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Vol. 8. Issue 6.
Pages 5057-5065 (November - December 2019)
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Vol. 8. Issue 6.
Pages 5057-5065 (November - December 2019)
Original Article
DOI: 10.1016/j.jmrt.2019.05.019
Open Access
A crystal plasticity FEM investigation of a Cu single crystal processed by accumulative roll-bonding
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Hui Wang
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hw737@uowmail.edu.au

Corresponding author.
, Cheng Lu, Kiet Tieu
School of Mechanical, Materials, Mechatronics and Biomedical Engineering, University of Wollongong, New South Wales 2522, Australia
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Tables (3)
Table 1. Notation of 12 slip systems.
Table 2. Material parameters used in the Bassani–Wu hardening model.
Table 3. Shear strain on slip systems, shear strain γ12, and crystal rotation at four points, where the values are those evolved in a cycle, not the cumulative ones.
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Abstract

In this study, texture modelling of a copper single crystal processed by accumulative roll-bonding (ARB) was conducted for the first time. A crystal plasticity finite element method (CPFEM) model was developed, and the predicated textures were validated by the experimental observations conducted by another research group. A heterogeneous through-thickness deformation was revealed. Two types of matrix bands formed through the thickness after the first ARB cycle, and different crystal rotation, slip system activation, and shear strain evolved in matrix bands. Due to the cutting, stacking and roll-bonding involved in ARB, transition of deformation behaviours has been found after a position being moved to a new position in the next cycle, and this transition was investigated based on the CPFEM model.

Keywords:
Accumulative roll bonding
Crystal plasticity FEM
Texture
Single crystal
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1Introduction

Severe plastic deformation (SPD) has been widely used to fabricate ultra-fine grained (UFG) materials by introducing a large strain [1,2]. Among various SPD techniques, accumulative roll bonding (ARB) attracts extensive research interests due to its effectiveness in manufacturing UFG materials [1–4] and composites [5,6] by repeating roll-bonding, cutting, and stacking. Copper polycrystals have been extensively used to study the deformation in ARB, and a heterogeneous through-thickness distribution of texture and microstructure in ARB-processed copper has been revealed [1,3,4]. Unlike polycrystals, single crystals have no pre-existed grain boundaries and thus no grain interaction, and possess an identical initial orientation. Single crystals are thus the ideal material for tracing texture evolution, microstructure refinement, and deformation of ARB process. Only one experimental study on an ARB-processed copper single crystal (15 12 5)[9 10¯3¯] has been reported [7]. Crystal orientations split during ARB, which is noticeably different from that in copper polycrystals, since single crystals are strongly anisotropic in texture. The texture evolution in this copper single crystal is also different from ARB-processed aluminium single crystals [8,9] due to the difference in initial orientations. How the material deformed and crystal orientation rotated during ARB in this copper single crystal need further study, especially the effect of cutting-stacking to transition of deformation.

Deformation history is critically important to study the transition of deformation. Texture modelling, unlike experimental techniques, can access the whole deformation history and provide more details on the deformation. Texture modelling has become a powerful method to study texture evolution, and various crystal plasticity models have been proposed. The Taylor model [10], ALAMEL model [2,11], and the viscoplastic self-consistent (VPSC) model [12] have been applied to investigate the texture evolution in ARB-processed aluminium. An iso-strain is assumed in the Taylor model and the stress equilibrium at grain boundaries is disregarded, while in the ALAMEL model strain compatibility and stress equilibrium are only met for paired grains. In the VPSC model, a grain is regarded as an inclusion in the homogenized matrix and thus grain interaction cannot be fairly accessed. Different from the above crystal plasticity models, no homogenization is assumed in the so-called ‘full-field’ crystal plasticity finite element method (CPFEM) model, and the stress equilibrium and strain compatibility are reached in each time increment [13]. The texture evolution is coupled with deformation in the CPFEM model. As suggested by Knezevic et al. [14], the CPFEM model is applicable for ARB because of its advantage on solving mechanical problems under complicated internal and external boundary conditions [15], where the roll-bonding, cutting and stacking in ARB result in complex deformation. The CPFEM model has been applied to ARB-processed aluminium single crystals [16]. However, texture modelling of ARB-processed copper has never been attempted.

In the current report, a CPFEM model was developed to investigate the deformation of an ARB-processed copper single crystal. The predicted textures was validated by comparison with experimental observations performed by Yoshida et al. [7]. The deformation in this single crystal was investigated, in terms of texture, slip activities, and shear strain.

2Texture modelling

The ARB experiment in Ref. [7] was followed, in which a copper single crystal with an initial orientation of (15 12 5)[9 10¯3¯] was deformed up to 4 cycles. The finite element method (FEM) model in Fig. 1 is two-dimensional under the plane strain conditions. The simulation was performed by commercial FEM code ABAQUS/Standard ver.6.9. After comparison of different methods, mapping solution, a built-in remeshing technique of ABAQUS [16], was used between cycles. After applying mapping solution, the deformed mesh was completely replaced by a new mesh, while the deformation solution was automatically transferred from the deformed mesh into the new mesh by interpolation. The deformation solution at nodes of the deformed mesh was firstly obtained by extrapolating the values of integration points and then the solution at nodes was averaged over all elements around the nodes in the new mesh. When performing mapping solution, it is necessary to ensure the equivalence of the deformation solution, i.e., solution continuity. The continuity of the deformation solution in this study was improved to an acceptable level, as examined later.

Fig. 1.

A schematic of ARB FEM model.

(0.09MB).

In the corresponding experiment, the initial sheet thickness was 2mm and was rolled to 1mm by 50% reduction in the first ARB cycle, i.e., using rolling to replace roll-bonding in this cycle. ARB (including cutting, stacking, and roll-bonding) was used from the second cycle onwards. In the simulation, the deformation solution of this deformed sheet after the first cycle was mapped into two separate undeformed ones (‘Cutting’ in Fig. 1), and then rigid translation along the rolling direction (RD) and normal direction (ND) was performed to realize the stacking of the two sheets (‘Stacking’ in Fig. 1). Mapping solution was performed again in the preparation of the next cycle. The two stacked sheets were treated as a single-layered one by omitting the interface. This treatment is reasonable based on the following considerations. In the experiment, the two stacked sheets were joined together at the four corners, and the friction between the two sheets is expected to be very high because the bonded surfaces were wire-brushed beforehand, which resulted in a negligible relative displacement between the two sheets [16]. Conventional rolling at 18% reduction was also conducted in the experiment, and this was also simulated. Plane strain four-node elements with reduced integration (element ID: CPE4R) were used, and after mesh calibration the number of elements was kept at 80 through the thickness. To match the dry rolling conditions [7], a coefficient of friction of 0.25 was chosen after testing various coefficients, since it provided the best match of textures. The sheet was deformed by the rotating rolls, which were considered as rigid bodies with a diameter of 310mm.

A well-recognized crystal plasticity model was adopted, which was developed by Peirce [17,18], and Asaro and Needleman [19]. The crystal plasticity model was implemented into ABAQUS/Standard via User-defined MATerial subroutine (UMAT). The slip plane and direction in FCC structured copper are {1 1 1} and <1 1 0>, respectively, and their combinations generate 12 potentially activated slip systems, as listed in Table 1.

Table 1.

Notation of 12 slip systems.

Slip plane  (1 1 1)(1¯ 1 1)(1 1¯1)(1 1 1¯)
Slip direction  [0 1¯1]  [1 0 1¯]  [1¯ 1 0]  [101]  [110]  [0 1¯ 1]  [0 1 1]  [110]  [1 0 1¯]  [0 1 1]  [101]  [1¯ 1 0] 
Slip system  a1  a2  a3  b1  b2  b3  c1  c2  c3  d1  d2  d3 

The used Bassani-Wu hardening model [20,21] is regarded as the best texture predictor after comparison of five different hardening models [22]. In this hardening model, the shear strain rate (γ˙(α)) of slip system α is determined by resolved shear stress (τ(α)) relative to its strength (τc(α)), as expressed by Eq.1, where γ˙0(α) is the reference value of the shear strain rate and n is the rate-sensitive exponent.

The τ˙c(α) is calculated by:

where hαβ is the hardening modulus. It is self-hardening when α=β(hαα), while it is latent hardening hαβ when α≠β. The hαα and hαβ are expressed by:
where h0 is the hardening modulus after initial yield, hs is the hardening modulus of easy slip, τ1 is the critical stress when plastic flow begins, τ0 is the initial critical resolved shear stress, q is the ratio between latent hardening modulus and self-hardening modulus, and fαβ represents the interaction between two slip systems α and β. The material parameters in Eqs. (1) and (3) are listed in Table 2, which were obtained by fitting the simulated stress-strain curve with the experimental results [23]. Three elastic moduli are C11=112,000 MPa, C12=66,000 MPaand C44=28,000 MPa, as also listed in Table 2.

Table 2.

Material parameters used in the Bassani–Wu hardening model.

n  γ˙0(s−1)  h0 (MPa)  hs (MPa)  τ1 (MPa)  τ0 (MPa)  C11 (MPa)  C12 (MPa)  C44 (MPa) 
20  0.0001  90  1.5  1.3  168,400  21,400  75,400 
3Results and discussion

The thickness location X is defined in terms of the total sheet thickness t, where X=0t and X=1t correspond to the upper and lower surfaces, respectively.

3.1Prediction validation

The predicted textures after conventional rolling (18% reduction) and 2-ARB are shown by {1 1 1} pole figures in Fig. 2, and the textures obtained by electron backscattered diffraction (EBSD) in the corresponding experiment [7] are also shown for comparison. The simulated pole figures were constructed from all elements in the same areas (central region) as those in the experiment. The predictions in Fig. 2 agree well with the experimental results at different strains. It should be noted that the element size in the simulation is much larger than the step size of EBSD maps, and thus compared to the experimental observation, the scatter of the predicted textures was obviously low. It can be seen that crystal orientations slightly rotated away from the initial orientation after 18% reduction (Fig. 2a), while they rotated evidently after 2-ARB (Fig. 2b).

Fig. 2.

{1 1 1} pole figures in the simulation and experiment [7] after (a) 18% rolling reduction, and (b) 2-ARB (upper panel: simulation, lower panel: experiment), where the initial and final orientations in the simulation are shown in red and black, respectively.

(0.12MB).
3.2Texture evolution

The crystal rotation angle, i.e., misorientation between the initial and final orientations, in each element was calculated according to the Angle/Axis method. The distribution of crystal rotation angles is shown in Fig. 3 and the through-thickness crystal rotation is also plotted in Fig. 4. Fig. 5 shows the {1 1 1} pole figures after all four cycles, which were constructed from all elements in steady state deformation region.

Fig. 3.

Distribution of crystal rotation angles and deformed FEM meshes (a) after 1-ARB, (b) before 2-ARB, (c) after 2-ARB, and (d) after 3-ARB. Red solid and dashed lines indicate the bonded interfaces formed in the current cycle and previous cycles, respectively.

(0.71MB).
Fig. 4.

Distribution of through-thickness crystal rotation (a) after 1-ARB and before 2-ARB, (b) before and after 2-ARB, (c) before and after 3-ARB. The angles in (a) are those along the line marked in Fig. 3a and b, while they are the average values (of all elements along the RD) in (b and c).

(0.19MB).
Fig. 5.

{1 1 1} pole figures after (a) 1-, (b) 2-, (c) 3-, (d) 4-ARB, where the red and black poles show the initial and final orientations, respectively.

(0.16MB).

After 1-ARB, the variation of crystal rotation along the RD is very low (Fig. 3a). In contrast, the through-thickness crystal rotation is non-uniform (Figs. 3a and 4a), and macro-subdivision occurred by forming four matrix bands, marked as M1a-M1b-M1c-M1d (‘M’ for matrix band, ‘1’ for 1-ARB), where the thickness centre is located in M1a. The four matrix bands can be categorized into two groups, where M1a and M1c are in the first group, and M1b and M1d in the second group. The crystal rotation angles are very high in M1b and M1d, even up to 33°, while they are relatively low in M1a and M1c, at about 20°. The misorientation angles change sharply between adjacent matrix bands (Figs. 3a and 4a). The splitting of the single crystal was also observed in the corresponding experimental research [7]. No shear band was observed in the simulation. The pole figure after 1-ARB (Fig. 5a) shows that the crystal orientations rotated away from the initial orientation mainly in two paths and the final textures are highly concentrated at two positions. One position corresponds to the crystal orientations in M1a and M1c, while the other position corresponds to those in M1b and M1d.

Mapping solution was applied at the beginning of 2-ARB. Two 1-ARB processed sheets (two layers) were stacked (Fig. 3b), and the number of matrix bands was doubled (M2a1-M2b1-M2c1-M2d1-M2a2-M2b2-M2c2-M2d2), where the subscripts indicate the layer of sheets. The stacking pattern is clearly reflected by the distribution of matrix bands (Fig. 3b) and crystal rotation (Fig. 4a) before 2-ARB. The accuracy of mapping solution is very high (Fig. 3a) after comparing the crystal rotation angles along the line (marked in Fig. 2a and b) after 1-ARB and before 2-ARB. The stacking at the beginning of 2-ARB resulted in a bonded interface, as marked by the red solid line in Fig. 3b and c. The change of crystal rotation is also high at the interface that connects M2a2 and M2d1, like those at the boundaries of matrix bands. The maximum increase of crystal rotation in 2-ARB is only about 10°, not as large as that in 1-ARB (Fig. 4b), since the crystal orientations rotated slowly as approached relatively stable positions. The crystal rotation increased slightly in M2a1 near the upper surface (X=0t to 0.3t), but it increased obviously in M2a2 (X=0.5t to 0.78t), and the increase is even higher in M2c2 (X=0.9t to 0.96t). In contrast, the increase of crystal rotation in M2b (M2b1 and M2b2) and M2d (M2d1 and M2d2) is slightly higher than that in M2a and M2c. Textures became more scattered after 2-ARB (Fig. 5b), and one texture component rotated toward (2 1 1)[1 1¯1¯], similar to that observed in the experiment [7]. The number of matrix bands was doubled again in 3-ARB due to the cutting–stacking (Fig. 3d), while the thickness of matrix bands was halved. The increase of crystal rotation in 3-ARB is very low by comparing with that before this cycle (Fig. 4c), which means the crystal orientations have almost reached stable positions. Texture scattered more after 3- and 4-ARB, as shown in Fig. 5c and d, and the definition of the two texture components became unclear with increasing cycles.

3.3Shear strain

The distorted FEM meshes in Fig. 3 also indicate the strain, and the through-thickness shear strain γ12 is plotted in Fig. 6. The distortion of FEM meshes in Fig. 3 and shear strain in Fig. 6 are those developed in a cycle, not cumulated with previous cycles, since the distorted FEM mesh in a cycle was replaced by an undeformed one in the next cycle during mapping solution. After 1-ARB, the direction of shear deformation in the first group of matrix bands (M1a and M1c) is opposite to that in the second group (M1b and M1d), where the shear strain is positive in the former and negative in the latter (Fig. 6). The through-thickness shear strain in Fig. 6 is quite different from that observed in ARB deformed polycrystals, where the shear strain in polycrystals increases from the centre to surface [11,24], as measured by the embedded-pin method. However, this kind of through-thickness shear strain (Fig.6) is characteristic in rolled single crystals due to their strong anisotropy [16,25]. Furthermore, the shear strain in single crystals is different from orientation to orientation [16]. Similar to the crystal rotation in Fig. 3a, the shear strain is almost uniform along the RD. The alternation of shear deformation between matrix bands was also observed after 2- and 3-ARB (Fig. 6). It is interesting that the net shear strain after 3-ARB is extremely low in the region from 0.76t to 0.9t, where it is close to the lower surface that is supposed to experience high surface friction.

Fig. 6.

Distribution of through-thickness shear strain γ12 after 1-, 2-, and 3-ARB, where the shear strain is that evolved in a cycle, but not cumulated with previous cycles.

(0.13MB).

The crystal rotation and shear strain are quite different from matrix band to matrix band, while they are almost uniform within a matrix band. In this study, slip activity was used to represent microstructural evolution, since slip system activation plays a key role in microstructure [26]. Fig. 7 shows the shear strain on slip systems within representatively selected matrix bands. Slip systems, a2-d1-a3, were mainly activated in M1a after 1-ARB (Fig.7a), while they are c2-b1-a1 in M1b (Fig.7b). The M1a was moved to M2a1 and M2a2 in 2-ARB. The activated slip systems in M2a1 (Fig. 7c) are a3-a2-d1, similar to those in M1a, since they both are near the upper surface. However, the activated slip systems in M2a2 are d1-a3-b1-a2 (Fig. 7d), different from those in M1a, due to the difference in thickness position. In 3-ARB, the M2a2 was moved to M3a2 and M3a4. The slip system activation in M3a2 (a3-a2-d1-b1 in Fig. 7e) is different from that in M2a2, but similar to that in M2a1, since both M2a1 and M3a2 are near the upper surface. The slip system activation in M3a4 (Fig. 7f) is similar to that in M2a2. It is obvious that multi-slip is the feature of 1-ARB (Fig. 7a and b), while only four slip systems are mainly activated in the following cycles. It has been observed that only four slip systems were generally activated in single crystals that have slip systems being symmetrical about the RD-ND plane, such as Cube {0 0 1} <1 0 0> [25,27], rotated-Cube {0 0 1} <1 1 0> [28], while more slip systems were activated in single crystals that have an asymmetrical distribution of slip systems, such as S {1 2 3} <6 3 4> [8,16]. During deformation, the crystal orientation rotated into a relatively stable position and it is also the direction lowering the resolved shear stress on slip systems, and thus the number of activated slip systems reduced. The transition of slip system activation is important to the textural and microstructural evolution [16,26,29].

Fig. 7.

Shear strain on slip systems in the matrix band of (a) M1a, (b) M1b, (c) M2a1, (d) M2a2, (e) M3a2, (f) M3a4.

(0.28MB).
3.4Deformation mechanism in CPFEM

In the adopted CPFEM model, the material spin Ω consists of lattice rotation Ω* and plastic spin ΩP, namely

The material spin can be represented by shear strain [11]. The crystal orientation changes as lattice rotates, and this is the reason for texture evolution. Crystal slip causes plastic spin, during which crystal orientation keeps unchanged. The plastic spin ΩP can be calculated according to

are the slip direction and slip plane normal, respectively. The simulation model is two-dimensional, and thus material spin about the RD (ΩRD) and ND (ΩND) is zero. However, the slip systems are oriented in all three dimensions, and their activation would result in ΩRDP and ΩNDP. To retain zero material spin about the RD and ND, the lattice spin has to develop in the opposite direction to compensate the plastic spin. This is why crystal rotation about all three directions can be predicted by the two-dimensional model, as shown in Fig. 5.

The deformation history (shear strain γ12, shear strain rates on slip systems, and crystal rotation) at two positions was traced and is shown in Fig. 8. The first point A in M1a of 1-ARB was moved to C in 2-ARB (Fig. 3), while the second point is B in 1-ARB and was moved to D in 2-ARB. The deformation history of point A can be divided into three periods (Fig. 8a). It is multi-slip from 0.0s to 0.4s (II), two slip systems (a2-c1) with relatively high shear strain rates were mainly activated from 0.4s to 0.8s, and three slip systems (a2-a3-c1) were still highly activated in the last period (0.8–1.0s). Multi-slip in the first period produced low plastic spin (ΩTDP) according to Eq. (5), since the activated slip systems are not co-directional and would counteract the plastic spin caused by each other, where the distribution of activated slip systems on the RD-ND plane is shown in (II) of Fig. 8. The material spin (ΩTD), represented by the shear strain in (I) Fig. 8, is also very low in this period (0.0–0.4s), so low lattice spin (Ω*) is required according to Eq. (4). In the second period (0.4–0.8s), the shear strain rates on a2 and c1 are obviously higher than the other slip systems, and low plastic spin evolved, since a2 and c1 are in different directions. The low plastic spin is not enough to meet the greatly increased material spin, so large lattice spin is required. In the last period, i.e., from 0.8s to 1.0s, the shear strain rates are not very high and produced low plastic spin. Lattice spin remained almost unchanged due to the low material spin. The deformation at A, B, C, and D is also summarised in Table 3. The shear strain and slip system activation at A and C are different, and a small increase of crystal rotation developed at point C. Similarly, the deformation behaviour can also be deduced at point B and D, though the shear strain, slip system activation and lattice spin are different from those at point A and C.

Fig. 8.

(I) Evolution of shear strain γ12 (or material spin ΩTD), (II) shear strain rates on four highly activated slip systems (or plastic spin ΩP), and (III) crystal rotation (or lattice spin Ω*) at points (a) A–C, and (b) B–D marked in Fig. 3. The inserted figures in (II) show the distribution of slip systems on the RD-ND plane.

(0.35MB).
Table 3.

Shear strain on slip systems, shear strain γ12, and crystal rotation at four points, where the values are those evolved in a cycle, not the cumulative ones.

  γa1  γa2  γa3  γb1  γb2  γb3  γc1  γc2  γc3  γd1  γd2  γd3  γ12  Crystal rotation 
A,1st  0.022  0.81  0.31  0.092  0.004  0.029  0.82  0.028  0.041  0.24  0.16  0.0002  0.19  19.1° 
C,2nd  0.47  0.46  0.82  0.003  0.0003  0.58  0.0005  0.001  0.095  6.8° 
B,1st  0.24  0.045  0.59  0.65  0.31  0.0045  0.073  0.14  0.13  0.43  0.022  0.054  −0.4  30.0° 
D,2nd  0.021  0.91  0.19  0.46  0.0006  0.077  0.03  0.0006  0.025  −0.23  10.9° 

The crystal rotation paths of A-C and B-D are shown by a pole figure in Fig. 9. Point A and B rotated away from the initial orientation in different directions, and point A rotated toward (2 1 1)[1 1¯1¯]. The rotation paths are not straight and crystal rotation about all three directions (RD, ND, and TD) developed. Basically speaking, the crystal orientation of point C continued to rotate following the rotation path of A, and it is the same for point B-D.

Fig. 9.

Crystal rotation paths of point A, B, C and D from the initial orientation.

(0.06MB).
3.5Texture evolution and slip activities in ARB

This transition of crystal rotation and slip activities in ARB is probably due to two reasons. The first reason is that the through-thickens deformation is non-uniform. According to the hardening model (Eq. (1)), the slip system activation is only determined by the resolved shear stress on slip systems relative to their strengths. The slip system activation at A and B is different though they have the same initial orientation, which means the imposed through-thickness stress is inhomogeneous. The through-thickness shear strain γ12 at A and B is different too, which suggests that the introduced through-thickness strain is non-uniform. The variation of stress and strain through the thickness is due to the surface friction and rolling bite geometry [30]. The imposed loading determines the slip system activation and shear strain, and also crystal rotation according to Eq. (4). The second reason is that the previously introduced deformation definitely influences the following deformation. For instance, the crystal rotation and slip system activation at A are different from those at D (Table 3) though they both are located at the same thickness position (centre), which indicates that the previously evolved crystal rotation at B influences the deformation at D. According to the Schmid's law, the previously developed crystal rotation would influence the resolved shear stress on slip systems and thus affects slip system activation. The deformation history that the material had previously experienced was brought into the new position by the FEM model (Fig. 1), and accordingly the deformation continuity was preserved. The coupled effect of these two reasons is responsible for the change of deformation in ARB.

4Conclusions

  • 1.

    For the first time, the CPFEM was used to study the deformation behaviours of an ARB-processed copper single crystal. The simulation successfully captured the texture experimentally observed by another research group.

  • 2.

    The through-thickness deformation is inhomogeneous, in terms of crystal rotation, shear strain and slip activities. Matrix bands formed after 1-ARB, and the number of matrix bands was doubled after each cycle.

  • 3.

    By tracing the deformation history at two points, the transition of crystal rotation and slip system activation was successfully revealed. It was found that the coupled effect of previously experienced deformation and thickness position change is the reason for this transition.

Conflicts of interest

The authors declare no conflicts of interest.

Acknowledgements

This work was supported by Australian Research Council Discovery Project (DP170103092).

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