Finite Element Analysis of Grain Size Effects on Curvature in Micro-Extrusion

The precision and accuracy of the final geometry in micro-parts is crucial, particularly for high-value-added metallic products. Micro-extrusion is one of the most promising processes for delivering high-precision micro-parts. The curving tendency observed in micro-extrusion parts is a major concern, significantly affecting the final part geometry. The purpose of this paper was to investigate the driving mechanism behind the curvature in micro-extrusion at room temperature. A finite element (FE) simulation was carried out to observe the influential primary factors: (1) grain size, (2) grain boundary, (3) grain orientation, and (4) bearing length of a 6063 aluminum alloy. The Extrusion Curvature Index (ECI) was also established to indicate the level of curvature in micro-extruded parts. The results showed that the grain boundary at the high strain and die opening area was the dominant factor for single-grain conditions. The interactive effects of the grain boundary and grain orientation also affected the curvature under single-grain conditions. If the number of grains across the specimen increased up to 2.7 (poly-grains), the curvature effect was dramatically reduced (the pins were straightened). For all conditions, the curvature in micro-extrusion could be eliminated by extending the bearing length up to the exit diameter length.


Introduction
The global demand for end-uses of aluminum products has been rising [1]. One of the potential areas where one can add value for aluminum is in micro-products development [2]. An example of a micro aluminum product is a micro-gear [3]. Not only are aluminum micro-parts suitable for lightweight miniaturized applications in the automotive and electronic industries, but they can also be produced for biomedical applications (rod, wire, ribbon, screw, and tube drawing shapes) [4]. Many of the processing techniques can be applied to produce aluminum micro-parts [5]. Machine developments and manufacturing systems were also developed in this area [6]. Recent trends in micro-forming processes that are particularly suitable for lightweight materials (aluminum, titanium, and magnesium alloys) have been reviewed [7]. Most of the macro-forming processes could be downsized to micro-forming

Micro-Extrusion Experiment and Simulation
The principle of the micro-extrusion process was similar to that of the traditional extrusion process but was carried out at room temperature, while also reducing the millimeter-sized billets to a few millimeter-sized or micro-sized parts. A horizontal type high-speed micro-extrusion machine (Figure 1) was used in this study to push the prepared 6.00-mm billets having a 1.70-mm diameter down to an extruded pin having a 1.14-mm diameter at room temperature. The punch and die material was SKD11. Aluminum alloy 6063 was selected as a billet material because it is a commonly used material in aluminum extrusion. The billet specimen was prepared by hot extrusion and machining, and the grain size of the non-annealed billets was 66 µm, with a hardness value of 30.7 HV. The material flow curve of the billet specimen obtained by the compression test is shown in Figure 2 (σ is stress, υ is Poisson's ratio, E is Young's moduli, G is shear moduli, and ε is strain).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 18 ( Figure 1) was used in this study to push the prepared 6.00-mm billets having a 1.70-mm diameter down to an extruded pin having a 1.14-mm diameter at room temperature. The punch and die material was SKD11. Aluminum alloy 6063 was selected as a billet material because it is a commonly used material in aluminum extrusion. The billet specimen was prepared by hot extrusion and machining, and the grain size of the non-annealed billets was 66 μm, with a hardness value of 30.7 HV. The material flow curve of the billet specimen obtained by the compression test is shown in Figure 2 (σ is stress, υ is Poisson's ratio, E is Young's moduli, G is shear moduli, and ε is strain).  A finite element (FE) model was developed to determine how the size effects (grain boundary and grain orientation) affected the curving behavior of the micro-extruded parts by using MSC.Marc 2019 (Lagrangian). The model was a 2D axisymmetric, four-node fully integrated element, with no Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 18 ( Figure 1) was used in this study to push the prepared 6.00-mm billets having a 1.70-mm diameter down to an extruded pin having a 1.14-mm diameter at room temperature. The punch and die material was SKD11. Aluminum alloy 6063 was selected as a billet material because it is a commonly used material in aluminum extrusion. The billet specimen was prepared by hot extrusion and machining, and the grain size of the non-annealed billets was 66 μm, with a hardness value of 30.7 HV. The material flow curve of the billet specimen obtained by the compression test is shown in Figure 2 (σ is stress, υ is Poisson's ratio, E is Young's moduli, G is shear moduli, and ε is strain).  A finite element (FE) model was developed to determine how the size effects (grain boundary and grain orientation) affected the curving behavior of the micro-extruded parts by using MSC.Marc 2019 (Lagrangian). The model was a 2D axisymmetric, four-node fully integrated element, with no heat transfer, as seen in Figure 3. A total number of 4978 nodes and 4810 elements were generated. A finite element (FE) model was developed to determine how the size effects (grain boundary and grain orientation) affected the curving behavior of the micro-extruded parts by using MSC.Marc 2019 (Lagrangian). The model was a 2D axisymmetric, four-node fully integrated element, with no heat transfer, as seen in Figure 3. A total number of 4978 nodes and 4810 elements were generated. The tools (punch and die) were set to be rigid bodies. The billet was set to be a deformable body having the power-law material model, as shown in Equation (1): where σ is the flow stress, k is the material constant, ε is the true strain, and n is the strain-hardening exponent. Since the maximum strain rate used in this work was 100 s −1 at room temperature, the effects of the strain rate on the flow stress were considered minimal, as presented in the work of Ye et al. [35]. Thus, the strain-rate sensitivity was not included in Equation (1) The tools (punch and die) were set to be rigid bodies. The billet was set to be a deformable body having the power-law material model, as shown in Equation (1): where σ is the flow stress, k is the material constant, ε is the true strain, and n is the strain-hardening exponent. Since the maximum strain rate used in this work was 100 s −1 at room temperature, the effects of the strain rate on the flow stress were considered minimal, as presented in the work of Ye et al. [35]. Thus, the strain-rate sensitivity was not included in Equation (1). The flow stress-true strain curves were calculated from the cold compression tests. The material constants for aluminum alloy 6063 having a 66-μm grain size were: k = 168.40 MPa and n = 0.30. The lubricant (2.606 mm 2 /s kinematic viscosity) was applied at the billet-tool interfaces. The FE simulation was estimated by comparing it with the three micro-extrusion experiments, as described in Table 1.  In the validation conditions, the 66-μm grain size billet was extruded from a diameter of 1.70 mm to 1.14 mm with the 2.00-mm punch stroke. The ASTM E112 standard was used to measure the average grain size of the billet material. The main differences among these conditions were the friction conditions of the tool-billet interfaces and the extrusion speeds. The combined friction models  In the validation conditions, the 66-µm grain size billet was extruded from a diameter of 1.70 mm to 1.14 mm with the 2.00-mm punch stroke. The ASTM E112 standard was used to measure the average grain size of the billet material. The main differences among these conditions were the friction conditions of the tool-billet interfaces and the extrusion speeds. The combined friction models (Coulomb and shear friction models) were used at the tool-billet interfaces, as shown in Equation (2): where f is friction force, µ s is the static friction coefficient, N is the normal load, and τ is the shear friction force. Note that the values of the coefficient of frictions were obtained by trials and errors in the FE simulations. The results comparison between the FE simulations and experiments is illustrated in Figure 4.
where f is friction force, µs is the static friction coefficient, N is the normal load, and τ is the shear friction force. Note that the values of the coefficient of frictions were obtained by trials and errors in the FE simulations. The results comparison between the FE simulations and experiments is illustrated in Figure 4. The considered tests (A, B, and C) had different friction pairs (different COF values). The FE results matched well with those of the experiments, implying that the material model Equation (1) and friction model Equation (2) were suitable for the micro-extrusion process.

Considered Grain Sizes, Grain Boundaries, and Grain Orientations
The initial billet textures resulted from the specimen preparation processes (hot extrusion at 460 °C, followed by annealing) for achieving the desired grain sizes (47 μm, 66 μm, and 97 μm), and their grain size distributions are illustrated in Figure 5. Three additional grain sizes were investigated, in order to look at the influences of poly-grain to single-grain conditions, as seen in Figure 6. The grain shapes used here were selected on purpose and were based on the study of Roters et al. [36]. Grain shapes with a face-centered cubic (FCC) crystal structure, clearly showing individual grains, were selected for this study. The grain sizes and shapes (boundaries) were modeled in the FE simulation based on the assumption that there were no annealing twins. The Average Grain Size (GA) was the average maximum width of each grain across the billet specimen. The Grain Size Ratio (GR) was the Average Grain Size (GA) over the billet diameter (1.70 mm). Since the material data of these set grains were not available, the prediction of the material flow curves was carried out based on the existing stress-strain curves (47 μm, 66 μm, and 97 μm) from the compression tests. The estimated material flow curves of these enlarged grains are shown in Figure 7, and their power-law model constants are described in Table 2.

Considered Grain Sizes, Grain Boundaries, and Grain Orientations
The initial billet textures resulted from the specimen preparation processes (hot extrusion at 460 • C, followed by annealing) for achieving the desired grain sizes (47 µm, 66 µm, and 97 µm), and their grain size distributions are illustrated in Figure 5. Three additional grain sizes were investigated, in order to look at the influences of poly-grain to single-grain conditions, as seen in Figure 6. The grain shapes used here were selected on purpose and were based on the study of Roters et al. [36]. Grain shapes with a face-centered cubic (FCC) crystal structure, clearly showing individual grains, were selected for this study. The grain sizes and shapes (boundaries) were modeled in the FE simulation based on the assumption that there were no annealing twins. The Average Grain Size (G A ) was the average maximum width of each grain across the billet specimen. The Grain Size Ratio (G R ) was the Average Grain Size (G A ) over the billet diameter (1.70 mm). Since the material data of these set grains were not available, the prediction of the material flow curves was carried out based on the existing stress-strain curves (47 µm, 66 µm, and 97 µm) from the compression tests. The estimated material flow curves of these enlarged grains are shown in Figure 7, and their power-law model constants are described in Table 2.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 18        The grain sizes in the simulation did not reflect typical microstructures. However, the large grain sizes and grain boundaries in this study were specifically chosen to determine their influences on the curvature and if they could be potentially produced by using the grain selector method in our parallel study (single-crystal parts manufacturing). A few examples of the grain selection method can be found in the cited papers [38][39][40][41]. Since the stress-strain curves strongly depend on the loading axis, single crystals with different orientations would yield different stress-strain curves. In this study, instead of changing the loading axis to obtain different grain orientations, the stress-strain curves were oriented in set degrees away from the loading axis. In Figure 8, the loading axis (extrusion direction) was the x-axis. Since the material model was elastic-plastic orthotropic, the orientation of the stress-strain curves could be varied by changing the orthotropic elastic principal direction (E11). If E11 of any particular grain rotated (or changed orientation), its stress-strain curve changed. The grain orientation was described by the plane angle subtended by an arc along the x-axis (radians). For instance, if E11 equals 0 radians, the grain orientation is aligned with the x-axis (extrusion direction). If E11 equals 0.79 radians, the grain orientation is rotated 45° counterclockwise from the x-axis. The grain sizes in the simulation did not reflect typical microstructures. However, the large grain sizes and grain boundaries in this study were specifically chosen to determine their influences on the curvature and if they could be potentially produced by using the grain selector method in our parallel study (single-crystal parts manufacturing). A few examples of the grain selection method can be found in the cited papers [38][39][40][41]. Since the stress-strain curves strongly depend on the loading axis, single crystals with different orientations would yield different stress-strain curves. In this study, instead of changing the loading axis to obtain different grain orientations, the stress-strain curves were oriented in set degrees away from the loading axis. In Figure 8, the loading axis (extrusion direction) was the x-axis. Since the material model was elastic-plastic orthotropic, the orientation of the stress-strain curves could be varied by changing the orthotropic elastic principal direction (E11). If E11 of any particular grain rotated (or changed orientation), its stress-strain curve changed. The grain orientation was described by the plane angle subtended by an arc along the x-axis (radians). For instance, if E11 equals 0 radians, the grain orientation is aligned with the x-axis (extrusion direction). If E11 equals 0.79 radians, the grain orientation is rotated 45 • counterclockwise from the x-axis.

Grain Orientations
Two cases that had rectangular grain sizes and shapes were established in order to only investigate the grain orientation effects, as illustrated in Figure 9. In the first case (EX1), all of the grains were oriented along the x-axis (0.00 radians). For the second case (EX2), the grain sizes and bounda-

Grain Orientations
Two cases that had rectangular grain sizes and shapes were established in order to only investigate the grain orientation effects, as illustrated in Figure 9. In the first case (EX1), all of the grains were oriented along the x-axis (0.00 radians). For the second case (EX2), the grain sizes and boundaries were the same as those of EX1, but the only difference was the orientation of the bottom grain (E11 rotated 75 • counterclockwise from the x-axis).

Grain Orientations
Two cases that had rectangular grain sizes and shapes were established in order to only investigate the grain orientation effects, as illustrated in Figure 9. In the first case (EX1), all of the grains were oriented along the x-axis (0.00 radians). For the second case (EX2), the grain sizes and boundaries were the same as those of EX1, but the only difference was the orientation of the bottom grain (E11 rotated 75° counterclockwise from the x-axis).

Grain Boundaries and Grain Orientations
In reality, those rectangular grain boundaries did not exist, so the referenced literature grain boundaries ( Figure 8) were investigated throughout this study. However, the systematic way of evaluating the interactive effects of grain boundaries and grain orientations was developed as follows.

Grain Boundaries and Grain Orientations
In reality, those rectangular grain boundaries did not exist, so the referenced literature grain boundaries ( Figure 8) were investigated throughout this study. However, the systematic way of evaluating the interactive effects of grain boundaries and grain orientations was developed as follows.
In Figure 10, all four cases (LG1 to LG4) had the same grain sizes and grain boundaries, but the only difference was the grain orientations of the bottom grains. The grain orientation was rotated 0 • (LG1), 45 • (LG2), 90 • (LG3), and −45 • (LG4) along the x-axis. In Figure 11, the influences of the top and bottom grain orientations were studied (LG5), where the grains were oriented −14.3 • along the x-axis. The influences of the top, middle, and bottom grain orientations were observed in LG6. The grain orientation effects of only the middle grain (LG7) were examined. In LG8, only the middle three grains were rotated by −14.3 • along the x-axis. In Figure 12, the bottom grains were turned by −14.3 • along the x-axis for all four cases. The main difference was the grain orientations of the middle grain (LG10), the middle three grains (LG11), and the top and middle three grains (LG12).

Grain Sizes and Grain Boundaries
The variations of the average grain sizes (G A ) were established in Figure 13 to observe the coupling effects of the grain sizes and grain boundaries. Note that all four cases had the same grain orientation (0 • with respect to the x-axis). Figure 14 displays the mesh elements of the enlarged grains in the finite element simulation.
only difference was the grain orientations of the bottom grains. The grain orientation was rotated 0° (LG1), 45° (LG2), 90° (LG3), and −45° (LG4) along the x-axis. In Figure 11, the influences of the top and bottom grain orientations were studied (LG5), where the grains were oriented −14.3° along the x-axis. The influences of the top, middle, and bottom grain orientations were observed in LG6. The grain orientation effects of only the middle grain (LG7) were examined. In LG8, only the middle three grains were rotated by −14.3° along the x-axis. In Figure 12, the bottom grains were turned by −14.3° along the x-axis for all four cases. The main difference was the grain orientations of the middle grain (LG10), the middle three grains (LG11), and the top and middle three grains (LG12).   In Figure 10, all four cases (LG1 to LG4) had the same grain sizes and grain boundaries, but the only difference was the grain orientations of the bottom grains. The grain orientation was rotated 0° (LG1), 45° (LG2), 90° (LG3), and −45° (LG4) along the x-axis. In Figure 11, the influences of the top and bottom grain orientations were studied (LG5), where the grains were oriented −14.3° along the x-axis. The influences of the top, middle, and bottom grain orientations were observed in LG6. The grain orientation effects of only the middle grain (LG7) were examined. In LG8, only the middle three grains were rotated by −14.3° along the x-axis. In Figure 12, the bottom grains were turned by −14.3° along the x-axis for all four cases. The main difference was the grain orientations of the middle grain (LG10), the middle three grains (LG11), and the top and middle three grains (LG12).

Grain Sizes and Grain Boundaries
The variations of the average grain sizes (GA) were established in Figure 13 to observe the coupling effects of the grain sizes and grain boundaries. Note that all four cases had the same grain orientation (0° with respect to the x-axis). Figure 14 displays the mesh elements of the enlarged grains in the finite element simulation.

Grain Sizes and Grain Boundaries
The variations of the average grain sizes (GA) were established in Figure 13 to observe the coupling effects of the grain sizes and grain boundaries. Note that all four cases had the same grain orientation (0° with respect to the x-axis). Figure 14 displays the mesh elements of the enlarged grains in the finite element simulation.

Bearing Lengths
The effects of the bearing length (BL) were also explored in order to determine how the tool geometry affects the curvature of the micro-extruded pins. Three bearing lengths varied as follows: 0.42 mm (1X), 0.84 mm (2X), and 1.26 mm (3X).

Extrusion Curvature Index
The Extrusion Curvature Index (ECI) was established in order to determine the amount of curvature of the micro-extruded pins, as illustrated in Figure 15. After each extrusion, the curvature point that could fit its inner radius and outer radius to the curvature of the micro-extruded pin was located. The radii were then normalized by the exit diameter (DE), and its inverse value in percentage was the Extrusion Curvature Index (ECI). High values of ECI implied a high degree of curvature. Low ECI values indicated that the micro-extruded pin was rather straight.

Bearing Lengths
The effects of the bearing length (B L ) were also explored in order to determine how the tool geometry affects the curvature of the micro-extruded pins. Three bearing lengths varied as follows: 0.42 mm (1X), 0.84 mm (2X), and 1.26 mm (3X).

Extrusion Curvature Index
The Extrusion Curvature Index (ECI) was established in order to determine the amount of curvature of the micro-extruded pins, as illustrated in Figure 15. After each extrusion, the curvature point that could fit its inner radius and outer radius to the curvature of the micro-extruded pin was located. The radii were then normalized by the exit diameter (DE), and its inverse value in percentage was the Extrusion Curvature Index (ECI). High values of ECI implied a high degree of curvature. Low ECI values indicated that the micro-extruded pin was rather straight.  Figure 16 shows a comparison of the micro-extrusion results between the two cases (EX1 and EX2) having the same grain sizes and grain boundaries, but having different grain orientations of the bottom grains. If all grains were oriented in the same direction (EX1), and the grain sizes and grain boundaries were symmetric along the extrusion direction, there should be no curvature at all. The results of EX1 demonstrated that the FE simulation was valid. The results of EX2 clearly showed that the bottom grain orientation that was rotated 75° with respect to the x-axis caused the micro-extrusion pin to bend (ECI = 4.45%). With no effects of grain sizes and grain boundaries under single-grain conditions, the change in grain orientation could cause the micro-extrusion pin to curve. Regarding the stress distributions, high stresses occurred at high deformation zones (reduction of billet diameter to the outlet diameter). These areas caused the grains to fragment, leading to smaller grain sizes and higher stress values. The simulation results provided effective stress (συ) values based on the following equation:

Effects of Grain Orientations
where σ1, σ2, and σ3 are the principal stresses.  Figure 16 shows a comparison of the micro-extrusion results between the two cases (EX1 and EX2) having the same grain sizes and grain boundaries, but having different grain orientations of the bottom grains. If all grains were oriented in the same direction (EX1), and the grain sizes and grain boundaries were symmetric along the extrusion direction, there should be no curvature at all. The results of EX1 demonstrated that the FE simulation was valid. The results of EX2 clearly showed that the bottom grain orientation that was rotated 75 • with respect to the x-axis caused the micro-extrusion pin to bend (ECI = 4.45%). With no effects of grain sizes and grain boundaries under single-grain conditions, the change in grain orientation could cause the micro-extrusion pin to curve. Regarding the stress distributions, high stresses occurred at high deformation zones (reduction of billet diameter to the outlet diameter). These areas caused the grains to fragment, leading to smaller grain sizes and higher stress values. The simulation results provided effective stress (σ υ ) values based on the following equation:

Effects of Grain Orientations
where σ 1 , σ 2 , and σ 3 are the principal stresses. Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 18 Figure 16. The micro-extrusion results comparison between the EX1 and EX2 conditions.       In other words, the average number of grains across the specimen was 1.3 (single-grain) under all the conditions. In LG1, all of the grain orientations were along the x-axis, which showed that the micro-extrusion pin was curved. This result indicated that, when there was no influence from the grain orientation, the grain boundaries alone could significantly affect the curvature. It could be observed that the grain boundaries were not symmetric along the x-axis, causing the specimen to flow nonuniformly into the extrusion die. This behavior implied that the asymmetric grain boundaries were the primary driving mechanism of the bending. In LG2, the effect of the grain orientation (+45 • from the x-axis) of the bottom grain was coupled with the influence of the asymmetric boundaries.

Effects of Grain Boundaries and Grain Orientations
The ECI values of both LG1 and LG2 were similar (same curvature); thus, the effect of grain orientation was not observed here. However, when the bottom grain orientation was rotated +90 • from the x-axis (LG3), the ECI values slightly increased from 6.90% to 7.17%. As a result, the grain orientation must be dramatically changed in order to see the curvature effect. When the bottom grain orientation was rotated−45 • from the x-axis (or −90 • opposite to LG2), the ECI values were reduced from 6.90% to 3.09%, which helped to straighten the micro-extrusion pin. This result also confirmed that the grain orientation must be significantly changed in order to see the curvature effect. In the direction of the curvature, the curvature increased with an increase in the amount of grain orientation mismatch. However, if the orientation mismatch was in the opposite direction to the curvature, the curvature decreased according to the amount of grain orientation mismatch.  In other words, the average number of grains across the specimen was 1.3 (single-grain) under all the conditions. In LG1, all of the grain orientations were along the x-axis, which showed that the micro-extrusion pin was curved. This result indicated that, when there was no influence from the grain orientation, the grain boundaries alone could significantly affect the curvature. It could be observed that the grain boundaries were not symmetric along the x-axis, causing the specimen to flow nonuniformly into the extrusion die. This behavior implied that the asymmetric grain boundaries were the primary driving mechanism of the bending. In LG2, the effect of the grain orientation (+45° from the x-axis) of the bottom grain was coupled with the influence of the asymmetric boundaries.
The ECI values of both LG1 and LG2 were similar (same curvature); thus, the effect of grain orientation was not observed here. However, when the bottom grain orientation was rotated +90° from the x-axis (LG3), the ECI values slightly increased from 6.90% to 7.17%. As a result, the grain orientation must be dramatically changed in order to see the curvature effect. When the bottom grain orientation was rotated−45° from the x-axis (or −90° opposite to LG2), the ECI values were reduced from 6.90% to 3.09%, which helped to straighten the micro-extrusion pin. This result also confirmed that the grain  In other words, the average number of grains across the specimen was 1.3 (single-grain) under all the conditions. In LG1, all of the grain orientations were along the x-axis, which showed that the micro-extrusion pin was curved. This result indicated that, when there was no influence from the grain orientation, the grain boundaries alone could significantly affect the curvature. It could be observed that the grain boundaries were not symmetric along the x-axis, causing the specimen to flow nonuniformly into the extrusion die. This behavior implied that the asymmetric grain boundaries were the primary driving mechanism of the bending. In LG2, the effect of the grain orientation (+45° from the x-axis) of the bottom grain was coupled with the influence of the asymmetric boundaries.
The ECI values of both LG1 and LG2 were similar (same curvature); thus, the effect of grain orientation was not observed here. However, when the bottom grain orientation was rotated +90° from the x-axis (LG3), the ECI values slightly increased from 6.90% to 7.17%. As a result, the grain orientation must be dramatically changed in order to see the curvature effect. When the bottom grain orientation was rotated−45° from the x-axis (or −90° opposite to LG2), the ECI values were reduced from 6.90% to 3.09%, which helped to straighten the micro-extrusion pin. This result also confirmed that the grain  Figure 18 shows the conditions (LG5 to LG8) that take into account the effects of the grain orientations from the middle and top grains. Considering LG5 and LG6, it could be noted that both the middle and top grain orientations did not affect the curvature. Similarly, the middle grain orientations (LG7 and LG8) did not affect the curvature when compared with LG1. As a result, the bottom grain orientations significantly affect the curvature. Since the bottom grain was located at the high deformation (reduced diameter) and opening area, this implied that only the grain orientations at the high strain and opening area influenced the curvature. This phenomenon could also be observed in Figure 19, where the middle and top grain orientations did not show any effects on the curvature.
Based on the results from Figures 16-19, it could be stated that the grain boundary of the high strain and die opening area was the dominating factor for single-grain billet conditions. Since the actual metallic grain shapes were not symmetric and the grain orientations were random, the grain boundary dominated. The combination of the grain shapes (grain boundaries), grain sizes, and grain orientations dictate the direction of the curvature. In the EX2 case, there was only one single grain at the bottom of the billet. As a result, the change of grain orientation (+75 • to the extrusion direction) of the bottom grain caused the extruded pin to curve to the right. In the other single-grain cases, the curvature was mainly influenced by the initial bottom grain boundaries. As seen in the LG1 case, the extruded pin was already curved to the left, even when all the bottom grain orientations were 0 • to the extrusion direction. When the grain orientations of the large bottom grains were changed in the single-grain cases, the curvature directions were still towards the left, but the amount of curvature (ECI values) changed according to the set grain orientations. The results clearly showed that the initial grain boundaries of the bottom grains significantly affected the curvature directions. Figure 20 shows the influences of the grain sizes and grain boundaries on the micro-extrusion, when the grain orientations did not influence any of the conditions. If the Grain Size Ratio (G R ) was reduced from 0.75 to 0.37 or below, the ECI values decreased dramatically. In other words, if the specimen grains moved from 1.3 to 2.7 grains or more, the micro-extrusion pin was straightened. The impact of the grain boundaries could be clearly noticed under the single-grain condition (G R = 0.75). The influence of the grain boundaries was dramatically reduced when the number of grains increased (G R = 0.37 and below). Even though the grain orientations were not considered here, it could be stated that the effects were negligible when the number of grains was increased from 1.3 to 2.7 or more, according to the previous results. The grain boundaries that were symmetrical along the extrusion axis were G R = 0.04 and G R = 0.37. Both G R = 0.25 and G R = 0.75 had asymmetric grain boundaries along the centerline. According to Figure 20, it could be observed that the increased number of grain sizes straightened the extruded pins. The results implied that the asymmetry of grain boundaries did not have an effect on the curvature in the multi-grain cases.

Effects of Grain Sizes and Grain Boundaries
Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 18 high strain and opening area influenced the curvature. This phenomenon could also be observed in Figure 19, where the middle and top grain orientations did not show any effects on the curvature. Based on the results from Figures 16-19, it could be stated that the grain boundary of the high strain and die opening area was the dominating factor for single-grain billet conditions. Since the actual metallic grain shapes were not symmetric and the grain orientations were random, the grain boundary dominated. The combination of the grain shapes (grain boundaries), grain sizes, and grain orientations dictate the direction of the curvature. In the EX2 case, there was only one single grain at the bottom of the billet. As a result, the change of grain orientation (+75° to the extrusion direction) of the bottom grain caused the extruded pin to curve to the right. In the other single-grain cases, the curvature was mainly influenced by the initial bottom grain boundaries. As seen in the LG1 case, the extruded pin was already curved to the left, even when all the bottom grain orientations were 0° to the extrusion direction. When the grain orientations of the large bottom grains were changed in the single-grain cases, the curvature directions were still towards the left, but the amount of curvature (ECI values) changed according to the set grain orientations. The results clearly showed that the initial grain boundaries of the bottom grains significantly affected the curvature directions. Figure 20 shows the influences of the grain sizes and grain boundaries on the micro-extrusion, when the grain orientations did not influence any of the conditions. If the Grain Size Ratio (GR) was reduced from 0.75 to 0.37 or below, the ECI values decreased dramatically. In other words, if the specimen grains moved from 1.3 to 2.7 grains or more, the micro-extrusion pin was straightened. The impact of the grain boundaries could be clearly noticed under the single-grain condition (GR = 0.75). The influence of the grain boundaries was dramatically reduced when the number of grains increased (GR = 0.37 and below). Even though the grain orientations were not considered here, it could be stated that the effects were negligible when the number of grains was increased from 1.3 to 2.7 or more, according to the previous results. The grain boundaries that were symmetrical along the extrusion axis were GR = 0.04 and GR = 0.37. Both GR = 0.25 and GR = 0.75 had asymmetric grain boundaries along the centerline. According to Figure 20, it could be observed that the increased number of grain sizes straightened the extruded pins. The results implied that the asymmetry of grain boundaries did not have an effect on the curvature in the multi-grain cases.  Figure 21 shows the effects of the bearing lengths (BL) on the curvature of single-grain conditions (GR = 0.75). The ECI values decreased when the bearing lengths increased. In other words, if the bearing length was extended further, it could help strengthen the micro-extruded pins. If the BL value was increased up to 1.26 mm, which was slightly larger than the exit diameter (1.14 mm), it could almost eliminate the curvature in the micro-extrusion. As a result, the extension of the bearing lengths  Figure 21 shows the effects of the bearing lengths (B L ) on the curvature of single-grain conditions (G R = 0.75). The ECI values decreased when the bearing lengths increased. In other words, if the bearing length was extended further, it could help strengthen the micro-extruded pins. If the B L value was increased up to 1.26 mm, which was slightly larger than the exit diameter (1.14 mm), it could almost eliminate the curvature in the micro-extrusion. As a result, the extension of the bearing lengths (B L ) from the high strain and opening area could help reduce the curvature effects influenced by the grain sizes, grain boundaries, and grain orientations.

Effects of Bearing Lengths
Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 18 (BL) from the high strain and opening area could help reduce the curvature effects influenced by the grain sizes, grain boundaries, and grain orientations. Based on the results, the following observations and analysis could be made: -For single-grain conditions (GR = 0.75), if the grain sizes and grain boundaries were symmetric along the extrusion direction, the grain orientation was the dominating factor for the curvature. If the grain sizes and grain boundaries were not symmetric along the extrusion direction, the grain boundary at the high strain and die opening area was the dominating factor for the curvature. The interactive influences of the grain boundary and grain orientation also affected the curvature.
-For poly-grain conditions (GR < 0.75), if the number of grains across the specimen increased up to 2.7, the curvature effects were dramatically reduced. The influences of the grain boundaries and grain orientations were not significant.
-For all conditions, if the bearing lengths (BL) were extended beyond the exit diameter (DE), the micro-extrusion pins were straightened. Although this work was carried out by using the numerical simulation method, the findings of this research work fundamentally helped explain how the micro-extrusion parts were curved. Future work within this research would apply the FE techniques to the designed grain sizes and orientations under controlled micro-extrusion process parameters. Higher-grade aluminum alloys and multi-material extrusion would also be considered by using this same technique. The results of all the studies could be integrated in order to design and develop single-crystal parts by using the grain selector. The insights gained from this series of ongoing studies could be applied to the control and optimization of lightweight multi-material micro-parts.

Conclusions
This research work investigated the driving mechanism behind the curvature in the micro-extrusion of a 6063 aluminum alloy at room temperature. An FE simulation was used to observe the influences of the grain sizes, grain boundaries, grain orientations, and bearing lengths. The FE simulations were validated with experiments using a 66-μm grain-size billet, and the results agreed well. The billet grain sizes were enlarged, ranging from poly-grain to single-grain conditions, and their Based on the results, the following observations and analysis could be made: -For single-grain conditions (G R = 0.75), if the grain sizes and grain boundaries were symmetric along the extrusion direction, the grain orientation was the dominating factor for the curvature. If the grain sizes and grain boundaries were not symmetric along the extrusion direction, the grain boundary at the high strain and die opening area was the dominating factor for the curvature. The interactive influences of the grain boundary and grain orientation also affected the curvature. -For poly-grain conditions (G R < 0.75), if the number of grains across the specimen increased up to 2.7, the curvature effects were dramatically reduced. The influences of the grain boundaries and grain orientations were not significant. -For all conditions, if the bearing lengths (B L ) were extended beyond the exit diameter (D E ), the micro-extrusion pins were straightened.
Although this work was carried out by using the numerical simulation method, the findings of this research work fundamentally helped explain how the micro-extrusion parts were curved. Future work within this research would apply the FE techniques to the designed grain sizes and orientations under controlled micro-extrusion process parameters. Higher-grade aluminum alloys and multi-material extrusion would also be considered by using this same technique. The results of all the studies could be integrated in order to design and develop single-crystal parts by using the grain selector. The insights gained from this series of ongoing studies could be applied to the control and optimization of lightweight multi-material micro-parts.

Conclusions
This research work investigated the driving mechanism behind the curvature in the micro-extrusion of a 6063 aluminum alloy at room temperature. An FE simulation was used to observe the influences of the grain sizes, grain boundaries, grain orientations, and bearing lengths. The FE simulations were validated with experiments using a 66-µm grain-size billet, and the results agreed well. The billet grain sizes were enlarged, ranging from poly-grain to single-grain conditions, and their material properties were extrapolated. The grain shapes (boundaries) were established, and the grain orientations were varied to determine the interactive effects on the curvature (Extrusion Curvature Index or ECI) of micro-extruded pins. The key finding was that the grain boundary was the dominating factor driving the mechanism of curvature in micro-extrusion under single-grain conditions.