## 3 Experimental Results and Discussion
### Controlled Curvature (\(\kappa_{c}\)/\(\kappa_{\rm o}\))-N\({}_{f}\) Relationships  
Figure 9a-d illustrate the experimental \(\kappa_{c}\)/\(\kappa_{\rm o}\)-\(N_{f}\) relationships for EMT carbon steel RHTs under cyclic bending with \(\phi\) = \(0^{\circ}\), \(30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\), and different \(d\). Here, \(\kappa_{\rm o}\) is used to nondimensionalize \(\kappa_{c}\), calculated as \(\kappa_{\rm o}\) = \(t/{D_{\rm o}}^{2}\). The experimental results indicate that, for fixed values of \(\phi\) and \(d\), larger values of \(\kappa_{c}\)/\(\kappa_{\rm o}\) correspond to a reduced \(N_{\rm f}\). Similarly, when \(\phi\) and \(\kappa_{c}\)/\(\kappa_{\rm o}\) are fixed, larger values of \(d\) result in a decreased \(N_{\rm f}\). Meanwhile, for fixed values of \(d\) and \(\kappa_{c}\)/\(\kappa_{\rm o}\), larger values of \(\phi\) lead to an increased \(N_{\rm f}\). Figure 10a-d depict the same data as Figure 9a-d using double logarithmic coordinates, with straight lines representing the results of least squares fitting. This phenomenon is consistent with the experimental results of [PERSON] et al. [27]. Figure 7: \(\Delta D/D_{\rm o}\)-\(\kappa\) relationships of EMT carbon steel RHTs under cyclic bending with \(\phi\) = \(0^{\circ}\) and \(d\) = (a) 2, (b) 4, (c) 6, (d) 8, (e) 10 mm. experimental results of [PERSON] et al. [27]. However, an inconsistency arises. For a certain \(\phi\), [PERSON] et al. [27] showed five almost-parallel straight lines, while this study found five non-parallel straight lines. In 1987, [PERSON] and [PERSON] [1] proposed the \(\kappa_{c}/\kappa_{\rm o}\)-\(N_{\rm f}\) relationship for smooth circular tubes under cyclic bending as follows:  
\[\kappa_{c}/\kappa_{\rm o}=C(N_{\rm f})^{-\alpha}\ {\rm or}\ {\rm log}\kappa_{c}/ \kappa_{\rm o}={\rm log}C-\alpha{\rm log}N_{\rm f}, \tag{1}\]  
where \(C\) and \(\alpha\) are material parameters associated with the mechanical properties of the material and the geometric shape. The value of \(\alpha\) represents the slope of the double logarithmic coordinate line's relationship between \(\kappa_{c}/\kappa_{\rm o}\) and \(N_{\rm f}\), while \(C\) indicates the corresponding value of \(\kappa_{c}/\kappa_{\rm o}\) when \(N_{\rm f}\) is 1. However, this formula cannot cover the \(N_{\text{f}}\) relationship for 6061-T6 aluminum alloy RHTs under cyclic bending with different \(\phi\) and \(d\). Therefore, [PERSON] et al. [27] proposed that \(C\) is a function of \(d/t\) as  
\[\text{log}C=-\beta(d/t)+C_{\text{o}}, \tag{2}\]  
where \(\beta\) and \(C_{\text{o}}\) are material parameters. Through experimental data in Figure 10a-d, the log\(C\) versus \(d/t\) relationships for different \(\phi\) are shown in Figure 11a. The straight lines are obtained by the least squares method. It can be found that \(\beta\) represents the slope of the straight line in Figure 11a, while \(C_{\text{o}}\) denotes the intercept of this straight line. Based on the straight lines depicted in Figure 11a, the corresponding values of \(\beta\) and \(C_{\text{o}}\) for each \(\phi\) are determined. These values are then plotted in Figure 11b,c, respectively. Since they are all linear relationships, this study proposes the following equation to express the relationships between \(\beta\) and \(\phi\) and \(C_{\text{o}}\) and \(\phi\):  
\[\beta=b_{1}\phi+b_{2},0\leq\phi\leq\pi/2, \tag{3}\]  
and  
\[C_{\text{o}}=c_{1}\phi+c_{2},0\leq\phi\leq\pi/2, \tag{4}\]  
where \(b_{1}\), \(b_{2}\), \(c_{1}\), and \(c_{2}\) are material constants and can be obtained as 0.0777, 0.2823, 0.006 and 0.0219, respectively, according to Figure 11b,c. Next, this paper continues to discuss the form of \(\alpha\). [PERSON] et al. [27] found that for a specific \(\phi\), the \(\kappa_{\text{c}}/\kappa_{\text{o}}\)-\(N_{\text{f}}\) relationship on double logarithmic coordinates showed five almost-parallel straight lines.