## 5 Tests at Different Temperatures: Results and Discussion  
In the following, the results obtained from the tests performed at different temperatures are presented and compared with the results of the same tests conducted at room temperature, which were taken from a previous work by some of the authors [31]. Again, three repetitions were performed, with very good repeatability. The acquired raw load-crosshead data for the different tests at 600 \({}^{\circ}\)C and 700 \({}^{\circ}\)C are reported in Figure 14.  
From the RB tests, the stress-strain constitutive behavior could be evaluated at different temperatures, as shown in Figure 15. An inverse procedure was adopted to extend the stress-strain data up to a high strain level. To this end, RB tests were simulated with ANSYS(r) implicit code, reproducing the exact specimen geometries and loading conditions up to fracture. An optimization algorithm was used to vary the material parameters of a modified [PERSON]'s formulation of the material constitutive law (Equation (17)), until the best match between the simulated and experimental global load-displacement data of the  
Figure 14: Load–displacement curves of RB, RN and PS tests at different temperatures. RB tests could be found. The procedure was repeated for each investigated temperature. The best fitting material parameters are summarized in Table 8.  
\[\sigma=\sigma_{0}+R_{\infty}\Big{(}1-e^{-b\epsilon_{p}}\Big{)}+R_{0}\epsilon_{p} \tag{17}\]  
From Table 8 and Figure 15, it can be seen how material yield stress and ultimate strength decreased with temperature, progressively. For all tests, increasing the temperature to 600 \({}^{\circ}\)C reduced material performance by about 50%, while at 700 \({}^{\circ}\)C, the mechanical properties decreased to about 30% of the corresponding property at room temperature. Even if ductility is expected to increase with temperature, this effect cannot be observed directly from the global data on elongation at fracture, presented in Figure 14. In fact, the elasto-plastic behavior varied with temperature, and global load-displacement curves changed accordingly; at the same time, the information on material ductility is related to the local strain at fracture of the critical points of the specimens, which cannot be directly measured, but must be derived from the FEM analysis of the tests, as shown in the following. Each experimental test was simulated via the FE code, using the constitutive law of Table 8, and the same displacement ramp of the experiments, which was stopped at the moment of fracture observation. The local stress and strain histories were extracted from the simulations, then used to calculate _T,X_ (see Equations (7) and (8)) and the equivalent strain at fracture \(\epsilon_{f}\). Table 9 reports the average \(T_{avg}\) and \(X_{avg}\) (which did not change appreciably with temperature) and the values of \(\epsilon_{f}\) retrieved for each geometry at different  
Figure 15: API X65 extended stress–strain curves, at different temperatures. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\sigma_{0}\) & \(R_{\infty}\) & \(b\) & \(R_{0}\) \\  & **(MPa)** & **(MPa)** & **(—)** & **(MPa)** \\ \hline
20 \({}^{\circ}\)C & 464 & 150 & 15 & 250 \\
600 \({}^{\circ}\)C & 230 & 160 & 100 & 220 \\
700 \({}^{\circ}\)C & 150 & 75 & 100 & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Material parameters of the linear–exponential formulation of the API X65 constitutive behavior, at different temperatures. temperatures. The CCF damage model's parameters, calibrated with the same minimization approach described in Section 3, are reported in Table 10.  
Figure 16 shows the fracture surfaces of the calibrated damage model at warm temperatures, using Equation (10), as are usually compared with the corresponding surface at room temperature. The calibration points of Table 9 are also reported. The effect of temperature on the increase in ductility is clear; in fact, the strain at fracture is seen to increase when moving from the lower temperature to the higher.