## Results and Discussion
### Simulation of the Hot Fracture  
Stress triaxiality criteria \(\sigma^{*}\) was used to analyze the fracture behavior of 13 CrMoNbV steel during tensile tests. Stress triaxiality may be determined as a ratio of hydrostatic pressure (\(\sigma_{m}\)) to the von Mises equivalent stress \(\overline{\sigma}\) :  
\[\sigma^{*}=\frac{\sigma_{m}}{\overline{\sigma}}=\frac{1/3(\sigma_{1}+\sigma_ {2}+\sigma_{3})}{\sqrt{\frac{\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left( \sigma_{2}-\sigma_{3}\right)^{2}+\left(\sigma_{3}-\sigma_{1}\right)^{2}}{2}}} \tag{10}\]  
where \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\) are the principal stresses. For the description of the steel fracture behavior, the Rice and Tracy ductile fracture criteria \(K\)[3] was modified by the addition of a parameter that depended on accumulated strain:  
\[K^{*}=\int_{0}^{\overline{\varepsilon}_{f}}e^{\sigma^{*}}e^{\gamma\overline{ \varepsilon}}\ d\overline{\varepsilon} \tag{11}\]  
where \(\gamma\) is a constant, \(\overline{\varepsilon}\) is the equivalent strain, and \(\overline{\varepsilon}_{f}\) is the critical strain, which can be determined by the reduction of area value of the samples after fracture:  
\[\overline{\varepsilon}_{f}=\ln\left(\frac{S_{0}}{S_{F}}\right) \tag{12}\]  
where \(S_{0}\) and \(S_{F}\) are the cross-section areas at the beginning of a tensile test and after failure, respectively. Figure 8: Microstructure of the tensile samples’ fractured surface after deformation at (**a**) 1100, (**b**) 1150 and (**c**) 1200 \({}^{\circ}\)C.  
The tensile tests with zero fracture criteria value were calculated using the finite element approach for the determination of the dependence of stress triaxiality on equivalent strain (Figure 9). The integration of obtained dependences using Equation (11) with different values of the \(\gamma\) coefficient give the critical values of the fracture criteria \(K^{*}\). To determine the correct values of the \(\gamma\) coefficient, it is necessary to compare the experimental force vs. displacement curves calculated using the FE approach curves obtained at different \(\gamma\) and \(K^{*}\). The comparison of the predicted and experimental curves is shown in Figure 10. As can be seen, the optimum value of the \(\gamma\) coefficient is changed from 0.1 at the temperature of 1100 \({}^{\circ}\)C to 0.3 at temperatures of 1200 and 1250 \({}^{\circ}\)C. The increase in the \(\gamma\) coefficient with the increase in temperature shows that recovery of the accumulated strain at a high temperature significantly prolongs deformation before fracture, even during extremely localized deformation. Optimized values of the fracture model coefficients \(\gamma\) and \(K^{*}\) are depicted in Table 3. The critical value of the ductile fracture criteria \(K^{*}\) is increased with temperature increase from 1.58 at 1100 \({}^{\circ}\)C to 2.20 at 1250 \({}^{\circ}\)C, which correlates with the tensile samples' reduction of area until fracture. Figure 10: Comparison between experimental and calculated tensile curves at (**a**) 1100, (**b**) 1150, (**c**) 1200 and (**d**) 1250 \({}^{\circ}\)C.  
Figure 9: Dependences of stress triaxiality on equivalent strain during tensile tests at (**a**) 1100–1175 \({}^{\circ}\)C and (**b**) 1200–1275 \({}^{\circ}\)C (the circles are the experimental values of \(\overline{\varepsilon}_{f}\)). The minor discrepancy between the predicted and experimental tensile curves can be seen in Figure 10, even for the optimized values of \(\gamma\) and \(K^{*}\) coefficients. It may be described by obtaining strain values up to 4.0 (Figure 9) during tensile tests due to deformation localization, while the constitutive model for the description of the steel deformation behavior was constructed for strain values up to 0.8 (Figure 5). The slight difference between experimental stress values and those calculated using Equations (1) and (4) and (6)-(9) may be present despite steady-state deformation mode after strain of 0.6.