## 3 Johnson-Cook Model
### Johnson-Cook Damage Model Parameters  
Substituting the material constants from Table 3 into Equation (1), the following relationships can be formed according to the JC model for stress, strain and strain deformation rate, as can be seen in Equation (6). When comparing experimental data with analytical prediction from Equation (6), good accuracy can be observed until the onset of damage and necking of the tensile specimen (Figure 10). \[\sigma_{y}=\left(933+1295e^{0.5376}\right)\left(1+0.0221\ln\left(\frac{\hat{ \epsilon}^{*}}{0.001}\right)\right)(MPa) \tag{6}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(A\) (MPa) & \(B\) (Mpa) & \(C\) & \(n\) \\ \hline
933 & 1295 & 0.0221 & 0.5376 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Johnson–Cook material model parameters for SBPR 930/1080 Type B No. 1. To accurately simulate damage that occurs in the material model with regard to the JC damage parameter setting, it is important to define at which point damage is calculated. In this study, after careful consideration (Appendix A), the authors decided to use the Damage Initiation point of Figure 10. With that, the JC damage model is used to relate fracture strain with stress triaxiality ratio, strain rate as well as temperature [11, 20], and it is expressed in Equation (7). \[\varepsilon_{f}=\left(D_{1}+D_{2}e^{D_{3}(\frac{\sigma_{m}}{\sigma_{q}})} \right)\left(1+D_{4}\ln(\varepsilon_{p}{}^{*})\right)\left(1+D_{5}T^{*}\right) \tag{7}\]  
\(D_{1}\) to \(D_{5}\) represent damage constants for the JC model, \(\sigma_{m}\) is the mean stress (hydrostatic) and \(\sigma_{eq}\) is the equivalent stress (von Mises). As damage occurs in an element governed by JC damage model, it is accumulated based on a damage law and can be represented by Equation (8) [21]. When damage occurs, during high levels of deformation, material strength is reduced [11] and the resulting relation for stress during this damage evolution step is presented in Equation (9). \[D=\sum\left(\frac{\Delta\varepsilon}{\varepsilon_{f}}\right) \tag{8}\]  
where \(\Delta\varepsilon\) is the equivalent plastic strain increment and \(\varepsilon_{f}\) is the equivalent strain to fracture under certain stress, strain rate and temperature conditions. \[\sigma_{D}=(1-D)\sigma_{eq} \tag{9}\]  
In Equation (9), \(\sigma_{D}\) is the resulting stress after damage in an element has been initiated, and \(D\) is a damage parameter with the following conditions (\(0\leq D\leq 1\)). In Equation (7), \(\frac{\sigma_{m}}{\sigma_{eq}}\) can also be defined as stress triaxiality ratio \(\eta^{*}\)[22, 23], and along with equivalent stress can be obtained from undamaged material, while considering plastic deformation up until the onset of necking. According to the work of [PERSON] [24], stress triaxiality values can be estimated from uniaxial tests of round specimens according to the analytical model which is presented in Equation (10). \[\eta^{*}=\frac{1}{3}+\ln\left(1+\frac{\alpha}{2R}\right) \tag{10}\]  
In the model, \(\eta^{*}\) is the stress triaxiality state value, \(R\) represents the radius of notch that the specimen is manufactured according to and \(\alpha\) represents the minimum cross sec  
Figure 10: Comparison of true stress–strain curve between experimental data and JC analytical prediction. tion's radius. Triaxialities calculated according to [PERSON]'s model for different notched specimens can be seen in Table 2.  
Neglecting the effects of strain rate and temperature, Equation (7) can be simplified representing fracture strain in terms of the aforementioned \(D_{1}\) to \(D_{3}\) damage parameters and stress triaxiality ratio effects. When plotting the fracture strain-stress triaxiality ratio (Figure 11), experimental tension data [25] is used, with 0.001 s\({}^{-1}\) strain rate for smooth and notched specimens from Figure 7b and Table 2, in the form of \(y=A+B\cdot\exp(R0\cdot x)\); \(D_{1}\) to \(D_{3}\) damage parameters can be calculated from the exponential coefficients of the equation similar in principal to the derivation of Equation (4).