## 4 Dynamic Tests: Results and Discussion  
The SHB facility described in Section 3.1 was used to conduct tensile tests at a high strain rate on the optimized sample geometries described in Table 2. Three repetitions were performed for each test condition, and excellent repeatability was observed. Figure 7 summarizes the typical results in terms of load and engineering strain rate versus time. The load is easily derivable using Equation (13). The engineering strain rate was computed by applying Equation (14) to the reflected waves measured experimentally during the tests; for comparison purposes, an initial gauge length L\({}_{\text{s}}\) of 3.0 mm was used for all three geometries. The experimental results in terms of load vs. time were processed using the procedure of Section 2.1, which aims at the accurate calibration of the JC and ZA plasticity models. The FE models are analogous to those used in the geometry optimization described in Figure 2, Section 3.1.1, with exceptions made for the PS geometry, which was extruded, along the width direction, to perform a more accurate 3D analysis. Moreover, the self-heating effect was considered, converting the plastic work into heat. Specifically, this conversion was done by imposing a Taylor-Quinney coefficient equal to 1, while the material density and specific heat were set at 7800 kg/m\({}^{3}\) and 486 J/(kg \({}^{\circ}\)C), respectively. The boundary conditions were applied to the numerical models replicating the real elongations measured in Hopkinson bar tests. ANSYS explicit code (Autodyn(r)) was used as the FE solver, while the optimization was driven by the ANSYS Workbench(r) optimization module with the response surface and genetic algorithm MOGA [43]. The material constants for the API X65, resulting from the inverse procedure, are reported in Tables 3 and 4. Figure 8 provides a comparison of the load-displacement experimental curves with those obtained by the FE simulations. In Figure 8, the numerical counterpart of the experimental results is also reported. A good agreement can be observed in all cases, which confirms that both the JC and ZA models are suitable for use to describe material behavior at high strain rates. Figure 7: Load vs. time and engineering strain rate vs. time for three different specimen geometries tested at high strain rates. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**Zerilli-Armstrong**} \\ \hline \(\sigma_{0}\) & \(C_{1}\) & \(C_{3}\) & \(C_{4}\) & \(C_{5}\) & \(n\) \\ (MPa) & (MPa) & (\(-\)) & (\(-\)) & (MPa) & (\(-\)) \\ \hline
363 & 668 & 0.004 & 0.001 & 401 & 0.439 \\ \hline \end{tabular}
\end{table} 
Table 4: Material constants, ZA model. Figure 8: Comparison between experimental and numerical load–displacement curves for RB, RN and PS, for both JC and ZA plasticity models. A very good match has also been obtained by analyzing the deformed shape of the samples. More specifically, Figure 9 compares the deformed shapes computed by the FE simulations at the onset of failure with the real observations made during the experimental tests by means of a high-speed camera (model Photron SA4(r), used at 100 kfps at resolution of 128 \(\times\) 192 pixel). Notice that the FE-predicted shapes also show the contour map of equivalent plastic strain. Even if only qualitatively, it can be seen that the profile predicted by the FE simulations is in excellent agreement with the real profile. It is worth noting that the profile of the deformed samples could be used quantitatively to extract the plastic flow stress directly, as is done analytically in [45] or numerically in [46]. However, for the sake of sensitivity, it was preferred here to use the more standard inverse FEM approach explained in Section 2.1. Then, the outcome of the numerical simulations performed with the optimized constitutive parameters was subsequently employed to extract the histories of triaxiality, Lode parameter and equivalent plastic strain for each test.