## 2 Materials and Hardening Model
### Hardening Models  
This study used [PERSON]'s material behavior model [39] to express the stress flow curves such as Equation (1) for isothermal environmental temperature conditions. To describe the V-bending process, both the pure isotropic and kinematic hardening laws were first used and then compared with corresponding experimental data. The Von-Mises yield surface, which represents the material's yielding behavior, translates and expands with plastic strain and is defined as follows (Equation (1)):  
\[\sigma_{iso}=\sqrt{3J_{2}} \tag{1}\]  
where:  
\(\sigma_{iso}\) is the uni-axial equivalent yield stress;  
\(J_{2}\) is the second invariant of the deviatoric stress tensor. The stress difference (\(\xi\)) is measured from the center of the yield surface and can be expressed as (Equation (2)):  
\[\xi=\sqrt{3J_{2}}-\sigma_{iso} \tag{2}\]  
In Equation (3), the deviatoric part of the current stress \((S_{j})\) is defined as:  
\[S_{j}=\sigma_{j}-\sigma_{m}I \tag{3}\]  
where:  
\(\sigma_{j}\) is the current stress;  
\(\sigma_{m}\) is the mean stress;  
\(I\) is the identity matrix. Now, let us discuss the implications of pure isotropic and kinematic hardening:  
Pure Isotropic Hardening: In the case of pure isotropic hardening, the yield locus only evolves in size, and there is no translation (Equation (4)):  
\[\sigma_{iso}=\sqrt{3J_{2}}+\xi \tag{4}\]  
Kinematic Hardening: For the kinematic hardening model, the size of the yield surface remains constant (\(\dot{\sigma}_{iso}=0\)), and the translation of the yield locus is determined by the back stress (\(\alpha\)) (Equation (5)):  
\[\sigma_{iso}=\sqrt{3J_{2}}+\xi-\alpha \tag{5}\]  
The evolution of kinematic hardening is described by the increment in back stress (\(\alpha\)) as a function of equivalent plastic strain, as shown in Equation (6):  
\[d\alpha_{j}=\frac{C}{\sigma_{iso}}\left(\sigma_{j}-\alpha_{j}\right)d\epsilon _{eq}^{pl}-\gamma\alpha_{ij}d\epsilon_{eq}^{pl} \tag{6}\]  
where \(C\) and \(\gamma\) are material coefficients related to kinematic behavior. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Levels** \\ \hline Temperature (\({}^{\circ}\)C) & 32; 300; 600 \\ \hline Bending speed (mm/s) & 1 \\ \hline Thickness sheet (mm) & 6 \\ \hline Poisson’s ratio & 0.3 \\ \hline Young’s modulus (MPa) & 213,000 (32 \({}^{\circ}\)C); 184,410 (300 \({}^{\circ}\)C); 107,640 (600 \({}^{\circ}\)C) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test conditions and mechanical properties of the material. In Equation (7), the back stress (\(\alpha\)) curve is obtained by offsetting tensile stress-strain curve data about the yield stress value (\(\sigma_{Y}\)) and fitting them to the back-stress evolution law:  
\[\alpha=C\bigg{(}\frac{\sigma_{j}-\sigma_{Y}}{\sigma_{iso}}\bigg{)}de_{eq}^{pl}- \gamma\alpha_{ij}d\epsilon_{eq}^{pl} \tag{7}\]  
These equations illustrate how isotropic and kinematic hardening variables influence the total flow stress and the yield surface in the context of V-bending simulations. To apply this to the FEM simulation, the hardening parameters of isotropic hardening law could be adopted according to Equation (8) and of the kinematic hardening law described by the increase in back stress (\(\alpha\)) as a function of the equivalent plastic strain, as shown in Equation (9). \[\overline{\sigma}=\sigma_{Y}+A(1-exp(-B\epsilon_{eq}^{pl})) \tag{8}\]  
where \(\sigma_{Y}\) is the yield strength, \(A\) and \(B\) are hardening parameters, \(\overline{\sigma}\) is the equivalent stress, and \(\epsilon_{eq}^{pl}\) is the equivalent strain. \[\alpha=\frac{C}{\gamma}(1-e^{-\gamma\epsilon_{eq}^{pl}}) \tag{9}\]  
where \(C\) and \(\gamma\) are material parameters that describe the kinematic behavior and can be determined using the least-squares method deviation with the SOLVER tool in Excel 2013 software. The implementation procedures and obtained parameters are shown in Figure 2 and Table 3, respectively. These material parameters will be used as input data in the FEM simulations performed using Abaqus software.