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Evolution of orientations and deformation structures within individual grains in cold rolled columnar grained nickel
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Acta Materialia 59 (2011) 5451–5461 www.elsevier.com/locate/actamat
Evolution of orientations and deformation structures within individual grains in cold rolled columnar grained nickel G.L. Wu a,b,c,⇑, A. Godfrey c, G. Winther b,d, D. Juul Jensen b,d, Q. Liu e a School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, People’s Republic of China Center for Fundamental Research: Metal Structures in Four Dimensions, Materials Research Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark c Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People’s Republic of China d Danish–Chinese Center for Nanometals, Materials Research Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark e School of Materials Science and Engineering, Chongqing University, Chongqing 400045, People’s Republic of China b
Received 2 April 2011; received in revised form 8 May 2011; accepted 9 May 2011 Available online 16 June 2011
Abstract Columnar grained Ni is used as a model material allowing simultaneous non-surface investigations of the evolution of crystallographic orientations and deformation microstructures within individual grains as a function of rolling strain up to e = 0.7. Electron channelling contrast and electron backscattered diffraction are used to visualise microstructures and crystallographic orientations. It is found that both the microstructural and the textural development depend strongly on the initial grain orientation. A grain size effect is observed on the deformation-induced orientation scatter within the grains. Large grains have microstructure and orientation scatters similar to those observed in single crystals of similar orientation. The observations are interpreted based on a slip system analysis, considering the relative effects of the initial grain orientation and the interaction between neighbouring grains as well. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Cold rolling; Electron backscattered diffraction; Nickel; Microstructure and texture; Modelling
1. Introduction Texture and microstructure are two key features of interest when relating the processing of metals to their final properties. A general framework has been established to describe the evolution of the deformation microstructure in facecentred cubic (fcc) single crystals and polycrystals of metals with medium to high stacking fault energies over a range of deformation modes [1]. Several levels of subdivision are possible. At the coarse scale level some grains subdivide into a banded structure comprising regions of approximately constant orientation with uniform deformation (matrix bands) ⇑ Corresponding author at: School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, People’s Republic of China. Tel.: +86 158 1157 9270; fax: +86 10 6891 3738. E-mail address: [email protected] (G.L. Wu).
and continuously varying (transition bands) crystallographic orientations. At the microscopic level a subdivision is observed in all cases into rotated volume elements, termed cell blocks, which are delineated by geometrical necessary boundaries and contain cells separated by incidental dislocation boundaries [1–3]. The textures developed in fcc single crystals and polycrystals during deformation have been extensively studied by X-ray and neutron diffraction, and more recently by the scanning electron microscopy (SEM) electron backscattered diffraction (EBSD) technique [4–10]. This experimental texture data has been used to test polycrystalline plasticity models, which are based on the prediction of active slip systems in individual grains and allow the calculation of consequent lattice rotations. Experimentally it is a challenge to follow the microstructural and textural development as a function of strain for
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.05.019
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individual grains. Much information has been gathered on single crystals [5,11–14], all of which shows that the lattice rotations and deformation microstructures strongly depend on the initial crystallographic orientations. However, uncertainty exists as to how directly these results can be applied to polycrystalline materials, where additional effects arising from constraints at grain boundaries may be present [9,15–19]. For polycrystals methods already attempted include examination after a given set of strains of the internal surfaces of split samples [4,10,19] or of the external surfaces of samples [6–8,20,21]. Recently, the three-dimensional X-ray diffraction (3-D XRD) method has been applied to this problem, and results on the evolution of orientation of individual grains during in situ low strain deformation in bulk aluminium have been reported [22–24]. However, this type of measurement does not allow a simultaneous investigation of the deformation microstructure. An alternative method is adopted in the present study, involving the use of a columnar grained sample. By taking slices perpendicular to the columnar grain direction a series of samples can be produced, each with the same initial grain structure and orientations. By deforming each sample to a different strain it is possible to follow the evolution of the deformation structure and the orientations of individual grains in a polycrystalline environment as a function of strain [25]. Materials with columnar grains are typically h0 0 1i fibre textures with h0 0 1i along the columnar direction. The samples in the present study were oriented so that h0 0 1i was parallel to the normal direction (ND) during rolling. This fibre is strongly symmetrical and as long as symmetrical slip systems are activated two of the orientations (the cube orientation, {0 0 1}h1 0 0i, and the 45°ND rotated cube orientation, {0 0 1}h1 1 0i, referred to in this paper as the ND45°RC orientation) are stable, and the rest rotate along the fibre. However, these two orientations are only metastable, implying that even small variations from the ideal fibre orientations are expected to cause rather large rotations away from the fibre. The experimental conditions are thus selected to give a scenario with initially equal grains, all of which at the same time have a potential for substantial rotations of their crystallographic orientations upon deformation. 2. Experimental A high purity (99.987%) columnar grained nickel sample was prepared by directional solidification. Four adjacent slices were cut perpendicular to the columnar grain axis, as shown schematically in Fig. 1. One sample was left undeformed; the other three specimens were cold rolled to reductions of 10%, 30% and 50% (true strains of 0.105, 0.357 and 0.693), respectively. In order to maintain homogeneous deformation the rolling geometry parameter l/dav (l is the contact length between the roll and samples, and dav is the average thickness of samples before and after
Fig. 1. Sketch of a columnar grained sample during cold rolling. RD, TD and ND are the rolling direction, transverse direction and normal direction, respectively.
rolling) was approximately 2.5 during each pass. The specimens were rolled in the same sense, without turning the samples end to end or up-side down between rolling passes. Except for a minor widening along the transverse direction (TD) (1%, 5% and 8% after the three rolling reductions, respectively) the samples appeared to undergo ideal plane strain compression, in particular no evidence of overall shear strain was found. During rolling the ND was parallel to the columnar grain axis (see Fig. 1), meaning that the ND is roughly parallel to the crystallographic h0 0 1i direction and that the rolling direction (RD) and TD take the forms hu v 0i and hv u 0i. For microstructural investigations the rolled samples were first ground parallel to the rolling plane to remove a layer of about one-quarter of the sample thickness. This was motivated by findings in previous studies of single crystals [26,27] and polycrystals [28] as well as modelling [29] that the e13 shear is typically near 0 at this sample location (the indices 1, 2 and 3 represent the RD, TD and ND, respectively). In order to investigate the through thickness behaviour of the grains the 50% rolled sample was subsequently sectioned along the RD to reveal the longitudinal plane in the centre of the sample. Electropolishing was carried out using a 20 vol.% sulphuric acid solution. The deformation structure was observed using electron channelling contrast (ECC) in an scanning electron microscope. Orientation measurements were obtained using a fully automatic EBSD system (HKL Technology) attached to a LEO1530 thermal fieldemission gun scanning electron microscope. By combining ECC montages with the EBSD data the grain structure and the corresponding crystallographic orientations within a 10 10 mm2 area of the undeformed sample were first determined. Grains were determined from the EBSD data using a reconstruction method with a grain boundary threshold of 5°. Mean orientations were calculated by the quaternion method described in Krieger Lassen et al. [30]. The grains in corresponding areas on the three deformed samples were investigated. For the undeformed, 10% and 30% rolled samples EBSD stage scanning with a 100 lm step size was used. For the 50% rolled sample beam scanning with a 10 lm step size was used: maps
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were “stitched” together to cover the complete area. The finer resolution for the 50% rolled sample was necessary because of the finer size scale of the deformation microstructure. 3. Results 3.1. Initial (undeformed) grain structure In total 259 grains were identified in the area investigated of the undeformed sample. After 50% rolling only 173 of these could be reliably identified. Those which could not be followed are coloured white in Fig. 2a and are not
Fig. 3. Examples of the different extents of deformation banding along ND in the deformed grains. The data shown are calculated from the misorientation between each point in a line scan and the initial grain orientation in samples rolled to 50% reduction. The misorientation is partitioned into TD and non-TD components according to the procedure described in Wert [27]: (a) banding; (b) no banding. The 3=4 and ½ layer thickness positions are marked by arrows.
Fig. 2. Sketch of initial grains within the rolling plane. (b) {1 1 1} Pole figures showing mean orientations of the individual grains before rolling; (c) {1 1 1} pole figure of the mean orientations of individual grains after 50% rolling. (Red, green and blue indicate grains with the cube, ND22.5°RC and ND45°RC orientations, respectively. In (a) white grains are those not followed in the present experiment.)
included in the analysis. The average cross-sectional grain area in the sample was 0.25 mm2. The initial grain orientations are shown in Fig. 2b, demonstrating the strong h0 0 1i fibre texture parallel to the ND. This study focuses on three orientations on this fibre, namely the two metastable ones {0 0 1}h1 0 0i (cube) and {0 0 1}h1 1 0i (ND45°RC) as well as the orientation {0 0 1}h0.92 0.38 0i (ND22.5°RC). The latter orientation has two symmetric variants, namely {0 0 1}h0.92 0.38 0i (ND22.5°RC) and {0 0 1}h0.38 0.92 0i (ND67.5°RC), each of which lies midway between a cube and ND45°RC orientation. Throughout most of the paper these two equivalents are treated simultaneously and referred to as the ND22.5°RC orientation. The fibre was therefore divided into three equally large orientation ranges, each spanning total ND rotation intervals of 30°, based on these orientations. This ensures that each class contains approximately the same number of grains, and the colours applied to the individual grains in Figs. 2a and b are determined accordingly. In practice, all orientation classes were assigned by partitioning the rotation between each orientation and a reference orientation (cube) on the fibre into two rotations (one about the ND and one about an axis perpendicular to the ND, where the non-ND rotation is minimised; see Wert et al. [31] for details of the procedure)
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and then assigning the orientation class based on the ND rotation component. The advantage of this procedure over using a simpler “deviation from ideal texture” criterion is that we are able to select for analysis only those grains with a given deviation from the ideal fibre. However, most of the grains shown in this paper have orientations within 5° of one of the three ideal orientations. 3.2. Through thickness deformation banding Most of the analysis in this paper is conducted in the rolling plane in the quarter thickness layer. In order to investigate the behaviour in other layers of the rolled sample, especially the possibility of through thickness deformation banding, line scans along the ND were conducted for eight large grains in the longitudinal plane. The eight grains had orientations representing the entire fibre well. Deformation banding along the ND was observed in less than half of the grains. Banding was observed in differently oriented grains and grains of similar orientation showed different banding behaviour. It was therefore concluded that the through thickness banding did not depend on the orientation of the grain on the h0 0 1i fibre. The orientation profiles from line scans along the ND are shown in Fig. 3 for a grain with banding and one without banding. The sample thickness after material removal to expose the quarter thickness layer (as described in Section 2) was about 1.5 mm. Measurements could not be conducted near the very edges of the sample, and hence in these profiles the three-quarter thickness and half thickness layers are found at distances of about 350 and 850 lm, respectively. The one-quarter thickness layer investigated in the rolling plane is about 100 lm outside the scan. The fact that the one-quarter thickness layer could not be investigated in the longitudinal plane is not considered important as the symmetrical three-quarter thickness layer is covered by the scans. Through thickness deformation banding due to e13 shear is typically associated with crystallographic rotations around the TD. This has, for example, been observed in single crystals of cube [32] and ND45°RC [26,27] orientation. As also illustrated in Fig. 3a, the banding involves large variations in TD rotations and pronounced changes in the rotation direction are observed near the three-quarter and half thickness layers. This is in agreement with the expected profile for the e13 shear and near 0 shears at the symmetrical one-quarter and three-quarter thickness layers. In conclusion, through thickness banding associated with variations in e13 shear was observed in some grains
3 Fig. 4. Example {1 1 1} pole figures showing the evolution of the orientation spread within individual grains during deformation: cube oriented grains 25 (a) and 257 (b); ND22.5°RC grain 170 (c), and ND45°RC grain 83 (d). In the pole figures black, red, blue and green represent the orientations after rolling to reductions of 0%, 10%, 30% and 50%, respectively.
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Fig. 6. Deformation structure in grain 25 after rolling to 10%, 30% and 50% reductions in thickness, respectively. The white box in the figure shows the region analysed further in Fig. 7.
Fig. 5. Average deviation of each point from the grain mean orientation versus grain cross-sectional area of individual grains after 50% rolling. (a) Individual grain values; (b) binned by grain cross-sectional area. In (b) single crystal data for two 50% cold rolled samples are also shown (these data were not included in the line fitting procedure).
observed that cube grains exhibit the smallest rotation of the mean orientation, of about 15°, whereas ND45°RC grains rotate by the largest angles, up to about 25°. Grains within each of the three initial orientation components rotate to three new orientation components after 50% rolling. These are approximately centred at {0 1 4}h1 0 0i, {1 1 6}h5 1 1i and {1 1 3}h3 3 2i, for initial orientations cube, ND22.5°RC and ND45°RC respectively, with some significant scatter around these mean values (see Fig. 2c).
but most grains did not develop such bands. Together with the observations of sample shape this implies that there was no substantial overall e13 shear component, and this is especially the case for the investigated one-quarter thickness layer. This does not, however, necessarily mean 0 shear components for all grains, rather that the many grains where no banding was observed did not have through thickness shear variations. The possibility of shear strains is analysed and discussed in much further detail in Sections 4 and 5, based on the detailed experimental observations in the rolling plane of the one-quarter thickness layer presented in Sections 3.3–3.5.
3.3.1. Cube grains The evolution of orientation of cube grains during rolling is illustrated by two examples, of grains 25 and 257, as shown in Fig. 4a and b. Before rolling both grains were within 5° of the ideal cube orientation, with cross-sectional grain areas of 0.83 and 0.1 mm2, respectively. During rolling both grains rotate about the RD and develop a large TD spread. After deformation grain 25 has a larger scatter in orientation while grain 257 is much less scattered. The extent of the scatter can be related to the pattern of grain subdivision during deformation. This is described further in Section 3.4.
3.3. Evolution of orientation of individual grains
3.3.2. ND22.5°RC grains The evolution of orientation of grains in the ND22.5°RC orientation class during rolling is illustrated in Fig. 4c for grain 170 (initial cross-sectional area 1.5 mm2) with an initial orientation within 5° of the ideal ND22.5°RC orientation. In general, grains with an ND22.5°RC orientation rotate around one of the h1 1 1i poles (as does grain 170 shown in Fig. 4c) in one direction. In a few cases grains rotate around two of the h1 1 1i poles in opposite directions, resulting in a large orientation spread around their initial orientations.
From an examination of the orientation data in the rolling plane gathered from the one-quarter thickness layer of each sample (0%, 10%, 30% and 50% reduction) it is possible to follow the evolution of orientation with strain on an individual grain basis. In general, for all grains the lattice rotations are small for rolling reductions up to 30%, however, large lattice rotations occur between rolling reductions of 30% and 50%. The mean orientations of all grains after the 50% rolling reduction are shown in Fig. 2c. It is seen that the grain rotation depends on the initial grain orientation. Cube grains rotate around the RD; ND22.5°RC grains rotate approximately around h1 1 1i poles; ND45°RC grains rotate about the TD. It was also
3.3.3. ND45°RC grains Fig. 4d gives an example of the evolution of orientations during rolling for grain 83 (initial cross-sectional area
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0.28 mm2) with an initial orientation within 5° of the ideal ND45°RC orientation. All grains of this orientation were found to rotate predominately about the TD. In some cases the rotations are only in one direction, although in most cases the rotations are in two opposite directions (as in Fig. 4d). 3.4. Orientation spread Beside net rotations of individual grains, another important parameter available from the experimental data is the
spread of orientation within each grain after deformation. The orientation spread of all grains after 50% rolling was analysed in the present results. For each grain the average misorientation of each point in the grain to the grain mean orientation (hhav,ii) was calculated. This parameter was chosen to quantitatively represent the orientation spread. Fig. 5a shows the average misorientation within grains close to the three main initial orientation components versus the initial cross-sectional grain area after 50% rolling. The average misorientations of grains with different areas exhibit a large spread, varying from a few degrees to more
Fig. 7. (a) {1 1 1} Pole figure and (b) EBSD orientation map of the area in grain 25 indicated by the white box in Fig. 6. (c and d) The misorientation angle versus distance relative to the first point along line 1 and line 2 in (b), respectively. In (a) the solid circles give the initial orientation of grain 25; the colours follow the scheme used in (b). (e) EBSD orientation maps of grain 108 with a near ND 22.5°RC orientation.
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Fig. 9. Distribution of calculated shears (a) e23 and (b) e13 based on the measured rotation during deformation of each grain. Black grains indicate those not followed in the experiment; white grains are those without e23 shear in (a) and without e13 shear in (b). Circles, triangles and squares represent the cube, ND22.5°RC and ND45°RC orientations, respectively. In (a) all grains without symbols have a ND45°RC orientation; in (b) all grains without symbols are cube grains. The asterisks mark 12 grains that do not behave as expected based on their orientation.
Table 1 Summary of numbers of grains of cube orientation. Deviation from ideal cube (°)
Total >0.16 mm2 Banded Fig. 8. Pole figures showing the initial and final (average) orientations after 50% rolling for grains with initial orientations within 5° of the h0 0 1i fibre with orientations nearest to (a) cube, (b) ND22.5°RC orientation and (c) ND45°RC orientations. Green and orange dots represent the initial orientations; red and blue dots represent the respective average grain orientations after 50% rolling, i.e. initial orientations shown in green (orange) rotate to final orientations shown in red (blue).
than 10°. No grain orientation effect can be found: grains with different orientations all show a similar range in the value of hhav,ii. However, hhav,ii increases with cross-sectional grain area, as can be seen more clearly by binning the hhav,ii data by the grain area irrespective of the crystal orientation (Fig. 5b). Fig. 5b also includes results from two
All
0–5
5–10
>10
8 5 5
22 16 6
20 20 8
50 41 19
single crystal orientations. These show good agreement with the present data from columnar grains. 3.5. Microstructural evolution in individual grains The microstructure developed in the rolling plane of each individual grain after rolling to 50% reduction was examined using the EBSD and ECC techniques, paying particular attention to the coarse pattern of subdivision (variations in orientation on a length scale similar to that of the cross-sectional grain diameter). Many grains showed no evidence for such subdivision. However, in some grains
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Table 2 Slip system in Bishop–Hill notation. Plane
Direction Notation
(1 1 1) ½1 0 1 ½1 1 0 ½0 1 1 a1 a2 a3
ð1 1 1Þ ½1 0 1 ½1 1 0 ½0 1 1 b1 b2 b3
either a regular or irregular pattern of coarse subdivision was observed. Regular coarse subdivision, characterised by the presence of deformation bands, was only observed in grains of cube orientation. An example is shown in Fig. 6 (grain 25), where two sets of bands can be seen: one set with a wide spacing approximately parallel to the RD, the other with a much narrower spacing roughly parallel to the TD. An EBSD scan with a finer step size (4 lm) taken in the banded region (indicated by the white box in Fig. 6) reveals that the bands of wide spacing have orientations with a pattern of alternating TD rotations (Fig. 7a and b). A calculation of the change in orientation across the wide bands (along “line 1” in Fig. 7b) reveals a pattern similar to that observed for matrix and transition bands in single crystal studies of the cube orientation [32] with some regions of approximately constant orientation and some regions with continuously varying orientations (Fig. 7c). A line scan across the narrow bands (“line 2”) shows that these have a spacing of 100 lm and also exhibit a pattern of alternating orientations (Fig. 7d), although with a smaller range of misorientations than what is seen in the wide bands. Not all cube grains were observed to develop regular deformation bands. Of the 50 grains classified as cube, regular deformation bands were only seen in 19. A correlation with the initial area of each grain was observed: all cube grains with cross-sectional areas larger than 1.3 mm2 contained regular deformation bands, whilst regular deformation bands were never seen in cube grains smaller than 0.16 mm2. For cube grains with areas between these two values grains within 5° of the ideal cube orientation all developed regular bands, while no strong correlation could be discerned in cube grains deviating by more than 5° from the ideal cube orientation. These data are summarised in Table 1. In a few cube grains an irregular pattern of subdivision was found, with different parts of the grain rotated in different crystal directions. In such cases the orientation differences were, however, still characterised by a “±” TD rotation pattern. Although a regular pattern of subdivision at the grain scale was not seen for grains of ND22.5°RC and ND45°RC orientation, many of these grains did display evidence of the mesoscale (narrow) deformation banding, with orientations alternating by up to 15° over distances of 5–20 lm. An example is given in Fig. 7e for a grain within 5° of the ND22.5°RC orientation. A similar pattern of mesoscale subdivision was found in the central region of rolled single crystals of the ND45°RC orientation [26], and
ð1 1 1Þ ½1 0 1 ½1 1 0 ½1 1 c1 c2 c3
ð1 1 1Þ ½ ½0 1 1 1 0 1 ½1 1 0 d1 d2 d3
across the thickness of channel die deformed samples of the same orientation [33]. 4. Theoretical analysis In the rolled sample, as no overall shear strains were detected from the sample shape and as the sample was investigated in the rolling plane of the one-quarter thickness layer, plane strain compression assumptions are employed to predict the slip systems. The dominant slip systems from such a prediction exhibit a systematic variation over the entire fibre with h0 0 1i parallel to the ND. These dominant systems are, furthermore, the same irrespective of whether they are deduced based on Schmid factors or calculated by the Taylor model. The Taylor factor for this plane strain compression is constant and equal to 2.45 over the entire fibre. Two systems (c2, d2) (see Table 2) are active over the entire fibre and are by far the dominant systems in the ND22.5°RC samples. In the ideal cube orientation four systems (a2, b2, c2, d2) are active, while the active systems in the ND45°RC orientation are (c1, c2, d1, d2). When the four systems in the ideal cube and in the ND45°RC orientations are equally active the orientations are stable. Equal activation of the two systems in the ND22.5°RC orientation gives rotation along the fibre towards the cube orientation. The experimentally observed rotations differ significantly from these predictions. A more detailed comparison of the experimentally observed rotations and the rotations produced by each of the expected slip systems leads to the conclusion that only subsets of the expected systems are active. In this way it is deduced: (i) that the observed rotations of the cube around ± RD are caused by the dominance of only two systems, either (b2, d2) or (a2, c2); (ii) that either c2 or d2 are more active than the other in the ND22.5°RC orientation to produce rotations around a crystallographic axis close to h1 1 1i; (iii) that either (c1, c2) or (d1, d2) are more active in the ND45°RC orientation to produce ± TD rotations. The strong orientation spread around the TD observed in cube oriented grains can furthermore be attributed to local dominance of only one of the two slip systems. When moving away from the ideal fibre orientations the Schmid factors change slightly. The dominance of either one subset or the other of slip systems deduced based on the observed rotations above is in agreement with these Schmid factor changes, so that the deduced subsets are
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those with the highest Schmid factors on the two sides of the fibre. Activation of the subsets only will, however, lead to local shear strain components for each grain. For the cube orientation strong dominance of only (b2, d2) or (a2, 2) gives rise to a e23 shear strain, the sign of which depends on which of the two slip system combinations dominates. It is emphasised that such a shear component is needed to produce a rotation of the observed magnitude. The local dominance of one of the systems in a subset giving rise to the TD spread is associated with local e12 shear strains. In contrast to the cube orientation, final grain orientations close to the observed may be obtained either with or without shears for grains initially slightly off the ideal ND22.5°RC or ND45°RC orientations; in all cases the deduced subset of slip systems are the most active systems. If the deduced subsets operate alone they will lead to e23 and e13 shear components for the ND22.5°RC orientation and e13 shear for the ND45°RC orientation. For the ideal orientations the Taylor factor is unaltered by the introduction of shears. For orientations slightly off the ideal fibre the Taylor factors for the near ND22.5°RC and near ND45°RC orientations decrease slightly (a few percents) while the Taylor factor for the near cube orientation is unaltered when they are allowed to deform with shear components. 4.1. Analysis of grain orientation effects If the small deviations from the ideal orientations control the selection of the active slip systems as described above then grains deviating from the ideal fibre by positive rotations around the RD should rotate in an opposite direction to grains deviating from the ideal fibre by negative rotations about the RD. This prediction is examined in Fig. 8. In this plot data are shown only for grains with orientations less than 5° from the ideal h0 0 1i fibre. Note that only one of the symmetric variants for the ND22.5°RC orientation is included for clarity. Results for the other variant (ND67.5°RC) are similar. The grains are coloured according to whether they rotate about the dominant rotation axis in a positive or negative direction during deformation: red and blue mark the final orientations for the two rotation directions. The initial orientations of the grains with red final orientations are marked in green while orange symbols are the initial orientations of the grains rotated to the blue symbols. The black lines indicate the locus of stability for non-ND rotations: initial orientations lying on different sides of this line are predicted to rotate in opposite directions. The lines therefore highlight the expected separation of initial orientations for oppositely rotating grain orientations (due to the existence of two ND22.5°RC variants this line is shorter in Fig. 8b (see Section 3.1)). The cube grains rotate horizontally in the pole figure around the RD, while the ND22.5°RC grains rotate diagonally (almost around the h1 1 1i pole in the lower right quarter) and the ND45°RC grains rotate vertically around the TD. It is seen that in most cases (79%) the orange and green colours lie on opposite sides of the black
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line, showing that most grains rotate in the direction expected based on the initial grain orientation. 4.2. Analysis of grain interaction effects From the analysis given above it can be concluded that the cube oriented grains must have local non-zero e23 shear strains in order to rotate as they do. Analogously, a local non-zero e13 shear strain is possible for the ND45°RC oriented grains, while the ND22.5°RC oriented grains may have local shear components of both types. Based on these observations a more detailed analysis has been carried out considering the potential shear strains for each grain followed to 50% reduction. The signs of the potential shear strains for each grain, derived from the observed grain rotation direction, were determined, resulting in the spatial maps of the distribution of potential e23 and e13 shear directions shown in Fig. 9. In the maps the two variants of the ND22.5°RC component were separated because the signs of e23 and e13 for one variant are identical while they are opposite for the other variant. Inspection of the strain maps reveals an overall clustering of grains with similar strains. In particular, for the e23 shear component there is a clear tendency for positive signs in the upper half of the map, while the sign is predominantly negative in the lower half. As previously stated, neither inspection of the sample shape after deformation nor analysis of the through thickness orientation profiles found significant overall shear components. This implies that the shear strains are accommodated locally by detailed interaction of the grains. It is further noted that the vast majority of cube oriented grains in a large cluster with negative e23 shear located in the lower left part of the map in Fig. 9a behave as expected based on their orientation deviation from the ideal fibre. The dominance of positive e23 shear in the upper part of the map in Fig. 9a involves both grains of cube and ND22.5°RC orientation, but again most of the grains behave as expected based on their initial deviation from the ideal fibre. A more detailed analysis was carried out for the grains shown in Fig. 8 with orientations lying within 5° of the ideal fibre. Of these 58 grains 46 behaved as expected from their orientation deviation from the ideal fibre. The remaining 12 grains are marked with an asterisk in the shear strain maps of Fig. 9. It is seen that except for two small grains all of these grains have at least one neighbouring grain with the same shear strain. It is therefore possible that the behaviour of these grains is controlled by interaction with their neighbours rather than by their orientation. 5. Discussion 5.1. Strain heterogeneities As previously mentioned, the sample dimensions and shape after rolling, as well as the through thickness orientation profiles, were in overall agreement with the plane
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strain conditions, especially in the investigated one-quarter thickness layer. However, a previous study of the same sample involving detailed characterisation of the area of each grain in the rolling plane before and after rolling revealed that only 41% of the grains changed their area as expected based on the overall strain of the sample [34]. The areas of the remaining grains were either smaller or larger, suggesting either variations at the local strain level or local deviations from ideal plane strain conditions. Neither the grain orientation nor the grain size was found to influence the strain variations. In the present analysis only variations in the shear strain components are considered. The shear strain variations must be accommodated locally, either within each grain or involving clusters of grains. A clustering of grains with similar deduced shear strains can indeed be clearly seen in Fig. 9. It is also possible that the columnar grain shape plays a role in promoting displacements along the grain boundaries, i.e. along the ND, giving rise to e13 and e23 shears, which may not occur as easily in conventional polycrystals with fairly equiaxed grains. At the same time these grain boundaries may suppress the through thickness banding commonly observed in single crystals. The finding that the grains rotate in directions which are in good agreement with subsets of the expected dominant slip systems both from a Schmid factor and Taylor analysis assuming plane strain compression suggests that these are indeed the dominant slip systems in all grains, although fluctuations in their relative activities as well as minor activity of other systems is certainly possible. Detailed analysis of the slip systems and strain components in each individual grain is, however, beyond the scope of this study. 5.2. Grain subdivision Deformation banding has been reported for single crystals of certain orientations, either in the longitudinal plane or in the compression plane. In the columnar grains used in this study deformation banding was also observed, although to a much weaker extent than seen in the single crystal studies. For the cube orientation it has been found that single crystals deformed by channel die compression have deformation bands parallel to the longitudinal plane which are visible in the compression plane [5,33], whereas for rolling the bands are formed parallel to the rolling plane and therefore visible in the longitudinal plane [32]. The differently oriented deformation bands in channel die deformed and rolled cube single crystals have been ascribed to differences in the boundary conditions giving rise to different shear components. In the present study banding in the longitudinal plane similar to that observed in rolled single crystals was sometimes observed and did not appear to correlate with the characteristics of the grain. The banding in the rolling plane was only seen for the largest grains with cube orientation. The overall RD rotations associated with e23 shears observed here for the cube orientation accompanied by
banding parallel to the RD in the rolling plane (accommodating e12 shear strain differences and giving rise to TD orientation spread) have, to the authors’ knowledge, not been observed before. They and the slip system combinations deduced in Section 4 are, however, within the possibilities suggested by others [33]. For single crystals of the ND45°RC orientation deformation banding parallel to the rolling plane has been observed for both rolling and plane strain compression [26,33,35]. These bands originate from activation of either the (c1, c2) or (d1, d2) system in different parts of the single crystal and result in bands with shears of opposite sign that rotate oppositely around the TD. The two subsets of slip systems that create these bands are the same as the two subsets deduced for the present grains. The less frequent observation of such bands in the present study may be due to the small initial deviations from the ideal orientation, which make one subset of slip systems sufficiently favoured from the beginning. Regardless of the extent of banding, for grains of all orientation classes the level of orientation spreading after 50% cold rolling does not depend on the initial orientation on the h0 0 1i fibre. However, the average orientation spread increases with increasing cross-sectional area of the grain, such that the spreads of the very coarse grains are quite close to those of single crystals with similar orientations cold rolled to the same reduction [26], as indicated in Fig. 5b. 5.3. Comparison with other studies The present study focuses on an initially metastable fibre texture and the effect of small orientation differences from the ideal fibre. While this is of course a special case, the study is an interesting supplement to other studies, which have often focussed on the evolution of the typical rolling texture fibre [4,36] or on the interaction between so-called hard and soft orientations (often measured in terms of Taylor factors) [9]. The findings of the present study that the initial orientation deviation from the ideal fibre is the dominant parameter controlling the rotation adds to the conclusion from other studies [24,36,37] that to a first approximation the orientation is the controlling factor. The study, however, also indicates that clusters of grains tend to exhibit similar shear strains, although this may to some extent be an effect of the an initial clustering of grains of similar orientations. In particular, clusters of cube oriented grains are found to exhibit substantial shear strains. A tendency for clusters of differently oriented grains to deform with similar shear strains has previously been observed [9], which was ascribed to a tendency to minimise the average Taylor factor for the cluster. As all grains in the present study had almost the same initial Taylor factor, which furthermore did not change significantly when introducing shear strains, the origin of the shear strains must be found elsewhere. The explanation seems to be the difference
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in Schmid factors, which promote activation of only those slip systems with the highest Schmid factors among a larger set of systems with almost equal Schmid factor values. For clusters grains with more frequent orientations appear to control the behaviour of grains with less frequent orientations. 6. Conclusions The evolution of the orientation and deformation microstructure of 173 individual grains in a columnar grained Ni sample with an initial fibre texture with h0 0 1i along the ND, i.e. spanning from the cube orientation to the 45° ND rotated cube orientation, were followed up to a rolling reduction of 50%. An analysis of the active slip systems and shear strains was conducted, from which the following was found. 1. The deformation structure within each grain is mainly determined by the grain orientation, with an additional influence of the cross-sectional grain area. The average orientation spread after 50% rolling within each grain increases with increasing grain area. Large-scale deformation banding in the rolling plane was only observed in cube oriented grains. Only weak banding was seen in the longitudinal plane, and the existence and extent of such banding was independent of grain orientation. 2. The initial grain orientations deliberately selected to be close to a metastable texture fibre all rotated substantially away from the fibre: (i) grains close to the cube orientation rotate around the RD; (ii) grains close to the ND22.5°RC orientations rotate to {1 1 6}h5 1 1i orientations around one of the h1 1 1i poles; (iii) grains close to the ND45°RC orientation rotate to {1 1 3}h3 3 2i orientations around the TD. The rotation amplitudes increase from the cube, to the ND22.5°RC to the ND45°RC orientation. 3. For the vast majority of grains (79%) the rotations were in agreement with those expected based on the deviation of the initial orientation from the ideal fibre. 4. In the one-quarter thickness layer local shears developed, as deduced from the rotation of each grain during deformation. Some clustering of grains with similar shears was observed, to a large extent correlated with clustering of the initial grain orientations.
Acknowledgements The authors wish to thank Drs. B. Ralph, N. Hansen, J. Wert and X. Huang for help comments and discussions. Dr. J. Wert (unfortunately deceased) kindly provided the data on orientation spreads in single crystals. The authors
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gratefully acknowledge financial support from the Danish National Research Foundation for the Center for Fundamental Research: Metal Structures in Four Dimensions, as well as from the Danish National Research Foundation and the National Natural Science Foundation of China (Grant No. 50911130230) for the Danish–Chinese Center for Nanometals. G.L. Wu would also acknowledge the financial support from the NSFC (Grant No. 51001016) and EYSRF of BIT. References [1] Hansen N. Metall Mater Trans A 2001;32A:2917. [2] Bay B, Hansen N, Hughes DA, Kuhlmann-Wilsdorf D. Acta Metall Mater 1992;40:205. [3] Hughes D, Hansen N. Acta Mater 1997;45:3871. [4] Panchanadeeswaran P, Doherty RD, Becker R. Acta Mater 1996;44: 1233. [5] Basson F, Driver JH. Acta Mater 2000;48:2101. [6] Delaire F, Raphnel JL, Rey C. Acta Mater 2000;48:1075. [7] Raabe D, Sachtleber M, Zhao Z, Roters F, Zaefferer S. Acta Mater 2001;49:3433. [8] Thorning C, Wert JA. Mater Sci Eng A 2005;397:215. [9] Kalidindi SR, Bhattachayva A, Doherty RD. Proc Roy Soc Lond A 2004;460:1935. [10] Quey R, Piot D, Driver JH. Acta Mater 2010;58:1649. [11] Driver JH, Juul Jensen D, Hansen N. Acta Mater 1994;42:3105. [12] Godfrey A, Juul Jensen D, Hansen N. Acta Mater 1998;46:823. [13] Godfrey A, Juul Jensen D, Hansen N. Acta Mater 1998;46:835. [14] Albou A, Driver JH, Maurice C. Acta Mater 2010;58:3022. [15] Gracio JJ. Mater Sci Eng 1995;A196:97. [16] Leffers T, Christoffersen H. Mater Sci Eng 1997;A234–A236:676. [17] Zolotorevsky N Yu, Titovets Yu F, Dyatlova G Yu. Scripta Mater 1998;38:1263. [18] Winther G. Mater Sci Eng 2001;A309–A310:486. [19] Barrett CS, Levenson LH. TMS AIME 1940;137:112. [20] Lebensohn RA, Brenner R, Castelnau O, Rollet AD. Acta Mater 2008;56:3914. [21] Skalli A, Fortunier R, Fillit R, Driver JH. Acta Metall 1985;33: 997. [22] Margulies L, Winther G, Poulsen HF. Science 2001;291:5512. [23] Poulsen HF, Margulies L, Schmidt S, Winther G. Acta Mater 2003;51:3821. [24] Winther G, Margulies L, Schmidt S, Poulsen HF. Acta Mater 2004;52:2863. [25] Wu GL, Godfrey A, Liu Q. Mater Sci Forum 2002;408–412:589. [26] Li ZJ, Godfrey A, Liu Q. Acta Mater 2004;52:149. [27] Wert J. Acta Mater 2002;50:3125. [28] Mishin O, Bay B, Winther G, Juul Jensen D. Acta Mater 2004;52: 5761. [29] Lee CS, Duggan BJ. Metall Trans A 1991;22:2637. [30] Krieger Lassen NC, Juul Jensen D, Conradsen K. Acta Crystallogr 1994;A50:741. [31] Wert J, Liu Q, Hansen H. Acta Mater 1997;45:2565. [32] Liu Q, Hansen N. Proc Roy Soc Lond A 1998;454:2555. [33] Akef A, Driver JH. Mater Sci Eng 1991;A132:245. [34] Wu GL, Godfrey A, Juul Jensen D, Liu Q. Scripta Mater 2005;53: 565. [35] Becker R, Butler JF, Hu H, Lalli LA. Met Trans A 1991;22A:45. [36] Quey R, Piot D, Driver JH. Acta Mater 2010;58:2271. [37] Winther G. Acta Mater 2008;56:1919.
Acta Materialia 59 (2011) 5451–5461 www.elsevier.com/locate/actamat
Evolution of orientations and deformation structures within individual grains in cold rolled columnar grained nickel G.L. Wu a,b,c,⇑, A. Godfrey c, G. Winther b,d, D. Juul Jensen b,d, Q. Liu e a School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, People’s Republic of China Center for Fundamental Research: Metal Structures in Four Dimensions, Materials Research Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark c Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People’s Republic of China d Danish–Chinese Center for Nanometals, Materials Research Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark e School of Materials Science and Engineering, Chongqing University, Chongqing 400045, People’s Republic of China b
Received 2 April 2011; received in revised form 8 May 2011; accepted 9 May 2011 Available online 16 June 2011
Abstract Columnar grained Ni is used as a model material allowing simultaneous non-surface investigations of the evolution of crystallographic orientations and deformation microstructures within individual grains as a function of rolling strain up to e = 0.7. Electron channelling contrast and electron backscattered diffraction are used to visualise microstructures and crystallographic orientations. It is found that both the microstructural and the textural development depend strongly on the initial grain orientation. A grain size effect is observed on the deformation-induced orientation scatter within the grains. Large grains have microstructure and orientation scatters similar to those observed in single crystals of similar orientation. The observations are interpreted based on a slip system analysis, considering the relative effects of the initial grain orientation and the interaction between neighbouring grains as well. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Cold rolling; Electron backscattered diffraction; Nickel; Microstructure and texture; Modelling
1. Introduction Texture and microstructure are two key features of interest when relating the processing of metals to their final properties. A general framework has been established to describe the evolution of the deformation microstructure in facecentred cubic (fcc) single crystals and polycrystals of metals with medium to high stacking fault energies over a range of deformation modes [1]. Several levels of subdivision are possible. At the coarse scale level some grains subdivide into a banded structure comprising regions of approximately constant orientation with uniform deformation (matrix bands) ⇑ Corresponding author at: School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, People’s Republic of China. Tel.: +86 158 1157 9270; fax: +86 10 6891 3738. E-mail address: [email protected] (G.L. Wu).
and continuously varying (transition bands) crystallographic orientations. At the microscopic level a subdivision is observed in all cases into rotated volume elements, termed cell blocks, which are delineated by geometrical necessary boundaries and contain cells separated by incidental dislocation boundaries [1–3]. The textures developed in fcc single crystals and polycrystals during deformation have been extensively studied by X-ray and neutron diffraction, and more recently by the scanning electron microscopy (SEM) electron backscattered diffraction (EBSD) technique [4–10]. This experimental texture data has been used to test polycrystalline plasticity models, which are based on the prediction of active slip systems in individual grains and allow the calculation of consequent lattice rotations. Experimentally it is a challenge to follow the microstructural and textural development as a function of strain for
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.05.019
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individual grains. Much information has been gathered on single crystals [5,11–14], all of which shows that the lattice rotations and deformation microstructures strongly depend on the initial crystallographic orientations. However, uncertainty exists as to how directly these results can be applied to polycrystalline materials, where additional effects arising from constraints at grain boundaries may be present [9,15–19]. For polycrystals methods already attempted include examination after a given set of strains of the internal surfaces of split samples [4,10,19] or of the external surfaces of samples [6–8,20,21]. Recently, the three-dimensional X-ray diffraction (3-D XRD) method has been applied to this problem, and results on the evolution of orientation of individual grains during in situ low strain deformation in bulk aluminium have been reported [22–24]. However, this type of measurement does not allow a simultaneous investigation of the deformation microstructure. An alternative method is adopted in the present study, involving the use of a columnar grained sample. By taking slices perpendicular to the columnar grain direction a series of samples can be produced, each with the same initial grain structure and orientations. By deforming each sample to a different strain it is possible to follow the evolution of the deformation structure and the orientations of individual grains in a polycrystalline environment as a function of strain [25]. Materials with columnar grains are typically h0 0 1i fibre textures with h0 0 1i along the columnar direction. The samples in the present study were oriented so that h0 0 1i was parallel to the normal direction (ND) during rolling. This fibre is strongly symmetrical and as long as symmetrical slip systems are activated two of the orientations (the cube orientation, {0 0 1}h1 0 0i, and the 45°ND rotated cube orientation, {0 0 1}h1 1 0i, referred to in this paper as the ND45°RC orientation) are stable, and the rest rotate along the fibre. However, these two orientations are only metastable, implying that even small variations from the ideal fibre orientations are expected to cause rather large rotations away from the fibre. The experimental conditions are thus selected to give a scenario with initially equal grains, all of which at the same time have a potential for substantial rotations of their crystallographic orientations upon deformation. 2. Experimental A high purity (99.987%) columnar grained nickel sample was prepared by directional solidification. Four adjacent slices were cut perpendicular to the columnar grain axis, as shown schematically in Fig. 1. One sample was left undeformed; the other three specimens were cold rolled to reductions of 10%, 30% and 50% (true strains of 0.105, 0.357 and 0.693), respectively. In order to maintain homogeneous deformation the rolling geometry parameter l/dav (l is the contact length between the roll and samples, and dav is the average thickness of samples before and after
Fig. 1. Sketch of a columnar grained sample during cold rolling. RD, TD and ND are the rolling direction, transverse direction and normal direction, respectively.
rolling) was approximately 2.5 during each pass. The specimens were rolled in the same sense, without turning the samples end to end or up-side down between rolling passes. Except for a minor widening along the transverse direction (TD) (1%, 5% and 8% after the three rolling reductions, respectively) the samples appeared to undergo ideal plane strain compression, in particular no evidence of overall shear strain was found. During rolling the ND was parallel to the columnar grain axis (see Fig. 1), meaning that the ND is roughly parallel to the crystallographic h0 0 1i direction and that the rolling direction (RD) and TD take the forms hu v 0i and hv u 0i. For microstructural investigations the rolled samples were first ground parallel to the rolling plane to remove a layer of about one-quarter of the sample thickness. This was motivated by findings in previous studies of single crystals [26,27] and polycrystals [28] as well as modelling [29] that the e13 shear is typically near 0 at this sample location (the indices 1, 2 and 3 represent the RD, TD and ND, respectively). In order to investigate the through thickness behaviour of the grains the 50% rolled sample was subsequently sectioned along the RD to reveal the longitudinal plane in the centre of the sample. Electropolishing was carried out using a 20 vol.% sulphuric acid solution. The deformation structure was observed using electron channelling contrast (ECC) in an scanning electron microscope. Orientation measurements were obtained using a fully automatic EBSD system (HKL Technology) attached to a LEO1530 thermal fieldemission gun scanning electron microscope. By combining ECC montages with the EBSD data the grain structure and the corresponding crystallographic orientations within a 10 10 mm2 area of the undeformed sample were first determined. Grains were determined from the EBSD data using a reconstruction method with a grain boundary threshold of 5°. Mean orientations were calculated by the quaternion method described in Krieger Lassen et al. [30]. The grains in corresponding areas on the three deformed samples were investigated. For the undeformed, 10% and 30% rolled samples EBSD stage scanning with a 100 lm step size was used. For the 50% rolled sample beam scanning with a 10 lm step size was used: maps
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were “stitched” together to cover the complete area. The finer resolution for the 50% rolled sample was necessary because of the finer size scale of the deformation microstructure. 3. Results 3.1. Initial (undeformed) grain structure In total 259 grains were identified in the area investigated of the undeformed sample. After 50% rolling only 173 of these could be reliably identified. Those which could not be followed are coloured white in Fig. 2a and are not
Fig. 3. Examples of the different extents of deformation banding along ND in the deformed grains. The data shown are calculated from the misorientation between each point in a line scan and the initial grain orientation in samples rolled to 50% reduction. The misorientation is partitioned into TD and non-TD components according to the procedure described in Wert [27]: (a) banding; (b) no banding. The 3=4 and ½ layer thickness positions are marked by arrows.
Fig. 2. Sketch of initial grains within the rolling plane. (b) {1 1 1} Pole figures showing mean orientations of the individual grains before rolling; (c) {1 1 1} pole figure of the mean orientations of individual grains after 50% rolling. (Red, green and blue indicate grains with the cube, ND22.5°RC and ND45°RC orientations, respectively. In (a) white grains are those not followed in the present experiment.)
included in the analysis. The average cross-sectional grain area in the sample was 0.25 mm2. The initial grain orientations are shown in Fig. 2b, demonstrating the strong h0 0 1i fibre texture parallel to the ND. This study focuses on three orientations on this fibre, namely the two metastable ones {0 0 1}h1 0 0i (cube) and {0 0 1}h1 1 0i (ND45°RC) as well as the orientation {0 0 1}h0.92 0.38 0i (ND22.5°RC). The latter orientation has two symmetric variants, namely {0 0 1}h0.92 0.38 0i (ND22.5°RC) and {0 0 1}h0.38 0.92 0i (ND67.5°RC), each of which lies midway between a cube and ND45°RC orientation. Throughout most of the paper these two equivalents are treated simultaneously and referred to as the ND22.5°RC orientation. The fibre was therefore divided into three equally large orientation ranges, each spanning total ND rotation intervals of 30°, based on these orientations. This ensures that each class contains approximately the same number of grains, and the colours applied to the individual grains in Figs. 2a and b are determined accordingly. In practice, all orientation classes were assigned by partitioning the rotation between each orientation and a reference orientation (cube) on the fibre into two rotations (one about the ND and one about an axis perpendicular to the ND, where the non-ND rotation is minimised; see Wert et al. [31] for details of the procedure)
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and then assigning the orientation class based on the ND rotation component. The advantage of this procedure over using a simpler “deviation from ideal texture” criterion is that we are able to select for analysis only those grains with a given deviation from the ideal fibre. However, most of the grains shown in this paper have orientations within 5° of one of the three ideal orientations. 3.2. Through thickness deformation banding Most of the analysis in this paper is conducted in the rolling plane in the quarter thickness layer. In order to investigate the behaviour in other layers of the rolled sample, especially the possibility of through thickness deformation banding, line scans along the ND were conducted for eight large grains in the longitudinal plane. The eight grains had orientations representing the entire fibre well. Deformation banding along the ND was observed in less than half of the grains. Banding was observed in differently oriented grains and grains of similar orientation showed different banding behaviour. It was therefore concluded that the through thickness banding did not depend on the orientation of the grain on the h0 0 1i fibre. The orientation profiles from line scans along the ND are shown in Fig. 3 for a grain with banding and one without banding. The sample thickness after material removal to expose the quarter thickness layer (as described in Section 2) was about 1.5 mm. Measurements could not be conducted near the very edges of the sample, and hence in these profiles the three-quarter thickness and half thickness layers are found at distances of about 350 and 850 lm, respectively. The one-quarter thickness layer investigated in the rolling plane is about 100 lm outside the scan. The fact that the one-quarter thickness layer could not be investigated in the longitudinal plane is not considered important as the symmetrical three-quarter thickness layer is covered by the scans. Through thickness deformation banding due to e13 shear is typically associated with crystallographic rotations around the TD. This has, for example, been observed in single crystals of cube [32] and ND45°RC [26,27] orientation. As also illustrated in Fig. 3a, the banding involves large variations in TD rotations and pronounced changes in the rotation direction are observed near the three-quarter and half thickness layers. This is in agreement with the expected profile for the e13 shear and near 0 shears at the symmetrical one-quarter and three-quarter thickness layers. In conclusion, through thickness banding associated with variations in e13 shear was observed in some grains
3 Fig. 4. Example {1 1 1} pole figures showing the evolution of the orientation spread within individual grains during deformation: cube oriented grains 25 (a) and 257 (b); ND22.5°RC grain 170 (c), and ND45°RC grain 83 (d). In the pole figures black, red, blue and green represent the orientations after rolling to reductions of 0%, 10%, 30% and 50%, respectively.
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Fig. 6. Deformation structure in grain 25 after rolling to 10%, 30% and 50% reductions in thickness, respectively. The white box in the figure shows the region analysed further in Fig. 7.
Fig. 5. Average deviation of each point from the grain mean orientation versus grain cross-sectional area of individual grains after 50% rolling. (a) Individual grain values; (b) binned by grain cross-sectional area. In (b) single crystal data for two 50% cold rolled samples are also shown (these data were not included in the line fitting procedure).
observed that cube grains exhibit the smallest rotation of the mean orientation, of about 15°, whereas ND45°RC grains rotate by the largest angles, up to about 25°. Grains within each of the three initial orientation components rotate to three new orientation components after 50% rolling. These are approximately centred at {0 1 4}h1 0 0i, {1 1 6}h5 1 1i and {1 1 3}h3 3 2i, for initial orientations cube, ND22.5°RC and ND45°RC respectively, with some significant scatter around these mean values (see Fig. 2c).
but most grains did not develop such bands. Together with the observations of sample shape this implies that there was no substantial overall e13 shear component, and this is especially the case for the investigated one-quarter thickness layer. This does not, however, necessarily mean 0 shear components for all grains, rather that the many grains where no banding was observed did not have through thickness shear variations. The possibility of shear strains is analysed and discussed in much further detail in Sections 4 and 5, based on the detailed experimental observations in the rolling plane of the one-quarter thickness layer presented in Sections 3.3–3.5.
3.3.1. Cube grains The evolution of orientation of cube grains during rolling is illustrated by two examples, of grains 25 and 257, as shown in Fig. 4a and b. Before rolling both grains were within 5° of the ideal cube orientation, with cross-sectional grain areas of 0.83 and 0.1 mm2, respectively. During rolling both grains rotate about the RD and develop a large TD spread. After deformation grain 25 has a larger scatter in orientation while grain 257 is much less scattered. The extent of the scatter can be related to the pattern of grain subdivision during deformation. This is described further in Section 3.4.
3.3. Evolution of orientation of individual grains
3.3.2. ND22.5°RC grains The evolution of orientation of grains in the ND22.5°RC orientation class during rolling is illustrated in Fig. 4c for grain 170 (initial cross-sectional area 1.5 mm2) with an initial orientation within 5° of the ideal ND22.5°RC orientation. In general, grains with an ND22.5°RC orientation rotate around one of the h1 1 1i poles (as does grain 170 shown in Fig. 4c) in one direction. In a few cases grains rotate around two of the h1 1 1i poles in opposite directions, resulting in a large orientation spread around their initial orientations.
From an examination of the orientation data in the rolling plane gathered from the one-quarter thickness layer of each sample (0%, 10%, 30% and 50% reduction) it is possible to follow the evolution of orientation with strain on an individual grain basis. In general, for all grains the lattice rotations are small for rolling reductions up to 30%, however, large lattice rotations occur between rolling reductions of 30% and 50%. The mean orientations of all grains after the 50% rolling reduction are shown in Fig. 2c. It is seen that the grain rotation depends on the initial grain orientation. Cube grains rotate around the RD; ND22.5°RC grains rotate approximately around h1 1 1i poles; ND45°RC grains rotate about the TD. It was also
3.3.3. ND45°RC grains Fig. 4d gives an example of the evolution of orientations during rolling for grain 83 (initial cross-sectional area
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0.28 mm2) with an initial orientation within 5° of the ideal ND45°RC orientation. All grains of this orientation were found to rotate predominately about the TD. In some cases the rotations are only in one direction, although in most cases the rotations are in two opposite directions (as in Fig. 4d). 3.4. Orientation spread Beside net rotations of individual grains, another important parameter available from the experimental data is the
spread of orientation within each grain after deformation. The orientation spread of all grains after 50% rolling was analysed in the present results. For each grain the average misorientation of each point in the grain to the grain mean orientation (hhav,ii) was calculated. This parameter was chosen to quantitatively represent the orientation spread. Fig. 5a shows the average misorientation within grains close to the three main initial orientation components versus the initial cross-sectional grain area after 50% rolling. The average misorientations of grains with different areas exhibit a large spread, varying from a few degrees to more
Fig. 7. (a) {1 1 1} Pole figure and (b) EBSD orientation map of the area in grain 25 indicated by the white box in Fig. 6. (c and d) The misorientation angle versus distance relative to the first point along line 1 and line 2 in (b), respectively. In (a) the solid circles give the initial orientation of grain 25; the colours follow the scheme used in (b). (e) EBSD orientation maps of grain 108 with a near ND 22.5°RC orientation.
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Fig. 9. Distribution of calculated shears (a) e23 and (b) e13 based on the measured rotation during deformation of each grain. Black grains indicate those not followed in the experiment; white grains are those without e23 shear in (a) and without e13 shear in (b). Circles, triangles and squares represent the cube, ND22.5°RC and ND45°RC orientations, respectively. In (a) all grains without symbols have a ND45°RC orientation; in (b) all grains without symbols are cube grains. The asterisks mark 12 grains that do not behave as expected based on their orientation.
Table 1 Summary of numbers of grains of cube orientation. Deviation from ideal cube (°)
Total >0.16 mm2 Banded Fig. 8. Pole figures showing the initial and final (average) orientations after 50% rolling for grains with initial orientations within 5° of the h0 0 1i fibre with orientations nearest to (a) cube, (b) ND22.5°RC orientation and (c) ND45°RC orientations. Green and orange dots represent the initial orientations; red and blue dots represent the respective average grain orientations after 50% rolling, i.e. initial orientations shown in green (orange) rotate to final orientations shown in red (blue).
than 10°. No grain orientation effect can be found: grains with different orientations all show a similar range in the value of hhav,ii. However, hhav,ii increases with cross-sectional grain area, as can be seen more clearly by binning the hhav,ii data by the grain area irrespective of the crystal orientation (Fig. 5b). Fig. 5b also includes results from two
All
0–5
5–10
>10
8 5 5
22 16 6
20 20 8
50 41 19
single crystal orientations. These show good agreement with the present data from columnar grains. 3.5. Microstructural evolution in individual grains The microstructure developed in the rolling plane of each individual grain after rolling to 50% reduction was examined using the EBSD and ECC techniques, paying particular attention to the coarse pattern of subdivision (variations in orientation on a length scale similar to that of the cross-sectional grain diameter). Many grains showed no evidence for such subdivision. However, in some grains
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Table 2 Slip system in Bishop–Hill notation. Plane
Direction Notation
(1 1 1) ½1 0 1 ½1 1 0 ½0 1 1 a1 a2 a3
ð1 1 1Þ ½1 0 1 ½1 1 0 ½0 1 1 b1 b2 b3
either a regular or irregular pattern of coarse subdivision was observed. Regular coarse subdivision, characterised by the presence of deformation bands, was only observed in grains of cube orientation. An example is shown in Fig. 6 (grain 25), where two sets of bands can be seen: one set with a wide spacing approximately parallel to the RD, the other with a much narrower spacing roughly parallel to the TD. An EBSD scan with a finer step size (4 lm) taken in the banded region (indicated by the white box in Fig. 6) reveals that the bands of wide spacing have orientations with a pattern of alternating TD rotations (Fig. 7a and b). A calculation of the change in orientation across the wide bands (along “line 1” in Fig. 7b) reveals a pattern similar to that observed for matrix and transition bands in single crystal studies of the cube orientation [32] with some regions of approximately constant orientation and some regions with continuously varying orientations (Fig. 7c). A line scan across the narrow bands (“line 2”) shows that these have a spacing of 100 lm and also exhibit a pattern of alternating orientations (Fig. 7d), although with a smaller range of misorientations than what is seen in the wide bands. Not all cube grains were observed to develop regular deformation bands. Of the 50 grains classified as cube, regular deformation bands were only seen in 19. A correlation with the initial area of each grain was observed: all cube grains with cross-sectional areas larger than 1.3 mm2 contained regular deformation bands, whilst regular deformation bands were never seen in cube grains smaller than 0.16 mm2. For cube grains with areas between these two values grains within 5° of the ideal cube orientation all developed regular bands, while no strong correlation could be discerned in cube grains deviating by more than 5° from the ideal cube orientation. These data are summarised in Table 1. In a few cube grains an irregular pattern of subdivision was found, with different parts of the grain rotated in different crystal directions. In such cases the orientation differences were, however, still characterised by a “±” TD rotation pattern. Although a regular pattern of subdivision at the grain scale was not seen for grains of ND22.5°RC and ND45°RC orientation, many of these grains did display evidence of the mesoscale (narrow) deformation banding, with orientations alternating by up to 15° over distances of 5–20 lm. An example is given in Fig. 7e for a grain within 5° of the ND22.5°RC orientation. A similar pattern of mesoscale subdivision was found in the central region of rolled single crystals of the ND45°RC orientation [26], and
ð1 1 1Þ ½1 0 1 ½1 1 0 ½1 1 c1 c2 c3
ð1 1 1Þ ½ ½0 1 1 1 0 1 ½1 1 0 d1 d2 d3
across the thickness of channel die deformed samples of the same orientation [33]. 4. Theoretical analysis In the rolled sample, as no overall shear strains were detected from the sample shape and as the sample was investigated in the rolling plane of the one-quarter thickness layer, plane strain compression assumptions are employed to predict the slip systems. The dominant slip systems from such a prediction exhibit a systematic variation over the entire fibre with h0 0 1i parallel to the ND. These dominant systems are, furthermore, the same irrespective of whether they are deduced based on Schmid factors or calculated by the Taylor model. The Taylor factor for this plane strain compression is constant and equal to 2.45 over the entire fibre. Two systems (c2, d2) (see Table 2) are active over the entire fibre and are by far the dominant systems in the ND22.5°RC samples. In the ideal cube orientation four systems (a2, b2, c2, d2) are active, while the active systems in the ND45°RC orientation are (c1, c2, d1, d2). When the four systems in the ideal cube and in the ND45°RC orientations are equally active the orientations are stable. Equal activation of the two systems in the ND22.5°RC orientation gives rotation along the fibre towards the cube orientation. The experimentally observed rotations differ significantly from these predictions. A more detailed comparison of the experimentally observed rotations and the rotations produced by each of the expected slip systems leads to the conclusion that only subsets of the expected systems are active. In this way it is deduced: (i) that the observed rotations of the cube around ± RD are caused by the dominance of only two systems, either (b2, d2) or (a2, c2); (ii) that either c2 or d2 are more active than the other in the ND22.5°RC orientation to produce rotations around a crystallographic axis close to h1 1 1i; (iii) that either (c1, c2) or (d1, d2) are more active in the ND45°RC orientation to produce ± TD rotations. The strong orientation spread around the TD observed in cube oriented grains can furthermore be attributed to local dominance of only one of the two slip systems. When moving away from the ideal fibre orientations the Schmid factors change slightly. The dominance of either one subset or the other of slip systems deduced based on the observed rotations above is in agreement with these Schmid factor changes, so that the deduced subsets are
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those with the highest Schmid factors on the two sides of the fibre. Activation of the subsets only will, however, lead to local shear strain components for each grain. For the cube orientation strong dominance of only (b2, d2) or (a2, 2) gives rise to a e23 shear strain, the sign of which depends on which of the two slip system combinations dominates. It is emphasised that such a shear component is needed to produce a rotation of the observed magnitude. The local dominance of one of the systems in a subset giving rise to the TD spread is associated with local e12 shear strains. In contrast to the cube orientation, final grain orientations close to the observed may be obtained either with or without shears for grains initially slightly off the ideal ND22.5°RC or ND45°RC orientations; in all cases the deduced subset of slip systems are the most active systems. If the deduced subsets operate alone they will lead to e23 and e13 shear components for the ND22.5°RC orientation and e13 shear for the ND45°RC orientation. For the ideal orientations the Taylor factor is unaltered by the introduction of shears. For orientations slightly off the ideal fibre the Taylor factors for the near ND22.5°RC and near ND45°RC orientations decrease slightly (a few percents) while the Taylor factor for the near cube orientation is unaltered when they are allowed to deform with shear components. 4.1. Analysis of grain orientation effects If the small deviations from the ideal orientations control the selection of the active slip systems as described above then grains deviating from the ideal fibre by positive rotations around the RD should rotate in an opposite direction to grains deviating from the ideal fibre by negative rotations about the RD. This prediction is examined in Fig. 8. In this plot data are shown only for grains with orientations less than 5° from the ideal h0 0 1i fibre. Note that only one of the symmetric variants for the ND22.5°RC orientation is included for clarity. Results for the other variant (ND67.5°RC) are similar. The grains are coloured according to whether they rotate about the dominant rotation axis in a positive or negative direction during deformation: red and blue mark the final orientations for the two rotation directions. The initial orientations of the grains with red final orientations are marked in green while orange symbols are the initial orientations of the grains rotated to the blue symbols. The black lines indicate the locus of stability for non-ND rotations: initial orientations lying on different sides of this line are predicted to rotate in opposite directions. The lines therefore highlight the expected separation of initial orientations for oppositely rotating grain orientations (due to the existence of two ND22.5°RC variants this line is shorter in Fig. 8b (see Section 3.1)). The cube grains rotate horizontally in the pole figure around the RD, while the ND22.5°RC grains rotate diagonally (almost around the h1 1 1i pole in the lower right quarter) and the ND45°RC grains rotate vertically around the TD. It is seen that in most cases (79%) the orange and green colours lie on opposite sides of the black
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line, showing that most grains rotate in the direction expected based on the initial grain orientation. 4.2. Analysis of grain interaction effects From the analysis given above it can be concluded that the cube oriented grains must have local non-zero e23 shear strains in order to rotate as they do. Analogously, a local non-zero e13 shear strain is possible for the ND45°RC oriented grains, while the ND22.5°RC oriented grains may have local shear components of both types. Based on these observations a more detailed analysis has been carried out considering the potential shear strains for each grain followed to 50% reduction. The signs of the potential shear strains for each grain, derived from the observed grain rotation direction, were determined, resulting in the spatial maps of the distribution of potential e23 and e13 shear directions shown in Fig. 9. In the maps the two variants of the ND22.5°RC component were separated because the signs of e23 and e13 for one variant are identical while they are opposite for the other variant. Inspection of the strain maps reveals an overall clustering of grains with similar strains. In particular, for the e23 shear component there is a clear tendency for positive signs in the upper half of the map, while the sign is predominantly negative in the lower half. As previously stated, neither inspection of the sample shape after deformation nor analysis of the through thickness orientation profiles found significant overall shear components. This implies that the shear strains are accommodated locally by detailed interaction of the grains. It is further noted that the vast majority of cube oriented grains in a large cluster with negative e23 shear located in the lower left part of the map in Fig. 9a behave as expected based on their orientation deviation from the ideal fibre. The dominance of positive e23 shear in the upper part of the map in Fig. 9a involves both grains of cube and ND22.5°RC orientation, but again most of the grains behave as expected based on their initial deviation from the ideal fibre. A more detailed analysis was carried out for the grains shown in Fig. 8 with orientations lying within 5° of the ideal fibre. Of these 58 grains 46 behaved as expected from their orientation deviation from the ideal fibre. The remaining 12 grains are marked with an asterisk in the shear strain maps of Fig. 9. It is seen that except for two small grains all of these grains have at least one neighbouring grain with the same shear strain. It is therefore possible that the behaviour of these grains is controlled by interaction with their neighbours rather than by their orientation. 5. Discussion 5.1. Strain heterogeneities As previously mentioned, the sample dimensions and shape after rolling, as well as the through thickness orientation profiles, were in overall agreement with the plane
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strain conditions, especially in the investigated one-quarter thickness layer. However, a previous study of the same sample involving detailed characterisation of the area of each grain in the rolling plane before and after rolling revealed that only 41% of the grains changed their area as expected based on the overall strain of the sample [34]. The areas of the remaining grains were either smaller or larger, suggesting either variations at the local strain level or local deviations from ideal plane strain conditions. Neither the grain orientation nor the grain size was found to influence the strain variations. In the present analysis only variations in the shear strain components are considered. The shear strain variations must be accommodated locally, either within each grain or involving clusters of grains. A clustering of grains with similar deduced shear strains can indeed be clearly seen in Fig. 9. It is also possible that the columnar grain shape plays a role in promoting displacements along the grain boundaries, i.e. along the ND, giving rise to e13 and e23 shears, which may not occur as easily in conventional polycrystals with fairly equiaxed grains. At the same time these grain boundaries may suppress the through thickness banding commonly observed in single crystals. The finding that the grains rotate in directions which are in good agreement with subsets of the expected dominant slip systems both from a Schmid factor and Taylor analysis assuming plane strain compression suggests that these are indeed the dominant slip systems in all grains, although fluctuations in their relative activities as well as minor activity of other systems is certainly possible. Detailed analysis of the slip systems and strain components in each individual grain is, however, beyond the scope of this study. 5.2. Grain subdivision Deformation banding has been reported for single crystals of certain orientations, either in the longitudinal plane or in the compression plane. In the columnar grains used in this study deformation banding was also observed, although to a much weaker extent than seen in the single crystal studies. For the cube orientation it has been found that single crystals deformed by channel die compression have deformation bands parallel to the longitudinal plane which are visible in the compression plane [5,33], whereas for rolling the bands are formed parallel to the rolling plane and therefore visible in the longitudinal plane [32]. The differently oriented deformation bands in channel die deformed and rolled cube single crystals have been ascribed to differences in the boundary conditions giving rise to different shear components. In the present study banding in the longitudinal plane similar to that observed in rolled single crystals was sometimes observed and did not appear to correlate with the characteristics of the grain. The banding in the rolling plane was only seen for the largest grains with cube orientation. The overall RD rotations associated with e23 shears observed here for the cube orientation accompanied by
banding parallel to the RD in the rolling plane (accommodating e12 shear strain differences and giving rise to TD orientation spread) have, to the authors’ knowledge, not been observed before. They and the slip system combinations deduced in Section 4 are, however, within the possibilities suggested by others [33]. For single crystals of the ND45°RC orientation deformation banding parallel to the rolling plane has been observed for both rolling and plane strain compression [26,33,35]. These bands originate from activation of either the (c1, c2) or (d1, d2) system in different parts of the single crystal and result in bands with shears of opposite sign that rotate oppositely around the TD. The two subsets of slip systems that create these bands are the same as the two subsets deduced for the present grains. The less frequent observation of such bands in the present study may be due to the small initial deviations from the ideal orientation, which make one subset of slip systems sufficiently favoured from the beginning. Regardless of the extent of banding, for grains of all orientation classes the level of orientation spreading after 50% cold rolling does not depend on the initial orientation on the h0 0 1i fibre. However, the average orientation spread increases with increasing cross-sectional area of the grain, such that the spreads of the very coarse grains are quite close to those of single crystals with similar orientations cold rolled to the same reduction [26], as indicated in Fig. 5b. 5.3. Comparison with other studies The present study focuses on an initially metastable fibre texture and the effect of small orientation differences from the ideal fibre. While this is of course a special case, the study is an interesting supplement to other studies, which have often focussed on the evolution of the typical rolling texture fibre [4,36] or on the interaction between so-called hard and soft orientations (often measured in terms of Taylor factors) [9]. The findings of the present study that the initial orientation deviation from the ideal fibre is the dominant parameter controlling the rotation adds to the conclusion from other studies [24,36,37] that to a first approximation the orientation is the controlling factor. The study, however, also indicates that clusters of grains tend to exhibit similar shear strains, although this may to some extent be an effect of the an initial clustering of grains of similar orientations. In particular, clusters of cube oriented grains are found to exhibit substantial shear strains. A tendency for clusters of differently oriented grains to deform with similar shear strains has previously been observed [9], which was ascribed to a tendency to minimise the average Taylor factor for the cluster. As all grains in the present study had almost the same initial Taylor factor, which furthermore did not change significantly when introducing shear strains, the origin of the shear strains must be found elsewhere. The explanation seems to be the difference
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in Schmid factors, which promote activation of only those slip systems with the highest Schmid factors among a larger set of systems with almost equal Schmid factor values. For clusters grains with more frequent orientations appear to control the behaviour of grains with less frequent orientations. 6. Conclusions The evolution of the orientation and deformation microstructure of 173 individual grains in a columnar grained Ni sample with an initial fibre texture with h0 0 1i along the ND, i.e. spanning from the cube orientation to the 45° ND rotated cube orientation, were followed up to a rolling reduction of 50%. An analysis of the active slip systems and shear strains was conducted, from which the following was found. 1. The deformation structure within each grain is mainly determined by the grain orientation, with an additional influence of the cross-sectional grain area. The average orientation spread after 50% rolling within each grain increases with increasing grain area. Large-scale deformation banding in the rolling plane was only observed in cube oriented grains. Only weak banding was seen in the longitudinal plane, and the existence and extent of such banding was independent of grain orientation. 2. The initial grain orientations deliberately selected to be close to a metastable texture fibre all rotated substantially away from the fibre: (i) grains close to the cube orientation rotate around the RD; (ii) grains close to the ND22.5°RC orientations rotate to {1 1 6}h5 1 1i orientations around one of the h1 1 1i poles; (iii) grains close to the ND45°RC orientation rotate to {1 1 3}h3 3 2i orientations around the TD. The rotation amplitudes increase from the cube, to the ND22.5°RC to the ND45°RC orientation. 3. For the vast majority of grains (79%) the rotations were in agreement with those expected based on the deviation of the initial orientation from the ideal fibre. 4. In the one-quarter thickness layer local shears developed, as deduced from the rotation of each grain during deformation. Some clustering of grains with similar shears was observed, to a large extent correlated with clustering of the initial grain orientations.
Acknowledgements The authors wish to thank Drs. B. Ralph, N. Hansen, J. Wert and X. Huang for help comments and discussions. Dr. J. Wert (unfortunately deceased) kindly provided the data on orientation spreads in single crystals. The authors
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gratefully acknowledge financial support from the Danish National Research Foundation for the Center for Fundamental Research: Metal Structures in Four Dimensions, as well as from the Danish National Research Foundation and the National Natural Science Foundation of China (Grant No. 50911130230) for the Danish–Chinese Center for Nanometals. G.L. Wu would also acknowledge the financial support from the NSFC (Grant No. 51001016) and EYSRF of BIT. References [1] Hansen N. Metall Mater Trans A 2001;32A:2917. [2] Bay B, Hansen N, Hughes DA, Kuhlmann-Wilsdorf D. Acta Metall Mater 1992;40:205. [3] Hughes D, Hansen N. Acta Mater 1997;45:3871. [4] Panchanadeeswaran P, Doherty RD, Becker R. Acta Mater 1996;44: 1233. [5] Basson F, Driver JH. Acta Mater 2000;48:2101. [6] Delaire F, Raphnel JL, Rey C. Acta Mater 2000;48:1075. [7] Raabe D, Sachtleber M, Zhao Z, Roters F, Zaefferer S. Acta Mater 2001;49:3433. [8] Thorning C, Wert JA. Mater Sci Eng A 2005;397:215. [9] Kalidindi SR, Bhattachayva A, Doherty RD. Proc Roy Soc Lond A 2004;460:1935. [10] Quey R, Piot D, Driver JH. Acta Mater 2010;58:1649. [11] Driver JH, Juul Jensen D, Hansen N. Acta Mater 1994;42:3105. [12] Godfrey A, Juul Jensen D, Hansen N. Acta Mater 1998;46:823. [13] Godfrey A, Juul Jensen D, Hansen N. Acta Mater 1998;46:835. [14] Albou A, Driver JH, Maurice C. Acta Mater 2010;58:3022. [15] Gracio JJ. Mater Sci Eng 1995;A196:97. [16] Leffers T, Christoffersen H. Mater Sci Eng 1997;A234–A236:676. [17] Zolotorevsky N Yu, Titovets Yu F, Dyatlova G Yu. Scripta Mater 1998;38:1263. [18] Winther G. Mater Sci Eng 2001;A309–A310:486. [19] Barrett CS, Levenson LH. TMS AIME 1940;137:112. [20] Lebensohn RA, Brenner R, Castelnau O, Rollet AD. Acta Mater 2008;56:3914. [21] Skalli A, Fortunier R, Fillit R, Driver JH. Acta Metall 1985;33: 997. [22] Margulies L, Winther G, Poulsen HF. Science 2001;291:5512. [23] Poulsen HF, Margulies L, Schmidt S, Winther G. Acta Mater 2003;51:3821. [24] Winther G, Margulies L, Schmidt S, Poulsen HF. Acta Mater 2004;52:2863. [25] Wu GL, Godfrey A, Liu Q. Mater Sci Forum 2002;408–412:589. [26] Li ZJ, Godfrey A, Liu Q. Acta Mater 2004;52:149. [27] Wert J. Acta Mater 2002;50:3125. [28] Mishin O, Bay B, Winther G, Juul Jensen D. Acta Mater 2004;52: 5761. [29] Lee CS, Duggan BJ. Metall Trans A 1991;22:2637. [30] Krieger Lassen NC, Juul Jensen D, Conradsen K. Acta Crystallogr 1994;A50:741. [31] Wert J, Liu Q, Hansen H. Acta Mater 1997;45:2565. [32] Liu Q, Hansen N. Proc Roy Soc Lond A 1998;454:2555. [33] Akef A, Driver JH. Mater Sci Eng 1991;A132:245. [34] Wu GL, Godfrey A, Juul Jensen D, Liu Q. Scripta Mater 2005;53: 565. [35] Becker R, Butler JF, Hu H, Lalli LA. Met Trans A 1991;22A:45. [36] Quey R, Piot D, Driver JH. Acta Mater 2010;58:2271. [37] Winther G. Acta Mater 2008;56:1919.