## 2 Materials and Methods  
### 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 \(\upmu\)m, 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 (\(\sigma\) is stress, \(\upsilon\) is [PERSON]'s ratio, \(E\) is [PERSON]'s moduli, \(G\) is shear moduli, and \(\varepsilon\) is strain). Figure 1: The micro-extrusion machine with the assembly model, and the micro-extrusion setup. Figure 2: The stress strain curve of the aluminum alloy 6063 having a 66-\(\upmu\)m grain size. 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):  
\[\sigma=k\cdot\varepsilon^{n}, \tag{1}\]  
where \(\sigma\) is the flow stress, \(k\) is the material constant, \(\varepsilon\) 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 [PERSON] 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-um 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-um 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 \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Condition** & **Lubrication** & **Tool Coating** & \begin{tabular}{c} **Coefficient of** \\ **Friction** \\ \end{tabular} & \begin{tabular}{c} **Extrusion** \\ **Speed** \\ \end{tabular} \\ \hline Test A & \
\begin{tabular}{c} PAO Lubricant (2.6 mm\({}^{2}\)/s \\ Kinematic Viscosity) \\ \end{tabular} & Uncoated & 0.45 & 0.10 mm/s \\ \hline Test B &
\begin{tabular}{c} PAO Lubricant (428.6 mm\({}^{2}\)/s \\ Kinematic Viscosity) \\ \end{tabular} & Uncoated & 0.25 & 0.10 mm/s \\ \hline Test C & Dry (No Lubricant) & DLC (PVD) & 0.11 & 100 mm/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental validation conditions for the finite element (FE) model. Figure 3: The finite element simulation setup of the micro-extrusion process. friction conditions of the tool-billet interfaces and the extrusion speeds.