Effect of External Constraints on Deformation Behavior of Aluminum Single Crystals Cold-Rolled to High Reduction: Crystal Plasticity FEM Study and Experimental Verification

Abstract

In this study, aluminum single crystals with a {1 0 0} <0 0 1> (Cube) orientation were rolled under two conditions: with external constraints imposed by an external aluminum frame (3DRC) and without external constraints (3DR). The crystal plasticity finite element method (CPFEM) was used to simulate texture evolution, and the results corresponded well with experimental observations. The minor discrepancies observed were primarily attributed to the idealized conditions in the simulation. The results demonstrate that in the 3DR model, crystal orientations predominantly rotate around the transverse direction (TD), with non-TD rotations playing a secondary role. In contrast, the 3DRC model exhibits similar rotation patterns to 3DR at lower reductions, but at higher reductions, non-TD rotations become comparable to TD rotations. This difference results in more concentrated orientations in 3DR and more dispersed orientations in 3DRC. Additionally, analysis reveals that external constraints cause deformation behavior to deviate from the plane strain condition rather than move closer to it. The presence of external constraints alters stress and strain states, modifying the activation of slip systems and crystal rotations, leading to significant variations in slip activity, shear strain, and crystal rotation along TD.

1. Introduction

Plastic-deformation-induced crystallographic texture has attracted extensive research attention due to its significant impact on material properties and deformation behavior, leading to the widespread study of techniques such as Equal Channel Angular Pressing [1], High-Pressure Torsion [2,3], and Accumulative Roll Bonding [4,5]. Among the various severe plastic deformation (SPD) techniques, rolling has been widely used for its simplicity and effectiveness [6,7]. In rolling, the lateral flow of a material leads to elongation along the transverse direction (TD), i.e., width spread [8]. This phenomenon is not obvious when the ratio of w/t is high, where w is the width and t is the thickness. Thus, the rolling process is usually assumed to be a plane strain (PS) deformation when w/t is high [9]. Among numerous materials, single crystals, characterized by the absence of pre-existing grain boundaries and grain interactions, as well as a uniform initial orientation, serve as an ideal material for tracing the texture evolution and deformation of the rolling process. However, the ratio of w/t is usually very low for a single crystal, resulting in a significant width spread, which causes the rolling process to deviate from PS deformation.

To reduce the lateral flow of the sample during rolling and to make the deformation process approximate the PS deformation as closely as possible, Kashihara et al. [10,11] have conducted experiments on aluminum single crystals {1 0 0} <0 0 1> (Cube), in which the single crystals were embedded into an external frame made of the same material. The observations indicate that the external constraints imposed by the applied frame significantly influenced texture evolution and plastic deformation behaviors, particularly in terms of shear strain and crystal rotation. In addition, Kashihara and Inagaki et al. [12], using a Cube single crystal deviated by 5° about the rolling direction (RD), found that under unconstrained conditions, the orientation mainly consists of two components. In contrast, when external constraints were applied, a single orientation emerged. Kamjo et al. [13] also conducted the same experimental process and found that texture evolution is predominantly influenced by inhomogeneous lateral material flow, which arises from the reaction of external constraints and leads to the inclination of the stress system. Furthermore, although previous studies have explored the influence of external constraints on the plastic deformation process and demonstrated that this approach offers valuable insights into the deformation behavior of polycrystalline materials, the texture evolution and slip system activation in different regions during the constrained rolling process have not yet been thoroughly investigated.

With the development of computer technology and crystal plasticity theory, texture modeling has emerged as a powerful method for investigating plastic deformation and texture evolution [14,15]. Numerous crystal plasticity models have been developed, including the Taylor model, the Alamel model, and the Visco-plasitic Self Consistant (VPSC) model. Among these models, the crystal plasticity finite element method (CPFEM) stands out due to integrating crystal plasticity (CP) into the finite element framework. By embedding CP theory within each element and treating the elements as local boundaries, CPFEM enables fully coupled simulations of texture evolution and plastic deformation [16]. Hu et al. [17] used CPFEM to simulate the deformation behavior of grains during rolling, and the simulation predictions corresponded well with the experimental observations from electron backscatter diffraction (EBSD). Wang et al. [18] successfully applied CPFEM to simulate the microstructural and crystallographic orientation evolution of {111} columnar grains under plane strain compression. Sahu et al. [4] employed CPFEM to investigate the effects of notch severity and initial texture on the micro-mechanisms governing low-strain deformation in commercially pure titanium. The simulations predicted pronounced prismatic slip traces and the early onset of prismatic slip activity in notched specimens, corresponding well with EBSD results.

In this paper, a Cube aluminum single crystal simulation with external constraints was conducted with up to 95% reduction by multi-pass rolling. In addition, another simulation was also conducted without external constraints for comparison. The CPFEM was used to predict the deformation in terms of shear strain, slip activity, and texture. The inhomogeneous deformation was successfully predicted, and experimental observations validated the predictions. The constrained condition exhibited a distinct texture compared to the unconstrained condition due to the external constraints altering the stress and shear conditions. The slip activity and texture evolution along the TD were analyzed. Furthermore, the effects of external constraints on texture, shear strains, and deformation bands were discussed.

2. CPFEM Modeling

2.1. CPFEM Model and Implementation

The crystal plasticity (CP) model follows the well-recognized kinematical scheme developed by Asaro [19] and Peirce [20,21]. In this model, the deformation gradient ($F$) is decomposed into two components, as shown in Equation (1).

$$F=F^{*}·F^P$$

(1)

where $F^{*}$ and $F^P$ are the deformation gradient caused by the rotation of the crystal lattices and plastic shearing, respectively.

The velocity gradient $L$ is derived from the deformation gradient by Equation (2)

$$L=\dot{F}{F}^{-1}=L^{*}+L^P$$

(2)

where $L^{*}$ and $L^P$ are due to lattice rotation and plastic shearing, respectively.

The velocity gradient $L$ is uniquely divided into a symmetrical component and a skewed-symmetrical component, as shown in Equation (3a–c).

$$L=D+\Omega$$

(3a)

$$D={\displaystyle \frac{1}{2}}\left(L+L^T\right)$$

(3b)

$$\Omega ={\displaystyle \frac{1}{2}}\left(L-L^T\right)$$

(3c)

where $D$ and $\Omega$ are the strain rate tensor and spin tensor, respectively. The spin tensor $\Omega$ can be represented by the rigid rotation of a finite region or redundant shear strain [22], and it can also be decomposed into plastic spin ${\Omega }^P$ and lattice spin ${\Omega }^{*}$ according to Equation (4)

$$\Omega ={\Omega }^P+{\Omega }^{*}$$

(4)

The plastic spin ${\Omega }^P$ arises from the motion of dislocation on slip planes and along slip directions, and is calculated based on Equation (5)

$${\Omega }^P={\sum }_{\alpha =1}^{12}{\displaystyle \frac{1}{2}}\left(s^{\left(\alpha \right)}{·m}^{\left(\alpha \right)}-m^{\left(\alpha \right)}{·s}^{\left(\alpha \right)}\right){\dot{\gamma }}^{\left(\alpha \right)}$$

(5)

where $s^{\left(\alpha \right)}$ and $m^{\left(\alpha \right)}$ are the slip direction and slip plane normal, respectively. ${\dot{\gamma }}^{\left(\alpha \right)}$ is the shear strain rate on the slip system $\alpha$. The plastic spin ${\Omega }^P$ does not lead to a change in crystal orientation. In contrast, the lattice spin ${\Omega }^{*}$ results from the distortion and rotation of the crystal lattice, driving texture evolution.

During the rolling process, the material undergoes work hardening, leading to nonlinear changes in the stress–strain curve. To accurately describe the stress–strain behavior in CPFEM, a hardening model is introduced to simulate the evolution of crystal slip systems and their impact on macroscopic mechanical properties. To more precisely capture the work-hardening phenomenon of Face-Centered Cubic (FCC) crystals in simulation, the rate-dependent Bassani–Wu hardening model [23,24] was adopted in the simulation, which most accurately predicted the texture evolution during uniaxial tension after comparing five different hardening models [25]. In this hardening model, the shear strain rate ${\dot{\gamma }}^{\left(\alpha \right)}$ is related to the resolved shear stress ${\tau }^{\left(\alpha \right)}$ on slip system $\alpha$, as expressed by Equation (6a–c)

$${\dot{\gamma }}^{\left(\alpha \right)}={\dot{\gamma }}_0^{\left(\alpha \right)}sgn\left({\tau }^{\left(\alpha \right)}\right){\left|{\displaystyle \frac{{\tau }^{\left(\alpha \right)}}{{{\tau }_c}^{\left(\alpha \right)}}}\right|}^n\mathrm{for} {\tau }^{\left(\alpha \right)}\ge {{\tau }_c}^{\left(\alpha \right)}$$

(6a)

$${\dot{\gamma }}^{\left(\alpha \right)}= 0 \mathrm{for} {\tau }^{\left(\alpha \right)}<{{\tau }_c}^{\left(\alpha \right)}$$

(6b)

$$\mathrm{The} sgn\left(x\right)=\left\{\begin{array}{c}1 \mathrm{for} x\ge 0\\ -1 \mathrm{for} x<0\end{array}\right.$$

(6c)

where ${\dot{\gamma }}_0^{\left(\alpha \right)}$ is the reference value of the shear strain rate, $n$ is the rate-sensitive exponent, and ${{\tau }_c}^{\left(\alpha \right)}$ is the critical resolved shear stress of the slip system $\alpha$.

The ${{\tau }_c}^{\left(\alpha \right)}$ represents the strength of activating the slip system $\alpha$, and its change rate ${{\dot{\tau }}_c}^{\left(\alpha \right)}$ is calculated by Equation (7)

$${{\dot{\tau }}_c}^{\left(\alpha \right)}= {\sum }_{\beta =1}^N{h}_{\alpha \beta }\left|{\dot{\gamma }}^{\left(\beta \right)}\right|$$

(7)

where $h_{\alpha \beta }$ is the hardening modulus. It is self-hardening, i.e., $h_{\alpha \alpha }$, when $\alpha$ is equal to $\beta$, while it is latent hardening, $h_{\alpha \beta }$, when $\alpha$ is not equal to $\beta$. The $h_{\alpha \alpha }$ and $h_{\alpha \beta }$ are calculated by Equation (8a,b)

$$h_{\alpha \alpha }=\left[\left(h_0-h_s\right){sech}^2\left({\displaystyle \frac{\left(h_0-h_s\right){\gamma }^{\left(\alpha \right)}}{{\tau }_1-{\tau }_0}}\right)+h_s\right]\left[1+{\sum }_{\begin{array}{c}\beta =1\\ \beta \ne \alpha \end{array}}^N{f}_{\alpha \beta }tanh\left({\displaystyle \frac{{\gamma }^{\left(\beta \right)}}{{\gamma }_0}}\right)\right], \alpha =\beta$$

(8a)

$${h_{\alpha \beta }=qh}_{\alpha \alpha }, \alpha \ne \beta$$

(8b)

where $h_0$ is the hardening modulus after initial yield, $h_s$ is the hardening modulus of easy slip, ${\tau }_1$ is the critical stress when plastic flow begins, ${\tau }_0$ is the initial critical resolved shear stress, $q$ is the ratio between the latent hardening modulus and self-hardening modulus, and $f_{\alpha \beta }$ refers to the interaction between slip system $\alpha$ and $\beta$.

The aluminum material parameters of the hardening model were evaluated by fitting the simulated stress–strain curve with the experimental results of an aluminum single crystal deformed by plane strain compression (PSC) [26,27], and they are listed in

Table 1. Parameters used in the Bassani–Wu hardening model.

$\mathit{n}$	$\dot{{\mathit{\gamma }}_0}\left({\mathit{s}}^{-1}\right)$	${\mathit{h}}_0$ (MPa)	${\mathit{h}}_{\mathit{s}}$ (MPa)	${\mathit{\tau }}_1$ (MPa)	${\mathit{\tau }}_0$ (MPa)	$\mathit{q}$
300	0.0001	100	0.01	6.3	6	1

.

The kinematical scheme and hardening model used in the CP model were implicitly incorporated into ABAQUS/Standard Ver2023 (Version 2023, Dassault Systèmes, Paris, France) by following the user-defined material (UMAT) subroutine framework initially developed by Huang [28]. In ABAQUS/Standard, the CPFEM variables were maintained as solution-dependent state variables (SDVs) and updates were conducted after each increment to maintain simulation integrity. According to Ref. [2], in an aluminum single crystal, the slip plane is {1 1 1} and the slip direction is <1 1 0>, and their combinations generate 12 potentially activated slip systems, as listed in

Table 2. Notation of slip systems.

Slip Plane

(111)

($\overline{1}$11)

(1$\overline{1}\overline{1}$)

(11$\overline{1}$)

Slip Direction

[0$\overline{1}$1]

[$\overline{1}$10]

[10$\overline{1}$]

[0$\overline{1}$1]

[110]

[101]

[>011]

[110]

[10$\overline{1}$]

[011]

[$\overline{1}1$0]

[101]

Slip System

a1

a2

a3

b1

b2

b3

c1

c2

c3

d1

d2

d3

Group

RD

TD

ND

RD

TD

ND

RD

TD

ND

RD

TD

ND

.

2.2. Simulation Conditions

Width spread, a common phenomenon during rolling, refers to the lateral flow of the material under rolling, as schematically illustrated in Figure 1a. To investigate its effects, researchers have conducted relevant experiments, as reported in Refs. [10,11,12,13]. To further investigate the effect of width spread, we also conducted simulations. Our simulations corresponded to the conditions of the experiments in Ref. [10]. The first experiment involved rolling with an external frame which featured a rectangular hole aligned with the sample. The sample was inserted into the frame to limit the TD elongation. In contrast, the second experiment was conducted without an external frame. In the simulation, constraints were modeled by restricting the displacements of the node on the outer surfaces along the TD, as shown in Figure 1b. Additionally, simulation without constraints was also conducted for comparison. The first and second simulations were termed 3DRC and 3DR, respectively.

The sample is an aluminum single crystal with a {1 0 0} <0 0 1> orientation with an initial size of 16.0 mm × 18.0 mm × 7.4 mm. The reduction is 30%, 50%, 70%, 80%, 90%, and 95%. Accordingly, the element size decreases as the reduction increases. To achieve the multi-pass rolling simulation, a mapping solution is used between passes. A mapping solution is a remeshing analysis technique which transfers the deformation solution and SDV values from the integration points of the deformed mesh to the integration points of the mesh in the next pass [19]. The element type is C3D8R and reduced integration occurs with 8 nodes, which provide a faster computation speed and higher accuracy. To balance computational accuracy and efficiency, the current mesh size is chosen, and results show that it is sufficient to capture the microstructural deformation. In the simulation, the roller was considered as an analytical rigid body with a diameter of 70mm. After calibration, a friction coefficient of 0.12 was used in the simulation to approximate the experimentally lubricated conditions.

The crystal rotation angle is defined as the misorientation between the final orientation and the initial orientation [29]. The rotation angle of each element was calculated based on Bunge’s convention, and it was further partitioned into rotations around the TD, RD, and normal direction (ND) according to the method proposed in Ref. [30]. The activation of a2, b2, c2, and d2 slip systems will result in crystal rotation about the TD, i.e., TD rotation [31]. Similarly, a1, b1, c1, and d1 are associated with RD rotation, and ND rotation is related to the other four slip systems [31], as shown in Figure 2a. The paths of TD, RD, and ND rotation from the initial Cube are marked in Figure 2b. Among the three rotation axes, TD rotation is dominant during rolling due to the instability of the Cube orientation [30,31].

3. Results

3.1. CPFEM Predictions vs. Experimental Observations

Figure 3 shows the predicted and experimental results after different reductions under two rolling conditions. As shown in Figure 3a,b, the CPFEM predictions and experimental observations exhibit the same trend of evolution in the 3DR model. After 30% reduction, crystal orientations mainly rotate around the TD axis, and remain closely distributed around the initial orientation. After increasing the reduction to 50%, the crystal orientations near the two components of {1 0 2} <2 0 1> undergo TD rotation. At 70% reduction, the orientation mainly rotates about the TD axis in both clockwise and counterclockwise directions relative to the initial orientation, resulting in the original orientation evolution into two primary orientations, which is consistent with a previous experiment in Ref. [22]. After 80% reduction, the crystal orientations reach {2 1 4} <9 2 5>, and the predicted results exhibit better symmetry, whereas the experimental observations show relatively poorer symmetry. This discrepancy is due to the experimental observation results from the mid-thickness along the ND by X-ray diffraction, while the simulated results are obtained from the entire thickness, which influence the results of the observation. At 95% reduction, the texture presents as homogeneous in both simulation and experiment.

As illustrated in Figure 3c,d, at up to 50% reduction, the crystal orientations are primarily distributed around the initial orientations in 3DRC model. After 70% reduction, the orientation begins to deviate from the initial state, and asymmetric phenomena are observed in the experiment. After further reduction from 80% to 95%, the pole figures exhibit a pattern similar to the 70% reduction. The primary orientations are concentrated near the {1 0 2} <2 0 1> orientation, with dispersion along the TD axis, and they present a homogeneous texture after 95% reduction.

3.2. Shear Strain and Texture Evolution

To better understand the underlying deformation mechanisms behind the observed texture evolution, the shear strain on the active slip systems was further examined, as illustrated in Figure 4. It was observed that TD rotation was dominant across all reductions, while non-TD rotation gradually developed as the reduction increased, which is consistent with findings reported in Ref. [30]. Although both 3DR and 3DRC exhibit similar rotation patterns, the presence of external constraints in the 3DRC results in distinct variations in non-TD rotation. At 30% reduction, shear strain acting on RD exceeds that on ND in 3DR, whereas this trend is reversed in 3DRC. After 50% reduction, TD is the dominant rotation in both 3DR and 3DRC. Additionally, RD remains secondary in 3DR, while ND is secondary in 3DRC, indicating that the rotation path is unchanged after 2-pass. As reduction reaches 70%, TD rotation remains dominant in 3DR, leading to a more dispersed orientation distribution. In contrast, in the 3DRC model, non-TD rotation becomes comparable to TD rotation, resulting in a more concentrated orientation distribution. At this stage, the differences between 3DR and 3DRC become more pronounced. In 3DR, crystal rotation spreads along both TD and RD, whereas in 3DRC, rotation is primarily distributed along TD and ND. Additionally, although RD remains the secondary rotation axis in 3DR, the shear strain acting on ND gradually increases. In contrast, the shear strain in 3DRC does not display significant changes. After 80% reduction, the shear strain acting on ND continues to increase in 3DR, and crystal rotation begins to exhibit ND rotation, while the 3DRC maintains its previous rotation path. This trend continues up to 95% reduction. Although the rolling conditions are identical in the simulation, differences in the shear strain acting on the slip system and crystal orientation rotation angles indicate that the initially formed texture affects the evolution of subsequent textures. This phenomenon has been observed in many experiments [31,32,33,34].

4. Discussion

4.1. Effect of External Constraints on Deformation Behavior

To investigate the effect of external constraints on texture evolution in different regions, three representative areas of the 3DRC model were selected, as illustrated in Figure 5. For comparison, the corresponding regions in the 3DR model were also selected.

As shown in Figure 6, in the 3DR model, the crystal orientations undergo significant rotation in the left and right regions, while the central area remains relatively stable with minor changes. In contrast, the 3DRC model exhibits the opposite behavior with the central region experiencing a larger rotation angle compared to the edge regions. Additionally, Figure 6a,b show that the shear strain on the slip system in the left area is symmetric to that in the right area, resulting in symmetric pole figures and rotation angles. This phenomenon pattern is also observed in the 3DRC model, as illustrated in Figure 6c,d. By comparing different regions in the 3DR and 3DRC models, it is found that the pole figures of the center area in the 3DR model after 1-pass are similar to those of the edge area in the 3DRC model. This similarity is due to the fact that the center area of the 3DR model, constrained by the edge area, is equivalent to the 3DRC. Furthermore, non-TD rotations are observed at the early stage of rolling. This is due to the imbalance between the shear strains acting on the slip systems, which induces additional rotation angles. This phenomenon has been discussed in detail in Ref. [35].

Figure 7 shows that the rotation pattern in the 3DR model remains similar between 1-pass and 2-pass, exhibiting larger rotations in the edge and smaller rotations in the center. Additionally, the rotations are primarily dominated by TD rotation, with non-TD rotations playing a secondary role. This leads to orientations separated into two components in the edge areas, while the center area has not been completely split. As for the 3DRC model, the center area undergoes significant rotation along the TD, resulting in a tendency for orientation separation. In contrast, the edge area exhibits comparable rotations along both the TD and non-TD, leading to more concentrated orientations. This heterogeneous texture evolution is associated with the localized strain heterogeneity induced by regional constraints [11].

After 4-pass, the rotation path remains similar to that observed after 2-pass, as illustrated in Figure 8. However, the difference between 3DR and 3DRC becomes pronounced. In 3DR, the rotation pattern is still predominantly characterized by TD rotation, with the non-TD rotation in the center region is greater than that in the edge areas. In contrast, the 3DRC model exhibits less non-TD rotation in the center region compared to the edge areas. In the center region, TD rotation is still dominant and ND rotation slightly exceeds RD rotation. The variance in textures between 1-pass and 4-pass suggests that the set of the activated slip system was altered when the reduction reached 80%. This phenomenon has also been observed in previous studies, such as in Ref. [31]. However, the presence of external constraints influences this transformation, leading to a significant increase in non-TD rotation angles.

After 6-pass, the texture gradually tends to become more uniform. By comparing the 3DR and 3DRC results in Figure 9, it is observed that in the 3DR model, the orientations exhibit a concentrated distribution, indicating that the unstable Cube orientations have transformed into relatively stable orientations. In contrast, in the 3DRC model, the orientations show a scattered distribution, suggesting that orientations have not yet reached stable orientations. The different texture evolutions after the same reduction indicate that the external constraints alter the stress and shear conditions, thereby slowing down the rate of texture evolution [13]. This can be further supported by another observation: when comparing the pole figures of different areas in the 3DR model, it is found that the texture evolution in the center region is slower than the edge region across all passes. This is attributed to the center region resembling the constrained state of the 3DRC model.

The uneven texture distribution has been widely reported in Ref. [32]. To further investigate the differences in texture evolution along TD, another experiment was conducted. In this experiment, a Cube aluminum single crystal without an external frame was used, and the partitioned rotation along TD was thoroughly examined. The results reveal that TD rotation is dominant in the center region, while non-TD rotation is prominent in the edge region, as shown in Figure 10.

According to the experimental observations in Ref. [11], in the 3DR model, the widening at the mid-thickness along the ND is lower than that at the surface, whereas in the 3DRC model, the widening at the mid-thickness is higher than the surface. The uneven distribution of width spread along the ND can be attributed to the different deformation conditions during rolling. In the 3DR experiment, the mid-thickness region is compressed by the surface, while in the 3DRC model, the surface is constrained by the external frame. Moreover, the external frame restricts displacement along the TD, thereby altering the stress state and causing the rolling deviation from PS deformation [13]. As a result, the shear strain distribution and width spread ratio vary along ND. In combination with the above analysis of the varying TD rotation tendency, it can be concluded that the amplitude of TD rotation is positively correlated with the widening ratio.

Since non-TD rotation increases with the degree of deviation from PS deformation [13], the above analysis reveals that the non-TD rotation in the 3DRC model is significantly larger than in the 3DR model. This phenomenon indicates that the application of external constraints does not bring the rolling process closer to the PS condition. On the contrary, it drives the deformation further away from the ideal PS state. Additionally, this conclusion is supported by the observation that the sample with external constraints shows slight elongation along the TD in the experiment, whereas TD elongation is completely constrained in the simulation. These relatively stronger external constraints in the simulation result in a larger non-TD rotation compared to the experiment.

This deviation also influences the evolution of texture. The experimental results of the 3DRC model exhibit significant TD rotation and asymmetry distribution at 70% reduction. This results from the reaction forces exerted by the external constraints, which induce an inclination in the shear strain conditions and consequently alter slip system activation [13]. Meanwhile, this asymmetric phenomenon is not observed in the simulation, which can be attributed to the differences between the simulation and experiment. In the simulation, rolling deformation occurs under idealized conditions, whereas full constraint is not achieved in the experiment due to the deformation of the external frame during rolling. As a result, the experimental results of the 3DRC model fall between those of the 3DR and 3DRC simulations, as illustrated in Figure 3.

4.2. Heterogeneity Deformation After Rolling

According to the strain field in the rolling gap, the uneven deformation in the rolling bite leads to the shear strain gradient along the ND [20,21,22,23]. During rolling, the shear strain along the ND results from surface friction (F_shear) and rolling bite geometry (G_shear) [36]. G_shear is related to the roll diameter, initial sample thickness, and reduction [37]. In the upper half of the sample, F_shear follows a nonlinear trend, while G_shear follows a linear form, both increasing from zero at mid-thickness to a maximum at the surface [38,39]. The lower half of the sample exhibits the opposite. According to the observation in Ref. [32], the G_shear primarily acts on the middle region, while F_shear is primarily dominant in the surface region. Thus, the shear strain varies along the ND, which leads to the inhomogeneous texture distribution [40].

In both the simulation and experiment, the primarily activated sets of slip systems are a2-d2 and b2-c2, where a2-d2 are in the same direction, while b2-c2 are aligned in another direction [11]. Owing to the uneven shear strain distribution, the shear strain on the two pairs of slip systems is imbalanced. This shear imbalance is related to crystal rotation, which in turn leads to changes in the Schmid factor and subsequently causes the alternating operation of two pairs of slip systems [30]. As the unidirectional rolling process continues, the shear stress acting on slip systems gradually accumulates, which causes a higher shear imbalance and larger rotation angle in subsequent passes. Liu et al. [32] found that TD rotation within ±22.5° is primarily due to the shear imbalance between a2-d2 and b2-c2. In the simulation, the range of TD rotation angles is mostly within ±22.5° before 4-pass (Figure 7), while TD rotation exceeds ±22.5° after 4-pass (Figure 8), indicating the operation of additional slip systems. This can be further supported by the fact that the shear strain imbalance between a2-d2 will cause RD rotation, while ND rotation requires the operation of additional slip systems [30].

The relationship between shear strain and TD rotation has also been reported in Ref. [32]. As shown in Figure 11, shear strain is highest at the surface during rolling, resulting in a concentrated rotation angle. In contrast, the relatively low shear strain at the mid-thickness along ND leads to a scattered texture distribution and manifests as an uneven distribution of rotation angles. A significant change in texture is typically observed when the shear strain exceeds a critical value, often occurring near the quarter-thickness position [39,40,41,42].

This phenomenon is consistent with the rotation patterns observed in the left and right areas in Figure 12a. However, the rotation angle in the center area of the 3DR model differs from that in the left and right areas, while closely resembling the result observed in the 3DRC model after 1-pass. This similarity arises from the fact that the center area of the 3DR model is equivalent to the constrained region in the 3DRC model. The effect of external constraints leads to a deviation from the PS deformation during rolling and results in an uneven deformation along the ND. Under the influence of inclined shear strain conditions, the unstable Cube orientation tends to deform in the form of deformation bands, which manifest as layered deformation rather than uniform deformation [43]. This non-uniform deformation mode lowers grain slip resistance, thereby enabling the material to accommodate deformation with fewer slip systems [43].

In the simulation, the deformation bands emerged as early as at a 30% reduction. Within deformation bands, the slip systems alternately operate along the ND, with only a single set of slip systems activated at a time. The rotation observed in the center area, as shown in Figure 12, is consistent with the deformation phenomenon illustrated in Figure 11. The rotation pattern observed in the left and right regions of the 3DR model further validates this phenomenon. In particular, the rotation behavior in the right region at mid-thickness along the ND exhibits a similar tendency to the center area (Figure 12a). This similarity can be attributed to the mid-thickness region being compressed by the upper and lower surfaces, causing it to deform in the form of a band. In contrast, the sample is constrained in the 3DRC model, causing the entire region to preferentially deform in the form of a deformation band.

5. Conclusions

1

CPFEM simulations of single crystals have been conducted up to 6-pass, and the predictions match well with the experimental observations. This confirms the capability of CPFEM in accurately capturing deformation behavior and texture evolution under different rolling conditions.

2

The external constraints make the deformation behavior far different from PS deformation rather than closer to it, leading to different non-TD rotation in the 3DRC model compared to the 3DR model. In the 3DR model, texture evolves along the RD, while in the 3DRC model, it follows the ND, with the non-TD rotation angle in the 3DRC model being larger than in the 3DR model. Moreover, the external constraints slow down the texture evolution process; the 3DR model exhibits a concentrated texture, while the 3DRC model shows a more dispersed distribution.

3

The uneven distribution of shear strain leads to inhomogeneous deformation, which predominantly manifests as deformation bands in Cube single crystals. In the 3DR model, deformation bands appear only in the central region, whereas in the 3DRC model, they are observed throughout the entire sample. Consequently, texture evolution varies depending on the location.

In summary, this study provides a comprehensive understanding of how external constraints influence deformation mechanisms and texture evolution in single-crystal aluminum during rolling. In the future, we plan to extend our investigation by replacing the Cube orientation with other crystallographic orientations and further exploring the effects of external constraints on polycrystalline materials. We believe these findings contribute to the advancement of rolling technology and lay a solid foundation for future innovations in material design and processing.