## 3 Results and Discussion
### Cu Tube Expansion Analysis  
The expansion force and Cu tube's inner groove deformation (\(\Delta\)P2 \(-\)\(\Delta\)P1) were 104.8 N and \(-\)0.007 mm, respectively, from the FE analysis of Cu tube expansion using the Type 1 ball. For Type 2, the expansion force and \(\Delta\)P2 \(-\)\(\Delta\)P1 were 270.5 N (158% higher than Type 1) and \(-\)0.003 mm (57% lower than Type 1), respectively. This result suggests that the contact area between the tube and the expansion ball plays an important role in reducing tube's inner groove deformation, because Type 2, which has expansion ball grooves, has larger contact area than Type 1 and hence was effective in dispersing the stress produced in the tube's inner grooves. However, the increased contact area produces the increased friction force between the tube and the expansion ball, which results in an increase in the expansion force. For the Type 3 ball, the expansion force and \(\Delta\)P2 \(-\)\(\Delta\)P1 were 138.3 N (49% lower than Type 2) and \(-\)0.004 mm (25% higher than Type 2), respectively. This results also could be explained with respect to the contact area between the tube and ball. The contact area between the tube and Type 3 ball was smaller than that between the tube and Type 2 ball, as the width (W) of the Type 3 ball groove decreased. This smaller contact area for Type 3 produced the expansion force lower and \(\Delta\)P2 \(-\)\(\Delta\)P1 higher than Type 2. For Type 4, the expansion force and \(\Delta\)P2 \(-\)\(\Delta\)P1 were 131.8 N (5% lower than Type 3) and \(-\)0.001 mm (75% lower than Type 3), respectively. This occurred because there was no contact between the expansion ball and the tube's inner groove peaks for Type 4, which led to a significant decrease in \(\Delta\)P2 \(-\)\(\Delta\)P1. The expansion force slightly decreased due to a decrease in the contact area between the tube and ball (Figure 8, Table 4). For Type 5, the expansion force and \(\Delta\)P2 \(-\)\(\Delta\)P1 were 115.5 N (30% lower than Type 3) and \(-\)0.005 mm (40% higher), respectively. Similar to the above results, this result also occurred because the contact area between the tube and Type 5 ball was smaller compared to Type 3 due to the fewer grooves in Type 5 than in Type 3. For Type 6, the expansion force and \(\Delta\)P2 \(-\)\(\Delta\)P1 were 91.6 N and \(-\)0.008 mm, respectively. For Type 7, the expansion force and \(\Delta\)P2 \(-\)\(\Delta\)P1 were 88.9 N (3% lower than Type 6) and \(-\)0.006 mm (25% lower than Type 6), respectively. This result was caused by the curved surface of Type 7 (\(R\) = 7.9 mm) which was smoother than that of Type 6 (\(R\) = 5.3 mm). Therefore, the Type 7 ball was effective in dispersing the load generated in the tube's inner grooves, resulting in a decrease in \(\Delta\)P2. For Type 6, the expansion force and \(\Delta\)P2 \(-\)\(\Delta\)P1 were 91.6 N and \(-\)0.008 mm, respectively. The \(\Delta\)P1 compared to Type 6 (Figure 9 and Table 4). Figure 7: Correlation between the expansion force and standard deviation of the expansion force rate from Al tube expansion experiments. Figure 8: (**a**) \(\Delta\)P2 \(-\)\(\Delta\)P1 of Type 1–3 expansion balls and (**b**) their expansion force profiles; and (**c**) \(\Delta\)P2 \(-\)\(\Delta\)P1 of Type 3 and 4 expansion balls and (**d**) their expansion force profiles. Figure 9: (**a**) \(\Delta\)P2 \(-\)\(\Delta\)P1 of Type 3 and 5 expansion balls and (**b**) their expansion force profiles; and (**c**) \(\Delta\)P2 \(-\)\(\Delta\)P1 of Type 6 and 7 expansion balls and (**d**) their expansion force profiles. The radial force was calculated from the analyses of the Cu tube expansion, and the expansion force was calculated in the axial direction. There is linear correlation between the ratio of the radial force to the expansion force and the tube's inner groove deformation. This suggests that the tube's inner groove deformation becomes larger when the difference between the radial and expansion forces is larger (Figure 10).