## 3 Results and Discussion  
### Al tube Expansion Experiments  
In the Al tube expansion experiments, the expansion force was \(\sim\)442 N for S1H with an expansion speed of 1 m/s. The expansion force for S05H was \(\sim\)439 N with a speed of 0.5 m/s. These results indicate that the speed had no influence on the expansion force (\(p\) = 0.58). The expansion force of G1N was \(\sim\)519 N, while that of S1N was \(\sim\)561 N (\(p\) = 0.27). Although there was no statistical difference in these results, there was a slight decrease in the expansion force with the lubricant groove on the expansion ball (G1N). This might have been caused by a decrease in the friction force due to the better lubricant supply. There was a significant difference in the expansion force between S1N (\(\sim\)561 N), which used no lubricant, and S1H (\(\sim\)442 N), which used the DH23H lubricant (\(p\cong\) 0.00). S1A used the DH173A lubricant, which has approximately three times lower viscosity than DH23H. The expansion force of S1A was \(\sim\)462 N, which was approximately 5% higher than that of S1H (\(p\cong\) 0.00). This result suggests the lubricants could decrease the friction between the expansion ball and tube, and lubricants  
Figure 5: FE models and boundary conditions for the analysis of (**a**) Al tube and (**b**) Cu tube expansions; (**c**) reference points P1 and P2 for calculating the tube’s inner groove deformation; and (**d**) axial and radial forces.  
with higher viscosity could stick to the ball surface for a longer time, resulting in more reduction in the expansion force (Figure 6a).  
The expansion force rate, which is differentiation of the expansion force with respect to time, can be used to represent the occurrence of adhesion in the tube's inner grooves [17]. The standard deviation of the rate was the highest for S1N (\(\sim\)815 N/s), which also produced the highest expansion force. However, the standard deviation of S05H, which had the lowest expansion force, was \(\sim\)568 N/s (Figure 6b and Table 3). According to our previous study [17], the adhesion of tube's inner grooves can be associated with the standard deviation of the expansion force rate, because the sudden increase in the expansion force has a close relation to the occurrence of adhesion in the tube's inner grooves during the expansion. In the current study, a linear correlation (R\({}^{2}\) = 0.984) was observed between the expansion force and the expansion force rate's standard deviations (Figure 7). This result indicates that because a large expansion force can lead to a large deformation of the tube's inner grooves, the tube's inner groove deformation, which is difficult to be experimentally measured, can be estimated by calculating the standard deviation of the expansion force rate.  
\begin{table}
\begin{tabular}{c c c} \hline
**Experimental Group** & **Expansion Force Rate (N/s)** & **STD of Expansion Force Rate (N/s)** \\ \hline G1N (\(n\) = 3) & 69.6 & 713.4 \\ S1N (\(n\) = 4) & 338.8 & 815.2 \\ S1A (\(n\) = 10) & 53.0 & 613.1 \\ S1H (\(n\) = 10) & 438.7 & 600.2 \\ S05H (\(n\) = 10) & 26.8 & 567.9 \\ \hline \end{tabular}
\end{table}
Table 3: Expansion force rate calculated by the average of each expansion force rate profile.  
Figure 6: Al tube expansion experiment results: (**a**) expansion force; and (**b**) expansion force rate.