## 4 Stress-Strain Behavior of High Silicon Steel  
The constitutive behavior of high silicon steel is expressed in a combination of exponential form and power law form. [PERSON] [17] reported that the predicted stress-strain behavior after yield as well as after strain localization was consistent with measurements when the [PERSON] model [18] and Vocc model [19] were properly combined, as shown in Equation (1):  
\[\overline{\sigma}_{c}(\overline{\varepsilon})=\alpha*\overline{\sigma}_{s}( \overline{\varepsilon})+(1-\alpha)*\overline{\sigma}_{v}(\overline{ \varepsilon}), \tag{1}\]  
\[\text{where }\overline{\sigma}_{s}(\overline{\varepsilon})=A_{1}*(\varepsilon_{0 }+\overline{\varepsilon})^{A_{2}}, \tag{2}\]  
\[\overline{\sigma}_{v}(\overline{\varepsilon})=B_{1}+B_{2}*(1-exp(-B_{3}* \overline{\varepsilon})) \tag{3}\]  
where \(A_{1}\), \(\varepsilon_{0}\), and \(A_{2}\) are the material constants of the Swift model (Equation (2)), and \(B_{1}\), \(B_{2}\), and \(B_{3}\) are the material constants of the Vocc model (Equation (3)). These constants were calibrated by fitting the equivalent stress (\(\overline{\sigma}^{Pred}\)) predicted from Equations (2) and (3) to the measured equivalent stress (\(\overline{\sigma}^{meas}\)). The calibration accuracy was assessed using the root mean squared error (RMSE) represented by Equation (4). The \(RMSE\) was 1.036 for the Swift model and 0.2835 for the Vocc model, respectively.  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(\overline{\sigma}^{meas}-\overline{ \sigma}^{Pred}\right)^{2}} \tag{4}\]  
Table 1 summarizes the values of the calibrated material constants. \(\alpha\) is a weighting factor with a value between 0 and 1. \(\alpha\) (=0.7) was determined by comparing the global force-displacement response measured from a uniaxial tensile test with the FE simulation of the tensile test.  
Figure 4(a) shows three equivalent stress-strain curves computed by the Swift, Vocc, and combined models. The equivalent stresses and strains measured before the onset of  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\epsilon_{0}\) & **A\({}_{1}\) (MPa)** & **A\({}_{2}\)** & **B\({}_{1}\) (MPa)** & **B\({}_{2}\) (MPa)** & **B\({}_{3}\)** & \(\alpha\) \\ \hline
0.003552 & 1009 & 0.1485 & 515.5 & 274.9 & 13.75 & 0.7 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the combined model for 3.0 wt% silicon steel.  
Figure 4: Types of specimens used for constructing 3D fracture locus of 3.0 wt% silicon steel: (**a**) NT10; (**b**) NT30; (**c**) SNT; (**d**) IPS. Unit is mm.  
necking are indicated by black dots. The necking point was determined by equivalent stress and strain at maximum load. It can be observed that the equivalent stress-strain curve generated by the combined model is between the curves generated by the Swift and Voce models. In this study, the equivalent stress-strain curve generated from the combined model was used in the FE analysis.  
Figure 5b shows the usefulness of the combined model used in the FE analysis. The global force and displacement curves predicted from the FE simulations were compared with the measurements. The combined model predicts the global force-displacement behavior much more accurately than the other two models. It should be emphasized that the reliability of the obtained global force and displacement response is strongly correlated with the reliability of the FE simulation [20].