## 4 Experiment and Simulation of V-Shaped Bending at Elevated Temperatures
### Simulation Procedure for V-Bending at Elevated Temperatures  
To simulate spring-back at elevated temperatures, this study used a proposed hardening law at room temperature combined with the determination of the back stress function at elevated temperatures. The method to determine the back stress function (\(\alpha_{n(T)}\)) at elevated temperatures is described in Equation (15).  
\[\alpha_{n(T)}=\left(\frac{\sigma_{T}}{\sigma_{R}}\right)\times\alpha_{(T)}= \left(\frac{\sigma_{yT}+A_{T}\times(1-exp(-B_{T}\times\varepsilon_{eqT}^{pl})) }{\sigma_{yR}+A_{R}\times(1-exp(-B_{R}\times\varepsilon_{eqR}^{pl}))}\right) \times\alpha_{(T)} \tag{15}\]  
where \(R\) and \(T\) are indices for the flow stress equation at room- and elevated temperatures, respectively, and \(\alpha(T)\) is the back stress function at elevated temperatures (see Equation (2) and Table 3).  
The obtained data on the back stress value (\(\alpha_{n(T)}\)) based on the equivalent strain (\(\varepsilon_{eq}^{pl}\)) at elevated temperatures were used to determine the parameters of the new combined hardening law for elevated temperatures, C1T and \(\gamma\)1T, using Equation (4) and shown in Table 8 and Figure 7a. The new parameters, A1T and B1T, in Equation (8) of the combined hardening law were determined using Excel 2013 calculation and are shown in Table 8. Figure 7b shows the simulation results for spring-back prediction based on the proposed combined hardening law. Table 9 compares the results between the experiment and simulation at 300 \({}^{\circ}\)C and 600 \({}^{\circ}\)C. The deviation angle (\(\Delta\theta\)) between the simulation and experiment is calculated using Equation (3). Table 9 shows that as the bending angle increases, the spring-back angle decreases, which can be determined by subtracting the bending angle from the simulation or experimentally measured angle. The comparison results indicate that the proposed method offers high predictive accuracy and good agreement with experimental values in describing the spring-back phenomenon after V-bending at elevated temperatures.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & \multicolumn{4}{c}{**Measure Angle (\({}^{\circ}\))**} \\ \cline{4-9} \multirow{3}{*}{**H (mm)**} & \multirow{3}{*}{**Bending Angle** (\({}^{\circ}\))} & \multicolumn{4}{c}{**300 \({}^{\circ}\)C**} & \multicolumn{4}{c}{**600 \({}^{\circ}\)C**} \\ \cline{3-9}  & & & **Level** & & & & **Level** & & \\ \cline{3-9}  & & **1** & **2** & **3** & & **1** & **2** & **3** & **Avg.** \\ \hline
10 & 136 & 139.1 & 140.15 & 138.9 & 139.38 & 137 & 137.45 & 137.2 & 137.22 \\ \hline
16 & 112 & 116.2 & 115.5 & 115.2 & 115.63 & 113.15 & 114 & 113.5 & 113.55 \\ \hline
22 & 88 & 93.5 & 93.0 & 93.0 & 93.167 & 90.5 & 90 & 90 & 90.167 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparison of V-bending results before and after spring-back at elevated temperatures of 300 \({}^{\circ}\)C and 600 \({}^{\circ}\)C.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(T\) (\({}^{\circ}\)C) & \(\sigma_{Y}\) (MPa) & \(A_{1}\) (MPa) & \(B_{1}\) & \(C_{1}\) (MPa) & \(\gamma_{1}\) \\ \hline
300 & 199.3 & 53.3422 & 6.9049 & 280.9023 & 2.0133 \\ \hline
600 & 72.43 & 29.2536 & 6.3814 & 36.5690 & 4.7288 \\ \hline \hline \end{tabular}
\end{table}
Table 8: The hardening parameters of combined hardening laws at elevated temperatures.  
Figure 7: Establishment of combined hardening law parameters at elevated temperatures (**a**), and simulation of V-bending with punching displacements of (**b**) 10 mm, (**c**) 16 mm, and (**d**) 22 mm at elevated temperatures.