## 3 Experiment and Simulation of V-Shaped Bending at Room Temperature
### Combined Hardening Law  
From the comparison data presented in Table 5, it was observed that the deviations between the simulation predictions using pure isotropic and kinematic hardening laws and the corresponding experiments were significant. The spring-back prediction using the pure isotropic hardening law was larger than the corresponding experiment by approximately 1.6\({}^{\circ}\), while the prediction using the pure kinematic hardening law was smaller than the corresponding experiment by approximately 2.8\({}^{\circ}\). Therefore, a new model was proposed by combining the isotropic and kinematic hardening laws as shown in Equation (11). The new parameters for this model were determined by using the data obtained from the V-bending simulations using both isotropic and kinematic hardening laws to calculate the values of back stress (\(\alpha\)) for various equivalent strains (\(e_{eq}^{pl}\)) as follows: at each equivalent strain position, the value of the new back stress (\(\alpha_{comb}\)) was calculated based on the previous back stress (\(\alpha\)) using Equation (12), in which \(\Delta\theta_{iso}\) and \(\Delta\theta_{kine}\) are the deviations of the measure angles after the simulation of V-bending compared with the corresponding experiments using pure isotropic and kinematic hardening laws, respectively (Table 5). The average value of the new back stress (\(\alpha_{n}\)) was calculated using Equation (13).  
\[\overline{\sigma}=\sigma_{Y}+A_{1}(1-exp(-B_{1}e_{eq}^{pl}))+\frac{C_{1}}{ \tau_{1}}(1-e^{-\tau_{1}e_{eq}^{pl}}) \tag{11}\]  
\[\alpha_{comb}=(\frac{\Delta\alpha_{iso}}{\Delta\alpha_{iso}-\alpha_{kine}}) \times\alpha=0.583\times\alpha \tag{12}\]  
The relationship between the new back stress (\(\alpha_{comb}\)) and equivalent strain (\(e_{eq}^{pl}\)) was used to determine the parameters of the new proposed hardening law \(C_{1}\) and \(\gamma_{1}\) using Excel 2013 calculation software based on the least-squares method. The values of \(C_{1}\) and \(\gamma_{1}\) were calculated as 3538.925 MPa and 33.526, respectively. The equivalent stress vs. equivalent strain value was then recalculated using Equation (14) and Figure 5a.  
\[\overline{\sigma}_{iso}(e_{eq}^{pl})=\overline{\sigma}(e_{eq}^{pl})-\alpha(e_ {eq}^{pl}) \tag{13}\]  
Figure 5: _Cont._The data obtained from Equation (13) are used to calculate the new parameters, \(A_{1}\) and \(B_{1}\), of Equation (14), using Microsoft Excel 2013. The resulting values are \(A_{1}\)= 55,004 MPa and \(B_{1}\) = 33,527, respectively.  
\[\overline{\sigma}_{iso}=\sigma_{\gamma}+A_{1}(1-exp(-B_{1}e_{eq}^{pl})) \tag{14}\]  
The results of a V-bending simulation performed on SS400 steel sheet using the proposed combined hardening law are depicted in Figure 5b-d and Table 6. Table 6 demonstrates that the proposed combined hardening law has higher accuracy in predicting spring-back in V-bending compared with the pure isotropic and kinetic hardening laws. The maximum deviation between the simulation and experimental results for the spring-back angle was found to be 0.668\({}^{\circ}\). Therefore, it can be concluded that the proposed combined hardening law can be utilized to predict spring-back in other cases involving SS400 steel sheets.