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Strain hardening and microstructure evolution of channel-die compressed aluminium bicrystals
Materials Science and Engineering A 477 (2008) 282–294
Strain hardening and microstructure evolution of channel-die compressed aluminium bicrystals H. Paul a,b,∗ , J.H. Driver c , W. Wajda a a
Polish Academy of Sciences, Institute of Metallurgy and Materials Science, 25 Reymonta St., 30-059 Krak´ow, Poland b University of Zielona G´ ora, Mechanical Department, 50 Podg´orna St., 65-246 Zielona G´ora, Poland c Ecole des Mines de Saint-Etienne, Centre SMS, 158 Cours Fauriel, 42023 Saint Etienne, France Received 23 February 2007; received in revised form 9 May 2007; accepted 11 May 2007
Abstract The different types of strain heterogeneities developed in pure aluminium bicrystals during plane strain compression have been characterized by advanced metallographic techniques over several length scales. Bicrystals, with three grain orientation combinations (one hard/soft, and two stable/unstable) containing grain boundaries perpendicular to the compression axis, were deformed up to strains of 1.5 at 77 K and their deformation substructure investigated by optical microscopy, high resolution EBSD and transmission electron microscopy. At the macroscopic scale the hard/soft combination develops major strain variations as the softer grain flows out over the harder grain. This is consistent with finite element simulations of bicrystal deformation using the appropriate individual grain hardening laws and leads to a large transition zone near the boundary. At the microscopic level, the grains of the other combinations behave essentially as single crystals under the same deformation conditions, in particular with respect to the deformation substructure and microtexture distributions. It is concluded that for most grain pair combinations of deformed bicrystals the individual grain orientations play a dominant role in deformation substructure development as a consequence of their dependence on slip system interactions. © 2007 Elsevier B.V. All rights reserved. Keywords: Aluminium; Bicrystals; Deformation; Microstructures; FEM; Strain hardening; Strain heterogeneities
1. Introduction As a natural consequence of crystal anisotropy, it is well known that individual grains in a polycrystalline aggregate exhibit different stress–strain behaviours. The total hardening of a deformed polycrystal then results from the hardening of the grains and a mutual interaction between grains in the material. As a consequence, neighbouring grains with different hardening rates tend to accumulate additional plastic (incompatibility) strains close to the grain boundaries. These highly deformed zones of the crystals may then become privileged sites for recrystallization nucleation during annealing. A knowledge of the influence of grain misorientation on both hardening and the development of localized deformation, e.g. deformation bands, ∗
Corresponding author at: Polish Academy of Sciences, Institute of Metallurgy and Materials Science, 25 Reymonta St., 30-059 Krak´ow, Poland. Tel.: +48 12 637 42 00; fax: +48 12 6372192. E-mail addresses: [email protected] (H. Paul), [email protected] (J.H. Driver), [email protected] (W. Wajda). 0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.05.044
close to the grain boundaries is therefore useful for understanding polycrystal plasticity and subsequent annealing effects. To study these complex phenomena, it is preferable to perform repeatable experiments with well-defined boundary conditions. In this context plane strain compression (PSC) tests on fcc bicrystals, with a centrally located grain boundary, can be particularly informative. In the present experiments, aluminium bicrystals, with high symmetry grain orientations, were chosen to represent different kinds of grain hardening and boundary misorientations. One of the grain orientations is the cube which is known to be very unstable during low temperature PS compression and forms deformation bands (DBs) along the extension direction [1,2]. The ‘shear’ orientation {1 0 0}0 1 1 (often denoted shear since it forms in the sheared zones of rolled fcc metals) is also very unstable and breaks up very rapidly during PSC into misoriented bands through opposite rotations around the transverse direction [1,3]. This forms transition bands (TB) at relatively low strains and eventually shear bands (SBs) at very large strains, e.g. [3,4]. On the other hand, the Goss orientation {1 1 0}0 0 1
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is known to be stable up to very high strains. The behaviour of crystallites with the hard {1 1 0}0 1 1 orientation is not so well characterized but is often considered moderately stable during compression. These crystal orientations can be combined into bicrystals composed of grains with very contrasting behaviours, e.g. unstable versus stable (‘cube/Goss’ or ‘shear/Goss’), unstable versus unstable (‘cube/shear’) and soft versus hard (‘cube/hard’). This paper describes an experimental study of the compression behaviour of such bicrystals and their microstructural evolution. Using channel-die compression experiments, stress–strain curves for single crystals have been established, giving the influence of grain misorientation on material hardening. This is then used as a basis for finite element (FE) simulations of the stress and strain distributions expected of such bicrystals. As shown below, large plastic strains of grains with very contrasting mechanical responses can create some major plastic heterogeneities in these bicrystals and FE simulations then indicate the local strains. The paper then concentrates on the deformation microstructure evolution of the grains using high resolution FEG-SEM EBSD and TEM, particularly for the microtextures. Some specific results for individual bicrystals have been previously reported [5–7] but the present analysis describes a wider set of bicrystals linking the stress–strain curves to microstructure development and the strain distributions. 2. Experimental procedures 2.1. Material and sample preparation The bicrystals with controlled orientations were grown by a modified Bridgman technique (horizontal solidification), using split graphite moulds from high purity Al (99.998%). The dimensions of the bicrystal bars were approximately 15 mm (thickness) × 22 mm (width) × 150 mm (length). Samples of 15 mm height, 15 mm length and 10 mm width were sectioned from a bar and plane strain compressed at a nominal strain rate of ∼10−4 s−1 . Teflon films were used as a lubricant and periodically replaced at strain intervals of about 0.2. To favour
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deformation banding and reduce dynamic recovery effects the tests were performed at 77 K by immersing the channel-die device in a reservoir with liquid nitrogen. This procedure gives, at relatively low strains, a work-hardened structure containing clearly defined bands. The following three types of grain configurations were used in channel-die compression test in order to study the effect of misorientation between grains on material hardening and strain localization: ‘cube/hard’ {1 0 0}0 0 1/{1 1 0}0 1 1; ‘cube/Goss’ {1 0 0}0 0 1/{1 1 0}0 0 1 and ‘shear/Goss’ {1 0 0}0 1 1/{1 1 0}0 0 1, where {h k l}u v w signify the compression plane of normal ND and the elongation direction ED, respectively. The position of the bicrystal in the channel-die device and the nominal orientations of the grains are shown in Fig. 1. The orientation relations between the grains of a bicrystal, could be described by the relations: {1 0 0}0 0 1 rotation 45◦ around 1 0 0 || TD → {1 1 0} 0 1 1 {1 0 0}0 0 1 rotation 45◦ around 1 0 0 || ED → {1 1 0} 0 0 1 {1 0 0}0 1 1 rotation 90◦ around 0 1 1 || TD → {1 1 0} 0 0 1 where TD is the transverse direction. The deformation geometry of these bicrystals was such that the bicrystal boundary separating the top and bottom crystallites at specimen mid-thickness lay parallel to the compression plane. 2.2. Microstructure and local orientation measurements The deformation microstructures were characterized over a wide range of scales using a combination of TEM, SEM and optical metallography. The samples were sectioned in the ND-ED plane by wire cutting. For thin foil preparation a twinset electropolishing technique was used in standard nitric acid–methanol solution. Optical microscopy was carried out using polarized light on electropolished and lightly anodized (in
Fig. 1. Schematic bicrystal configurations during channel-die compression showing the grain boundary location and the grain orientations: (a) ‘Goss/shear’, (b) ‘cube/Goss’ and (c) ‘cube/hard’.
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2% fluoroboric acid solution) sections. The majority of the metallographic observations were made on the longitudinal plane, i.e. ND-ED. The orientations of the crystallites composing the bicrystals were checked by the X-ray back reflexion Laue method. The deformed specimens were mostly examined by scanning electron microscopy (SEM) in a JEOL JSM 6500F, equipped with a field emission gun (FEG) and electron backscattered diffraction (EBSD) facilities, in which microscope control, pattern acquisition and solution were performed with the HKL Channel 5TM system. For more detailed analyses transmission electron microsopy (200 kV Philips TEM CM200) with semi-automatic Kikuchi pattern analysis, was employed. Local orientation data, obtained by SEM and TEM techniques on the ND-ED section were transformed to the standard ED-TD reference system and presented in the form of {1 1 1} pole figures.
A frequently used mixed Coulomb and Tresca friction law was used in following form: τ = μσn σo τ = m√ 3
σp μσn < √ 3
if if
σp μσn > m √ 3
(3)
where √ τ is the friction shear stress, σ n the normal stress, σp / 3 the shear yield stress and μ, m are the Coulomb and Tresca friction coefficients, respectively. In the simulations both friction coefficients were set to values of 0.05 or 0.2. A friction coefficient of about 0.05 or less is expected for Teflon in the low temperature range. 3. Results 3.1. Stress–strain behaviour
2.3. Simulation method For the FE simulations the bicrystal behaviour was described using the stress–strain curves obtained on single crystals with the same orientations compressed under the same conditions in a channel-die. This is in accordance with the general assumption that aluminium bicrystals deformed up to medium strains behave to a large extent like two individual single crystals [8]. The simulation was also simplified to a 2D plane strain problem as would be expected if adequate lubrication is used. The bicrystal deformation simulation was carried out using the Forge2 software in the Department of Computer Methods in Metallurgy in Krakow. The finite element model used in the FORGE code is based on the Norton-Hoff constitutive law and is described by Chenot and Bellet [9]. The law is generally written in the form of the following relation between deviatoric stress tensor (s- ) and the strain rate tensor (˙ε- ): √ m−1 s- = 2K( 3˙εi ) ε˙-
Two sets of experiments were carried out. First, plane strain compression of single crystals with different orientations was performed at 77 K: (hard {1 1 0}0 1 1 with a Taylor factor √ of 2 6, and Goss {1 1 0}0 0 1, cube {1 0 0}0 √0 1 and shear {1 0 0}0 1 1 with nominal Taylor factors of 6) in order to establish the stress–strain curves for use in the finite element simulations. The experimentally measured true stress–strain curves of the four crystal orientations are shown in Fig. 2 using standard definitions, σ = F/S and ε = ln(to /t), where F, S and t are, respectively, the current compression load, compression surface and specimen thickness. They can be approximated by Eq. (2) and these fitting flow curves are also shown in Fig. 2. To attain large deformations, specimens were deformed in two or three stages. After each stage the specimen was removed from the die, the ends of the specimen cut off and the Teflon films replaced before redeforming. This trimming gave rise in Fig. 2 to some discontinuity
(1)
The value of m = 1 corresponds to a Newtonian fluid with a viscosity η = K, m = 0 is the plastic flow rule for a material obeying √ the Huber–Mises yield criterion with a yield stress σp = 3K. For most metals m lies between 0.1 and 0.2. The ε˙ i symbol denotes equivalent strain rate. The influence of different orientations in bicrystal on material hardening was taken into account by applying different flow curves for each part of the bicrystals. The flow curves obtained from single crystal plane strain compression tests for each orientation were approximated by the following equation: σ = Aεm2 em4 /ε
(2)
where σ is the flow stress, ε the strain and A, m2 , m4 are the material parameters. An unstructured mesh of triangles with average edge length 0.25 mm was used giving about 4000 elements per sample. Due to the large deformation an automatic remeshing procedure was applied.
Fig. 2. Stress–strain curves obtained by plane strain compression tests at 77 K on single crystals with different orientations (hard, Goss, cube and shear). The fitting curves of Eq. (2) are also shown.
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in the experimentally measured stresses where each continuous section represents a single deformation stage. The crystals with √ cube, shear and Goss orientations, with a low Taylor factor ( 6) and low work hardening rates, are the ‘softest’ orientations, as expected. However, it should be noted that the latter exhibited significant differences in work hardening (i.e. shear > Goss > cube) implying that a simple analysis based on the initial Taylor factors is insufficient. The hard √ {1 1 0}0 1 1 orientation with the highest Taylor factor (2 6) is nevertheless always substantially harder. A second set of experiments was performed for the following bicrystals: ‘cube/hard’ {1 0 0}0 0 1/{1 1 0}0 1 1, ‘cube/ Goss’ {1 0 0}0 0 1/{1 1 0}0 0 1, ‘shear/Goss’ {1 0 0} 0 1 1/{1 1 0}0 0 1 with the grain boundary situated parallel to the compression plane (see Fig. 1). To obtain large deformations the ends of the bicrystal samples were periodically cut off as for the single crystals. 3.2. Shape changes and FE simulations The macroscopic structures, in the ED-ND longitudinal section, of the three deformed bicrystals at two different strains
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are presented in Fig. 3. These macroscopic shape changes correspond approximately to plane strain compression, i.e. compression, elongation and shear in the ED-ND plane, but the relative hardness of the grains has a clear effect on the strain distributions as seen in this section. The Goss/shear combination (Fig. 3a and b) appears to deform relatively homogeneously at the macroscopic scale although the grain boundary bends somewhat. The cube/Goss bicrystal initially appears to undergo larger strains in the cube part (Fig. 3c at 36% reduction) but then at higher strains the two zones even out (Fig. 3d at 69% reduction). The cube/hard combination is very heterogeneous as the cube grain undergoes large strains while the much harder {1 1 0}0 1 1 orientation scarcely deforms (Fig. 3e and f). Visual inspection suggests that the softer ‘cube’ grain undergo strains that are up to twice these of the global or nominal thickness reduction while the harder grain deforms significantly less. Exact values are difficult to give since the cube grain deform very inhomogeneously. As a result of the continuity requirement at the grain boundary the cube grain tends to flow over the harder grain and eventually at 59% reduction almost wraps around it (Fig. 3f). In the latter case there are large transition zones
Fig. 3. Macroscopic changes of the sample shape on the ED-ND longitudinal section. Bicrystals with: ‘Goss/shear’ orientation compressed (a) 48% and (b) 64%, ‘cube/Goss’ orientation compressed (c) 36% and (d) 69%, ‘cube/hard’ orientation compressed (e) 43% and (f) 59%.
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shapes, the distributions are shown at the same nominal deformations as in Fig. 3. However, it should be pointed out that in the simulation the sample was deformed continuously, i.e. without trimming the specimen ends; this may explain small differences in shape between the simulation and the experiment. ‘Goss’ and ‘shear’ orientations are characterized by similar Taylor factors and similar flow curves. As a consequence the bicrystals behave as a solid body without discontinuity of properties as depicted in Fig. 4. Compared with the experiment there is overall agreement although the simulated boundary position and its curvature near the extremities appears to be a little high. This is probably caused by too high values of the extrapolated stress taken from the compression test of the ‘shear’ crystal. The calculated strain distributions for the ‘cube/Goss’ bicrystal are presented in Fig. 5. Here again the orientations are characterized by the same nominal Taylor factor. The shape change at the moderate strain agrees quite well with experiment in that the lower Goss grain deforms less than cube which then tends to bend at the ends around Goss. This behaviour appears to be reduced at the higher strain so that the crystal boundary in the specimen centre than straightens up as seen experimentally. As pointed out above the Taylor factors and the flow stresses of the ‘cube’ and ‘hard’ orientations are significantly different. The shapes of the deformed specimens (Figs. 3e and f, and 6a and b) clearly indicate that specimen behaves almost like ‘two materials’ with very different mechanical properties. In the simulations the softer cube grains flow around the hard grain orientation exactly as in the experiments. Both grains undergo large strain heterogeneities, particularly at the lower strain where the central regions of the grains are highly strained (cube) or lightly strained (hard) whereas the extremities exhibit the opposite behaviour (Fig. 6a). There is a wide transition zone on both sides of the grain boundary. Further deformation tends to increase the strain gradients somewhat. The approximate values of strain near the central parts of the grain for the three bicrystal combinations are presented in Table 1.
Fig. 4. Calculated strain distribution in ‘Goss/shear’ bicrystal deformed (a) 48% and (b) 63% (scale in von Mises equivalent strain).
Fig. 5. Calculated strain distribution in ‘cube/Goss’ bicrystal deformed (a) 36% and (b) 69% (scale in von Mises equivalent strain).
Fig. 6. Calculated distribution strain in ‘cube/hard’ bicrystal deformed (a) 43% and (b) 59% (scale in von Mises equivalent strain).
near the grain boundary that are described in more detail below. It is also clear from the macrographs that macroscopic deformation bands are formed in some of the grains, particularly the cube orientation. To analyse the shape changes more quantitatively, a numerical analysis of bicrystal plane strain compression was applied as described in Section 2.3. The calculated strain distributions are presented in Figs. 4–6, assuming friction coefficients of 0.05. To facilitate comparison of the calculated and experimental sample
3.3. Microstructure evolution, optical microscopy The macroscopic structures of the deformed bicrystals in Fig. 3 give the sample shape changes but also an insight into the more localized deformation heterogeneities. Especially interesting is the behaviour of a crystallite of ‘hard’ orientation, which at the point of contact with a ‘cube’ grain orientation forms a strongly deformed transition zone (Fig. 3e and f). It is also appar-
Table 1 Experimental and FE simulated strains near the centre of bicrystal grains (values in percentage of thickness reductions) Bicrystal
{1 0 0}0 0 1/{1 1 0}0 1 1
{1 0 0}0 0 1/{1 1 0}0 0 1
{1 1 0}0 0 1/{1 0 0}0 1 1
Global thickness reduction (%)
43
36
48
Grain orientation
Cube
Hard
Cube
Hard
Cube
Goss
Cube
Goss
Goss
Shear
Goss
Shear
Experimental (%) Simulated (%)
47.0 68.1
38.6 17.9
87.1 75.5
38.1 42.5
42.4 56.5
29.6 15.5
69.0 76.6
58.0 61.4
31.7 46.3
64.2 46.7
52.0 65.5
78.2 60.5
59
69
63
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Fig. 7. (a) Orientation map (in grey scale) at the boundary of a bicrystal with ‘cube/Goss’ orientation configuration and the corresponding orientation distribution in the form of {1 1 1} pole figures; (b) ‘cube’-oriented grain; (c) ‘Goss’-oriented grain. Orientation measurements by SEM-FEG EBSD with 200 nm step size. Sample deformed 69%.
ent that the transition deformation zone between the two grains is mostly situated in the ‘hard’ grain. A crystallite with the ‘Goss’ orientation does not reveal the formation of band-like strain inhomogeneities. This behaviour is quite different from that of the other crystallites which show a strong tendency for strain localization from the early stages of deformation. The formation of band-like strain inhomogeneities in the unstable orientations (‘cube’ and ‘shear’) increases greatly with increasing deformation. Microscopic observations on the ND-ED plane reveal that these bands are inclined about 0–35◦ to ED, and their distribution in the sample section is strongly inhomogeneous. They usually penetrate the whole volume of the grain, separating areas with large lattice rotation, but do not show any tendency to intersect the grain boundaries. 3.4. Texture/microstructure evolution at the mesoscale; local orientation measurements by FEG-SEM EBSD The orientation maps were made by FEG-SEM EBSD using a step size of 200–500 nm. This enabled a detailed analysis of the deformation band distributions and their associated orientation spreads together with an analysis of the substructure development (via the spatial distributions of low-angle <2◦ boundaries). As is well known, the initial orientation of the crystallites and the configuration of the active slip systems clearly govern the different paths of the crystal lattice rotation.
Both {1 1 0}0 0 1/{1 0 0}0 0 1 and {1 0 0}0 1 1/ {1 1 0}0 0 1 bicrystals represent unstable versus stable grain orientation configurations. The orientation maps for the cube/Goss bicrystal in the area near the grain boundary (Fig. 7a) show differences in the morphology of the strain inhomogeneities that are formed. In the case of the ‘cube’oriented grain the crystal lattice undergoes major rotations, especially inside the elongated areas (Fig. 7a and b). This process leads finally to the formation of a coarse micro-band structure aligned approximately parallel to the compression plane. The crystallites with the ‘Goss’ orientation demonstrate only a weak tendency for lattice rotation, occurring mainly around an axis close to ND (Fig. 7c). The deformation banding tendency is more strongly marked in the case of the ‘shear/Goss’ configuration (Fig. 8a). Crystallites with the ‘shear’ orientation decompose very early on forming transition bands. This ‘decomposition’ process of the initial orientation leads to the formation of areas with approximately complementary {1 1 2}1 1 1 orientations (Fig. 8b), in which symmetrically situated slip systems operate, as described by Akef and Driver [1]. The orientation maps of the ‘hard/cube’ configuration indicate the formation of band-like strain inhomogeneities in the ‘cube’ grain (Fig. 9a), and an almost homogeneous substructure distribution in the ‘hard’ grain (Fig. 9b). Microstructure observations after 59% deformation in a cube crystallite clearly
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Fig. 8. (a) Orientation map (in grey scale) at the boundary of a bicrystal with ‘Goss/shear’ grains configuration and the corresponding orientation distribution as {1 1 1} pole figures; (b) grain with ‘shear’ orientation; (c) grain with ‘Goss’ orientation. Orientation measurements by SEM-FEG EBSD with step size 200 nm. Sample deformed 64%.
show that the inclination of the deformation bands depends on the location on the sample and is usually within the range 0–35◦ to ED. A large part of the crystal near the boundary undergoes a strong rotation towards {1 1 0}0 1 1 (by ±45◦ TD rotations), corresponding to the orientation of the neighbouring grain (Fig. 9c). This global rotation tendency, clearly observed in the area of the bands, also induces the matrix rotation outside the bands. The lattice rotation in the {1 1 0}0 1 1 crystallite of this combination (Fig. 9d) is relatively small and usually less than 10◦ . This small rotation is reflected in the {1 1 1} pole figure of Fig. 9d measured on the longitudinal section of the grain centre which only shows a small deviation from the initial position. However, in the transition zone near the grain boundary the orientation change attains values of about 35◦ . Cumulative misorientations, along the line scans drawn parallel to ND (marked in Figs. 7–9), reflect the differences in substructure that are formed. In the ‘Goss’ orientated crystallites of the ‘cube/Goss’ configuration, the misorientation is significantly less than 10◦ (Fig. 10b). This behaviour is slightly different from that observed in the same crystal orientation in the ‘shear/Goss’ configuration, where the misorientation values attain 20◦ (Fig. 10d). Higher values of misorientation are observed in crystallites with the ‘cube’ orientation, especially at the intersection of the bands of localized strain, where the measured misorientation values can go up to 35◦ (Fig. 10a and e). The greatest orientation changes (50◦ ), however, occur in crystallites with the ‘shear’ orientation and are associated with the TBs separating bands of different operative slip systems (Fig. 10c), as described by Driver et al. [10]. In the case of the ‘hard’-oriented crystalline the misorientation profile (Fig. 10f) shows a relatively
low amplitude and high frequency variation of the misorientation angle, which only occasionally attains 10◦ . 3.5. TEM microstructure and microtexture The TEM observations were carried out on samples taken from longitudinal sections of the deformed bicrystals. Although each TEM sample had a very large thin area no grain boundary could be found so that only separate single crystal areas of the deformed bicrystal specimens were observed. Nevertheless this gives information about the deformation mechanism in each grain during bicrystal compression. In the ‘cube’ grain of the ‘cube/hard’ bicrystal the deformation bands run across the whole volume of the crystal and show different inclinations with respect to ED. At relatively low strains most of the substructure consists of elongated areas, with a low density of misoriented sub-boundaries, and separated by areas of high dislocation density. At higher strains there are further lattice rotations associated with the gradual disappearance of the less misoriented volumes. After 59% reduction the substructure is finer, and the areas saturated by sub-grain boundaries tend to occupy a significant part of the ‘cube’ crystallite. Fig. 11a shows a TEM microstructure of a ‘cube’-oriented crystallite observed along the ED-ND plane after a thickness reduction of 59%; it is composed of regular layers of deformation bands separated by less dislocated matrix. The typical deformation bands observed here have widths in the range 1–10 m. The {1 1 1} pole figures, constructed from local orientations measured along two-line scans across the band, illustrate a characteristic scattering of the microtexture (Fig. 11b). The orientation gradients established in these bands, about 20◦ /m are very high for aluminium, prob-
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Fig. 9. Typical orientation maps (in grey scale) at the grain boundary of a bicrystal with ‘cube/hard’ orientation configuration for (a) cube and (b) hard orientation and the corresponding orientation distribution as {1 1 1} pole figures; (c) grain with ‘cube’ orientation, (d) grain with ‘hard’ orientation. Orientation measurements by SEM-FEG EBSD with step size 200 nm. Sample deformed 59%.
ably as a consequence of the low deformation temperature as noted by Paul and Driver [6]. The bands of localized strain, separated by areas of relatively weakly deformed matrix, clearly tend to form groups of compact bundles. For the ‘hard’-oriented crystallite of the same ‘cube/hard’ bicrystals, the crystallite does not show any significant tendency to produce strain heterogeneities (excluding the transition zone at the grain boundary). The orientation spread is nearly the same over a range of global deformation up to 59%. The TEM microstructure from the middle layer of the grain, presented in Fig. 11c shows a nearly uniform checkerboard pattern of roughly equiaxed dislocation cells with a diameter of 0.3–0.5 m. This structure is characterised by a relatively small orientation spread (Fig. 11d) such that the misorientation between particular sub-cells only rarely exceeds 10◦ . Generally, the {1 1 0}0 1 1-oriented grains deformed to the maximum applied strain did not exhibit any form of deformation banding. The Goss {1 1 0}0 0 1-oriented crystallite from the ‘cube/Goss’ bicrystal exhibits a typical, well-defined cellular dislocation structure, generated within this orientation (Fig. 12a). This microstructure consists of two complementary sets of elongated bands, inclined at ±35◦ to ED. Both sets of dislocation walls marking the bands lie very close to the expected
traces of the active {1 1 1} slip planes for this orientation. As a result of their intersection rectangular cells, with dimensions of 0.2–0.8 m, are formed. This microstructure is characterized by a relatively small orientation spread depicted in Fig. 12b with adjacent sub-cell misorientation angles which rarely attain 10◦ . Each set of dislocation bands occupies nearly the same volume of the crystal, but usually one set of the walls is more well defined. The cube {1 0 0}0 0 1-oriented crystallite TEM microstructure (Fig. 12c) from the ‘cube/Goss’ bicrystal obtained after 69% reduction is composed of regular layers of deformation bands separated by less dislocated matrix. It is apparent that the typical deformation bands observed here have a width in the range of 1–5 m, and are composed of slightly elongated and ill-defined ‘diffuse’ sub-cells. The latter are probably due to the low temperature deformation (77 K). These bands have different angles with respect to ED, typically ∼35◦ . They do not run completely through the crystals, but terminate in relatively broad zones (blocks). The {1 1 1} pole figure (Fig. 12d) illustrates the microtexture scattering, mainly as a result of an approximate TD (||1 0 0) rotation. The strongly dislocated bands are associated with rotations of 25–30◦ with respect to the neighbouring, only slightly dislocated matrix. The bands of localized strain, separated by areas of relatively weakly deformed matrix, tend to form
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Fig. 10. Cumulative misorientations (with respect to the first measured point) along ND direction; (a) grain with ‘cube’ and (b) grain with ‘Goss’ orientation from ‘cube/Goss’ bicrystal; (c) grain with ‘shear’ and (d) grain with ‘Goss’ orientation from ‘Goss/shear’ bicrystal; (e) grain with ‘cube’ and (f) grain with ‘hard’ orientation from ‘cube/hard’ bicrystal. Orientation measurements by SEM-FEG EBSD. Samples deformed globally: 69% (a and b), 64% (c and d), 59% (e and f).
groups of compact bundles, as reported by Paul and Driver [6] in cube grains from {1 0 0}0 0 1/{1 1 0}0 1 1-oriented bicrystals. The shear {1 0 0}0 1 1-oriented grain in the ‘shear/Goss’ bicrystal starts to form TBs from the early stages of deformation. They are the boundaries which separate sample volumes in which the crystal lattice rotates in opposite directions as a result of the different slip systems. TEM microstructure observation shows a strongly elongated cell substructure within these areas (Fig. 13a). The process of TB formation is associated with the decomposition of the initial crystal orientation resulting from the TD rotation towards two nearly complementary orientations of {1 1 2}1 1 1 type although with a strong scattering of the crystal lattice (Fig. 13b). The Goss {1 1 0}0 0 1-oriented grain in the ‘shear/Goss’ grain configuration appears a little less stable when compared to the above behaviour in the ‘cube/Goss’ grain configuration. Generally, two sets of dislocation cells dominate the TEM microstructure but their pattern in this case is more, ‘wavy’ (Fig. 13c). This leads to a wider visible scattering of the initial crystal orientation (Fig. 13d). 4. Discussion The major results of this experimental study of the large strain deformation of Al bicrystals is that the behaviour of both grains of the bicrystal is essentially very similar to that of individual crystals deformed under the same conditions (here low tem-
perature channel-die compression). The influence of the grain boundary appears to be limited to local zones along some boundaries although this is strongly amplified in the extreme case of the cube/hard bicrystal combination. The similarity of the grain deformation modes is found here for virtually all the grain orientations in the crystals. Thus: (i) The Goss grains remain very stable up to large strains with a uniform dislocation cell substructure as observed for example by Ferry and Humphreys [11] on similarly oriented Al–0.05 at% Si crystals and by Borbely et al. [12] on Cu crystals. (ii) The shear orientation breaks up rapidly during PSC into thin, alternating, layers of near {1 1 2}1 1 1 bands separated by transition bands as found on Al crystals by Akef and Driver [1] and Becker et al. [3]. The near “Cu” orientations then develop fine shear bands as observed many times for this orientation. (iii) Most of the cube grains break up by deformation banding creating high local misorientations and transition bands aligned along ED as previously documented in some detail by Akef and Driver [1], Liu et al. [13] and Basson and Driver [14]. (iv) The microstructures of the hard {1 1 0}0 1 1 orientation have not, to our knowledge, been previously described in detail but their relative homogeneity at moderate strains is consistent with the usual stability of the {1 1 0}u v w
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Fig. 11. TEM microstructures observed in crystallites forming ‘cube/hard’ bicrystals deformed 59%. (a) Deformation band in the cube grain, and (b) corresponding {1 1 1} pole figure, (c) homogeneous cell substructure in the ‘hard’ grain orientation’ and (d) corresponding {1 1 1} pole figure.
Fig. 12. TEM microstructures observed in crystallites forming ‘cube/Goss’ bicrystals deformed 69%. (a) Nearly perfect homogeneous cell substructure in a ‘Goss’oriented crystallite and (b) corresponding {1 1 1} pole figure. (c) Formation of deformation bands in a ‘cube’ grain and (d) corresponding {1 1 1} pole figure.
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Fig. 13. TEM microstructures observed in crystallites forming ‘Goss/shear’ bicrystals deformed 64%. (a) Transition bands (TBs) in a crystallite with ‘shear’ orientation, and (b) corresponding {1 1 1} pole figure. (c) Slightly disturbed cell substructure in a ‘Goss’ orientation and (d) corresponding {1 1 1} pole figure.
orientations in PSC. In accordance with Stanford et al. [15] at a strain of 0.5 these crystallites developed only a small amount of orientation spread, and at a strain of 1.0 slightly more spread occurred about TD. However, no deformation bands were detected up to deformations of 1.0. These observations on deformed bicrystals of controlled grain orientations are in basic agreement with Risø studies on the influence of grain orientation on substructure development in deformed polycrystals, as reviewed by Hansen [16]. The influence of the grain boundary only appears significant in the case of the transition region of the cube/hard bicrystal. This general behaviour is worth comparing with that of a recent study of the channel-die compression of Al bicrystals of other orientations by Zaefferer et al. [17]. The latter authors compressed three bicrystals with boundary misorientations defined by 1 1 2 tilt axes and misorientation angles of about 9, 15 and 31◦ . In this work ¯ which the grain orientations appear to be close to {1 1¯ 0}1 1 1 are non-symmetrical orientations leading to shape changes with macroscopic shears, of opposite sign, along ED (Skalli et al. [18]). The resulting deformation of the bicrystal is very heterogeneous. Zaefferer et al. [17] also find that in general the crystal orientation has a stronger influence on the grain behaviour than the structure of the grain boundary. Using the same high resolution EBSD technique, some slightly higher misorientations (by
1–2◦ ) were, however, observed near the grain boundaries in the 15 and 31◦ misoriented bicrystals and attributed to dislocation pile-ups at the grain boundaries. This type of macroscopic shear strain incompatibility is not expected in our more symmetrically oriented crystals which should not undergo macroscopic shearing (although microscopic shearing occurs extensively in the initial cube and shear orientations). In general, one therefore expects to find the substructure developments characteristic of similarly oriented single crystals. In all the cases examined here the finite elements simulations predict symmetrical strain profiles with strong maxima at the corners. Also, experimentally, all bicrystals show strong deformation heterogeneities in the form of macroscopic shear localizations, symmetrical with respect to the ND-TD plane. If the ‘hardness’ of the crystallites is nearly the same (as in the case of ‘Goss/shear’ grain configuration) there is a clear tendency to form zig-zag shaped localized strains. This is not so pronounced in the bicrystals with very different hardnesses. In accordance with Raabe et al. [8] and Zaefferer et al. [17] the strong heterogeneity of strain as predicted by modeling can be attributed to the combined influence of macromechanical effects, such as friction, grain hardening, grain interactions, etc. The behaviour of the ‘cube/hard’ bicrystal is surprising in that a transition zone of width about 1.5 mm is developed from the
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Fig. 14. (a) Optical micrograph of the transition zone near the grain boundary in ‘cube/hard’ bicrystal sample deformed 59%. (b) {1 1 1} pole figure showing orientation changes across the grain boundary and transition zone (scan area of 200 m × 5000 m). (c) Cumulative misorientation profile (with respect to the first point) lying within the ‘cube’ grain across the transition zone; SEM-FEG EBSD measurements in the ND-ED plane with step size 200 nm.
initial hard orientation. This is shown in more detail in Fig. 14a and is probably due to the very high-localized shear of the ‘hard’ {1 1 0}0 1 1 grain as it deforms near the ‘cube’ {1 0 0}0 0 1. The volumes of the hard crystallite lying near the grain boundary must accommodate the incompatibilities between the neighbouring grains, leading to a strongly deformed (and elongated) ‘cube’ grain and the slightly elongated ‘hard’ crystallite. As shown in Fig. 14b this induces a broad orientation spread across the grain boundary and the transition zone (near the free surfaces of the sample). The figure shows a strong, continuous dispersion of the 1 1 1 poles characterized by strong scattering by (±)TD rotation with some additional rotations around the ED and ND axes. The misorientations, calculated with respect to the first measured point lying within the ‘cube’ grain across the grain boundary, reflect the difference in the structure that is formed (Fig. 14c). There is only about 12–15◦ between the ‘hard’ grain and the first measured point in the ‘cube’ grain as a result of the rapid rotation of the ‘cube’ grain near the transition zone towards the {1 1 0}0 1 1 orientation. This very particular behaviour needs to be verified, for example by FE simulations incorporating crystal plasticity relations as opposed to the standard plasticity simulations used here. The areas lying near the middle layers are very homogeneous as observed at both the TEM and SEM scales. This is in accordance with earlier work by Stanford et al. [15] showing that this orientation is generally stable and did not exhibit significant deformation banding up to strains of 1.0. However, after higher strains (>1.4), Ferry and Humphreys [11] observed a well-developed band structure in aluminium. This implies a critical strain for bands to form in Al of this orientation between 1.0 and 1.4.
5. Conclusions Both macro- and micro-mechanical effects in three types of plane strain compressed aluminium bicrystals have been investigated by FE simulation and high-resolution electron metallography. The bicrystals were composed of grain pairs which are hard/soft and stable/unstable combinations. The main conclusions are: (i) The overall strain profiles and macroscopically observed grain shapes of the deformed bicrystals are quite well predicted by FE simulations (taking the appropriate single crystal hardening curves and assuming reasonable boundary conditions). In particular the extreme case of the bicrystal composed of the hard {1 1 0}0 1 1 and soft cube grains where the latter flows over the hard grain is well represented. (ii) At the microscopic level, the grains of the other two bicrystal combinations behave essentially as single crystals under the same deformation conditions with respect to the deformation substructure and microtexture distributions. The unstable cube and shear orientations break up into deformation and transition bands as previously reported for single crystals. The stable Goss orientation develops a typical uniform deformation substructure albeit with somewhat higher misorientations in the Goss/shear combination. It is concluded that for most grain pair combinations of the bicrystals the individual grain orientations play a dominant role in deformation substructure development as a consequence of their dependence on slip system interactions.
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Acknowledgements The authors wish to thank Prof. M. Pietrzyk from AGH University of Science and Technology, Krak´ow for providing access to Finite Element software and Dr. A. Pi˛atkowski (IMIM PAS, Krak´ow) who have helped in the deformation and optical metallography of the bicrystals. References [1] [2] [3] [4] [5]
A. Akef, J.H. Driver, Mater. Sci. Eng. A132 (1991) 245–255. C. Maurice, J.H. Driver, Acta Met. Mater. 41 (1993) 1653–1664. R. Becker, J.F. Butler, H. Hu, L.A. Lalli, Metall. Trans. 22A (1991) 45–58. H. Paul, M. Darrieulat, A. Pi˛atkowski, Z. Metallkd. 92 (2001) 1213–1221. H. Paul, J.H. Driver, C. Maurice, M. Bijak, Arch. Metall. Mater. 51 (2006) 43–53. [6] H. Paul, J.H. Driver, Arch. Metall. Mater. 50 (2005) 209–218. [7] H. Paul, J.H. Driver, A. Lens, Mater. Sci. Forum 495–497 (2005) 1249–1254.
[8] D. Raabe, M. Sachtleber, Z. Zhao, F. Roters, S. Zaefferer, Acta Mater. 49 (2001) 3433–3441. [9] J.-L. Chenot, M. Bellet, in: P. Hartley, I. Pillinger, C.E.N. Sturges (Eds.), The Viscoplastic Approach for the Finite-Element Modelling of Metal Forming Processes, Numerical Modelling of Material Deformation Processes, Springer-Verlag, London, Berlin, 1992, pp. 179–224. [10] J.H. Driver, M.C. Theyssier, C. Maurice, Mater. Sci. Technol. 12 (1996) 851–858. [11] N. Ferry, F. Humphreys, Mater. Sci. Forum 493 (1996) 1–6. [12] A. Borbely, Cl. Maurice, D. Piot, J. Driver, Acta Mater. 55 (2007) 487– 496. [13] Q. Liu, Cl. Maurice, J. Driver, N. Hansen, Metall. Trans. 29A (1998) 2333–2344. [14] F. Basson, J.H. Driver, Acta Mater. 48 (2000) 2101–2115. [15] N. Stanford, D. Dune, M. Ferry, Acta Mater. 51 (2003) 665–676. [16] N. Hansen, Met. Mater. Trans. 32A (2001) 2917–2935. [17] S. Zaefferer, J.-C. Kuo, Z. Zhao, M. Winning, D. Raabe, Acta Mater. 51 (2003) 4719–4735. [18] A. Skalli, J.H. Driver, M. Wintenberger, M´emoires et Etudes Sci. de la Rev. M´etall. 50 (1983) 293–302.
Strain hardening and microstructure evolution of channel-die compressed aluminium bicrystals H. Paul a,b,∗ , J.H. Driver c , W. Wajda a a
Polish Academy of Sciences, Institute of Metallurgy and Materials Science, 25 Reymonta St., 30-059 Krak´ow, Poland b University of Zielona G´ ora, Mechanical Department, 50 Podg´orna St., 65-246 Zielona G´ora, Poland c Ecole des Mines de Saint-Etienne, Centre SMS, 158 Cours Fauriel, 42023 Saint Etienne, France Received 23 February 2007; received in revised form 9 May 2007; accepted 11 May 2007
Abstract The different types of strain heterogeneities developed in pure aluminium bicrystals during plane strain compression have been characterized by advanced metallographic techniques over several length scales. Bicrystals, with three grain orientation combinations (one hard/soft, and two stable/unstable) containing grain boundaries perpendicular to the compression axis, were deformed up to strains of 1.5 at 77 K and their deformation substructure investigated by optical microscopy, high resolution EBSD and transmission electron microscopy. At the macroscopic scale the hard/soft combination develops major strain variations as the softer grain flows out over the harder grain. This is consistent with finite element simulations of bicrystal deformation using the appropriate individual grain hardening laws and leads to a large transition zone near the boundary. At the microscopic level, the grains of the other combinations behave essentially as single crystals under the same deformation conditions, in particular with respect to the deformation substructure and microtexture distributions. It is concluded that for most grain pair combinations of deformed bicrystals the individual grain orientations play a dominant role in deformation substructure development as a consequence of their dependence on slip system interactions. © 2007 Elsevier B.V. All rights reserved. Keywords: Aluminium; Bicrystals; Deformation; Microstructures; FEM; Strain hardening; Strain heterogeneities
1. Introduction As a natural consequence of crystal anisotropy, it is well known that individual grains in a polycrystalline aggregate exhibit different stress–strain behaviours. The total hardening of a deformed polycrystal then results from the hardening of the grains and a mutual interaction between grains in the material. As a consequence, neighbouring grains with different hardening rates tend to accumulate additional plastic (incompatibility) strains close to the grain boundaries. These highly deformed zones of the crystals may then become privileged sites for recrystallization nucleation during annealing. A knowledge of the influence of grain misorientation on both hardening and the development of localized deformation, e.g. deformation bands, ∗
Corresponding author at: Polish Academy of Sciences, Institute of Metallurgy and Materials Science, 25 Reymonta St., 30-059 Krak´ow, Poland. Tel.: +48 12 637 42 00; fax: +48 12 6372192. E-mail addresses: [email protected] (H. Paul), [email protected] (J.H. Driver), [email protected] (W. Wajda). 0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.05.044
close to the grain boundaries is therefore useful for understanding polycrystal plasticity and subsequent annealing effects. To study these complex phenomena, it is preferable to perform repeatable experiments with well-defined boundary conditions. In this context plane strain compression (PSC) tests on fcc bicrystals, with a centrally located grain boundary, can be particularly informative. In the present experiments, aluminium bicrystals, with high symmetry grain orientations, were chosen to represent different kinds of grain hardening and boundary misorientations. One of the grain orientations is the cube which is known to be very unstable during low temperature PS compression and forms deformation bands (DBs) along the extension direction [1,2]. The ‘shear’ orientation {1 0 0}0 1 1 (often denoted shear since it forms in the sheared zones of rolled fcc metals) is also very unstable and breaks up very rapidly during PSC into misoriented bands through opposite rotations around the transverse direction [1,3]. This forms transition bands (TB) at relatively low strains and eventually shear bands (SBs) at very large strains, e.g. [3,4]. On the other hand, the Goss orientation {1 1 0}0 0 1
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is known to be stable up to very high strains. The behaviour of crystallites with the hard {1 1 0}0 1 1 orientation is not so well characterized but is often considered moderately stable during compression. These crystal orientations can be combined into bicrystals composed of grains with very contrasting behaviours, e.g. unstable versus stable (‘cube/Goss’ or ‘shear/Goss’), unstable versus unstable (‘cube/shear’) and soft versus hard (‘cube/hard’). This paper describes an experimental study of the compression behaviour of such bicrystals and their microstructural evolution. Using channel-die compression experiments, stress–strain curves for single crystals have been established, giving the influence of grain misorientation on material hardening. This is then used as a basis for finite element (FE) simulations of the stress and strain distributions expected of such bicrystals. As shown below, large plastic strains of grains with very contrasting mechanical responses can create some major plastic heterogeneities in these bicrystals and FE simulations then indicate the local strains. The paper then concentrates on the deformation microstructure evolution of the grains using high resolution FEG-SEM EBSD and TEM, particularly for the microtextures. Some specific results for individual bicrystals have been previously reported [5–7] but the present analysis describes a wider set of bicrystals linking the stress–strain curves to microstructure development and the strain distributions. 2. Experimental procedures 2.1. Material and sample preparation The bicrystals with controlled orientations were grown by a modified Bridgman technique (horizontal solidification), using split graphite moulds from high purity Al (99.998%). The dimensions of the bicrystal bars were approximately 15 mm (thickness) × 22 mm (width) × 150 mm (length). Samples of 15 mm height, 15 mm length and 10 mm width were sectioned from a bar and plane strain compressed at a nominal strain rate of ∼10−4 s−1 . Teflon films were used as a lubricant and periodically replaced at strain intervals of about 0.2. To favour
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deformation banding and reduce dynamic recovery effects the tests were performed at 77 K by immersing the channel-die device in a reservoir with liquid nitrogen. This procedure gives, at relatively low strains, a work-hardened structure containing clearly defined bands. The following three types of grain configurations were used in channel-die compression test in order to study the effect of misorientation between grains on material hardening and strain localization: ‘cube/hard’ {1 0 0}0 0 1/{1 1 0}0 1 1; ‘cube/Goss’ {1 0 0}0 0 1/{1 1 0}0 0 1 and ‘shear/Goss’ {1 0 0}0 1 1/{1 1 0}0 0 1, where {h k l}u v w signify the compression plane of normal ND and the elongation direction ED, respectively. The position of the bicrystal in the channel-die device and the nominal orientations of the grains are shown in Fig. 1. The orientation relations between the grains of a bicrystal, could be described by the relations: {1 0 0}0 0 1 rotation 45◦ around 1 0 0 || TD → {1 1 0} 0 1 1 {1 0 0}0 0 1 rotation 45◦ around 1 0 0 || ED → {1 1 0} 0 0 1 {1 0 0}0 1 1 rotation 90◦ around 0 1 1 || TD → {1 1 0} 0 0 1 where TD is the transverse direction. The deformation geometry of these bicrystals was such that the bicrystal boundary separating the top and bottom crystallites at specimen mid-thickness lay parallel to the compression plane. 2.2. Microstructure and local orientation measurements The deformation microstructures were characterized over a wide range of scales using a combination of TEM, SEM and optical metallography. The samples were sectioned in the ND-ED plane by wire cutting. For thin foil preparation a twinset electropolishing technique was used in standard nitric acid–methanol solution. Optical microscopy was carried out using polarized light on electropolished and lightly anodized (in
Fig. 1. Schematic bicrystal configurations during channel-die compression showing the grain boundary location and the grain orientations: (a) ‘Goss/shear’, (b) ‘cube/Goss’ and (c) ‘cube/hard’.
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2% fluoroboric acid solution) sections. The majority of the metallographic observations were made on the longitudinal plane, i.e. ND-ED. The orientations of the crystallites composing the bicrystals were checked by the X-ray back reflexion Laue method. The deformed specimens were mostly examined by scanning electron microscopy (SEM) in a JEOL JSM 6500F, equipped with a field emission gun (FEG) and electron backscattered diffraction (EBSD) facilities, in which microscope control, pattern acquisition and solution were performed with the HKL Channel 5TM system. For more detailed analyses transmission electron microsopy (200 kV Philips TEM CM200) with semi-automatic Kikuchi pattern analysis, was employed. Local orientation data, obtained by SEM and TEM techniques on the ND-ED section were transformed to the standard ED-TD reference system and presented in the form of {1 1 1} pole figures.
A frequently used mixed Coulomb and Tresca friction law was used in following form: τ = μσn σo τ = m√ 3
σp μσn < √ 3
if if
σp μσn > m √ 3
(3)
where √ τ is the friction shear stress, σ n the normal stress, σp / 3 the shear yield stress and μ, m are the Coulomb and Tresca friction coefficients, respectively. In the simulations both friction coefficients were set to values of 0.05 or 0.2. A friction coefficient of about 0.05 or less is expected for Teflon in the low temperature range. 3. Results 3.1. Stress–strain behaviour
2.3. Simulation method For the FE simulations the bicrystal behaviour was described using the stress–strain curves obtained on single crystals with the same orientations compressed under the same conditions in a channel-die. This is in accordance with the general assumption that aluminium bicrystals deformed up to medium strains behave to a large extent like two individual single crystals [8]. The simulation was also simplified to a 2D plane strain problem as would be expected if adequate lubrication is used. The bicrystal deformation simulation was carried out using the Forge2 software in the Department of Computer Methods in Metallurgy in Krakow. The finite element model used in the FORGE code is based on the Norton-Hoff constitutive law and is described by Chenot and Bellet [9]. The law is generally written in the form of the following relation between deviatoric stress tensor (s- ) and the strain rate tensor (˙ε- ): √ m−1 s- = 2K( 3˙εi ) ε˙-
Two sets of experiments were carried out. First, plane strain compression of single crystals with different orientations was performed at 77 K: (hard {1 1 0}0 1 1 with a Taylor factor √ of 2 6, and Goss {1 1 0}0 0 1, cube {1 0 0}0 √0 1 and shear {1 0 0}0 1 1 with nominal Taylor factors of 6) in order to establish the stress–strain curves for use in the finite element simulations. The experimentally measured true stress–strain curves of the four crystal orientations are shown in Fig. 2 using standard definitions, σ = F/S and ε = ln(to /t), where F, S and t are, respectively, the current compression load, compression surface and specimen thickness. They can be approximated by Eq. (2) and these fitting flow curves are also shown in Fig. 2. To attain large deformations, specimens were deformed in two or three stages. After each stage the specimen was removed from the die, the ends of the specimen cut off and the Teflon films replaced before redeforming. This trimming gave rise in Fig. 2 to some discontinuity
(1)
The value of m = 1 corresponds to a Newtonian fluid with a viscosity η = K, m = 0 is the plastic flow rule for a material obeying √ the Huber–Mises yield criterion with a yield stress σp = 3K. For most metals m lies between 0.1 and 0.2. The ε˙ i symbol denotes equivalent strain rate. The influence of different orientations in bicrystal on material hardening was taken into account by applying different flow curves for each part of the bicrystals. The flow curves obtained from single crystal plane strain compression tests for each orientation were approximated by the following equation: σ = Aεm2 em4 /ε
(2)
where σ is the flow stress, ε the strain and A, m2 , m4 are the material parameters. An unstructured mesh of triangles with average edge length 0.25 mm was used giving about 4000 elements per sample. Due to the large deformation an automatic remeshing procedure was applied.
Fig. 2. Stress–strain curves obtained by plane strain compression tests at 77 K on single crystals with different orientations (hard, Goss, cube and shear). The fitting curves of Eq. (2) are also shown.
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in the experimentally measured stresses where each continuous section represents a single deformation stage. The crystals with √ cube, shear and Goss orientations, with a low Taylor factor ( 6) and low work hardening rates, are the ‘softest’ orientations, as expected. However, it should be noted that the latter exhibited significant differences in work hardening (i.e. shear > Goss > cube) implying that a simple analysis based on the initial Taylor factors is insufficient. The hard √ {1 1 0}0 1 1 orientation with the highest Taylor factor (2 6) is nevertheless always substantially harder. A second set of experiments was performed for the following bicrystals: ‘cube/hard’ {1 0 0}0 0 1/{1 1 0}0 1 1, ‘cube/ Goss’ {1 0 0}0 0 1/{1 1 0}0 0 1, ‘shear/Goss’ {1 0 0} 0 1 1/{1 1 0}0 0 1 with the grain boundary situated parallel to the compression plane (see Fig. 1). To obtain large deformations the ends of the bicrystal samples were periodically cut off as for the single crystals. 3.2. Shape changes and FE simulations The macroscopic structures, in the ED-ND longitudinal section, of the three deformed bicrystals at two different strains
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are presented in Fig. 3. These macroscopic shape changes correspond approximately to plane strain compression, i.e. compression, elongation and shear in the ED-ND plane, but the relative hardness of the grains has a clear effect on the strain distributions as seen in this section. The Goss/shear combination (Fig. 3a and b) appears to deform relatively homogeneously at the macroscopic scale although the grain boundary bends somewhat. The cube/Goss bicrystal initially appears to undergo larger strains in the cube part (Fig. 3c at 36% reduction) but then at higher strains the two zones even out (Fig. 3d at 69% reduction). The cube/hard combination is very heterogeneous as the cube grain undergoes large strains while the much harder {1 1 0}0 1 1 orientation scarcely deforms (Fig. 3e and f). Visual inspection suggests that the softer ‘cube’ grain undergo strains that are up to twice these of the global or nominal thickness reduction while the harder grain deforms significantly less. Exact values are difficult to give since the cube grain deform very inhomogeneously. As a result of the continuity requirement at the grain boundary the cube grain tends to flow over the harder grain and eventually at 59% reduction almost wraps around it (Fig. 3f). In the latter case there are large transition zones
Fig. 3. Macroscopic changes of the sample shape on the ED-ND longitudinal section. Bicrystals with: ‘Goss/shear’ orientation compressed (a) 48% and (b) 64%, ‘cube/Goss’ orientation compressed (c) 36% and (d) 69%, ‘cube/hard’ orientation compressed (e) 43% and (f) 59%.
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shapes, the distributions are shown at the same nominal deformations as in Fig. 3. However, it should be pointed out that in the simulation the sample was deformed continuously, i.e. without trimming the specimen ends; this may explain small differences in shape between the simulation and the experiment. ‘Goss’ and ‘shear’ orientations are characterized by similar Taylor factors and similar flow curves. As a consequence the bicrystals behave as a solid body without discontinuity of properties as depicted in Fig. 4. Compared with the experiment there is overall agreement although the simulated boundary position and its curvature near the extremities appears to be a little high. This is probably caused by too high values of the extrapolated stress taken from the compression test of the ‘shear’ crystal. The calculated strain distributions for the ‘cube/Goss’ bicrystal are presented in Fig. 5. Here again the orientations are characterized by the same nominal Taylor factor. The shape change at the moderate strain agrees quite well with experiment in that the lower Goss grain deforms less than cube which then tends to bend at the ends around Goss. This behaviour appears to be reduced at the higher strain so that the crystal boundary in the specimen centre than straightens up as seen experimentally. As pointed out above the Taylor factors and the flow stresses of the ‘cube’ and ‘hard’ orientations are significantly different. The shapes of the deformed specimens (Figs. 3e and f, and 6a and b) clearly indicate that specimen behaves almost like ‘two materials’ with very different mechanical properties. In the simulations the softer cube grains flow around the hard grain orientation exactly as in the experiments. Both grains undergo large strain heterogeneities, particularly at the lower strain where the central regions of the grains are highly strained (cube) or lightly strained (hard) whereas the extremities exhibit the opposite behaviour (Fig. 6a). There is a wide transition zone on both sides of the grain boundary. Further deformation tends to increase the strain gradients somewhat. The approximate values of strain near the central parts of the grain for the three bicrystal combinations are presented in Table 1.
Fig. 4. Calculated strain distribution in ‘Goss/shear’ bicrystal deformed (a) 48% and (b) 63% (scale in von Mises equivalent strain).
Fig. 5. Calculated strain distribution in ‘cube/Goss’ bicrystal deformed (a) 36% and (b) 69% (scale in von Mises equivalent strain).
Fig. 6. Calculated distribution strain in ‘cube/hard’ bicrystal deformed (a) 43% and (b) 59% (scale in von Mises equivalent strain).
near the grain boundary that are described in more detail below. It is also clear from the macrographs that macroscopic deformation bands are formed in some of the grains, particularly the cube orientation. To analyse the shape changes more quantitatively, a numerical analysis of bicrystal plane strain compression was applied as described in Section 2.3. The calculated strain distributions are presented in Figs. 4–6, assuming friction coefficients of 0.05. To facilitate comparison of the calculated and experimental sample
3.3. Microstructure evolution, optical microscopy The macroscopic structures of the deformed bicrystals in Fig. 3 give the sample shape changes but also an insight into the more localized deformation heterogeneities. Especially interesting is the behaviour of a crystallite of ‘hard’ orientation, which at the point of contact with a ‘cube’ grain orientation forms a strongly deformed transition zone (Fig. 3e and f). It is also appar-
Table 1 Experimental and FE simulated strains near the centre of bicrystal grains (values in percentage of thickness reductions) Bicrystal
{1 0 0}0 0 1/{1 1 0}0 1 1
{1 0 0}0 0 1/{1 1 0}0 0 1
{1 1 0}0 0 1/{1 0 0}0 1 1
Global thickness reduction (%)
43
36
48
Grain orientation
Cube
Hard
Cube
Hard
Cube
Goss
Cube
Goss
Goss
Shear
Goss
Shear
Experimental (%) Simulated (%)
47.0 68.1
38.6 17.9
87.1 75.5
38.1 42.5
42.4 56.5
29.6 15.5
69.0 76.6
58.0 61.4
31.7 46.3
64.2 46.7
52.0 65.5
78.2 60.5
59
69
63
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Fig. 7. (a) Orientation map (in grey scale) at the boundary of a bicrystal with ‘cube/Goss’ orientation configuration and the corresponding orientation distribution in the form of {1 1 1} pole figures; (b) ‘cube’-oriented grain; (c) ‘Goss’-oriented grain. Orientation measurements by SEM-FEG EBSD with 200 nm step size. Sample deformed 69%.
ent that the transition deformation zone between the two grains is mostly situated in the ‘hard’ grain. A crystallite with the ‘Goss’ orientation does not reveal the formation of band-like strain inhomogeneities. This behaviour is quite different from that of the other crystallites which show a strong tendency for strain localization from the early stages of deformation. The formation of band-like strain inhomogeneities in the unstable orientations (‘cube’ and ‘shear’) increases greatly with increasing deformation. Microscopic observations on the ND-ED plane reveal that these bands are inclined about 0–35◦ to ED, and their distribution in the sample section is strongly inhomogeneous. They usually penetrate the whole volume of the grain, separating areas with large lattice rotation, but do not show any tendency to intersect the grain boundaries. 3.4. Texture/microstructure evolution at the mesoscale; local orientation measurements by FEG-SEM EBSD The orientation maps were made by FEG-SEM EBSD using a step size of 200–500 nm. This enabled a detailed analysis of the deformation band distributions and their associated orientation spreads together with an analysis of the substructure development (via the spatial distributions of low-angle <2◦ boundaries). As is well known, the initial orientation of the crystallites and the configuration of the active slip systems clearly govern the different paths of the crystal lattice rotation.
Both {1 1 0}0 0 1/{1 0 0}0 0 1 and {1 0 0}0 1 1/ {1 1 0}0 0 1 bicrystals represent unstable versus stable grain orientation configurations. The orientation maps for the cube/Goss bicrystal in the area near the grain boundary (Fig. 7a) show differences in the morphology of the strain inhomogeneities that are formed. In the case of the ‘cube’oriented grain the crystal lattice undergoes major rotations, especially inside the elongated areas (Fig. 7a and b). This process leads finally to the formation of a coarse micro-band structure aligned approximately parallel to the compression plane. The crystallites with the ‘Goss’ orientation demonstrate only a weak tendency for lattice rotation, occurring mainly around an axis close to ND (Fig. 7c). The deformation banding tendency is more strongly marked in the case of the ‘shear/Goss’ configuration (Fig. 8a). Crystallites with the ‘shear’ orientation decompose very early on forming transition bands. This ‘decomposition’ process of the initial orientation leads to the formation of areas with approximately complementary {1 1 2}1 1 1 orientations (Fig. 8b), in which symmetrically situated slip systems operate, as described by Akef and Driver [1]. The orientation maps of the ‘hard/cube’ configuration indicate the formation of band-like strain inhomogeneities in the ‘cube’ grain (Fig. 9a), and an almost homogeneous substructure distribution in the ‘hard’ grain (Fig. 9b). Microstructure observations after 59% deformation in a cube crystallite clearly
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Fig. 8. (a) Orientation map (in grey scale) at the boundary of a bicrystal with ‘Goss/shear’ grains configuration and the corresponding orientation distribution as {1 1 1} pole figures; (b) grain with ‘shear’ orientation; (c) grain with ‘Goss’ orientation. Orientation measurements by SEM-FEG EBSD with step size 200 nm. Sample deformed 64%.
show that the inclination of the deformation bands depends on the location on the sample and is usually within the range 0–35◦ to ED. A large part of the crystal near the boundary undergoes a strong rotation towards {1 1 0}0 1 1 (by ±45◦ TD rotations), corresponding to the orientation of the neighbouring grain (Fig. 9c). This global rotation tendency, clearly observed in the area of the bands, also induces the matrix rotation outside the bands. The lattice rotation in the {1 1 0}0 1 1 crystallite of this combination (Fig. 9d) is relatively small and usually less than 10◦ . This small rotation is reflected in the {1 1 1} pole figure of Fig. 9d measured on the longitudinal section of the grain centre which only shows a small deviation from the initial position. However, in the transition zone near the grain boundary the orientation change attains values of about 35◦ . Cumulative misorientations, along the line scans drawn parallel to ND (marked in Figs. 7–9), reflect the differences in substructure that are formed. In the ‘Goss’ orientated crystallites of the ‘cube/Goss’ configuration, the misorientation is significantly less than 10◦ (Fig. 10b). This behaviour is slightly different from that observed in the same crystal orientation in the ‘shear/Goss’ configuration, where the misorientation values attain 20◦ (Fig. 10d). Higher values of misorientation are observed in crystallites with the ‘cube’ orientation, especially at the intersection of the bands of localized strain, where the measured misorientation values can go up to 35◦ (Fig. 10a and e). The greatest orientation changes (50◦ ), however, occur in crystallites with the ‘shear’ orientation and are associated with the TBs separating bands of different operative slip systems (Fig. 10c), as described by Driver et al. [10]. In the case of the ‘hard’-oriented crystalline the misorientation profile (Fig. 10f) shows a relatively
low amplitude and high frequency variation of the misorientation angle, which only occasionally attains 10◦ . 3.5. TEM microstructure and microtexture The TEM observations were carried out on samples taken from longitudinal sections of the deformed bicrystals. Although each TEM sample had a very large thin area no grain boundary could be found so that only separate single crystal areas of the deformed bicrystal specimens were observed. Nevertheless this gives information about the deformation mechanism in each grain during bicrystal compression. In the ‘cube’ grain of the ‘cube/hard’ bicrystal the deformation bands run across the whole volume of the crystal and show different inclinations with respect to ED. At relatively low strains most of the substructure consists of elongated areas, with a low density of misoriented sub-boundaries, and separated by areas of high dislocation density. At higher strains there are further lattice rotations associated with the gradual disappearance of the less misoriented volumes. After 59% reduction the substructure is finer, and the areas saturated by sub-grain boundaries tend to occupy a significant part of the ‘cube’ crystallite. Fig. 11a shows a TEM microstructure of a ‘cube’-oriented crystallite observed along the ED-ND plane after a thickness reduction of 59%; it is composed of regular layers of deformation bands separated by less dislocated matrix. The typical deformation bands observed here have widths in the range 1–10 m. The {1 1 1} pole figures, constructed from local orientations measured along two-line scans across the band, illustrate a characteristic scattering of the microtexture (Fig. 11b). The orientation gradients established in these bands, about 20◦ /m are very high for aluminium, prob-
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Fig. 9. Typical orientation maps (in grey scale) at the grain boundary of a bicrystal with ‘cube/hard’ orientation configuration for (a) cube and (b) hard orientation and the corresponding orientation distribution as {1 1 1} pole figures; (c) grain with ‘cube’ orientation, (d) grain with ‘hard’ orientation. Orientation measurements by SEM-FEG EBSD with step size 200 nm. Sample deformed 59%.
ably as a consequence of the low deformation temperature as noted by Paul and Driver [6]. The bands of localized strain, separated by areas of relatively weakly deformed matrix, clearly tend to form groups of compact bundles. For the ‘hard’-oriented crystallite of the same ‘cube/hard’ bicrystals, the crystallite does not show any significant tendency to produce strain heterogeneities (excluding the transition zone at the grain boundary). The orientation spread is nearly the same over a range of global deformation up to 59%. The TEM microstructure from the middle layer of the grain, presented in Fig. 11c shows a nearly uniform checkerboard pattern of roughly equiaxed dislocation cells with a diameter of 0.3–0.5 m. This structure is characterised by a relatively small orientation spread (Fig. 11d) such that the misorientation between particular sub-cells only rarely exceeds 10◦ . Generally, the {1 1 0}0 1 1-oriented grains deformed to the maximum applied strain did not exhibit any form of deformation banding. The Goss {1 1 0}0 0 1-oriented crystallite from the ‘cube/Goss’ bicrystal exhibits a typical, well-defined cellular dislocation structure, generated within this orientation (Fig. 12a). This microstructure consists of two complementary sets of elongated bands, inclined at ±35◦ to ED. Both sets of dislocation walls marking the bands lie very close to the expected
traces of the active {1 1 1} slip planes for this orientation. As a result of their intersection rectangular cells, with dimensions of 0.2–0.8 m, are formed. This microstructure is characterized by a relatively small orientation spread depicted in Fig. 12b with adjacent sub-cell misorientation angles which rarely attain 10◦ . Each set of dislocation bands occupies nearly the same volume of the crystal, but usually one set of the walls is more well defined. The cube {1 0 0}0 0 1-oriented crystallite TEM microstructure (Fig. 12c) from the ‘cube/Goss’ bicrystal obtained after 69% reduction is composed of regular layers of deformation bands separated by less dislocated matrix. It is apparent that the typical deformation bands observed here have a width in the range of 1–5 m, and are composed of slightly elongated and ill-defined ‘diffuse’ sub-cells. The latter are probably due to the low temperature deformation (77 K). These bands have different angles with respect to ED, typically ∼35◦ . They do not run completely through the crystals, but terminate in relatively broad zones (blocks). The {1 1 1} pole figure (Fig. 12d) illustrates the microtexture scattering, mainly as a result of an approximate TD (||1 0 0) rotation. The strongly dislocated bands are associated with rotations of 25–30◦ with respect to the neighbouring, only slightly dislocated matrix. The bands of localized strain, separated by areas of relatively weakly deformed matrix, tend to form
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Fig. 10. Cumulative misorientations (with respect to the first measured point) along ND direction; (a) grain with ‘cube’ and (b) grain with ‘Goss’ orientation from ‘cube/Goss’ bicrystal; (c) grain with ‘shear’ and (d) grain with ‘Goss’ orientation from ‘Goss/shear’ bicrystal; (e) grain with ‘cube’ and (f) grain with ‘hard’ orientation from ‘cube/hard’ bicrystal. Orientation measurements by SEM-FEG EBSD. Samples deformed globally: 69% (a and b), 64% (c and d), 59% (e and f).
groups of compact bundles, as reported by Paul and Driver [6] in cube grains from {1 0 0}0 0 1/{1 1 0}0 1 1-oriented bicrystals. The shear {1 0 0}0 1 1-oriented grain in the ‘shear/Goss’ bicrystal starts to form TBs from the early stages of deformation. They are the boundaries which separate sample volumes in which the crystal lattice rotates in opposite directions as a result of the different slip systems. TEM microstructure observation shows a strongly elongated cell substructure within these areas (Fig. 13a). The process of TB formation is associated with the decomposition of the initial crystal orientation resulting from the TD rotation towards two nearly complementary orientations of {1 1 2}1 1 1 type although with a strong scattering of the crystal lattice (Fig. 13b). The Goss {1 1 0}0 0 1-oriented grain in the ‘shear/Goss’ grain configuration appears a little less stable when compared to the above behaviour in the ‘cube/Goss’ grain configuration. Generally, two sets of dislocation cells dominate the TEM microstructure but their pattern in this case is more, ‘wavy’ (Fig. 13c). This leads to a wider visible scattering of the initial crystal orientation (Fig. 13d). 4. Discussion The major results of this experimental study of the large strain deformation of Al bicrystals is that the behaviour of both grains of the bicrystal is essentially very similar to that of individual crystals deformed under the same conditions (here low tem-
perature channel-die compression). The influence of the grain boundary appears to be limited to local zones along some boundaries although this is strongly amplified in the extreme case of the cube/hard bicrystal combination. The similarity of the grain deformation modes is found here for virtually all the grain orientations in the crystals. Thus: (i) The Goss grains remain very stable up to large strains with a uniform dislocation cell substructure as observed for example by Ferry and Humphreys [11] on similarly oriented Al–0.05 at% Si crystals and by Borbely et al. [12] on Cu crystals. (ii) The shear orientation breaks up rapidly during PSC into thin, alternating, layers of near {1 1 2}1 1 1 bands separated by transition bands as found on Al crystals by Akef and Driver [1] and Becker et al. [3]. The near “Cu” orientations then develop fine shear bands as observed many times for this orientation. (iii) Most of the cube grains break up by deformation banding creating high local misorientations and transition bands aligned along ED as previously documented in some detail by Akef and Driver [1], Liu et al. [13] and Basson and Driver [14]. (iv) The microstructures of the hard {1 1 0}0 1 1 orientation have not, to our knowledge, been previously described in detail but their relative homogeneity at moderate strains is consistent with the usual stability of the {1 1 0}u v w
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Fig. 11. TEM microstructures observed in crystallites forming ‘cube/hard’ bicrystals deformed 59%. (a) Deformation band in the cube grain, and (b) corresponding {1 1 1} pole figure, (c) homogeneous cell substructure in the ‘hard’ grain orientation’ and (d) corresponding {1 1 1} pole figure.
Fig. 12. TEM microstructures observed in crystallites forming ‘cube/Goss’ bicrystals deformed 69%. (a) Nearly perfect homogeneous cell substructure in a ‘Goss’oriented crystallite and (b) corresponding {1 1 1} pole figure. (c) Formation of deformation bands in a ‘cube’ grain and (d) corresponding {1 1 1} pole figure.
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Fig. 13. TEM microstructures observed in crystallites forming ‘Goss/shear’ bicrystals deformed 64%. (a) Transition bands (TBs) in a crystallite with ‘shear’ orientation, and (b) corresponding {1 1 1} pole figure. (c) Slightly disturbed cell substructure in a ‘Goss’ orientation and (d) corresponding {1 1 1} pole figure.
orientations in PSC. In accordance with Stanford et al. [15] at a strain of 0.5 these crystallites developed only a small amount of orientation spread, and at a strain of 1.0 slightly more spread occurred about TD. However, no deformation bands were detected up to deformations of 1.0. These observations on deformed bicrystals of controlled grain orientations are in basic agreement with Risø studies on the influence of grain orientation on substructure development in deformed polycrystals, as reviewed by Hansen [16]. The influence of the grain boundary only appears significant in the case of the transition region of the cube/hard bicrystal. This general behaviour is worth comparing with that of a recent study of the channel-die compression of Al bicrystals of other orientations by Zaefferer et al. [17]. The latter authors compressed three bicrystals with boundary misorientations defined by 1 1 2 tilt axes and misorientation angles of about 9, 15 and 31◦ . In this work ¯ which the grain orientations appear to be close to {1 1¯ 0}1 1 1 are non-symmetrical orientations leading to shape changes with macroscopic shears, of opposite sign, along ED (Skalli et al. [18]). The resulting deformation of the bicrystal is very heterogeneous. Zaefferer et al. [17] also find that in general the crystal orientation has a stronger influence on the grain behaviour than the structure of the grain boundary. Using the same high resolution EBSD technique, some slightly higher misorientations (by
1–2◦ ) were, however, observed near the grain boundaries in the 15 and 31◦ misoriented bicrystals and attributed to dislocation pile-ups at the grain boundaries. This type of macroscopic shear strain incompatibility is not expected in our more symmetrically oriented crystals which should not undergo macroscopic shearing (although microscopic shearing occurs extensively in the initial cube and shear orientations). In general, one therefore expects to find the substructure developments characteristic of similarly oriented single crystals. In all the cases examined here the finite elements simulations predict symmetrical strain profiles with strong maxima at the corners. Also, experimentally, all bicrystals show strong deformation heterogeneities in the form of macroscopic shear localizations, symmetrical with respect to the ND-TD plane. If the ‘hardness’ of the crystallites is nearly the same (as in the case of ‘Goss/shear’ grain configuration) there is a clear tendency to form zig-zag shaped localized strains. This is not so pronounced in the bicrystals with very different hardnesses. In accordance with Raabe et al. [8] and Zaefferer et al. [17] the strong heterogeneity of strain as predicted by modeling can be attributed to the combined influence of macromechanical effects, such as friction, grain hardening, grain interactions, etc. The behaviour of the ‘cube/hard’ bicrystal is surprising in that a transition zone of width about 1.5 mm is developed from the
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Fig. 14. (a) Optical micrograph of the transition zone near the grain boundary in ‘cube/hard’ bicrystal sample deformed 59%. (b) {1 1 1} pole figure showing orientation changes across the grain boundary and transition zone (scan area of 200 m × 5000 m). (c) Cumulative misorientation profile (with respect to the first point) lying within the ‘cube’ grain across the transition zone; SEM-FEG EBSD measurements in the ND-ED plane with step size 200 nm.
initial hard orientation. This is shown in more detail in Fig. 14a and is probably due to the very high-localized shear of the ‘hard’ {1 1 0}0 1 1 grain as it deforms near the ‘cube’ {1 0 0}0 0 1. The volumes of the hard crystallite lying near the grain boundary must accommodate the incompatibilities between the neighbouring grains, leading to a strongly deformed (and elongated) ‘cube’ grain and the slightly elongated ‘hard’ crystallite. As shown in Fig. 14b this induces a broad orientation spread across the grain boundary and the transition zone (near the free surfaces of the sample). The figure shows a strong, continuous dispersion of the 1 1 1 poles characterized by strong scattering by (±)TD rotation with some additional rotations around the ED and ND axes. The misorientations, calculated with respect to the first measured point lying within the ‘cube’ grain across the grain boundary, reflect the difference in the structure that is formed (Fig. 14c). There is only about 12–15◦ between the ‘hard’ grain and the first measured point in the ‘cube’ grain as a result of the rapid rotation of the ‘cube’ grain near the transition zone towards the {1 1 0}0 1 1 orientation. This very particular behaviour needs to be verified, for example by FE simulations incorporating crystal plasticity relations as opposed to the standard plasticity simulations used here. The areas lying near the middle layers are very homogeneous as observed at both the TEM and SEM scales. This is in accordance with earlier work by Stanford et al. [15] showing that this orientation is generally stable and did not exhibit significant deformation banding up to strains of 1.0. However, after higher strains (>1.4), Ferry and Humphreys [11] observed a well-developed band structure in aluminium. This implies a critical strain for bands to form in Al of this orientation between 1.0 and 1.4.
5. Conclusions Both macro- and micro-mechanical effects in three types of plane strain compressed aluminium bicrystals have been investigated by FE simulation and high-resolution electron metallography. The bicrystals were composed of grain pairs which are hard/soft and stable/unstable combinations. The main conclusions are: (i) The overall strain profiles and macroscopically observed grain shapes of the deformed bicrystals are quite well predicted by FE simulations (taking the appropriate single crystal hardening curves and assuming reasonable boundary conditions). In particular the extreme case of the bicrystal composed of the hard {1 1 0}0 1 1 and soft cube grains where the latter flows over the hard grain is well represented. (ii) At the microscopic level, the grains of the other two bicrystal combinations behave essentially as single crystals under the same deformation conditions with respect to the deformation substructure and microtexture distributions. The unstable cube and shear orientations break up into deformation and transition bands as previously reported for single crystals. The stable Goss orientation develops a typical uniform deformation substructure albeit with somewhat higher misorientations in the Goss/shear combination. It is concluded that for most grain pair combinations of the bicrystals the individual grain orientations play a dominant role in deformation substructure development as a consequence of their dependence on slip system interactions.
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Acknowledgements The authors wish to thank Prof. M. Pietrzyk from AGH University of Science and Technology, Krak´ow for providing access to Finite Element software and Dr. A. Pi˛atkowski (IMIM PAS, Krak´ow) who have helped in the deformation and optical metallography of the bicrystals. References [1] [2] [3] [4] [5]
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