## 4 Dynamic Tests: Results and Discussion
Tables 5 and 6 summarize the average value of such parameters according to the different constitutive laws of the material employed, i.e., JC and ZA; the quantities \(\epsilon_{f}^{model}\) represent the plastic strain under incipient failure conditions predicted by the calibrated damage model. Figure 10 shows the temporal evolution of triaxiality and equivalent plastic strain, as computed by FE analyses up to fracture initiation. The trend of triaxiality can be seen to be very similar for the two calibrated models. The equivalent plastic strains predicted by the two models are identical for the RN configuration, while a slight overestimation of the strain to failure occurred with the JC model when compared with the ZA model used for the RB and PS configurations. Figure 10: Evolution of triaxiality and equivalent plastic strain computed with the ZA and JC models. As a result, the points in space (\(T_{avg}\), \(X_{avg}\), \(\varepsilon_{f}\)) of both plasticity models were identified, and, by means of the procedure described in Section 2.2, the parameters granting the best match between the fracture surfaces of the damage models CCF and BW and the experimental points were derived (see Table 7). The resulting errors (SD) were slightly higher for the ZA plasticity model than for the JC. In particular, the fracture surfaces obtained with the CCF damage model exhibited a better match with the experimental data than the fracture surfaces of the BW model, as expressed by the low standard deviation values. Figure 11 illustrates the dynamic fracture surfaces predicted by the BW and CCF damage models via the JC and ZA plasticity models. In addition to the 3D graphs, a 2D view of the \(\varepsilon_{f}\) behavior as a function of the \(T\) along curves at constant \(X\) has been reported. Overall, all surfaces showed a good match with the experimental points. It is worth noting that the BW model tended to predict a slightly higher ductility than the CCF model. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & **CCF** & & & **BW** & \\ \hline
**Plasticity** & \(C_{1}\) & \(C_{2}\) & \(\beta\) & \(\gamma\) & SD & \(C_{1}\) & \(C_{2}\) & **SD** \\
**Model** & (\(-\)) & (\(-\)) & (\(-\)) & (\(-\)) & (\(-\)) & (\(-\)) & (\(MPa\)) & (\(-\)) \\ \hline JC & 0.44 & 0.89 & 0 & 1.00 & \(1.1\times 10^{-2}\) & 0.49 & 470.24 & 0.28 \\ ZA & 0.56 & 0.63 & 0 & 1.00 & \(4.1\times 10^{-2}\) & 0.41 & 428.17 & 0.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Values of the material damage parameters and the standard deviation (SD) of the minimization. Figure 11: Comparison between the fracture surfaces of BW and CCF, JC and Za plasticity models, and positions of calibration points. Figures 12 and 13 present the comparison between the fracture surfaces obtained under static and dynamic conditions (BW and CCF models), obtained by applying as plasticity model the JC and ZA, respectively. All four comparisons suggest an increase in the ductility of the API grade X65 as the strain rate increased. Regarding the BW model, the dynamic fracture surfaces were slightly below the static fracture surfaces with high triaxialities (T > 1); however for lower triaxialities (T < 1), which are most commonly found in actual applications, the dynamic fracture surfaces were clearly higher than the static ones, showing a globally higher ductility. With the combination of the JC-CCF damage model, the increase in ductility with strain rate was very evident, with most of the dynamic fracture surface being above the static one. Only a very small portion of the dynamic surface lay below the static one with a Lode parameter close to 1. For the last combination of the ZA and CCF models, the dynamic fracture surface exhibited a limited portion below the static one, but globally, an increase in ductility was evident. To summarize, in the above comparisons, there were limited areas wherein ductility under static conditions appeared to be higher than under dynamic conditions.