °C for 7.2 ks before rolling. Specimens with 30 m m in width and 300 mm in length were cut from the sheets and subjected to the rolling experiment. Prior to the rolling, a specimen was hold for 900 s at 100. 150. 2 0 0 or 350 °C in an electric tube furnace, then supplied to the rolling mill through the pinch roller. Alternatively cold rolling w a s also performed without heating samples and rolls. The rolling conducted in single-pass operation with reduction in thickness from 30 % to 60 %. The peripheral speed of the rolls w a s 1000 and 2000 m/min. The mean strain rate έ was estimated as follows. 34
Figure 1. Illustration of high-speed experimental rolling mill.
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έ = \·εΙΙ, s = -2\n(l-r)/JX L = -jRÄh where ε is an equivalent strain, ν is a roll peripheral speed. r(=0.5 in the present experiment) is a reduction in thickness, L is a length of contact arc, R is a roll radius and Ah is a thickness change. T h e estimated strain rate during 5 0 % rolling reduction is 7.5 χ 1 0 s*' and 1.5*10 s' respectively. The specimen w a s quenched immediately (precisely, 7 m s for 2000 m/min and 14 m s for 1000 m / m i n ) after rolling by the water spray closely attached to the exit of the mill. The rolls were neither heated nor lubricated. 2
3
1
Mierostructures on longitudinal section ( T D plane) were observed by an optical microscope. The (0001) incomplete pole figures were measured by Schulz reflection method using Cu-K.ct radiation. The contours in pole figures were expressed in units of the intensity from the standard sample prepared by p o w d e r compaction and sintering. RESULTS AND DISCUSSION High Speed Rolling o f A Z 3 1 Β Occurrence o f defects at edges of the sheets rolled at high speed of 2 0 0 0 m/min is summarized in Fig.2. It is possible to reduce the thickness u p to 6 0 % without fracture at both 200 °C and 350 °C. Heavy reduction up to 60 % is attained in single pass by high-speed rolling. Minor edge cracks are observed at 200 °C. H o w e v e r , these cracks may not be detrimental practically because much w i d e r sheets are rolled in industry and e d g e s are usually trimmed after rolling. Higher reduction than 6 0 % may be applicable. The m a x i m u m reduction attainable in the present study is not clear because further reduction w a s not applicable due to elastic deformation o f the rolls and the housing. At lower temperature ( R T and 100 °C). it is found that 6 0 % reduction is applicable without fracture even at r o o m temperature. However deeper edge-cracks arranged regularly are observed. The cracks are d u e to the conjugated shear bands, i.e., shear bands running along 4 5 and 135 degrees to the rolling direction, which are arranged by turns periodically. In addition, at 200 °C with small reduction (34 % ) , cracks initiated at
Figure 2. Defects o f A Z 3 1 B sheets rolled at 2000m/min.
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T e x t u r e of M a g n e s i u m Alloy S h e e t s H e a v i l y Rolled b y H i g h S p e e d W a r m Rolling
edges propagates to longitudinal fracture at the center (so-called "scissors c r a c k s ' ) are observed. N o transverse fracture occurs under all the conditions performed. Sufficiently sound sheets are obtained above 200 ° C . It is concluded that high-speed rolling significantly improves workability of m a g n e s i u m alloy sheets and that heavy reduction (> 50 % ) is applicable by singe-pass rolling even at room temperature. Figure 3 s h o w s the microstructure of the sheets rolled 6 0 % at R T , 100. 150. 200 and 350 °C. T h e microstructures of the sheets rolled above 100 °C w e r e covered by fine equiaxed grains attributed to the dynamic recrystallization. The size of grains (= the m e a n intercept length) increases with the rolling temperature. While, in the case of the sheets rolled at R T , twins were observed in grains. In addition, the shear bands run through the thickness with the angle of 45 degrees to the rolling plane, only in which very fine grains are formed by dynamic recrystallization. The m e a n grain size of the sheet rolled at 150 °C w a s 2.9 μηι w h i c h is m u c h smaller than the m i n i m u m grain size commercially available (β-ΐμπί) . 1
Figure 3 Microstructure of A Z 3 1 Β rolled 6 0 % at 2000m/min.
Texture of Rolled A Z 3 1 Β Sheets Effect of rolling temperature on the texture measured at the midthickness of the sheets rolled to the reduction of 6 0 % is s h o w n in Fig.4. The texture is represented by (0001) incomplete pole figures. A sheet rolled at room temperature which has the deformation microstructure s h o w s a typical basal texture. The (0001) peak at the center of the pole figure elongates along the rolling direction. At the rolling temperature of 100 °C, recrystallization occurred throughout the thickness of the sheet. A double peak texture is formed, in which the position of strong (0001) pole intensity tilts ± 1 0 to 15 degrees away from the N D position towards R D around the T D axis. Both the sheets rolled at 200 °C and at 300 °C also show the double peak texture. The distance between t w o peaks (the sum of tilting angle of (0001) peaks from N D position) increases with rolling temperature. The texture measured at the midthickness of the sheet rolled 3 6 % at 200 °C is shown in Fig.5. It also has double peak texture while the peak intensity is higher than that of the 60 % rolled sheet. Recrystallization is not completed in
604
Materials Processing and Texture
T e x t u r e of M a g n e s i u m Alloy S h e e t s H e a v i l y R o l l e d b y H i g h S p e e d W a r m Rolling
Figure 4 Effect of rolling temperature on the texture at the midthickness of the sheets rolled to 6 0 % reduction in single pass rolling. the sheet rolled 3 6 % at 200 °C. T h e retained deformation microstructure increases the (0001) peak intensity. From these figures, it is indicated that the deformation texture and the recrystallization texture are almost similar in rolled magnesium alloys. Figure 6 s h o w s the effect of rolling temperature on the texture measured beneath the surface of the sheets rolled to the reduction of 60 %. A sheet rolled at room temperature shows basal texture. The (0001) peak at the N D position elongates along the T D axis while it elongates along R D axis at the midthickness. With increasing rolling Figure 5 Texture at the midthickness temperature, elongated (0001) peak begins to split of the sheet rolled to 3 6 % to form double-peak texture. However, the direction reduction at 200°C. of split is T D axis which is perpendicular to that at the midthickness. In the present experiment, lubricant is not applied to the roll surface and large shear strain is introduced near the surface of the rolled sheets during rolling due to the high
Figure 6 Effect of rolling temperature on the texture beneath the surface of the sheets rolled to 6 0 % reduction in single pass rolling.
Materials Processing and Texture
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T e x t u r e of M a g n e s i u m Alloy S h e e t s H e a v i l y R o l l e d b y H i g h S p e e d W a r m Rolling
friction. T h e texture variation t h r o u g h the thickness is probably caused by the variation of shear strain. The effect of shear strain o n the deformation texture of h e p metals h a s not been reported in detail.
ZK60, W=1000m/min, Solution treatment(5O0t2h—WQ)
H i g h Speed Rolling of Z K 6 0 A Among magnesium alloys, A Z 3 1 is k n o w n as the most ductile alloy. It remains uncertain w h e t h e r the above-mentioned high limiting reduction by h i g h speed rolling can be attained for other m a g n e s i u m alloys that are less ductile but stronger than A Z 3 1 B . Figure 7 s h o w s the m a p p i n g of defects observed on edges of ZK60A
Figure 7 Effects of rolling temperature and reduction on defects observed on Z K 6 0 A m a g n e s i u m alloy sheets rolled at sheets rolled at 1000 m / m i n . 1000 m/min. Z K 6 0 A alloy, w h i c h is normally a g e d after extrusion, s h o w s the highest room temperature strength o f the c o m m o n l y used wrought magnesium alloys. Z K 6 0 A sheets are commercially manufactured by extrusion and understood that it is m o r e difficult to b e rolled. H o w e v e r , as is s h o w n in Fig.7, Z K 6 0 A alloy sheet is successfully rolled to high reduction by h i g h speed rolling. T h e tendency of the occurrence of edge cracks in Z K 6 0 A sheets is almost similar to that of A Z 3 1 B . It can be concluded that the high speed rolling is effective for single p a s s large draught rolling of t h e w i d e range of Figure 8 Microstructure of Z K 6 0 A sheets q u e n c h e d after 6 0 % m a g n e s i u m alloys. Effect of rolling at lOOOm/min. rolling temperature o n the microstructure of the Z K 6 0 A sheets rolled to 6 0 % at 1000 m/min is s h o w n in Fig.8. Similar to that of A Z 3 1 B , the mierostructures o f the sheets rolled a b o v e 2 0 0 °C are covered by fine equiaxed grains attributed to the d y n a m i c recrystallization. T h e size of grains increases with the rolling temperature.
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Materials Processing a n d Texture
T e x t u r e of M a g n e s i u m Alloy S h e e t s H e a v i l y R o l l e d b y H i g h S p e e d W a r m Rolling
Texture of Rolled Z K 6 0 A Sheets T h e texture of the extruded and solution treated Z K 6 0 A sheet is shown in Fig.9. The texture before rolling consists of a basal component and (0001) peaks on T D axis resulting from extrusion texture. Effect of rolling temperature on the texture measured at the midthickness of the Z K 6 0 A sheets rolled to the reduction of 6 0 % at 1 0 0 0 m / m i n is shown in Fig. 10. A sheet rolled at room temperature w h i c h h a s the deformation microstructure s h o w s typical double-peak texture and weak extrusion texture characterized by (0001) peaks on T D axis. Regardless of rolling temperature, a double peak texture is formed, in which the position of strong (0001) pole intensity tilts ± 1 0 to 15 degrees away from the N D position towards rolling direction around
Figure 9 Texture of the sheet before rolling,
Figure 10 (0001) pole figures measured at the midthickness of Z K 6 0 A sheets rolled to 6 0 % at lOOOm/min.
the T D axis. The texture of rolled and quenched sheets of Z K 6 0 A closely resembles to that of A Z 3 1 B except for retained extrusion texture. The texture beneath the surface of 60 % rolled Z K 6 0 A sheets also had a (0001) peak at N D position with elongated shape along T D direction. However, the peak did not split to double peaks, w h i c h is different from that of A Z 3 1 B . Textures of rolled hep metals have attracted significant interest because of the use of titanium alloys for light structural materials. The rolling textures of h e p metals are classified in three groups according to their c/a r a t i o . According to these reported results, magnesium alloys with c/a ratio approximately equal to the ideal value of 1.633 should exhibit (0001) basal texture. H o w e v e r , in the present study, both A Z 3 1 B and Z K 6 0 A m a g n e s i u m alloy sheets form double-peak texture which is typical in hexagonal metals with c/a ratio greater than ideal value. The deformation mechanism in high speed and one pass large draught rolling, especially the role of twinning is unclear and other factors affecting texture evolution during rolling should be investigated, because the double peak texture seems to be advantageous to plastic workability and to clarify' the mechanism of evolution of double peak texture may have significant industrial meanings. 67
Materials Processing and Texture
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T e x t u r e of M a g n e s i u m Alloy S h e e t s H e a v i l y R o l l e d b y H i g h S p e e d W a r m Rolling
CONCLUSION The limiting reduction in thickness of m a g n e s i u m alloy sheets achieved by single pass rolling is surprisingly increased by high speed rolling faster than 1000 m/min. Reduction larger than 6 0 % can be applied to A Z 3 1 B and Z K 6 0 A alloy sheets by single pass even at room temperature by high speed rolling. A Z 3 1 Β and Z K 6 0 A alloy sheets rolled to 60 % reduction by one pass high speed rolling has double-peak texture in which (0001) peak at N D position splits to t w o peaks along R D direction. T h e (0001) peak elongates to T D direction beneath the surface due to shear deformation caused by high friction between a sheet and rolls during rolling. REFERENCES ' C . S. Roberts, Magnesium and Its Alloys, (John Wiley and Sons, Inc., N J , U S A ) , 171-177(1960). T . Sakai. H. Utsunomiya. S. Minamiguchi, and H. K o h , Single Pass Large Draught Rolling o f Some Magnesium Alloys b e l o w 4 7 3 K by High Speed Rolling, Proc. 7th Int. Conf. on Magnesium Alloys and Their Applications, 370-376 (2006). T . Sakai, Y. Saito. and K. Kato, Recrystallization and Texture Formation in High Speed Hot Rolling of Austenitic Stainless Steel, Transactions ISIJ, 27, 520-525 (1987). T . Sakai, Y. Saito, K. Hirano, and K. Kato, Deformation and Recrystallization Behavior of L o w Carbon Steel in High Speed Hot Rolling, Transactions ISIJ, 28, 1028-1035 (1988). M . Sato. T. Kajiya, and T. Yashiro, Commercial Manufacturing Processes of Rolled Thin Magnesium Alloy Coils and Their Properties, Journal of Japan Institute of Light Metals, 54, 465-471 (2004), (in Japanese). U . F. Kocks, C. N . T o m e and H. R. W e n k : Texture and Anisotropy (Cambridge University Press, U K ) , 2 0 3 - 2 0 9 ( 1 9 9 8 ) M . J. Philippe, Texture Formation in Hexagonal Materials, Materials Science Forum, 157-162, 1337-1350 (1994) 2
3
4
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M I C R O S T R U C T U R E A N D T E X T U R E G R A D I E N T IN T I T A N I U M S E V E R E L Y STRAINED BY FRICTION ROLL PROCESSING A N D S U B S E Q U E N T ANNEALING Meiqin Shi, T o m o h i r o U m e t s u , Y o s h i m a s a T a k a y a m a , H a j i m e K a t o and H i d e o W a t a n a b e Department o f M e c h a n i c a l S y s t e m s E n g i n e e r i n g , U t s u n o m i y a University, U t s u n o m i y a , Japan [email protected], [email protected], [email protected], [email protected], [email protected]
ABSTRACT T h e m i c r o s t r u c t u r e and texture evolution in t i t a n i u m after severe plastic deformation using friction roll p r o c e s s i n g ( F R P ) followed by a n n e a l i n g h a s been investigated. FRP, which is a p r o m i s i n g m e t h o d to give severe strain in surface layers, w a s e x p e c t e d to obtain the grain refinement with preferred orientation. Significant c h a n g e s o f strain w e r e imposed into from the surface t o w a r d t h e interior o f s a m p l e s , and then t h e m e c h a n i s m of deformation w a s suggested. A few rotation speeds of roll w e r e used as a p a r a m e t e r for finding the o p t i m u m operating c o n d i t i o n s . S a m p l e s w e r e annealed for v a r i o u s t i m e s at 8 2 3 K and then e x a m i n e d by scanning electron m i c r o s c o p y / electron b a c k scatter diffraction pattern ( S E M / E B S D ) . In as processed s a m p l e , material flow w a s observed at the surface layer, w h i l e both tensile and c o m p r e s s i o n t w i n s w e r e m a i n l y found b e t w e e n the subsurface and center layers. After annealing for a short time, fine grains with t r a n s v e r s e t e x t u r e w e r e observed only at the surfaces o f the s a m p l e s . With the increase of a n n e a l i n g t i m e both microstructure and texture gradient can be formed on t h e s a m p l e through the t h i c k n e s s . Different deformation m o d e s a c c o m p a n i e d to strain gradients
w e r e discussed
as the reason
for formation
of these
m i c r o s t r u c t u r e and texture gradient. K e y w o r d s : Texture gradient, Titanium, Severe plastic deformation, Annealing, F R P INTRODUCTION M e c h a n i c a l properties of m e t a l s depend both on the m i c r o s t r u c t u r e and on the texture developed
u n d e r p r o c e s s i n g . Therefore, a c o m p l e t e u n d e r s t a n d i n g of the physical
and
metallurgical p h e n o m e n a o c c u r r i n g u n d e r all o v e r t h e p r o c e s s i n g is necessary to control their final p r o p e r t i e s . In t h e last d e c a d e , severe plastic deformation ( S P D ) has occupied great interest d u e t o t h e attractive prospect to i m p r o v e m e c h a n i c a l properties. U p to n o w , m i c r o s t r u c t u r e a n d t e x t u r e evolution during S P D has been studied m a i n l y for face-centered cubic ( F C C ) and b o d y - c e n t e r e d c u b i c ( B C C ) m e t a l s w h i c h w e r e subjected to equal channel [l
3
4
angular pressing ( E C A P ) ~ ', a c c u m u l a t i v e roll b o n d i n g ( A R B ) ' ' , high pressure torsion (HPT)
151
6
and rolling process ' ' . N e v e r t h e l e s s , the specific deformation m e c h a n i s m s in
h e x a g o n a l close p a c k e d ( H C P ) m e t a l s are less well u n d e r s t o o d than t h o s e in cubic metals w h i c h usually h a v e a large n u m b e r o f i n d e p e n d e n t slip s y s t e m s . F o r instance, in pure titanium, not only slip, but t w i n n i n g p l a y s an essential role in d e f o r m a t i o n . T h e effect o f this special d e f o r m a t i o n m o d e on t e x t u r e evolution still n e e d s a further investigation. In the present study, t i t a n i u m w a s c h o s e n to be t h e material, and a n e w p r o c e s s i n g n a m e d as Friction Roll
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M i c r o s t r u c t u r e a n d T e x t u r e G r a d i e n t in T i t a n i u m S t r a i n e d b y Friction Roll P r o c e s s i n g
Processing ( F R P ) , o n e o f S P D t e c h n i q u e s , h a s been p r o p o s e d for controlling m i c r o s t r u c t u r e .of the materials. In principle, F R P is a c o n t i n u o u s sliding with very large shear strain i m p o s e d into t h e surface layer, w h i c h similar with a s y m m e t r i c rolling, a p r o c e s s i n g m e t h o d that e n a b l e s 71
the generation o f shear strain t h r o u g h o u t the sheet t h i c k n e s s ' . T h i s microstructural control includes not o n l y grain refinement but also formation o f crystallographic t e x t u r e . D u r i n g this surface friction p r o c e s s , a large a m o u n t o f plastic deformation is imparted into s a m p l e s by the m e c h a n i c a l rolling action o f a high speed rotating t o o l . T h e r e exists a strain g r a d i e n t from t h e surface layer to center part o f t h e F R P e d s a m p l e s . O u r study will be helpful for u n d e r s t a n d i n g the formation o f m i c r o s t r u c t u r e and t e x t u r e in H C P m e t a l s by m o r e e x p e r i m e n t a l results and their analysis.
EXPERIMENTAL PROCEDURE C o m m e r c i a l purity ( C P ) titanium g r a d e I with a n a v e r a g e grain size o f 3 0 p m
and
containing impurities ( m a s s % ) including 0.041 O, 0.002 N , 0.0024 H, 0.031 F e and 0.009 C was
used
as
the
starting
material.
For
FRP
pressing,
titanium
sheet
samples
of
6 0 m m x 2 0 m m x I m m w e r e p r e p a r e d . S c h e m a t i c illustration o f F R P is d i s p l a y e d as Fig. I. Fixed roll with d i m e n s i o n of
Φ 7 0 x 1 0 m m _ w h i c h w a s m a d e o f tool steel S K 3 , rotated at t w o
s p e e d s o f 4 8 and 2 4 0 r p m . T h e indentation, defined as the d e p t h w h i c h the rotating roll w a s pressed d o w n into the s a m p l e , w a s selected as 0.1 and 0 . 2 m m . T h e feeding speed o f s a m p l e w a s set as 14 and 80 m m / m i n . F o r e x a m p l e , one o f t h e F R P e x p e r i m e n t a l c o n d i t i o n s w a s m a r k e d as 4 8 r p m (rotation s p e e d ) - 0 . 1 m m (indentation)-14 m m / m i n (feeding s p e e d ) . T h e directions feeding
of
rolfs
rotation
w e r e sketched
performed
at
room
and
sample's
as F i g . l .
FRP
was
temperature;
first,
the
position w h e r e the roll and the s a m p l e j u s t touched w a s located, and r e g a r d e d as reference plane w h i c h indentation is z e r o . T h e n , the roll w a s put d o w n to t h e given indentation w i t h o u t contacting with t h e s a m p l e . After starting to rotate the roll, t h e s a m p l e w a s m o v e d forward until the entire surface w a s treated. It m a d e the w o r k i n g plane lower than t h e reference plane and led to a great a m o u n t of friction. After FRP,
^g-
the F R P e d s a m p l e s w e r e a n n e a l e d at 8 2 3 K for
( )
b
1
a
( )- Schematic illustration of FRP and
C r o s s
section for investigation,
various t i m e s in A r g a s and air cooled to study the texture evolution d u r i n g a n n e a l i n g . M i c r o s t r u c t u r e w a s e x a m i n e d on the N D - R D plane (i.e. rolling direction ( R D ) - n o r m a l direction ( N D ) plane). T o be consistent w i t h rolling, t h e shear d e f o r m a t i o n direction o f F R P w a s parallel to R D , illustrated as Fig. 1(b). M i c r o s t r u c t u r e o f F R P e d s a m p l e w a s o b s e r v e d by optical m i c r o s c o p y . E v o l u t i o n s o f crystallographic t e x t u r e o f titanium s a m p l e s after the F R P e d and a n n e a l i n g w e r e e v a l u a t e d using s c a n n i n g electron m i c r o s c o p e / electron back scatter diffraction pattern ( S E M / E B S P ; H I T A C H I S - 3 5 0 0 H , T S L Orientation I m a g e M i c r o s c o p y system) after e l c c t r o p o l i s h i n g . In order to depict the gradient of m i c r o s t r u c t u r e and texture
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Materials Processing and Texture
M i c r o s t r u c t u r e a n d T e x t u r e G r a d i e n t in T i t a n i u m S t r a i n e d b y Friction Roll P r o c e s s i n g
through t h e t h i c k n e s s , t h e N D - T D plane (i.e. t r a n s v e r s e direction ( T D ) - n o r m a l direction ( N D ) plane) w a s s c a n n e d c o n t i n u o u s l y every 2 0 0 μιη from the processed surface to center part with step size 1 μιη and then O I M m a p s w e r e s u p e r p o s e d . RESULT A N D DISCUSSION Strain distribution F o r the p u r p o s e o f e v a l u a t i n g the strain distribution t h r o u g h t h e t h i c k n e s s o f the F R P e d 4)
samples, shear strain m e a s u r e m e n t w a s carried out by the e m b e d d e d - p i n m e t h o d ' . T h e method is s c h e m a t i c a l l y illustrated in Fig. 2(a). A c y l i n d e r - s h a p e d pin 2 m m in d i a m e t e r and I m m thick, w h i c h w a s m a d e from t h e s a m e material, w a s e m b e d d e d at the center o f the s a m p l e . The pin w a s s u p p o s e d to b e d e f o r m e d to the s a m e extent as the w h o l e s a m p l e . After FRP, the longitudinal p l a n e o f the pin w a s o b s e r v e d by optical m i c r o s c o p y and strain w a s calculated using the definition o f shear strain ( S h e a r strain = tan θ = Deformation / Original Length), showed as Fig. 2(b). T h e result is displayed in Fig.3 and listed in T a b l e 1. T h e optical m i c r o g r a p h o f pin interface gives a p r o o f that strain gradient o c c u r through the thickness and very severe strain is stored near t h e surface after F R P .
Deformation
Original Length
Fig.2. Schematic illustration showing the shear strain measurement by embedded-pin method: (a) Experimental method; (b) Definition of shear strain.
RD
<
Feeding direction Table 1 Shear strain through the thickness.
D i s t a n c e from ND
surface 0
γ= t a n Ö 9.35
• 50 μιη
1.12
100 μηι
0.95
2 0 0 μπι
0.30
Fig.3. Optical micrograph of pin interface in the titanium sheet processed after FRP.
Starting material T h e m i c r o s t r u c t u r e o f the as received s a m p l e consisted of e q u i a x e d grains with grain size a b o u t 30 μηι. Its (0001 ) pole figure (as s h o w n in Fig. 6(a) b e l o w ) indicated that the basal poles
Materials Processing and Texture
·
611
M i c r o s t r u c t u r e a n d T e x t u r e G r a d i e n t in T i t a n i u m S t r a i n e d by Friction Roll P r o c e s s i n g
were found
tilted 30
4 0 ° away
from
t h e N D t o w a r d s the T D , d i s p l a y i n g T D
split
texture { Ï 2 Î 4 } < lOTO > o r { 0 2 2 5 } < 2 Ϊ Τ θ > ({rolling p l a n e } < r o l l i n g d i r e c t i o n > ) .which is c o m m o n l y o b s e r v e d in rolled titanium sheet after a n n e a l i n g . D e f o r m e d material T h e optical m i c r o g r a p h o f the N D - R D plane o f a s - F R P e d s a m p l e is s h o w n in Fig. 4 . Deformation h e t e r o g e n e i t y a p p e a r s in the m i c r o s t r u c t u r e s c o r r e s p o n d i n g to the strain gradient in titanium. Different d e f o r m a t i o n m o d e s w e r e o b s e r v e d a c c o r d i n g to the d i s t a n c e from t h e processed surface. Tn the surface layer of a b o u t 50μιη thick, shear b a n d s with an inclination about 2 0 - 2 5 ° (This inclination w a s not consistent with the calculation in strain test, s h o w e d in Table. 1 a b o v e , it w a s b e c a u s e o f the scattering during e x p e r i m e n t o p e r a t i o n s ) with regard to R D arc found to be a p r o o f o f material flow. In this surface layer, slip could b e c o m e m o r e p r e d o m i n a n t m e c h a n i s m to a c c o m m o d a t e to a m o d e r a t e strain. At the position with a distance o f a b o u t 100 μιη from the surface, m a n y deformation t w i n s , w h i c h imply low strain o f deformation, w e r e formed. In rolled Ti d e f o r m a t i o n t w i n i n g is found in all g r a i n s at a very early stage. T h e a b s e n c e o f additional t w i n n i n g d u r i n g the large deformation o f titanium has been discussed as t h e result of the reduced grain size introduced by prior t w i n n i n g
For the center
layer, e q u i a x e d g r a i n s with n o twin w e r e o b s e r v e d . For processed s a m p l e . N D - T D plane w a s scanned using S E M / E B S D t h r o u g h the w h o l e t h i c k n e s s . Fig. 4(b) is an i m a g e quality m a p ( I Q m a p ) o f E B S D in N D - T D plane of t h e F R P e d o n e . T h e dark field indicates large strain c o v e r the z o n e closed t o the F R P e d surface, and it reaches the center with gradual increase in brightness. A l m o s t to the center layer, half parts of the s a m p l e s are covered by the black points, w h i c h are the p o i n t s failed to be indexed. C o n t r a r y to this, in t h e region w i t h o u t F R P effect, the p o i n t s with high C o n f i d e n c e Index (CI) data displayed that the original e q u i a x e d g r a i n s still exist. It revealed that F R P w a s a straining technique that could g i v e a strain gradient t h r o u g h the t h i c k n e s s .
Fig.•
Microstructures in cross section after FRP (a) Optical micrograph o f N D - R D plane, (b)
IQ map o f N D - T D plane o f FRPed samples.
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Materials Processing and Texture
M i c r o s t r u c t u r e a n d T e x t u r e G r a d i e n t in T i t a n i u m S t r a i n e d b y Friction Roll P r o c e s s i n g
A n n e a l e d material Microstructure development
Fig.5 I Q M A P s o f mierostructures in the N D - R D planes o f the surface layers o f samples after annealed at 823k. FRP condition: 4 8 r p m - 0 . 2 m m - 1 4 m m / m i n : annealing time: (a) l h , (b) 2h, (c) 4h, (d) 8h.
I Q m a p s in Fig. 5 illustrate the N D - T D planes o f s a m p l e s after a n n e a l e d for different t i m e s at 8 2 3 K . T h i s t e m p e r a t u r e w a s d e t e r m i n e d after the analysis o f strain by
intragranular
[9
misorientation after a n n e a l i n g ' . T h e area a b o u t 3 0 0 p m * 3 0 0 p m from the processed surface w a s m e a s u r e d with step size 1 p m . G r a y scale in each I Q m a p c o r r e s p o n d s to different strain level a c c o r d i n g to m e a s u r i n g c o n d i t i o n . M i c r o s t r u c t u r e in this area constitutes differently characterized features t h r o u g h the t h i c k n e s s . T h e fine g r a i n s with grain size about 5 p m a p p e a r e d at the region within depth a b o u t 50 p m in the first recrystallization state, as s h o w e d in Fig.5 (a). T h e fine grain m i c r o s t r u c t u r e w a s c o m p o s e d o f inconlinuously recrystallized grains within a matrix of deformed elongated cells. Shear b a n d s w h e r e the high strain stored are s u p p o s e d to be t h e favorable nucleation sites for the a p p e a r a n c e o f t h e first recrystallized grains [ 1 0
l T h e r e m a i n e d d a r k parts in t h e m a p indicate the i m p o s e d strains had not been released
c o m p l e t e l y during a n n e a l i n g for such a short t i m e . After a n n e a l i n g for 2 h . it gave rise to a fairly uniform e q u i a x e d grain structure at the top layer o f s a m p l e ; see Fig.5 (b); h o w e v e r the m i c r o s t r u c t u r e of other part can be characterized by strain r e m a i n e d in t h e grains. With the increase o f a n n e a l i n g time, strain w a s annihilated, but a large n u m b e r o f t w i n s still can be o b s e r v e d . T h e a n a l y s i s revealed that t w o kinds o f t w i n s existed, that is. both c o m p r e s s i v e
Materials Processing and Texture
613
M i c r o s t r u c t u r e a n d T e x t u r e G r a d i e n t in T i t a n i u m S t r a i n e d b y Friction Roll P r o c e s s i n g
twins {1122} < 1123 > and tensile twins {101 2} < 101 1 > ({twinning
plane}
-
direction> boundaries were observed in the samples. Therefore, FRP can be described to have the same strain component as rolling which is the combination of extension along the rolling direction (RD) and compression perpendicularly to the rolling plane (ND). The whole view of microstructure development during long time annealing suggests that recrystallization is only partial and recovery seems to be the predominant softening mechanism.
Fig.6 Pole figures showing the texture gradient through thickness, FRP condition: 48rpm-0.2 mm -14 mm/min: annealing condition, at 823K -1h, (a)As received sample, (b) 0-200μιτι , (c)200-40ttym, (d) 400-600μπι.
Texture evolution In this study, every 200 μιη depth layer from the FRPed surface to the central layer of the sample was scanned by SEM/EBSP and the (0001) pole figures of three layers are presented in Fig. 6. By comparing Fig.6 (b) (c) and (d), texture gradient was found along the thickness, i.e. ND direction. After FRP and subsequent annealing at 823K for lh, transverse texture (T-texture) which is defined as and {10 1 0} < 1120 >appeared with relatively high intensity in the FRPed surface layer (see Fig. 6(b)). In the area 200-400μηι, displayed as Fig. 6(c), the intensity of T-texture is reduced and TD-split component appears to be the main component in
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M i c r o s t r u c t u r e a n d T e x t u r e G r a d i e n t in T i t a n i u m S t r a i n e d b y Friction Roll P r o c e s s i n g
the figures. Finally, T-texture t u r n s to be fairly a m b i g u o u s in Fig. 6 ( d ) , w h i l e the typical rolling texture T D - s p l i t o c c u p i e s t h e high fraction. It d i s p l a y s t h e center layer of the sample has the similar t e x t u r e to the as received o n e . In t h e surface layer w h e r e shear band is the primary deformation structure, high density o f a c c u m u l a t e d dislocation resulted in the a p p e a r a n c e of n e w nuclei. After a n n e a l i n g at 8 2 3 K for l h , refined g r a i n s with T-texture a p p e a r e d with higher fraction. It can b e p r o p o s e d that T-texture is the favor t e x t u r e after F R P and subsequent short t i m e a n n e a l i n g . On the other hand, in the area o f 200-400μιτι from the surface, w h e r e twin is the m a i n d e f o r m a t i o n m o d e , texture c h a n g e is hardly detected. H o w will the twin m a k e an effect on t e x t u r e evolution if given a long t i m e a n n e a l i n g is expected to be fully investigated.
CONCLUSIONS F R P used as a n e w t e c h n i q u e give a strain gradient t h r o u g h the thickness of c o m m e r c i a l purity t i t a n i u m s a m p l e s . Different deformation m o d e s w e r e o b s e r v e d to a c c o m m o d a t e t o the strain g r a d i e n t a l o n g the N D direction of s a m p l e s . C o n s e q u e n t l y , grain refinement only can be obtained at t h e surface layer w h e r e the high density o f dislocation w a s a c c u m u l a t e d . T e x t u r e gradient w a s also c o n f i r m e d a l o n g N D direction, and T-texture a p p e a r e d at the surface layer w a s s u p p o s e d to be the favor orientation after F R P and s u b s e q u e n t short t i m e annealing.
REFERENCE ' C . P i t h a n , T. H a s h i m o t o . M . K a w a z o e , J. N a g a h o r a and K. Higashi, Microstructure and texture evolution in E C A E p r o c e s s e d A 5 0 5 6 , Mater. Sei. Eng A, 2 8 0 . 6 2 - 6 8 (2000). 2
Y.T. Z h u and T . C . L o w e , O b s e r v a t i o n s and issues on m e c h a n i s m s o f grain refinement during
E C A P process, Maler. Sei. Eng A, 2 9 1 , 4 6 - 5 3 ( 2 0 0 0 ) . 3
I . J . B e y e r l e i n , R . A . L e b e n s o h n , C . N . T o m e , M o d e l i n g texture and microstructural evolution in
the equal c h a n n e l a n g u l a r extrusion p r o c e s s . Mater. Sei. Eng A, 3 4 5 , 122-138 ( 2 0 0 3 ) . 4
N . K a m i k a w a , T.Sakai, N.Tsuji, Effect of r e d u n d a n t shear strain on microstructure and texture
evolution during a c c u m u l a t i v e r o l l - b o n d i n g in ultralow carbon IF steel, Acta.Mater.,
55,
5873-5888 (2007). 5
E.Schafler,
Μ.Β.Kerber,
Microstructural
investigation
of the
annealing
behaviour
of
h i g h - p r e s s u r e torsion ( H P T ) d e f o r m e d copper, Mater. Sei. Eng A, 4 6 2 , 139-143 (2007). 6
J . A . d e l Valle, M . T . P e r e z - P r a d o and O . A . R u a n o , Texture evolution during large-strain hot
rolling o f the M g A Z 6 1 alloy, Mater. Sei. Eng A, 3 5 5 , 6 8 - 7 8 ( 2 0 0 3 ) . 7
K . H . K i m and D . N . L e e , A n a l y s i s o f deformation t e x t u r e s o f a s y m m e t r i c a l l y rolled a l u m i n u m
sheets, Acta Mater, 8
Y.B.Chun,
49, 2583-2595 (2001).
S.H.Yu, S.L.Semiatin
and
S.K.Hwang,
Effect
o f deformation
twinning
on
m i c r o s t r u c t u r e and t e x t u r e e v o l u t i o n during cold rolling of C P - t i t a n i u m , Maler. Sei. Eng A, 398,
Materials Processing and Texture
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M i c r o s t r u c t u r e a n d T e x t u r e G r a d i e n t in T i t a n i u m S t r a i n e d by Friction Roll P r o c e s s i n g
209-219(2005). 'M.Q.Shi,
Y.Takayama,
H . K a t o , M i c r o s t r u c t u r e and M e c h a n i c a l
Severely Strained by m e a n s o f Friction Roll P r o c e s s i n g
Properties of Titanium
P r o c e e d i n g s of T h e Sixth Pacific R i m
International C o n f e r e n c e on A d v a n c e d Materials and P r o c e s s i n g , 2007.Mater.
Sei.
For,
561-565,909-912 (2007). l0
A . O . F . H a y a m a , H . R . Z . S a n d i m , A n n e a l i n g b e h a v i o r o f coarse-grained titanium d e f o r m e d by
cold rolling, Maler. Sei. Eng A, 4 1 8 , 182-192 ( 2 0 0 6 ) .
616
Materials Processing and Texture
EVOLUTION
OF
TRANSFORMATION
TEXTURE
IN
A
METASTABLE
B
-
TITANIUM ALLOY S a t y a m S u w a s , N i l e s h P . G u r a o , A s h k a r Ali D e p a r t m e n t of Materials E n g i n e e r i n g , Indian Institute of S c i e n c e . Bangalore, India.
ABSTRACT T e x t u r e e v o l u t i o n in h . c . p . ( a ) p h a s e d e r i v e d f r o m a g i n g o f a differently p r o c e s s e d m e t a s t a b l e b . c . c . ( β ) t i t a n i u m a l l o y w a s i n v e s t i g a t e d . T h e s t u d y w a s a i m e d at e x a m i n i n g (i) t h e effect o f different b . c . c . c o l d r o l l i n g t e x t u r e s a n d (ii) t h e effect o f different d e f e c t structures on the h.c.p transformation texture. T h e alloy metastable β alloy Ti-10V-4.5Fe1.5A1 w a s r o l l e d at r o o m t e m p e r a t u r e by u n i d i r e c t i o n a l ( U D R ) a n d m u l t i - s t e p c r o s s r o l l i n g ( M S C R ) . A p i e c e o f t h e a s - r o l l e d m a t e r i a l s w e r e s u b j e c t e d t o a g i n g in o r d e r t o derive the h.c.p. (a) phase. In the other route, the as-rolled materials were recrystallized and then aged. T e x t u r e s w e r e m e a s u r e d using X-ray as well as Electron Back Scatter Diffraction. Rolling texture of β p h a s e , as characterized by the presence of a strong γ
fibre,
w a s f o u n d s t r o n g e r in M S C R c o m p a r e d to U D R , a l t h o u g h t h e y w e r e q u a l i t a t i v e l y similar. The
stronger texture of M S C R
sample
could
be attributed
to the
inhomogeneous
d e f o r m a t i o n t a k i n g p l a c e in t h e s a m p l e t h a t m i g h t c o n t r i b u t e t o w e a k e n i n g o f t e x t u r e . U p o n r e c r y s t a l l i z a t i o n in β p h a s e
field
c l o s e to ß - t r a n s u s . t h e t e x t u r e s
qualitatively
r e s e m b l e d t h e c o r r e s p o n d i n g β d e f o r m a t i o n t e x t u r e s ; h o w e v e r , t h e y got s t r e n g t h e d . T h e aging of differently
β r o l l e d s a m p l e s r e s u l t e d in t h e p r o d u c t α - p h a s e w i t h
different
textures. T h e ( U D R + Aged) sample had a stronger texture than ( M S C R + Aged) sample, which could be due to continuation
of defect
accumulation
in U D R
sample,
thus
p r o v i d i n g m o r e p o t e n t i a l s i t e s for t h e n u c l e a t i o n o f α p h a s e . T h e t r e n d w a s r e v e r s e d in samples recrystallized prior to aging. T h e ( M S C R + Recrystallized + A g e d )
sample
s h o w e d stronger texture of α p h a s e than the ( U D R + Recrystallized + A g e d ) sample. This could
be
attributed
to
extensive
defect
annihilation
in
the
UDR
sample
on
recrystallization prior to aging. T h e ( M S C R + A g e d ) sample exhibited m o r e α variants w h e n c o m p a r e d to ( M S C R + R e c r y s t a l l i z e d + A g e d ) s a m p l e . T h i s h a s b e e n a t t r i b u t e d t o t h e a v a i l a b i l i t y o f m o r e p o t e n t i a l s i t e s for n u c l e a t i o n o f α p h a s e in t h e f o r m e r . It c o u l d b e c o n c l u d e d that α transformation texture d e p e n d s mainly o n the defect structure of the parent phase.
INTRODUCTION T i t a n i u m a n d its a l l o y s find e x t e n s i v e a p p l i c a t i o n in a e r o s p a c e a n d b i o m e d i c a l a p p l i c a t i o n s d u e t o t h e i r h i g h s p e c i f i c s t r e n g t h , e x c e l l e n t f a t i g u e a n d fracture r e s i s t a n c e a n d g o o d c o r r o s i o n r e s i s t a n c e . T i t a n i u m e x i s t s in H C P α p h a s e at r o o m t e m p e r a t u r e a n d u n d e r g o e s a l l o t r o p i e t r a n s f o r m a t i o n t o B C C β p h a s e at h i g h e r t e m p e r a t u r e ( 8 8 2 ° C ) . O w i n g to H C P s t r u c t u r e , oc-Ti h a s l i m i t e d d u c t i l i t y at r o o m t e m p e r a t u r e . B y a d d i t i o n o f a l l o y i n g e l e m e n t s , t h e β p h a s e c a n b e s t a b i l i z e d at r o o m t e m p e r a t u r e a n d v a r i o u s ( α + β )
617
Evolution of T r a n s f o r m a t i o n T e x t u r e in a M e t a s t a b l e B - T i t a n i u m Alloy
and metastable
β alloys are obtained
w h i c h exhibit the desired level of
plasticity.
V a n a d i u m , c h r o m i u m , n i o b i u m e t c a r e t h e m o s t s u i t a b l e e l e m e n t s for s t a b i l i z i n g β p h a s e . H o w e v e r , these alloying e l e m e n t s like are quite costly. T h i s restricts the application base o f t h e s e a l l o y s to a e r o s p a c e i n d u s t r y o n l y . In o r d e r t o e x p a n d t h e a p p l i c a t i o n b a s e , c o s t l y c h r o m i u m h a s b e e n r e p l a c e d b y c o m m o n l y a v a i l a b l e a n d c h e a p f e r r o c h r o m e t h a t is c o m m o n l y u s e d in steel i n d u s t r y [ 1 ] . O n e s u c h a l l o y t h a t h a s b e e n d e v e l o p e d for l o w c o s t a u t o m o b i l e a p p l i c a t i o n is t h e T i - 1 0 V - 4 F e - 1 . 5 A l a l l o y , w h i c h is c o m m o n l y k n o w n a s L o w C o s t B e t a Ti a l l o y ( h e r e a f t e r r e f e r r e d a s L C B Ti a l l o y ) . T h e t e x t u r e e v o l u t i o n in v a r i o u s ( α + β) a n d β t i t a n i u m a l l o y s h a s b e e n i n v e s t i g a t e d in d e t a i l [ 2 - 7 ] . H o w e v e r , t h e r o l e o f v a r i o u s p a r a m e t e r s o f d e f o r m a t i o n a n d p o s t d e f o r m a t i o n a n n e a l i n g is n o t y e t i n v e s t i g a t e d . It is n o w w e l l e s t a b l i s h e d t h a t t h e m e t a s t a b l e ß-Ti a l l o y s a r e p r o c e s s e d b y d e f o r m a t i o n a n d r e c r y s t a l l i z a t i o n a n n e a l d u r i n g s h a p e m a k i n g o f c o m p o n e n t s a n d a g i n g t r e a t m e n t is g i v e n t o p r e c i p i t a t e α in t h e β m a t r i x in o r d e r t o i m p a r t s t r e n g t h . . In t h e p r e s e n t w o r k , e v o l u t i o n o f t e x t u r e a n d m i c r o s t r u c t u r e in L C B Ti a l l o y h a s b e e n s t u d i e d for b o t h t h e aforesaid conditions, namely deformation and recrystallization as well as aging. The effect o f g r a i n s i z e , d i f f e r e n t s t r a i n p a t h s in t e r m s o f u n i d i r e c t i o n a l a n d m u l t i d i r e c t i o n a l r o l l i n g w e r e m a d e t h e a d d i t i o n a l v a r i a b l e s for s t u d y o f e v o l u t i o n o f t e x t u r e in t h e β p h a s e . F u r t h e r , t h e i n f l u e n c e o f d i f f e r e n t t y p e s o f t e x t u r e in β p h a s e o n t r a n s f o r m e d α p h a s e w a s investigated.
EXPERIMENTAL Material and Processing: T h e low cost β Ti-alloy w a s obtained from Laboratory,
Hyderabad.
The composition
Defence Metallurgical
of the alloy w a s examined using
Research Energy
Dispersive Analysis of X-rays ( E D A X ) . T h e results of this analysis indicated the actual c o m p o s i t i o n t o b e 8 4 . 1 5 % T i , 9 . 8 2 % V , 4 . 2 2 % F e a n d 1.8% A l . T h e initial m a t e r i a l w a s c a s t a n d h o t f o r g e d in β p h a s e field at 9 0 0 ° C t o g e t a n a v e r a g e g r a i n s i z e o f a b o u t 175 p m . T h e m a t e r i a l w a s further h e a t t r e a t e d at 9 0 0 ° C for 4 h r s to o b t a i n h i g h e r g r a i n s i z e s w i t h a v e r a g e g r a i n size o f a b o u t 2 5 0 p m . T h e s e t w o g r a i n s i z e s w e r e s u b j e c t e d i n v e s t i g a t i o n . All t h e s a m p l e s w e r e r o l l e d at r o o m t e m p e r a t u r e t o 8 0 % r e d u c t i o n . s c h e m e o f r o l l i n g is s h o w n subjected
to recrystallization.
in F i g . l . P i e c e s f r o m Recrystallization
differently
w a s carried
rolled materials
o u t at 8 0 0 ° C for
to The
were Ihr.
S p e c i m e n s f r o m e a c h o f t h e r o l l e d a n d r e c r y s t a l l i z e d m a t e r i a l s w e r e h e a t t r e a t e d for a g i n g at 5 0 0 ° C for 2 4 h r s . in o r d e r t o t r a n s f o r m t h e m a t e r i a l to α p h a s e .
618
Materials Processing a n d Texture
Evolution of T r a n s f o r m a t i o n T e x t u r e in a M e t a s t a b l e B - T i t a n i u m Alloy
Unidirectional Rolling ( U D R )
Multi-step Cross rolling ( M S C R )
F i g . 1 S c h e m e o f r o l l i n g u n d e r t a k e n in t h e p r e s e n t s t u d y
Characterization: Microstructural Characterization M i c r o s t r u c t u r a l c h a r a c t e r i z a t i o n ( b o t h o p t i c a l a n d S E M ) w a s c a r r i e d out as a f u n c t i o n o f v a r i o u s h e a t t r e a t m e n t c o n d i t i o n s , g r a i n size a n d r o l l i n g c o n d i t i o n s . T h e longitudinal plane o f the s a m p l e s w a s observed. For optical microscopy, the samples w e r e p r e p a r e d b y c o n v e n t i o n a l g r i n d i n g a n d p o l i s h i n g u p t o 2 5 0 0 grit p a p e r f o l l o w e d b y mirror polishing using A l u m i n a (0.05μ), and then etching. For A g e d samples, the etchant u s e d w a s K r o l l ' s r e a g e n t ( 2 % H F , 4 % HNO3 a n d 9 4 % w a t e r ) for a b o u t 5-8 s e c o n d s . F o r all t h e R o l l e d , R e c r y s t a l l i z e d a n d S t a r t i n g s a m p l e s , g r a i n b o u n d a r y e t c h a n t
(2%HF,
5%H2C>2 a n d r e s t w a t e r ) w a s u s e d for a b o u t 3 0 - 4 0 s e c o n d s . M i c r o l c x t u r e e v o l u t i o n in t h e a g e d s a m p l e s w a s s t u d i e d u s i n g E l e c t r o n B a c k s c a t t e r e d diffraction ( E B S D ) a t t a c h e d t o a F E G - S E M S I R J O N . D a t a a c q u i s i t i o n a n d a n a l y s i s w a s c a r r i e d out u s i n g O r i e n t a t i o n Imaging Microscopy ( O I M ) software. For E B S D , the mirror polished samples
were
electropolished using standard A 3 electrolyte provided by Stuers.
Characterization of Texture T e x t u r e s from t h e m i d t h i c k n e s s s e c t i o n s o f t h e s p e c i m e n s w e r e m e a s u r e d u s i n g X-ray diffraction g o n i o m e t e r based on Schultz reflection geometry. T h e texture of β
(110), (200) a n d (211). (10Î0), (0002), (10Î1), (10Î2), (11 2 0)
phase w a s d e t e r m i n e d by m e a s u r i n g three pole figures, n a m e l y w h i l e for t h e t e x t u r e for α p h a s e six p o l e f i g u r e s a n d (112
2) w e r e m e a s u r e d . T h e d a t a so o b t a i n e d w a s e m p l o y e d for t h e c a l c u l a t i o n o f
Orientation Distribution Functions (ODFs). The O D F s were calculated using LaboTex software and no restriction of specimen symmetry w a s imposed while calculating the ODFs.
Materials Processing and Texture
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Evolution of T r a n s f o r m a t i o n T e x t u r e in a M e t a s t a b l e B - T i t a n i u m Alloy
A l t h o u g h c o m p l e t e O D F s w e r e d e t e r m i n e d , s i n c e m o s t o f t h e i m p o r t a n t ideal o r i e n t a t i o n s for α ( h e p ) a r e l o c a t e d o n (p2=0° a n d (p2=30° s e c t i o n s o f t h e O D F o n l y t h e s e s e c t i o n s w e r e u s e d for a n a l y s i s . S i m i l a r l y for β ( b c c ) , φ2=0° a n d φ2=45° a s w e l l a s ipi=0° and φ ι = 9 0 ° are important O D F sections, therefore only these sections w e r e used
for
evaluation of texture.
RESULTS AND DISCUSSION Microstructure T h e initial h i g h e r a n d l o w e r g r a i n s i z e s a m p l e w i t h s i n g l e β p h a s e h a d a n a v e r a g e g r a i n size o f 2 5 0 a n d
175 μ ι η r e s p e c t i v e l y . U p o n
rolling, the microstructure
was
c h a r a c t e r i z e d b y f l o w l i n e s in b o t h g r a i n s i z e m a t e r i a l s . T h e u n i d i r e c t i o n a l r o l l e d s a m p l e s s h o w e d a s i n g l e set o f f l o w l i n e s w h i l e m u l t i s t e p r o l l e d s a m p l e s s h o w e d t w o s u c h set o f f l o w l i n e s in b o t h g r a i n s i z e d m a t e r i a l s . U p o n r e c r y s t a l l i z a t i o n , t h e f l o w l i n e s d i s a p p e a r e d a n d t h e d e f o r m e d g r a i n s w e r e r e p l a c e d b y c o m p l e t e l y r e c r y s t a l l i z e d e q u i a x e d g r a i n s in all t h e c a s e s . U p o n a g i n g , α l a t h s w e r e f o r m e d w i t h d i f f e r e n t lath o r i e n t a t i o n s p r e s e n t i n s i d e a s i n g l e p r i o r β g r a i n a l o n g w i t h g r a i n b o u n d a r y a. T h e m i c r o s t r u c t u r a l e v o l u t i o n a s a f u n c t i o n o f t h e r m o m e c h a n i c a l t r e a t m e n t o n l o w e r g r a i n s i z e s a m p l e is s h o w n in F i g . 2 .
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Eig. 2 M i c r o s t r u c t u r a l e v o l u t i o n in l o w e r g r a i n size L C B Ti a l l o y Initial T e x t u r e F i g . 3 s h o w s t h e ( 1 1 0 ) p o l e figure for t h e s t a r t i n g m a t e r i a l w i t h h i g h e r a n d l o w e r grain
size respectively.
The
higher
grain
size
sample
has a strong
{141}
<113>
c o m p o n e n t while the lower grain size sample has a strong {110} < 2 2 1 > component. The o v e r a l l i n t e n s i t y l e v e l s a r e l o w i n s a m p l e s for b o t h t h e g r a i n s i z e s i n d i c a t i n g w e a k e r initial t e x t u r e .
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F i g . 3 ( 1 0 1 ) p o l e f i g u r e for h i g h e r a n d l o w e r g r a i n s i z e s t a r t i n g m a t e r i a l
Deformation Texture The m a j o r t e x t u r e c o m p o n e n t s in d e f o r m a t i o n o f B C C m e t a l s a n d a l l o y s a r c t h e a l p h a fiber { 0 0 1 } < 1 1 0 > t o { 1 1 1 } < 1 1 0 > ( a p p e a r s in t h e
O D F ) [ 8 - 1 0 ] . In t h e p r e s e n t i n v e s t i g a t i o n , a l p h a fiber d i d n o t e v o l v e in a n y o f t h e s a m p l e s a n d h e n c e o n l y t h e (p2 = 4 5 ° s e c t i o n s a r e p r e s e n t e d ( F i g . 4 ) t h a t s h o w s t h e g a m m a fiber. T h e h i g h e r g r a i n s i z e s a m p l e s s h o w a s t r o n g e r g a m m a fiber i r r e s p e c t i v e o f t h e m o d e o f r o l l i n g w i t h t h e U D R s a m p l e s h o w i n g s t r o n g e r g a m m a fiber t h a n M S C R s a m p l e . T h e s i t u a t i o n is h o w e v e r r e v e r s e d in c a s e o f l o w e r g r a i n s i z e s a m p l e w h e r e g a m m a f i b e r is m i s s i n g in U D R s a m p l e w h i l e a w e a k fiber is s e e n in M S C R s a m p l e . In g e n e r a l , t h e higher grain size sample p r o d u c e s a stronger texture w h e n c o m p a r e d to the lower grain size m a t e r i a l . T h e p r e s e n c e o f G o s s { 1 1 0 } < 0 0 1 > c o m p o n e n t in t h e l o w e r g r a i n s i z e U D R sample
can
contradiction
be
attributed
to
the
with the generally
formation accepted
of
shear
bands.
These
fact t h a t t h e t e n d e n c y
results
of shear
are
in
banding
i n c r e a s e s w i t h i n c r e a s e in g r a i n s i z e . Recrystallization texture T h e d e f o r m a t i o n t e x t u r e d o e s n o t c h a n g e s u b s t a n t i a l l y after r e c r y s t a l l i z a t i o n for U D R a n d M S C R s a m p l e s for t h e h i g h e r a s w e l l a s l o w e r g r a i n s i z e s a m p l e s ( F i g . 5 ) . In higher grain size sample, the g a m m a while the intensity of g a m m a
fiber
fiber
a l m o s t d i s a p p e a r s in U D R + R e x s a m p l e ,
d e c r e a s e s s u b s t a n t i a l l y in c a s e o f M S C R
sample.
A l o n g w i t h t h e g a m m a fiber, t h e M S C R + R e x s a m p l e s h o w s a s t r o n g { 1 1 2 } < 0 2 1 > c o m p o n e n t for t h e h i g h e r g r a i n s i z e s a m p l e . T h e w e a k t e x t u r e o b s e r v e d in U D R s a m p l e
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q u a l i t a t i v e l y r e m a i n s s i m i l a r b u t s t r e n g t h e n s w i t h v e r y s t r o n g ( 5 5 6 ) [11 5 5 ] c o m p o n e n t after r e c r y s t a l l i z a t i o n ; w h i l e in t h e c a s e o f M S C R t h e r e is a s u b s t a n t i a l d e c r e a s e in i n t e n s i t y for l o w e r g r a i n s i z e s a m p l e . T h e e v o l u t i o n o f r e c r y s t a l l i z a t i o n t e x t u r e c a n b e e x p l a i n e d o n t h e b a s i s o f s t o r e d e n e r g y o f d e f o r m a t i o n in t h e m a t e r i a l . D u r i n g p l a s t i c d e f o r m a t i o n , p a r t o f t h e e x p e n d e d e n e r g y is s t o r e d in t h e c r y s t a l in t h e f o r m o f elastic e n e r g y a s s o c i a t e d w i t h s t r a i n field o f d i s l o c a t i o n s . D e p e n d i n g o n t h e o r i e n t a t i o n , different grains h a v e different a m o u n t of stored energy. V a r i o u s analysis carried out on rolled B C C m e t a l s w i t h v a r i o u s t e c h n i q u e s h a v e s h o w n t h e h i e r a r c h y o f s t o r e d e n e r g y in t h e f o l l o w i n g o r d e r { 1 1 1 } < 1 1 2 > f o l l o w e d w i t h { 1 1 1 } < 1 1 0 > a n d { 0 0 1 } < 1 1 0 > [111. T h e d e c r e a s e in g a m m a fiber i n t e n s i t y o n r e c r y s t a l l i z a t i o n c a n t h u s b e a t t r i b u t e d t o t h e h i g h e r t e n d e n c y o f t h e s e o r i e n t a t i o n s t o r e c r y s t a l l i z e . In h i g h e r g r a i n size s a m p l e t h a t h a s b e e n r e c r y s t a l l i z e d after U D R , t h e r e is a s h a r p d e c r e a s e in t h e g a m m a fiber i n t e n s i t y w h i l e t h e i n t e n s i t y o f g a m m a fiber r e d u c e s s l i g h t l y in M S C R s a m p l e o n r e c r y s t a l l i z a t i o n . T h i s c a n b e a t t r i b u t e d to t h e h i g h e r a m o u n t o f s t o r e d e n e r g y in c a s e o f U D R t h a n M S C R s a m p l e in case of higher grain size samples.
F i g . 4 O D F ψ2 = 4 5 ° s e c t i o n s for a ) h i g h e r g r a i n s i z e U D R b ) l o w e r g r a i n s i z e U D R c) h i g h e r g r a i n size M S C R d ) l o w e r g r a i n s i z e M S C R
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Evolution of T r a n s f o r m a t i o n T e x t u r e in a M e t a s t a b l e B - T i t a n i u m Alloy
F i g . 5 O D F
figures
for t h e h i g h e r a n d l o w e r g r a i n s i z e s t a r t i n g
m a t e r i a l after a g i n g . Λ c o n s i d e r a b l e o v e r l a p is s e e n b e t w e e n t h e 101 p o l e figure ( F i g . 3 ) o f t h e s t a r t i n g m a t e r i a l p r i o r to a g i n g . H o w e v e r , t h e t e x t u r e o f α p h a s e is q u i t e w e a k l i k e that o f β p h a s e o n a g i n g . T h e t h r e e m a j o r f i b e r s p r e s e n t in t h e p r i o r r o l l e d a n d a g e d s a m p l e s a r e t h e { 0 0 0 1 J, { 1 0 1 0 } a n d ( 1 1 2 0 ) fiber. D e s p i t e o f v e r y d i f f e r e n t β t e x t u r e , α t e x t u r e in all t h e c a s e s is f o u n d t o b e s i m i l a r w i t h m i n o r v a r i a t i o n in i n t e n s i t i e s ( F i g . 7 ) . T h e α t e x t u r e o f t h e h i g h e r g r a i n s i z e d s a m p l e s is s h o w n in F i g . 7 a s a r e p r e s e n t a t i v e o f all t h e s a m p l e s . T h u s t h e m a c r o t e x t u r e o f all t h e s a m p l e s is t h e s a m e o n a g i n g . A r e p r e s e n t a t i v e I P F m a p is s h o w n in F i g . 8. It is g e n e r a l l y f o u n d t h a t t h e r e is m o r e c o i n c i d e n c e in t h e { 1 0 1 } ρ a n d { 0 0 0 1 }„ p o l e f i g u r e o f t h e d e f o r m e d s a m p l e s t h a n t h e r e c r y s t a l l i z e d o n c e . T h i s is in a c c o r d a n c e w i t h e a r l i e r r e s u l t s p r o p o s e d b y G e y et al [12J t h a t s l i p a c t i v i t y is e s s e n t i a l for v a r i a n t s e l e c t i o n t o o c c u r in m e t a s t a b l e ß - T i a l l o y . T h e a c c u m u l a t i o n o f d e f e c t s p r o v i d e s p o t e n t i a l s i t e s for n u c l e a t i o n o f α d u r i n g p h a s e t r a n s f o r m a t i o n a n d l e a d to v a r i a n t s e l e c t i o n [ 1 3 - 1 4 ] . It is w e l l k n o w n t h a t t h e β p h a s e o f m e t a s t a b l e ß-Ti a l l o y s t r a n s f o r m to ( α + β) s t r u c t u r e . D u r i n g p h a s e t r a n s f o r m a t i o n from β t o α p h a s e , B u r g e r ' s o r i e n t a t i o n r e l a t i o n s h i p is g e n e r a l l y f o l l o w e d i.e. { 0 0 0 1 }ajj < 1 1 0 > β
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Evolution of T r a n s f o r m a t i o n T e x t u r e in a M e t a s t a b l e B - T i t a n i u m Alloy
and { 1 1 2 0}a || <11 l>p, however due to symmetry of the crystal, a single β orientation gives rise to 12 equivalent α orientations with equal probability. The analysis of EBSD data showed that there was strong variant selection in case of samples aged after deformation as against rolled samples subjected to recrystallization prior to aging in higher as well as lower grain size material. This was evident from the overlapping of orientations in the {0001 } pole figure and {101 }p pole figure obtained from the IPF map. A representative IPF map and the corresponding pole figures for UDR + Age sample are shown in Fig. 8. The stronger variant selection in rolled samples is attributed to the existence stronger defect structure while variant selection is minimal in case of recrystallized samples. The defect structure could get annihilated during recrystallization and hence the potential sites for nucleation of α lath might get reduced. This argument is supported by the observation that α lath size is larger in case of recrystallized sample when compared to the rolled sample. a
Fig. 6 (0001) pole figure for higher and lower grain size sample after aging
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Evolution of T r a n s f o r m a t i o n T e x t u r e in a M e t a s t a b l e B - T i t a n i u m Alloy
25.0 20.0 13.0 10.0 5.0 1.0
Fig. 7 O D F φ
2
- 0° a n d φ
2
= 30° section o f h i g h e r g r a i n size a) U D R b) M S C R c) U D R +
R e x d) M S C R + R e x s a m p l e on a g i n g
626
·
Materials Processing a n d Texture
Evolution of T r a n s f o r m a t i o n T e x t u r e in a M e t a s t a b l e B - T i t a n i u m Alloy
F i g . 8 I P F m a p a n d c o r r e s p o n d i n g p o l e f i g u r e s for l o w e r g r a i n s i z e a ) U D R + A g e d b) UDR + Rex + Aged sample
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Evolution of T r a n s f o r m a t i o n T e x t u r e in a M e t a s t a b l e B - T i t a n i u m Alloy
CONCLUSIONS 1.
C o l d r o l l i n g o f L C B Ti a l l o y is c h a r a c t e r i z e d b y t h e f o r m a t i o n o f g a m m a fiber. In higher grain size s a m p l e . M S C R sample has a w e a k e r texture than U D R sample w h i l e t h e t r e n d is r e v e r s e d in l o w e r g r a i n s i z e s a m p l e . T h e r e v e r s e d t r e n d is e x p l a i n e d o n t h e b a s i s o f f o r m a t i o n o f s h e a r b a n d s in t h e latter c a s e .
2.
Recrystallization
texture
in
LCB
Ti
alloy
is
qualitatively
similar
to
the
deformation texture. 3.
D e s p i t e o f different initial β t e x t u r e , t h e b u l k α t e x t u r e o n a g i n g is q u a l i t a t i v e l y s i m i l a r for all s a m p l e s . T h e m i c r o t e x t u r e is h o w e v e r , q u i t e d i f f e r e n t d e p e n d i n g o n the defect structure and strain path.
4.
A s t r o n g e r d e f e c t s t r u c t u r e a s in c a s e o f r o l l e d s a m p l e s l e a d t o s t r o n g v a r i a n t s e l e c t i o n w h e n c o m p a r e d to t h e i r r e c r y s t a l l i z e d c o u n t e r p a r t s . T h e y a l s o
show
s m a l l e r α lath size i n d i c a t i n g t h e a b u n d a n c e o f n u c l e a t i o n sites for α in t h e p r i o r b e t a g r a i n . S t r a i n p a t h h a s a s i m i l a r effect w i t h U D R s a m p l e s s h o w i n g s t r o n g e r variant selection than M S C R samples. 5.
It is f o u n d t h a t t h e β a s w e l l a s α t e x t u r e is s t r o n g e r in h i g h e r g r a i n s i z e s a m p l e w h e n c o m p a r e d to l o w e r g r a i n s i z e s a m p l e .
REFERENCES: [1 ]
A . B h a t t a c h a r j e e , N . A g a r w a J , S. V . K a m a t , P. K. S a g a r a n d A . K. G o g i a , T i t a n i u m 4 , ( 1 9 9 9 ) 1.
[2]
M . H u m b e r t and N . Gey, Material Sei. F o r u m , 273 (1998) 163.
[3]
N . Gey, M. Humbert and H. Moustahfid, Scripta Mater. 42 (2000) 525.
[4]
N . Gey and M . Humbert, Acta Mater., 50 (2002) 277.
[5]
L. G e r m a i n , N . G e y , M . H u m b e r t , P. B o c h e r a n d M . Jahazi. A c t a Mater., 53 (2005) 3535.
[6]
M . H u m b e r t . L. G e r m a i n , N . G e y , P. B o c h e r and M . Jahazi, M a t . Sei. Engg., A 4 5 0 ( 2 0 0 6 ) 157.
[7]
S. S u w a s a n d A . K . S i n g h , M e t a l l . M a t e r . T r a n s . , 3 5 A , ( 2 0 0 4 ) 9 2 5 .
[8]
A . B o c k e r , H. K l e i n a n d H . J. B u n g e , T e x t u r e s a n d M i e r o s t r u c t u r e s 12, ( 1 9 9 0 )
[9]
H. I n o u e , S. F u k u s h i m a , N . I n a k a z u , M a t . T r a n s . , 3 3 ( 1 9 9 2 ) 1 2 9 .
103. [10] A . K . S i n g h , A . B h a t t a c h a r j e e a n d A . K. G o g i a , M a t . S e i . & E n g g . A . 2 7 0 ( 1 9 9 9 ) 225. [11] A . S a m e t - M e z i o u , A . E t t e r , T. B a u d i n a n d R. P e n e l l e , M a t . S e i . F o r u m , 5 5 8 - 5 5 9 (2007) 323. [12] N . G e y . M . H u m b e r t , M . J. P h i l i p p e a n d Y . C o m b r e s , M a t . S e i . E n g g . , A 2 3 0 (1997) 68. [13] N . S t a n f o r d a n d P. B a t e , A c t a M a t e r . , 5 2 ( 2 0 0 4 ) 5 2 1 5 . [14] D . B h a t t a c h a r y y a , G. B . V i s w a n a t h a n , R. D e n k e n b e r g e r , D . F u r r e r a n d H . L. Fraser, Acta Materialia 5 1 , (2003) 4 6 7 9 .
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D I S L O C A T I O N S IN T E X T U R E D M g - 4 . 5 % A l - l % Z n A L L O Y : T E M A N D S T A T I S T I C A L STUDIES V.N. Timofeev, V.N. Serebryany, Yu.A. Zaliznyak A.A.Baikov Institute of Metallurgy and Material Science. R A S M o s c o w , Russia ABSTRACT Basal-textured samples of M g - 4 . 5 A l - l Z n alloy cut at right angle to rolling direction were examined to define the density of different types of dislocation after w a r m rolling of the plates. T h e 22 different grains and their orientations were investigated by the transmission electron microscope J E M - 1 0 0 0 at accelerating voltage 750kV using a dark field - weak beam method of observation and the gb = 0 invisibility criterion as a basic method of the analysis of Burgers vectors. Dislocations with Burgers vectors . [c] and < a + c > have been identified. The reaction representing decomposition c+a dislocation into a and c has been observed experimentally and it has been found that [c] dislocations are the result of this reaction. The density of (basal and non-basal) dislocations does not depend on grain orientations. The rare [c] and
T h e aim of the present study is to investigate the dislocation structure of the warm rolled M g - 4 , 5 A l - l Z n alloy by transmission electron microscopy ( T E M ) . MATERIAL AND EXPERIMENTAL PROCEDURES M g - 4 . 5 % A l - l % Z n alloy w a s selected as model material because it is one of the most c o m m o n wrought m a g n e s i u m alloys. An initial state of alloy w a s a plate obtained by hot rolling with the deformation temperature of 420°C and with 7 8 . 5 % total deformation. The hot rolled plate had the microstructure consisted of the equiaxed fine (10-H5 μηι) recrystallized grains and the texture exhibited a strong basal type assembly. T E M analysis of dislocation structure of this state showed, that there were no dislocation tangles in majority of grains, and only in several grains separate dislocations were observed \ The hot rolled samples cut from the core part of the plate were warm rolled on the laboratory' rolling mill up to 2 + 3 % deformations. The w a n n rolling direction coincided with the hot rolling one. T h e rolling temperature was about 180°C. The mean strain velocity was about 1,85 s-'. The samples for T E M study were cut from the planes normal to rolling direction. The procedure of preparation of the TEM-specimens consisted of the following steps; •
cutting of the initial samples into strips (approximately "diamond coated wire"
1mm thickness) by the
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Dislocations in T e x t u r e d M g - 4 . 5 % A I - 1 % Z n Alloy: T E M a n d Statistical S t u d i e s
• • • •
preliminary grinding of strips to a thickness of approximately 0.6mm punching out 3 m m discs by the "ultrasonic cutter" polishing of the disks to - 3 0 0 p m thinning of disks using a double-jet electropolisher (Struers T e n u p o l - 2 . 15 V. 30 m A ) with a solution ( C2H5OH-45O ml. H C L O - 5 0 m l , butoxyethanol - 25 ml) cooled to 30°C.and rinsing of foils in an alcohol bath and flowing cold water. 4
•
In the present investigation, the J E M - 1 0 0 0 transmission electron microscope with accelerating voltage of 7 5 0 k V was used. The basic method of observation w a s dark field - weak beam ( W B DF) (g. ng) (n = 2,3...). The basic method of the analysis of Burgers vectors was the g b = 0 invisibility criterion: - the dislocations of
Figure 1. T E M image of a fragment of a grain (and its orientation) with
dis!ocations.
The
majority of them are basal ones. The dark-field image is m a d e in a reflex ( 2 1 1 0 ) . T h e plane of a picture is perpendicular to a direction [ 0 1 1 0 ] .
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D i s l o c a t i o n s in T e x t u r e d M g - 4 . 5 % A I - 1 % Z n Alloy: T E M a n d Statistical S t u d i e s
The density of basal dislocations e x c e e d s density- of non-basal dislocations in the half of grains. T E M image of a fragment of a grain with significant excess of basal dislocations is shown in figure 1. T h e orientation of C axis of this grain is s h o w n here. In the presented image parallel direct lines correspond to dislocations laying in basal planes, a n d considerably smaller quantity of the bent loop segments of lines of dislocations of non-basal slip are also visible. In this grain the ratio of density o f basal a n d non-basal dislocations w a s defined as 2 : 1 . In second half of grains the density of non-basal dislocations exceeds density of basal dislocations. T h e T E M image o f a fragment o f such grain and its orientation on a stereographic projection are presented in figure 2 . In a micrograph it is clearly seen that . non-basal dislocations are the bent lines, a n d there is an insignificant quantity of the parallel direct lines appropriate to dislocations o f basal slip. In this grain the ratio of density of non-basal and basal dislocations w a s defined a s 4 0 : 1 . Alongside with dislocations in some grains dislocations of type [c| and
Figure 2. T E M image of a fragment of a grain (and its orientation) with , dislocations. The majority of them is non-basal ones. T h e dark-field image is made in a reflex ( 0 1 1 0 ) . The plane of a picture perpendicular t o a direction [ 2 1 1 0 ] . alloy after compression at room t e m p e r a t u r e \ Similar results w e r e presented for other m a g n e s i u m alloys in other p a p e r s ' . Authors of these papers observed decomposition of
5
various reflexes ( 1 1 0 0 ) , ( 2 112) and (0002) are presented in figure 3(a,b,c). In the upper part of a picture (figure 3a) it is possible to see a fragment o f this reaction as decomposition o f a and a+c dislocations ([c] dislocations are suppressed in the given reflex). In the upper part of a picture (figure 3b) a fragment o f this reaction as c a n d a dislocations is presented (< a+c >
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dislocations in this reflex are suppressed). I n the upper part of a picture (figure 3c) a fragment of this reaction as c and a+c dislocations is presented ( < a > dislocations in this reflex are suppressed). For presentation in the increased scale this reaction is given completely in figure 4 by superposition of the figures 3a. 3b and 3c.
Figure 3 . A fragment of a grain at which there are , |c) and
Figure 4. Reaction of decomposition of a dislocation a+c o n c and a in the increased scale, obtained by superposition of figures 3a, 3b and 3c. The T E M data presented in figures 3 and 4, and also results of the above mentioned papers can specify the m e c h a n i s m of origin of the ..sessile"[c] edge dislocations, as a result of given dislocation reaction behavior. In this case the density o f [c] edge dislocations can serve as a parameter of
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D i s l o c a t i o n s in T e x t u r e d M g - 4 . 5 % A I - 1 % Z n Alloy: T E M a n d Statistical S t u d i e s
the first exceeds density of the second. T h e designation • (die shaded square) corresponds to grain orientations where along with basal and non-basal dislocations (the density of the first is more than density o f the second) [c] a n d
Figure 5. Grain orientations designated • . c . • . · and constructed on a stereographic projection of the [0002] crystallographic direction in sample coordinate system combined with a {0002} pole figure o f researched material obtained by a X-ray method. It is visible, that the majority of grain orientations is located in the field of the increased pole density (area of texture m a x i m a ) that confirms type of the given structure by T E M measurements. H o w e v e r , • and ο are equiprobably located in the field of texture maxima. It shows that grain orientations do not influence on a ratio o f density of basal and non-basal dislocations. At the same time all • and · are placed in the field of texture maxima. It confirms our earlier s u g g e s t i o n s that in the grains focused along an axis of a basal texture.
STATISTICAL EVALUATION OF T E M RESULTS Dislocation density for the separate grain is evaluated by chord m e t h o d . It is based o n the analysis o f the function of frequency spacing distribution on the image between the points of intersection of dislocations with the lines o f a certain grid (in the photograph). Then integral of the distribution density7
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F{R)=\f(r)dr
(1)
gives the estimation of that part of the entire sample, in which the distance between the dislocations
r < R.
Integration
from
zero to
R
m
( the m a x i m u m
distance between
the
dislocations ) m a k e s it possible to estimate general dislocation density. The general characteristic of frequency spacing distribution between the points of intersection of dislocations with the test grid escapes from the histograms of the results of measurements. The integral curve of frequency distribution is built on this density F ( / ) o f spacing distribution. Using these curves, general dislocation density finds
where F(R ,)\s the asymptotic limit of the integral curve of frequency distribution F(r'); L is the overall length of all lines of grid; t is the thickness of foil. T h e precision determination of the thickness of foil is a c o m p l e x and labor-consuming problem. Therefore we evaluated the ratios of densities of different types of dislocations instead of the determination of their densities. For M grains the average value of ratios of dislocation densities and its variance are calculated from the formulas n
8
(3)
We made the appropriate statistical evaluations for the dislocations such as basal and nonbasal ones separately, using results of the ratios of dislocation densities of various types for separate 22 grains received by chord method( see Table 1). T h e results represented in Table 1 show that the distribution of basal and non-basal dislocations in the grains is extremely i n h o m o g e n e o u s . The high values of standard deviation and variance in the ratio of densities of basal and non-basal dislocations indicate this. The high heterogeneity of rarely meeting
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Table 1. Experimental data and statistical evaluations of the different types of dislocations
Grain number
Grain square μηι 2
α degrees
ß degrees
P<«>b/p<»nb 2.77
Pt/p
5620
27,61
18
78
5622
27,89
21
73
1,44
5634
31,47
6
85
7,79
16.44
5647
30,66
16
81
2.19
19,67 0,25
5652
27,99
12
82
0.15
5660
30,73
29
76
1,08
5664
31.23
16
82
4
5666
30,84
20
74
1.66
5666
30.52
32
77
3.63
5672
30,28
13
88
3,68
5680
29.62
29
85
0.06
5684
30,12
30
85
0,13
5688
27,25
21
87
0,18
0,75
5743
27,44
18
89
0.12
0,58
5751
30,07
26
89
2
5754(1)
29,19
56
89
0.28
5 7 5 4 (2)
29,79
56
89
0,17
5758
29,69
15
85
0,14
5768
30,5
36
66
0,02
5649
26,27
5
85
0,18
5676
28,76
17
88
0,08
5677
29,75
55
85
3,6 1,44
Average value Standard deviation Variance
Pb/p
1,25
1,98 1.37
CONCLUSION •
T h e 22 grains with the different orientations were investigated by T E M . Dislocations with < a > (the basal and non-basal ones), [c] and < a + c> type Burgers vectors were found.
•
T h e density of (basal and non-basal) dislocations does not depend on the grain orientations. The rare [c] and < a + c > dislocations arrange in the field of texture maxima.
•
The reaction showing decomposition
•
W e a s s u m e that the presence of
•
The statistical estimation of the ratio of the densities o f basal and non-basal dislocations w a s carried out.
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REFERENCES S.R.Agnew, J.A.Horton and M . N . Y o o . Transmission electron microscopy investigation of
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3
4
5
6
7
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MICROSTRUCTURE
AND
MICRO-TEXTURE
EVOLUTION
OF
COMPRESSION
T W I N S IN M A G N E S I U M
P. Yang*, L. Meng, X . Li, W. M a o , L. Chen School of Materials Science and Engineering. University of Science and Technology Beijing. Beijing, China, 100083 ABSTRACT In magnesium, both basal slip and tension twinning lead to basal texture which promotes compression twinning during compression or rolling. The fracture originates mainly compression
twins
or
shear
bands
developed
from
compression
twins.
from
However,
the
recrystallization nucleation at compression twins or shear bands w a s observed to be more effective than that at tension twins or grain boundaries. Thus, the understanding of the evolution of compression twins has the significance of both preventing crack formation and optimizing annealing processing. In this work, the mierostructures and (mis)orientations of compression twins with matrices during their evolution into shear bands and the recrystallization nucleation within them were analyzed by E B S D technique. K E Y W O R D S : M a g n e s i u m , Twins, Deformation, E B S D , Micro-texture INTRODUCTION Strong anisotropy exists inevitably in polycrystalline wrought magnesium and a mere 8 % deformation can produce strong basal texture. Further deformation will initiate compression twins 1
followed by shear band formation which promotes either fracture at low temperature "
3
or
4
recrystallization nucleation at high temperature . 3
In a previous s t u d y w e determined the special misorientation of compression twins with their matrices, which was in most cases attributed to be of {10 11} type. Based on E B S D measurement and Schmid factor calculation a clear dependence of compression twins on grain orientations was determined.
The
distinctive
differences
in
morphology
and
boundary
compression twins and tension twins were discussed. In a separate work
mobility
between
w e found that the
orientation o f compression twins is very unstable and misorientations of ~40°<11 20> were often detected and w e determined the orientational conditions for basal slip and tension twinning which will be active within compression twins during further deformation. W e noticed that at early stages the shear bands developed from a group of parallel compression twins containing also the matrix orientations. In addition, a preliminary study on misorientations induced in matrix grains by compression twins or shear banding w a s made. In this paper w e will further investigate the microtextures evolved during the evolution of compression twins into shear bands and the orientations of new grains nucleated at shear bands/compression twins. This information will be helpful to optimize microstructure and to prevent fracture formation as well as to explain the general observation o f retained deformation texture after recrystallization in magnesium. 1. E X P E R I M E N T A L Commercial magnesium alloys AZ31 were used. T o facilitate the occurrence of compression twins, rectangular samples with initial basal or TD-rotated basal textures were prepared from hot rolled plates or extruded bars. These samples w e r e either rolled or plane-strain compressed (in
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Microstructure and Micro-Texture Evolution of Compression Twins in Magnesium
channel die) below 200°C with a strain rate of O.Ol/s. An HKL-EBSD system mounted on an FEM-SEM of model Zeiss-Supra 55 was used to determine grain orientations. Samples wcre polishcd in commercial AC-2 electrolyte solution and etched in picric acid. 2. RESULTS AND ANALYSES 2. I (Mis) orientations during the transformation from compression twins to shear bands kig.1 shows an orientation map on compression twins formed at -100°C. Although orientations of most regions within twins were dificult to be detected due to high strain, usehl information on misorientation can be found. In Fig.1 (a) two types of twin variants were seen. The Kikuchi pattern quality was shown in Fig. I (b) in the form of band contrast and misorientation data were also provided. I'he twin bands A on the upper-left region produced a misorientation angle of 7.3" in the matrix: the two twin bands A and B induced a misorientation angle of 9.8" totally in the matrix nhen they were apart; a misorientation angle of 12.5' was detected when two twin bands merged each other. It is clear that misorientation angles produced by each t\cinning can be roughly accumulated. A characteristic rotation axis of 4 1 TO> type between matrix and compression twins was detected. This means that rotation axis changes more sluggish than rotation angles. In Fig.l(c) the orientation No.4 is typical twin orientation of upper-left twin bands A, whereas the orientation No.7 is the typical twin orientation of upper-right &\in bands B, they are mainly -40°<1 1 20> relationship. According to the {0002} pole figure of Fig.l(c), orientation spread nearly around sample coordinate TD can be seen.
Fig. I Orientation map of compression twins formed at -1 00°C for a true strain of 0.15 (a) micrograph; (b) band contrast distribution; (c) pole figure; (d) orientation map
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Fig.2 shows another example of orientation mapping on compression twins. T w o types of twin variants were detected within a matrix grain with orientations l~3 in Fig.2 (a), (b). T h e micrograph of Fig.2 (a) demonstrates that the upper-right twin bands A w e r e blocked by the upper-left twin bands B. The color and the numbers of grain orientations in Fig.2 (d) are correlated with those in Fig.2 (b). [Tie pattern quality is shown in Fig.2(c) together with s o m e misorientation data. From Fig.2(b) it is seen that large strain presented mainly in the left twin bands Β leading to high "blind'" regions, while small misorientations o f less than 2° were detected effectively in the upper-right twin bands A indicating that these twins were not severely strained. The matrix orientation N o . l located in the upper hemisphere of {0002} pole figure in Fig.2(d) and it produced the twin orientations No.4, 6 situated at the lower hemisphere. T h e orientation No.9 may be a double twinning orientation. Several E B S D orientation m a p s were examined and it is observed that the orientations of matrices and their twins generally locate at two hemispheres o f pole figure. This feature of twinning is in contrast to the effect o f basal slip which leads to a gradual rotation toward the center of pole figure but never goes across it.
Fig.2 Orientation m a p o f compression twins, 1 5 % rolled, at 100°C (a) micrograph; (b) orientation m a p ; pink lines:
~40°
2 0 > ; green lines: 56°<11 20>; (c) pole
figure; (d) band contrast distribution Fig.3 shows an orientation map on shear bands evolved at relative high
deformation
temperature of 185°C at a strain of 0.6. It is seen that at higher temperature high strain can be accomplished without fracture formation and shear band became much wider. Fig.3(a) shows that t w o twin variants were merged and tended to lay parallel to the compression plane. Grain fragment proceeded apparently and recovery in this region took place. From Fig.3(b) and Fig.3(c) it is seen
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that green color corresponds to initial twin orientations located in the upper hemisphere of pole figure and matrix orientation situated in the lower part. At this strain and temperature, the orientations are m u c h scattered in comparison with Fig.l and Fig.2.
Fig.3 Orientation m a p o f shear bands formed at 185°C for a true strain of 0.60 (a) S E M micrograph: (b) orientation m a p ; (c) {0002} pole figure Generally speaking, as deformation temperature increases the width of compression twins or shear bands b e c o m e also large, therefore, orientations of new grains should also change accordingly. At low temperature, in contrary, high misorientation angle of nearly ideal compression twins and low strain difference between twin and matrix will be dominant. A t high strain and at high temperature grain orientations within compression twins/shear bands will approach to basal orientation in a low speed.
2.2 Recrystallization nucleation at compression twins It is well k n o w n that magnesium can only suffer from a compression of ~ 1 5 % at room temperature and the strain is mainly concentrated at compression twins/shear bands. Therefore, the dominant nucleation sites will be at these places. Fig.4 shows s o m e n e w grains nucleated at compression t w i n s during annealing at 280°C. It is further observed that new grains grew normally along twin bands at early stage and it is difficult for them to grow out o f twin bands.
Fig.4 Optic microstructure o f nucleation at compression twins, ( a ) 1 2 % rolled, 280°C,10s ;(b), 1 0 % rolled, 350°C, 30s
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Fig.5 shows an orientation m a p of the sample rolled for 1 2 % and annealed at 280°C for 10sec. It is seen that n e w grains formed within compression twins and grew mainly along twin bands. It is difficult for them to grow out of twin bands. T h e reason is attributed to rather low stored strain energy in matrix where only basal slip proceeded and lamellar substructure was stable. Sometimes only partially recrystallized structure was resulted. In this case the static recrystallization at compression twins in magnesium can be roughly divided into t w o stages, namely, the growth within compression twins and the growth out of twin bands. T h e inhomogeneities in grain sizes and grain morphology are often the features of recrystallized m a g n e s i u m . B y referring to the Fig.5 (b) and Fig.5(c) it is seen again that the orientations of matrix and new grains located in different hemispheres of {0002} pole figure, i.e. they are mainly initial compression twin orientations. A s compression twins did not evolve well into shear bands, the orientation distribution of mapped region is similar to that before annealing.
Fig.5 Recrystallization nucleation at compression twins, rolled 12%, annealed at 280°C for 10s. (a) micrograph; (b) orientation m a p ; (c) pole figure; (d) band contrast distribution Fig.6 is another example of orientation mapping on annealed sample. T h e nucleation within compression twin bands was well developed (Fig.6 (b)). According to Fig.6(c) new grains (pink color) and matrix (blue color) show their orientations on both hemispheres reflecting typical twin orientations. A s the compression twins were not in their simple morphology like that in Fig.5, the microtexture reveals a large orientation scattering. The band contrast distribution in Fig.6(d) shows the high quality of some n e w grains.
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Fig.6 Orientation map o f nucleation at compression twins, rolled 12%, annealed at 280°C for 10s. (a) micrograph: (b) orientation m a p ; (c) pole figure; (d) band contrast distribution 3. C O N C L U S I O N S Based on the E B S D orientation m a p s the characteristic (mis)orientations related with compression twins as well as those of n e w grains nucleated at compression twins can be summarized as follows: 1)
The misorientations
in
matrix
were
also
increased
due
to twin
formation.
Typical
misorientations o f 3 5 ~ 4 2 ° < 1 1 2 0 > and their orientations on both hemispheres of pole figures are frequently detected. Due to the high strain concentration compression twins are the preferred nucleation sites for fracture and recrystallization. 2)
T h e interaction of compression twin variants or shear bands when they are merged leads to a complicated orientation evolution near shear bands and enhances the misroeintation in the neighboring matrix.
3)
During annealing the nucleation within compression twins did not change
apparently
micro-texture, i.e. they show the initial twin orientation distribution. At this stage the growth of grains out of twin bands is difficult. ACKNOWLEDGEMENT: This work w a s funded by the National Nature Science Foundation o f C h i n a ( N o : 50571009). 'CORRESPONDING AUTHOR Tel: + 8 6 10 8 2 3 7 6 9 6 8 ; Fax: + 8 6 10 6 2 3 3 2 3 3 6
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E-mail address: [email protected] REFERENCES 1
P. Klimanek, A. Poetzsch, Microstructure evolution under compressive plastic deformation of magnesium at different temperatures and strain rates, Mater Sei Eng. 145-150, A324 (2002)
2
E. W . Kelley, W . F. Hosford, Jr. Plane-strain compression of magnesium and magnesium alloy crystals, Trans A I M E , 5-13, 242(1968)
3
B . C. Wonsiewicz, W. Α. Backofen, Plasticity of magnesium crystals. Trans A I M E , 1422-1431, 239(1967)
4
S. Ε. Ion, F. J. Humphreys and S. H. White, Dynamic recrystallisation and the development of microstructure during the high temperature deformation of magnesium, Acta Metall, 1909-1919, 30(1982)
5
P. Yang, L. Meng, Q. G. Xie, and F. E. Cui, A Preliminary Analysis on Compression Twins in Magnesium, Materials Science Forum, 546-549, 297-300 (2007).
6
L. M e n g , P. Yang, Q. G. Xie, and W. M . Mao, Analyses on Compression Twins in Magnesium, Materials Transaction (Japan), accepted.
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Texture in Materials Design
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A C C E S S I N G T H E T E X T U R E H U L L A N D P R O P E R T I E S C L O S U R E BY R O T A T I O N A N D L A M I N A T I O N : R E S U L T S IN T H E P R I M I T I V E B A S I S O F D I L A T I O N F U N C T I O N S Brent L. A d a m s , David T, Fullwood Brigham Young University Department of Mechanical Engineering, 435 C T B P r o v o , U T 84602 ABSTRACT Recent w o r k of A d a m s and c o w o r k e r s ' described the theory of Rotations and Laminations as an approach to altering the texture and properties of crystalline materials. The original work used Fourier representations of the Orientation Distribution Function based in the classical generalized spherical harmonic functions. Lamination by ultrasonic consolidation of thin sheet c o m p o n e n t s was demonstrated for t w o F C C materials. The results were presented in terms of the Texture Hull and the Properties Closure, with uniaxial yield strength versus Y o u n g ' s modulus selected as the particular example. In the present paper the theory of rotations and laminations is extended to a non-classical Fourier basis, consisting of selected dilation functions on the fundamental zone of orientations. The particular approach taken is shown to be an e x a m p l e of a representation of the group of rotations. Special properties of this representation are highlighted. In particular, an exponential form of the representation is explored, including its potential applications to texture design. INTRODUCTION Recent advances in representation of polycrystalline textures include the concept of a microstructure hull . For a selected homogenization model (i.e., texture-properties relation) and its associated representation of microstructure (here the focus is on the Orientation Distribution Function, O D F ) , the microstructure hull is the set of all possible textures, or O D F s . This hull can be represented by a compact, convex set of points in a multidimensional Fourier s p a c e ' . Each point in the hull represents a possible O D F . Having constructed the texture hull, it is then possible to enumerate all possible combinations of properties that can be achieved, theoretically, by the textures belonging to the hull. This complete set of properties combinations is called the properties c l o s u r e ' . Regions of this closure are of particular interest to designers, particularly in the preliminary stages of design where the design space can be expanded to the designer's advantage to explore texture variables , along with the traditional exploration of geometry. A previous paper' focused on elastic-plastic closures that can b e obtained by rotations and laminations of stock materials using new processing technology k n o w n as ultrasonic consolidation. Theoretical developments were presented there in terms of the classical generalized spherical harmonic functions, popularized by B u n g e . Changes in the Fourier coefficients for the O D F upon rotation were formulated using the Legendre addition theorem. Motions within the texture hull and the elastic-plastic closure were described as orbits. Laminations (by ultrasonic consolidation) were shown to comprise convex combinations of any combination of points lying upon the orbits affected by rotations. In this first paper' it was illustrated that a substantial portion of the theoretical properties closure can be realized by rotation and lamination of highly textured thin foil materials (e.g., strongly cube-oriented Cu). Rolled materials of weaker texture also expand the range of properties available to the designer, but to a smaller degree. 2
2
3
3
4
5
6
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A c c e s s i n g t h e T e x t u r e Hull a n d P r o p e r t i e s C l o s u r e b y R o t a t i o n a n d L a m i n a t i o n
In this paper the theory of Rotations and Laminations is re-formulated in terms of selected dilation functions. T h e O D F is expressed by a set of real numbers weighting these dilation functions. This set of weighting factors comprises a vector in a linear space. Twisting of the sample by rotation R can be expressed as a transformation Γ« in the linear space. It is shown that transformations of this type comprise a representation of the rotation group. Further, it is shown that any Tr can be expressed in terms of a set of 3 constant matrices, A\, A2, and A3 that represent small rotations about the 3 fixed sample coordinate axes. These constructs follow previous work on Lie G r o u p s described by Gelfand, Minlos and S h a p i r o . S o m e potential implications of this representation on texture design are discussed. In the sections that follow we first describe notation for the rotations. A description of approximations to the O D F by a weighted set of dilation functions follows. Transformations as a representation of the rotation g r o u p are then introduced. These describe orbits of the O D F . This is followed by the description of infinitesimal transformations, and m e a n s whereby an arbitrary Τ of the O D F can be expressed in terms of infinitesimal transformations. The process of lamination is described as an augmentation of transformations by rotation; this results in a richer set of possible O D F s and properties. In the final section a discussion of the implications of the theory for texture design is provided. 7
κ
DEFINITIONS AND OTHER PRELIMINARIES Rotation here refers to any twist of the polycrystalline sample relative to a selected coordinate system fixed relative to the c o m p o n e n t or sample:
!
jef,e,63J. 2
This frame comprises a
right-handed, orthonormal coordinate system; it is considered fixed and immobile; and the macroscopic properties are always referred to this frame.
T h e local crystal coordinate system,
also right-handed and orthonormal, is labeled as: j ê f . ê ^ ' . ê j j . upon position within the component sample. crystallographic axes of the material. êf //| 1001. ii'//|010|,
This coordinate frame depends
Its c o m p o n e n t s are fixed relative to specified
As an e x a m p l e , for cubic lattices it is convenient to set
êy //|0011. Finally, we consider a rotated set of coordinate axes: jêf , ^ , ^ ΐ } ·
These are related to the sample frame by the rotation R by the relations: ê[ = R-ê[.
(1)
If the component or sample is un-rotated with respect to the sample frame, j ê / j , then R = I, and ê[ =è[.
For any physical twist of the component relative to its reference frame, however,
will not be equivalent to
je'}
. T h e effective properties, at any location within the component,
however, will always be expressed in the fixed sample frame,
jê/J.
The orientation of the local crystal coordinate system, fixed on the lattice, is described relative to the current (twisted) sample frame by the rotation G: êf = Gif.
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Materials Processing and Texture
(2)
A c c e s s i n g t h e T e x t u r e Hull a n d P r o p e r t i e s C l o s u r e b y R o t a t i o n a n d L a m i n a t i o n
What is generally k n o w n experimentally is the O D F , / ( G ) , which will be defined more precisely in the next section.
This O D F describes the volume-fraction
density of lattice =
orientations at an orientation ϋ in the component. It is generally measured at jê/"j { ^ / } < or R = I. What is of interest in this paper is the transformed O D F , Rf(G),
describing the volume-
fraction density o f lattice orientations in the twisted sample, relative to the fixed sample frame ê/J.
The relationship between / ( G ) and Rf(G)
is evident from (1, 2):
Rf(GR) = f(G).
(3)
Thus, if / ( G ) is k n o w n , and R is fixed, Rf(G) is easily determined from (3). A P P R O X I M A T I O N S T O T H E O D F BY W E I G H T E D D I L A T I O N F U N C T I O N S T h e O D F and transformed O D F will be approximated by a finite series of dilation functions weighted by appropriate real coefficients. The d o m a i n of these functions is the fundamental zone of orientations, FZ. It is known that the FZ is a h o m o g e n e o u s space of physically-distinctive rotations of the crystal lattice relative to a selected reference coordinate frame. Here the sample frame jê/J
is the most convenient choice of reference frame. Thus, the
reference lattice of the pertinent crystal phase is fixed relative to the sample frame.
The FZ is
expressed as the left coset of the set of all proper rotations 5 0 ( 3 ) with respect to the symmetry subgroup o f the lattice, Γ : FZ = SO(3)l
Γ where Γ is the set of rotations belonging to SO(3) that 0,8
leave the lattice physically u n c h a n g e d . Partition the FZ
into a
finite
set of orientation bins,
a>\,ω ,...,ω„ 2
ω
Ν
with the
following properties: Ν
[}ω = FZ, η
ω,
n(Oj
=
0 (/ * j , i,j = 1,2
N)
(4)
n=\
where 0
is the empty set. Also, each orientation bin is defined such that its measure, m(to,), is
equivalent to every other bin:
m(e>,) = \\\dG = \l Ν,
(5)
6
8
Here dG is the invariant measure on the orientation space " . S u m m i n g relation (5) over the entire set of orientation bins defines the measure of the FZ to be 1 :
Σ »>(ω )= 1.
(6)
π
Next, define Ν dilation functions, x„(G) (n = l,2,...,W)of the form:
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\l if Gzco„ ^"
[0
_
otherwise
It is evident that these functions obey the following orthogonality relations: ]\\x„(G) (G)dC
= 5 lN.
Xm
(8)
nm
FZ
Any function on the FZ can be approximated as a weighted sum of the dilation functions. This includes the O D F and the transformed O D F . The O D F is defined by the expression:
^ ^ -
= f(G)dG,
(9)
where dV(G)IV\s the v o l u m e fraction of the material (for example, a material point) that associates with a neighborhood of infinitesimal measure dG containing orientation G. Similarly, the transformed O D F is defined by the expression:
^-^-=Rf(G)dG.
(10)
It is evident that these functions must satisfy a volume (mass) conservation relation of the form:
JlJf(G)dG
= \ = j\JRf(G)dG.
(11)
rz
1-7.
These two functions will be approximated by the expressions:
/ ( C ) = X U ( C ) ,
(12)
n=l
and Σ RF x (G).
Rf(G)=
The
(column)
iV-vectors
[RF ,RF ,...,RF ,...,RF .,] l
2
a
N
r
comprising
n
the
(13)
n
Fourier
coefficients
[F ,F ,...,F„,...,F ,]' I
2
N
and
represent the O D F and the transformed O D F , respectively, in the
Fourier space delineated by the dilation functions. Combining relations (6, 9-13) reveals the following constraint on the Fourier coefficients of the O D F s : Ν
Ν
2lF
n
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Evidently, the individual components of the Λ'-vectors representing the O D F and the transformed O D F represent volume fractions o f the material that have orientations that occupy the associated orientation bins, ω „ , a n d since these volume fractions must b e positive, it follows that the coefficients must also satisfy the relations: 0 < F ,RF n
<1.
n
(15)
T R A N S F O R M A T I O N S O F T H E O D F AS A R E P R E S E N T A T I O N ROTATIONS; ORBIT OF T H E ODF Our purpose is to define a Ν χ Ν
transformation
matrix
ON THE GROUP OF T
R
that transforms t h e
coefficients o f the O D F , F , into those o f the transformed O D F , RF , upon twisting the sample n
n
by a rotation R. This transformation will be expressed a s RFf
F
RF
R\N
—
M
TRNI
' '
TRNN
.
FN.
RFN-
The individual c o m p o n e n t s , T ,
}
T
''
describe the fraction o f invariant volume belonging to bin ω
Rmn
η
that transfers into bin ω„,. W h e n multiplied by F , T n
Rmn
signifies the fraction of the initial O D F
that contributes t o the component RF o f the n e w O D F . m
The coefficients o f 7« must satisfy
'conservation of m a s s ' relationships of the form:
lT =l Kmn
(forall« = 1,2
N),
(17)
(forall m = 1 , 2
Ν).
(18)
m=l
and lT =\ Rmn
Relation (17) means that all of invariant measure m(
ηι
rotation. terms.
with T
Ν
Rmn
having the meaning o f the fraction o f m(co„) moving into a> upon m
If R is an infinitesimally small rotation then T
Rml
will be much larger than the other
Relation (18) means that all o f the destination bins, ω„,, will receive from other bins
ω, ,...,a>„ ,...,ω
Ν
a n amount o f invariant measure equal to what it had before the sample twist,
which w a s m(co ) = \lΝ. m
This can be understood in terms o f a random O D F , for which F„ =1
for all n. Since any rotation of the material must preserve the r a n d o m n e s s of the O D F , it follows that after rotation all of the bins contain the same invariant measure that they contained before rotation. It must also be true that
0 < 7 > „ < l (forallm,n m
= \.2
N).
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That Tu forms a representation of the ^ - d i m e n s i o n a l real space (in this case, the space of Ndimensional approximations of the O D F ) is evident from t w o basic properties of the transformations. T h e first is: T T =T . Rl
Kï
(20)
RlRl
The second property is the presence of the identity transformation, dimensional identity matrix: 1 0
ρ
0 0 1 0
0 0 0
T , which is the jVx/V e
0 0
ρ
0
0 1
o'
Clearly the effect of T is to leave the O D F unchanged. define TR to be a representation of the rotation g r o u p . E
T h e properties expressed in (20, 21)
7
7
Following the traditional approach of Lie group t h e o r y , it is convenient to parameterize the rotations that twist the s a m p l e using the axis-angle parameters.
Let ξ =ξ^1
represent the rotation ξ = | | | in a right-handed sense about an axis ή = ξ^ξ\. the rotation is restricted to the range 0<ξ<π. 7 8
sphere . T
In this approach,
=T
a
Rmn Rmn(ÇL42<Ç?,)<
r
e
κ
ί
2 2
T h u s , the set of all rotations belongs to the π -
Τ =Τ (ξ ,ξ ,ξτ,), κ
+ξ ί +ξτ,ίι
T h e magnitude of
and elements o f the transformation
1
continuous functions of the rotation variables ξ ,ξ ,ξ . ι
will shorten the notation for the transformation matrix to Τ^ξ ,ξ ,ξ ) ί
2
2
= Τξ =
3
matrix,
Hereafter w e
?
Τ(ξ ,ξ ,ξ ). [
2
?ι
The orbit of the O D F / ( G ) is the set of all O D F s derivative of / ( G ) by all possible rotations ξ s π-sphere.
Expressed in the finite Fourier basis associated with the dilation
functions, the orbit of the O D F is the set of all O D F - v e c t o r s ^ F ] , . . . 4 F ] W
by linear transformations of the starting O D F - v e c t o r [F , . . . , F ] t
W
7
that can be obtained
for all rotations ξ belonging to
the π-sphere. Formally:
Orbi([F
T
F]
t
N
)=|[ξ>,
ξ>Λ,]
Γ
1 [ξ>ι
7
« F j v ] " =^[F ...,F ] -
V§ Ε π-sphere^
T
h
N
(22)
From a practical viewpoint, the orbit of any particular O D F is the set of all n e w O D F s that can be achieved by any twist of the sample. A few w o r d s about the calculation of the transformation matrix Τξ·. T h e c o m p o n e n t s of the transformation
matrix
are dependent
upon
the particular
choices
in partitioning the
fundamental zone. FZ, into bins ω „ . This is a limitation of the discrete representation described in this paper.
For any particular choice of binning, however, a relatively simple procedure can
be followed to determine the c o m p o n e n t s of Τξ for any choice o f ξ.
T h e FZ is filled randomly
with a large n u m b e r of points, representing a random distribution of orientations. Given that the invariant measure for each ω„ is equivalent in our procedure, t h e n u m b e r of r a n d o m points
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Materials Processing a n d Texture
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found in each bin will be about the same. The sample twist is then chosen and each orientation point is changed according t o G—>G£ . If the n u m b e r of random points found in
T,
(23)
The quality of the approximation achieved by relation (23) is governed by the size of the set of random points placed in the fundamental zone, and by the quality of randomness achieved by the placement scheme. T h e reader is reminded that relations (17-19) must hold in constructing (23). INFINITESIMAL ROTATIONS A N D THEIR TRANSFORMATIONS Expand the continuous function Τ{ξ\,ξ2,ξτ,) about 7(0,0,0) = T using T a y l o r ' s theorem: e
Tfr&&)
= T + A&+A& + A& + ... .
(24)
e
In (24) + . . . indicates that we have omitted the higher-order terms in the T a y l o r ' s series. These 1
higher-order terms will be negligible in comparison with -^ξ + ξ +ξ 2
2
. The terms Ai, A2, and
A3, called t h e infinitesimal rotation matrices, are defined as
Λ-(ξ,,ο,ο)
_Λ·(ο,ο.ξ ; 3
•
(25)
3ξι 7
c
It is k n o w n that Γ ^ , ^ . ξ . ι )
a
n
be obtained explicitly in terms of Αι, A2, and A3 alone. The
expression is A
Τ(ξ ,ξ ,ξ ) ί
2
x
+Α
ξ2
= e <S< *
3
2
The reader will recall that e = I +x + -^x
+ Α } ξ }
.
(26)
+ ...; thus, the right hand side of (26) should be
interpreted as Τ + Α ξ + Λ ξ + Λ ξ3 +.... T h e rate of convergence of (27) must be characterized ε
ί
]
2
2
3
in order to make effective use of the convenient form of (26); to the author's knowledge the order of the infinitesimal rotation matrices that must b e considered, in order to achieve sufficient convergence in the transformation matrix, remains to be explored. Further conditions on the infinitesimal rotation matrices are k n o w n . T h e most important are the c o m m u t a t o r relations that interconnect them: 7
[ A , , A ] = A , [ A , A ] = A,, [ A , A , ] = A , 2
3
2
3
3
2
(27)
where \A,B] = AB-BA.
LAMINATIONS OF ORBITS
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A c c e s s i n g t h e T e x t u r e Hull a n d P r o p e r t i e s C l o s u r e b y R o t a t i o n a n d L a m i n a t i o n
Lamination refers to the mixture of two or more O D F s , say R^f(G),
Ä / ( G ) , to obtain a 2
n e w O D F . The method p r e s u m e s that no additional rotations, beyond Ri, and R2, are allowed to occur. If a „ a n d a, are the v o l u m e fractions of R f(G), t
Ä / ( G ) that are laminated, then the new, 2
laminated O D F , say Lf[G) is given by Lf(G) = a R f(G) i
i
+ a R]f(G)
(28)
( α , + α = 1; 0 < α , , ο _ < 1 ) .
2
2
Extension of the lamination procedure to more c o m p l e x cases is easily accomplished using the basic ideas presented in relation (28). Feasibility for lamination, via ultrasonic consolidation, was considered in prior w o r k ' . T h e important characteristic of lamination is that it e x p a n d s the range of possible O D F s and their derivative properties combinations relative to the set of O D F s achievable purely by rotations. Figure 1 illustrates the range of combinations of yield strength versus Young" s m o d u l u s predicted to be achievable for two F C C materials by rotations and laminations. The principal conclusion is that stronger textures give rise to larger accessible portions of the c o m p l e t e (theoretical) properties closure relative to weak starting textures. D I S C U S S I O N O F T H E R E S U L T S IN T E R M S O F T E X T U R E D E S I G N Relation (26) is the central result of the paper, as it constitutes an explicit expression for the transformation Τξ associated with an arbitrary rotation ξ.
These results are to be c o m p a r e d
with similar results obtained for texture evolution during plastic deformation, which were shown to have similar f o r m ' . Once the infinitesimal rotation matrices. A/, A2, and A3, have been calculated, and it is verified that they satisfy the c o m m u t a t i o n relationships (27), then an arbitrary Τξ can be calculated. T h e important caveat to be applied to these statements, however, is that the practical implementation of relation (26) involves an infinite series; and the rate of convergence of this series is not k n o w n . With arbitrarily large twists of the sample, one must be concerned with rotations that cause a lattice orientation G, which is initially located in the FZ, to b e c o m e ΰξ outside the FZ.
Of course ϋξ
that might lie
can a l w a y s be mapped back into the FZ by applying rotations that
belong to the symmetry subgroup of the lattice. This re-binning procedure can be numerically costly, however; and any approach whereby it can be avoided would be attractive.
Assuming
that the infinitesimal rotation matrices have been correctly determined, with transformations that remain entirely within the FZ, the implementation of rotations by relation (26) will always remain in the FZ. Texture design seeks to identify specific textures that give rise to an optimal set of properties for that design " . In previous e x a m p l e s of optimal design of O D F s , it is not clear h o w 2
5
the optimal O D F s could be obtained by processing. Here w e pose a limited texture design problem. W e assume that a target O D F is k n o w n from the m e t h o d s of texture design. Let this γ ODF
be expressed
f(G) = [F ...,F ] h
N
T
as
®/(G) = [ ^ 1 , . . . , * / > ]
as starting material.
.
Let us assume that w e begin with
ODF
W e shall allow this starting O D F to be rotated arbitrary
by twisting, with the result that the O D F c h a n g e s to
£f(G)=m
654
.ζ*
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200
-
/
180
-
f
«" CL 180 2 . 140
-
c S _ 3
\\
ΊΓ
-
\
120
-
\
100
*
80 Γ , 0.1
0.2
0.3 S\,
0.4
0.5
Ο.β
0.7
1
= Ι/Ε*,
u
(0 01 G P a " )
120
-
110
*Ξ 0
•
•
CD Ο
\
\ \
tO Ο
(MPa)
100
\\
70 60
-
\
-
w
.—-—-" ^~~ .
— '
SO 40 I
0.2
L· 04
1
0.6
s;
t 1 1
1
0.8
= 1/E;
ι 1
1
1-2
ι 1.4
1
(0.01 GPa" )
Figure 1. Elastic (inverse Y o u n g ' s modulus)-plastic (yield strength) properties closures for a weakly textured Ni 201 sheet material (top) and a strongly cube textured Cu 10100 material bottom. T h e outer envelope (black) is the theoretical closure for all possible textures of these materials. T h e triangle indicates the position of the isotropic texture. The asterisk * indicates the predicted location of the starting texture. The inner-envelope describes the "primary' orbit" and its laminations, which consist of rotations about the nomia! to the thin sheet material. The intermediate envelope (blue) comprises the complete orbit of the material and its laminations.
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Design for rotation can be posed as a minimization problem of the form
m
\nW
where
ι
Thus, the rotation parameters ξ ,ξ ,ξ ι
between the target O D F and the
@
Ψ(ξ 4 &)
2
3
φ
2
= ^ξΡ - Ρ )\... (ξΡ - Ρ ) ^.
2
ι
ι
+
Ν
Ν
(30)
are found that minimize the least-squares differences OrbiKfiG)).
ACKNOWLEDGEMENTS This work w a s supported at Brigham Young University by a grant from the U S Army Research Office, Metallurgy P r o g r a m , Dr. David Stepp, Program Director. REFERENCES 'B.L. A d a m s , C. Nylander, B. Aydelotte, S. A h m a d i , C. Landon, B . Stucker, and G.D. Janaki-Ram, "'Accessing the Elastic-Plastic Properties Closure by Rotation and Lamination," Acta Materialia 56 (2008) 128-139. B.L. A d a m s . A.Henrie, B. Henrie, M. Lyon, S.R. Kalidindi, and H. Garmestani, "'Microstructure sensitive design of a compliant b e a m , " J. Mechanics and Physics of Solids, 49 (2001) 1639-1663. B.L. A d a m s , M. Lyon, and B . Henrie, "Microstructures by design: linear p r o b l e m s in elastic-plastic design," Int. J. Plasticity, 2 0 (2004) 1577-1602. B.L. A d a m s , X. G a o , and S.R. Kalidindi, "Finite approximations to the second-order properties closure in single-phase polycrystals," Acta Materialia, 53 (2005) 3563-3577. S.R. Kalidindi, J.R. H o u s k a m p , M. Lyons, and B.L. A d a m s , "Microstructure sensitive design of an orthotropic plate subjected to tensile load," Int. J. Plasticity, 2 0 (2004) 1561-1575. H.-J. Bunge, " T e x t u r e Analysis in Materials Science," Butterworths, L o n d o n (1982). I.M. Gelfand. R.A. Minlos. and Z.Ya. Shapiro, "Representations of the rotation and Lorentz groups and their applications," P e r g a m o n Press, Oxford (1963). A. M o r a w i e c , "Orientations and rotations," Springer-Verlag, Berlin (2004). D.S. Li, H. Garmestani and B.L. A d a m s , " A texture evolution model in cubicorthotropic polycrystalline s y s t e m , " Int. J. Plasticity, 2 1 (2005) 1591-1617. 2
3
4
5
6
7
8
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CORRELATION RELATIONSHIP BETWEEN YIELD STRENGTH AND THE ELASTIC COMPLIANCE FOR THE POLYCRYSTALLINE NICKEL 2 0 1
3
Sadegh Ahmadi" , Brent L. A d a m s , David T. Fullwood", and Bradley S. Fromm" Department of Mechanical Engineering, Brigham Y o u n g University, Provo, U T 84602.USA
a
ABSTRACT: Theoretical relationships for the variance, covariance and the correlation coefficient of yield strength and the elastic compliance are presented. Variance of interest here is due to the ' w i n d o w size" of a representative statistical element, randomly taken from the material. A Taylor viscoplastic theory was used to predict the yield strength in deviatoric form. To compare the predicted correlation relationships for yield strength and the elastic compliance, a polycrystalline hot-rolled and annealed Nickel 201 sheet was used. Tensile testing along the rolling direction was carried out and yield strength and Y o u n g ' s m o d u l u s were measured. Pair correlations were recovered by orientation image microscopy (OIM). Results for the variance, covariance and the correlation coefficient based on variation of the w i n d o w size are introduced. Variance is predicted to reach a m a x i m u m w h e n the w i n d o w size shrinks to zero; it tends to zero when the w i n d o w size exceeds the coherence limit. An anti-correlation linear relationship between the elastic compliance and yield strength along the rolling direction is observed. The implication of these results on the accuracy and reliability of texture-based properties predictions is discussed. 1. I N T R O D U C T I O N Dealing with variation in material properties and performance has always been part of modern engineering practice. M u c h of this variation is a result of the heterogeneity in the microstructure and nanostructure of materials due to defects, texture, mixed phases, precipitates, etc. On larger scales, the effects of microstructure variance are usually ignored because short wavelength variations of behavior average out over large volumes. Such homogenization assumptions take place on scales that are larger than the size of a representative volume element ( R V E ) , which is defined as the smallest representative volume that is statistically indistinguishable from any other similarly sized volume sampled in the bulk material. In contrast, v o l u m e s less than the R V E exhibit statistical scatter in the microstructure from one sampled volume to the next and in the literature are referred to as statistical volume elements ( S V E ) . T h e statistical scatter between sampled S V E s logically tends to zero as the size of the SVE approaches that of a R V E . Thus to consider the variation in material response relative to variations in the microstructure. volumes less than a R V E must be considered. Previous work focused on estimating the variance of an arbitrary property of a given material. The derivation of a texture-based property variance relationship was first presented in abbreviated form at N U M I F O R M 2 0 0 4 ' . Gao et al. used a method to analyze two-point pair correlation functions, which are also used in variance relationship, and coherence length in polycrystalline materials. Afterwards, a methodology to determine the variance of an arbitrary material property based on the statistics of the texture of polycrystalline materials for a specific volume w a s introduced by Przybyla et al. . They evaluated the connection between the variance of the r-value and variance of the Taylor factor for Iron Silicon 3 % steel, and it was observed that only a weak connection was found between these t w o properties. They also concluded that small variations in the Taylor factor yield large variance in the r-value. 2
3
657
C o r r e l a t i o n R e l a t i o n s h i p B e t w e e n Y i e l d S t r e n g t h a n d t h e Elastic C o m p l i a n c e for Ni 2 0 1
In this paper, some relationships - especially covariance and correlation relations - are introduced that will be helpful to c o m p a r e two arbitrary material properties due to the morphology and texture of the microstructure in SVEs taken from polycrystalline materials. By definition, covariance or correlation between two properties is an indication of h o w they are linearly correlated relative to one another. The scope of this paper is to define new covariance and correlation relations, and to illustrate their use, along with variance relations, by application to a 201 nickel material. The use of these relations, as a function of w i n d o w size, will also be used to obtain estimates of the coherence length that define the m i n i m u m size of an R V E in the sample. 2. T H E O R Y 2.1. Calculation of Yield Strength by Using P o w e r - L a w Viscoplasticity A Taylor-like viscoplastic model has been used here to estimate the yield strength under prescribed loading conditions. The model differs from the traditional approach in that local grain size D is considered, in addition to orientation g. This new formulation is described in the paper of F r o m m et al* elsewhere in this proceedings, and is not repeated here. Following the Taylor assumption, it is considered that everywhere local and macroscopic strain rates are equal, such that the local stress σ = σ(ε, g, D). 2.2. Elastic C o m p l i a n c e Tensor The linear elastic relationship between the stress and strain components is expressed as
W =^S, a, a , , =S„ a lkl
where
S
u
is the
fourth-rank
elastic
l
kl e l
kl
(1)
kl
compliance
tensor.
This
matrix possesses
certain
symmetries and for the cubic crystals can be expressed in the sample frame as
s, = W « ltl
+
W+{s -s u
n
-%)Xg„g, g„g . r
ip
(2)
where g is the matrix used to relate the crystal frame to the sample or global frame. δ represents the Kronecker delta, and S,,, S and 5 are the fundamental elastic compliances. tJ
l2
4 4
2.3. Texture-Based Covariance and Correlation Relationships of T w o Properties Let p'(Cl)
and
S V E within region Ω . Let Q equal the total n u m b e r of sample S V E s in the ensemble. Then the covariance of the two properties is defined by
c o v ( p , q I Ω) = 1 £ ( ρ ' ( Ω ) - p ) ( q ' ( Q ) - q ) .
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C o r r e l a t i o n Relationship B e t w e e n Yield S t r e n g t h a n d t h e Elastic C o m p l i a n c e for Ni 2 0 1
N o t e that if small/large values of one property, e.g. p. tend to b e associated with small/large values o f another property, e.g. q, then ρ'(Ω)-ρ and q'(ÇÏ)-q will typically have the same algebraic signs, resulting in a positive covariance. If the reverse is true and small/large values of first property associate with large/small values of second property then this results in a negative covariance. Therefore, the covariance is an indication of h o w t w o properties vary relative to one another. T h e variance of an arbitrary property is introduced as
y
=^Ι>'(Ω) y i.i
2
/.ι
1
Γ
-ρ
.
(4)
For more information about the derivation o f Eq. (4), the reader can reference the work of Przybyla et al. . In both equations, ρ and q are assumed to be equivalent to the volume average of the property values associated with an RVE. A n ergodic hypothesis is invoked here, equating volume and ensemble averages. Assuming the material to be statistically h o m o g e n e o u s one can readily derive the variance and covariance equations based o n pair correlation functions (PCF) from relations (3,4) . 3
3
σ
«°> =
r T ^ i J J J J J i i i I np(g)-P(g')&(r !/(")!
I Ciidrdgdg- -
f
(5)
i-z 17 «'
and
cov(p,
\ d)dgdg'
2
I r).p(g).q(g')<ä{7 \ Q)drdgdg'-pq
FZ
(6)
R>
is the two-point probability density function of a randomly placed
vector r - having its head at orientation g and tail at the orientation g'. T h e function Θ(Ρ | Ω) is the volume intersection of t w o three-dimensional regions of identical size and shape Ω and 2
3
displaced by vector r . This volume has been previously described ' . Another useful connection is the so-called correlation coefficient, which w e will denote as ç(p,q\Ω). T h e correlation coefficient has the form
^|Ω)
where again σ
2 [ £ ϊ
and σ
2 Ω
=
£
0
ν
(
^
|
Ω
)
=
(
Ω)
7 /'^ .
(7)
are the variance for t w o properties, ρ'(Ω) and (?'(Ω). which have
the correlation relationship in the w i n d o w s Ω . Unlike the covariance which can assume any real value, the correlation coefficient is bounded; it must lie in the range o f - 1 < ç(p,q j Ω) < 1. For large positive values of the correlation coefficient w e expect a strong linear correlation between properties ρ and q. For large negative values - a strong linear anti-correlation is expected. When the correlation coefficient is zero, no correlation is expected.
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Correlation Relationship B e t w e e n Y i e l d S t r e n g t h a n d t h e Elastic C o m p l i a n c e for Ni 2 0 1
2.4. Validation of Variance/Covariance Calculations using Fast Fourier Transforms The variance/covariance discussed earlier can be calculated by brute force using the original definitions. However, a rapid method of calculating these for validation purposes is available by rewriting the equations in terms of convolutions, and exploiting the efficiencies of Fast Fourier Transforms ( F F T s ) . 5
Let ρ ( Ω ) be the value of property ρ associated with the ith sampled SVE within region Ω . Then ρ ( Ω ) may be rapidly calculated for every point in Ω by taking the convolution of a function. ^ ( Ω ) . that takes the value one within a w i n d o w Ω
about the origin, and zero
elsewhere,
p\Q)
= W(Q)*p 1
then the variance of each property. ρ ' ( Ω ) or q (p.)
(8)
is determined using Eq. (4). To calculate
covariance of t w o properties Eq. (3) is used. And finally after calculating the variance and covariance of both properties using FFTs, the correlation coefficient is calculated from Eq. (7). 3. E X P E R I M E N T A L W O R K Hot-rolled and annealed Ni 201 was used for this study. To find the yield strength, Y o u n g ' s modulus and other mechanical properties, tensile testing based on A S T M E8 was conducted for three sub-size specimens. Characterized average values are presented in Table 1. Table. 1. Mechanical properties of Ni 201 selected for this study. Y o u n g ' s M o d u l u s (GPa)
Yield Strength (MPa)
Elongalion in 1 inch (%)
Tensile Strength (MPa)
205
256
22
435
In order to find the texture effect on simulations, the orientations of the microstructure are measured using orientation image microscopy ( O I M ) . The electron backscatter patterns were collected with a Philips S F E G - X L 3 0 scanning electron microscope. O n e scan w a s taken for the sample with an area of approximately 2.5x5 m m and step-size of 10 microns. Fig. 2 shows the complete scan area of the sample. The w i n d o w , Ω , placed in the scan defines the area in which the two-point statistics are sampled. The m a x i m u m size of Ω must not exceed half of the dimensions of the O I M scan so the statistics sampled by the vector 7 are reasonably equally weighted for all choices of the sampling vector 7.
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C o r r e l a t i o n R e l a t i o n s h i p B e t w e e n Yield S t r e n g t h a n d t h e Elastic C o m p l i a n c e for Ni 2 0 1
Fig. 1. OIM scanned surface of Ni 201. The black region indicates the maximum size of a SVE/RVE
Total n u m b e r of grains w a s 1004, of which 138 of grains were on the edges of scan area and so a total of 866 grains w e r e chosen to find average grain size based on Chord-Length Distribution ( C L D ) function method. Further details of the application of the C L D function to the problem of grain size characterization are described elsewhere in this proceeding by Fromm et a l . . T h e average grain size for all grains calculated by C L D method was found to be 74.7 microns. 4
Experimental tensile tests were aligned with the rolling direction ( R D ) . and we require the yield strength and the elastic compliance c o m p o n e n t in rolling direction, σ,', and S
.
u u
for
comparison. In order to theoretically calculate the yield strength using power-law viscoplasticity ( P L V P ) relations, the rate sensitivity parameter w a s chosen to be 0.02,
the reference shear rate
w a s selected to be O.OOI.f'. the intrinsic frictional shear stress, 63.5MPa, y 2
slope, 1.91x10'' Nm~ ' .
and the Hall-Petch
To calculate the compliance in the rolling d i r e c t i o n , S , w h i c h is the m t
inverse of the familiar elastic Y o u n g ' s modulus in the 1- material direction. £ , , the compliance material constants 5 . S . and S . for pure nickel w e r e used; these are 0.734. M
2
[
0.802 xl0~ GPa~ ,
1 2
u
-0.274,
and
6
respectively .
4. R E S U L T S A N D D I S C U S S I O N 4 . 1 . C o m p a r i s o n between Experimental and Numerical Calculations The local elastic compliance and yield strength in the 1-direction, S '
1
i m
and σ, , were
calculated for all O I M points. Fig. 2. As it can be seen, by changing the orientation of O I M points, the values of yield strength and the elastic compliance are varied. The m a x i m u m and m i n i m u m values of calculated yield strength are 206 and 127 MPa. compliance in the 1-direction varies from 0.329 to
2
respectively. The elastic
l
0.734x\0- GPa .
Materials Processing and Texture
•
661
Correlation Relationship B e t w e e n Y i e l d S t r e n g t h a n d t h e Elastic C o m p l i a n c e for Ni 2 0 1
Fig. 2. Local yield strength vs. the elastic compliance (the inverse of Young's modulus) in 1-direction calculated for each OIM scan points. The symbol • shows the average of all OIM points for both properties, errand S
n u
.
The symbol A marks the experimental (measured) point.
The ensemble average for both properties was also calculated. Average for the yield strength,
ëf,',.
0.4~5χ\0^Ρα~'.
is
1~4.5
MPa,
and for the elastic
compliance
in
1-direction.S
IMI
,
is
If w e convert the value o f the elastic compliance into the Y o u n g ' s modulus
along 1-direction, £, = l / . S ' , w e would get l m
202.2
GPa which is very close to
205
GPa
measured from experimental work (see Table. 1 ). 4.2. Variance of the Yield Strength a n d the Elastic Compliance The effect of w i n d o w size on the variance o f yield strength a n d the elastic compliance was investigated, F i g . 3 . F o r simplicity in calculations, the shape of the w i n d o w Ω w a s taken to be square. By varying the size o f window Ω from 1 micron to 125 microns, variance of both properties w a s obtained. A s it can be seen, variance of yield stress in the 1-direction, Fig. 3(a), and variance of the elastic compliance in the 1-direction, Fig. 3(b). is decreased as the w i n d o w size of a S V E increases. Furthermore, for the m a x i m u m available w i n d o w size the variance for both properties converges to zero. This implies that for a larger size S V E the value of each property, for instance ρ ' ( Ω ) , is really close to the e n s e m b l e average, ρ. and therefore the variance is very small. It is also found that for w i n d o w sizes smaller than a R V E t h e variance o f t h e elastic compliance is much smaller than the variance of yield strength.
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Materials Processing a n d Texture
C o r r e l a t i o n R e l a t i o n s h i p B e t w e e n Y i e l d S t r e n g t h a n d t h e Elastic C o m p l i a n c e for Ni 2 0 1
Fig. 3. Variance of two properties, (a) yield strength and (b) the elastic compliance, along the rolling direction as a function of window size. The window Ω is taken to be square.
4.3. Covariance / Correlation relationship of Yield Strength and the Elastic Compliance Covariance of yield strength and the elastic compliance were estimated for polycrystalline 201 nickel material. Calculated covariance as a function of w i n d o w size is presented in Fig 4(a). It is observed that covariance between yield stress and the elastic compliance is negative and tends to zero as the window size increases. Covariance relationships m a y b e especially important in Microstructure Sensitive Design ( M S D ) w here ratios of two properties are k n o w n to govern the performance of the design. ;
Fig. 4. (a) Covariance and (b) Con-elation of yield snength and the elastic compliance along the rolling direction as a function of window size. The window Ω is taken to be square.
If the variance of the properties with w i n d o w size is not the issue then the Correlation coefficient can be employed to give a normalized measure of distance from random (or no correlation between the properties). Fig. 4(b) s h o w s the stability of the correlation coefficient with w i n d o w size of the S V E . with values close to - 1 , indicating a high degree of correlation. Negative values of the correlation coefficient indicate an anti-correlation relationship between yield stress and the elastic compliance for this material. Upon increasing the yield strength the
Materials Processing and Texture
•
663
C o r r e l a t i o n Relationship B e t w e e n Y i e l d S t r e n g t h a n d t h e Elastic C o m p l i a n c e for Ni 2 0 1
inverse of Y o u n g ' s modulus decreases. As a result, we can conclude that for this material Y o u n g ' s modulus and yield stress have direct relationships. This is not surprising. 5. C O N C L U S I O N S Theoretical relationships for the variance, covariance and the correlation coefficient of yield strength and the elastic compliance w e r e presented. Effect of the w i n d o w size of a S V E on the linear correlation relationship between yield strength and the elastic compliance of a polycrystalline nickel 201 material was studied. It is concluded that: 1. Variance of yield stress and the elastic compliance in the 1 -direction is decreased as the w i n d o w size of a SVE increases, approaching the size of a R V E . 2. Covariance between yield stress and the elastic compliance for this material is negative and for larger w i n d o w size of a S V E tends to zero. 3. Use of the linear correlation coefficient was shown to maintain the expected anticorrelation of yield strength and inverse Y o u n g ' s modulus over the entire range of w i n d o w sizes examined. These observations are not surprising; the main result of this paper is the development of a formulation that enables direct calculation of these statistical measures from the pair correlation statistics. The measures m a y then form the basis for evaluation of sample size necessary for quality control operations, and variance analysis for small components. Correlations between local states (e.g. orientation) and resulting properties may also be examined in m o r e detail. ACKNOWLEDGEMENT The authors wish to acknowledge the support of this work by the Army Research Office, Dr. David Stepp, Program Manager. REFERENCES C P . Przybyla. B.L. A d a m s , and M.P. Miles, A Method for Determining Property Variance in Polycrystalline Materials, N U M 1 F O R M 2 0 0 4 : 8th International Conference on Numerical Methods in Industrial Forming Processes, S. Ghosh et al., eds., C o l u m b u s , O H , 1 7 6 0 - 1 7 6 4 . X. Gao. C P . Przybyla, and B.L. A d a m s , Methodology for Recovering and Analyzing T w o Point Pair Correlation Functions in Polycrystalline Materials, Metall. Mater. Trans. A, 37A, 1279-2387(2006). C P . Przybyla, B.L. A d a m s , and M . P . Miles, Methodology for Determining the Variance of the Taylor Factor: Application in F e - 3 % S i , Journal of Engineering Materials and Technology, vol. 1 2 9 ( 1 ) , 82-93 (2007). B.S. F r o m m , B.L. A d a m s , S. A h m a d i , and M . Knezevic, Grain Size and Orientation Distribution Function of High Purity α-Titanium, Submitted to I C O T O M 15: The 15th International Conference on the Textures of Materials, T h e American Ceramic Society, Pittsburgh, PA, (2008). E.O. Brigham, The Fast Fourier Transform and Applications, Englewood Cliffs: Prentice Hall, (1988). H.B. Huntington. Solid State Physics, 7, 213 (1958). 1
2
3
4
5
6
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Materials Processing a n d Texture
M O D E L I N G R O L L I N G T E X T U R E O F T W I N N E D CU S I N G L E C R Y S T A L S Khaled Al-Fadhalah
| a )
. Armand Beaudoin
l h l
, and Marek N i e w c z a s
> c }
(a) Mechanical Engineering Department. Kuwait University, Safat. Kuwait (b) Mechanical Science and Engineering Department. University of Illinois at UrbanaC h a m p a i g n , Illinois. USA. (c) Materials Science and Engineering Department. M c M a s t e r University. Ontario, Canada
ABSTRACT A polycrystal plasticity model is proposed to predict the rolling texture of twinned Cu single crystals. Prior to rolling, the Cu crystals were tested in tension at 4.2 Κ to induce extensive twinning, w h i c h resulted in fine structure of twin-matrix lamellae. T w o twinned Cu samples have been examined in this study: one sample is as deformed that has a texture of strong twin component, and the other is recrystallized at 4 0 0 Κ that has balanced texture of twin and matrix components. T h e samples were rolled to about 7 0 % reduction in thickness. X-ray texture m e a s u r e m e n t s indicate that the rolling texture of the deformed sample preserves to a large degree the initial texture, suggesting a strong tendency for codeformation between the twin and matrix phases. For the recrystallized sample, the rolling texture partially preserves the initial texture, implying a weaker correlation between the recrystallized layers. T o model the microstructural effect, each lamella is regarded as a composite grain consisting of twin and matrix layers. The interaction at the lamella interface is accounted for by satisfying compatibility and equilibrium requirements across the interface, and imposing corotation of twin and matrix layers. The aggregate response of composite grains is computed using a viscoplastic self-consistent (VPSC) scheme. With this continuum approximation, the simulated rolling textures show large similarities with the measured ones, keeping in mind no length scale has been considered in the model. INTRODUCTION T w i n n i n g in C u is k n o w n to rarely occur at room temperature as a result of having a moderate value of stacking fault energy. However, it w a s discovered in the pioneering work of Blewitt et al. that Cu can undergo twinning deformation at 4.2 Κ due to the limited dynamic recovery at this low temperature allowing material to achieve high stress required to nucleate t w i n n i n g . . A recent study by N i e w c z a s et al. o n C u single crystals indicated that extensive twinning occurred during tensile testing along [541] at 4.2 Κ . T h e tests were carried out to large deformation up to fracture. Quantitative X-ray texture m e a s u r e m e n t s carried out on deformed crystals revealed that the amount of twinning phase is higher in continuously deformed samples than that in samples deformed with intermittent annealing. Results from the intermittent annealing samples suggest that annealing removes large density of sources, which provide Shockley dislocations required to produce twinning deformation. This reduces the volume of the parent crystal transformed to twin lattice during the onset twinning from about 6 0 % to about 10%. The twinning deformation in copper single crystals at 4.2K results in very fine lamellar structure, w h i c h consists of twinned and untwined (matrix) layers of an average thickness of 100 nm. The twinning process creates additional parent-twin interfaces in the microstructure that act as barriers to dislocation motion in a composite-like material. Moreover, observations by T E M 1
2
665
M o d e l i n g Rolling T e x t u r e of T w i n n e d C u S i n g l e C r y s t a l s
indicated that the dislocation mobility is largely affected by twinning and results in the forming of immobile (sessile) dislocations inside the twin-transformed volumes of the material. Subsequent studies of the recrystallization of C u single crystals deformed at 4.2 Κ ' have revealed that samples deformed to a point before t h e occurrence of twinning yield randomly oriented grains whereas the twinned samples produce t w o c o m p o n e n t s of the recrystallization texture, which represents the crystallographic dependence o f the lamellar (twinned) structure. Modeling the deformation behaviour of a material with lamellar microstructure h a s been a challenging task. Several investigators have examined the deformation of twinned (γ-TiAl +a T Î 3 A I ) single crystals using crystal plasticity " . T o model the microstructure, each lamella was regarded as a composite grain. T h e plastic deformation o f composite is computed by averaging the response of γ-TiAI and a2-Ti Al phases presented in the lamella. It w a s concluded that the plastic deformation and texture development is best described by codeformation of the t w o phases \ In addition. I.efiers'"* used a relaxed-constraints crystal plasticity model to study (he effect o f grain subdivision into flat long b a n d s on the texture of roiled a l u m i n u m . By having different lattice rotations o f the bands, it w a s found that the Taylor version of the model predicts the experimental copper-type texture better than that obtained from the simple Taylor model. 2
4
6
3
In this work, a crystal plasticity m o d e l , considering deformation o f composite grains, is proposed to study texture development during rolling o f twinned copper single crystals. A brief description o f single crystal behavior is provided in the next section, followed by reviewing the composite grain model and its incorporation in the viscoplastic self-consistent ( V P S C ) scheme . Details of crystal plasticity a n d applications to texture studies are available in another work . 9
1 0
MODELING COMPOSITE GRAINS To c o m p u t e the plastic deformation of a crystal, o n e can relates the plastic shearing rate 1
γ" on the a' ' slip system t o the resolved shear stress r " by a p o w e r law:
sgn(r")
where γ
i(
(1)
τ" = Ρ"-a'
(2)
is the reference shear rate, τ" is the slip system hardness, η is the inverse of rate sensitivity
(taken to be 20 for rate insensitivity), σ' is the crystal deviatoric stress, and P" is the Schmid tensor. Using crystal kinematics, w e can define the crystal deformation rate D and spin M ' a s :
D = 2Zr" L P
and
i/ = l
W=
(3) a-\
A viscoplastic constitutive relation can be written as:
θ4±^Ρ"Ρ«\.σ'=Μ:σ'
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·
Materials Processing a n d Texture
(4)
M o d e l i n g Rolling T e x t u r e of T w i n n e d C u Single Crystals
where M is the 4 " order plastic compliance tensor. The lamellar structure is regarded as composite grains. T h e twin and matrix phases represent a single lamella, i.e. a composite. Each phase has uniform deformation response but not necessarily equal to each other. It is assumed that the slip system hardness is the same for all slip systems in matrix and twin. Following the schematic illustration of the microstructure in Fig. l a , w e can average the matrix (T) and twin (M) responses using their volume fractions ( / & / ' ) to find the deformation rate D and spin W for the composite grain (eg): Λ 1
1
D - I'D
J D
and
Η - / »
· ( W
(5)
1
Similarly, the deviatoric stress CJ " is written as:
(6)
ff'-'=/V+/V
H a v i n g thin flattened layers requires consideration of compatibility and equilibrium at the twin boundary separating twin and matrix layers. Compatibility requires the following deformation c o m p o n e n t s to be continuous across the interface: M
D^ = D , = Dl,
α, β
= 1,2
(7)
O n the other hand, the equilibrium requirement on the continuity of traction across the interface entails that only the following shear stress c o m p o n e n t s have to be continuous:
σ[" =
c\l = σ\1
a = 1,2
(8)
The heterogeneous interaction between individual composites (lamellae) and the medium (twinned C u single crystal) is accounted for using V P S C scheme. Following the Eshelby solution for a heterogeneous m e d i u m , each composite grain is treated as a viscoplastic ellipsoid embedded in a H o m o g e n e o u s Equivalent Medium ( H E M ) , representing the average response of all composite grains. For the case of a composite grain, the plastic compliance M ' is expressed by": s
M - = (/" M " : A + / ' M •):(/" :A + f : I Μ
(9)
where matrix A depends on Λ ί , M , and boundary conditions applied at the interface. 1
Figure 1. (a) Schematic illustration of twin-matrix lamellar structure, (b) C u single crystal showing the lamellar structure developed by twinning.
Materials Processing and Texture
·
667
M o d e l i n g Rolling T e x t u r e of T w i n n e d C u S i n g l e C r y s t a l s
RESULTS The Cu single crystals were provided from the work of N i e w c z a s and coworkers, w h o studied the effect of twinning and subsequent recrystallization on the microstructural development in Cu single crystals . Figure l b shows the topography of the twinned crystal, suggesting extensive twinning and lamellar structure. T w o types of twinned crystals are investigated for rolling; one sample w a s in as deformed state (no annealing), and the other was recrystallized at 400 Κ after it w a s deformed. T h e samples were rolled and reduced in thickness to 7 0 % , corresponding to 1.38 of strain. The initial and rolling textures of the samples were measured by Philips x-ray diffractometer, and are shown in Fig 2a-b. The initial texture of the deformed sample has a sharp twin c o m p o n e n t (T) compared to the matrix component (M), while balanced texture c o m p o n e n t s ( R l & R2) are presented for the recrystallized sample. T h e rolling texture of the deformed sample indicate that the initial texture is largely preserved, while for the recrystallized sample rolling caused noticeable loss of the initial texture (see Fig 2b). Electron Back Scattered Diffraction ( E B S D ) measurements show that most of the boundaries created by twinning in the Cu single crystals are Σ3 type, i.e. high symmetric twin boundaries. In the model, this boundary orientation between the layers was imposed into each composite grain to account for the twinning crystallography. In addition, the composite grains were assigned different volume fractions for matrix and twin phases. This m i m i c s the geometry of layers of different thickness in the microstructure. Corotation is imposed by assigning the average spin W of each composite grain to the corresponding matrix and twin. Using a total of 100 composite grains, the macroscopic deformation w a s c o m p u t e d in the V P S C code. The simulated textures, shown in Fig. 2, identify similar texture development but with higher intensities of texture components. 3
08
DISCUSSION The current work investigates modeling the deformation and texture development of a lamellar microstructure resulted from extensive twinning of the C u single crystals. The initial texture for the deformed sample indicates a strong twin texture, consistent with the large v o l u m e of twin phase in the sample ( 8 0 % ) . On the other hand, the initial texture for recrystallized sample has has more balanced c o m p o n e n t s R l and R2 (volume of R l is about 6 0 % ) . Previous work indicated that the c o m p o n e n t s R l and R2 of the recrystallization texture is obtained from the deformed texture component Τ and Μ , respectively, by 40° rotation around the normal of twinning plane, i.e. ( I l l ) plane as s h o w n in Fig 2. It w a s also suggested that the grains (layers), grown during recrystallization, were formed along the original matrix-twin interfaces. This developed a lamellar structure similar to that in the deformed sample. After recrystallization, the size of layers is at least ten times bigger than twin and matrix layers produced in the deformed state. Consequently, the deformation of coarse dislocation-free layers in the recrystallized sample is expected to be different from that in the fine lamellar structure found in the deformed state, where the latter contains many obstacles such as sessile dislocations and narrowly-spaced twinmatrix interfaces. The above findings are consistent with the texture results for rolled samples, which show a strong tendency for twin and matrix layers to codeform during rolling of the deformed sample. This justifies the large preservation of the initial M and Τ texture. In the recrystallized sample, a weaker correlation between layers is found that is explained by the spreading of the R l and R2 c o m p o n e n t s after rolling. 3
668
Materials Processing a n d Texture
M o d e l i n g Rolling T e x t u r e of T w i n n e d C u Single Crystals
Figure 2. Pole figures of initial and rolling textures of C u single crystals for (a) deformed and (b) recrystallized samples. Results are obtained using X-ray measurements and V P S C simulation. T h e simulated textures have provided good assessment of the applicability of the composite grain model to the case of rolling twinned Cu single crystals. It was shown that equilibrium is imposed by relaxing t w o shear c o m p o n e n t s across the interface between layers. This procedure is similar to that in the relaxed constraint model, commonly used when grains b e c o m e long and flat during large deformation, which requires relaxation of the shear c o m p o n e n t s of deformation to satisfy continuity of zero shear stresses across the interface . In addition, crystal corotation sufficiently signifies the codeformation between layers in the lamellar microstructure. In particular, corotation preserves the initial boundary orientation, which is possibly favorable for the case of high symmetric twin boundaries such as Σ 3 . Moreover, it should b e mentioned that the effect of length scale (spacing between layers) has not been considered, which can manifest the dislocations impediment at interfaces and give a better prediction of texture development. 12
CONCLUSIONS A composite-grain based plasticity model was utilized to study thé effects of prior deformation and recrystallization on the rolling texture of twinned C u single crystals (deformed and recrystallized samples). Results from X-ray measurements of the initial and rolling textures suggest that the microstructure of deformed Cu sample, consisting of very fine twin-matrix lamella, is responsible for the large preservation of the initial texture. This slow development in rolling texture w a s modeled by imposing the following on each twin-matrix composite grain: 1) compatibility and equilibrium at the twin-matrix interface, and 2) corotation of the twin and matrix phases. O n the other hand, the rolling texture of the recrystallized sample has been shown to slightly maintain the initial texture, indicating that the coarsening of the lamellar microstructure due to recrystallization had weakened the codeformation of the twin and matrix
Materials Processing and Texture
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M o d e l i n g Rolling T e x t u r e of T w i n n e d C u S i n g l e Crystals
layers. For the recrystallized sample, it w a s found that the rolling texture is best described by imposing compatibility and equilibrium without considering phase corotation. ACKNOWLEDGMENT The authors would like to thank J.D. Embury for his support and helpful di scussions. Xray and E B S D measurements were performed at the Center of the Microanalysis of Materials at the University of Illinois at U r b a n a - C h a m p a i g n . REFERENCES 1. T. Blewitt, R. Coltman. and J. R e d m a n , Proceedings of Defect in Crystalline Solids, Physical Society, London, 369 (1955). 2. M . N i e w c z a s . Z. Basinski, S. Basinski, and J. E m b u r y , Deformation of copper single crystals to large strains at 4.2 K, Part I: Mechanical response and electricity resistivity, Philosophical Magazine A, vol. 81, 1121-1142 (2001). 3. M. N i e w c z a s , O. Engler, and J. E m b u r y , The recrystallization of copper single crystals defomred at 4.2 K, Acta Materialia, 52, 539-552 (2004). 4. B. Lee, S. Ahzi, B. Kad, and R. Asaro, O n the deformation m e c h a n i s m s in lamellar Ti-Al alloys, Scripta Metallurgical, 29, 823-828 (1993). 5. R., Lebensohn, and G. C a n o v a , A self-consistent approach of modeling texture development of two-phase polycrystals: Application to titanium alloys, Acta Materialia, 45, 3687-3694(1997). 6. M. Grujicic. and S. Batchu, A crystal plasticity materials constitutive model for polysynthetically-twinned γ-TiAI + 02-TÎ3A1 single crystals, Journal of Materials Science. 3 6 , 2851-2863 (2001). 7. T. Leffers, A model for rolling deformation with grain subdivision. Part 1: The initial stage, Int. J. Plasticity. 17. 4 6 9 - 4 8 9 (2001). 8. T. Leffers, A model for rolling deformation with grain subdivision. Part II: T h e subsequent stage, Int. J. Plasticity, 17, 491 -511 (2001). 9. R. Lebensohn, and C. T o m é , A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys, Acta Melallurgica, 4 1 , 2611-2624 (1993). 10. U. K o c k s , C. T o m é , and H. W e n k . H , Texture and Anisotropy, Cambridge University Press, 2nd edition (2000). U . R . Lebensohn, Modeling the role of local correlations in polycrystal plasticity using viscoplastic self-consistent s c h e m e s , Modeling and Simulation in Materials Science and Engineering. 1, 739-746 (1999). 12. H. Honneff, and H. Mecking, A method for the determination of the active slip systems and orientation during single crystal deformation, Texture of Materials, editors: G. Gottstein and K. Lucke, 265-275 (1978).
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Materials Processing and Texture
MULTISCALE MODELING OF ALUMINIUM WITH STRAIN PHENOMENOLOGICAL MODELS :
5
EQUAL CHANNEL ANGULAR EXTRUDED GRADIENT CRYSTAL PLASTICITY AND
8
L. D u c h ê n e , M . G . D . G e e r s , W . A . M . B r e k e l m a n s , E. Chen*, B . Verlinden* and A.M. Habraken* * A R G E N C O Department, University of Liège, Belgium § Dept. of Mechanical Engineering, Eindhoven University of Technology, The Netherlands • Dept. of Materials Engineering, Katholieke Universiteit Leuven. Belgium ABSTRACT The Equal Channel Angular Extrusion process is used to modify the microstructure of an A A 1 0 5 0 a l u m i n u m alloy in order to produce an ultra fine grained material. Due to the severe plastic deformation undergone by the material during the E C A E process, the subsequent behavior of the material is non-conventional and difficult to model with classical constitutive laws (e.g. E C A E aluminum presents a large initial back-stress which must be adequately incorporated in the model). In this study, the evolution of the back-stress during the E C A E process is analyzed. T w o different numerical m o d e l s were investigated in this respect. The first one is a single crystal strain gradient plasticity model based on dislocation densities. The second model is the Teodosiu and H u ' s hardening model, which is a microstructuraly based phenomenological model at the macroscale. The results provided by the two models are obviously distinct. Nevertheless, some c o m m o n trends can be pointed out. among which the amplitude of the back-stress that is similar. In agreement with the cyclic deformation m o d e of the studied route C E C A E process, the evolution of the predicted back-stress is also cyclic in both models. INTRODUCTION T h e Equal Channel Angular Extrusion ( E C A E ) process is used in the present study to produce ultra fine grained a l u m i n u m . It is well-known that a decrease in the grain size of a material is accompanied by an increase of the yield strength as can be represented by the HallPetch relation [1]. Due to their particular mechanical properties, an increasing interest is currently dedicated to the study of ultra fine grained materials. However, the modeling of these materials is not straightforward. The material deformed by the E C A E process is far from its virgin state. Very large plastic strains (in the studied case, the plastic equivalent strain is 1.15 per pass) are imposed to the material. In the framework of this study, several mechanical tests were performed on the aluminum produced by E C A E in order to assess its mechanical behavior [2]. It appeared that a significant kinematic hardening was observed. Furthermore, the E C A E aluminum presented an initial backstress resulting from the deformation that occurred during the E C A E process. This contribution assesses the performance of t w o different numerical models for an accurate modeling of E C A E processed aluminum. A single crystal strain gradient plasticity model was scrutinized in this respect. This model uses as internal variables the densities of statistically stored dislocations ( S S D ) and geometrically necessary dislocations ( G N D ) . The evolution of the S S D densities is based on a balance between dislocation accumulation and annihilation rates depending on the slip rates. The G N D densities on the other hand result from the incompatibilities in the crystal lattice due to gradient of the dislocation slip. Both G N D and
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M u l t i s c a l e M o d e l i n g of Equal C h a n n e l A n g u l a r E x t r u d e d A l u m i n u m
S S D densities are taken into account for the isotropic hardening of the material. The G N D densities naturally induce a physically based kinematic hardening through their internal stresses (i.e. the back-stress, c o m p u t e d as a function of the G N D densities gradient). In addition, a macroscopic phenomenological hardening model w a s investigated. The Teodosiu and Hu's hardening model [3;4] is a physically-based microstructural model. Basically, it is able to describe both kinematic and isotropic hardening, reflecting the influence of the dislocation structures and their evolutions, at a macroscopic scale. It permits to describe complex hardening behaviors induced by strain-path changes. The goals of this research are multiple. First, different numerical models were used to predict the evolution of the back-stress during an E C A E process. This yields n e w insights and more accurate material parameters as used in phenomenological models adapted to E C A E materials (mainly a reasonable initial back-stress). The m o d e l i n g of the E C A E process also results in an improved k n o w l e d g e o f the m e c h a n i s m s involved during this forming process. Finally, this research permits the c o m p a r i s o n of two promising yet different models with very different approaches on a relevant application. THE MODELS Strain gradient crystal plasticity model The constitutive framework
o f this model departs from the classical
decomposition of the deformation gradient F into a plastic part F = F-F F
multiplicative
and an elastic part F : c
(1)
p
is assumed to be achieved only by dislocation slip.
F
e
includes small
lattice
deformation and (possibly) large rigid body rotations. T h e elastic behavior o f the material is determined by equation (2), w h e r e S is the second Piola-Kirchhoff stress tensor expressed in the stress-free intermediate configuration (the material deformed by F
4
only), C is the fourth order
anisotropic elasticity tensor (it is defined by three material parameters, which are three of its components: C\],Ci2.C44). T
obtained by E = i ( F c
E
p
is the Green-Lagrange elastic strain (work-conjugated to
S),
·F -1) . 4
S = C : E„
(2)
The plastic deformation is due to dislocation glide on slip systems. Therefore, the plastic velocity gradient tensor (related to the plastic part of the deformation gradient by F = L · F ) is expressed in the form: L
p
= f >
a
p
o
a
(3)
a=l
In equation (3), γ are defined by ( P ° )
α
is the slip rate on slip system a, the non-symmetric Schmid tensors
= ( i " ) . ( « " ) , with
and « " the slip direction and the slip plane normal
for the slip system α (expressed in the undeformed configuration). A visco-plastic slip law relates the slip rates to the effective stresses τ " , :
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(
κ, exp
s"
r kT
1 -
s
(4)
a
v.
)J
γ , m, G are material parameters; k is the B o l t z m a n n ' s constant and Τ is the absolute 0
0
temperature. In equation (4), the exponential term represents the thermally induced dislocation motion. T h e effective stresses τ °
;
are obtained from equation (5). where τ
α
are the Schmid
stresses defined as the resolved part of the second P-K stress on the slip systems τ " = S : P°.
τ " is the back-stress related to the kinematic hardening included in this model. It is the resolved part of the internal stresses induced by the edge and screw dislocation densities present on all slip systems according to equation (6). (6)
The presence of dislocations in the crystal creates a deformation of the lattice. Assuming (at this stage only) that the material is an isotropic elastic continuum medium, the internal stresses resulting from distributions of dislocation densities take the form of equation (7) for edge dislocations and equation (8) for screw dislocations. GbR,
+ 4v« p V5) s
0
8(l-v)fGbR
- Σ VOPLD
w
·
{-»O4PO
-
J
ξ
o Po
(7)
0
ξχ
•
w
i
t
h
Pi = ?ο "ο
(8)
5=» In equations (7) and (8), G is the (isotropic) shear modulus, ν is P o i s s o n ' s ratio, b is the length of the Burgers vector and R
e
and
are the radii of the fictitious spherical domain
(around the point where the internal stresses are computed) limiting the dislocations taken into consideration. It should be noted at this stage that the dislocation densities are separated in two categories: the geometrically necessary dislocations ( G N D ) and the statistically stored dislocations (SSD). The G N D s are generated during the plastic deformation by dislocation glide in order to maintain the compatibility of the crystal lattice. They have a specific orientation (depending on the slip on each slip system) and, they therefore have a net contribution to the internal stresses. On the other hand, the S S D have a random orientation such that their influence on the internal stresses vanishes. It appears in equations (7) and (8) that the gradient of the G N D is the governing quantity for the internal stresses. The present crystal plasticity densities p\ Nn
model considers face centered cubic metals having 12 slip systems and, consequently, 12 edge dislocations and 6 screw dislocations families are considered. In equation (4), the slip resistance s" is still not defined. It is related to the local slip system hardening. The slip resistance evolves as a function of G N D and S S D densities through equation (9). , ç = cGb a
Pc.vo
(9)
t-i
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M u l t i s c a l e M o d e l i n g of E q u a l C h a n n e l A n g u l a r E x t r u d e d A l u m i n u m
c is a material constant and
are interaction parameters between slip system α and
dislocation ξ. T h e evolution rule for the S S D densities results from a competition b e t w e e n an accumulation rate and an annihilation rate: P L =^ ^ - 2 V P L J | I
Ç
|
Î
with
Γ =
(10)
,
(11)
V ϊ=ι and an initial condition:
p\
SD
In equations (10) to ( 1 2 ) , y ,
5-1
(t = 0 ) = ( p | ) s o
and ( P S S D )
Κ
e
| 2
( )
0
a
r
e
0
material parameters;
Z/"
4
is an
interaction matrix (similar to Α°*· ) and t is the time. The G N D densities evolution is based on the geometrical compatibility of the crystal lattice during plastic deformation. T h e G N D are introduced to maintain the lattice continuity in the crystals in spite of the dislocation slips. T h e evolution rules are: P L , = ( P L ; )
~ V
0
Ç O
Y
- ^
(13)
α
= ( P c . v o ) + ^ ( ^ ο Υ ' • Po"' + ν „ γ
P\so
0
α ι
· PÔ' )
for e d g e G N D densities (ξ=Τ . . . 12) and screw G N D densities (ξ= 1 3 . . . 18), respectively. Further details about the strain gradient crystal plasticity model can b e found in [5-8]. Teodosiu and Flu's hardening model Teodosiu and H u ' s hardening model is described by 13 material parameters: C
p-
c
C
R
C
- sd' s ,C .f.n .n ,r,Y ,R L
x
L
p
0
S j l
,S , ,X s
l
0
(15)
and d e p e n d s on four state variables: P,S,X,R
(16)
Variable Ρ is a second-order tensor that depicts the polarity o f the persistent dislocation structures ( P D S s in [9]) and S is a fourth-order tensor that describes the directional strength of the P D S s . Scalar R represents the isotropic hardening d u e to randomly distributed dislocations and the second-order tensor X is the back-stress. These state variables evolve with respect to the plastic strain rate f
and the equivalent plastic strain rate ρ as given by Y = fv(Y^)p
(17)
A precise description of these evolution equations can b e found in [3; 10-11 ]. It should h o w e v e r be noticed that the fourth-order tensor S must be d e c o m p o s e d into So and Si. according to Equation (18). S is the strength of the dislocation structure associated with the currently D
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Aluminum
active slip systems and S| is the latent part of S related to the persistent dislocation structure associated with the latent slip systems.
S = S N^ ®N.„ + S D
where Ν
p
(18)
L
is the plastic strain rate direction. T w o distinct evolution equations (with the
form of Equation (17)) are applied to S and S . The yield condition is given by Equation (19). D
L
σ = σ = Y + R + f |S|
(19)
t l
where σ is the equivalent stress, function of dev(o) - X . σ
is the current elastic limit,
Y is the initial size of the yield locus and R + f | S j represents the evolution of the isotropic 0
hardening. The expression of σ depends on the definition of the yield locus. MODELING THE ECAE PROCESS The E C A E process is schematically represented in Figure 1. In this study, the material (one material point is illustrated by the square in Figure 1) was assumed to be submitted to simple shear in the intersection plane of the t w o channels. In this respect, the numerical simulations were defined in a reference system attached to that plane. The material was submitted to the E C A E process following route C (180° rotation of the sample between each pass) for 4 passes. This route can be modeled by simple shear (with a shear strain ranging from 0 to 2 for the studied E C A E geometry) for the first pass and reversed shear for the second pass. The initial shape of the material is recovered every 2 passes. For the strain gradient crystal plasticity law, the orientations of the individual grains are required as input data. The texture of the aluminum before the E C A E process was used to extract a set of 8 representative orientations. T h e grains with these orientations were submitted to simple shear separately (Taylor analysis). The material parameters for the strain gradient crystal plasticity m o d e l were largely extracted from [6] and [12]. Only the initial S S D densities ( ρ ^ > ) „ were adjusted according to the actual hardening behavior of the studied aluminum. A value of 400 μτη' w a s adopted. For the Teodosiu and H u s model, the material parameters for the aluminum were obtained from [13]. ?
+ RD
Figure 1. Schematic representation of the E C A E process (RD is the rolling direction of the initial material).
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M u l t i s c a l e M o d e l i n g of E q u a l C h a n n e l A n g u l a r E x t r u d e d A l u m i n u m
NUMERICAL RESULTS Figure 2 presents the evolution of the shear stress as a function of the shear strain with the strain gradient crystal plasticity model during the E C A E process following route C for 4 passes. The Bauschinger effect can be observed at the load reversal between successive passes. T h e kinematic hardening (linked to the back-stress) is at the origin of this effect in the model. Shear stress (MPa)
Shear strain Figure 2. Shear stress - shear strain during E C A E process for 4 passes (route C). The evolution of the back-stress during the 4 passes is plotted in Figure 3. The back-stress at the centre of each grain is next considered in more detail. T h e mean value over the 8 grains of the representative set is shown. The m a x i m u m values for the back-stress were obtained after the first and the third passes. L o w e r values were noticed after 2 and 4 passes. This result is confirmed by the amplitude of the Bauschinger effect between the successive passes in Figure 2. Due to the repetitive aspect of the route C, it w a s expected (and confirmed) that the material behavior during the first and the second passes w a s similar to the one during the third and fourth passes. By c o m p a r i n g Figures 2 and 3, it appears that the amplitude of the back-stress is a fraction (around a fifth) of the shear stress. Figure 4 presents the evolution of the back-stress calculated with the Teodosiu and H u ' s hardening model. The overall amplitude of the back-stress as predicted by Teodosiu and Hu"s model is similar to the amplitude with the crystal plasticity model. O n the other hand, a very' different evolution of each c o m p o n e n t w a s observed, i.e. the orientation of the back-stress is very different. Again, a repetition (more strict in this case) of the two first passes appeared during the t w o last passes. Besides, an abrupt evolution of the back-stress w a s observed during the first stage of the reversed shearing (beginning of passes 2 and 4 ) . At the end of passes 2 and 4 , only the shear c o m p o n e n t ( 12) of the back-stress remains. CONCLUSIONS D u r i n g this study, two different m o d e l s (microscopic and macroscopic) w e r e investigated for the numerical prediction of the back-stress (kinematic hardening) during the E C A E process following route C for 4 passes. The t w o m o d e l s yield different results in terms of the evolution of the back-stress while a similar amplitude was observed.
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MPa
Pass number Figure 3. Evolution of the back-stress (X) during the E C A E process for 4 passes (route C) with the strain gradient crystal plasticity model. The c o m p o n e n t s of X are expressed in the reference frame of Fig. 1. MPa
Pass number Figure 4. Evolution of the backstress (X) during the E C A E process for 4 passes (route C) with the Teodosiu and Ffu's hardening model. T h e c o m p o n e n t s of X are expressed in the reference frame of Fie. 1.
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The difference between the two models can be partly explained by the microscopic and macroscopic aspects of these models. On the one hand, for the strain gradient crystal plasticity model, the back-stress originates from a gradient of the G N D densities inside each grain. This gradient is mainly due to the presence of the grain boundaries. On the other hand, a h o m o g e n e o u s behavior is assumed with the Teodosiu and H u ' s model. More consistently, both models predicted a larger overall back-stress after passes 1 and 3 and a lower value after 2 and 4 passes. This observation should be checked experimentally. ACKNOWLEDGEMENTS The authors acknowledge the Interuniversity Attraction Poles P r o g r a m m e - Belgian State - Belgian Science Policy (Contract P6/24). A . M . H . and L.D. also acknowledge the Belgian Fund for Scientific Research F R S - F N R S for its support. REFERENCES 1. Hall, E.O.. The deformation and aging of mild steel: iii. Discussion of results. Proc. Physical Society of London. B 6 4 , 747-753 ( 1951). 2. Poortmans, S. and Verlinden, Β.. Mechanical properties of fine-grained A A 1 0 5 0 after E C A P . Materials Science Forum 5 0 3 - 5 0 4 , 847-852 (2006). 3. Teodosiu, C , Hu, Z., Evolution of the intragranular microstructure at moderate and large strains: modeling and computational significance. In Proc. NUMIFORM 95, edited bv Shen, S.F. and D a w s o n , P.R.. pp. 173-182 (1995). 4. Bouvier, S., Teodosiu, C , H a d d a d i . H. and Tabacaru, V.. Anisotropic W o r k - H a r d e n i n g Behaviour of Structural Steels and A l u m i n i u m Alloys at Large Strains. In: Proc. of the Sixth European Mechanics of Material Conference (EMMC6), S. Cescotto Eds, 3 2 9 - 3 3 6 (2002). 5. Evers. L.P., Brekelmans, W.A.M., Geers, M.G.D., Non-local crystal plasticity model with intrinsic SSD and G N D effects. J. of the Mechanics and Physics of Solids. 5 2 , 2379-2401 (2004). 6. Evers, L.P., Brekelmans, W . A . M . , Geers, M.G.D., Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of solids and structures. 4 1 , 5 2 0 9 - 5 2 3 0 (2004). 7. Bayley, C.J.. Brekelmans, W . A . M . , Geers, M.G.D., A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. International Journal of Solids and Structures. 4 3 , 7 2 6 8 - 7 2 8 6 (2006). 8. Bayley, C.J., Brekelmans, W . A . M . , Geers, M.G.D., A three-dimensional field crystal plasticity approach applied to miniaturized structures. Philosophical Mag., 8 7 (8-9), 1361-1378 (2007). 9. Li. S., Hoferlin, E „ Van Bael, A „ Van Houtte, P., Teodosiu, C . Int. J. Plast. 1 9 , 647-674 (2003). 10. Bouvier. S., Alves, J.L., Oliveira, M.C., Menezes, L.F., Comp. Mater. Sei. 3 2 . 301-315 (2005). 11. Alves. J.L., Oliveira, M.C., M e n e z e s . L.F., An advanced constitutive model in sheet metal forming simulation: the Teodosiu microstructural model and the Cazacu Barlat yield criterion. In Proc. NUMIFORM, edited by Ghosh, S. et al., AIP Conf. Proc. 712, pp. 1645-1650 (2004). 12. Fülöp. T., Brekelmans, W . A . M . . Geers, M.G.D., Size effects from grain statistics in ultrathin metal sheets. Journal of Materials Processing Technology 174, 233-238 (2006). 13. Flores, P., D u c h ê n e , L., Bouffioux, C , Lelotte, T., Henrard, C , Pernin, N . , Van Bael, Α., He, S.. Duflou, J., Habraken, A . M . , Model Identification and F E Simulations: Effect of Different Yield Loci and Hardening L a w s in Sheet Forming. Int. J. of Plasticity, 2 3 (3), 4 2 0 - 4 4 9 (2007).
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EFFECT O F D E F O R M A T I O N C O N S T R A I N T S ON T H E T E X T U R E F O R M A T I O N IN Al-5mass%Mg S O L I D S O L U T I O N A T H I G H T E M P E R A T U R E S Kazuto Okayasu, Masayuki Sakakibara* and Hiroshi Fukutomi Graduate School of Engineering. Y o k o h a m a National University 79-5, Tokiwadai. Hodogaya-ku Yokohama 2 4 0 - 8 5 0 1 , Japan ABSTRACT In order to examine the mechanism of the texture change from {011} (compression plane) to {001} in the A l - M g solid solution with an increase in strain during the uniaxial compression deformation at high temperatures, the effect of deformation m o d e on texture evolution is investigated. Plane-strain compression deformation is conducted and the characteristics of the textures are compared with those constructed by uniaxial compression. Similar to the texture change appeared during uniaxial compression, texture change with an increase in strain is found in plane-strain compression deformation. N a m e l y . {011}
1
The authors have been investigating the behavior of deformation and texture formation on Al-Mg solid solution by the high temperature uniaxial compression. It was found that the texture change occurs in the alloys with solute concentration between 3 m a s s % and 10mass%, and the sharpness of {001} texture is enhanced by an increase in solute concentration and grain size at the s a m e deformation condition. Examination on the lattice rotation of Al-3mass%Mg single crystal indicates that the orientation close to < 0 0 1 > (compression direction) is stable for compression. It w a s also suggested that preferential growth of {001} grains contributes to the development of {001} texture . It is considered that the solute atoms retard the formation of subgrains in the crystal grains, resulting in the enhancement of the orientation dependence of stored energy, which accelerates the growth of grains with specific orientations. In this study, in order to examine the mechanism of texture change in the uniaxial compression deformation, plane-strain compression deformation is conducted and the formation behavior of texture is studied. 1
EXPERIMENTAL PROCEDURE A l - 5 m a s s % M g alloy (hereafter expressed as Al-5Mg) ingots were prepared by melting
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Effect of D e f o r m a t i o n C o n s t r a i n t s o n t h e T e x t u r e F o r m a t i o n in A I - 5 m a s s % M g Solid Solution
and casting in air. A l - 3 m a s s % M g a n d A l - 3 m a s s % M g - 0 . 2 m a s s % S c were also prepared to clarify the role of grain boundary migration by comparing with the results on AI-5Mg. T h e surface layers of the ingots were removed by machining. After the h o m o g e n i z a t i o n . the ingots were cold rolled to 2 0 % of their original thickness. Cylindrical specimens with various sizes were prepared by spark erosion technique for compression tests. Aspect ratio of specimens is 1.5 for all sizes of specimens. Rectangular specimen with the dimension o f 1 5 x 1 5 x 1 0 m m is produced also by the spark erosion method for the plane-strain c o m p r e s s i o n deformation. The specimens were annealed at 753K for 2 0 min before the deformation. Uniaxial compression tests and plane-strain c o m p r e s s i o n tests were conducted under constant crosshead speed conditions at 6 7 3 K and 7 2 3 K in air u p to strains between - 0 . 6 and - 1 . 4 . Final true strain rates were in the range between l . O x l O V and 5 . 0 1 0 " V for uniaxial compression and 5.0 10" s"' for the plane-strain c o m p r e s s i o n deformation. The infrared-ray furnace was used for heating. Immediately after the deformation, the infrared-ray furnace w a s opened and the power supply w a s switched off simultaneously. Then specimens were quenched into water quickly, in order to avoid the change in microstructure after the deformation. Texture m e a s u r e m e n t s were conducted on the mid-plane sections by the Schulz reflection method using copper Κ α radiation. [ 1 " } . {001 j and { 0 1 1 } pole figures were constructed. Based on these three pole figures, orientation distribution function ( O D F ) w a s calculated by the D a h m s - B u n g e m e t h o d . Textures were examined on the basis of pole figures, inverse pole figures and ψ2=0° and 4 5 ° cross sections derived by O D F . E B S D m e a s u r e m e n t s were conducted to evaluate the spatial distribution of the texture c o m p o n e n t s . The m e a s u r e m e n t s were performed with 4 p m intervals. W h e n the m i n i m u m rotation angle between the neighboring positions w a s greater than 15°, it w a s j u d g e d that a high angle grain boundary exists. L o w angle grain boundaries were j u d g e d w h e n m i n i m u m rotation angle w a s between 5°and 15°. 1
x
X
4
4
RESULTS Stress-Strain C u r v e s Figure 1 shows the true stress-true strain curves for uniaxial compression and plane-strain compression at 673K under strain rate o f 5 . 0 x l 0 " V . (a) and (b) are results for plane-strain compression and uniaxial compression, respectively. T h e curve for uniaxial compression s h o w s the high temperature yielding while no o b v i o u s decrease in stress is seen on the stress strain curve for plane-strain compression.
Fig. 1 True stress-true strain curves examined at 673K with the strain rate of 5 . 0
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10"V
Effect of D e f o r m a t i o n C o n s t r a i n t s o n t h e T e x t u r e F o r m a t i o n in A I - 5 m a s s % M g Solid Solution
Texture Formation Figures 2(a) and (b) are the {111) pole figures for uniaxial compression and plane-strain compression examined after the deformation, respectively. T h e stress-strain curves are given in Fig. 1. In both pole figures, pole densities are projected onto the compression plane. Mean pole density is used as a unit for the contours. In the case of uniaxial compression, the pole densities are distributed in a concentric circular manner, namely a fiber texture is formed. The texture w h i c h is formed by the plane-strain compression is different from the case of uniaxial deformation. The distribution of pole density has a mirror symmetry instead of the axisymemtric distribution s h o w n by (a). In order to identify the m a i n c o m p o n e n t of the texture for uniaxial compression given in Fig. 2(a), an inverse pole figure is constructed from O D F . The result is given in Fig. 3.
(a)
(b)
Fig. 2 {111} pole figures taken after the deformation at 673K. with the strain rate of 5.0x1 O'V up to -1.0 in true strain, (a) uniaxial compression, (b) plane-strain compression .
Ill Figure 3 shows the density distribution of poles for compression plane. Mean pole density is used as a unit. The figure clearly shows that the main component of the fiber texture is (001). It is seen that the pole densities are not continuous along (001)(011) line. N a m e l y , t w o peaks exist: (011) and (001). The texture constructed by the plane-strain compression deformation is shown in Fig. 4(a) and (b) by the ip2=0°(a) and 45°(b) cross sections. Both the ψ2 cross sections s h o w that the texture mainly consists o f (110)[uvw] and the cube orientations. Accumulation of orientation density is also seen around (112)[-1-11] and (hkl)[100]. T h e density at (110)[uvw] is about 5-6. while that of (001)[0-10] is about 8. Namely, the cube orientation is the major component of this texture.
001
Oil
Fig. 3 Inverse pole figure showing the distribution of poles for compression plane measured after the uniaxial compression at 673 Κ up to -1.0 in true strain.
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(a) (b) Fig. 4 (p2=0° (a) and 45° (b) cross sections examined after the plane-strain compression at 673K. with a strain rate of 5.Οχ 10 V up t o - 1 . 0 in true strain. _
The development process of the texture w a s examined by measuring the texture in t h e early part of deformation. Figure 5 s h o w s the (02=0° cross section of the specimen after the deformation u p to a strain of - 0 . 6 with the same temperature a n d strain rate as the case o f Fig. 4 . It is seen that the cube c o m p o n e n t already develops. At t h e same time, it is seen that (hkl)[001 ] develops. Figure 6(a) and (b) s h o w the texture measured after the plane-strain compression at 723K with a strain rate o f 5.0xlO"V up to a strain of - 0 . 6 . Different from Fig. 4 ( b ) , orientation density is quite l o w at ( 1 1 0 ) [ u v w ] , while the orientation density at (001)[ 100] is a b o v e 2 2 . N a m e l y , a sharp cube texture is formed at this strain.
Fig. 5 φ2=0° cross section after the deformation at 6 7 3 K with a strain rate of 5.0x10"V 'up to - 0 . 6 in true strain.
Effect of A l i S c Precipitates on the Texture Formation Microstructure observation after deformation at 673K and 723K showed that grain boundary migration occurred during the high temperature deformation. Therefore the effects of lattice rotation due to crystal slip a n d dynamic recrystallization should be taken into account to understand the texture development. In this study, in order to reduce the effect of grain boundary migration, A l 3 m a s s % M g with AljSc precipitates w a s prepared and the formation behavior of texture is investigated. Examination on t h e relationship between strain rate and t h e steady state flow stress showed that the dominant deformation mechanism is the dislocation motion, which m o v e s with the solute atmosphere.
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(a) (b) Fig. 6 φ2=0 (a) and 4 5 ° (b) cross sections showing the texture after the plane-strain compression at 7 2 3 K with a strain rate of 5 . 0 χ 10"V up to - 0 . 6 in true strain. ο
1
Figure 7(a) and (b) s h o w the mierostructures of A l - 3 M g - 0 . 2 S c and A l - 3 M g after deformation at 723K with a strain rate S . O x l O ^ s u p to a strain of - 1 . 0 . Observation is conducted on the mid-plane sections parallel to the compression plane. The crystal grains in Fig.7(a) is extended along the elongation direction (RD) at the plane-strain compression, while grains are extended parallel to the transverse direction (TD) in the alloy without precipitates. This m e a n s that the grain boundary migration extensively occurs in the binary alloys (b) and the migration is well suppressed by the AfjSc precipitates (a). T h u s it is expected that texture formed primarily by slip deformation can be estimated by A l - 3 M g - 0 . 2 S c . The result of the texture measurement is given in Fig. 8(a) and (b). -1
(a) (b) Fig. 7 Microstructure examined by E B S D after the plane-strain compression at 723K with a strain rate o f 5.Οχ 1 0 " V ' up to - 1 . 0 . Observation was conducted on the compression plane, (a) and (b) show the results on with and without AljSc precipitates, respectively
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(a)
(b)
Fig. 8 ψ2=0° (a) and 45° (b) cross sections examined after the plane-strain compression of AI-3Mg-0.2Sc at 723K with a strain rate of 5.0><1 O ' V up to a strain o f - 1 . 0 .
DISCUSSION The deformation characteristics of A l - M g alloys at high temperature are the simultaneous occurrence of grain boundary migration, formation of n e w grains (dynamic recrystallization) and the dislocation motion which is dominated by the dragging of solute atmosphere. Comparison of Fig. 4 with Fig. 6 s h o w s that the texture change, which has been found in the uniaxial compression, also occurs in the case of plane-strain compression deformation. Since the grain boundary migration as well as the formation of new grains is driven by the stored energy, it is considered that the accumulation of dislocations should proceed prior to the grain boundary migration or the formation of n e w grains. The q>2 cross sections given in Fig. 4 for the deformation at low temperature and Fig. 8 where grain boundary migration w a s suppressed, indicate that the texture consisting of (110)[uvw] and (112)[-1-11] is formed in the early stage of deformation. A s well k n o w n , these are the orientations frequently reported on the rolling texture of aluminum alloys. However, with these usual components, {001J<100> exists as the major component. Texture measurement after the deformation up to a strain o f - 0 . 3 showed no strength at {001 ] < 1 0 0 > but at around (110)[1-12]. N a m e l y , {001 ) < 1 0 0 > component arises in the later stage of deformation. This probably means that the major mechanism for the development of {001 }<100> is not the lattice rotation caused by crystal slip deformation. As s h o w n in Fig. 7(b), grain boundary migration extensively occurs in A l - M g binary alloy during plane-strain compression deformation. This m e a n s that preferential growth of the grains with (001 j < 1 0 0 > orientation contributes to the development of {001 }<100> texture. In Fig. 6, no strong c o m p o n e n t s other than {001}<100> are seen. This indicates that the orientations observed at the early stage of deformation is not formed after the development of cube texture, although deformation always proceeds. W h e n deformation is conducted at high temperatures, dislocation microstructure develops depending o n the deformation mechanism. In the case of pure metals, subgrains are formed due to the free flight motion of dislocations. Since the strain field of dislocations relax each other in the subgrain wall, it is expected that the dependence of the stored energy o n orientation is reduced.
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Different from pure metals, development of subgrains is retarded in the high temperature deformation of A l - M g solid solution, as far as dislocations can m o v e with solute atmosphere^. In other words, the dislocations are isolated, and homogeneously distributed. In this case, the relaxation of strain fields of dislocations do not proceed extensively, hence it is expected that the stored energies of crystal grains depend on orientations; the number of slip systems and their activity vary depending on orientations. Both {001} orientation in uniaxial compression and { 0 0 1 ] < 1 0 0 > orientation in planestrain compression are the orientations with low Taylor factors. This suggests that these orientations are advantageous for the growth, which coincides with the present results. However, the development of a sharp {001 }<100> texture cannot be attributed only to the growth, because deformation proceeds simultaneously. O b v i o u s development of orientation components other than {001 }<100> is not seen in Fig. 6. This m e a n s that the orientation stability for plane-strain compression deformation also plays an important role in the formation of {001 }<100> texture. CONCLUSION In order to elucidate the m e c h a n i s m of texture change in A l - M g solid solution with increasing strain at high temperatures, plane-strain compression deformation and uniaxial deformation are conducted and the characteristics of the textures are studied. It w a s found that texture consists of several components in the early part of deformation, followed by the development of {001 }<100> in plane-strain compression. It is concluded that the development of {001 }<100> texture is attributed to the preferential growth of {001 }<100> grains as well as its stability for plane-strain compression deformation ACKNOWLEDGMENTS T h e authors greatly appreciate Miss H. Shimada for her assistance in conducting experiment. T h e Japan Light Metal Educational Foundation is also appreciated for the financial support. FOOTNOTES Graduate student. Y o k o h a m a National University. Present affiliation: IHI Corporation REFERENCES 1
S. R. Chen and U. F. K o c k s , Texture and Microstructure Development in A l - 2 % M g during High-Temperature Deformation, Hot Deformation of Aluminum Alloys. Edited by T. G. Langdon, H. D. Merchant. J. G. Morris and M. A. Zaidi. The Minerals, Metals & Materials Society, 89104(1991). ' K . O k a y a s u and H. Fukutomi, Texture Formation during High Temperature Deformation of AJ3 m a s s % M g Solid Solution, Mat. Sei. Forum. 4 9 5 - 4 9 7 , 579-84 (2005). " K. O k a y a s u . Hiroki Takekoshi and Hiroshi Fukutomi. Influence of Grain Boundary Migration on {001} Texture Formation in A l - 3 m a s s % M g Based Alloys during High Temperature Compression Deformation. Mat. Trans., 48, 2002-07 (2007). M . D a h m s , H.J. B u n g e . T h e Iterative Series-Expansion Method for Quantitative Texture Analysis. I. General Outline, J. Appl. Cryst., 22. 439-47 (1989). R . Horiuchi and M. Otsuka. Mechanism of High Temperature C r e e p of A l u m i n u m Magnesium Solid Solution Alloys. Trans. Japan Inst. Metals. 13. 284-93 (1972). !
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S P E C T R A L M E T H O D S IN T H E S T A T I S T I C A L D E S C R I P T I O N A N D D E S I G N O F MICROSTRUCTURE D Τ Fullwood, Β L A d a m s , Κ A Stevens Mechanical Engineering Department, Brigham Y o u n g University, Provo, Utah, 84602 S R Niezgoda, S R Kalidindi Department of Materials Science and Engineering, Drexel University Philadelphia, Pennsylvania, 1 ABSTRACT Various tools are useful in the design of materials from texture and related higher order structure statistics. This paper presents recent results in two key areas - homogenization methods linking the material structure to its properties, and reconstruction techniques that produce material realizations from the statistics. Homogenization techniques that are based upon material statistics captured by correlation functions have been available for several decades. With recent increases in computational power, and with the added efficiencies of spectral methods, attention has returned to these methods as viable inputs to multiscale models. One such method that involves a more complex integration scheme than the typical Kroner-type approach, is the strong contrast formulation pioneered by Brown, and more recently developed by Torquato. In this paper we utilize spectral techniques to apply the strong contrast formulation to the computationally d e m a n d i n g case of polycrystalline materials, and discuss the potential benefits to material design. Another important capability in the statistical modeling, analysis and design of materials involves the reconstruction of statistical data in order to recover detailed realizations that can be used in deterministic models. Popular reconstruction techniques such as simulated annealing take large amounts of computational p o w e r and time to create relatively small realizations. Recent work, based upon image analysis methods, has resulted in dramatic increases in speed. These advances in reconstruction capabilities are presented and some of the issues inherent in the approach are discussed as they relate to material design. HOMOGENIZATION RELATIONS Introduction Scale-bridging relationships between local properties, microstructure, and global properties of a polycrystal may be achieved through perturbation expansions. The approach pioneered by Kroner , is sometimes referred to as a 'weak-contrast' expansion. While the integral series are formally exact, their convergence depends upon the 'contrast' in the properties exhibited between the phases, and hence the polarization of the local state from the chosen reference state. Furthermore, the integral equations present in the terms of the expansion are generally conditionally convergent, and hence depend upon the shape of the sample region. Since an effective property is not dependent upon the shape of the sample, absolutely convergent integrals should be used. In 1955 Brown " suggested an expansion for effective electrical conductivity that resulted in absolutely convergent integrals. Torquato subsequently expanded upon the technique, and 3
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applied it to elasticity. The resultant e x p a n s i o n s have been termed 'strong contrast' due to their ability to achieve good c o n v e r g e n c e for materials with a high degree of contrast between the properties of their phases. In both the weak and strong contrast formulations, the numerical burden of calculating the integrals involved in the series is large. T h e s e integrals are often written in terms of statistical measures of increasing order, which m a y in turn b e calculated a s convolutions. One method of efficiently evaluating such integrals makes use of Fast Fourier Transforms (FFTs). In this paper we describe the F F T approach to both t h e weak and strong contrast solutions, and look at some of the issues involved. The W e a k and Strong Contrast Formulations The G r e e n ' s function solution often used in perturbation m e t h o d s provides a natural link between terms of the series and different orders of geometrical information. In keeping with standard notation w e will write the stress, strain and displacement within the material as σ, ε and u, respectively; the local stiffness tensor is C ( x ) , and the reference stiffness tensor for the perturbation analysis is C * . Fourth order tensors will be indicated b y bold upper case letters, and second order tensors (as well as vectors) b y bold lower case. Introduce the polarization field: p(x) = l c ( x ) - C j ( x )
(1)
o(x) = C(x)e(x) = C V x ) + P(x)
(2)
E
so that:
Then using ε
ή
= («, , · + « , • , )/ 2 the conservation principle V · σ = 0 implies:
This is solved using G r e e n ' s functions, leading to t h e usual solution: ε(χ) = ε„ - E p ( x ) + [ o ( x , x ' ) p ( i ' ) Ä
(4)
where Ε is the contribution from the singularity, and ( - Φ ) is the symmetrized double gradient of the G r e e n ' s function. Note that these terms depend upon the choice of reference tensor. ε „ is the strain field at infinity. This leads to the usual Kroner-type expansion for the effective elastic stiffness tensor C in terms of the local property, C , and often written as ' :
c* = (c) - (ère) + (crère) -...
(5) R
where C is defined b y C = C " + C for some reference tensor C , and Γ refers to the appropriate G r e e n ' s function operator (combining the terms Ε and Φ in Eq. (4)) associated with the governing differential equations and the given reference tensor. T h e angled brackets indicate
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ensemble averaging, which is taken to be equivalent to volume averaging using the ergodic assumption for h o m o g e n e o u s material. If the series is truncated after the first term, we have the usual upper bound theory in terms of the volume fraction of the local states. The subsequent terms introduce increasingly higher order geometrical statistics. For a discrete state space, with different states enumerated by A, the different statistical functions on the structure may be written Ρ * , P"' (r), etc. The first order statistics, P , evaluate the probability of finding material of state h at a random point in the sample. The second order statistics, P (r), determine the probability of finding state h at the head and state h' at the tail of a vector r thrown randomly into the sample. h
hh
U s i n g this notation, writing C ' for the stiffness tensor associated with discrete local state h and splitting the G r e e n ' s function operator into a volume integral with integrand Φ , and a principal value, E , at the origin as in Eq. (4), we m a y rewrite Eq. (5) as (applying a correction term for a non-spherical domain, as derived in ) : 4
5
,
l{p (r)-P P lc' }jf(ric")dT + ... hh
C"=P*C*-P*(c')E(c*)+
h
h
(6)
«(n).r>o
where r = (χ - x ' ) and for an isotropic reference tensor defined by parameters μ and λ, Φ and Ε 6
7
are given by in pg 5 3 3 , pg 260. W e will refer to this equation as the weak contrast formulation. W e now briefly outline the strong contrast expansion for multiple phases. While the derivation fundamentally follows the work of Torquato ' ' , we provide an outline of the approach to both introduce notation suitable for the formulation of the spectral framework. The derivation follows that for the weak contrast solution up to Eq. (4). Define individual tensors L for each phase, h: 3
6
8
[+ E C
(7)
Then the effective (global) version of this tensor may be determined: i
(vY ={xy + j o ( r ) * 12
>
il
i t v
(r ,r l 2
12
,
ρ»\Γ, )$)-'νΦ(τ )ν {ϊ.γ άτ ...
- j
1
,
2 3
Ι2
ι
ι1
(8)
,
)(L)"'L^(r,,)L''^(r )L " (Lr-... 2 1
dr dr„ +... n
ρ* )P« (Γ|2
(
and we subsequently find C
r )([)-'L"Φ(Γ )L"(l)"'VΦ(Γ )L' Ι 2
2 3
E
2 3
(Ε)
using:
C'=[(l')~'-E]~'+C
(9)
R
Equations (8) and (9) represent the heart of the strong contrast calculation. The Spectral Framework T h e integrals involved in calculating Eqs. (6) and (8) c o n s u m e a large amount of computing power. Furthermore, if the calculation is required for several material samples, the
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full integrals must be performed each time. A spectral formulation may be used to take advantage of the efficiencies of F F T s for repeated calculation of correlation functions during the analysis process, for both the weak and the strong contrast frameworks. For the sake of this exercise a single-phase polycrystalline material is assumed; the global properties will be determined by the orientation of the anisotropic crystallites throughout the sample (i.e. the local state space of interest is the orientation space; the space of all such distinct orientations is known as the fundamental zone, FZ). For the Fourier-type formulation that we adopt here, both the real material space and the orientation space must be discretized into small volumes, or tessellations, within each of which the position / orientation are approximated by single values. If the material domain is defined on a discrete rectangular lattice the correlation functions in Eq. (6) may be determined using FFTs. For example, on a 1-D space where the vectors are evenly spaced from r to Γ _ , , one may w r i t e 9
0
C
λ
=^3'((3k))*3k))
(10)
where 3 refers to the FFT of a function (over the spatial domain), the asterisk denotes complex conjugate, subscript η refers to vector τ , and we write m* for the structure function; m* takes η
the value 1 if the material is of state h in cell t of the sample, and 0 otherwise. T h e values of x, are the same as those for r„, and ; take the values from 0 to N-\. Note that this formulation implicitly assumes a periodic microstructure. Different boundary conditions may be applied by using padding in the definition of the F F T . From this calculation we also derive the 1-point 9
statistics P"=P ""
(11)
0
Similarly we may obtain higher order statistics using FFTs. Each subsequent set of functions fully contains the correlation functions of lower order. The third order statistics are given by
h
Hence the calculation burden for a given 3-point function, P^
is not much greater than
that for a 2-point function. However, the number of functions is potentially much larger, depending upon the size of the local state space. Consider the weak contrast solution given by Eq. (6). T h e correlation functions, P, are given by terms of the type given in Eq. ( 12). For the other terms, the idea is to gather function values into arrays that may take s o m e time to compute, but once available lead to rapid computation of relations, hence making optimization algorithms (with repeated functions calls) possible. This facilitates determination and search of the property closure. In this paper we use the 'primitive' Fourier-type basis to capture the functions (based upon the indicator functions). For the tensor, Φ , we derive a spectral form (normalized to remove the requirement for an extra normalization term on the integrals):
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ΦίΟ-^Φ,/,Μ, where χ {τ) η
Φ„ = [ Φ(τ)ζ„(τ)αΓ
(13)
{ίι
is the indicator function for the spatial cell around discrete point r„, taking the
value I for vectors lying in the cell, and 0 for vector lying outside of it; and S one of these uniformly sized cells. Then
c* = P"C"
h
,
i
is the volume of
h
-p"(c*^:(c")+{^"'' -p"p 'fc' ^, (c ')+...
(i4)
l
W e may collect the terms that are constant for the given material: Λ*
= (è" )E(C"
} nj"' = (è* )φ„ (è"' )
(15)
Then w e h a v e what a m o u n t s to an algebraic expression for C " with the material information effectively separated from the geometry data contained in the correlation functions, P: H
H
C" = P C
-Ρ'Ά''
+{Ρ„''* -
Ρ''Ρ'''}π;'''
+ ..
(16)
This is the spectral form of the weak contrast series. l For the strong contrast case, Eqs. (8) and (9) involve inverse tensors, and hence there wil not result a simple algebraic expression for the effective stiffness tensor. Nevertheless, the approach taken for the weak contrast case will enable Eq. (8) to be calculated in an efficient m a n n e r appropriate for searches of the material design space. Firstly w e may write: K
L=L P
H
(17)
with the usual summation convention on the h's. In a similar m a n n e r to that shown in Eq.
(15)
w e may define a 'Fourier' expansion for various of the functions involved in determining
L".
ψ*""
= Ι_,'Φ„Ι/Φ,,Ι/
(18)
Then the final form of Eq. (8) is:
1
( L - V ={LV + Υ Φ
4
4
1
-P *'(ΕΓΨ "(LV ...
- ( / > * ' * • ( Ε ) " ' Ψ^:*" ( Ε Γ - ρ?'ρ?'$ΐ'
(19)
Ί · Ι * '(LT
1
Τ " ' (L)" ) + ...
And C is once again obtained from Eq. (9). These equations form the basis for the efficient numerical calculations used in the following sections.
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Results and Discussion The F F T formulation described a b o v e has been tested on various material combinations and structures. In general the strong contrast formulation is significantly m o r e robust than the weak contrast formulation. Nevertheless, the accuracy for low and medium contrast materials is not dramatically better than the weak contrast approach. This is demonstrated in Fig. I which shows similar convergence for the weak and strong contrast formulations as contrast increases (note that the strong contrast formulation gave significantly more stable results in other examples not discussed here).
Figure 1: Comparison of weak (dashed line) and strong (solid line) formulations with varying contrast for 5 0 % mixture of two isotropic phases. The stiff phase (with L a m e constant X ) is arranged as fibers in the softer phase. The error is calculated from finite element results . 2
5
It is well known that the integration of the G r e e n ' s function must be carefully performed in the region of the singularity at the origin. Hence the effect of using the grid (FFT) scheme in this region was evaluated by c o m p a r i n g an integral performed on a grid with an accurate integral performed by a Monte Carlo method. The calculated value vs grid spacing for a high contrast between two phases is shown if Fig. 2. The integral is calculated on the grid by s u m m i n g over all points except the origin - where the singularity is (this is the technique used for the results in Fig. 2). While the values from this e x a m p l e are highly dependent upon the choice of reference tensor and contrast, they demonstrate the significant error involved for an integral which should be zero. However, if the same integral is now performed with a fixed cubic region about the origin excluded (rather than a single point), the integral correctly converges to zero as the grid spacing decreases (the dashed line in Fig.2). This example only demonstrates one part of the error that results from the integral on the grid. T h e second issue involves the fact that the principal value calculated for the G r e e n ' s function integral assumes that the volume excluded about the origin is a sphere. If a cube is excluded then a correction must be m a d e for the shape difference. For the particular example shown above the correction required was a value of 2 0 - thus introducing a further significant error into the calculation. These errors will affect both the weak and strong contrast methods as
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described above. T h e error correction methods have n o w been developed, and will be applied to the methods for presentation at the I C O T O M 2008 conference.
Figure 2: Integral value vs 1 /(grid spacing) for G r e e n ' s function integral in second term of weak contrast expansion in a cube of a single material about the origin with only the center point excluded (solid line) and with a small cube excluded about the origin (dashed line). The integral should be zero. PHASE RECOVERY RECONSTRUCTION ALGORITHM Introduction Correlation functions are a key method of representing material structure within a statistical framework. However, designers often require a method for producing statistically accurate material realizations for input into deterministic models, such as finite element tools. The action of producing such realizations from statistical data is termed reconstruction. Of particular interest in this paper are 2-point correlation functions. These provide a balance between detail and efficiency in the material description. The 2-point functions are used extensively in fields such as image analysis where the intensity is naturally related to 2-point information. In such areas as astronomy. X-ray crystallography, image processing and electron microscopy, much effort has been applied to reconstructing the original images from the 2-point data. In such cases, the intensity of incoming radiation is easily measured but it is often impractical or impossible to record the phase. Phase recovery algorithms have been successfully used for rapid reconstruction of images/signals in various applications . In this paper, we consider the application of these techniques to recovery of material realizations where only second order statistical information is available, this information being analogous to the intensity measurements available for an image. 10
T h e sample space is divided into S cells, and state space is divided into Ν cells (enumerated by η in this section of the paper). T h e numbering of the sample space starts at 0 rather than 1 for convenience in the numerical algorithms. M o v i n g from the probability view described in the first part of this paper to the distribution view, the 2-point function may be defined as:
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(20) 1=0
The enumerator 's+f indicates the cell reached from cell s by adding the central vector of cell t. It should be noted that in 1-D the number of this cell will take the actual number s+t; however, in higher dimensions the numbering scheme will not be so simple. Nevertheless, we trust that there will be no confusion introduced by using the same notation for arbitrary dimensions. For later use, we define the F F T of the microstructure function as
Ml = 3 «
!
)= X m > * "
s
=
\Ml\e'
K
(21)
The term Λί j'j will be referred to as the amplitude of the Fourier transform and θ[ as the phase. The F F T of f""
F
becomes
t " = 3
ZxisklS ST
(22) 0
- AT; e-"' '' \M
ιγ''
If we take n=n ' in Eq. (22), w e arrive at a special set of correlations termed the autocorrelations. T h e functions involving ηφ ri are generally termed cross-correlations. It is important to note that the F F T of the autocorrelation of any η is simply the square of the amplitude of the Fourier modulus of the microstructure function evaluated in cell n, and that all phase information is lost.
e:
ΡΓ =\\M:\\Ml\e' e-'
e;
=~\M^
(23)
As seen from Eq. (23), the Fourier transform of the autocorrelation is real, relating to the fact that f (h,h\r) 2
is an even function.
Phase Recovery Algorithm W e now describe the algorithm for reconstructing material realizations with these statistics. Let us first consider the case of a t w o state material. In this case, the autocorrelation function of the first state fully defines the 2-point function. It is clear from Eq. (23) that the information lost during the convolution process that defines the 2-point function is that of the phase of the Fourier transform of the microstructure function. T h e amplitude of the function is immediately recoverable. Thus the reconstruction process involves recovery of the phase information for each sub-function of the 2-point statistics. Phase recovery from intensity data is a long studied problem in electron microscopy, xray crystallography, and for deconvolving astronomical images. While there are many approaches to the problem, the algorithm presented here loosely follows the Gerchberg-Saxton algorithm " . In the reconstruction process, recovery of the original microstructure, m " , is
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equivalent to determining its Fourier transform M' '. C h o o s e material state 1 as the state that will k
be used in the reconstruction algorithm. T o reduce numerical error, this should be chosen as the state with the highest overall volume fraction (determined by the value of F" ). A trial microstructure is chosen, and the amplitude of the Fourier transform for state 1 is replaced with the known amplitude from the 2-point function from Eq. (23). (24) Thus, in effect our trial function supplies only the phase at this step. Once the phase for state 1 is determined in this manner, the phase for the other states may be determined using Eq. (22), with the amplitude for each state retrieved using Eq. (24):
e
(25)
*=r^-\
Κ IK I The phase retrieval algorithm subsequently proceeds along the lines of an input-output algorithm subject to the constraint that m" have positive values in each cell. For reconstruction of only eigen-microstructures, the constraint may be modified such that m" must take the value of either 1 or 0. This briefly describes the basic algorithm. In order to improve the convergence of the algorithm the 'step length' between iterations is adjusted using a 'hybrid' approach that keeps the trial function well inside the feasible (positive) region at each step. To summarize the method more concisely, the basic algorithm consists of four steps: 1) guess a trial microstructure, m" ; 2) Replace the modulus of the guess with the square root of the autocorrelation; 3) Inverse Fourier transform the results; 4) Apply the known constraints in real space. This then b e c o m e s the next guess to be fed back into step 1. W e wish to reconstruct the microstructure variables corresponding to state n=\. The initial guess is termed m | , and at each iteration we let γ be the set of spatial cells that take a 0
value of less than zero. At the j'
h
iteration the algorithm proceeds as follows: (26) (27)
where 0.5 < β < 1 is a constant that may be modified for optimal convergence. T h e algorithm is terminated when an error metric defined by:
(28)
reaches s o m e arbitrarily small value (typically 10
12
in the examples below).
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Once the microstructure function for the first state. n-\,
has been obtained, the rest of the
microstructure function may be resolved using Eq. (22). O n e should bear in mind that the information that we require for m" is the phase of its Fourier transform. By setting «=1 and rearranging the terms of Eq. (22) we obtain:
(29)
This may be repeated for all states «' to recover the full microstructure function. Results and Discussion In the preceding discussion of the reconstruction algorithm it is assumed that the full 2point correlation function is available for a material. In general, this will not be the case. In this paper we do not attempt to complete partial 2-point functions; we assume that a full 2-point function is available, and demonstrate use of the reconstruction algorithm in this case. W e first demonstrate the method using a 2-D microstructure. The advantage of the 2-D e x a m p l e s are that one may immediately see that the reconstructed image is simply a translation (with a possible inversion) of the original image. W e demonstrate this with a 128x 128 structure of 100 local states. The reconstruction was achieved in 1.3 seconds, and both the original and reconstructed images are shown in Fig. 3.
Figure 3 : 2-D microstructure with 100 local states (orientations) prior to reconstruction (left) and after reconstruction (right). T h e reconstructed image is a translation of the original image, with periodic boundary conditions. It should be noted at this point that when we employ F F T s to calculate the 2-point function we implicitly assume that the structure is periodic. This is discussed in m o r e detail in . A corollary of this is that two periodic microstructures that are translated from each other are indistinguishable in terms of their 2-point function. Thus it is not surprising that the reconstructed structures are translated from the original. T h e relationship between the original a = + 1; />e {l,2,...,.S} In the and reconstructed microstructures is given by: m" = m' , 9
lsth
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microstructure shown in Fig. 3, the two mierostructures are identical to a shift of 38 units in the χ direction and 53 units in the y direction. W e have similarly demonstrated these issues further using m u c h larger datasets in 3-D. It should be appreciated that such large reconstructions using simulated annealing or gradient methods are infeasible using standard computing power. However, the exciting advances with reconstruction of 3-D polycrystal material structure nevertheless have s o m e interesting limitations when applied to simple two phase structures. For example, in the case of a multiphase material, the volume fraction of any one phase is small, and this appears to significantly affect the speed and accuracy of the reconstruction when c o m p a r e d to attempts with a two phase material of approximately 5 0 % v o l u m e fraction. Simple shapes of low volume fraction reconstruct rapidly for t w o phase materials, as do highly structured shapes such as checkerboard patterns; but less structured images - such as micrographs of carbon-epoxy composites - do not converge readily. In order to study these issues, it is necessary to first be able to compare the initial and reconstructed images accurately. As noted earlier, even if the reconstruction algorithm has converged perfectly, the resultant image will be a translation, combined with a possible inversion, of the original image. W e employ an approach c o m m o n to image processing '". Suppose that the original image is defined by the matrix / , , and the reconstructed image is defined by l . Then the translation required to optimally line up the images is given by the peak of the convolution of the images: 3~' {(3(/, )) • 3(7, )} 2
T h e convolution is applied to both the reconstructed image, and the inversion of the reconstructed image, and the m a x i m u m value of the t w o defines the optimal positioning. With this capability in place, we demonstrate the issues mentioned above with a small ( 100x 100) image taken from a carbon e p o x y micrograph, after converting it to a black and white image. Figure 4a shows the original image; Fig. 4 b shows an image that has been reconstructed and optimally translated. T h e error calculated by the reconstruction algorithm reduces steadily for approximately the first 2400 iterations (as averaged over several attempts), after which it does not reduce significantly, indicating a local m i n i m u m . As observed by Feinup, this type of algorithm is based upon a steepest descent approach and is therefore prone to find local minima.
a Figure 4: a) the original image, taken from a carbon-epoxy composite micrograph, b) the partially reconstructed and optimally aligned image. O n e method of escaping from local minima is to use a genetic algorithm. The drawback is generally an inefficient method, but one that is compatible with parallel processing, thus compensating s o m e w h a t for the high n u m b e r of computations required. A genetic algorithm was implemented after each 2400 iterations using the phase retrieval method, with 20 parents per
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generation, tournament size of two, a crossover probability of 0.9, a low mutation rate, and elitism (to ensure that the best parents from each generation passed to the next generation). The result was a much m o r e stable convergence, but at the cost of significantly greater computation time. Other avenues are being explored to improve the reconstruction algorithm for composite materials of the type mentioned above, and these will be reported at the I C O T O M conference. CONCLUSIONS In this paper, both homogenization techniques and reconstruction methods have benefited from spectral approaches that are based upon FFTs. In the case of the homogenization techniques, it is hoped that composite materials of higher contrast m a y be successfully modeled (perhaps by adding higher order terms in the integral series) once the issues of integrating on a rectangular grid have been fully resolved. In the case of the reconstruction methods, the dramatic reduction in computer p o w e r required to arrive at full 3-D structure is encouraging, but more work is required to optimize the approach for general composite materials. ACKNOWLEDGEMENTS T h e authors acknowledge financial support for this work from the A r m y Research Office under David Stepp, and the Office of Naval Research, under Julie Christodoulou. S R N has been supported by the National Science Foundation Drexel/UPenn I G E R T Program in Nanoscale Science & Engineering. REFERENCES Έ . Kröner, "Elastic moduli of perfectly disordered composite materials," J. Mech. Phys. Solids, 15 3 1 9 - 3 2 9 ( 1 9 6 7 ) . "W.F. Brown, "Solid Mixture Permittivities," The Journal of Chemical Physics, 33[8] 1514-1517 (1955). S. Torquato, "Effective Electrical Conductivity of T w o - P h a s e Disordered Composite Media," Journal of Applied Physics, 58 (1985). D . T . Fullwood, B.L. A d a m s , and S.R. Kalidindi, "Generalized Pareto front methods applied to second-order material property closures," Computational Materials Science, 38[4] 7 8 8 799 (2007). S . R . Kalidindi, et al., "Elastic properties closures using second-order homogenization theories: Case studies in composites of two isotropic constituents," Acta Materialia, 54[11] 31173126 (2006). S . Torquato, "Random Heterogeneous Materials," Springer-Verlag, N e w York, 2 0 0 2 . E . Kröner, "Statistical modelling," in Modeling small deformation in polycrystals. Edited by J. Gittus and J. Zarka. Elsevier, 1986. D . C . Pham and S. Torquato, "Strong-Contrast Expansions and Approximations for the Effective Conductivity of Multiphase Composites," Journal of Applied Physics, 94 6591 -6602 (2003). 4
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D . T . Fullwood, et al., "Gradient-based Microstructure Reconstructions from Distributions Using Fast Fourier Transforms," Materials Science & Engineering A: Structural Materials: Properties, Microstructure and Processing, (In Press). J . R . Fienup, "Phase retrieval algorithms: a comparison," Applied Optics, 21[15J 2758-2769 (1982). " R . W . Gerchberg and W . O . Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik, 35 237-246 (1972). J . P . Lewis, "Fast T e m p l a t e Matching," Vision Interface, 120-123 ( 1995). 10
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A COMPARISON OF DEFORMATION TEXTURES A N D M E C H A N I C A L PROPERTIES P R E D I C T E D BY D I F F E R E N T C R Y S T A L P L A S T I C I T Y C O D E S Craig S. Hartley El Arroyo Enterprises L L C Sedona, A Z U S A Paul R. D a w s o n , Donald E. Boyce Sibley School of Mechanical and Aerospace Engineering, Cornell University Ithaca, N Y U S A Surya R. Kalidindi, M a r k o Knezevic Department of Materials Science and Engineering, Drexel University Philadelphia, P A U S A Carlos T o m é , Ricardo Lebensohn M S T - 8 , Los A l a m o s National Laboratory Los A l a m o s , N M U S A S. Lee Semiatin, Todd J. Turner Air Force Research Laboratory, Materials and Manufacturing Directorate Wright-Patterson A F B , O H U S A A y m a n A. Salem Universal Technology Corporation Dayton, O H U S A ABSTRACT Four crystal plasticity codes, the viscoplastic Material Point Simulator (MPS) developed at Cornell and the ViscoPlastic Self-Consistent code ( V P S C 7 b ) developed at L A N L , and two elastic-viscoplastic codes developed at Drexel University, were employed to calculate deformation textures and mechanical properties of model polycrystalline specimens by simulating isochoric. free upsetting. Uniaxial compression of a model sample with a starting r a n d o m texture of 5000 grains was carried out at a constant true strain rate of 0.001/s to a true strain of 1.0 with 0.02 strain increments. Material properties simulated a face-centered cubic (FCC) alloy, Type 304 Stainless Steel, and a hexagonal close-packed (HCP) material, unalloyed Ti. Both non-hardening and linear hardening conditions were investigated. Different strain-rate sensitivities simulated deformation conditions appropriate to ambient and elevated temperature conditions. All codes permitted use of the Taylor homogenization hypothesis, resulting in an upper bound for the mechanical properties. All codes produce essentially identical results for the same input material, homogenization hypothesis and deformation conditions. For comparison, one alternative homogenization hypothesis to model grain interactions was examined for each of the M P S and V P S C 7 b codes.
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INTRODUCTION Over the last two decades, significant progress in the field of crystal plasticity modeling has led to the development of many codes with proven success in simulating various aspects of the mechanical behavior of both F C C and H C P metals. This proliferation m a k e s it difficult for researchers to decide which code is best suited for a particular application. This study aims to provide an unbiased study of the capabilities and limitations of three different types of codes by examining the results obtained from each using identical input conditions. The codes employed are 1) the ViscoPlastic Self-Consistent c o d e ' ( V P S C 7 b ) developed and maintained by C. T o m é and R. Lebensohn at Los A l a m o s National Laboratory, L o s A l a m o s , N M , hereafter referred to as V; 2) the Material Point Simulator ( M P S ) c o d e developed by P. R. Dawson, D. E. Boyce and associates at Cornell University, hereafter referred to as C; and 3) t w o elastic-plastic codes, one each for face-centered cubic and hexagonal close-packed metals, developed by S. Kalidindi and associates at Drexel University ' , hereafter referred to collectively as D. 2
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All of the codes are capable of calculating effective stress-effective strain (SS) curves, the final orientation of each grain in the ensemble (TEX), from w h i c h pole figures can be calculated, and the effective Taylor factor (M) for the ensemble. All employ a form of the Voce hardening law for hardening on the active slip systems and the V and D codes permit the inclusion of latent hardening in cases where more than one type of slip system is active. Although all codes are capable of including both slip and twinning in the calculations, only slip was considered in this study to reduce the n u m b e r of permutations of material variables. A homogenization assumption specifies the relationship between the state of deformation for individual grains and the global state of deformation for the ensemble. The Taylor hypothesis, in which the strain or strain rate in each grain is the same as that for the ensemble, is c o m m o n to all three codes and gives an upper bound to the resulting SS curve. In addition the C code provides an option for the Sachs hypothesis, in which the state of stress in each grain is the same as for the ensemble, giving a lower bound to the SS curve. Although this code also permits selected weighted averages of the Taylor and Sachs hypotheses, these were not investigated in this study. The Affine linearization hypothesis of the V c o d e was employed in this work for comparison. 5
MATERIAL PARAMETERS AND COMPUTATIONAL DETAILS Material properties employed in the calculations correspond to austenitic stainless steel, a face-centered cubic (FCC) avatar, and unalloyed Ti, a hexagonal close-packed ( H C P ) avatar. Crystal deformation was confined to {111 }<110> slip for the F C C material and to the basal , first order prismatic and pyramidal
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where
is the i j
th
c o m p o n e n t of the deviatoric strain rate due to slip on the k
m j is the Schmid tensor on that system, o exponent, γ
r s
th
slip system,
is the local deviatoric stress tensor, η is the stress
is a reference strain rate, and and T„is a reference stress for slip on the k
0
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slip
system, respectively. The strain rate due to slip o n all systems is obtained by summing equation (1) over all active slip systems. While the V code e m p l o y s this form of the rate-dependent law, the C and D codes use the form in which the resolved shear stress is given in terms of the strain rate, using a strain-rate sensitivity exponent, m = 1/n. Both hardening and non-hardening conditions were investigated using an empirical Vocetype hardening law. On each active slip system these relationships have the form Β
τ(γ) = τ „ + ( τ , - τ ) ( ΐ - β - ' )
(2)
0
where τ(γ) is the resolved shear stress at a resolved shear strain of γ, τ and τ are the critical resolved shear stress ( C R S S ) and saturation stress, respectively on the slip system and α is a hardening parameter. The V code employs a version of equation (2) that uses the accumulated shear strain in a grain instead of γ for the strain and permits hardening to approach a non-zero asymptotic value, introducing a fourth parameter into the e x p r e s s i o n . The present study treats only linear hardening, simulated over the range of strain for which calculations were performed by setting t = 1000 t„ and the initial hardening rate θ = α(τ -τ ) = 0.2τ . For the H C P material the initial hardening rates on the active slip systems were selected in the same ratios as the CRSSs. The effective Taylor factor, M, for the ensemble of grains was calculated at each increment of effective strain for the V and D codes. In the former case M is given by σ έ 0
5
1
s
0
M=
,
5
0
ο
"
(3)
where summation over repeated indices is implied and the deviatoric stress and strain rate components refer to global values. The SS curves were normalized by dividing stresses by the C R S S on the reference slip system (basal slip for H C P ) . Equation (3) and the use of 1.0 as the C R S S on the reference slip system results in the value of M being numerically equal to the von Mises effective stress at e a c h increment of strain for both F C C and H C P materials. The D code uses a definition for M that replaces individual stress c o m p o n e n t s in equation (3) with the C R S S on the reference system. C o m p u t a t i o n s of the SS curves were performed at effective strain intervals of 0.02 from an initial true strain of 0.0 to a final value of 1.0 under uniaxial compression along the X3 axis at a rate of 10" s" . However, because of the elastic-viscoplastic nature of the D codes, when the instantaneous total strain was less than 0.05 it was necessary to perform calculations at a smaller increment of strain than for the rest of the curve to achieve agreement with the purely viscoplastic codes in the vicinity of initial yielding, an ensemble of 5000 randomly oriented grains was selected as the starting material. 3
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RESULTS Face-Centered Cubic The SS curves tor the F C C calculations produced by all codes are shown in Figure 1. The legend identifying data sets specifies the code (C. D , V ) , crystal structure (F or H), temperature as high (H) or ambient (R), hardening (H) or non-hardening ( N ) . and the homogenization assumption. T h e latter is represented by U for the Taylor hypothesis, L for the Sachs and A for Affine linearization. The effective stress is numerically equal to M for the V code.
Figure 1. F C C U p p e r B o u n d Effective Stress-Strain Curves from All C o d e s Pole figures from the calculations shown in Figure 1 are given in Figures 2a and 2b for hardening and non-hardening, respectively, under ambient temperature deformation conditions. Orientations of all grains in the ensemble at an effective true strain of 1.0 were e m p l o y e d to construct pole figures for the deformed materials using T S L software marketed by E D A X . All pole figures have the compression ( χ ) axis at the center of the figure and intensity contours represent multiples of the random probabilities for the indicated poles. 6
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Figure 2a. F C C Pole Figures for Ambient Temperature, Hardening Deformation Condition
Figure 2b. F C C Pole Figures for A m b i e n t Temperature, N o n - Hardening Deformation Condition Figures 3a and 3b show similar pole figures for high temperature conditions.
Figure 3a. F C C Pole Figures for High Temperature. Hardening Deformation Condition
Figure 3b. F C C Pole Figures for High Temperature. N o n - H a r d e n i n g Deformation Condition Figure 4 compares the SS curves calculated with the C code using the U and L assumptions.
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Figure 4. F C C U p p e r and L o w e r B o u n d Effective Stress-Strain Curves from C code Figure 5 s h o w s pole figures associated with the lower bound stress-strain curves in Figure 4.
Figure 5. F C C Pole Figures for Lower Bound calculations with C code Figure 6 compares stress-strain curves obtained with the V code using the Taylor (T) model with the s a m e deformation conditions calculated using the Affine (A) option of the code.
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Figure 6. F C C Effective Stress-Strain Curves from Taylor and Affine options of V code Figure 7 gives pole figures obtained using the Affine option of the V code.
Figure 7. F C C Pole Figures from Affine option calculated with V code
Hexagonal Close-Packed Figure 8 shows SS curves obtained from all codes for H C P material.
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Figure 8. H C P Upper B o u n d Effective Stress-Strain Curves from all codes Pole figures for the stress-strain curves in Figure 8 are s h o w n in Figure 9a, 9b, 10a and 10b.
Figure 9a. H C P Pole Figures for A m b i e n t Temperature. Hardening Deformation Condition
Figure 9b. H C P Pole Figures for Ambient Temperature, N o n - H a r d e n i n g Deformation Condition
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Figure 10a. H C P Pole Figures for High Temperature, Hardening Deformation Condition
Figure 1 Ob. H C P Pole Figures for High Temperature. N o n - H a r d e n i n g Deformation Condition
Figure 11. H C P Upper and Lower Bound Effective Stress-Strain Curves from C code
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Figure 11 compares lower bound and upper bound SS curves for H C P material. Pole figures from the low er bound calculations are shown in Figure 12. :
Figure 12. H C P Pole Figures for Lower B o u n d calculations with C code Figure 13 compares S S results from the Affine and Taylor options of the V code.
Figure 13. H C P Effective Stress-Strain Curves from Taylor and Affine options of V code Pole figures obtained with the Affine option are shown in Figure 14.
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Figure 14. H C P Pole Figures from Affine option calculated with V code
DISCUSSION AND CONCLUSIONS The ratio of initial flow stresses for the polycrystalline effective stress-strain curves under high and ambient temperature deformation conditions w a s nearly equal to the strain rate raised to a power [ ( l / n O - (1/nn)], where nn and ni. refer to the stress exponents for high (H) and ambient (L) temperature deformation. Thus the difference in initial flow stresses is accentuated by the use of a relatively slow strain rate in the calculations. If the strain rate employed for the calculations were 1.0 s"'. there would be virtually no difference in the initial flow stresses of the polycrystalline effective stress-strain curves obtained for the t w o deformation conditions. The differences that do exist can be attributed to a small amount of texture hardening. Hardening exhibited by the effective stress-strain curves calculated using the Taylor assumption for the polycrystalline materials is virtually linear, as is the input constitutive equation. However, the intensity of hardening is enhanced in the polycrystalline aggregate as a result of texture formation. While the input hardening rates are 0.2(CRSS) on each active slip system, the apparent linear hardening rates of the F C C polycrystal curves are 0.7 and 0.6 times the initial flow stress for the ambient and high temperature deformation conditions, respectively. The corresponding factors for the H C P polycrystal curves are 0.46 and 0.17. This enhanced hardening of the polycrystalline aggregate may also be attributed to texture hardening effects. Texture differences between upper and lower bound calculations obtained with the C code can be attributed to t w o characteristics of the lower bound assumption: 1) in the lower bound approximation fewer grains a c c o m m o d a t e m o r e deformation in contrast to the upper bound w h e r e all grains experience the same deformation and 2) since all grains experience the same stress, hardening of a grain renders it less capable deforming further forcing other grains to a c c o m m o d a t e more of the deformation. These effects cause the texture to be m o r e strongly developed in the hardening cases for both high and low temperature deformation. With no hardening, all grains remain active and reorient as deformation progresses, while hardening causes decreased slip activity in some grains, forcing the activation of other grains. All three codes produced excellent agreement a m o n g calculated effective stress-strain curves for all material and deformation conditions investigated. Pole figures produced by all codes w e r e essentially identical for the Taylor homogenization hypothesis. The principal differences in pole figures for both F C C and H C P material occurred as a result of different strain-rate sensitivities.
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ACKNOWLEDGEMENTS One of the authors ( C S H ) wishes to acknowledge the support of the Materials Processing G r o u p , Air Force Materials and Manufacturing Directorate for the conduct of the research. REFERENCES 'R. Lebensohn and C. T o m é , A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals - Application to zirconium alloys, Ada Metall. Mater.. 4 1 . 2611-2624 (1993). P . R . Dawson and E.B. Marin. Computational mechanics for metal deformation processes using polycrystal plasticity. Advances in Applied Mechanics, 3 4 . edited by E. van der Giessen and T. Y. Wu, Academic P r e s s , 7 8 - 1 6 9 ( 1998). S . R. Kalidindi, C. A. Bronkhorst and L. Anand, Crystallographic Texture Evolution in Bulk Deformation Processing of F C C Metals. J. Phys. Mech. Sol., 4 0 , 537-569 (1992). X . Wu, S. R. Kalidindi, C. N e c k e r and A. A. Salem, Prediction of crystallographic texture evolution and anisotropic stress-strain curves during large plastic strains in high purity [alpha]titanium using a Taylor-type crystal plasticity model, Acta Mater.. 5 5 , 423-432 (2007). R. A. Lebensohn, C. N. T o m é and P J . Maudlin, A selfconsistent formulation for the prediction of the anisotropic behavior of viscoplastic polycrystals with voids, J. Mech. Phvs. Solids, 5 2 , 249-278 (2004). B . L. A d a m s , S. I. Wright and K. K u n z e , Orientation imaging: The emergence of a n e w microscopy, Metall. Trans. A, 2 4 A , 819-31 (1993). 2
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S I M U L A T I O N O F T E X T U R E D E V E L O P M E N T IN P U R E A L U M I N U M D E F O R M E D BY EQUAL CHANNEL ANGULAR PRESSING 1
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Majid H o s e i n i ' , M a h m o o d M e r a t i a n , H u a l o n g L i ' and Jerzy Szpunar
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1
Department of Mining, Metals and Materials Engineering. McGill University, Montreal, Q C . Canada H 3 A 2B2 2
Department of Materials Engineering, Isfahan University of Technology, Isfahan, Iran
ABSTRACT A texture simulation procedure for multi-pass equal channel angular pressing ( E C A P ) was studied by using a computer texture simulation based on Visco Plastic Self Consistent modeling. The program inputs were the initial texture and the loading condition and the output was the texture after deformation. The simulation w a s done for each individual pass of E C A P and the results were compared with experimental texture results of commercially pure aluminum processed by E C A P . For simulation of the first pass, X-ray measured texture of the aluminum rod before E C A P w a s used as input data. Simulation of each pass after that was carried out by using input data from previous simulation and rotating this texture for different routes in ECAP. There w a s a good agreement in peak position between experimental and simulated pole figures after first and second passes. 1. I N T R O D U C T I O N Severe Plastic Deformation (SPD) techniques have attracted much attention in the last decade because of their capability to produce sub-micron and nanostructured in polycrystalline materials with superior mechanical properties [ I ] . The techniques are characterized by very high plastic strains applied on solid bulk materials meanwhile there appears no change in the overall s p e c i m e n s ' dimensions. Equal Channel Angular Pressing, ( E C A P ) is one of the S P D processing techniques which was first introduced by Segal [2] in the early 70s but extensive works have been done on this process from the 90s [3]. A bulk solid rod or bar is pressed trough a die consisting of two channels in the E C A P process. T h e channels have a constant cross section meeting each other at an abrupt bent through a certain angle Φ (usually close to 90 or 120 degrees), called the die angle which is an important parameter in the process [1-3]. Large plastic strains taking place in a narrow region at the intersection of two channels, cause microstructure and grain size to evolve during E C A P [4]. Like in other S P D techniques, since the material's cross section remains the same, E C A P process can be repeated several times. This is how large plastic strains can be applied to the material. The other important parameter in the E C A P process is the rotation of the billet (processing routes) between passes [5], In general, deformation m e c h a n i s m s in E C A P depend on material properties and processing parameters. The deformation process and the microstructure are affected by the n u m b e r of passes (amount of induced strain), the die angle (Φ), and the processing routes for a certain material, which are all controllable parameters with large effects [3-5], Large plastic strains and frequent changes in strain paths are responsible for complex changes in microstructure and crystallographic texture of material. To understand the deformation mechanism and the grain refinement during this process texture investigation is used as an essential way [1 ]. Research has shown that texture developed during E C A P is almost compatible with the texture obtained by other deformation processes like torsion, which are characterized by simple
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shear [3, 6-10]. A simple shear model for the deformation m e c h a n i s m during E C A P is proposed by Segal [4]. Many recent studies have applied computer modeling of the developed texture during E C A P . C o m p u t e r simulations based on a Taylor model [11] are used by Gholinia et al. [9] to predict the deformation texture after several passes. They considered two different die angles (90 and 120 degrees) regardless of the rotation of the billet. Li et al. [12-15] in systematic progressive investigation used texture simulation to evaluate the effect of processing parameters and also to improve deformation texture modeling in different materials. They compared the results for simple shear, fan shape [16] and finite element m o d e l s to simulate plastic deformation. They implemented Visco Plastic Self Consistent ( V P S C ) [11, 17] and Taylor models for texture simulation obtained from deformation. This work presents a general procedure for texture simulation during E C A P . The procedure is characterized by the capability to simulate texture after a multi-pass process. All process routes may be applied as well. The procedure lets experimentally measured textures to be used as input data and all the above mentioned models will be applied based upon. Texture of pure aluminum on subsequent passes was predicted by the proposed method. Simple shear deformation model and V P S C texture model was then applied. Comparisons are made between simulation results with different initial texture inputs and experimental x-ray texture measurements results. 2. C O N S E C U T I V E M O D E L I N G O F E C A P 2.1 Modeling procedure Fig. 1 illustrates the E C A P process schematically. Similarly to Segal [4], simple shear deformation at the intersection plane of the channels is considered. T w o coordination systems are then considered for the die geometry and the applied deformation. Coordination system xyz, considered as reference, corresponds to the d i e ' s exit channel. The y axis is along the s p e c i m e n ' s length and the texture is defined in the xz plane. Deformation coordination system xy'z', includes shear deformation in the xy' plane along - y ' direction. The deformation plane is perpendicular to the yz plane and the shear direction m a k e s an angle of Φ/2 with the y direction. A model with respect to the xyz system is proposed for texture simulation in E C A P process. It a s s u m e s xz plane (sample cross section) as a reference for measured and simulated textures. Material flows along y direction in this model (Fig. 2) and deformation is applied on it according to the plastic deformation model. For example in the simple shear model deformation occurs in a plane that is inclined by Φ/2 degrees with respect to the xy plane.
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Fig. I. Schematic of the E C A P process. In each pass, simulation is conducted by applying deformation condition on the input texture data and the output texture is predicted by using texture simulation models (VPSC). The simulated output texture of each pass can be put as an input texture for the next pass. The texture in multi-pass E C A P could then be predicted. Sample rotation b e t w e e n consecutive passes ( E C A P routes) could also be applied in this model by rotation of texture results around y axis before using it as an input texture for the next pass. In order to verify the m o d e l , a simple shear was used for the deformation model, while other m o d e l s like fan shape model or deformation data obtained from finite element simulation could also be applied. 2.2 Simulation with Simple Shear Deformation T h e magnitude of shear strain depends on the d i e ' s intersection angle (Φ) and the arc of outer corner (Ψ) in the intersection. This is formulated as in Eq. (1) [18]. Φ Ψ Φ Ψ y = 2cot(— + — Η Ψ ο ο β ε ο ί — + — ) ' 2 2 2 ~>
(1)
The displacement gradient tensor in the xy'z' system is shown by Eq. (2). The minus sign before γ indicates that the shear strain occurs in - y' direction.
T„v
0
0
0
0
0
- γ
0
0
0
(2)
As seen in Fig. 1, T in the xyz system can be derived by a 0= Φ/2 clockwise rotation of T „ v around χ axis. Eq. (3) s h o w s the displacement gradient tensor of E C A P process in the xyz reference system. v y 7
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0
0
0
0
ysinöcosf?
- ycos θ
0
/ s i r r t9
:
γύη
(3)
t9cosé>
Texture simulation was carried out using a computer program based on V P S C model. In this model each grain is considered as an ellipsoidal visco-plastic inclusion interacting with effective media [ 1 1 . 1 7 ] , The shape and orientation of grains are characterized by length and orientation of the ellipse main axes which can be mathematically described by the eigenvalues and eigenvectors of the displacement tensor. General fee structure with 12 slip systems and constant critical resolved shear stress ( C R S S ) were assumed in the calculations. Texture results were calculated in the Orientation Distribution Function ( O D F ) represented in xz-plane of xyz system. The simulation of the first pass w a s run with two kinds of texture as the initial texture; the O D F of 5000 grains with random texture (created by TextTools program) and the texture measured from the sample before E C A P . Texture simulation was d o n e for the first and second E C A P passes assuming Φ = 9 0 ° and Ψ = 2 0 ° to compare with experiments.
Fig. 2. Modeling procedure used for multi-pass texture simulation of E C A P process
3. E X P E R I M E N T A L The experiments were conducted using commercially pure a l u m i n u m rod with chemical composition of A l - 0 . 1 l % S i - 0 . 4 6 % F e - 0 . 1 2 % C u - 0 . 0 1 % M n - 0 . 0 2 % T i . Cylindrical samples of 12 mm in diameter and 70 mm in length w e r e cut for E C A pressing. Each sample was annealed for 2 hours at 4 0 0 °C that caused the average grain size of samples after annealing to be about 70 p m . E C A P process was carried out using a Φ = 9 0 ° die with Ψ = 2 0 ° outer arc of curvature. The samples were subjected to E C A P with carbon oil based lubricant at room temperature for one and two passes via route Be (clockwise 90 degrees rotation of sample between t w o passes). The textures of the samples were measured in a cross-section cut from the middle of the specimens using S I E M E N S D X - 5 0 0 X-ray diffractometer. O D F ' s and Pole figures were analyzed using TextTool software.
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4. R E S U L T S A N D D I S C U S S I O N S 4.1 initial texture The {111} and {200} pole figures of the initial texture measured from the cross section of specimen before E C A P are s h o w n in Fig. 3. The initial texture shows a {200} fiber texture parallel to the longitudinal axes of the specimen and a relatively weak {111} fiber with the fiber axes tilted by 50 to 60 degrees from the s p e c i m e n ' s longitudinal axis.
Fig. 3. {111} and {200} pole figures measured from a l u m i n u m specimen before E C A P . 4.2 Texture results after the first pass The pole figures of simulated and measured textures after one pass E C A P are illustrated in Fig. 4. Texture results are represented as three pole figures ( { 1 1 1 } , {200} and {220}) projected on the plane perpendicular to the longitude axes of specimen at the exit channel of the E C A P die (xz-plane). Position and strength of peaks in these three pole figures will be used to evaluate the accuracy of the simulation results. Comparing the corresponding pole figures shows that the simulations with a r a n d o m initial texture (Fig. 4.a) could not perfectly match the experiments (Fig. 4.c). The peaks in the simulated pole figures are quite lower (about half) than the experimental peaks. Although some of the peaks w e r e simulated in the same positions as the measured peaks, at least one strong peak in each pole figure was predicted in a totally different position. For e x a m p l e in the {111} and {220} pole figures, one pair of measured peaks appeared as a single but wider peak at the middle position in the simulated pole figures.
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Fig. 4. Texture results showing pole figures after first pass of E C A P (a) simulated by using random texture as an input, (b) simulated by using initial measured texture as an input, (c) x-raymeasured aluminum rod.
Fig. 4.b s h o w s the results which w e r e obtained from simulation using the actual measured texture as an input. These pole figures have a good similarity to the measured textures in Fig. 4.c, such that almost all the peaks were predicted in the right positions with similar strengths, except only in | 2 0 0 j pole figure where strength of the peak in the simulation is higher than on the measured pole figure. This is probably caused by the relatively strong {200} fiber texture existing in the initial texture. It is worth to note that the simulated results shown in Fig.4.a are in good agreement with the results found in the literature [8. 9, 13-15] for fee metals processed under similar conditions. In this simulation a polycrystalline material model with a random texture was used as an initial texture. As shown in Fig. 4. texture simulation during E C A P using a random initial texture could not match experimental results. 4.3 Texture results after the second pass Fig. 5 s h o w s the texture results after two passes of E C A P using route Be. Since in the route Be the specimen rotates 90 d e g r e e s between t w o passes, significant changes occurred in the texture during the second pass. Three sets of pole figures including measured and simulated textures are shown in this picUire. The random and the measured initial textures w e r e used as the input data for the simulation of the first pass in Fig. 5.a and Fig. 5.b respectively. For the simulation of the second pass, results of the first pass w e r e used as input. There still exist differences between the simulation results using a random initial texture and the measured ones
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(Fig. 5.c) while the differences are less severe c o m p a r i n g to the results after the first pass. The measured and simulated textures fit perfectly regarding positions and strengths of the peaks. The match between measured and simulated pole figures after the second pass shows that the simulation model h a s predicted well experimental results.
Fig. 5. Texture results showing pole figures after t w o passes of E C A P (a) simulated by using random texture as an input, (b) simulated by using initial measured texture as an input, (c) x-ray measured a l u m i n u m rod.
5. C O N C L U S I O N S A general procedure proposed for computer simulation was used to successfully predict deformation texture of aluminum rod after two passes of E C A P . The initial texture of aluminum rod before E C A P must be used in the simulation. Significant differences were observed between the experimental and simulated textures w h e n an unrealistic r a n d o m input texture was used. The effectiveness of the proposed procedure was proved by excellent agreement between texture results from experiments and simulation after one and two passes although simple shear model was used for deformation.
REFERENCES [1] R. Z. Valiev. R. K. Islamgaliev, I. V. Alexandrov, Progress in Materials Science 45 (2000) 103-189. [2] V . M . Segal, A.E. Drobyshevski. V. 1. Kopylov, Russ Metal (Eng trans) 1 (1981) 99.
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[3] R. Z. Valiev, T. G. Langdon, Progress in Materials Science 51 (2006) 8 8 1 - 9 8 1 . [4] V. M. Segal, Materials Science and Engineering A 197 (1995) 157-164. [5] K. Nakashima, Z. Horita, M . N e m o t o , T. G. Langdon, Materials Science and Engineering A 281 (2000) 82-87. [6] S. S u w a s . L. S. Toth, J.-J. Fundenberger, A. Eberhardt, W. Skrotzki, Scripta Materialia 49 (2003) 1203-1208. [7] J.-Y. Suh, J.-H. Han, K.-H. Oh, J.-C. Lee, Scripta Materialia 4 9 (2003) 185-190. [8] 1. J. Beyerlein, R. A. Lebensohn, C. N. T o m e , Materials Science and Engineering A 345 (2003) 122-138. [9] A. Gholinia, P. Bate, P. B . Prangnell, Acta Materialia 50 (2002) 2121-2136. [10] W. H. Huang, L. Chang, P. W. K a o , C. P. Chang, Materials Science and Engineering A 307 (2001) 113-118. [11] U. F. Kocks, C. N. T o m e , H. R. Wenk, Texture and Anisotropy, Cambridge University Press. 2000. [12] S. Li, A. A. Gazder, I. J. Beyerlein, Ε. V. Pereloma, C. H. J. Davies, Acta Materialia 54 (2006) 1087-1100. [13] S. Li. I. J. Beyerlein. M. A. M. Bourke, Materials Science and Engineering A 394 (2005) 66-77. [14] S. Li, 1. J. Beyerlein, D. J. Alexander, S. C. Vogel, Acta Materialia 53 (2005) 2 1 1 1 - 2 1 2 5 . [15] S. Li, I. J. Beyerlein, D. J. Alexander, S. C. Vogel, Scripta Materialia 52 (2005) 1099-1104. [16] V. M . Segal. Materials Science and Engineering A 345 (2003) 36-46. [17] R. A. Lebensohn, C. N. T o m e , Acta Metall. Mater 41 ( 1 9 9 3 ) 2 6 1 1 . [18] Y. Iwahashi. J. W a n g . Z. Horita, M . N e m o t o , T. G. Langdon, Scripta Materialia 35 (1996) 143-146.
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P L A S T I C H E T E R O G E N E I T Y D U E T O G R A I N B O U N D A R I E S A N D ITS I N F L U E N C E ON GLOBAL DEFORMATION TEXTURES Anand Κ Kanjarla ', Paul Van Houtte ', Department of Metallurgy and Materials Engineering Katholieke University- Leuven (K.UL) Leuven, Belgium 2
Laurent Delannay Department of Mechanical Engineering Université catholique de Louvain ( U C L ) Louvain-la-Neuve, Belgium ABSTRACT Plastic deformation in metals is heterogeneous in nature. This heterogeneity is induced by variety of factors at different length scales. The present study involves understanding the contribution of grain boundaries towards such a heterogeneous deformation field at mesoscopic scale. Finite element method coupled with crystal plasticity theory is used to deform a simple set-up of four grains. Each grain is discretized into large n u m b e r of elements to effectively capture the complex nature of the plastic fields in the vicinity of the grain boundaries. Distribution of strain rates, both along and perpendicular to the grain boundaries is analyzed. The observations from above simulations are then used to enrich the microstructural description and grain interaction effects in a Taylor-type micromechanical model. A new model, aimed at predicting deformation textures in face centered cubic materials, is presented. It takes into account the near neighbor interaction as well as the influence of global texture on the deformation behavior of an individual grain. This is achieved by an iterative scheme for solving the field equations for a system consisting of a grain boundary segment (and the associated grain boundary zones) e m b e d d e d in a h o m o g e n o u s media. Deformation textures predicted by the new model are presented along with a quantitative comparison with both experimental and simulated textures from other micromechanical models. INTRODUCTION Micromechanical models capable of predicting deformation textures are imperative for predicting global anisotropy induced in a material during processing. But very few models succeed in capturing both quantitative and qualitative aspects of experimentally measured textures'. Traditionally texture m o d e l s were based on the original "Isostrain" idea of Taylor. The microstructure is assumed to be a discrete set of grains, with different crystallographic orientations, and each individual grain takes the same microscopically applied deformation. Taylor m o d e l , in spite of capturing qualitatively some of the features of experimental textures, is far from reality w h e n quantitative comparison is made. Initial developments to Taylor model consisted in relaxing the strict Isostrain condition, but no improvement in the definition of microstructure was included. O f late, models are incorporating a better representation of microstructures. As a consequence of which the texture predictions have improved. Most of these m o d e l s introduce a n e w length scale into Taylor analyses, a mesoscale, where in a part of a microstructure, or as was mentioned in ', a cluster is defined. Such cluster contains a group of grains with laws defining their interactions, or as in A L A M E L model, it consists of a Grain
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boundary and a related regions. N o w , the Isostrain hypothesis links the Macro-Mesoscale instead of the Macro-Microscale. The final strain applied to individual crystal depends on the interaction laws at the mesoscale. Experimentally it is difficult to observe directly the strain patterns in a microstructure, what one mostly sees is the footprint of the already applied deformation . Although owing to recent techniques, one is capable of studying deformation, in-situ, both at the surface and in the bulk , but these are relatively n e w techniques at earlier stages. 7
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On the other hand. Crystal Plasticity Finite element method ( C P F E M ) proved to be a valuable tool to study deformation of a material both at the m a c r o and microstructural s c a l e s . The current work explores this technique to study the deformation in a model microstructure somewhat resembling the meso-scale, in the above mentioned scheme of things. In the current article, emphasis is given only on the first part of the work described in the abstract; the remainder is work in progress. 4 , 1
The scheme of the paper is as follows: (i) The A L A M E L model is briefed with emphasis on the strain partitioning assumptions involved. (ii) A model microstructure is set-up. (iii) C o m p a r i s o n is made between the strain distribution patterns obtained from C P F E M and the ones assumed in the A L A M E L model. ALAMEL MODEL The advanced L A M E L model is a Taylor-based micromechanical model taking into account near neighbor grain interactions. The main principle of the model consists in (i) Defining a infinitesimally thin cluster consisting of a grain boundary segment ( G B S ) and the corresponding grain boundary zones (ii) Ensuring that the cluster achieves the macroscopic applied strain while the constituting grains undergo heterogeneous strain which satisfies relaxed kinematical constraints and minimizes the deformation energy of the cluster. (iii) After each strain increment, the n e w deformation texture is assumed to be a weighted average of the textures in the thin zones at both sides of the G B S . Each of the zones are assigned a weighting factor estimated to be a segment of grain (extending from G B S halfway into the grain) An assumption regarding the stress and strain fields is m a d e in the interaction equations between the two zones. It was shown mathematically that in a reference f r a m e ' attached to the grain boundary, if both regions are relaxed in an equal and opposite way, then the corresponding c o m p o n e n t s of the stress tensor are equilibrated. In other words, if d ] 2 and d 3 are relaxed in a complementary way (equal in magnitude but opposite in direction), then the shear stresses are same in both the regions ( 1 - R D , 2 - N D , 3-TD). In s u m m a r y , A L A M E L model assumes (i) that the average strain rate of the t w o thin regions on either side of the G B S equals the macroscopic strain rate (ii) there is a strain rate j u m p across the G B D corresponding to opposite shear strain relaxations and that the shear stress c o m p o n e n t s are in equilibrium at the G B S . In the following section C P F E M is used to study the above assumptions made regarding the plastic heterogeneity in the A L A M E L model. 1
1
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2
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CPFEM STUDY To study the effect of grain boundaries in partitioning the macroscopically applied strain at the microstructural level, a model microstructure consisting of 4 hexagonal shaped grains is set up. The material behavior is mimicked by implementing a single crystal plasticity routine in to A B A Q U S through U M A T , the details of which can be found in . It suffices here to say, that a rate dependent formulation having the following expression is used for slip system activity. 3
X, γ=Υο
In the above equation, the reference slip rate γ resolved shear stress r
0
and the exponent m are constant. The critical
is assumed to be same on all slip systems and it's evolution with
c
deformation is formulated by a work hardening law, w h o s e expression is as follows:
In the above equation, r o, η, Γ are material parameters. The values for the material parameters c
0
used for the current study are shown in table I. Table I. Material properties used in the current study. M
Yo 0.001 sec"
1
0.012
η 16.0 M P a
0.5
r„ 0.0275
C,,
C,2
C44
189 GPa
99 GPa
45GPa
Since very few grains are deformed here, it is necessary to avoid the surface effects owing to the applied boundary conditions. Hence, Periodic Boundary Conditions (PBC) are implemented in all the 3 directions, thus the microstructure behaves as if it was in the bulk of the material and the heterogeneity in the deformation is purely because of the crystallographic orientation effects instead of surface effects. Each grain is constituted of 648 second order elements, each element with 27 integration points, totaling up to 17496 integration points per grain. Since P B C are implemented and only one row of elements is used, effectively only 5832 calculation points are considered for current study. The set up is deformed up to 6 0 % rolling reduction, in three steps in plane strain compression (PSC). It is also deformed to 6 0 % shear in three steps in simple shear (SSh).
In the current study the focus in on the grain boundaries and the plastic heterogeneity in their vicinity. This is carried out in the following manner. A grain boundary is chosen (in the current study the boundary between grains 1, 2) and two trajectories are defined in the initial undeformed state. The strain rates and stresses along these trajectories, in the final deformed state are then studied. Figure 2 shows the schematic of the trajectories mentioned.
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(i)
(ii)
Trajectory A B starts from the centre of grain! till the centre of grain2, perpendicular to the grain boundary between the t w o grains. Trajectory is made up of discrete integration points where the material constitutive law is implemented. Trajectory C D for each grain starts from one end of the grain boundary' to other end. In this case, trajectory consists of the integration points closest to the grain boundaries.
Figure 1: (a) Initial undeformed mesh, s h o w i n g four hexagonal shaped grains, (b) Deformed mesh after simple shear ( S S h ) of 4 0 % along direction 1. (c) Deformed mesh after 4 0 % rolling reduction in plane strain compression ( P S C ) . Thick black lines indicate the continuum grain boundaries between the grains. ( 1 - R D , 2 - N D , 3-TD)
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Figure 2: Schematic of the Trajectories used for studying the strain rate distribution in the current study. RESULTS Analysis of heterogeneity perpendicular to the Grain boundary':
Figure. 3(a)
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Figure 3 (a) & (b) distribution of dn and d component along the trajectory A B perpendicular to the grain boundary-' after Plane Strain Compression. Legends indicate the thickness reduction in direction 2. ( 1 - R D , 2-ND.3-TD) 2 3
Figure 3a s h o w s the distribution of the ( R D - N D ) strain rate c o m p o n e n t d | along the trajectory A B . for three different height reductions in plane strain compression ( P S C ) . Since the grain boundary plane is not distorted m u c h w.r.t global reference frame, it is assumed for the current study that the global and the grain boundary reference frame coincide. It is clear from the graph that, at the grain boundary, there is a discontinuity in the d12 component. It is the nature of the discontinuity which is more interesting. At the grain boundary, if one of the grains exhibits a positive shear, then the other exhibits a negative shear, effectively compensating it and resulting in the net near zero shears at the grain boundaries. For the shears to be completely zero, the magnitude of the t w o shears on either side must be exactly the same, as is the case in the A L A M E L model. However, what one sees here is that though strict nature of the conditions applied on relaxations are not satisfied, A L A M E L model d o e s seem to imbibe the characteristics of overall deformation behavior of t w o regions in the vicinity of a grain boundary. It is also interesting to see that there is no steep gradient observed in the immediate vicinity of the grain boundary. This was also recently confirmed experimentally by detailed Transmission electron microscopy work by H u a n g and W i n t h e r , who clearly demonstrate that when global structural evolution within individual grains, is considered, the effect of grain boundaries tend to extend well into the interior of the grain, irrespective of the grain orientation. 2
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Figure 4 (a) & (b) distribution of d,2 and da component along the trajectory A B perpendicular to the grain boundary after simple shear. Legends indicate the percentage of shear along direction 1. (1 - R D . 2 - N D . 3 - T D )
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Figure 4 shows the distribution of shear strain rate c o m p o n e n t s along the same trajectory A B . as in figure 3 . but in case of the simple shear. Both the c o m p o n e n t s exhibit a discontinuity at the grain boundary. H o w e v e r the nature of discontinuity is rather different. In case of d j component, it resembles the earlier case of P S C , where in the shears are of opposite nature, and the net resulting shears are close to zero, di2 component, on the other hand, d o e s n ' t exhibit opposing shears (i.e positive, negative shear). H o w e v e r w h a t ' s interesting is that at the grain boundary itself, the shears on either side tend to relax in opposite direction compared to the average applied shear. This is consistent with the applied deformation m o d e (simple shear along direction 1, with 1-RD, 2 - N D . 3-TD). 2
Analysis of heterogeneity parallel to the Grain boundary: The earlier analysis was carried out on a trajectory which meets the grain boundary only at one point in its centre, away from the influence of the other grains. Here the analysis will be performed on a trajectory parallel to the G B plane to see if that trends observed earlier case are valid along the length of the grain boundary plane. Figure 5 s h o w s the distribution of the d and di? c o m p o n e n t s on either sides of the G B . all through the length of the G B plane, in case of P S C . The general trend seems to be consistent with the earlier analysis, in that the shears exhibits opposite nature all along the grain boundary plane. It is also observed that in general, behavior is m u c h more c o m p l e x near the end of the grain boundaries, c o m p a r e d to the centre. This is probably an effect of the third interacting grain at the triple junction. ) 2
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Figure 5 (a) & (b) distribution of d | 2 and d 2 j component along the trajectory C D parallel to the grain boundary after plane strain compression. L e g e n d s indicate the thickness reduction in direction 2. (1 - R D . 2 - N D . 3-TD)
It should also be mentioned that, in actual materials, w h e n dislocations approach a grain boundary, they can traverse across the boundary, form a pile up or get absorbed into the boundary depending on the type of grain boundary (e.g. the misorientation) and the relative orientation of the slip s y s t e m ' ' " . The present study overlooks this fact considering that all dislocations are absorbed into the boundary. The authors believe that one can still improve such m o d e l s aimed at predicting the kinetics of grain interaction by relying on continuum mechanics and on a representation of dislocation behaviour by means of a simple Schmid law. However, comparison between plastic strain fields obtained in the current study and those obtained from the m o d e l s incorporating the dislocation-grain boundaries will be done in the future, to check the importance of such interactions on local heterogeneity. It is expected that incorporating the dislocations-grain boundary interaction is very important in simulations of the work hardening behaviour. A c c o u n t i n g for the plastic strain j u m p s and their effects at the boundaries by means of excess or geometrically necessary dislocations'" did not result in significantly improved predictions of macroscopic deformation textures . Although the results presented here are for a particular set of randomly chosen grain orientations, a statistical study reveals that the general trends presented here are also observed in a large number of different grain orientations as w e l l . " 9
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CONLCUSIONS A model microstructure consisting of j u s t four grains w a s set up using micro-mechanical Crystal Plasticity Finite Element Method. It w a s subjected to two deformation m o d e s . Particular emphasis was given to the distribution of applied strain rate in the vicinity of grain boundaries. Assumptions pertaining to such distributions in the A L A M E L have been validated. ACKNOWLEDGEMENTS LD is mandated by the National Fund for Scientific Research ( F N R S , Belgium). Authors would like to a c k n o w l e d g e the Funding from Belgian Science Policy IAP/06-24. REFERENCES 'P.Van Houtte, S.Li, M.Seefeldt and L.Delannay, Deformation texture prediction: from the Taylor model to the advanced L a m e l model, International Journal of Plasticity, 21, 589-624 (2005). 2
P . V a n Houtte, L.Delannay and I.Samajdar, Quantitative prediction of cold rolling textures in low-carbon steel by means of the L A M E L model, Textures and Microstructures, 31, 109149(1999). L . D e l a n n a y , P.Jacques and S.R.Kalidindi, Finite element modeling of crystal plasticity with grains shaped as truncated octahedrons, International Journal of Plasticity, 22, 18791898(2006). A . Musienko, A. Tatsehl, K. S c h m i d e g g , O. Kolednik, R. Pippan and G. Cailletaud, Threedimensional finite element simulation of a polycrystalline copper specimen, Acta Materialia, 55, 4121-4136(2007). "X. H u a n g . G. Winther, Dislocation structures. Part I.Grain orientation d e p e n d e n c e " , Philosophical Magazine, 8 7 , 5 1 8 9 - 5 2 1 4 ( 2 0 0 7 ) . H . F. Poulsen, L. Margulies, S. Schmidt, and G. Winther, Lattice rotations of individual bulk grains. Part 1: 3D X-ray characterization. Acta Materialia, 51, 3821-3830(2003). S.R.Kalidindi. A. Bhattacharya and R . D . Doherty, Detailed Analyses of Grain-Scale Plastic Deformation in C o l u m n a r Polycrystalline A l u m i n u m Using Orientation Image M a p p i n g and Crystal Plasticity M o d e l s , Proceedings of Royal Society of London A, 460, 1935-1956(2004). P . V a n Houtte, A. K. Kanjarla, A . V a n Bael, M.Seefeldt and L.Delannay, Multiscale modelling of the plastic anisotropy and deformation texture of polycrystalline materials, European Journal of Mechanics A, 25. 634-648(2006).
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^W. M . A s h m a w i and M. A. Zikry, Grain boundary effects and void porosity evolution, Mechanics of Materials, 35, Pages 537-552(2003). A . M a . F. Roters and D. Raabe, On the consideration of interactions between dislocations and grain boundaries in crystal plasticity finite element modelling - T h e o r y , experiments, and simulations, Acta Materialia 54, 2181 - 2 1 9 4 ( 2 0 0 6 ) . " K . S . C h e o n g and E.P. Busso, Effects of lattice misorientations on strain heterogeneities in F C C polycrystals. Journal of the Mechanics and Physics of Solids, 54, 671-689 (2006). 0 . Engler, M . C r u m b a c h and S. Li, Alloy-dependent rolling texture simulation of aluminium alloys with a grain-interaction model, Acta Materialia, 53, 2241-2257(2005). A.K.Kanjarla, L.Delannay and P.Van Houtte, Mesoscale plastic heterogeneity in polycrystalsinfluence of grain boundaries, to be submitted to International Journal of Plasticity (2008). I 0
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ON T H E C O R R E L A T I O N O F S U R F A C E T E X T U R E A N D S T R A I N I N D U C E D S U R F A C E R O U G H N E S S IN A A 6 X X X A L U M I N U M S H E E T u
S. Küsters \ M. Seefeldt ', P. Van Houtte ' ' K.U. Leuven, Department of Metallurgy and Materials Engineering Kasteelpark Arenberg 44 - box 2 4 5 0 , B-3001 Heverlee (Leuven), Belgium N I M R , Netherlands Institute for Metals Research M e k e l w e g 2 - P.O. Box 5008, 2600 G A Delft, The Netherlands
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ABSTRACT T h e dependence of the strain induced surface roughness on the surface texture was investigated for finally annealed and subsequently stretched A A 6 x x x aluminum sheet. To characterize the surface texture, E B S D scans (performed at TUDelft by T. Bennett and L. Kestens) were analyzed. A statistical approach, based on two-dimensional correlation functions, w a s introduced to obtain quantitative information on the spatial arrangement of orientations and the possible clustering of similarly oriented grains. Clusters of grains exhibiting the Cube, the RD-rotated Cube or the Goss orientation are observed to align in the R D . Various approximating micromechanical m o d e l s were implemented and validated with experimental data to acquire k n o w l e d g e on which texture c o m p o n e n t s can cause surface roughness under which loading conditions. T h e developed micromechanical m o d e l s differ in the way the boundary conditions are implemented, to be able to investigate the influence of the mechanical constraints on the predicted surface roughness. Fourier analysis was applied to discriminate between low and high frequency surface roughness c o m p o n e n t s and statistical m e t h o d s were developed to correlate texture and predicted surface roughness profiles. It is shown that the a-SRM model yields the best correlation between the surface texture and simulated surface roughness. INTRODUCTION The recent expansion in the application of Al alloys in the automotive industry requires a combination of high strength together with an improved formability and excellent surface properties from the material. The development of rough bands on the surface of high strength A l M g S i sheets, A A 6 x x x , after stretching, also k n o w n as ridging or roping, imposes a serious restriction on the application of these alloys for pressing outer panel parts in the automotive industry. Serious efforts have already been made both to understand and to characterize the cause of this surface roughness, with variable success. It is generally accepted that the cause of surface roughness is a complex interaction of material properties, microstructural properties and forming parameters during deformation. It is possible to characterize and to quantify the material and microstructural properties by performing standard experimental procedures like tensile tests, E B S D scans and X R D measurements. However, the forming parameters, and especially the parameters involving the interaction between forming tools and the formed part, can only be estimated due to their complex nature. Surface roughness in general can be subdivided into three groups, depending on the length scale of the o b s e r v a t i o n ' . The smallest surface roughness, the one with the shortest wavelength, can be observed at the microscale, the so-called in-grain dislocation patterns and slip lines. At the mesoscale or grain scale, shear bands and heterogeneous grain rotations can occur, leading to orange peel, which is typically observed in F C C materials. Finally, the surface roughness of the longest wavelength is called roping and can extend over several grain 2
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boundaries. In this research, the dislocation patterns and slip lines inside the grains are ignored because they have a negligible contribution to the total surface roughness w h e n superimposed on orange peel and roping. The subdivision of surface roughness into three groups is consequently not exclusive, but j u s t a method to distinguish between the different features involved. By using a low-pass filter, the short-wave contributions can be removed, resulting in a surface profile which is exclusively related to r o p i n g . 3
N u m e r o u s investigators have already studied the existence of roping in A A 6 x x x alloys. Roping is described as a set of parallel surface roughness lines which develop in the rolling direction if an as-received sheet is stretched in the transverse direction, figure la. The depth of the surface profile is in the range 10-30 p m , and the shape is one out of three possibilities, related to the correspondence between the profiles of the two sides of the sheet . The first roughness profile type exhibits the most symmetry and is called the ribbed profile. Thinning at one side of the sheet equals thinning at the opposite site and the midplane of the plate is the symmetry plane of the profile. A corrugated profile h o w e v e r is characterized by antisymmetry around the midplane of the plate. The third profile and the most c o m m o n , is the irregular one, in which there is no correlation between the surface roughness at the t w o sides of the plate. The symmetrical shapes identified here as a ribbed profile are very rare in A A 6 x x x , even after significant experimental effort. Also the corrugated profile is not reported in literature to date. 4
1
Figure 1. a. Roping lines at the surface of the sheet', b. T h e difference in thinning when bands of Cube, Goss and X texture c o m p o n e n t s , lying in the rolling direction, are stretched in die transverse direction.
The influence of the initial texture on the roping behavior is of crucial importance. In literature, it is e v e n claimed that the relative contributions of the texture evolution, the strain rate sensitivity, crystal elasticity and i n h o m o g e n e o u s deformation inside the grains are negligible . In order to understand roping, it is important to realize that the spatial distribution of grains with a specific orientation is not h o m o g e n e o u s . N u m e r o u s authors see roping as a consequence of the clustering of grain orientations into zones which have a different plastic response than the surrounding m a t r i x ' ' 1 Especially the existence of b a n d s of grains having the { 0 0 1 ) < 1 0 0 > (Cube), { O i l } < 100> (Goss) and {112}<110> (X) orientation tend to promote the roping features of the deformed sheet. T h e C u b e texture component is highly symmetric, has a low Taylor factor and consequently, a low resistance to thinning when strained in the transverse direction. Both the Goss and X texture c o m p o n e n t s are m u c h harder, their Taylor factor is high and their resistance to thinning is much higher in comparison with the Cube orientation, figure l b . 4
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TEXTURE CHARACTERISTICS T h e first step in analyzing the texture properties of a sheet of material, is applying an X R D or E B S D measurement and calculating the relative amount of the m a i n texture components which are present in the sheet. T h e main texture c o m p o n e n t s in cold-rolled aluminum are the C u b e and Goss crystallographic orientations, accompanied by the texture components belonging to the so-called ß-fiber. This ß-fiber is a result of the cold rolling process and runs through Euler space from the S-orientation, through the C-orientation towards the B-orientation. Additionally, also the presence of the X orientation is studied in s o m e m o r e detail. Table 1. T h e m a i n texture c o m p o n e n t s of a sheet of aluminum, exhibiting strong roping behavior.
Texture component
Miller indices
Relative amount (%)
Cube Goss X C
{001}<100> {011}<100> {112}<110> (112}<111> |123}<634> {011}<211>
8,03 4.52 1,54 1.93 1,39 1.64
S
Β
T h e main texture components of a sheet of aluminum, exhibiting strong roping behavior, are presented in table I. The material s h o w s a low fraction of the X texture component and the c o m p o n e n t s belonging to the ß-fiber, in contrast to the relative big amount of Cube and Goss oriented grains. Since it is claimed in literature that bands of these components are the main cause of roping, their spatial distribution on the surface of the sheet is studied in more detail.
Figure 2. Spatial distribution of the datapoints exhibiting the C u b e (red) and Goss (blue) orientation, within a 15° misorientation spread. Starting from an E B S D measurement, obtained by TUDelft, and in order to be able to study the possible banding of C u b e and Goss oriented grains, it is advisable to select all the datapoints with an orientation close to these reference orientations. Figure 2 shows the spatial distribution of all the datapoints which have an orientation close to the C u b e and Goss orientation. It is clear that some clustering of C u b e oriented grains exists at the surface of the material. However, the existence of bands which extend in the rolling direction, as claimed in literature, is not proven at this point. The same results can be drawTi for the Goss oriented grains.
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IMPLEMENTATION OF CRYSTAL PLASTICITY MODELS O v e r v i e w of the developed m o d e l s The aim of the modeling part o f this research is to be able to predict the surface roughness of a stretched sheet. At this point, the first results of four crystal plasticity m o d e l s will be presented and their output will be interpreted as being only a qualitative prediction o f the roping p h e n o m e n a . The aim of this first modeling step is to investigate the influence o f the mechanical constraints on the predicted surface roughness in order to trace back the physical origin of the roping behavior. In a later stage, other more precise m o d e l s will be developed. Figure 3 gives an schematic o v e r v i e w of the four developed m o d e l s .
Figure 3. Schematic o v e r v i e w of the four developed m o d e l s .
Difference b e t w e e n the a-xxx and b-xxx models Figure 4 is a schematic representation of the difference b e t w e e n the a-xxx and b-xxx models. Both models use an E B S D m e a s u r e m e n t as main input, in the figure indicated by the colourful figure at the left hand side. Each grid point has a specific orientation and colour in the sample reference frame. In order to model the surface roughness, an external stress tensor, o , is applied in the transverse direction of the sheet. Each crystallographic orientation is assumed to behave as a free standing single crystal during plastic deformation, in contrast to most C P F E M models. Let 0" en e be the stress and strain tensor of a specific grid point respectively. e x l
gp
g p
In the a-xxx m o d e l s , it is a s s u m e d that each grid point feels the same external applied stress tensor, o = σ ' . In this assumption, major strain incompatibilities m a y arise b e t w e e n the different orientations, as can be seen on the center figure o f figure 4. Although this is far from the real behavior of a polycrystalline material, it is claimed in literature that even in this way, it is possible to m o d e l the surface r o u g h n e s s at least in a qualitative w a y ' \ T h e external stress tensor is chosen in such a w a y that the total strain in the transverse direction is about 1 5 % . g p
οχ
In the b-xxx m o d e l s , the plastic deformation of the polycrystalline material is simulated using a different point of view. In order to minimize the strain incompatibilities b e t w e e n neighboring orientations, an attempt has been m a d e to reduce the total degrees of freedom during plastic deformation by setting the strain in the loading direction o f each grid point equal to a user-defined external applied strain ε 22 = ε 2 . The total strain tensor of each single crystal is calculated as in the a-xxx m o d e l s , but using a different stress tensor for each grid point. This individual stress tensor is related to the external applied stress by a parameter called the stress = . σ " . T h e stress factor o f a specific factor, which is unique for each gird point: o orientation is related to the resistance of this orientation to deformation in the loading direction. ?Ρ
ί Μ
2
s p
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Figure 4. Schematic output of both the a-xxx and b-xxx m o d e l s Difference between the x - S R M and x - V P M m o d e l s Both x - S R M , Simplified Roping Model, and x - V P M , Visco-Plastic M o d e l , models are based on micromechanical crystal plasticity. Before going m o r e into detail about the differences and their consequences, a few c o m m o n properties can be mentioned. Plastic deformation is assumed to occur by crystallographic slip of dislocations on each of the twelve slip systems. T h e a m o u n t of slip is proportional to the value of the resolved shear stress on that slip system. T h e resolved shear stress is related to the external applied stress and the Schmid tensor w h i c h contains all the necessary geometrical information of a specific slip system. T h e shear strain on each slip system is some function of the resolved shear stress, depending strongly on the choice of crystal plasticity formulation. This crystal plasticity formulation is one of the major differences between the t w o models. T h e x - S R M model is a very fast 1 -step algorithm, in w h i c h the shear on each slip system is related to the resolved shear stress by one simple mathematical expression : 5
In this formula, γ is the slip on a certain slip system a . t the resolved shear stress on that slip system and η and γο t w o material parameters. In literature, this model is claimed to be in good approximation with C P F E M . Crystal Plasticity Finite Elements, at least in a qualitative w a y . In contrast, the x - V P M model is a time incremental model in which the slip on a certain slip system is a function of the resolved shear stress and the time. This model is able to include both strain hardening and texture development during plastic deformation. It is expected to be more accurate. The slip rates at a certain m o m e n t in time are closely related to the resolved shear stress at that time: α
a
4
(2) In this formula, γ„" is the slip rate at slip system α and time tj, τ„" and g„" the resolved shear stress and flow stress respectively o n that slip system and at that time and y and m two material parameters. 0
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V A L I D A T I O N OF CRYSTAL PLASTICITY M O D E L S
Figure 5. The main texture c o m p o n e n t s of cold rolled aluminum and their representation in a n u m b e r of sections through Euler space. Output for the complete Euler space Prior to simulating the surface roughness starting from a experimentally obtained E B S D scan, it is advisable to validate the four implemented crystal plasticity m o d e l s by calculating the output for a set of standard crystallographic orientations. During this validation, the plastic deformation of for instance the C u b e or the Goss orientation can be studied in m o r e detail. In this paragraph, the output of the full set of crystallographic orientations is calculated and the acquired data is plotted as sections of Euler space. In the next paragraph, the focus will lie on the behavior of the main texture c o m p o n e n t s w h i c h are present in cold-rolled aluminum, particularly the Cube and Goss orientation. Figure 6 s h o w s the strains £33 and £22. simulated by the a - S R M model, for a few sections of Euler space. The main texture c o m p o n e n t s of cold-rolled aluminum can also be represented on these sections, figure 5. The deformation behavior, simulated by the a - S R M model, seems to be a smooth and continuous function of the crystallographic orientation, as expected. Both the simulated 633 and ε strains s h o w a big spread, ranging from 0 till -0.14 and from 0.037 till 0.278 respectively. In contrast to the big spread in £33 values, which can be a reflection of the possible development of surface roughness, the big spread in ε values is not that realistic, due to the strain incompatibilities which m a y arise in the loading direction, figure 4. In order to account this problem, the b - S R M model is implemented, for which the output is plotted in figure 7. The crystallographic orientations which exhibit a low ε value in the a - S R M model are m o r e resistant to straining in this loading direction and consequently, need a higher stress factor to reach the applied strain of 1 5 % . The spread in £33 values is comparable, but it is expected to be more realistic. 2 2
2 2
2 2
The simulated ε and £33 strains for the a - V P M model are plotted in figure 8. In this case, the spread in ε 2 values is very high, ranging from 0.011 till 0.693, which is not acceptable. The output of the b - V P M model is represented in figure 9. D u e to strain hardening and texture evolution, the transition between different crystallographic orientations is less smooth, as can be seen in the plot of the stress factor. This stress factor ranges from 6.69 till 80.11 to m a k e sure that the strain £33 is m u c h m o r e realistic than in the a - V P M model, ranging from 0 till -0.177. 2 2
2
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Figure 6. Strains £33 and £22. simulated by the a - S R M model, for a n u m b e r of sections through Euler space.
Figure 7. Stress factor and strain 633. simulated by the b - S R M model, for a n u m b e r of sections through Euler space.
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Figure 8. Strains ε and ε ι , simulated by the a - V P M model, for a n u m b e r of sections of Euler space. 1 3
2
Figure 9 . Stress factor and strain £33, simulated by the b - V P M model, for a n u m b e r of sections of Euler space.
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Output for the m a i n texture c o m p o n e n t s of cold-rolled a l u m i n u m Figure 10 shows a few bar graphs of the strains £ n . ε 2 and ε « for the main texture components of cold-rolled aluminum, as output of the implemented crystal plasticity models. In this paragraph, these strains will be studied in some more detail. 2
Figure 10. Bar graph showing the strains ει ι, ε and En for the main texture c o m p o n e n t s , as output of the four implemented models. 2 2
Considering the C u b e texture component first, it is clear that the strain in the loading direction, ε . simulated by the a - S R M model, is quite high and amounts to 2 8 % . Bearing in mind the fact that the average strain of all the crystallographic orientations is set to 1 5 % , the C u b e texture component seems to be rather soft. Further, since this texture component is highly symmetric, the strains ειι and ε ^ . which are both perpendicular to the loading direction, are equal. D u e to the condition that the volume d o e s n ' t change during plastic deformation, they must be equal to one half of the strain in the loading direction. - 1 4 % . The same conclusions can be drawn for the a - V P M model, but in this case, the C u b e texture component seems to be harder than the average due to the strain hardening effect. Finally, the deformation of the Cube texture component seems to be identical for the two b-xxx models which are two intermediate cases. T h e Goss texture c o m p o n e n t s h o w s a completely different deformation behavior. All the implemented models predict the strain in the normal direction of the sheet. ε , to be equal to zero. The strain in the loading direction is fully compensated by a strain in the rolling direction, ει ι = - ε . In the a-xxx models, the Goss texture component seems to be harder than the average. 2 2
53
2 2
The deformation of the S and Β c o m p o n e n t s of the ß-fiber are comparable with the Goss texture component although the C component shows m o r e or less the opposite behavior.
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SIMULATION OF THE SURFACE ROUGHNESS USING THE CRYSTAL PLASTICITY MODELS Selection of the most appropriate model As stated before, the difference in thinning behavior of bands of Cube and G o s s texture components at the surface of the sheet, w h e n straining in the transverse direction, is expected to be the major cause of roping. Consequently, the most appropriate model to simulate roping can be selected bearing in mind the output for the C u b e and Goss texture c o m p o n e n t s individually, as presented in figure 10. At this point however, only a first qualitative expression of the surface roughness is presented. Therefore, although the b - V P M model is based on the most realistic deformation behavior of polycrystalline materials, the a - S R M model seems to be the most appropriate to visualize the surface roughness in a qualitative way since the difference in thinning behavior between the C u b e and Goss texture c o m p o n e n t s is most pronounced in this case.
Figure 11. Strain in the normal direction of the sheet. ε , simulated by the a - S R M model, after straining in the transverse direction for 1 5 % . 3 3
Figure 11 s h o w s the surface roughness of a sheet of a l u m i n u m exhibiting strong roping behavior, simulated by the a-SRM model. The plot consists of a grayscale plot of the strain in the normal direction of the sheet, after straining for 1 5 % in the transverse direction. A first qualitative impression of the surface r o u g h n e s s plot will result in the conclusion that the plastic deformation of grains depends on their crystallographic orientation which w a s already demonstrated during the validation of the model. Also some zones with both a higher and lower strain in the normal direction of the sheet can be deduced and they can be linked to the bands of C u b e and Goss texture c o m p o n e n t s respectively, figure 2. Due to the difference in thinning behavior, it seems that b a n d s of the C u b e texture component tend to form valleys after deformation while bands of the G o s s texture component tend to form ridges. Applying Fourier analysis to the surface roughness plots A s mentioned before, the surface roughness of a sheet of a l u m i n u m can be subdivided into three groups depending on the wavelength of observation. In order to investigate the surface roughness features which are exclusively related to roping, one has to discriminate between low
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Roughness
and high frequency surface roughness components. For this purpose, the two dimensional fast Fourier transform of the grayscale plot can be calculated. Λ/-1 .V-l F[P<Ù-
Σ
Σ.
1
f "'•"'> ' '
•r ~ " . ρ
- il. 1
U -
I.9-U.I
(3)
V - l
In the frequency domain, a filter can be designed to delete all the c o m p o n e n t s with a frequency higher than a cut-off value, both in the rolling and in the transverse direction. By calculating the inverse Fourier transform afterwards, one is able to plot only the low frequency surface roughness variations of the stretched sheet, figure 12 and 13.
Figure 12. Strain in the normal direction of the sheet, £33, simulated by the a - S R M model, after applying Fourier analysis. Cut-off frequencies in the rolling and transverse direction are both equal to 1/25 . 2π.
Figure 13. Strain in the normal direction of the sheet, εΐ3, simulated by the a - S R M model, after applying Fourier analysis. Cut-off frequencies in the rolling and transverse direction are equal to 1/1000 . 2 π and 1/100 . 2 π respectively.
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Figure 12 presents the surface roughness of the same sheet, after applying t w o dimensional Fourier analysis. In this case, the cut-off frequencies in the rolling and transverse direction are both equal to 1/25 . 2π. The short wave surface roughness variations are suppressed and the roping lines become more visible. In contrast to these symmetric cut off frequencies, it is also possible to design a filter with different cut off frequencies in the rolling and transverse direction, figure 13. For this plot, the cut off frequencies in the rolling and transverse direction were chosen to be 1/1000 . 2 π and 1/100 . 2 π respectively. As a result, almost all the fluctuations in the rolling direction are suppressed. Although very clear roping lines seem to appear in this case, one has to be aware of the fact that these lines can also be a mathematical result of the Fourier analysis instead of real roping features. Nevertheless, in all figures 1 1 , 1 2 and 13, the same zones can be distinguished which exhibit a different thinning behavior than the surrounding material. Fourier analysis seems to be a promising mathematical tool to emphasize these zones. CONCLUSIONS In order to correlate the initial surface texture and strain induced surface roughness for finally annealed and subsequently stretched A A 6 x x x a l u m i n u m sheet, four crystal plasticity models were implemented and validated. The a - S R M model seemed to be the best model to simulate surface roughness in a qualitative way. Due to the difference in thinning behavior of the Cube and Goss texture c o m p o n e n t s , it was shown that bands of these texture c o m p o n e n t s in the rolling direction of the sheet seemed to be the major cause of roping. Fourier analysis was successfully applied to suppress the high frequency surface roughness components in order to be able to visualize only the low frequency characteristics which are predominantly related to roping. However, further research is necessary to explore the possibilities of this two dimensional fast Fourier transform. ACKNOWLEDGEMENT This work is performed within the framework of the scientific research program of the Netherlands Institute of Metals Research ( w w w . n i m r . n n . Project M C 4 . 0 5 2 3 8 . The authors would like to thank T. Bennett and L. Kestens for performing the E B S D measurements. REFERENCES ' P . D . Wu, D.J. Lloyd. A. Bosland, H. Jin and S.R. M a c E w e n , "Analysis of roping in A A 6 1 1 1 automotive sheet". Acta Materialia, 5 1 , 1945-1957 (2003). P . S . Lee, H.R. Piehler, B.L. A d a m s , G. Jarvis, H. Hampel and A.D. Rollett, "Influence of surface texture on orange peel in aluminum", Journal of Materials Processing Technology, 8 0 - 8 1 , 3 1 5 - 3 1 9 (1998). 0 . Engler and E. Brünger, "On the Correlation of Texture and Ridging in A A 6 0 1 6 Automotive Alloys", Materials Science Forum, 3 9 6 - 4 0 2 , 345-350 (2002). G . J . Baczynski, R. G u z z o , M . D . Ball and D.J. Lloyd, "Development of roping in an aluminum automotive alloy AA6111 ". Acta Materialia, 4 8 , 3361-3376 (2000). P . D . Wu, D.J. Lloyd and S.R. M a c E w e n , "A simple model describing roping in Al sheet". Scripta Materialia, 4 8 , 1243-1248 (2003). Z . Z h a o . R. Radovitzky and A. Cuitino, "A study of surface roughening in fee metals using direct numerical simulation", Ada Materialia, 52, 5 7 9 1 - 5 8 0 4 (2004). 2
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THE VARIATION RANGE FOR THE RESULTS OF FCC T E X T U R E SIMULATIONS BASED ON {111}<110> SLIP Torben Leffers Center for Fundamental Research: Metal Structures in 4 Dimensions Materials Research Department Riso National Laboratory for Sustainable Energy Technical University of Denmark DK-4000 Roskilde, Denmark ABSTRACT The simulated textures for {111 }<110> slip in fee materials resulting for a variety of (simple) models are described and compared with experimental textures. The range of models covers variations in the degree of constraint relaxation and in the rules used for the calculation of lattice rotations. INTRODUCTION The aim of the present work is to demonstrate the variation range (and variation limit) for the fee deformation textures resulting from modelling of plastic deformation with {111}< 110> slip, depending on the mathematical procedures or rather on the conceptual models behind the mathematics. There is not going to be any discussion of the underlying physics. The aim is a strictly empirical description of the simulated textures and their relation to experimental textures. The types of deformation to be considered are rolling/plane strain and axi-symmetric elongation. The emphasis is going to be on simple (1-point) models since they in practice cover the whole variation range. This implies that self-consistent models only play a minor role. Unfortunately the available space only allows the presentation of one of the many illustrations needed to tell the story. The lecture, on the other hand, will concentrate on illustrations. THE T A Y L O R AMBIGUITY There is a theoretical ambiguity in the selection of the active slip systems in the rateindependent Taylor model - an ambiguity which is eliminated if rate sensitivity is introduced, e.g. Canova et al. . The present author organized a round-robin investigation of the actual variation in the simulated textures resulting from this ambiguity (for the full-constraint Taylor model, see later). The investigation included three cases with different procedures for solving the actual ambiguity problem and one case with rate sensitivity and hence no ambiguity. For all four cases two deformation schemes were considered: rolling/plane strain to 9 5 % reduction and axi-symmetric elongation to a strain of 0.7. The rolling and the elongation "experiments" included 5000 grains with initially random orientations and 15 grains with specific initial orientations, respectively. For rolling the resulting textures were practically identical, i.e. the ambiguity is without any practical significance. A similar conclusion was reached by Hirsch and Lücke . The similarity with experimental textures will be discussed later. For elongation all four cases showed that most grains rotated towards orientations with <111> in the direction of elongation and the other grains rotated towards orientations with <100> in the direction of elongation. There was some variation between the four cases when it came to the exact orientation path followed by the individual grains, including variation in the rate of orientation change. However, the details of the experimental texture development during axi-symmetric elongation are not very well known (partly because it is difficult to establish ideal axi-symmetric elongation conditions), and compared to this uncertainty about the experimental texture the variation in the simulated texture due to the Taylor ambiguity is hardly significant - as it was not for rolling. The experimental textures are dual <111>/<100> fibre textures dominated by the <111> texture (Stout et al. ). 1
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For rate-insensitive Taylor models with relaxed constraints (which we shall return to later) there is still some ambiguity, but it is less than for the full-constraint Taylor model, and hence it has even less effect on the simulated texture. THE H O S F O R D AMBIGUITY Another ambiguity, which was first pointed out by Hosford , has attracted far less attention in the texture community than the Taylor ambiguity. The Hosford ambiguity is about the calculation of the lattice rotation for a given slip pattern. Hosford listed three different procedures for the calculation of the lattice rotations: (i) mathematical analysis. MA, (ii) Schmid tension analysis and (iii) Taylor compression analysis. Procedure (i) is the one normally used in solid mechanics, calculating the rotations from the skevvsymmetric component of the plastic distortion. Procedures (ii) and (iii) are the procedures to be used for 6
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Figure I. Simulated { 1 1 1 ) and {200} pole figures for 5 0 % rolling reduction with M A lattice rotation rules (a,b) and PSA lattice-rotation rules (c,d). the calculation of the lattice rotations in single crystals deformed in tension and compression, respectively. Hosford suggested that (i) is the correct procedure for polycrystals with equiaxed grains whereas (ii) and (iii) are the correct procedures for polycrystals with elongated and flat grains, respectively. In accordance with the declared aim of the present work as outlined in INTRODUCTION the results for the different procedures will just be quoted without any references to the underlying physics. It should be mentioned, however, that Lebensohn and Leffers have demonstrated such a transition from MA to PSA (to be described later) with change in grain shape. Leffers and Lebensohn showed that the Hosford ambiguity is relevant only for deformation models which deviate from the (full-constraint) Taylor model (models where the strain in the individual grains is different from the macroscopic strain), and they added a fourth procedure, (iv) plane strain analysis. PSA, for rolling/plane strain combining (ii) and (iii). In this section we shall look at the effects of the Hosford ambiguity for Sachs-type models (Sachs'') which stand for an extreme deviation from the Taylor model, for an extreme degree of relaxed constraints (as already mentioned we shall return to the concept of relaxed constraints later). We shall use the simple Sachs model and 7
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the "modified Sachs model" (Leffers ). which is a simple Sachs model with random stresses added. Here we can take the random stresses to be an empirical procedure used to smear out the simulated texture. For a description of the philosophy behind the modified Sachs the reader is referred to Pedersen and Leffers". For rolling to 5 0 % reduction with the modified Sachs model Leffers and Lebensohn showed that mathematical analysis, MA, leads to a simulated texture similar to the experimental copper-type texture while plane strain analysis, PSA, leads to a texture similar to the experimental brass-type texture - see Figure 1. With the simple Sachs model MA and PSA lead to simulated textures which may be seen as extreme versions of the copper and the brass texture, respectively. For MA there is a very high density of orientations close to the copper orientation {211}<111> (Leffers and Lebensohn ), and for PSA the orientations are narrowly concentrated in the range from the brass orientation {110}<112> to the Goss orientation {110}<001> ( L e f f e r s ) . It is repeatedly stated in literature, e.g. Wierzbanowski et a l . and K o c k s , that in order to simulate the brass-type texture (with its high density of the brass orientation) with {111 }<110> slip without deformation twinning one must relax the third shear constraint which should not be relaxed according to the standard relaxed-constraint models to be described later, i.e. one must approach a Sachs deformation pattern. Thus, all these references must refer to PSA even though the MA/PSA ambiguity was not mentioned. Leffers and Lebensohn also showed {111} pole figures simulated with one of the standard relaxed-constraint models, the "pancake model" (Van Houtte ), for MA and PSA. Now. when there is less deviation from the full-constraint model, the difference between the textures for MA and PSA is hardly significant. For axi-symmetric elongation with Sachs-type models the present author is only aware of few references dealing with the effect of the lattice-rotation rule. Lorentzen et a l . showed the lattice rotations for the different orientations in the unit triangle for the simple Sachs model combined with MA. The great majority of the grains rotated towards a <111> orientation while a very small fraction rotated towards a <100> orientation. Winther et a l . showed the lattice rotation for the modified Sachs model combined with MA. N o w a significant, but still minor, fraction of the grains rotate towards <100>, the majority of the grains rotating towards <111>. This is in agreement with the experimental results of Stout et al. for various fee metals and alloys. For the simple Sachs model combined with Schmid tension analysis the result is well known from single crystals, and computer experiments are not necessary: first the grains rotate towards the <100>/<111> side of the unit triangle, and then they rotate along the side towards <211>. The <211> fibre texture is not a texture observed experimentally. Without going into any details the present a u t h o r " mentioned that the addition of random stresses, which in the author's present terminology would mean an approach to modified Sachs, led to a change in simulated texture in the direction of a dual <111>/<100> fibre texture. 8
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R E L A X E D - C O N S T R A I N T MODELS The term "relaxed constraint" was introduced in connection with the suggestion by Mecking and Kocks and C a n o v a that some of the constraints in the Taylor model should be relaxed when the grains are flat (pancake-shaped) or elongated (cigar-shaped). As opposed to this physical argument for relaxing specific constraints the present author had earlier, on a purely empirical basis, investigated the effect of the various constraints on the simulated rolling texture. While Mecking and Kocks and Canova started from the Taylor model and relaxed some constraints, the present author started from the Sachs model and added constraints (not using the word constraint). Obviously it does not make any difference whether it is done one way or the other, e.g. K o c k s . With reference to the declared empirical approach in the present work we are going to focus on the results of the present author and those of Ahzi and c o - w o r k e r s ' based on a similar empirical attitude. They are all (in the present author's interpretation) based on PSA lattice rotation rules (and not MA) which, as shown earlier. 20
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makes a difference for heavily relaxed constraints but not for moderately relaxed constraints. As a matter of fact we are not, in this section, going to deal with completely relaxed constraints as in the Sachs model. That was dealt with in the section about the Hosford ambiguity. For rolling we take the rolling direction to be the χι axis, the transverse direction to be the x axis and the normal direction to be the x axis. There is one normal constraint to consider, which maintains ε at zero, and three shear constraints, which maintain ει 2, ε and ε ι at zero. When Leffers' introduced these constraints one at a time, it was found that the introductions of the constraints keeping ε and ε ι at zero made very little difference for the simulated texture - the texture was still close to a brass-type texture. A significant change appeared when the constraints keeping ε υ and ε at zero were introduced either one by one or together - a change towards a copper-type texture. Adding the third shear constraint to the two others made some difference, but the simulated texture was still of the copper type. Neither when combined with the shear constraints did the addition of the normal constraint make any significant difference for the simulated texture. The three combinations of constraints for which Leffers concluded that the simulated textures were copper-type textures are the ones which Van H o u t t e ' later called the "lath model" (the ε ι and ε constraints added), the "pancake model" (the ε ι constraint added) and the "full-constraint model" (all constraints added). Leffers compared the various simulated textures with the experimental copper textures and concluded that the combination of constraints corresponding Van Houtte's (later) lath model gave the best agreement (see also Leffers ) which was also Van Houtte's conclusion. This is surprising in the sense that the philosophy behind the grain-shape-based relaxed-constraint model would favour the pancake model. Kallend and Davies" claimed that the full-constraint Taylor model provides simulated textures (with the maximum orientation density at {11 4 4} <8 11 11>) which agree very well with the experimental copper textures - a claim which in the present author's opinion represented wishful thinking. Kallend and Davies did not by then accept the idea of relaxed constraints. Van Houtte concluded that the rolling texture in aluminium is in closest agreement with full-constraint simulated textures which apparently is correct, at least for the aluminium texture he had selected but also for instance for the aluminium alloys investigated by Juul Jensen et a l . Ahzi et a l . made a different type of empirical investigation of the effect of relaxation of constraints based on their elastic-viscoplastic approach. They defined a parameter φ which governed a proportional relaxation of all constraints. φ=0 corresponds to the full-constraint Taylor model, and φ=1 corresponds to the (zero-constraint) Sachs model. For 5 0 % rolling reduction they found that φ=0.3 (moderately relaxed constraints) produced the simulated texture closest to the experimental coppertype texture and that φ=0.7 (heavily but not fully relaxed constraints) produced the simulated texture closest to the experimental brass-type texture. 2
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For axi-symmetric elongation Ahzi and M ' G u i l considered the same values for φ as above: 0, 0.3, 0.7. 1. Only for φ=0. the full-constraint Taylor model, do the simulated textures reproduce the experimental dual <111>/<100> fibre texture with < 111 > as the predominant component. For all the other values of φ, representing increasing deviation from the Taylor model in the direction of the Sachs model, there are only very few grains ending up near <100>, and those ending up in the <111> corner of the unit triangle tend to concentrate near <211>, increasingly so for increasing φ values. Wierzbanowski in his texture simulations for axi-symmetric elongation only allowed relaxation of the normal constraint (not the shear constraints). This relaxation had very little effect on the simulated texture; it was still the dual <111>/<100> fibre texture with the <111> fibre as the predominant one. It should be mentioned that for instance the present author has questioned the physical basis for constraint relaxation in elongated grains. 27
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It is very well known within the field of texture simulation that the simulated textures normally are significantly sharper than the experimental textures. The reason is obvious: while each individual grain in real materials interacts with a different population of neighbouring grains, the individual
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grains in the simple models as those described so far (with the exception of the modified Sachs model) interact with the same continuum matrix - in the terminology of Molinari et a l . they are 1-point models. The present a u t h o r ' ' used random stresses as an integral part of the modified Sachs model. Such random stresses may also be introduced in full-constraint and relaxed-constraint models, and they do serve the purpose of smearing out the texture, but Leffers and Juul J e n s e n observed an unexpected side effect of the random stresses. They compared the experimental copper texture with a texture simulated with the lath model with random stresses added. Because of the random stresses there was a very good agreement between the experimental and the simulated orientation densities for the main texture components (the copper component, the brass component, the S component, the Goss component). However, the orientation density of the cube component in the simulated texture was about three times higher than in the experimental texture. Without random stresses the simulated orientation density of the cube component in the experimental and the simulated textures were about the same, but now the other orientation densities in the simulated texture were significantly higher than in the experimental texture. Used together with the Full-constraint model the random stresses did not introduce extra orientation density of the cube component; they just smoothed the texture. The random stresses change the simulated texture by changing the pattern of slip leading to the lattice rotations. There is also the possibility to smooth the texture after the lattice rotations have finished, e.g. by a Gaussian procedure (Wierzbanowski et al. ), but there are no detailed reports about the effects of such procedures. 28
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2-POINT AND N-POINT MODELS The main effect of letting two or more grains interact during deformation is that the sharpness of the simulated textures decreases approximately to the level of sharpness observed experimentally. This is observed already in the 2-point models of Van Houtte et a l . and Lee et a l . But of course npoint models, in practice normally finite-element models, are the radical solution. The use of such FEM models in texture simulation has been reviewed by Dawson and Beaudoin . Obviously n-point models rest on a healthier physical basis than 1-point models, and they do catch the details of the experimental textures better than 1-point models. But as to the aim of the present work, an empirical description of the range of textures simulated with {111}<110> slip, they do not add much to the results of 1-point models. As a matter of fact practically all the η-point models for rolling produce simulated textures of the copper type. Recently Miraglia et a l . attempted to get an FEM-simulated texture of the brass type by adding latent hardening. They did manage to increase the fraction of brass component but not to get a proper transition to the brass-type texture. 30
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CONCLUSIONS For rolling/plane strain all the textures simulated with simple models, even when the constraints are added/subtracted arbitrarily, correspond roughly either to the experimental copper-type or to the experimental brass-type texture. The rules applied for the calculation of the lattice rotations are essential for the resulting simulated textures. Only the combination of PSA lattice-rotation rules and heavily relaxed constraints as in Sachs-type models leads to simulated textures of the brass type. MA lattice-rotation rules always lead to copper-type simulated textures. The exact similarity to the experimental copper-type textures in various materials is open to discussion. The textures simulated with simple models are sharper than the experimental textures unless special additions like random stresses or Gaussian procedures are made. More complex models with two or more grains interacting during deformation produce copper-type simulated textures with sharpness closer to that of the experimental textures. For axi-symmetric elongation it should be noticed initially that the experiments of Stout et a l ; seem to overrule the normally cited results of English and C h i n showing a strong variation with 34
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stacking fault energy of the relative strength of the two fibres in the dual <111>/<100> fibre texture. Stout et al. found that the <111> fibre is always predominant. Of the Sachs-type models only the modified Sachs model combined with MA lattice rotation rules leads to a texture similar to the experimental dual <111>/<100> fibre. Otherwise only the full-constraint Taylor model or models very close to the full-constraint Taylor model (for which there is no difference between M A and Schmid tension analysis) produce simulated textures similar to the experimental textures. It is an interesting observation that for rolling textures deviation from the MA lattice-rotation rule to PSA is necessary in order to get a simulated texture of the brass type whereas deviations from MA (to Schmid tension analysis) do not lead to experimentally relevant simulated textures for axisymmetric elongation. REFERENCES 'G.I. Taylor. Plastic Strain in Metals,./. Inst. Metals 62, 307-24 (1938). 'G.R. Canova, C. Fressengeas, A. Molinari, and U.F. Kocks, Effect of Rate Sensitivity on Slip System Activity and Lattice Rotation. Acta Metall. 36, 1861-70 (1988). T. Leffers. Deformation Textures: Simulation Principles, in: Proceedings ICOTOM 8 (J.S. Kallend and G. Gottstein, eds.), The Metallurgical Society, Warrendale, 273-84 (1988). J . Hirsch and K. Lücke, Mechanism of Deformation and Development of Rolling Textures in Polycrystalline F.C.C. Metals - I, Acta Metall. 36, 2863-82 (1988). M . G . Stout. J.S. Kallend. U.F. Kocks, M.A. Przystupa, and A.D. Rollett, in: Proceedings ICOTOM 8 (J.S. Kallend and G. Gottstein, eds.), The Metallurgical Society, Warrendale, 479-84 (1988). W . F . Hosford, The Orientation Changes Accompanying Slip and Twinning, Text. Cryst. Solids 2, 17582(1977). R . A . Lebensohn and T. Leffers, The Rules for the Lattice Rotations Accompanying Slip as Derived from a Self-Consistent Model, Textures and Mierostructures 3 1 , 217-30 (1999). T . Leffers and R.A. Lebensohn, Ambiguities in the Calculation of Lattice Rotations for Plane-Strain Deformation, in: Proceedings ICOTOM 11 (Z. Liang et al., eds.), International Academic Publishers, Beijing, 307-14(1996). Ό . Sachs, Zur Ableitung einer Fliessbedingung, Ζ. Verein, deut. Ing. 72, 734-36 (1928). T . Leffers, A Modified Sachs Approach to the Plastic Deformation of Polycrystals as a Realistic Alternative to the Taylor Model, in: Proceedings ICSMA J (P. Haasen et a l , eds.), Pergamon Press, Oxford, 769-74(1979). " O . B . Pedersen and T. Leffers. Modelling of Plastic Heterogeneity in Deformation of Single-Phase Materials, in: Constitutive Relations and Their Physical Basis (S.I. Andersen et al., eds.), Riso National Laboratory, Roskilde, 147-72 (1987). T . Leffers, Computer Simulation of the Plastic Deformation of Face-Centred Cubic Polycrystals and the Rolling Texture Derived, Rise Report No 184 (1968). T . Leffers, Computer Programs for Texture Simulation, Rise Report No. 283 (1973). K . Wierzbanowski, J. Jura, W.G. Haije, and R.H. Helmholdt, F.C.C. Rolling Texture Transition in Relation to Constraint Relaxation, Cryst. Res. Technol. 27, 513-22 (1992). U . F . Kocks, Simulation of Deformation Texture Development for Cubic Metals, in: Texture and Anisotropy (U.F. Kocks et a l , eds.), Cambridge University Press, Cambridge, 391-418 (1998). P . Van Houtte, Adaptation of the Taylor Theory to the Typical Substructure of some Cold Rolled FCC Metals, in: Proceedings ICOTOM 6 (S. Nagashima, ed.), The Iron and Steel Institute of Japan, Tokyo. 428-37(1981). T . Lorentzen, T. Leffers, and B. Clausen, Polycrystal Models and Intergranular Stresses, in: Modelling of Structure and Mechanics of Materials from Microscale to Product (J.V. Carstensen et al., eds.). Riso National Laboratory, Roskilde, 345-54 (1998). 3
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G. Winther, T. Leffers, and B. Clausen, Revision of the Concept of 1 -Site Models, in: Proceedings ICOTOM 12 (J.A. Szpunar, ed.), NCR Research Press, Ottawa, 399-404 ( 1999). T . Leffers, A Kinematical Model for the Plastic Deformation of Face-Centred Cubic Polycrystals, Ris0 Report No. 302 (1975). H . Mecking, Deformation of Polycrystals, in: Proceedings ICSMA 5 (P. Haasen et al.. eds.), Pergamon Press, Oxford, 1573-94 (1980). U . F . Kocks and G.R. Canova, How Many Slip Systems, and Which?, in: Deformation of Polycrystals: Mechanisms and Microstructures (N. Hansen et a l , eds.), Riso National Laboratory. Roskilde, 35-44 (1981). S . Ahzi, S. M'Guil, and A. Agah-Tehrani, A New Formulation for the Elastic-Viscoplastic Lower Bound and Intermediate Modeling for Polycrystalline Plasticity, Mater. Sei. Forum 408-412, 463-68 (2002). S . Ahzi and S. M'Guil, Simulation of Deformation Texture Evolution Using an Intermediate Model. Solid State Phenomena 105, 251-58 (2005). T . Leffers, The Concept of Relaxed Constraints Reconsidered, Problems of Materials Science 3 3 . 243-50 (2003). J . S . Kallend and G.J. Davies, A Simulation of Texture Development in F.C.C. Metals, Phil. Mag. 2 5 , 471-90(1972). D . Juul Jensen, N . Hansen, and F.J. Humphreys, On the Textural Development in FCC Metals and Alloys, in: Proceedings ICOTOM 8 (J.S. Kallend and G. Gottstein, eds.). The Metallurgical Society, Warrendale, 431-44(1988). K . Wierzbanowski, Etudes de la Formation des Textures de Fibre des Métaux à Réseau Cubique. Bulletin de l'Académie Polonaise des Sciences, Serie des Sciences Techniques X X V I , 7-11 (1978). A . Molinari, G.R. Canova, and S. Ahzi, A Self-Consistent Approach of the Large Deformation Polycrystal Viscoplasticity, Acta Metall. 3 5 , 2983-94 (1987). T . Leffers and D. Juul Jensen, Quantitative Simulation of the Copper-Type Rolling Texture, in: Modelling of Plastic Deformation and Its Engineering Applications (S.I. Andersen et al., eds.), Riso National Laboratory, Roskilde, 323-29 (1992). P . Van Houtte, S. Li, L. Delannay, and P. Van Bael, Modelling Deformation Textures and Plastic Anisotropy, Problems of Materials Science 3 3 , 225-33 (2003). B . J . Lee, S. Ahzi, and D.M. Parks, Bicrystal-Based Modeling of Plasticity in FCC Metals, J. Eng. Mater. Technol. 1 2 4 , 2 7 - 4 0 (2002). P . R . Dawson and A.J. Beaudoin, Finite-Element Modeling of Heterogeneous Plasticity, in: Texture and Anisotropy (U.F. Kocks et al., eds.), Cambridge University Press, Cambridge. 512-31 (1998). M . Miraglia, P. Dawson, and T. Leffers, On the Influence of Mechanical Environment on the Emergence of Brass Textures in FCC Metals, Acta Mater. 5 5 , 799-812 (2007). A . T . English and G.Y. Chin, On the Variation of Wire Texture with Stacking Fault Energy in F.C.C. Metals and Alloys, Acta Metall. 13, 1013-16(1965). I9
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DEVELOPMENT OF LOW MAGNETISM A N D STRONGLY CUBE-TEXTURED COMPOSITE Ni-7%W/Ni-10%W SUBSTRATE FOR COATED SUPERCONDUCTOR a
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Dan-min Liu , Fei H a o , Jiu-xing Z h a n g , Yan-cao H u , Mei-ling Z h o u , Wei-peng Liu
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"Institute of Microstructure and Properties of Advanced Materials College of Materials science and Engineering Beijing University of Technology, Beijing 100022, P. R. of China b
ABSTRACT N i - 7 a t . _ W / N i - 1 0 a t . _ W composite alloys were fabricated by powder metallurgy process. After heavy cold-rolling and recrystallization annealing, sharp cube-textured composite foils with high strength and low magnetism were obtained. Textures of composite foils were studied. The magnetic and mechanical properties of N i - W composite foils were also reported. 1. I N T R O D U C T I O N The rolling assisted biaxially textured substrates ( R A B i T S ) method is a very desirable approach for the preparation of the substrate for high temperature superconducting YBa2Cu307.e. ( Y B C O ) tapes with long lengths . In order to get Y B C O tapes with high critical current densities, Y B C O films should have strong texture with c axis perpendicular to the surface and a-b-axial in-plane alignment. Using R A B i T S method, this condition can be achieved by the epitaxial growth of the buffers and Y B C O film on a highly cube-textured substrate. It has been shown that Ni and Ni-based alloy are suitable as the substrate materials because they form very strong cube texture after heavy rolling and recrystallization. In addition its oxidation resistance and the small lattice mismatch allow epitaxial growth of buffer layers and Y B C O film. However, for A C applications, hysteretic losses caused by the ferromagnetism of Ni must be avoided. N i - W alloys are preferred as substrates for the preparation of Y B C O tapes due to its reduced magnetization losses and improved mechanical strength. It has been confirmed that the magnetization intensity decreases strongly with the addition of W. Curie temperature is suppressed to below 77 Κ at W content m o r e than about 9.5at.% and hysteretic losses are negligible at W content about 7.5at.% at 77 Κ ' " . H o w e v e r , the cube fraction in structure was found to decrease with W content >5at.%. V. Subramanya Sarma et al. reported that about 7 7 % and 4 5 % cube texture could be obtained in Ni-7.5at.% W and Ni-9at.% W alloys, respectively' . This is attributed to the transition in the rolling texture from copper type to brass type caused by the stacking fault energy decreasing with increasing W content ' . In addition, the low tensile yield strength of N i - W alloy with low W content (3~5at.%) limits the processing to produce thin tapes. For these reasons the preparation of substrate material with low magnetism and high strengthen is important. V.Subramanya Sarma et al. developed the Ni-4.5at.%W/Ni-15at.%Cr composite substrates with strong cube texture, high yield strength and reduced magnetization losses . In this paper, we studied on the development of N i - 7 a t . % W / N i - 1 0 a t . % W composite substrates fabricated by Spark Plasma Sintering, a new kind of powder metallurgy technique. 1 U 1
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2. E X P E R I M E N T A L Fig.l s h o w s the schematic of powder metallurgy process for the Ni-7at.%W/Ni-10at.%W composite tapes. The Ni p o w d e r (Alfa Aesar Co. 99.9%) with size of 3 ~ 7 p m was mixed with W p o w d e r (Alfa Aesar Co. 99.9%) with size of l ~ 5 p m . The mixed p o w d e r s were then sintered and
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D e v e l o p m e n t of Low M a g n e t i s m a n d Strongly C u b e - T e x t u r e d C o m p o s i t e N i - 7 % W / N i - 1 0 % W
compacted in a Spark Plasma Sintering (SPS) furnace. The S P S process simultaneously applies a low-voltage, high-intensity pulsed direct current and uni-axial pressure, which offers the possibilities of using rapid heating rates and very short holding times to obtain highly dense samples ' "'' . Besides, it makes the fabrication of composite easier than other methods. 7
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Fig. 1 Schematics of p o w d e r metallurgy process for the N i - W composite tape. The pressure used was 3 0 M P a . the sintering temperature was 9 0 0 - 1 0 0 0 " C . After sintering, homogenizing annealing carried out at 1200°C for 10 hours with flowing A r - 4 % H . The sintered compacts were rolled at room temperature to thin tapes using m a n y times intermediate annealing. After the rolling procedure, the deformed tapes were annealed at 1200-Ί 300°C for 2 hours which was performed using a tube furnace in flowing A r - 4 % H 2 . T h e X-ray pole figures were measured in a D 8 B r u k e r / A X S texture goniometer with two dimension Detector ( G A D D S ) . E B S D technique ( E D A X / T S L instrument in a J E O L - 6 5 0 0 S E M ) was used to investigate the microstructure of the samples. Magnetization measurements were m a d e in a M P M S - X L S Q U I D of Q u a n t u m Design C o . with the magnetic field being parallel to normal direction of the foil. The yield and tensile strength of the foils were measured with an 810 Material Test System of M T S from the foils, prepared to dog bone geometry with a g a u g e section of 10 m m w i d e and 25 cm long. 2
3. R E S U L T S A N D D I S C U S S I O N 3.1 Texture Fig.2 s h o w s the three-layer structure sketch of composite N i - 7 a t . % W / N i - 1 0 a t . % W . T h e thickness of each layer is nearly the same and total thickness of the sintered composite is 8.98mm. Fig.3 s h o w s the process of cold-rolling, intermediate annealing was performed for 4 times at 600°C for 2 hours, the total thickness reduction was .,
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Fig.3 Process of cold-rolling It was conformed in our study that intermediate annealing during cold-rolling process is a key factor to obtain strong cube texture in N i - 7 a t . % W surface layer. The increase of W content in
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D e v e l o p m e n t of L o w M a g n e t i s m a n d S t r o n g l y C u b e - T e x t u r e d C o m p o s i t e N i - 7 % W / N i - 1 0 % W
Ni-W alloy lowers the stacking fault energy and e n h a n c e s the resistance to slip during the process of rolling. This effect causes twinning. Hence twinning may b e c o m e a major deformation mode in N i - W alloy with high W content and the deformation process is different from that in the N i - W alloy with low W content, in w h i c h slip is the m a i n deformation m o d e and the copper type rolling textures having strong C orientation {112}<111> and S orientation {123}<634> can be formed after large reduction. H o w e v e r in N i - W alloy with high W content brass type rolling texture having very strong Β orientation {110}<112> and G [ 1 1 0 ) < 1 0 0 > orientation can be formed which prevent the formation of a sharp cube texture after a n n e a l i n g ' . The intermediate annealing can eliminate the stress produced by deformation and hence reduce the slip resistance during further deformation. After intermediate annealing slip might b e c o m e the main deformation mode. We h a v e used 600°C as the intermediate annealing temperature. At this temperature the deformation stress can be eliminated but primary recrystallization does not begin. Therefore we annealed the sample after several passes w h e n the sample reduced to every half thickness. This procedure is also necessary for obtaining copper type rolling texture and hence cube recrystallization texture in N i - 7 a t . % W surface layer. Fig.4 s h o w s the pole figures of the surface of the N i - 7 a t . % W / N i - 1 0 a t . % W composite foils after cold-rolling and annealing at 1200°C and 1300°C for 2 hours respectively. It reveals that very strong cube texture has been formed in the foils. 1
01
Fig.4 Pole figures of the surface of N i 7 a t . % W 7 N i - 1 0 a t . % W 7 N i - 7 a t . % W composite foil after annealing at 1200°C a) or 1300°C b) for 2 hours
Materials Processing and Texture
753
D e v e l o p m e n t of L o w M a g n e t i s m a n d S t r o n g l y C u b e - T e x t u r e d C o m p o s i t e N i - 7 % W / N i - 1 0 % W
Fig.6 Pole figures of N i 9 . 5 a t . % W monolithic foil We have also studied the textures in N i - 7 a t . % W and N i - 9 . 5 a t % W monolithic tapes systematically. Fig.5 and fig.6 s h o w the pole figures of N i - 7 a t . % W and N i - 9 . 5 a t % W monolithic foils after cold-rolling and annealing respectively. It is noted that very strong cube texture formed in the Ni-7at.%W foil but not in Ni-9.5at%W foil, which has i001 ) < 1 0 0 > . ( 112> <111 >.{112 j < 1 3 2 > and {221}<122> orientations. Such textured N i - 9 . 5 a t % W substrate is not suitable for the epitaxial g r o w t h of Y B C O film with high critical current densities. Fig.7 is the E B S D grain orientation m a p of the surface of N i - 7 a t . % W / N i - 1 0 a t . % W composite foil annealed at 1300°C. After these treatments, fraction of 98.1 % cube texture was obtained in the foil. The v o l u m e fraction of the [221}<122> component, which is the twinning orientation of cube, was calculated to be 0 . 0 1 % . Fig.8 s h o w s the grain boundary misorientation distribution calculated from the E B S D measurement. It reveals that very' strong cube texture with a negligible fraction of twin and other mis-oriented grains w a s formed in the surface of the composite foil.
754
Materials Processing a n d Texture
D e v e l o p m e n t of Low M a g n e t i s m a n d Strongly C u b e - T e x t u r e d C o m p o s i t e N i - 7 % W / N i - 1 0 % W
Fig.7 E B S D image of the surface of the composite substrate 0.40
Misorientation Angel(degreee)
Fig.8 Grain boundary mis-orientation distribution of the surface of the composite substrate Fig.9 s h o w s the E B S D grain orientation map of the cross-section of Ni-7at.%W/Ni-10at.%W composite foil annealed at 1300°C. T h e left and right edges of the micrograph represent the surfaces of the composite N i - W alloy foil. It reveals that the core of the composite has m a n y non-cube oriented grains including twins. This reveals that the texture in the core does not affect o n the formation of cube grains on the surface of the composite foil during recrystallization.
Fig.9 E B S D image of the cross-section of the composite substrate
3.2 Magnetization results The M - H hysteresis loops at 77 Κ for the N i - 7 a t . % W and Ni-7at.%W7Ni-10at.%W composite substrate with two layers are shown in Fig. 10(a). It reveals that the hysteresis loss is reduced strongly in the composite w h e n compared with the N i - 7 a t . % W foil. T h e temperature dependence
Materials P r o c e s s i n g a n d Texture
755
D e v e l o p m e n t of L o w M a g n e t i s m a n d S t r o n g l y C u b e - T e x t u r e d C o m p o s i t e
Ni-7%W/Ni-10%W
of the mass magnetization M ( T ) of the composite m e a s u r e d in an applied magnetic field of 5 0 0 0 O e is shown in Fig.lO(b) along with the data for N i - 7 a t . % W and N i - 7 a t . % W / N i - 1 0 a t . % W composite substrate. It can be seen that the saturation magnetization o f the c o m p o s i t e is suppressed strongly w h e n c o m p a r e d with that of N i - 7 a t . % W substrate.
z-
Ni-7%W Ni-7%W/Ni-1<
ο c
Ni-7%W Ni-7%W/Ni-10%W
D> 5
100 -2000
-1000
0
1000
150
2000
a) M - H Fig. 10 M - H results a) and M - T results b) o f Ni-7atW_ foil 3.3 Tensile properties The r o o m temperature stress-strain response of the N i - 7 a t . % W / N i - 1 0 a t . % W composite foil recrystallized at 1300°C is given in Fig. 11. Stress-strain data of N i - 7 a t . % W and N i - 9 . 5 a t . % W are also included for comparison. It can be seen that the composite exhibits increased yield strength and tensile strength. T h e yield strengths of N i - 7 a t . % W , N i - 9 . 5 a t . % W and Ni-7at.%W/Ni-10at.%W substrates are 196MPa. 2 6 3 M P a and 3 1 3 M P a respectively. And the tensile strengths o f these foils are 4 9 3 M P a , 6 7 0 M P a and 6 9 0 M P a respectively. It is noted that the strength of Ni-7at.%W/Ni-10at.%W is higher than N i - 9 . 5 a t . % W which can be explained that, the cross-sectional interface structure of the composites is also another important factor of strengthening, except for content difference.
200
250
300
Temperature(K)
Magnetic Fleld(oe)
b) M - T foil and N i - 7 a t _ W / N i - 1 0 a t _ W composite
ο
Ni-7%W/Ni-10%W/Ni-7%W
ο
Ni-9.5%W
Λ
Ni-7%W
[0.2% ot(Mc$ ~l r
±1
CO
8
10
Strain(%) Fig.l l Stress-strain curves of recrystallized N i - 7 a t . % W / N i - l 0at.%W. N i - 9 . 5 a t . % W and N i - 7 a t . % W substrates
CONCLUSION A p o w d e r routine for N i - W composite ingots with metallurgical bonding w a s developed. Sharp cube-textured N i - 7 a t . % W / N i - l 0 a t . % W c o m p o s i t e foils with high strength and low m a g n e t i s m were obtained by R A B i T S technique. It is noted that the texture in the core does not
756
Materials Processing a n d Texture
D e v e l o p m e n t of L o w M a g n e t i s m a n d S t r o n g l y C u b e - T e x t u r e d C o m p o s i t e N i - 7 % W / N i - 1 0 % W
affect the formation of cube grains on the surface of the composite foil during recrystallization. Compared with N i - 7 a t . % W and N i - 9 . 5 a t . % W substrates, the N i - 7 a t . % W / N i - 1 0 a t . % W composite possesses better mechanical properties and lower hysteretic losses. REFERENCES Ά. Goyal, D. R Norton, J. D. Budai, M. Paranthaman et al.. High Critical Current Density Superconducting Tapes by Epitaxial Deposition of YBa2Cu307. Thick Films o n Biaxially Textured Metals, Appl. Phys. Lett., 6 9 , 1795-97(1996). D . P. N o r t o n , A. Goyal, J. D. Budai, D. K. Christen et al., Epitaxial Y B a C u 0 on Biaxially Textured Nickel (001): A n Approach to Superconducting Tapes with High Critical Current Density, Science, 2 7 4 , 755-57(1996). V . S u b r a m a n y a Sarma, J. Eickemeyer, L. Schultz and B . Holzapfel. Recrystallisation Texture and Magnetisation Behaviour of Some F C C N i - W Alloys, Scripta Mater., 5 0 , 953-57(2004). .I.V.Gervasyeva,D.P.Rodionov,B.K.Sokolov and Y.V.Khlebnikova. Effect of Deformation Texture C o m p o n e n t Composition on C u b e Texture Formation during Primary Recrystallization in Ni-Based Alloys, Mater.Sci.Forum, 4 9 5 - 4 9 7 , 1213-18(2005). x
2
2
3
7
3
4
5
A . O . Ijaduola, J.R. T h o m p s o n , A. Goyal, C.L.H. T h i e m e et al.. Magnetism and Ferromagnetic Loss in N i - W Textured Substrates for Coated Conductors, Physica C, 4 0 3 , 163-71(2004). V. S u b r a m a n y a Sarma, J. Eickemeyer, A . Singh, L. Schultz et al.. Development of high strength and strongly cube textured N i - 4 . 5 % W / N i - 1 5 % Cr composite substrate for coated conductor application, Acta Mater., 5 1 , 4919-27(2003).
6
7
J . X . Zhang, X.Y. Song, D.M. Liu and M. Yue, In P o w d e r metallurgical high performance materials, 16th International Plansee Seminar 2 0 0 5 , Reutte, Austria, 2 0 0 5 . p 9 7 . X . Y . Song, J.X. Zhang, M . Yue, E.D Li, H. Z e n g , N . D . Lu, M.L. Z h o u and T.Y. Zuo,Technique for Preparing Ultrafine Nanocrystalline Bulk, Materialof Pure Rare-Earth Metals, Adv. Mater. 1 8 , 1210-12(2006). D . M . Liu, F. H a o , M.J. Li, Y C . Hu, F. Gao and M.L. Zhou, Study on the texture of N i - W substrates for high temperature superconductor, Materials Science and Technology, 2 1 . 1387-91(2005). D . M . Liu, F. H a o , J.X. Z h a n g , Y C . Hu, and M . L . Z h o u , Investigation of cube-textured N i - 7 a t . % W alloy substrates for YBa2Cu307.5 coated superconductors, submitted to "Frontiers of Materials Science in China". 8
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Materials Processing and Texture
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TWO-POINT ORIENTATION COHERENCE FUNCTIONS: AN ORTHONORMAL SERIES SOLUTION Peter R. Morris 1276 Oakmont Dr., Hamilton, OH 45013, USA ABSTRACT An orthonormal series solution for the two-point orientation coherence function is proposed using functions orthonormal on the range r = 0,1 with weight function corresponding to the distribution of r in a typical experimental procedure. INTRODUCTON In 1960, A.S. Viglin suggested representation of the distribution of crystallite orientations in a series of generalized spherical functions (harmonics). In 1965, H. J. Bunge and R J. Roe explained how coefficients of generalized spherical harmonics of even-order L could be obtained from those of spherical surface harmonic expansions of metallurgical pole figures. Bunge subsequently published a text in German on his method This text has been translated into English . In 1987, Mamies, Vine! and Helming noted that the coefficients of generalized spherical harmonics of odd-order L could not be obtained from spherical surface harmonic expansions of metallurgical pole figures, the absence of such coefficients leading to "ghosts." The coefficients of odd- order L may be obtained with single-crystal orientation measurements using an electron microscope. See, e.g., Wang'. While the ODF (Orientation Distribution Function) has met with some success in relating single and polycrystal elastic and magnetic properties, it has not done as well where plastic properties are concerned. This may be due to the lack of position information It might seem that position information could be incorporated into the ODF by 3 additional coordinates This assumption proves incorrect, as shift of origin generally results in a different distribution 1
2
3
4
5,6
7
Two-point Orientation-cohérence (correlation) Function The two-point orientation coherence (correlation) function (OCF), f(g,r,g') describes the probability that g and g' are orientations associated with the end points (tail and head) of a vector r, randomly located within a potycrystal. This function is similar to the two-point probability density described by Kröner . 9
Series Representations Following Bunge and Roe , generalized spherical harmonics seem appropriateforrepresenting the orientations g and g', and spherical surface harmonics for the angular dependence of r. The choice of functions for representation of r = | r | is less obvious. Description of fl(g,r,g') within a spherical sample volume is assumed Current techniques involve circular sections along successive parallels of longitude. We seek functions of r orthonormal on the range 0, 1, with weight function corresponding to the frequency of occurrence of r within the circular sample section. Such functions were considered in 2000 by Morris . 2
3
10
Weight Function If we denote the longitude of the vector r by ij, and the colatitude by χ, η is determined by the angle of x' from the χ toward the y sample direction of the particular section, whose surface is determined by x> and ζ. χ is determined by the angle from the common ζ axis to the vector r. x, y, and ζ are presumed uniquely defined by sample symmetry. For given η, χ, we seek to characterize g and g'
759
T w o - P o i n t Orientation C o h e r e n c e F u n c t i o n s : A n O r t h o n o r m a l S e r i e s Solution
as a function of r. To do mis, we inscribe a square grid, within a circular section of longitude n, with lines perpendicular and parallel to the direction of r specified by η and χ. We may consider circular sections of unit diameter, without loss of generality. The results may be scaled to conform to actual dimension. We consider a vector of length r, where 0 < r < 1. We consider one-fourth of a circle, bounded by radii perpendicular and parallel to r. Initially the head of the vector is placed at X„ Y,, where r is parallel to the Y axis and Y, = r/2. (We consider only vectors of length corresponding to an integral multiple of the distance between successive grid points.) X and Y are local coordinates corresponding to a particular choice of η and χ and should not be confused with χ and y sample coordinates. For given r, we wish to determine the number of grid points which the head of the vector r may occupy within the quarter circle. In this process, we restrict Y to grid points > r/2 (Y < r/2 can be considered by equivalent movement of the tail of the vector r in the lower quarter circle.) If we introduce sucessively finer grids, in the limit, where the separation between grid points becomes irifinitessimaL the positions which the head of the vector r may occupy within the quarter circle describe the area A defined by r/2 < Y < 1/2, i.e., f
2
1/2
(
2 112
A = j'^(l/4 - Y ) d Y = [π/2 - r(l - τ )
- sin-'(r)]/8
(1)
The probability of finding a vector of length r is thus proportional to the quantity in rectangular parentheses. We seek to normalize this probability so that the integral over all possible r is unity. To this end,
J^-rO-r^-sinHryjdr = [πτ/2 + (1 - r ) /3 - r sin" (r) - (1 - r )" ] J = 2/3. 2
3/2
1
2
2
(2)
Dividing the integrand by this result gives the probability, P(r)dr, offindinga vector of length r between r and r + dr as 2
2
P(r)dr = [3/2][π/2 - r(l - r )" - sin'(r)]dr.
(3)
We desire functions, orthonormal on the range r = 0, 1, with weight function P(r)dr given by Eq.(3). It is possible to employ a change of variable in functions orthonormal on the range r = 0,1 with weight function dr, to produce functions orthonormal on the range r = 0,1 with weight function P(r)dr given by Eq.(3). If we set 2
2
dw = [3/2][π/2 - r(l - r )" - sin-'(r)]dr, then 2
2
w = [3/2]f [π/2 - r(l - r ) " 2
3/2
(4)
sin-'iOJdr + C 2
,/2
1
= 3*r/4 + (1 - r ) /2 - 3(1 - r ) / 2 - 3r s i n ^ + C.
(5)
Let C = 1 Then w = 0 whenr = 0 and w = 1 when r = 1. Thus, if we have a series of functions orthonormal on the range r = 0, 1, with weight function dr, replacing r by w and dr by dw yields a series of functions orthonormal on the range r = 0, 1 with the desired weight function. In a previous paper , the spherical Bessel functions were selected as functions orthonormal on the range r = 0,1 with weight function dr. The normalized Legendre polynomials, J* (u), more familiar and accessible", may be made to serve this purpose by substitution of u = 2r -1, and multiplication by ( 2 ) , i.e., = 2" P (2r -1). 7? (r) may be written 10
L
w
2
L
760
L
Materials Processing and Texture
T w o - P o i n t O r i e n t a t i o n C o h e r e n c e Functions: A n O r t h o n o r m a l S e r i e s Solution
2
L
S (r) = ( 2 L + l ) > ' S „ a u r
0
(6)
L
The a are given in Table I through L = 16. u
Effects of Weight Function and Change of Variable Consider the expansion of a Dirac delta function, 5(r - r ), in terms of the Ä (r) defined above. 0
L
*(r-ro) = Z L b i A ( r ) .
(7)
Multiplying both sides of Eq. (7) by S (r)dr and integrating from r = 0tor =1, gives N
* n 0 - o ) = <5ln°l>
(8)
where 5 is the Kronecker delta, i.e., 5 = 1 if L = Ν and 5 = 0 if L Φ Ν. Thus, b = S (r ). We next expand a Dirac delta function, iS(r - r ), in terms of R [w(r)]. L N
L N
L N
0
N
N
0
L
« ( ' - O )
=Z l M i N ' »
(9)
Multiplying both sides of Eq. (9) by R [w(r)]dw and integrating from w = 0 to w =1, gives N
>Uw(r )]=<S b . 0
LN
(10)
L
10
Thus b = 7? [w(r )]. [The integration in a previous paper should have been carried out with respect to w, rather than r, which ledtothe factor P(r) on the left hand side of Eq. (17).] The peak height (ph.) and half width (h.w.) of $ (r) and ( ) expansions of Dirac delta functions, <5(r - r„), where r = 0.1, 0 . 2 , a n d Ν = 0,1 16 are compared in Table Π. Peak height and half width, h.w.(w) are fairly consistentforr„ = 0.1 to 0.5. N
N
0
w
N
0
Orthonormal Series Solution The proposed orthonormal series solution is given (in Roe notation) by:
^^7.w#r'f)-{Zi'..Zl. Ei. z -(i)e^e^}{^. ^.,^,0{) J
e-^ [w(r)]} {Σ1 r
0
4
0
l
Σ Ι . . E ; . , Z J Î ' ) e ^ e^ W ^ .
(11)
Braces have been used to separate representations of g, r, and g'. The W coefficients are complex, of form A + i B. A general termforrepresentation of g has the form ' w(g) = 2Z (i5)[A (cos my cos - sin my sin ηφ) + B ^ s i n my cos ηφ + cos my sin n^) + 2(-l)' "Zi-iiiXA^icos my cos n^ + sin my sin n^) - B^isin my cos n<> - cos my sin ηφ)], (12) 3 12
taoi
llI1I1
+
ly2
s
5
where Α μ ^ ο ο ^ , , , = (2) /(128)r ). "Times random" = 256jr w(g,r,g'). For representation of g', Imn are replaced by stu, and ξψφ by ξ'ψ'φ'. A general term for representation of r has the form
Materials Processing a n d Texture
·
761
T w o - P o i n t Orientation C o h e r e n c e F u n c t i o n s : A n O r t h o n o r m a l S e r i e s Solution
Table I Coefficients a
11
0
-1
6
-12012
1
132
9
355655300
2
7
3432
2
-4290
10
-327202876
0
1
3
60060
11
194699232
-72
4
-450450
12
-67603900
13
10400600
1
1
0
1 0
1
1
3
4
5
6
7
•
16632
0
2
762
5
0
7
L
au
au
η
of r° in R ( r ) η
η
L
u
L
L
8
1
-6
a
u
L
n
-1
13
8
2
6
2
1260
5
2018016
0
-1
3
-9240
6
-5717712
1
12
4
34650
7
2
-30
5
-72072
3
20
6
84084
0
1
7
-52480
8
12870
0
-1
1
-20
2
90
9
12
a
u
-261891630
0
1
10501920
1
-210
8
-12471030
2
10920
9
9237800
3
-247520
10
-3879876
4
3063060
11
705432
5
-23279256
0
1
6
116396280 -3399072960
14
3
-140
1
90
1
-156
7
4
70
2
-1980
2
6006
8
960269310
0
-1
3
18480
3
-100100
9
-1636014380
1
30
4
-90090
4
900900
10
1963217256
2
-210
5
252252
5
-4900896
11
-1622493600
3
560
6
-420420
6
17153136
12
878850700
4
-630
7
411840
7
-39907296
13
-280816200
5
252
8
-218790
8
62355150
14
40116600
9
48620
9
-64664600
0
-1
0
1
10
42678636
1
240
0
1
1
-42
10
15
2
420
1
-110
11
-16224936
2
-14280
3
-1680
2
2970
12
2704156
3
371280
4
3150
3
-34320
0
-1
4
-5290740
5
-2772
4
210210
1
182
5
46558512
6
924
5
-756756
2
-8190
6
-271591320
0
-1
6
1681680
3
160160
7
1097450640
1
56
7
-2333760
4
-1701700
8
-3155170590
13
2
-756
8
1969110
5
11027016
9
6544057520
3
4200
9
-923780
6
-46558512
10
-9816086280
4
-11550
10
184756
7
133024320
11
10546208400
Materials Processing and Texture
T w o - P o i n t O r i e n t a t i o n C o h e r e n c e F u n c t i o n s : A n O r t h o n o r m a l Series Solution
Table I (continued) L n au L η -542640 16 10 15 12 -7909656300 16 3 L
η
3931426800
4
8817900
14 -1163381400
5
-88884432
12
155117520
6
597500904
13 -38003792400
0
1
7
-2804596080
14
17450721000
1
-272
g
9465511770
15
-4808643120
2
18360
9 -23371634000
16
601080390
13 15 16
>u 42536373880
11 -56949525360 55367594100
Table H. ß (r) and S (w) expansions of Dirac delta functions, δ(τ - r ), where r =0.1,0.2,... and Ν = 0, 1,.... 16. N
N
0
0
»n(w)
18.9
.064
o p.h. h.w.(w) .221 13.2 .092
0.2 13.0
.093
.411 11.3
0.3 11.9
.102
.573 11.0
.109
0.4 11.3 0.5 11.1
.107
.706 11.8
.102
.108
.811 13.6
.089
0.6 11.3
.107
.891 18.0
.067
0.7 11.9
.102
.946 25.7
.047
0.8 13.0
.093
.980 44.0
.027
0.9 18.7
.064
h 01
p.h. h.w.(r)
w
.107
w(r) = 2 ^(ORrMrMtA^cos q„ + B „ s i n qn]
(13)
The implementation of Eq. (11) seems a daunting task. To achieve resolution comparable to that of the ODF, the numbers of coefficients and terms are estimated to be the cube of those required for the ODF. The interpretation of a nine dimensional function is not at all obvious. The coefficients of the series form a sort of one dimensional array. Perhaps a search for the largest coefficients and development of the terms associated with those coefficients might offer some understanding. CONCLUSIONS An orthonormal series solution is proposed for the two-point orientation coherence (correlation) function, f{g, r, g'), where g and g' are orientations at the tail and head of a vector r randomly located within a polycrystal. Generalized spherical harmonics are suggested for representation of g and g' and spherical surface harmonics for the direction of r. For representation of r = |r |, functions are suggested which are orthonormal with weight function corresponding to the probability of the occurrence of a vector of length r in a typical experimental procedure.
Materials Processing and Texture
763
T w o - P o i n t Orientation C o h e r e n c e Functions: A n Orthonormal Series Solution
REFERENCES Ά S Viglin, Fizika Tverdogo Tela 2, (I960) 2463-2476. H.J. Bunge, Ζ. Metallkde 56, (1965) 872-874. 2
3
R J Roe, J Appl. Phys.36, (1965) 2024-2031.
4
H.J. Bunge, "Mathematische Methoden der Texturanalyse", Akademie-Verlag, Berlin (1969). H.J. Bunge, "Texture Analysis in Materials Science: Mathematical Methods" Butterworths, London(1982) H. J. Bunge, "Texture Analysis in Materials Science: Mathematical Methods," Cuvillier Verlag, Crörangen (1993). S. Matthies, G.W. Vinel and K. Helming, "Standard Distributions in Texture Analysis," Akademie-Verlag, Berlin (1987). T.T. Wang, "Evolution of the Two-Point Orientation Coherence Function in Plane-Strain Deformation," Phd. Dissertation, Brigham Young University (1988). Έ . Kröner, "Statistical Modelling," in "Modelling Small Deformations of Polycrystals," J. Gittus and J, Zarka, Eds., Elsevier Applied Science, London (1986) 229-291. P.R- Moms, Textures and Microstructures, 3 4 (2000) 233-241. "J. Pospiech and J. Jura, Kristall und Technik, 10(1975)783-787 5
6
7
8
10
12
764
P. R. Morris andD. C. Koskenmaki, Textures ojCrystalline
·
Materials Processing a n d Texture
Solids,!
(1977) 253.
T H E I N F L U E N C E O F S H E A R B A N D S ON M I C R O T E X T U R E E V O L U T I O N IN POLYCRYSTALLINE COPPER
H. P a u l '
2
'institute of Metallurgy and Materials Science, P A S , Krakow, Poland 2
University of Zielona Gora, Mechanical Department, Poland
J.H. Driver 3
3
E c o l e des M i n e s de Saint Etienne, Centre S M S , France
ABSTRACT Periodic crystal lattice rotations within compact clusters of shear bands (SB), developed in pure copper, have been characterized to examine the role of lattice re-orientation within grains on slip propagation across grain boundaries. Polycrystalline pure copper (grain size 50pm) was deformed 5 0 % in plane strain compression at room temperature to form two sets of well-defined macroscopic shear bands ( M S B ) . T h e deformation-induced sub-structures and local changes in crystallographic orientations were investigated by F E G - S E M , equipped with high resolution E B S D . In all the deformed grains examined (within M S B s ) a strong tendency to strain-induced re-orientation could be observed. Their crystal lattice rotated in such a way that one of the {111} slip planes b e c a m e nearly parallel to the direction of m a x i m u m shear. A natural consequence of this rotation is the formation of a specific M S B microtexture {100}<110> which facilitates slip propagation across grain boundaries without any visible variation in the slip direction although the slip plane did not coincide exactly in the adjacent grains.
INTRODUCTION It has been well k n o w n for decades that plastic deformation of metals is not homogeneous but concentrated in different kinds of inhomogeneities of plastic flow. Shear bands (SB) or their compact clusters, called macroscopic shear bands ( M S B ) , are frequently observed examples of unstable behavior of fee metallic materials at large strains, e.g. [1], However, their formation and development within the as-deformed structures and their influence on the overall texture evolution are still not completely understood. The crystallographic aspects of shear banding in single crystals of medium-high stacking fault energy (SFE) metals have been analyzed in the past, e.g. by Wagner et al. [2] using X-ray diffraction and for low SFE metals work by Paul and Driver [3, 4] using local orientation measurements in T E M and S E M . The generally accepted facts associated with shear banding within fee single crystals are as follows: • Shear banding is closely related to the mechanical anisotropy of the pre-existing microstructure. This suggests that shear banding is preceded by the formation of obstacles to
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homogeneous dislocation glide in the crystallite. The formation of these obstacles is strongly influenced by the crystallography and SFE. • T w o groups of SB can be distinguished. If the obstacles are fine twin-matrix lamellae, typical for metals with low SFE [3], the SBs are classified as brass-type. If the precursory obstacles are the elongated dislocation walls of a cell block structure the shear bands are of the copper-type. They are typically observed in materials with high or medium SFE [5], • In the both cases rotation-induced mechanical instability within narrow areas of the anisotropic structure of elongated cells or twin-matrix layers (and kink-type bands formation), leads to the formation of SBs. The orientation changes induced by shear banding have been studied in fee polycrystals by both X-ray diffraction, e.g. Weider & Klimanek [6], D a Costa Viana et al. [7] and E B S D technique, e.g. Inagaki et al [8] or Huot et al [9]. However, in polycrystalline metals the situation can be quite complicated. In this case macroscopic shear b a n d s very often cross the grain boundaries without any significant change in shear direction. F r o m the point of view of crystallography the requirement of slip propagation across grain boundaries leads to some basic questions about the m e c h a n i s m s responsible for slip system organization along traces of the shear plane within differently oriented grains situated inside the sheared zone. In this context, the main debate centers on whether slip within an MSB is crystallographic or non-crystallographic, i.e. whether the shear occurs on {111 {-type planes and in <110>-type directions. A second question relates to how the SB can influence the global texture development. In this work, the (micro)texture development in pure polycrystalline copper has been investigated in order to characterize the influence of local lattice re-orientations within particular grains on slip propagation and the formation of macroscopically visible clusters of copper-type shear bands. Computer-automated electron backscattered diffraction ( E B S D ) is a particularly suitable method of investigating the p h e n o m e n o n .
MATERIAL AND METHODS 3
Polycrystalline copper (99.98%) samples of size 1 0 x 1 0 x 1 0 m m were deformed in plane strain compression to a thickness reduction of 5 0 % at a strain rate of ~ 1 0 " V . The experiment was performed in two stages using markers to locate the M S B s . The initial sample with dimensions was channel-die compressed 2 4 % . Then, on the longitudinal face ( N D - E D plane, in which: N D and ED were normal and extension directions, respectively) three scratches along ED were made ( Fig. 1 a). The sample was further compressed up to - 5 0 % , at which clearly defined M S B were visible ( F i g . l b ) . The deformation-induced dislocation structures and local changes in crystallographic orientations were investigated by S E M - J E O L J S M 6500F equipped with a field emission gun and facilities for electron backscattered diffraction ( E B S D ) . To reveal the crystallographic contrast backscattered electrons operating at 20kV were used. The microscope control, pattern acquisition and solution were carried out with the H K L Channel 5 system. For m o r e global (i.e. sample) scale microstructure observations an optical microscopy w a s used on mechanically and chemically polished samples. Additionally, the texture variation in the global scale were analysed by X-ray diffraction from the as-received to final as-deformed states.
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Figure 1. M S B formation and shear deformation in copper samples deformed in channel-die. (a) initial position of scratches after 2 4 % reduction, (b) bending of the scratches within MSBi and M S B 2 after further 2 5 % deformation up to a final 5 0 % reduction. Optical micrographs on longitudinal ( N D - E D ) plane.
RESULTS AND DISCUSSION Macroscopic changes at the sample scale The microstrucUire of the as-received copper consisted of nearly equiaxial grains with an average grain size of ~ 5 0 p m . T h e orientation distribution, determined by X-ray diffraction, revealed a relatively w e a k texture without any significant tendency for peak texture components. After deformations of 2 4 % and 5 0 % , the global texture m e a s u r e m e n t s clearly indicated a systematic formation of stronger c o m p o n e n t s close to the two complementary positions of brass,' 110) < 1 1 2 > with a scattering t o w a r d s S{ 123}<634> and copper{112}<111> (Fig.2).
Figure 2. X-ray diffraction {111) pole figures showing global texture development measured on the compression plane: (a) in initial material, (b) after 2 5 % and (c) 5 0 % compression in channel die.
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After the higher deformation, clearly visible differences were observed in the intensity of plastic flow due to successive strain localizations within M S B s . This w a s especially well pronounced at a final reduction of 5 0 % . where the M S B s formed a characteristic V shaped set of two families (Fig 1 b). The width of each set w a s 1.5-2mm and they w e r e positively and negatively inclined at - 4 5 " to E D . T h e scratches made on the longitudinal plane (along E D ) showed a well-defined rotation (Fig.3a), of opposite sign within each set of bands. This rotation occurs with the increasing inclination of the line segments crossed by M S B s . T h e values of the rotation angles inside M S B i and M S B attain ~ ± 20° (Fig.3b). The inclination of the lines decreased outside the bands. Clearly the macroscopic rotation observed within both families of bands influences the crystal lattice rotations of grains situated within the M S B volumes. 2
Figure 3. (a) C h a n g e s in the inclination angle (a) of the scratches m a d e on longitudinal plane and (b) values of tg(a) along E D for particular lines. X axis is distance in m m . Sample deformed 50%.
Slip propagation across grain boundaries Figure 4 s h o w s the E B S D grain boundary map from a representative region of localized shear in the compressed 5 0 % copper sample . M o r e or less parallel bands are the important features of the M S B deformation microstructure, with a crystallographic orientation different from the surrounding matrix. The microtexture analysis of the whole m a p shows the formation of two nearly complementary brass) 110}<112> orientations with scattering towards S { 1 2 3 } < 6 3 4 > and Cu {112}<111> c o m p o n e n t s . However, this explains neither the m e c h a n i s m s responsible for slip propagation across grain boundaries nor slip organization inside the M S B . Figure 5a shows the orientation m a p taken within the deformed microstructure of an M S B . in which microbands cross a grain boundary. As a result, characteristic steps are formed on the boundary indicating large shear strains due to localized slip associated with microbands. The height of each step in the band direction might be different, depending mainly on the width
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of the microband, and the values attain in some cases l - 2 p m . H o w e v e r , an obvious question concerns the way in which the slip propagates across grain boundaries, i.e. whether the stress concentration near the grain boundary leads to slip o n {111} plane in < 1 1 0 > directions in the neighbouring grain.
Figure 4. (a) Grain boundary' orientation m a p showing microstructure within highly localized regions of M S B and (b) corresponding {111} pole figure. S E M - F E G / E B S D measurements with step size of 2 0 0 n m .
Figure 5. (a) Orientation map showing microbands crossing grain boundary, (b) and (c) {111} pole figures corresponding to orientations of grains A and Β and (d) and (e) stereographic projections presenting situation of the active slip systems within both grains. S E M - F E G / E B S D measurements with step size of 200nm.
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A detailed analysis of the slip traces within both grains (A and B) s h o w e d that they could be related with { 1 1 1 ! planes. An ' a v e r a g e ' orientation of grain A could be described as near (4 5 1 )[ 111 ] or in Euler angles ( 144, 8 1 , 141 ), whereas grain Β is near (2 7 5)[ 111 ] ( 13 5. 5 5 . 164). The grain orientations were misoriented by a ~33°<111> rotation. Although the traces of the bands observed in the longitudinal plane were nearly parallel, in fact the {111} planes, w h i c h were important for the analysis in each grain were only slightly misoriented (Fig.5b and c). The situation can be illustrated by stereographic projections for the ' a v e r a g e ' orientations of the grains in the ( 2 1 3 ) plane for grain A and in the ( 4 1 3 ) plane for grain B . It is clearly visible that band formation resulted from the operation of two pairs of co-planar slip systems: ( Ϊ 1 Ϊ )[ 110]+[0ï Î ] (A) and ( 1 ί 1 )[ 110]+[0î Î ] (B). In the both cases the [ 121 ] axis ('direction of the resultant slip s y s t e m ' ) coincided with the shear direction. In the longitudinal section the traces of the ,111} planes for both grains were situated along one line and corresponded to the observed traces of the bands. T h i s situation is schematically presented in Fig.6. A similar case is shown in Fig.7a. where adjacent grains were penetrated by S B s and formed compact clusters of MSB.
Figure 6. Schematic presentation of situation analysed in Figure 5.
Local lattice rotation vs. slip organization within macroscopic shear bands The a b o v e mechanism of slip propagation across grain boundaries is essential for the organization of microshear bands within M S B s . T h e accumulation of SBs into bundles and their propagation through grain boundaries is an important problem in the process of M S B formation. The sharp crystallographic texture development, observed at increasing deformation, favors the penetration of slip in the M S B area through the neighboring grains. The situation is simple when neighboring grains have a similar orientation, and the {111} planes coincide with the plane of m a x i m u m shear stress. Slip penetration, however, occurs in regions of quite different orientations. Nevertheless, from the crystallographic point of view, the existence of a c o m m o n plane for both areas is required; it is along this plane that slip can penetrate the boundary. This was clearly visible within the grains lying inside the M S B . In all analyzed grains a strong
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tendency to grain subdivision and strain-induced re-orientation w a s observed. Their crystal lattice rotated in such a w a y that one of the {111} slip planes b e c a m e nearly parallel to the direction of the m a x i m u m shear. A natural consequence of this rotation is the formation of a specific M S B microtexture which facilitates slip propagation across grain boundaries along the shear direction without any visible variation in the slip direction. The possibility of local reorientation of the crystal lattice as a result of SB formation in single crystals of metals with fee lattice and low S F E has been demonstrated earlier, e.g. by Paul et al [3].
Figure 7. (a) Orientation m a p showing compact clusters of shear bands crossing grain boundaries, (b) - (d) {111 j pole figures corresponding to the orientations of grains 1 - 3 . (e) - (g) stereographic projections presenting the situation of the active slip systems within all grains. S E M - F E G / E B S D m e a s u r e m e n t s with step size of 200nm.
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Figure 7a. s h o w s three adjacent grains penetrated by b a n d s of strongly localized strain. The orientations of particular grains are shown in Figs. 7(b-d). It is again clearly visible that their crystal lattice rotated in such a way that one of the [111} slip planes became nearly parallel to the direction of m a x i m u m shear although the orientations of grain 1 and 2 were quite different from that of grain 3. Additionally, in each case one of the <01 l > - t y p e directions, lying in these planes, systematically tended to coincide with shear direction. This leads to the important conclusion that macroscopically observed shear plane in fact consists of small parts limited to particular grains (or their fragments) (Fig.8). These parts were only slightly deviated from the macroscopic shear plane ( M S P ) . It is important to note that the higher the strain the smaller the deviations from the M S P s (built up compact clusters of S B ). In Fig.7a the SBs are indicated by black arrows and are inclined 3 5 - 4 5 " to E D . Across these shear bands the crystal orientation changed periodically and the accumulated point-to-origin misorientations varied by 35-40° but their axes were close to one of the < 112> poles.
T h e change of the dominating slip system The dominant slip system often changed within the highly localized areas of S B . The main reason for this phenomenon is the systematic crystal lattice rotation. This process quite often led to the appearance of orientations which are symmetrical with respect to the external axes. In the case of metals with middle-high SFE they could be described by orientations close to {100} < 0 1 1 > . identified as a main c o m p o n e n t of the so-called copper-type S B [2,5]. In these orientations the old slip and the new one were nearly symmetrical. In the case presented in Fig.7a. significant parts of grains 1 and 2 represented these orientations. Therefore, a characteristic (of two sets) microband substructure in the form of low angle boundaries was observed. The traces of such slips lie symmetrically with respect to the shear direction.
Figure 8. Schematic presentation of the situation of active { 1 1 1 ] - type slip planes in relation to a macroscopically observed shear plane ( M S B plane).
Orientation changes within microband structures as a result of new system activation has also been observed by Dorner et al [10. 11] in bcc coarse-grained Fe-3%Si. In the case of M S B s this p h e n o m e n o n w a s observed by Paul et al [3.4] in highly deformed silver single crystal with
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{112}<111> orientation. However, in the case of low S F E metals this characteristic position was described by orientations close to Q o s s j l 10}<001>. T h e activation of n e w slip systems strongly disturbed the microbands or SB microstructures and is responsible for a n e w strong (although strictly defined) crystal lattice rotation. This is in agreement with the Schmid-Boas relationship for the activation of new slip systems.
CONCLUSIONS Microtexture measurements on polycrystalline Cu samples deformed in channel-die compression have been used to analyse the propagation of macroscopic shear bands. They showed that well defined crystal lattice re-orientations occurred in s o m e grains situated within the area of the broad M S B , although those grains initially had quite different crystallographic orientations. Their crystal lattice rotated in such a way that one of the {111} slip planes became nearly parallel to the direction of m a x i m u m shear. A natural consequence of this rotation is the formation of specific M S B microtextures which facilitates slip propagation across grain boundaries along the shear direction without any visible variation in the slip direction. It was thereby established that shear banding occurred across grain boundaries by the continuity of slip direction although the slip plane did not coincide exactly in the adjacent grains.
REFERENCES Έ . El-Danaf, S.R. Kalidindi, R.D. Doherty, C. Necker, "Deformation texture transition in α-brass: critical role of micro-scale shear bands", Acta Materialia, 4 8 , 2665-2673 (2000). 2
P . Wagner, O. Engler, K. Lücke, "Formation of Cu-type shear bands and their influence on deformation and t e x t u r e o f r o l l e d f . e e . {112}<111> single crystals", A eta Metall. Mater. 4 3 , 3799-3812(1995). 3
H . Paul, J.H. Driver, C. Maurice, Z. Jasienski, "The Structure of Shear Bands in Twinned FCC Single Crystals", Mat. Sei. Engn., A 3 5 9 , 178-191 (2003). 4
H . Paul, J.H. Driver, C. Maurice, A. Pia&owski, " T h e role of shear banding on deformation texture in low stacking fault energy metals as characterized on model A g crystals", Acta Materialia, 5 5 , 833-847 (2007). 5
H . Paul, M . Darrieulat, A. Pia.tkowski, "Local Orientation Changes and Shear Banding in {112}
A . Weider, P. Klimanek, "Shear banding and texture development in cold rolled α-brass", Scripta Materialia, 3 8 , 851-856 (1998). 7
C . S . Da Costa Viana, J.C. Parades, A.L. Pinto, Α . M Lopez, " E B S D analysis of shear banding in α-brass", in: J. Szpunar (Ed.), Proceeding of the 12th International Conference on Textures of Materials, Trans. Tech. Publ., Toronto, Canada. 671-676 (1999).
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H. Inagaki. M. K o i z u m i . C.S.T. C h a n g , B.J. Duggan, "Orientation imaging microscopy of the shear bands formed in A l - 5 % M g alloys during cold rolling", Mater. Sei. Forum, 5 8 7 - 5 9 2 , 396-402 (2002). 9
A . Huot, R.A. Schwarzer, J.H. Driver, "Texture of shear bands in A l - M g 3 % ( A A 5 1 8 2 ) measured by B K D " . Mat. Sei. Forum, 2 7 3 - 2 7 5 , 319-326 (1998). I 0
D . Dorner. Y. Adachi, K. Tsuzaki, "Periodic crystal lattice rotation in microband groups in a bcc metal", Scripta Materialia. 5 7 , 775-778 (2007). " D . Dorner. S. Zaefferer, D. Raabe, "Retention of the Goss orientation between microbands during cold rolling of an F e 3 % S i single crystal", Acta Mater., 5 5 , 2519-2530 (2007).
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I N F L U E N C E O F D I S L O C A T I O N S T R E S S ON T H E C O R R O S I O N B E H A V I O R O F (111) A N D (123) S U R F A C E P L A N E O F A L U M I N U M S I N G L E C R Y S T A L Dashi Pei, Weimin M a o , Huiping Feng Department of Materials, State Key Laboratory for A d v a n c e d Metals and Materials, University of Science and Technology Beijing Xue-Yuan Road 30, 100083 Beijing, China, (e-mail: [email protected]) ABSTRACT The corrosion behaviors of (111) and (123) surface plane of high purity aluminum single-crystal were observed after chemical etching. The amount of etching pits on (111) plane was higher than that on (123) plane, and the average distance of the etching pits were 29.13μιη and 3 7 . 3 7 p m on (111) and (123) plane respectively which should be larger than the average distance between dislocation ends where they emerged at the surface. The tensile stresses on (111) and (123) plane were calculated according to the stress fields of different edge and screw dislocations. It is deduced that the etching pits nucleate at the sites where the dislocations meet the surface, which is promoted by the tensile stresses around the dislocations. The tensile stress fluctuation induced by dislocation stress fields is stronger on ( 111 ) plane than that on ( 123), which induces higher nucleation rate on (111) surface plane. KEY W O R D S aluminum single crystal, corrosion behavior, dislocation stress, etching pits INTRODUCTION The surface area of a l u m i n u m foils for electrolytic capacitors can be largely increased by electrochemical corrosion, which leads to m u c h higher specific capacitance of the foils . The < 1 0 0 > etching tunnels will form during direct current corrosion, while the facet of the tunnel walls are c o m p o s e d of {100} planes ' . The surface energy Ε of aluminum crystals varies in the order E,no)> E ioo)> E n i ) , while the pitting potentials E , are in the order as Ε ;„ιοο) E i ) > Epjt( 11 D . Therefore, it w a s observed that the indices of surface planes have also certain effect on the corrosion b e h a v i o r . It is generally believed that the pit corrosion is initiated preferentially in the area around grain boundary, dislocation, or segregation of microelements ' , and factors including trace elements, grain boundary characteristics, dislocation density etc., would influence the corrosion behaviors of aluminum foils. However, the dislocation stress fields around the surface ends of the dislocations, where the corrosion begins, depend also on the miller indices of surface planes and should exhibit different corrosion behaviors, which have not investigated in details so far. High-purity single-crystal a l u m i n u m is used in present work to investigate the influence of dislocation stress field on the surface corrosion behavior of aluminum samples with different surface indices, while the interference of grain boundaries is avoided. u
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CORROSION EXPERIMENT OF SINGLE-CRYSTAL A L U M I N U M A l u m i n u m samples with (111) and (123) surface plane were cut from annealed single-crystal a l u m i n u m of 9 9 . 9 9 7 5 % in purity provided by the institute of physical metallurgy and metal physics of R W T H Aachen university, Germany. T h e miller index of surface planes were
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determined by m e a n s of the [111] pole figures measured by D-5000 X ray diffractometer. Firstly, the samples were electrolytically polished in order to remove the surface oxide film. T h e polished samples were put into the 1st erosion solution ( 5 0 % hydrochloric acid, 1 5 % nitric acid, 10% hydrofluoric acid and 2 5 % distilled water) for one minute, and cleaned with distilled water. Then, they were put into the 2nd erosion solution ( 5 % hydrofluoric acid and 9 5 % distilled water) for one minute and cleaned with distilled water. The mierostructures of the corrosion surfaces were observed under the scanning electron microscope ( S E M ) . The amount, distance and size of the etching pits w e r e counted or measured statistically in the same observation area (0.52 m m ) under the S E M . Figure 1 (a) and (b) give the [ 1 1 1 ! P ° l ° figures of the two samples, indicating the miller index of the surface plane with very' small deviations from the exact (111) and (123) (filled symbols). Figure 1 (c) and (d) give the S E M microstructure of the samples. The basic forms of the etching pits are indicated in the bottom right corners of figure 1 (a) and (b). 2
Figure 1 Pole figure and corrosion pit observation of the experiment samples
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It can be found according to the pole figures and the corrosion microstructure. that the pit walls are c o m p o s e d of {100} planes because of their low surface energy ' . The average distance of the etching pits are 2 9 . 1 3 p m and 37.37pm on (111) and (123) plane respectively. The corresponding average pit sizes are 184.2pm and 212.9μηι" respectively. Figure 2 gives the statistic analysis of the etching pits on (111) and (123) surface. The statistics data show that the total counts and therefore the density of etching pits o n (111) surface plane are higher than those on (123) surface plane, w h i c h mainly concentrated in the small size area (less than 300 μην ). The pit size distribution implies that the etching pits kept nucleating easily on (111 ) surface than those on (123) surface. H o w e v e r , the growth rate of the etching pits seems similar in the t w o samples under the identical corrosion condition, since no obvious difference on the counts of large size pits of the samples are observed. 2
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-Δ-(111) -•-(123) Average size: (111): 184.2 μ m (123): 212.9 μηΤ
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Size^m ) Figure 2
Statistic analysis of etching pits on (111 ) and (123) surfaces
CALCULATION OF STRESS FIELD A R O U N D THE SURFACE ENDS OF THE DISLOCATIONS A s s u m i n g that all dislocations in a l u m i n u m crystals are homogeneously distributed straight screw dislocations and straight edge dislocations, the screw dislocation lines will distribute in six different < 1 1 0 > directions and the edge dislocation lines will distribute in twelve different <112> directions. The average distance of dislocation ends on any surface plane (hkl) will be d /cos θ and cL/cosO for screw and edge dislocation respectively if the average distance of parallel screw dislocations and edge dislocations are d and d respectively, while θ is the angle between the dislocation lines and the [hkl] direction. It can be deduced diat the density of dislocation ends on the (hkl) surface plane will be s
s
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for screw dislocations and
Σ for edge dislocations, which is independent o n the miller index (hkl). The dislocation density of , while the average distance between annealed a l u m i n u m is approximately 1 0 ~ 1 0 / m dislocations falls into the range of O . l - l O p m , 10
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Figure 3 Distribution of the tensile stress around screw dislocation on (111) and (123) plane (Stress levels in M P a : 2 0 , 4 0 , 80. T h e sticks indicate directions of the stress gradient and the arrow indicates direction of the highest stress gradient around the dislocation) Only the tensile stresses in surfaces of die aluminum single crystal are considered to promote the nucleation of etching pits. The three-dimensional stress field of straight dislocations in dislocation coordinate system can be conversed in arbitrary coordinate system, while the tensile stress distribution on any (hkl) plane can b e calculated. T h e 6 straight screw dislocations can only induce one kind of tensile stress field and the 12 straight edge dislocations two kinds of tensile stress fields on (111) plane. H o w e v e r , there are 5 kinds of tensile stress fields for screw dislocation and 8 kinds for edge dislocations on (123) plane. Figure 3 and 4 give some examples of the calculations in (111) and (123) plane for straight screw and straight edge dislocations, where the interactions a m o n g the dislocations are neglected in the calculation. It is observed that the tensile stress decreases m o r e rapidly around screw dislocations with increasing distance to the dislocation cores.
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Figure 4 Distribution of the tensile stress around edge dislocation o n (111) and (123) plane (Stress levels in M P a : 20, 4 0 . 80. T h e sticks indicate directions of the stress gradient and the arrow indicates direction of the highest stress gradient around the dislocation) The stress components concerning the [hkl] direction will be reduced down to 0 at the (hkl) plane w h e n it b e c o m e s the surface of the samples, while the other stress components ate largely relaxed. Supposing that the relaxation a m o n g the stress c o m p o n e n t s are proportional each other, the distributions of the tensile stresses around dislocations would have the same forms shown in figure 3 and 4. but with m u c h lower stress levels. I N F L U E N C E O F D I S L O C A T I O N O N N U C L E A T I O N O F C O R R O S I O N PITS It h a s been observed in figure 1 and 2 that the density of etching pits in (111) and (123) plane is different from each other, though the density of dislocation ends should be theoretically the s a m e . T h e average distance 29.13pm and 37.37pm of the etching pits for sample (111) and (123) are higher t h a n the average dislocation distance 0 . 1 - 1 0 p m in annealed aluminum. That implies the etching pits should initiate mainly at the positions of dislocation ends on the sample surfaces. On the other hand, not all the dislocation ends could initiate etching pits. It may be deduced that the formation of etching pits is related with the stress field of dislocation ends on the sample surface. It is k n o w n , that tensile stress will promote the surface corrosion ' ' . T h e tensile stress distribution of edge dislocation with higher elastic energy covers larger area around the dislocation ends, which may lead to m o r e active effect of the edge dislocations on the etching pits nucleation. Higher tensile stress drop in same distance should promote the stress induced pit corrosion m o r e actively. Figure 5 s h o w s the tensile stress 10
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distributions around edge dislocations along the highest stress gradient lines, e.g. indicated in a i T o w directions in figure 4. The tensile stress of some edge dislocations on (123) plane could not drop d o w n to very low level at the distance of" 5μηι to the dislocation cores, if the average dislocation distance of 0.1~10μηι in annealed aluminum is considered. Therefore, the tensile stress drop around many ends of edge dislocations on (123) plane might be not high enough to promote the nucleation of etching pits. That could be the reason that the (123) exhibited lower pit density (figure 2).
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Figure 5 Tensile stress distributions around edge dislocations along the highest stress gradient lines on (111) and (123) plane It should be noticed that the image force will induce the configuration rearrangement of edge or screw dislocations near surface, after which they may b e c o m e mixed dislocation " . Therefore detailed analysis in this aspect as well as the relaxation levels of the surface stresses need to be conducted. CONCLUSIONS The density of dislocation ends where they meet crystal surface is the s a m e for all {hkl} surface planes of a l u m i n u m single crystals. The etching pits induced by chemical corrosion should initiate mainly at the sites where the dislocation ends meet the surfaces. H o w e v e r not all dislocation ends will become nucleation sites. The formation of etching pits is related with the tensile stress level and the stress fluctuation o n the sample surface. T h e dislocation stress field on (111) surface indicates stronger fluctuation than that on (123) surface, which would result in larger potential difference and higher nucleation rate of etching pits o n (111) surface than that on (123) surface. The study can be extended to other crystal planes which may lead to more general conclusion.
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Influence of D i s l o c a t i o n S t r e s s o n t h e C o r r o s i o n B e h a v i o r of A l u m i n u m Single Crystal
It is indicated that the tensile stress fields of dislocations are related to the miller indices of crystallographic planes. Therefore, textures of polycrystalline a l u m i n u m will also influence the surface corrosion behaviors. The research works in this aspect need to be done. ACKNOWLEDGEMENT This study w a s supported by the National Natural Science Foundation of China (Grant N o . 5 0 5 7 1 0 2 0 ) and the Doctoral Foundation of the Education Ministry of China (Grant N o . 2 0 0 4 0 0 0 8 0 1 0 ) . They also like to thank Prof. G. Gottstein and Dr. W. Hu of R W T H Aachen Germany for supplying the a l u m i n u m single crystal sample. REFERENCES N . O s a w a , and K. Fukuoka, Pit nucleation behavior of a l u m i n u m foil for electrolytic capacitor during early stage of D C etching, Corrosion Science, 4 2 , 5 8 5 - 5 9 7 ( 2 0 0 0 ) . B. W. D a v e s , P. J.Moran, and P. M. Natishan, Metastable pitting behavior of aluminum single crystal. Corrosion Science, 4 2 , 2 1 8 7 - 2 1 9 2 ( 2 0 0 0 ) . J. H. Seo, J. H. Ryu, and D. N . L e e , Formation of Crystallographic Etch Pits during A C Etching of A l u m i n u m . Journal of The Electrochemical Society, 1 5 0 ( 9 ) , B 4 3 3 ~ B 4 3 8 ( 2 0 0 3 ) . J. Jeong, C H . Choi, and D. Y. Lee D Y. A model for the < 100 > crystallographic tunnel etching of A l u m i n u m , Journal of Materials Science, 3 1 , 5 8 1 1 - 5 8 1 5 ( 1 9 9 6 ) . G. M. Treacy, and C. B. Breslin, Electrochemical studies on single-crystal aluminum surfaces, Electrochemical Acta, 4 3 , 1 7 1 5 - 1 7 2 0 ( 1 9 9 8 ) . L. W. Shu, W. M. M a o , H. P. Feng, Y. N . Yu, and J. Xu. Recrystallization and grain growth during annealing of cold rolled low voltage electronic a l u m i n u m foil. The Chinese Journal of Nonferrous Metals, S I . 1 7 8 - 1 8 2 ( 2 0 0 2 ) . 1
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W. M . M a o , L. Chen, L. and M . Sa. Influence of grain boundaries on corrosion structure of low voltage a l u m i n u m foil, The Chinese Journal of Nonferrous Metal, 1 4 ( 1 ) , 1-5(2004). J. B. Song, W. M. M a o , H. Y a n g , and H. P. Feng. Influence of Micro-additive Sn on Recrystallization Texture of High Voltage A n o d e A l u m i n u m Foil. Heal Treatment of Metals, 3 2 ( 7 ) , 1-4(2007). 8
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L. J. Qiao, Mechanics of stress etching, Science Press, 1993. L. F. M o n d o l f o , A l u m i n u m alloys, Metallurgical Industry Press, 1988. " D. Li, W. M. M a o , and Y. N.Yu, Stable dislocation configuration in the surface layer of metals, Chinese Science Bulletin, 5 2 , 1 8 6 4 - 1 8 6 6 ( 2 0 0 7 ) . 1 0
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T E X T U R E R A N D O M I Z A T I O N BY I N T E N S E S H E A R I N G IN L A Y E R E D Al A N D Al (0.3%Sc) A R B - P R O C E S S E D S H E E T
M.Z. Quadir 1, O. A l - B u h a m a d 2, L. B a s s m a n 2 , 3 and M . Ferry 2 Electron Microscope Unit, University Analytical Centre, University of N e w South Wales, Sydney, Australia. 1 A R C Centre of Excellence for Design in Light Metals, School of Materials Science and Engineering, University of N e w South Wales, Sydney, Australia. 2 Department of Engineering, Harvey M u d d College, Claremont, C A , U S A . 3 ABSTRACT High purity Al sheet w a s roll bonded with Al containing 0 . 3 % Sc in the supersaturated solid solution (plus aged) condition, to up to five A R B cycles. The layers in the roll bonded samples remained reasonably straight w h e n bonding w a s achieved in the supersaturated solid solution condition of Al(Sc), but became curved by intense shearing w h e n pre-aged Al(Sc) was roll bonded. This behaviour was explained by differential strengthening characteristics of the candidate materials. In both Al and Al(Sc) layers, the deformation substructures are dominated by well-developed lamellar bands of different fineness; in Al and Al(Sc) layers the band thickness are 0.7-1.00pm and 0.2-0.4pm, respectively. Extended annealing for 6h at 350 °C led to recrystallization and growth to full layer thickness in the Al layers and extended recovery to generate a 0.5-0.8 p m relaxed subgrain structure in the Al(Sc) layers. The recrystallization textures in the Al layers in the supersaturated A R B sheet was determined to be Brass-free β fibre, tending t o w a r d s a r a n d o m distribution in the aged A R B sheet. INTRODUCTION Accumulative roll bonding ( A R B ) is a promising sheet metal fabrication technique for achieving severe plastic deformation (SPD). The essential accomplishment of this method is to synthesize nanostructured martial through an inexpensive and technologically feasible technique. The aspiration of this method is two fold; advancement in material technology and scientific explorations. The primary technological motivation is to achieve novel combination of properties as demonstrated by W a n g et al. in nanostructured copper, while the scientific motivations are wide spread. Because A R B is easy to accomplish using an ordinary rolling mill, this method has been well studied since the outline proposed by Saito et al. . To date, the majority of investigations have been dedicated to bonding of a single alloy up to six A R B cycles with a few recent articles describing bonding of dissimilar metals. A m o n g those, Ohsaki et al. demonstrated mechanical alloying, by ten A R B cycles, of C u and Z r and Quadir et al. showed processing of layered A R B sheet of recrystallized Al and recovered Al(Sc) in alternating sequences. In the first example the deformation w a s extremely severe (true strain = 15.8), where the harder material w a s fragmented and dispersed into the matrix of softer material. In other w o r k it was shown that after a controlled heat treatment the Al layer recrystallized while the Al(Sc) layer only recovered to submicron scale because of particle pinning. In both works the differences in hardness of the candidate material w a s exploited either to achieve layer structures or breaking the lamellar structures by shearing . In this investigation it is demonstrated that by using the strengthening property of the layers, the structures and crystallographic textures can be tailored in A R B bonded layered sheets. 1
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Fig 1 : R D - N D section S E M images of SSSS A R B and Aged A R B s a m p l e s s h o w i n g straight and curved layers. T h e white boxes are regions where E B S D scanning w a s carried out. The higher magnification ICC and T E M images from Al and Al(Sc) layers are s h o w n in the insets.
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T e x t u r e R a n d o m i z a t i o n b y I n t e n s e S h e a r i n g in Al a n d Al ( 0 . 3 % S c ) A R B - P r o c e s s e d S h e e t
EXPERIMENTAL PROCEDURE Commercial purity aluminium (99.8 w t . % purity) (Al) and an Al-0.3 wt.% Sc alloy (Al(Sc)) sheet were processed to supersaturated solid solution (SSSS) and plus aged ( S S S S + A g e d ) condition. After thorough cleaning and brushing, sheets of 1 χ 50 χ 100 m m were mechanically stacked and held for 5 min at 200 °C in a preheated air furnace, and then rolled without lubrication to 50 % reduction in a single pass using a two high rolling mill. The rolled sheet w a s cut into t w o pieces and roll bonded in the identical manner while maintaining the alternating sequence of Al and AI(Sc) layers. After five cycles plus final rolling to 50 % reduction, the processing produced 0.5 m m thick sheet containing 32 alternating layers. In the A R B routes, w h e n the Al w a s bonded with supersaturated solid solution Al(Sc) it was named as " S S S S A R B " and w h e n the Al was bonded with supersaturated solid solution plus aged Al(Sc) it was n a m e d as " A g e d A R B " sheet, in the following of this article. Both the A R B sheets were annealed up to 6 h at 350 °C and cooled in stagnant air.
Fig 2: Vickers hardness as a function of true strain of Al and Al(Sc) layers of SSSS and Aged A R B samples measured after each A R B cycle.
Fig 3: In plane shear stresses operate between Al and Al(Sc) layers shown against the hardness ratio of Al and AI(Sc) in SSSS and Aged A R B samples at each A R B cycles. The data points from Aged A R B and SSSS A R B samples remain above and b e l o w the yield strength (in pure shear condition) of Aged A l ( 0 . 3 % S c ) measured by conventional tensile testing.
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Microscopy investigations were conducted in normal direction (ND)-rolling direction (RD) sections using an FEI D u a l B e a m ™ platform comprised of ion channeling contrast (ICC) and electron channeling contrast imaging facilities. E B S D w a s carried out using a TSL O I M ™ system operating with 10 keV accelerating voltage and 10 m m working distance. Site specific T E M foils were prepared using the FEI D u a l B e a m ™ machine and examined in a Philips C M 200 machine operates at 2 0 0 keV.
Fig 4 : φ O D F section at 5° intervals s h o w i n g E B S D measured (from the marked boxes s h o w n in Fig 1 ) orientation distributions of S S S S and Aged A R B samples. C = C u b e . B = B r a s s . RC^rotated Cube. G=Goss. Co=Copper. 2
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RESULTS A N D DISCUSSION Deformed structures The roll bonding micrographs in R D - T D sections of S S S S - A R B and A g e d - A R B sheets are shown in the low magnification S E M images in Figs l a and b. Oxide debris at the interfaces between Al and Al(Sc) layers appeared brighter. T h e Al layers appeared darker and thinner than the AI(Sc) layers, in both samples. The main difference between these two micrographs is the waviness of layers in A g e d - A R B sheet, which w a s produced by the operation of intense shear banding, inclined at ± 1 7 ± 3 ° with R D in Fig lb. The Al(Sc) layers became thinner at those points where shear b a n d s operated and there the neighbouring Al layers were brought closer. This implies that shear banding w a s prompted in Al(Sc) layers and forced the softer Al layers to follow the deformation trajectories created by them. The substructures in Al and Al(Sc) layers in both S S S S and Aged A R B sheets are comparable, comprising of 0.7-1.00 p m and 0.2-0.4 p m thick lamellar bands in Al and Al(Sc) layers, respectively. This is shown in the insets (taken from SSSS A R B ) in Fig l a and b. M o r e refinement in Al(Sc) layers occurred due to solute-dislocation interactions in SSSS A R B and particle-dislocation interactions in Aged A R B samples . 4
Fig 5: E B S D micrograph of R D - N D section of (a) S S S S A R B and (b) Aged A R B samples after six hours annealing at 350 °C. A T E M image from Al(Sc) layer in the inset of (a) showing substructures. T h e differential hardening characteristics of the bonding layers, shown in Fig 2. is the origin of intense shearing in the Aged A R B sample. In both S S S S and Aged A R B samples the hardening curves of Al layers followed comparable trajectories w h i c h leveled off at ~ 4 4 VHN after the 2nd A R B cycle, corresponds to 1.4 true strain. A hardness value 44 VHN is relatively
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lower than the hardness measured at such high strain and this is attributed from dynamic and static recoveries occurred during deformation and intermediate heating time at a high = 0.5. T h e starting hardness of Al(Sc) layers in S S S S A R B h o m o l o g o u s temperature (T/T ) sample was - 3 0 V H N which leveled off at - 6 8 V H N . while in A g e d A R B sample the starting hardness w a s as high as - 7 8 V H N w h i c h increased to - 8 8 V H N after t w o cycles and then leveled off. That means in the hardening track the hardness ratio of the candidate materials varies between 1.3-1.5 and 2.0-2.6 in S S S S A R B and Aged A R B samples, respectively. A n in-plane shear stress operates between the layers during rolling t w o materials of different h a r d n e s s . D u e to this shear stress the Al(Sc) layers experiences tensile dragging by the easily elongating neighbouring Al layers. The magnitude of the force can be approximately estimated using the formulae derived by Semiatin and Piehler , used for Al cladding on steel sheets, and later used by Yeung and D u g g a n for determining the direction of m a x i m u m shear in softer Al cladding on harder Cu sheet. T h e results are s h o w n in Fig 3. where the upper set of data with open circles corresponds to Aged A R B sample and the lower set of data corresponds to S S S S A R B sample. Significantly, for the Aged A R B sample, the Al(Sc) layers experienced shear stresses greater than the yield strength at pure shear condition, while this in S S S S A R B sample merely reached the yield strength. Therefore from this data it is understandable why shearing in the form of necking is severe in Al(Sc) layers of Aged A R B sample while it is subtle in S S S S A R B sample. m
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region in the next cycle. H o w in plane shear stresses get distributed throughout the thickness was schematically described by Lee et a l . The influence of in-plane shear stresses and the resulting curving of Al layers in modifying textures can be realized by comparing data from S S S S and Aged A R B samples. As per deformation texture concern the authors are interested only on Al layers since these layers recrystallized during subsequent annealing. The texture w a s measured by EBSD scanning "4 p m χ full layer thickness" areas from all the Al layers (except first t w o near surface layers), as marked as small rectangles in Figs 1 a and b. Twelve such E B S D files were combined separately from both S S S S - A R B and A g e d - A R B sheets and plotted in φ sections at 5° intervals in Fig 4a and b . In Fig 4a the texture is typical of Al rolling textures comprising β fibre components i.e. Brass {110}<112>, S3 {123}<634> and Copper {112}<111>. The intensity of Brass is lower than Copper as this happens after high rolling reductions, > 9 5 % . 9
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The orientation distribution of A g e d - A R B sheet is distorted and yielded relatively lower intensities of different degree as a function of orientation. Concerning two primary orientation c o m p o n e n t s , Brass and Copper, intensity reduction of Brass is more than the magnitude of 9.6x Random reduced to BrassAged A R B = 3.6x reduction occurred to Copper. Brassssss A R B 15x R a n d o m reduced to C o p p e r e d A R B 12x Random. R a n d o m , whereas Copperssss A R B Operation of in-plane shear stress at the interfaces intends to rotate Brass towards rotated cube R C , {100}<110>, as predicted in relaxed Taylor type simulations by Hölscher et a l . . Thus a weak fibre extended from Brass to rotated cube in φ ι = 0 found in Aged A R B sample. A higher intensity at Goss is also a shear texture component. However, the rotation from Copper cannot be observed in a single φ section, since it is extended to φ2=15° in Aged A R B sample and Φ2 = 30° in SSSS A R B sample. This is indeed a relative small rotation as compared to the 45° rotation towards rotated cube from Brass in symmetry. A lesser reduction of Copper intensity can be explained based on the finding of Wagner et a l . " , showing Copper tends to rotate to the Dillamore orientation {4 4 11 )<11 11 8> at high strain, but again rotates back to Copper by T D rotation induced by the operation of shear banding. Their analogy w a s supported by experimental evidences using single crystals of Al and Cu. In another investigation Copper was also found stable under shear strains. Therefore combination of high strains, shear strain and shear banding stabilizes C o p p e r orientations in the notion of orientation oscillations around stable end orientations. =
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Annealed mierostructures and textures Extended annealing for 6 h at 3 5 0 C w a s carried out to complete recrystallization (plus grain growth) in the Al layers and recovery in the Al(Sc) layers. The mierostructures of R D - N D sections are presented as orientation imaging microscopy ( O I M ) in Figs 5a and b, showing recrystallized grains in straight R D aligned Al layers in SSSS A R B sample and in curved Al layers in Aged A R B sample. T h e Al(Sc) layers in either samples remained unrecrystallized. Site specific T E M investigation showed (shown in index) they are recovered to 0.5-0.8 p m equiaxed substructures because of particle pinning. Turning to the orientations of recrystallized grains of Al layers, this was measured by taking data from 130 grains from both samples and plotted in Figs 6b and c as <111> pole figures, where each grain contributed one data point. In SSSS A R B sample the rolling texture is largely retrained after recrystallization, except that the Brass c o m p o n e n t has been eliminated. This was explained in ref. [4] as a stored energy driven phenomenon. In contrast, the orientations of the recrystallized grains in Aged A R B sample are quite randomized. This randomization primarily occurred because of shear banding as it was also
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found by KJiizumi et a l . in A l - M g alloys. Texture randomization is beneficial to enhance drawability of Al (towards Lankford parameter of unity) and. therefore, this material might be of special interest in connection to the other properties, such as. it is also expected that the soft Al layer will contribute to improve ductility while submicron structured Al(Sc) will contribute to enhance strength. CONCLUSIONS High purity Al was roll bonded to Al(Sc) in the supersaturated solid solution plus aged conditions to produce lamellar sheet. A n extended annealing produced soft recrystallized Al layers and hard recovered Al(Sc) layers. Because of differential hardening between the layers during rolling, the Aged A R B sheet deformed by intense shear banding; this altered the recrystallization texture towards randomization. ACKNOWLEDGEMENTS It is a pleasure to a c k n o w l e d g e Australian Research Council ( A R C ) for supporting this work through the A R C Centre of Excellence for Design in Light Metals ( C E 0 5 6 1 5 7 4 ) , and Electron Microscope Unit of Analytical centre of University of N e w South Wales. REFERENCES Y. W a n g , M. Chen. F. Zhou and E. Ma, " H i g h Tensile Ductility in a Nanostructured M e t a l . " Nature. 4 1 9 . 912-15(2002). Y. Saito, H. Utsunomiya. N . Tsuji and T. Sakai. "Novel Ultra-High Strengthening Process for Bulk Materials Development of T h e Accumulative Roll B o n d i n g ( A R B ) Process". Ada Mater.. 4 7 , 5 7 9 - 8 3 (1999). ' S. Ohsaki, S. Kato. N. Tsuji. T. O h k u b o , K. H o n o . "Bulk Mechanical Alloying of C u - A g and Cu/Zr T w o - P h a s e Microstructures by Accumulative Roll-Bonding Process", Ada Mater.. 55.2885-2895 (2007). M . Z. Quadir. O. A l - B u h a m a d , L. Bassman, M . Ferry. " D e v e l o p m e n t of a Recovered/Recrystallized Multilayered Microstructures in Al Alloys by Accumulative Roll B o n d i n g " . Ada Mater.. 55. 5438-48 (2007). F. J. H u m p h r e y s , M . Hatherly, "Recrystallization and Related Annealing P h e n o m e n o n " . 2nd Edition, Elsevier (2004). A. Kelly, "Strong Solids". 2nd edition. Clarendon Press. Oxford (1973). S. L. Semiatin and H. R. Pieliler, "Formability of Sandwich Sheet Materials in Plane Strain C o m p r e s s i o n and Rolling". Metall Trans. A. 10, 97-107 (1979). W. Y. Y e u n g and B. .1. D u g g a n . " S h e a r Band Angle in Rolled F C C Materials". Acta Metall.35. 541-48(1987). S. H. Lee. Y. Saito. N. Tsuji. H. Utsunomiya and T. Sakai. " R o l e of Shear Strain in Ultragrain Refinement by Accumulative R o l l - B o n d i n g ( A R B ) Process", Scripta Mater.. 4 6 . 2 8 1 - 8 5 ( 2 0 0 2 ) . M . Hölscher. D. Raabe, K. Lücke, "Relationship Between Rolling Textures and Shear Textures in F C C and B C C Metals". Acta Metall. Mater., 4 2 , 8 7 9 - 8 6 ( 1 9 9 4 ) . 1
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" P . Wagner. O. Engler and K. L ü c k e . "Formation of Cu-type Shear Bands and Their Influence on Deformation and Texture of Rolled F C C ( 1 1 2 j < l l l > Single Crystals", Acta Metall. Mater., 43.3799-812(1995). M . K o i z u m i . H. Okudaira and H. Inagaki, " D e v e l o p m e n t of Annealing Textures in A l - M g Alloys", Zeitschrift Fur Metallkunde. 89,424-32(1998). I 2
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TEXTURE DEVELOPMENT HEAT TREATMENT
OF M O L Y B D E N U M
S H E E T S D U R I N G L A S T STEP O F
C.-G. Oertel, I. Hünsche, W. Skrotzki Institut für Strukturphysik, Technische Universität Dresden. D-01062 Dresden, G e r m a n y A. Lorich. W. Knabl P L A N S E E Metall G m b H , Technologiezentrum, Entwicklung Hochschmelzende Metalle A-6600 Reutte, Austria
ABSTRACT M o l y b d e n u m is a refractory metal with special properties, like excellent high-temperature strength, low coefficient of thermal expansion, good electrical and thermal conductivity and high Y o u n g ' s modulus. In sheet or ribbon form molybdenum is widely applied as electrodes, heating elements and ideal substrate material for many technological applications, such as electronics, electric power, lighting technology, nuclear energy and even aerospace engineering. T o produce work pieces the material has to be thermomechanically processed producing characteristic deformation and recrystallization textures depending on the deformation and/or annealing conditions. The present work concentrates on the influence of the last step of annealing on the texture development of selected sheets and strips of molybdenum. Usually, unidirectional cold rolling in the last stage of production leads to a strengthening of the main texture component, which for m o l y b d e n u m is the rotated cube component {100} <110>). This c o m p o n e n t leads to a strong anisotropy of the mechanical properties in the sheet plane. Therefore, specific annealing stages have been tested in order to influence the texture. The texture development during these annealing stages is discussed on the basis of microstructure investigations, texture measurements and electron backscatter diffraction (EBSD).
INTRODUCTION M o l y b d e n u m is a refractory metal with wide use. The high melting point, hightemperature strength, low vapor pressure and relative low density are the basis for applications in lighting, glass, furnace and nuclear industry. Due to a weak oxidation behavior, especially in the range of 4 0 0 - 8 0 0 ° C , an oxygen-free atmosphere is a necessity. As the coefficient of thermal expansion is similar to that of glass, m o l y b d e n u m generally is used for current feed into glass bulbs. For industrial production of molybdenum work pieces, sheet rolling and deep-drawing of compacted and sintered m o l y b d e n u m powders are important processes. For hot and cold rolling, in general annealing treatment is necessary between the rolling passes. As a consequence, specific crystallographic textures and microstructures are produced depending on the thermomechanical process applied. The sheets investigated were produced by multiple unidirectional hot rolling steps accompanied by reheating and unidirectional cold rolling with annealing periods between the rolling passes. The as-rolled state is characterized by the main rolling texture c o m p o n e n t s of body centered cubic (bcc) metals {100} < 1 1 0 > and {112} <110> located on the α-fiber (<110> || rolling direction). Additionally, a weak γ-fiber (<111> || normal
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direction) is observed. High intensities of the γ-fiber are important for good deep drawing properties of bee metal sheets. This is because on the one hand they produce an in-plane mechanical isotropy and on the other hand the Lankford parameter b e c o m e s m u c h larger than one [1]. To change the microstructure and texture of materials, recrystallization is an important process. It is the aim of the present work to influence the microstructure and texture by variation of the temperature of the final heat treatment.
EXPERIMENTAL Samples The m o l y b d e n u m sheets were produced by hot rolling of sintered plates heated to 1000°C - 1400°C and additional cold rolling. Steps of hot and cold rolling were unidirectional. To preserve the ductility of the sheets, annealing treatments in hydrogen atmosphere between the steps were necessary. To investigate the microstructure and texture development during the final heat treatment, as-rolled sheets were annealed at 500, 700, 750, 800, 850, 1 1 0 0 X for 2 hours under Ar atmosphere in a laboratory furnace. For comparison with the production process, a sheet annealed at 780°C under H atmosphere w a s used, too. 2
Microstructure The microstructure of the sheets w a s investigated by orientation contrast imaging and E B S D ( H K L software) in a scanning electron microscope ( S E M , Zeiss D S M 962). The samples were grinded, mechanically polished and finally electropolished at a voltage of 18 V using LectroPol-5 (Struers A/S). The electrolyte consisted of 20 v o l % HCl. 20 v o l % H S 0 and 60 v o l % C H j O H . 2
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Texture measurements To characterize the texture development in detail, texture measurements were done on the surface and center of the sheets prepared by a precision mechanical grinding procedure. Three pole figures ( 1 1 0 . 200 and 211), measured by X-ray diffraction using a Θ - 2 Θ goniometer ( H Z G 4. FPM Freiberg, Cu Κα-radiation) with a Euler cradle in back reflection m o d e [2], were used for calculating the orientation distribution function ( O D F ) with a computer program developed by D a h m s [3]. The degree of series expansion used is 22, the Euler angles used are in the Bunge notation [4]. The O D F is represented by the
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Fig. 1: Location of the main texture components of rolled bcc metals in the φ , = 45° O D F section.
RESULTS AND DISCUSSION Figure 2 s h o w s = 45° ODF-sections illustrating the texture development in the surface and in central layer of the sheets as a function of final annealing temperature. For a more detailed discussion, the intensity distribution of the a - and γ-fiber in the central layer as a function of annealing temperature is s h o w n in Fig. 3.
Fig. 2: (f>2 = 45° ODF-sections illustrating the texture development in the surface and central layer of the sheets as a function of final annealing temperature. Intensities are given in m.r.d. = multiples of a random distribution. T h e as-rolled state is characterized by an incomplete α-iiber dominated by the {112}<110> component in the center of the sheet. In the surface, additionally the rotated cube component exists and the γ-fiber only contains a weak {111 ) < 1 1 2 > component. In the central layer of the sheet the γ-fiber is weak but homogeneous. The through-thickness texture gradient results from changes in the deformation m o d e . N e a r the sheet surface shearing processes take place because of differences in velocity between the rolls and the sheet surfaces. The deformation m o d e in the center of the sheets is almost plane strain.
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The main texture c o m p o n e n t s of the sheets independent of final heat treatment are 1 1 0 0 | < 1 1 0 > and ( 112}<110>. both c o m p o n e n t s being located o n the α-fiber. The intensity of the γ-fiber orientations is quite weak. This is in agreement with the literature [5-7]. L o w degrees of straight rolling produces an incomplete α-fiber with the m a x i m u m at the rotated cube component [100} <110>. The m a x i m u m shifts to {112}<110> at higher degrees of rolling. T h e intensity of the m a i n texture c o m p o n e n t s does not change drastically up to annealing temperatures of 8 0 0 ° C . A w e a k e n i n g of the γ-fiber, especially of the {112} < 1 1 1 > component, is noticed in the central layer of the sheets. A b o v e an annealing temperature of 750°C recrystallization starts (Fig. 4). Figure 4a shows the as-rolled microstructure. At an annealing temperature of 700°C nucleation and/or beginning of recrystallization is observed (Fig. 4b). With increasing annealing temperature the recrystallized v o l u m e fraction continuously increases. The sheet annealed at 780°C under hydrogen atmosphere is less recrystallized than expected from the course of Xv(T) in Fig. 5. This may be caused by differences in heating up the sheets during annealing under argon and hydrogen atmosphere yielding different effective annealing times. Figure 4d s h o w s the partially recrystallized microstructure after a final heat treatment at 800°C. It is clearly seen that recrystallized grains form larger colonies. The sample annealed at 1100°C is completely recrystallized (Fig. 4e) and also shows grain growth. The recrystallized grains are
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always flattened in the rolling plane with elongation in the rolling direction. With increasing annealing temperature the preferred orientation of the grain boundaries parallel to the rolling direction disappears.
Fig. 4: Typical microstructures observed after a n n e r o r g at different temperatures a) as-rolled, b) 700°C, c) 7 5 0 ° C . d) 8 0 0 ° C . e) 1100°C.
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Figure 5 s h o w s the development of the volume fraction recrystallized and the texture index ( as defined in [8]) as a function of final annealing temperature. T h e decrease of the texture index up to an annealing temperature of 700°C (regime I) may be correlated with recover}' of the microstructure and the formation of recrystallization nuclei. The following rise from 750°C till 800°C (regime II) is due to partial recrystallization. Annealing between 850°C and 1100°C (regime III) leads again to a decrease of the texture index correlated with colony formation of recrystallized grains up to complete recrystallization combined with grain growth. In the following the microstructure and texture development within the three annealing regimes will be discussed in more detail.
Fig. 5: Development of the recrystallized volume fraction X and the texture index as a function of final annealing temperature in the surface and the central layer of the sheets. Regime 1 In regime 1 the microstructure is changed by recovery processes and the formation of recrystallization nuclei. E B S D mapping of an annealed sample with beginning recrystallization shows that grains belonging to the a - and γ-fiber contain a low and high density of subgrain boundaries (misorientations <15°), respectively (Fig. 6). This agrees with E B S D investigations on rolled steel by T h o m a s [9] and m e a s u r e m e n t s of the stored energy by Novillo [10]. The highest stored energy is found in γ-fiber oriented steel grains followed by grains with [112}<110> orientation. The rotated cube component has the lowest stored energy. Simlar observations have also been m a d e by other authors [ 1 1 . 12. 13. 14].
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Fig. 6: E B S D mapping of the central layer of a sheet with beginning of recrystallization (annealing temperature: 750°C). The different texture c o m p o n e n t s are distinguished by different colors: dark grey = α-fiber, light grey = γ-fiber and white = other orientations. Thin and thick lines are grain boundaries with misorientations smaller and larger than 15°. respectively. The stored strain energy determines the stability of particular orientations during annealing. Highly strained grains have the tendency to nucleate new grains and those with low stored energy (less deformed) have the tendency to g r o w at the expense of their neighbours. Therefore, the texture developing during recrystallization will depend on the balance between nucleation (defined by the probability and "critical strain" for nucleation) and the boundary mobility [15]. According to [16] the latter mechanism is called strain induced boundarymigration ( S I B M ) . A s will be shown below, S I B M takes place at higher annealing temperatures in regime II. T h e formation of recrystallization nuclei in γ-grains leads to a randomization of the texture (Fig. 5) and to a decrease of the intensity of the γ-fiber components. Similar observations are described by G u t t m a n n on rolled m o l y b d e n u m (purity > 99.95%) after 8 0 % cold rolling and annealing in a temperature range of 650° - 1250°C for l h under v a c u u m [17]. In the as-rolled state G u t t m a n n using transmission electron microscopy observed a low dislocation density in a grains and a dislocation cell/subgrain structure in γ-grains. Recrystallization starts at 950°C and is dominated by S I B M from a - into γ-grains. R e g i m e II Regime II is characterized by partial recrystallizion with Xv approaching 5 0 % (Figs. 4c, 5). T h e texture strength given by the texture index increases. Simultaneously, the intensity of the α-fiber c o m p o n e n t s increases, while that of the γ-fiber stays low. This observation may be explained by S I B M of α-fiber grains. Similar observations are m a d e by Raabe [18] on rolled tantalum during annealing in a temperature range of 1000°C - 1200°C. In contrast, a transformation of the rotated cube
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component into a γ-fiber texture with a weak m a x i m u m near {111}<112> takes place during recrystallization at 1300°C after cold rolling with a thickness reduction of 7 0 % . With increasing rolling deformation the reversal of the rotated cube c o m p o n e n t into the γ-fiber texture by annealing is shifted to lower temperatures. The rotated cube c o m p o n e n t as annealing texture at low temperatures is explained by R a a b e by continuous recrystallization, i.e. extended recovery. This process produces large angle grain boundaries by incorporation of dislocations into subgrain boundaries and/or coalescence of subgrain boundaries. T h e change to the γ-fiber texture at high annealing temperatures is explained by growth selection [19,]. T h e development of γ-fiber orientations by recrystallization of bcc metals has also been observed in chrome steels [14], steels with low carbon content [10, 20] and interstitial free steels [21], Resume 111 In this regime the texture strength decreases and minor changes of the texture type take place. The rotated cube c o m p o n e n t deviates from the ideal position in φι direction up to 10° describing a rotation around the normal direction. M o r e and more recrystallized grains g r o w together to form colonies until complete recrystallization is reached. This fact is well documented in Figs. 4d, e. Moreover, a clear increase of the grain size of the recrystallized grains is observed with increasing annealing temperature. A c c o m p a n y i n g grain growth leads to a randomization of the texture. As can be seen in Fig. 3 the intensity of the α-fiber drastically decreases. Fujii [7] observed a weakening of the texture in recrystallized m o l y b d e n u m sheets, too. These results are confirmed by Park [22] and agree with the observations made here.
CONCLUSIONS Three different temperature regimes have been distinguished during annealing of cold rolled m o l y b d e n u m sheets. In the low temperature regime I (up to 700°C) extended recovery takes place. The medium temperature regime II (750°C - 800°C) is characterized by partial recrystallization by S I B M . In the high temperature regime III (above about 850°C) the sheets totally recrystallize. Simultaneously, grain growth takes place. In all regimes the type of texture does not change drastically, but the relative intensities of the different texture components are shifted. The heat treatments applied did not produce a strong and h o m o g e n e o u s γ-fibre favourable for good d e e p drawing.
ACKNOWLEDGMENTS Sample preparation and technical support of A. Jurck. A. Lankau. T. Reiter and W. Tirschler are gratefully acknowledged.
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REFERENCES ' P . Juntunen, D. Raabe. P. Karjalainen, T. Koio and G. Bolle. "Optimizing Continuous Annealing of Interstitial-Free Steels for Improving D e e p Drawability", Met. Mat. Trans., A 32 1989-1995 (2001). W . Skrotzki, R. T a m m , C.-G. Oertel, B . Beckers, H.-G. Brokmeier and E. Rybacki. „Texture Induced Plastic Anisotropy of NiAl Polycrystals", Mater. Sei. Eng., A319-321, 364367 (2001). M . D a h m s and H.J. B u n g e , " T h e Iterative Series-Expansion Method for Quantitative Texture Analysis. I. General Outline", / Appt. Cryst.. 22, 439-447 (1989). H . J . Bunge, " Z u r Darstellung allgemeiner Texturen", Ζ. Metallkd., 56, 872-875 (1965). Y . B . Park, D.N. L e e and G. Gottstein, " T h e Evolution of Recrystallization Textures in Body Centred Metals", Acta Mater., 46-10, 3371-3379 (1998). D . Raabe and K. Lücke, "Rolling Textures of Niobium and M o l y b d e n u m " , Ζ. Metallkd.. 85, 3 0 2 - 3 0 6 ( 1 9 8 4 ) . T . Fujii and R. Watanabe, "Effects of Rolling Procedures on the Development of Annealing Textures in M o l y b d e n u m Sheets", J. Less-Common Met., 97, 163-171 (1984). H.J. B u n g e , „Texture Analysis in Materials Science", Cuviller Verlag, Göttingen (1995). I . T h o m a s , S. Zaefferer, F. Friedel and D. Raabe, "High-Resolution E B S D Investigations of Deformed and Partially Recrystallized IF Steel", Adv. Eng. Mal., 1, 566-570 (2003). E , Novillo, M . M . Petite, J.L. Bocos, A. Iza-Mendia and I. Gutierrez, "Texture and Microtexture Evolution in an Ultra-Low Carbon Steel During Recrystallization", Adv. Eng. Mat. 5, 575-578 (2003). " G . M o h a m e d and B. Bacroix, "Role of Stored Energy in Static Recrystallization of Cold Rolled Copper Single and Multicrystals". Acta Mater, 48, 3295-3302 (2000). B . Bacroix, A. Miroux and O. Castelnau, "Simulation of the Orientation Dependence Stored Energy During Rolling Deformation of L o w Carbon Steels", Modelling Sim. Mater. Set. Eng., 7, 8 5 1 - 8 6 4 ( 1 9 9 9 ) . D . Raabe and K. Lücke, "Texture and Microstructure of Hot Rolled Steel". Scripta Metall. Mater., 26, 1221-1226 (1992). D . Raabe and K.. Lücke, "Textures of Ferritic Stainless Steels", Mater. Sei. Technol, 9, 302-312 (1993). H . - R . Wenk, G. Canova, Y. Brechet and L. Flandin. " A Deformation Based Model for Recrystallization o f Anisotropic Materials", Acta Mater., 45, 3283-3296 (1997). F . J . H u m p h r e y s and M . Hatherly, "Recrystallization and Related Annealing Phenomena, Elsevier, Oxford (2004). V . Guttmann, ,.Keimbildung bei der Rekristallisation von Molybdän", J. Less-Common Mer., 21, 51-61 (1970). D . Raabe and K. Lücke, "Annealing Textures of bec Metals". Scripta Metall. Mater.. 27, 1533-1538 (1992). G . Ibe and K. Lücke, "Orientierungszusammenhänge bei der Rekristallisation von Einkristallen einer Eisen-Silizium-Legierung mit 3 % S i " . Archiv Eisenhüttenwesen 39. 693-702 (1968). 2
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I. Samajdar. B. Verlinden, L. Kestens and P. Van Houtte, "Physical Parameters Related to the Development of Recrystallization Textures in an Ultra L o w Carbon Steel", Acta Mater., 4 7 , 5 5 - 6 5 (1999). 1 . Samajdar, B. Verlinden, P. Van Houtte and D. Vanderschueren, "γ-Fibre Recrystallization Texture in IF-steel: An Investigation on the Recrystallization M e c h a n i s m " , Mater. Sei. Eng.,A238, 343-350 (1997). 22 Y . B . Park, D.N. Lee and G. Gottstein, " T h e Evolution of Recrytallization Textures in Body Centred Metals", Acta Met., 10, 3371-3378 (1998). 2 I
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S O L U T E , S U P E R P L A S T I C I T Y A N D T E X T U R E C H A N G E IN A L U M I N I U M A L L O Y S K. Sotoudeh, P.S. Bate, F.J. H u m p h r e y s The University of Manchester Manchester, U.K. ABSTRACT There are several aluminium alloys which exhibit high tensile ductility at slow strain rates and elevated temperatures. This superplasticity is due principally to a high sensitivity of flow stress to strain rate. It is well k n o w n that a fine grain size is needed for this to occur, but in addition it appears that a relatively high level of solute, such as copper or magnesium, is also required. This has been investigated in Al-Cu-Zr alloys with different copper contents and AlM g - M n alloys with different m a g n e s i u m contents. In both cases, l o w solute materials were not superplastic. They also exhibited texture change during uniaxial tension essentially consistent with that expected with simple octahedral slip as the deformation mechanism. In the high solute, superplastic, alloys there w a s a significant divergence of preferred orientation. This orientation divergence is the main microstructural characteristic associated with the superplasticity: other factors such as dynamic grain growth occurred in non-superplastic alloys. The degree of boundary sliding and relative grain translation was very limited and did not differentiate the superplastic alloys either. The possibility that rate sensitive dislocation slip can explain the texture changes is discussed. INTRODUCTION Superplasticity, the ability of some crystalline solids exhibit very high tensile ductilities, is mainly due to a high sensitivity of stress to strain rate. It occurs at relatively high temperatures and slow strain rates, and is coupled with a low overall flow stress . The ability of some aluminium alloys to behave in this way is exploited commercially in superplastic forming, which is used in transportation industries. T h e main microstructural factor in superplasticity is a fine grain size (typically < ΙΟμιτι). For this type of grain size to be stable at the high temperatures involved (> 450°C for aluminium alloys), Zener p i n n i n g is used, involving fine dispersoids, such as A l Z r and A l M n in Al-CuZr and A l - M g - M n alloys, respectively. This pinning slows static grain growth, though grain growth during d e f o r m a t i o n - dynamic grain growth - u s u a l l y occurs, and because the flow stress is sensitive to grain size, gives strain hardening " . Several aspects of superplasticity, especially microstructural evolution during deformation, have led to the commonly held view that relative grain translation via grain boundary sliding ( G B S - together with dislocation creep and/or diffusion creep as a c c o m m o d a t i n g mechanisms) is the principal m e c h a n i s m " . This mechanism is taken to explain the apparent retention of nearly equiaxed grains after large strains and also the general w e a k e n i n g of crystallographic texture. However, some mechanical and microstructural aspects cannot be demonstrated by those models. Elongated grains after superplastic deformation and shifts in preferred orientations are m o r e consistent with models based on rate sensitive s l i p " " ' . Texture studies correlated with mechanical properties and microstructural changes may give further insight into dislocation slip and G B S mechanisms and their interaction. Work by Bate on plastic anisotropy and texture evolutions in several superplastic and c o - w o r k e r s " aluminium alloys suggests that dislocation slip m a y have a primary and fundamental role in 1
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superplastic behaviour. Texture divergence was found to be a c o m m o n feature for superplastic alloys tested within the superplastic regime. Recent w o r k on two Al-Cu-Zr alloys containing different amount of copper solute, with initially banded mierostructures, also showed that texture weakening is associated with the superplasticity (strain rate sensitivity index, m ~ 0.4 and - 3 5 0 % elongation) observed in the alloy with a higher amount of Cu. The present work e x a m i n e s the evolution of microstructure and texture, in parallel with studies of the mechanical characteristics of the A A 3 0 0 3 and A A 5 0 8 3 alloys with nearly equiaxed grains. The texture results of these alloys are presented and discussed together with texture results obtained from Al-Cu-Zr a l l o y s . 17
17
MATERIALS AND EXPERIMENTAL PROCEDURE The details of the experiments and results for the Al-2Cu-0.3Zr and Al-4Cu-0.4Zr alloys are given e l s e w h e r e . The two other materials involved were commercial A A 3 0 0 3 ingot supplied by Sapa G r o u p - S w e d e n , and as-rolled A A 5 0 8 3 sheet of about 1.6mm thickness supplied by S K Y Aluminium C o . - J a p a n . T h e chemical analysis of the alloys is given in Table 1. 17
Table I. Chemical Compositions of the Alloys Alloys
Chemical Composition, [wt.%] Mg
Mn
Cr
Fe
Si
Cu
other
AA3003
0.0
1.0
0.0
0.58
0.14
0.10
<0.02
AA5083
4.67
0.70
0.11
0.03
0.02
0.10
< 0.01
Al Balance
Thermo-mechanical Processing A block of 7 0 m m x 7 0 m m x 4 2 m m was cut from the middle part of the A A 3 0 0 3 ingot. A microstructure containing fine precipitates w a s produced by heating at 600°C for 3 hours and quenching in water to room temperature. The material w a s then hot rolled at 400°C and finally cold rolled to a thickness of about 1.8mm. T h e A A 5 0 8 3 alloy is proprietary material which recrystallises prior to superplastic deformation. Hot Tensile Deformation Details of the tensile specimen geometry have been provided p r e v i o u s l y . The A A 3 0 0 3 and A A 5 0 8 3 specimens were annealed in a purpose-built tensile machine, at 530°C, for Ihr and 2hrs (including a heating-up and stabilisation time), respectively, to develop nearly equiaxed and similar mierostructures prior to straining. Uniaxial tensile testing was performed, both to failure and to intermediate strains of 0.3 for A A 3 0 0 3 and 0.45 for A A 5 0 8 3 . The specimens were tested with a strain rate alternating between 4 . 5 x 1 0 " V and 5 . 5 x 1 0 " V , with the rate switched at 0.1 strain intervals. The resulting change in stress allowed the strain rate sensitivity index, m, to be determined as explained e l s e w h e r e . 1417
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E B S D Mapping and Texture Determination Samples were taken from the gripped ends (representing non-deformed, statically annealed material) and gauges (representing strained material) of the tensile specimens. Because of inhomogeneous deformation, dimensional measurements were performed prior to sample collection to estimate the true strain ( ε ) in the gauges. Samples were then sectioned, mechanically polished and electro-polished to show the transverse (TD) section. These were examined using an F E I - S i r i o n - F E G S E M equipped with H K L Channel 5 electron back-scattered τ
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diffraction ( E B S D ) acquisition software. E B S D m a p s w e r e taken from regions of about 0 . 4 m m x 0 . 4 m m using a step size of 0.5μηι, near the sheet mid-plane of both grip and gauge samples. The data were used to plot and analyse the microstructures using in-house software V M A P - V 8 . E B S D m e a s u r e m e n t s w e r e also carried out using a coarser step size of 4 p m over the complete thickness of the alloys, excluding a distance of lOOpm from the edges, in the transverse plane. The data collected from approximately 2 5 0 0 0 orientations were used to generate inverse = 22. pole figures of the tensile axis using harmonic series m e t h o d s , with series truncation at l m a x
Grid Milling by FIB microscopy and S E M Imaging In order to asses the inhomogeneity of deformation. 2 0 0 μ ι η χ 2 0 0 μ π ι square grids with a line spacing, depth and width of 2 5 p m . 300nm and 330nm, respectively, were milled on the sheet plane surface of annealed and mechanically polished tensile specimens. This was done using a F E I - N o v a 6 0 0 - F I B microscope using an ion b e a m with a voltage of 30kV and current of 6.5nA. Specimens were then deformed by hot tensile straining to a true stain of 0.3. Backscattered images were subsequently taken from the deformed grids using an F E I - S i r i o n - F E G S E M . RESULTS AND OBSERVATIONS Mechanical Properties The results of perturbed strain rate tension tests of A A 3 0 0 3 and A A 5 0 8 3 alloys including stress-strain curves and strain rate sensitivity index variations are shown in figure 1. Differences in the flow stresses, m values and total elongations can be clearlydetected. T h e m a x i m u m flow stress did not exceed l O M P a in the A A 5 0 8 3 alloy, while it was 14MPa in the A A 3 0 0 3 alloy. The A A 5 0 8 3 alloy with about 5wt.% m a g n e s i u m solute gave an average m value of ~0.42 with a total elongation of approx. 3 0 0 % . whereas these values were about 0.15 and 130%. respectively, for the A A 3 0 0 3 alloy with no M g solute content. The ductility difference is aligned with differences in the m values. The m value did not change significantly with deformation in the A A 3 0 0 3 alloy, however a notable decrease w a s observed in the A A 5 0 8 3 alloy which m a y be linked to dynamic grain growth occurred during deformation. Microstructure Examination T h e c h a n g e s of H A G B spacing in N D (λΝϋ) and aspect ratio with straining, obtained by the quantitative analysis of experimental results (Exp.) are plotted in figure 2. These have been also c o m p a r e d to a condition where
Figure 1. σ-ε curves from tensile tests of A A 3 0 0 3 and A A 5 0 8 3 alloys, and the m values derived from the curves are shown.
Figure 2. G r a p h s showing changes in aspect ratio ( A R - dashed lines) and Xmi of the alloys, in the presence (continuous lines) and in the absence (dotted lines) of D G G .
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deformation occurs only by the simple geometric effect of straining ( S G E S - grain thinning) in the absence of dynamic grain growth. Grain structures of the A A 3 0 0 3 and A A 5 0 8 3 alloys are illustrated as E B S D m a p s , both unstrained and after deformation to a strain close to failure, in Fig.3a-d. Thick black lines indicate high angle grain boundaries ( H A G B ' s ) with misorientations >15° and thin grey lines indicate low angle grain boundaries ( L A G B ' s ) with misorientations > 2 ° .
Figure 3. E B S D m a p s - with grey scale depending on orientation - o f the mid-plane layer of the A A 3 0 0 3 alloy, a) ε = 0, annealed and b) ε = 0.85, as well as the A A 5 0 8 3 alloy, c) ε = 0. annealed and d ) ε = 1. The initial microstructure of the alloys consisted of a nearly equiaxed grains with a λκυ of about 9 p m , as s h o w n in figure 2, 3a and 3c. The development of a coarser microstructure of elongated grains aligned with the N D can be seen in figure 3b. 3d for the A A 3 0 0 3 and A A 5 0 8 3 alloys after deformation, respectively. Microstructure Examination by Grid Deformation In spite of straining the materials to a considerable strain of 0.3 (about 3 5 % elongation), the grids did not become obscured. Grid offsets and rotations, as well as the distorted appearance of boundary regions, all indicate shear at grain boundaries, as shown in figure 4. Localised disruption in markers on the AA3003 and AA5083 specimens appeared to be qualitatively very similar.
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Figure 4. B S E micrographs of a) the A A 3 0 0 3 alloy and b) the A A 5 0 8 3 alloy s h o w i n g inhomogeneous deformation and grain boundary displacements after deformation to a true strain of 0.3.
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T h e t w o classes of material reported here, A l - C u - Z r and A l - M g - M n , had very different mierostructures and crystallographic textures. T h e mierostructures of the Al-Cu-Zr alloys have been reported e l s e w h e r e " , together with s o m e texture results. These materials had a banded microstructure, and their textures prior to hot tensile deformation w e r e dominated b y the 'brass' texture, centred on {110}<112>. T h e development of texture during tension is shown, using inverse pole figures, in figure 5.
Figure 5. Inverse pole figures, for the specimen tensile direction, of A l - C u - Z r alloys deformed at 4 5 0 ° C with a strain rate of 10"V. In both alloys, there w a s a shift of the major axial alignment from < 1 1 2 > in the tensile direction to < 1 1 1 > . This shift is expected with octahedral slip. T h e main difference is the intensity of texture that develops. In the low copper, l o w m, alloy the texture intensity b e c a m e m u c h higher than in the high copper, high m, alloy. In the alloys used to demonstrate the effect o f m a g n e s i u m solute, the textures present prior to tensile deformation were m u c h weaker. In the A A 3 0 0 3 the m a x i m u m orientation density w a s only about 3 times r a n d o m , and in t h e A A 5 0 8 3 , the m a x i m u m orientation density less t h a n 5 times r a n d o m . T h e r e w a s a significant difference between the texture evolutions in the two alloys. T h i s is s h o w n in figure 6.
Figure 6. Inverse pole figures, for the specimen tensile direction, of A l - M g - M n alloys deformed at 5 3 0 ° C with a strain rate of 10"V .
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For the AA30O3. with no m a g n e s i u m solute and low m value, the near-random texture developed into a clear <111> alignment with the tensile axis. For A A 5 0 8 3 . with high m a g n e s i u m solute and high m. the texture intensity reduced, and although some trace of a weak <111> alignment developed (in addition to the < 0 0 1 > / / σ present from the start) in the highest strain sample measured, the dominant trend was texture weakening. The behaviours, in terms of m a x i m u m inverse pole figure density, are s h o w n in figure 7.a. In both cases, then, higher solute levels are associated not only with higher m values but also with texture weakening. Modelling of Texture The development of deformation textures via intragranular slip is a well k n o w n phenomenon. However, models such as those based on T a y l o r ' s uniform deformation assumption predict deformation textures w h i c h are m u c h stronger than those observed in almost every case. O n e important reason for this is that deformation is i n h o m o g e n e o u s , and a significant part of that inhomogeneity is due to the mechanical interaction between differently oriented regions. There are several w a y s of modelling this: the one adopted here uses the finite element method with crystal plasticity constitutive equations ( C P F E M ) . It is usual in C P F E M to assume rate-sensitive slip, such that the slip stress d e p e n d s on the current slip rate on that system according to a power law. If the overall 'instantaneous' rate sensitivity is solely due to the slip rate sensitivity, then the index m in the slip power law will be the same as that measured in tensile tests by changing strain rate. In the majority of w o r k that has used C P F E M , m is assumed to be very low. representing l o w temperature deformation. The rate sensitivity has a major effect on the development of texture. T h e distribution of slip activity on different systems becomes more uniform as m increases. In a simple Sachs-type model, with no rate sensitivity only the highest Schmidt factor system is active, whereas with rate sensitive slip all systems are. in principal, active. This reduces the rate of crystallographic orientation change, and with {111}<110> slip, and other cases, the rate b e c o m e s zero when m = 1. The plastic anisotropy also reduces but to a m u c h lower degree. The overall effect is that deformation textures for a given total deformation are weaker, i.e. they have a greater spread, as the slip rate sensitivity increases. This effect is s h o w n in figure 7.b.
Figure 7. The m a x i m u m axis correlation density on the tensile direction inverse pole figure. A , as functions of strain a) for the A A 3 0 0 3 and A A 5 0 8 3 alloys and b) from simulations of texture development starting from a random texture. Except for the m — 0.8 results at strains >0.6. where < 0 0 1 > / / σ was dominant, the m a x i m u m alignment was <111>//σ. m a x
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S o l u t e , S u p e r p l a s t i c i t y a n d T e x t u r e C h a n g e in A l u m i n u m Alloys
A domain of 5832 ( 1 8 x 1 8 x 1 8 ) 20-node hexahedral quadratic elements was used to simulate uniaxial deformation, with free translation in transverse directions. The initial element orientations were chosen at random. Linear strain hardening, of a degree similar to that shown by the A A 5 0 8 3 , w a s used, although this aspect of the material behaviour has little effect other than to reduce macroscopic 'necking'. The orientations of the 4 6 6 5 6 integration points were used to generate harmonic coefficients from which tensile axis inverse pole figures were produced. All these showed development of the typical <111> and <001> alignments. It is clear from figure 7.b that the rate of texture development with tensile strain is m u c h lower with increasing rate sensitivity, and certainly lower than with a Taylor model. H o w e v e r , as with previous w o r k using a commercial Al-Cu-Zr a l l o y , the rate sensitivity required to match the texture development is much greater than that measured in tensile tests. What is of particular interest here is that appears to be true for the material with relatively low m as well as in true superplastic materials. In order for the texture evolution in the A A 3 0 0 3 to correlate with the C P F E M results, a value of slip rate sensitivity of m ~ 0.6 would be required; considerably greater than the value from tensile testing, m = 0.2. 14
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It was suggested in previous w o r k that this discrepancy may indicate an additional source of texture divergence. It is interesting that in modelling the development of low temperature deformation textures, C P F E M gives quite good predictions of the rate of development with straining. There is an important difference between those cases and the present ones, however. Figure 8 s h o w s v i e w s of the surfaces of FE domains at a tensile strain of 0.3. In contrast to the real surfaces s h o w n in figure 4, the inhomogeneity in the FE model exists within the grains.
Figure 8. Views of the central regions of surfaces of finite element meshes following deformation to ε = 0.3. These show element boundaries superimposed on a grey scale representing effective plastic strain. There is some reduction in the degree of heterogeneity as the rate sensitivity increases, though significant heterogeneity remains at very high m values. This might be reasonable for low temperature deformation, but clearly differs from observation with elevated temperature deformation of fine grained materials. Heterogeneity under conditions of extensive dynamic recovery' is effectively quantised, with a structure consisting of subgrains and subgrain boundaries. In the materials involved here, the typical subgrain size is about the same, or larger than, the grain size. Under these conditions, the heterogeneity must be a c c o m m o d a t e d at the grain boundaries. The hypothesis is, then, that restriction of inhomogeneity within grains leads to greater intergranular inhomogeneity and greater divergence of texture. The possibility that the n u m b e r of slip systems m a y be restricted within a given grain may also need to be considered.
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CONCLUSION Measurements of texture changes due to hot tensile deformation of A l - M n and A l - M g Mn alloys are consistent with results from Al-Cu-Zr alloys, and show that higher solute content is associated with higher rate sensitivity. In turn, higher rate sensitivity is associated with a greater component of divergence in the texture development. S o m e of this may be due to a reduction in the convergent c o m p o n e n t - crystallographic slip rotations - w i t h increasingly rate sensitive intragranular slip. There appears to be an additional contribution to deformation inhomogeneity, and so texture spreading, in both low and high rate sensitivity fine-grained materials in hot deformation. This view is reinforced by similarities in the appearance of surface grids after deformation in the t w o alloys. REFERENCES ' j . Pilling and N. Ridley: in Superplasticity in Crystalline Solids, The Institute of Metals, (1989). C . Zener, quoted in C S . Smith, Grains Phases, and Interfaces: An Interpretation of Microstructure, Trans. Met. Soc. AIME, 1 7 5 , 15-51, (1948). B . M . Watts, M.J. Stowell, B.L. Baikie and D.G.E. O w e n , Superplasticity in Al-Cu-Zr Alloys, Part I: Material Preparation and Properties, J. Met. Sei., 1 0 , 189-197 (1976). J . S . Vetrano, C.A. Lavender, C H . Hamilton, M.T. Smith and S.M. B r u e m m e r , Superplastic Behavior in a Commercial 5083 A l u m i n u m Alloy, Scripta Metall. Mater., 3 0 , 565-570 (1994). P . A . Friedman, A.K. Ghosh, Microstructural Evolution and Superplastic Deformation Behavior of Fine Grain 5083A1. Metall. Mater. Trans., 2 7 , 3827-3839 (1996). 2
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B . Z h a n g . P.S. Bate and N.Ridley, The Effect of Strain Rate History on the D y n a m i c Grain Growth Behaviour of A A 5 0 8 3 , Mat. Sei. Forum, 5 5 1 - 5 5 2 , 627-632 (2007). K . Karman. C H . Johnson, and C H . Hamilton, A Study of Superplasticity in a Modified 5083 A l - M g - M n Alloy, Metall. Mater. Trans. A, 2 9 , 1211-1220 (1998). 7
J . W . Edington, K.N. Melton and C P . Cutler, Superplasticity, Prog. Mal. Sei., 2 1 , 61-158 (1976). ' R . C . Gifkins, M e c h a n i s m s of Superplasticity, Superplastic Forming of Structural Alloys; ed. by: N . E . Paton and C H . Hamilton, A publications of the Metallurgical Society of A I M E , 3-26 (1982). T.J. Nieh, J. Wadsworth and O.D. Sherby, Superplasticity in Metals and Ceramics, Cambridge University Press, C a m b r i d g e (1997). " V . S . Levchenko, O.V. Solovjeva, V.K. Portnoy and Yu.V. Shevnuk, Superplasticity of commercial A l - C u - M g - M n alloy A 1 9 , Mat. Sei. Forum, 1 7 0 - 1 7 2 , 2 6 1 - 2 6 6 (1994). F . Li, W.T. Roberts and P.S. Bate. Superplasticity and the D e v e l o p m e n t of Dislocation Structures in an A l - 4 . 5 M g Alloy, Acta Mater., 4 4 , 217-233 (1996). 8
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P . L . Blackwell and P.S. Bate, Superplastic Deformation Without Relative Grain Translation?, Mat. Sei. Forum, 3 0 4 - 3 0 6 , 189-194 (1999). P . S . Bate. F.J. H u m p h r e y s , N. Ridley, B . Z h a n g , Microstructure and Texture Evolution in the Tension of Superplastic Al-6Cu-0.4Zr, Acta Mater., 5 3 , 3 0 5 9 - 3 0 6 9 (2005). " P . S . Bate, K . B . Hyde, S.A. Court and F.J. H u m p h r e y s , D y n a m i c Grain Growth in Superplastic and Non-Superplastic A l u m i n i u m Alloys, Mat. Sei. Forum, 4 4 7 - 4 4 8 , 61-66 (2004). P . S . Bate, N . Ridley and B. Z h a n g , Mechanical Behaviour and Microstructural Evolution in Superplastic A l - L i - M g - C u - Z r A A 8 0 9 0 , Acta Mater., 5 5 , 4 9 9 5 - 5 0 0 6 (2007). K . Sotoudeh, P.S. Bate and F.J. H u m p h r e y s , The Effect of Copper Content on the D y n a m i c Grain Growth of Al-Cu-Zr systems, Mat. Sei. Forum, 5 5 8 - 5 5 9 , 803-809 (2007). 14
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Materials Processing a n d Texture
T E X T U R E - B A S E D P L A S T I C P O T E N T I A L S IN S T R E S S S P A C E Albert V a n Bael, Sampath K. Yerra, Paul Van Houtte Department M T M , Katholieke Universiteit Leuven Kasteelpark Arenberg 44 Bus 02450, BE-3001 Leuven, Belgium Albert Van Bael Departement I W T , K H L i m , C a m p u s Diepenbeek Agorialaan G e b o u w Β Bus 3, B E - 3 5 9 0 Diepenbeek, Belgium
ABSTRACT The Facet method is a new approach to describe the plastic anisotropy of textured materials. It has been developed for implementation into finite-element models for the simulation of metal forming processes. The method uses analytical expressions of plastic potentials in strain rate space and/or stress space. T h e expressions contain parameters that are derived from the texture and a multilevel model for the plastic deformation of the polycrystalline material. Three strategies are presented here for the identification of the parameters of the plastic potentials in case of the Facet method in stress space. Examples of anisotropic yield loci as obtained with the Facet method in stress space combined with the Taylor theory are shown for a model texture and for three industrial materials. The results are compared with those obtained directly from the Taylor theory. INTRODUCTION The strategies of Van Houtte et a l . ' to derive constitutive models for materials with texture-induced anisotropy and their implementation into finite element codes for the simulation of metal forming processes are based on the theory of dual plastic potentials in stress and strain rate space " . The constitutive models consist of analytical expressions describing the stressstrain relations at the macro-scale. Furthermore, the parameters in the analytical expressions are obtained from virtual tests using a macro/meso multilevel model to calculate the anisotropic response of the material. In previous work, the authors have used the Taylor theory as multilevel model. Their first expressions for potentials in strain rate space sufferred from artificial non-convexities of the corresponding yield loci in stress space in case of sharp textures, and this could cause divergence problems during finite-element simulations . Thanks to a modified expression for the plastic potential in strain rate space, and an additional iterative modification of the fitted parameters, strict convexity of the yield surfaces has been achieved . This latter approach is called the "Quantic m e t h o d " hereafter. It has been implemented in both implicit and explicit finite-element software p r o g r a m s through user material routines . A n e w approach, the "Facet m e t h o d " has been designed to o v e r c o m e several limitations of the Quantic method. T h e theoretical framework of the Facet method, its advantages over the Quantic method and some first results obtained with the Facet method in strain rate space are presented in another paper in the proceedings of the I C O T O M 15 c o n f e r e n c e " . The current paper focuses on the Facet method in stress space. More in particular, three strategies will be detailed for the identification of the parameters in the expressions in stress space. Results for yield locus sections obtained with the Facet method in stress space will be given and discussed. 1
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T e x t u r e - B a s e d Plastic P o t e n t i a l s in S t r e s s S p a c e
D E S C R I P T I O N OF T H E F A C E T M E T H O D IN S T R E S S S P A C E As a special case of the more general expressions presented in other p a p e r s ' , the following ones with 6 order terms are adopted here for the yield surface of a plastically incompressible material: 10
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,h
G'(s)=l
(1)
4 0 2
/ ^ ( G ' ( S ) = Σ K [d S ) K
Kp
\
(2)
p
«-=1
The 5,, (with p= 1.. .5) are the c o m p o n e n t s of the 5-dimensional vector representation of a deviatoric stress tensor S that belongs to the yield locus. AVand d ( i . e . plastic strain rate tensors with the components d ) are the parameters. To comply with the Facet method in strain rate s p a c e ' " , the set of plastic strain rates d should ideally correspond to yield stresses s* that approximately have equidistant directions covering the complete stress space. Unfortunately, the multilevel models are usually strain driven. This means that they provide the stress for a given plastic strain rate, but not the other way around. It is therefore not obvious to find the yield stress and the plastic strain rate that correspond to a given stress mode. Three strategies have been tested to identify the parameters d* and λ\, and these will be explained next. r
Kp
10
K
The first strategy uses a set of strain rates D that define a nearly equidistant grid of points on a hypersphere in 5-dimensional strain rate space. The 402 strain rates adopted in the Facet method in strain rate s p a c e have been used. The m a c r o / m e s o multilevel model is used for each strain rate D , producing the corresponding yield stress s and rate of work per unit volume D s (no summation over κ). A higher density of such yield stresses s may be expected in yield locus regions with a sharp curvature (i.e. near vertices). Furthermore, each tensor D is re-scaled into a tensor d^ such that the corresponding rate of work per unit v o l u m e is equal to unity: v
11
K
Kp
K
Kp
v
K
d
s p=l
Kp
(3)
K
The method for estimating the parameters λ is the same as the one for the parameters λ in the Facet method in strain rate s p a c e " , h o w e v e r using the value 1 for the constant C in the equations of that paper. Accordingly, the first guess for λ' is obtained assuming that for S=sK parameters other than λ\ and d would not contribute to the overall value of G'(S) in Eq. (2) and may thus be set to zero. This leads to: κ
κ
κ
K
Λ
λ
κ
=\
(4)
Using this estimate of λ\- a value of G'(s )=G'' is calculated and the new estimate for is then found as: v
G
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Materials Processing and Texture
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T e x t u r e - B a s e d Plastic P o t e n t i a l s in S t r e s s S p a c e
In the second strategy, a set of 402 deviatoric stresses that define a nearly equidistant grid of points on a hypersphere in 5-dimensional deviatoric stress space has been considered. The grid is constructed in the same way as the nearly equidistant grid in strain rate space. In order to find the corresponding strain rate directions, the Facet method in strain rate space is used to derive the corresponding dual potential first as described elsewhere". A Legendre transformation is then applied for each stress S* using an existing iterative procedure described by Van Houtte , giving the corresponding plastic strain rate D (with unit length) and also the scaling factor a such that s»- = a S lies on the yield surface. Each tensor D is re-scaled into a tensor d«so that Eq. (3) is satisfied, and the parameters AV are determined in the same way as in the first strategy. In the third strategy, the plastic strain rates are obtained as in the second one, i.e. by iterative Legendre transformations for an equidistant set of stresses S*- using the Facet method in strain rate space. But then the macro/meso multilevel model is applied for each resulting strain rate D*. This produces more accurate values for the corresponding yield stress s„- and rate of plastic work per unit volume. The identification of the parameters d and Althen proceeds as in the two other strategies. 6
K
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v
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Fig. 1 π-plane sections of yield surfaces obtained with the Facet method in stress space, using strategy 2. Model texture with a strong (001)[110] component (rotated cube). Texture index: 30.0. The thinnest Une is reached after 1 A'^-iteration step, the thicker after 2 steps and the thickest after 4 steps. Open dots: directly calculated by the Taylor model. FIRST RESULTS The first example to assess the performance of the Facet method in stress space (Fig. 1) illustrates the iteration algorithm for the parameters λ' (Eqs. 4-5). The considered material has a very sharp model texture with a Gaussian spread of 11° around the (001)[110] orientation (rotated cube), and a texture index (referred to as T.I. in the remainder of this paper) of 30. The second strategy was followed to identify the parameters d^and AV, using the Taylor model with κ
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{110} and {112} slip planes, < 1 1 1 > slip directions and a critical resolved shear stress of 1. The π-plane sections of the yield surfaces are obtained using Eqs. (1) and (2) with the AVvalues after 1, 2 and 4 iteration steps. N o c h a n g e s were observed after m o r e iteration steps. T h e superimposed open markers are stresses that have been obtained directly with the Taylor model. It should be noted that most of these points have not been used for the identification of the p a r a m e ters for the Facet expression in stress space. Actually it was attempted to generate points on the yield surface in 48 equidistant directions in stress space. The appropriate strain rate m o d e s were looked for by using the Facet expression in strain rate space and Legendre transformations for these directions in stress space. Then the Taylor model w a s applied for each of these strain rate modes. This procedure has been adopted because a grid based on equidistant orientations in strain rate space gave lots of stresses clustered near the vertices, but hardly one or two in the flat parts. The effects of the different strategies to identify the parameters d - and λ' are illustrated in Fig. 2. It s h o w s π-plane sections of yield surfaces obtained for the same model texture, using the three strategies of the Facet method in stress space. For comparison also the results obtained with the Quantic method (a 6 order expression in strain rate space) and superimposed points calculated directly with the Taylor model are included. Apparently for this sharp texture with four vertices in the observed π-plane section the first strategy is rather poor, which should be linked to the anticipated concentration of data points near the vertices. The second strategy improves both the flat parts and the corners drastically. The third strategy (necessitating 402 additional Taylor simulations) for the parameter identification only slightly improves the flat parts. In contrast with the Quantic result, none of the three strategies succeeds to reproduce the high curvatures predicted by the Taylor model in the vicinity of the vertices. Basically the same effect can be observed for the reproduction of vertices in the equipotential surface in strain rate space when the Facet method in strain rate space is used with a 6 order e x p r e s s i o n " . This limitation is linked to the order of the analytical expression (Eq. 2). A higher (even) order than 6 would be needed for sharper corners. However, also a larger set of parameters to be determined from a denser grid in strain rate and/or stress space m a y be required in this case. k
κ
t h
l h
The next e x a m p l e s (Fig. 3) are obtained for the same three industrial materials as used in another paper for the Facet method in strain rate s p a c e " : an aluminium alloy and a low-carbon steel with textures of moderate intensity (T.I. of 2.31 and T.I. 1.85, resp.), and an interstitial free (IF) steel with a strong texture (T.I. 4.84). Fig. 3 shows the π-plane sections of the yield surfaces obtained with the 6 order Facet expression in stress space (using the first two strategies) and with the Quantic method. The open markers represent stresses that are calculated directly with the Taylor model. These are obtained for 48 strain rate m o d e s with equidistant orientations in strain rate space. Identical Taylor model assumptions have been used for the two steels as for the model texture above whereas {111} slip planes and < 1 1 0 > slip directions have been assumed for the aluminium. The figures at the left-hand side are obtained with the first strategy, those at the right-hand side with the second strategy. A close look at the figures reveals that the flat parts are better reproduced by the second strategy (especially for the aluminium alloy), but also that the corners connecting the flat parts are still s o m e w h a t rounded off. The results obtained with the third strategy are not shown, since visually these can hardly be distinguished from those obtained with the second strategy. lh
The C P U times required for the determination of the parameters d and λ' depend on the chosen strategy. In each case, the texture is discretised first into 5000 discrete orientations using the "statistical m e t h o d " " . T h e n the Taylor model is performed for 402 strain rate m o d e s and at K
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T e x t u r e - B a s e d Plastic Potentials in Stress S p a c e
most 5000 different crystal orientations. Depending on the sharpness of the texture (and thus the precise n u m b e r of different crystal orientations), this takes about 30 minutes (for the sharp model texture) to 1 hour (for the industrial textures) o n a standard PC (2 GHz, 256 M B R A M ) . The iterative identification of the A V v a l u e s only requires a fraction of a second on the same PC. In the second strategy about half a minute more is needed for the Legendre transformations. Finally, the third strategy is the most computationally intensive, because it requires an additional set of Taylor simulations for 4 0 2 strain modes and the complete texture.
Fig. 2 π-plane sections of yield surfaces obtained with the Quantic method and the Facet method in stress space. M o d e l texture with a strong (001)[110] component (rotated cube). Texture index: 30.0. Thin lines: Quantic method. Thick lines: Facet method with strategy 1 in (a), 2 in (b) and 3 in (c). O p e n dots: yield stresses directly calculated by the Taylor model.
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T e x t u r e - B a s e d Plastic P o t e n t i a l s in S t r e s s S p a c e
Fig. 3 it-plane sections of yield surfaces obtained with the Quantic method and the Facet method in stress space. Materials: (a, d) Aluminium 6016-T4 alloy (TT. 2.31): (b, e) Low carbon steel (T.I. 1.85); (c, f) IF steel (T.I. 4.84). Thin lines: Quantic method. Thick lines: Facet method with strategy 1 in (a-c), 2 in (d-f). Open dots: directly calculated by the Taylor model.
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CONCLUSION The yield surfaces obtained with the Facet method in stress space are almost identical to those obtained with the Quantic method, except for yield loci with rather sharp corners. In this case the 6 * order Facet expressions in stress space lead to yield loci that are more rounded off than the Quantic ones. Further research is required to find out to which extent sharper corners can be better reproduced by higher order expressions. Also, aspects such as r-values (or qvalues) and variations of the uniaxial yield stress as a function of the orientation of the tensile specimen need further investigation. Apart from the several advantages of the Facet method over the Quantic method as discussed in another paper in the I C O T O M 15 p r o c e e d i n g s " , the m a i n advantage of the Facet method in stress space probably resides in C P U savings during finite-element simulations. ACKNOWLEDGEMENT Financial support from the Interuniversity Attraction Poles Program from the Belgian State through the Belgian Science Policy agency, contract IAP6/24, is gratefully acknowledged. REFERENCES ' P . Van Houtte, A. Van Bael, J. Winters, The incorporation of texture-based yield loci into elastoplastic finite element programs. Textures and Mierostructures, 2 4 , 2 5 5 - 2 7 2 (1995). P . V a n Houtte, A. Van Bael, Convex Plastic Potentials of 4th and 6th Rank for Anisotropic Materials, Int. J. Plasticity, 2 0 , 1505-1524 (2004). R . Hill, Constitutive Dual Potentials in Classical Plasticity, J. Mech. Phys. Solids, 3 5 , 23-33 (1987). H . Ziegler, An Introduction to Thermomechanics, North Holland Publishing C o m p a n y , A m s t e r d a m (1977). J . Lemaitre and J. L. Chaboche, Mechanics of Solid Materials, Cambridge University Press (1990). P . Van Houtte, Application of Plastic Potentials to Strain Rate Sensitive and Insensitive Anisotropic Materials. Int. J. Plasticity, 1 0 , 719-748 (1994). M . Arminjon, Β. Bacroix, D. Imbault, J.L. Raphanel, " A fourth-order plastic potential for anisotropic metals and its analytical calculation from the texture function", Acta Mechanica, 1 0 7 , 33-51 (1994). G . I . Taylor, Plastic strain in metals, J. Inst. Metals, 6 2 , 307-324 (1938). S . Ristic, S. He, A. Van Bael, P. V a n Houtte, "Texture-based explicit finite-element analysis of sheet metal forming", Materials Science Forum 4 9 5 - 4 9 7 , 1535-1540 (2005). P . Van Houtte, S.K. Yerra and A. Van Bael, The Facet method: a hierarchical multilevel modelling scheme for anisotropic convex plastic potentials, accepted for publication in Int. J. Plasticity (2007). " P . Van Houtte, S. K. Yerra and A. V a n Bael, Hierarchical Multi-Level Modelling of Plastic Anisotropy using C o n v e x Plastic Potentials, submitted for the Proceedings of ICOTOM 15 (2008). H . J . Bunge, Texture Analysis in Materials Science, Butterworth, London (1982). L . S . Toth and P. V a n Houtte, Discretization Techniques for Orientation Distribution Functions, Textures and Mierostructures, 1 9 , 229-244 (1992). 2
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HIERARCHICAL MULTI-LEVEL MODELLING CONVEX PLASTIC POTENTIALS
OF
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ANISOTROPY
USING
Paul Van Houtte, Sampath K. Yerra, Albert V a n Bael Department M T M , Katholieke Universiteit Leuven Kasteelpark Arenberg 44 Bus 02450, BE-3001 Leuven, Belgium Albert Van Bael Departement I W T , K H L i m , C a m p u s Diepenbeek Agorialaan G e b o u w Β B u s 3, B E - 3 5 9 0 Diepenbeek, Belgium
ABSTRACT A n e w approach, the "Facet" method, is proposed to implement the anisotropic plastic behaviour of textured materials in finite element m o d e l s for the simulation of metal forming. It is based on an analytical expression of plastic potentials in strain rate space and/or stress space. The method has been designed to be used in combination with a multilevel model for the plastic deformation of the polycrystalline material. The parameters of the expressions for the plastic potentials are identified by fitting to the predictions of the multilevel models instead of being fitted to the results of mechanical tests. The n e w formulation has the advantage that it automatically ensures convexity of the anisotropic yield loci. Moreover, it is more flexible than the previous method, and can in principle be used with more advanced multilevel models than the Taylor theory. In contrast to most existing methods, the new method can also be applied to materials with a stress differential effect. For n o w anisotropic equipotential surfaces in strain rate space will be shown. They are obtained by the Facet method combined with the Taylor theory, for a model texture and for three industrial materials, and will be compared with results directly obtained from the Taylor theory, without passing through an analytical model. INTRODUCTION There are roughly two ways of using an "engineering" finite element (FE) model for the simulation of forming processes of materials with texture-induced plastic anisotropy: (i) each time the FE model at the engineering length scale needs the stress-strain relationship at a "point", it m a k e s a call to macro/meso multilevel model for the material behaviour at macro-scale. The latter model calculates the average stress-strain relationship of a representative volume element ( R V E ) at the macro-scale, which corresponds to the "point" at the engineering length scale and consists of several hundreds or thousands of grains. In this, it takes account of the interactions between R V E s at the meso-scale which each consist of a single grain or a small cluster of grains; (ii) the finite element model at the engineering length scale uses a closed form formulation of a constitutive model which is formulated by analytical expressions (such as yield locus formulations) describing the anisotropic stress-strain relations at macro-scale; this is called the "hierarchical method". There are two approaches to identify the parameters of analytical expressions which constitute the constitutive model in the hierarchical method. In the first, the parameters are identified from the results of mechanical tests'. In the second, the parameters are obtained from virtual tests using a multilevel modelling scheme where the anisotropic response of the material
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Hierarchical Multi-level M o d e l l i n g of Plastic A n i o s t r o p y U s i n g C o n v e x Plastic
Potentials
is calculated by a macro/meso multilevel model. The latter must take the crystallographic texture of the material into account. Depending on the model chosen, some other aspects of the microstructure also may need to be known. The advantages of the second approach are its capability to embody complex deformation m o d e s (including some which are impossible to achieve in the laboratory testing environment) and the less work-intensive nature compared to the practical route. Disadvantage is that it is not very useful unless the multilevel model is reliable. Previous work of Van Houtte and Van Bael " includes the application of the second approach for the development of an analytical expression which m a k e s use of average Taylor factors calculated by the Taylor t h e o r y as multilevel model for several thousands of deformation modes. Despite its successful implementation in FE codes, this method, however, features nonconvexity for materials with very sharp textures in some vertices of the stress space leading to divergence of the F E solution . This led to a development of a new method (called "Quantic method" hereafter) for which at least a mathematical criterion for overall convexity could be formulated . In case the parameter identification led to a non-convex analytical expression, an additional algorithm was proposed which slightly modifies the parameters to surpass this problem . Although reliable, the procedure is not very elegant and quite complex. Another drawback of the Quantic method is that it was defined in strain rate space but not in stress space, which m a k e s it computationally intensive for determining the onset of yielding in both implicit and explicit FE simulations. The novel expression proposed in this paper not only overcomes these limitations but is also is well-suited for materials with stress differential effect . The analysis given below is rigid-plastic. Materials are plastically incompressible. 2
3
4
2
3
3
5
D E S C R I P T I O N OF T H E N E W M E T H O D Theory of plastic potentials It is possible to describe the plastic anisotropy of polycrystalline materials with texture by means of dual plastic potentials " . In their implementation. Van Houtte et a l . ' consider the well-known yield locus in stress space and a function (*(D) in strain rate space which is identified as Ρ , the plastic work dissipated per unit volume. D is the macroscopic strain rate tensor. ψ(Ό) has the following property: 6
8
S=-
5
1 \+μ
di//(D) 3D
9
(1)
S is the deviatoric stress tensor derived from the plastic flow stress, μ is the strain rate sensitivity. It is usually set to zero for room temperature deformation of metals. Eq. 1 shows, that if the function (*(D) were known under an analytical form, the flow stress can be calculated for every strain rate tensor D. This would effectively model the plastic anisotropy, and would m a k e the construction of the yield locus possible when needed. In a similar way, a plastic potential also exists in stress space. Not to make the text too lengthy, only the formulas for the case that strain rate sensitivity can be neglected are given. More general formulas can be found e l s e w h e r e . The plastic potential is described by: 5,9
G'„(S) = 1
818
Materials Processing and Texture
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Hierarchical Multi-level M o d e l l i n g of Plastic A n i o s t r o p y U s i n g C o n v e x Plastic
Potentials
with
λ should be non-negative. Eq. (3) describes a surface in stress space known as the yield locus. Note that the functions (*(D) and G' (S) can be derived from each other. n
Quantic method. T h e idea now is to choose analytical expressions for i/KP) and G\ (S) to be used at the macro-scale, and to identify their coefficients by means of "virtual mechanical tests" performed by a m e s o / m a c r o multilevel model, such as the Taylor theory , which must have sufficient information about the microstructure and/or the texture of the material. First, the meso/macro multilevel model will be asked to compute the values of Ρ for a series of strain tensors D . 4
v
The following generic expression is n o w proposed for ψ(Ώ):
ψ(Ό)=[β,(ρ)Υ
(4)
G„(D) is a h o m o g e n o u s polynomial of degree η in terms of the components of the tensor D. In the present work, we will consider the values 6 and 8 for n. In case strain rate sensitivity can be neglected, m=\/n; otherwise, it is m =
—
(5)
η T h e Quantic m e t h o d
G„(D)=
a' D D D Dß,O nmu
p
tl
r
u
3
uses the following sum as expression for G„(D) (case n=6):
with
1< ρ
(6)
The usual indices / and j of the components of strain rate tensors have been replaced by a single contracted index p, which ranges from 1 to 5 in case of plastically incompressible materials. This notation will also be used below for deviatoric stress tensors. The indices q, r, s, t, u have the same nature as p. A h o m o g e n e o u s algebraic expression as given here is called a "quantic" in mathematics. In previous w o r k the option had been taken to use the full constraint Taylor t h e o r y in its most basic form as the meso/macro multilevel model to perform the "virtual mechanical tests". It is then possible, in imitation of the pioneering work by B u n g e , to maximally exploit the possibilities of the harmonic analysis of functions in Euler space and calculate the average Taylor factor M for 35000 directions in strain rate space in a fraction of a second on a standard P C . In this, a library of Taylor factors which have been pre-calculated by the Taylor model is used. However, as soon as another multilevel is used, it is no longer possible to use this fast method. 3
4
10
1 1
N o t e that Ρ can be computed from M :
Materials Processing a n d Texture
819
Hierarchical Multi-level Modelling of Plastic Aniostropy Using C o n v e x Plastic Potentials
c
P=T D M(a)
(7)
M
with
(8) and (direction in strain rate space, also called strain m o d e ) : a = D/VD:D
(9)
The parameters oi ,u in Eq. (6) are then identified by m e a n s of a least squares method (LSQ). A problem then is, that it is not guaranteed that a surface of the type (*(D)=constant would be convex. This also holds for the corresponding surface G'„(S) = 1. Examples of the types of non-convexities which can occur (especially if the textures are sharp) have been described in earlier w o r k . Experience has shown that they cannot be tolerated when the plastic potentials are used as constitutive model of the material in finite element simulations. Yet it had turned out to be possible to derive a criterion for global convexity as well as an algorithm which iteratively modifies the parameters already obtained by a L S Q until they are convex. The changes are rather modest; however, it was felt that an analytical expression which would automatically guarantee convexity would be preferable. pqn
2
Facet method in strain rate space. The Facet method m a k e s use of a particular class of quantics of degree n, namely those given by: κ (10)
Λ,-and s ( i . e . . the tensor with the components ί ) are parameters, η is an even natural number which is at least 2. Accordingly, G„ is never negative. In addition, it can be shown that surfaces (i
φ
5
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Materials P r o c e s s i n g a n d T e x t u r e
Hierarchical Multi-level M o d e l l i n g of Plastic A n i o s t r o p y U s i n g C o n v e x Plastic Potentials
space is shown in a stereographic projection o f a unit sphere (Fig. 1) and consists of the pseudocrystallographic directions [001], [ O i l ] , [111] and [113] (cubic lattice). T h e average distance between neighbouring points is 3 0 ° . This choice o f directions in strain rate space has the merit, that the five co-ordinate axes all play exactly the same role in t h e grid: the grid would not change if t w o of them would be interchanged. T h e 402 directions in strain rate space are described by strain rate tensors w h i c h are called D . T h e stresses calculated by the multilevel model deliver the values o f the s -parameters used in Eq. (10). These parameters are no more modified from then on. T h e further fitting procedure is described here for the case μ=0. It consists in an iteration to find the values of t h e parameters λ„ In a first step, the results of the macro/meso multilevel model are used to re-scale the tensors Y) so that Ρ , as calculated by the multilevel model (and given by Eq. 7), is exactly equal to C, a chosen value for the work rate per unit volume. In this way, the multilevel model generates a series of points through w h i c h ideally the surface (*(D)=C as obtained from t h e analytical expression, should go. Next step is the choice of the initial values of λ . They are chosen assuming that for D=D , other λ w o u l d not contribute to the overall value o f G„(D) (Eq. (10)) and may be set to zero. This leads to: r
K
K
κ
V
G (D ) n
r
κ
(Eq. 10) is n o w calculated. A correction factor is n o w be applied on λ , leading κ
to a n e w estimate:
(12)
T h i s can be repeated several times. It will be s h o w n below (Fig. 2) h o w this algorithm performs. Facet method in stress space. Eqs. (2-3) are the formulation of the plastic potential (yield locus) in stress space w h e n strain rate sensitivity is neglected. G'„ (S) is constructed using an expression similar to Eq. (10):
(13)
Ideally one should n o w use a meso/macro multilevel model which calculates the strain rate for a given stress direction. It would then be possible to get the parameters A for a m o r e or less equidistant set o f equi-angular stress directions. (In the previous section, the a i m w a s to get s,- values for a set of equi-angular strain rate directions). Unfortunately, the multilevel models usually obtain the stress from the strain rate, not the other way around. Yet the simplest way to identify the parameters d i s t i l l is to use the multilevel model a s before; rescale the tensors D,- as explained in the previous section, and use these then as d„-. However, the stress directions which values w o u l d most often not be equi-angular. So other correspond to the resulting set of d K
K
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Hierarchical Multi-level M o d e l l i n g of Plastic A n i o s t r o p y U s i n g C o n v e x Plastic P o t e n t i a l s
strategies for the choice o f uV have been devised as well, and are discussed in another paper in the proceedings of the I C O T O M 15 c o n f e r e n c e . The method for estimating the parameters λ\is exactly as the one for the parameters λ described in the previous section. Results are given in a the other paper'". 12
κ
Fig. 1 Stereographic projection s h o w i n g a set of equi-angular directions in 3 D-space with a resolution of about 30°. Inspired by a (OOl)-standard projection of a cubic crystal.
Fig. 2 Equipotential surface calculated by the Facet method with n~ 6. π-plane section of strain rate space. Model texture with a strong (001 )[110] component (rotated cube). Texture index: 30.0. The thin line is reached after 1 Aj-iteration step, the thick line after 4 steps. Open dots: directly calculated by the Tavlor model.
822
Materials Processing a n d Texture
H i e r a r c h i c a l Multi-level M o d e l l i n g of Plastic A n i o s t r o p y U s i n g C o n v e x Plastic Potentials
FIRST RESULTS T h e p o w e r of the new method w a s examined, both qualitatively and quantitatively, for four examples. T h e first (Fig. 2) is intended to show the effect of the / ^ a l g o r i t h m (Eqs. 11-12). A model texture is used with a single texture component around the (001 )[ 110] orientation (rotated-cube). This texture component is very c o m m o n l y observed in ferritic steels. The texture c o m p o n e n t is Gaussian with a spread of 11° (Φο defined by B u n g e ) . The texture index is 30.0. The next three e x a m p l e s (Fig. 3) are industrial materials: an aluminium 6016-T4 alloy and a lowcarbon steel, both with a texture of moderate intensity (T.I is 2.31 and 1.85, respectively), and an interstitial-free steel with a strong texture (T.I. is 4.84). The O D F s of the crystallographic texture of these materials have been calculated from measured X R D pole figures, and have then been discretised into sets of 5000 discrete orientations using the "statistical m e t h o d " . The discrete textures have been fed to the conventional Taylor theory with 24 slip systems: {110} and {112} slip planes and < 1 1 1 > slip directions, in order to identify the parameters of the Facet model in strain rate space, as explained above. Figs. 2-3 show a section of the equipotential surfaces in the π-plane section of these four materials. T h e points of these sections represent strain rates for which all strain c o m p o n e n t s are zero except Du, D22 and D33, while maintaining the constancy of v o l u m e . The figures also show the projections of these axes on the plane of the section. The open circles represent points in strain rate space which have been directly calculated with the Taylor model. These points are not necessarily those w h i c h had been used for the identification procedure; rather, they are additional points only needed for the assessment of the quality of the reproductions m a d e by the analytical models (Facet and Quantic). Fig. 2 demonstrates that after four iteration steps for λ , the Facet curves pass almost right through the Taylor points, except at the vertices visible in Fig. 2. It is seen in Fig. 3 that the curves of the Facet method after four iteration steps (thick line) and the Quantic method (thin line) almost coincide. 13
14
κ
DISCUSSION AND CONCLUSION T h e first results clearly show, that the new Facet method can reproduce the plastic anisotropy of cubic metals as predicted by the Taylor method very well, leading to convex plastic potentials, equally accurate in all parts of strain rate space. The only deviations are observed in regions of very high curvature (Fig. 2). However, the older Q u a n t i c method was also capable of doing that. It seems then necessary to highlight the advantages of the n e w method: (i) the theory of the n e w Facet method is m u c h simpler and elegant than that of the Quantic method, and therefore much m o r e accessible to other researchers; the same holds for the software implementation. (ii) the parameter identification procedure of the Quantic method does not m a k e use of the stress tensors which are obtained by the Taylor theory, only of the average Taylor factors . The main reason is that it would take an inconvenient non-linear L S Q fitting procedure to identify the parameters on the basis of the stress tensors. In contrast to this, the n e w Facet method manages easily to extract useful information from the stress tensors; (iii) as a result, 4 0 2 estimations of these stress tensors are vastly enough to accurately identify the Facet model. H e n c e the latter can be done without m a k i n g use of the powerful tools offered by harmonic analysis of texture functions w h e n combined with the simplest possible version of the Taylor model. This m e a n s that the Facet method can be used in combination with more 3
3
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Hierarchical Multi-level M o d e l l i n g of Plastic A n i o s t r o p y U s i n g C o n v e x Plastic P o t e n t i a l s
advanced macro/meso multilevel models. Future work will include the use of (a) a self-consistent m o d e l , (b) the A L A M E L - m o d e l , (c) a crystal-plasticity finite element model ( C P F E M ) . (iv) it has been s h o w n that the Facet method can be adapted easily for materials with a stress differential effect, i.e. the flow stress is not simply inverted upon inversion of the deformation m o d e . Thus it will be possible to apply it on hep alloys, or to use it in combination with a macro/meso/micro 3-level model which takes substructure-induced anisotropy into a c c o u n t . 15
16
1 7
5
18
Fig. 3 Equipotential surfaces calculated by the Quantic and the Facet methods (the latter with 4 iteration steps for Aj-and n= 6). π-plane section of strain rate space, a: A l u m i n i u m 6016-T4 alloy (T.I=2.31). b: Low Carbon Steel (T.I=1.85). c: IF steel (T.I=4.84). Thin lines: Quantic method. Thick lines: Facet method. O p e n circles: strain rates (belonging to the equipotential surface) directly calculated by the Taylor model. ACKNOWLEDGEMENT Financial support from the Interuniversity Attraction Poles Program from the Belgian State through the Belgian Science Policy agency, contract I A P 6 / 2 4 , is gratefully acknowledged.
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Materials Processing and Texture
H i e r a r c h i c a l Multi-level M o d e l l i n g of Plastic A n i o s t r o p y U s i n g C o n v e x Plastic Potentials
REFERENCES ' M . Z y c z k o w s k i , Anisotropic Yield Conditions. In: J. Lemaitre (Ed.), Handbook of Materials Properties, vol. L Deformation of Materials. A c a d e m i c Press, N e w York, pp. 155-165 (2001). P . Van Houtte, A. Van Bael, J. Winters, The incorporation of texture-based yield loci into elastoplastic finite element programs. Textures and Mierostructures, 2 4 , 2 5 5 - 2 7 2 (1995). P . V a n Houtte, A. Van Bael, C o n v e x Plastic Potentials of 4th and 6th Rank for Anisotropic Materials, Int. J. Plasticity, 2 0 , 1505-1524 (2004). G . I . Taylor, Plastic strain in metals, J. Inst. Metals, 6 2 , 307-324 (1938). P . Van Houtte, S.K. Yerra and A. Van Bael, The Facet method: a hierarchical multilevel modelling scheme for anisotropic convex plastic potentials, submitted for publication to the International Journal of Plasticity, 2007. R . Hill, Constitutive Dual Potentials in Classical Plasticity, J. Mech. Phys. Solids, 3 5 , 23-33 (1987). H . Ziegler, An Introduction to Thermomechanics, North Holland Publishing Company, A m s t e r d a m (1977). J . Lemaitre and J. L. C h a b o c h e , Mechanics of Solid Materials, Cambridge University Press (1990). P . V a n Houtte, Application of Plastic Potentials to Strain Rate Sensitive and Insensitive Anisotropic Materials. Int. J. Plasticity, 1 0 , 719-748 (1994). H . J . B u n g e , S o m e applications of the Taylor theory of Polycrystal Plasticity, Kristall und Technik, 5, 145-175 (1970). " P . Van Houtte, Fast calculation of average Taylor factors and Mandel spins for all possible strain m o d e s , International Journal of Plasticity, 1 7 , 807-818 (2001). A . Van Bael, S. K. Yerra and P. Van Houtte, Texture-Based Plastic Potentials in Stress Space, submitted for the Proceedings of ICOTOM 15 (2008). H . J . B u n g e , Texture Analysis in Materials Science, Butterworth, London (1982). L . S . Toth and P. V a n Houtte, Discretization Techniques for Orientation Distribution Functions, Textures and Mierostructures, 19, 229-244 (1992). " R . A . L e b e n s o h n , C.N. T o m é , A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys, Acta Metall. Mater, 4 1 , 2 6 1 1 - 2 6 2 4 (1993). P . Van Houtte, S. Li, M . Seefeldt and L. Delannay, Deformation texture prediction: from the Taylor model to the advanced Lamel model, Int. J. Plasticty 2 1 , 589-624 (2005). L . Delannay, P. J. Jacques and S. R. Kalidindi, Finite element modeling of crystal plasticity with grains shaped as truncated octahedrons, International Journal of Plasticity, 22, 1879-1898 (2006). B . Peeters, S.R. Kalidindi, C. Teodosiu, P. Van Houtte, E. Aernoudt, A theoretical investigation of the influence of dislocation sheets on evolution of yield surfaces in single-phase b.c.c. polycrystals, Journal of the Mechanics and Physics of Solids, 5 0 , 783-807 (2002). 2
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8
9
I0
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I 6
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Author Index
Adams, Β. Α., 687 Adams, B. L , 509, 647, 657 Ahmadi, S., 657, 5 0 9 Al-Buhamad, O., 7 8 3 Alexander, D. J., 5 6 3 Al-Fadhalah, K., 6 6 5 Ali, Α., 6 1 7 Arai, S„ 2 7 5 Azzi, M„ 6 3 , 4 4 3 Bacaltchuk, C. M. B., 4 0 5 Bacroix, B., 243, 3 4 9 Banarjee, S., 5 3 3 Banerjee, S., 5 9 3 Barbé, L , 71, 3 3 3 Barbier, D., 79, 87 Barnett, M. R„ 451 Bassman, L, 7 8 3 Bate, P. S„ 801 Baudin, T., 123, 251 Beaudoin, Α., 6 6 5 Beyerlein, I. J., 5 6 3 Bhanumurthy, K., 29, 3 5 Bhattacharjee, D., 103 Bingert, J. F., 17 Bolle, B„ 87 Böttcher, Α., 143 Boyce, D. Ε., 701
Bozzolo, Ν„ 461 Bracke, L, 341 Brekelmans, W. A. M„ 671 Brewitt, N., 521 Brown, D. W., 5 6 3 Cabanas-Moreno, J. G., 123 Carpenter, J. O., 2 8 5 Castello-Branco, G. Α., 4 0 5 Castelnau, O., 3 4 9 Cernik, M., 9 5 Chakraborty, Α., 103 Chauveau, T., 3 4 9 Chen, E„ 671 Chen, L, 6 3 7 Cho, J„ 3, 4 7 3 Choi, J. K., 4 8 3 Choi, S.-H., 4 8 3 Chosh, C , 161 Cizek, P., 451 Coghe, F., 491 Colligan, K. J., 17 Cong, D. Y., 397 Cottam, R„ 501 Cruz-Gandarilla, F., 123 Dahlman, P., 301 Davis, B„ 501
827
Dawson, P. Ρ,., 701 Decroos, Κ., 1 1 5 Delannay, L , 721 Derose, D. J., 3 5 Dey, D. K„ 29, 3 5 Dey, G. Κ., 533, 5 9 3 Driver, J. H„ 7 6 5 Duchêne, L, 671 Elia, J., 131 Elwazri, Α., 577 Esling, C , 3 8 1 , 389, 397, 421 Etter, A. L, 251 Faghihi, S „ 4 4 3 Fang, W„ 3 6 5 Feng, H„ 3 7 1 , 7 7 5 Ferry, M., 7 8 3 Fonda, R. W„ 17 Fromm, B. S., 509, 657 Frommert, M., 143 Fu, Ζ., 3 6 5 Fukutomi, Η., 6 7 9 Fullwood, D. T . , 647, 657, 687 Fundenberger, J. J., 8 7 Furrer, D., 521 Garmestani, H., 405, 4 1 3 Gay, B„ 131 Geers, M. G. D., 671 Gey, N., 79 Ghosh, P., 151 Godet, S., 577 Gourgues-Lorenzon, A.-F., 2 4 3 Gurao, N., 5 8 5 Gurao, N. P., 617 Habraken, A. M., 671 Haldar, Α., 161 Hänninen, H., 53 Hao, F„ 751 Hartley, C , 701 Hatano, M., 2 2 3 He, C , 389, 4 3 5 Hilinski, E., 9 5 Hiwarkar, V. D., 533, 5 9 3
828
·
Materials Processing and Texture
Homma, H., 2 7 5 Hoseini, M., 7 1 3 Hou, C.-K., 173, 183, 231 Houbaert, Y., 115 Hsu, E„ 571 Hu, X„ 3 6 5 Hu, Y., 751 Humbert, M., 79 Humphreys, F. J., 265, 801 Hünsche, I., 791 Imai, N„ 3 2 5 Inagaki, H., 193 Isaenkova, M., 5 3 9 J a c q u e s , P. J., 577 Jiang, L , 547, 555, 577 Jonas, J. J„ 547, 577 Kalidindi, S. R„ 687, 701 Kang, S.-B., 4 7 3 Kanjarla, A. K„ 721 Kassner, M. E., 5 5 5 Kato, H., 6 0 9 Kestens, L A. I„ 207, 217, 341 Khatirkar, R., 2 1 7 Kim, D. H„ 4 8 3 Kim, H.-W., 4 7 3 Kimura, K., 2 2 3 Klein, H., 381 Knabl, W„ 791 Knutson Wedel, M., 301 Koh, H„ 601 Kokawa, H. 4 3 Knezevic, M., 509, 701 Knipling, K. E., 17 Krishnan, J., 3 5 Kumbhar, N. T., 29, 3 5 Küsters, S., 731 Lahn, L , 143 Laik, A„ 3 5 Lamelle, C , 87 Lebensohn, R., 701 Lecomte, J-S., 131 Lee, C.-W., 4 3
A u t h o r Index
Lee, H. W., 4 8 3 Lee, T.-H., 173 Leffers, T., 7 4 3 Lesko, Α., 95 Li, Η., 5 7 1 , 713 Li, L , 3 6 5 Li, S., 5 6 3 Li, Χ., 6 3 7 Liao, C . - C , 173, 231 Liu, C . - C , 183 Liu, D., 751 Liu, W., 751 Lorich, Α., 791 Lorimer, G., 501 Lubin, S., 2 4 3 Maazi, N., 251 Manikrishna, Κ. V., 5 3 3 Mao, W., 3 7 1 , 637, 7 7 5 Martin, E., 5 7 7 Mendoza Leon, H., 123 Meng, L, 6 3 7 Meratian, M., 7 1 3 Minamiguchi, S., 601 Miracle, D. B., 5 8 5 Mishra, S. K., 257 Mononen, J., 53 Montheillet, F., 2 4 3 Morris, P. R., 7 5 9 Nakamichi, H., 2 6 5 Namba, E., 2 7 5 Narasimhan, K., 257, 5 3 3 Niewczas, M., 6 6 5 Niezgoda, S. R., 687 Nowell, M. M., 285, 3 5 7 Nyborg, L , 301 Oertel, C.-G., 791 Ohms, C , 115 Okayasu, K., 6 7 9 Oliveira, T. R., 3 1 7 Pant, P., 257, 5 9 3 Paul, H., 7 6 5 Pei, D., 7 7 5
Penelle, R., 123, 2 5 1 , 2 9 3 Penning, J., 341 Pérez-Prad, M. T., 5 5 5 Perlovich, Y., 5 3 9 Petit, Ε., 131 Petrov, R., 207, 2 1 7 Pitchon, V., 131 Predmersky, M., 95 Quadir, M. Z., 7 8 3 Raabe, D., 143, 3 3 3 Rabet, L, 491 Ray, R. K., 103, 151, 311 Réglé, H., 2 4 3 , 3 4 9 Reynolds, A. P., 17 Robson, J., 501 Rouag, N., 2 9 3 Roy, S., 5 8 5 Ruan, Ο. Α., 5 5 5 Rugg, D., 521 Ryttberg, K., 301 Saha, R., 311 S a h o o , S. K., 29, 35, 533, 5 9 3 Sakai, T., 601 Sakakibara, M., 6 7 9 Salem, Α. Α., 701 Samajdar, I., 29, 35, 217, 257, 533, 5 9 3 Sandim, H. R. Z., 3 1 7 Sato, Y. S., 4 3 Saukkonen, T., 53 Savolainen, K., 5 3 Sébastien, Α., 79 Seefeldt, M., 731 Sekkak, C , 2 9 3 Semiatin, S. L., 701 Serebryany, V. N., 6 2 9 Sheikh-Ali, A. D., 4 1 3 Shi, M., 6 0 9 Siqueira, R. P., 3 1 7 Skrotzki, W., 791 Sotoudeh, K., 801 Srinivasan, R., 5 8 5 Srivastav, D., 533, 5 9 3 Stevens, Κ. Α., 687
Materials Processing and Texture
·
829
A u t h o r Index
Sugimoto, Α., 4 3 Suthar, R. L, 35 S u w a s , S., 585, 617 Szpunar, J. A„ 6 3 , 4 4 3 , 5 7 1 , 7 1 3 Tabrizian, M., 4 4 3 Takahashi, Α., 2 2 3 Takayama, Y., 6 0 9 Tamirisakandala, S., 5 8 5 Tang, W„ 17 Tewari, R., 29, 593 Tidu, Α., 87 Timofeev, V. N., 6 2 9 Tiwari, R., 5 3 3 Tomé, C , 701 Tomida, T., 3 2 5 Tsurekawa, S., 421 Turner, T. J., 701 Umetsu, T., 6 0 9 Ushigami, Y., 2 7 5 Utsunomiya, H., 601 Van Bael, A„ 809, 817 Van Houtte, P., 4 9 1 , 7 2 1 , 7 3 1 , 809, 817 Verbeken, K., 7 1 , 3 3 3 , 341 Verlinden, Β., 671 Wagner, F., 461
830
Materials Processing and Texture
Wakita, M., 3 2 5 Wang, Y., 3 6 5 Wang, Y. D., 397 Watanabe, H., 6 0 9 Watanabe, T., 421 Wauthier, A„ 3 4 9 Wert, J. A„ 17 Wright, S. I., 285, 357 Wu, Q., 3 6 5 Wu , Y., 389, 4 3 5 Xu, L, 371 Yang, P., 3 7 1 , 637 Yerra, S. K., 809, 817 Yoshida, M., 3 2 5 Zaefferer, S., 143 Zaliznyak, Y. Α., 6 2 9 Zhang, Β., 3 6 5 Zhang, J., 751 Zhang, Y„ 389, 421 Zhang, Y. D., 3 8 1 , 3 9 7 Zhao, H., 3 6 5 Zhao, X., 3 8 1 , 389, 397, 4 2 1 , 4 3 5 Zhou, M., 751 Zobrist, C , 143 Zuo, L„ 3 8 1 , 389, 397, 4 2 1 , 4 3 5
