# Discontinuous Yielding of Fe E420 under High Strain Rate Loading
## 2 Materials and Methods  
The investigated Fe E420 was in form of 1 mm thick laminated sheet for industrial applications. The experimental campaign was conducted on both the as-is and thermally treated material. The thermal treatment was a water quenching (WQ) process. It consists in heating the material to a temperature over the 800 \({}^{\circ}\)C and a subsequent cooling performed in a water bath. The treatment effectiveness was assessed through the execution of Vickershardness tests both before and after the treatment. In Table 1, the measured treatment temperature is reported together with the Vickers hardness values. The quenching treatment effectiveness was confirmed by the resulting 40% Vickers hardness increment. The experimental campaign consisted in quasi-static and dynamic tensile tests, with the specimens machined from the metal sheets. Quasi-static tensile tests, performed according to the ASTM E8 standard, were conducted under displacement controls using an electro-mechanical testing machine Instron 5566 equipped with a 100 kN load cell. The specimens were flat dog bone-shaped, with a gauge length equal to 80 mm. To measure the strain during the tensile tests, a 25 mm gauge length extensometer was adopted. The quasi-static testing conditions were ensured by applying a strain-rate equal to \(2\times 10^{-4}\) s\({}^{-1}\). The dynamic tests were conducted via a direct tensile split Hopkinson pressure bar (DT-SHPB) [20] at a nominal strain-rate of (500 and 1000) s\({}^{-1}\). A specific geometry for the specimens was designed, since the standard Hopkinson bar specimens had a round section, while the studied material was in the form of a sheet. A specific threaded fixture was designed to lodge a dog bone-shaped specimen with a gauge length equal to 10 mm. To guarantee the elastic wave transmission, the specimen was pinned and glued to the fixture. The specimen and the fixture system are here reported in Figure 1.  
As schematically reported in Figure 2, the DT-SHPB system consists of two metallic bars connected by the specimen with its fixture system. At the initial section of one bar, referred to as the incident bar, a stress wave is generated by instantly releasing a pre-strained portion of the bar itself. The resulting stress wave has an amplitude equal to half of the pre-tension stress, and moves toward the specimen. Due to difference in mechanical impedance of the specimen with respect to the bars, part of the wave is reflected back into the incident bar and the reminder is transmitted through the specimen to the other bar, referred to as the transmitting bar. The propagating waves within the bars were measured through a strain gauge located in the middle of each bar. The resulting strength curve of the specimen, i.e., the stress-strain curve, can be obtained from the analysis and manipulation of the incident and transmitted waves. The strain-rate \(\hat{\epsilon}(t)\) can be calculated according to \begin{table}
\begin{tabular}{c c c} \hline \hline
**Max Temp [\({}^{\circ}\)C]** & **HV before** & **HV after** \\ \hline \(906\pm 4\) & \(141\pm 2\) & \(239\pm 3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quenching maximum temperature and the Vickers hardness before and after the treatment. Figure 1: Hopkinson bar flat dog bone-shaped specimen and fixture system. Equation (1), where \(C_{0}\) is the wave propagation speed within the bars, \(L_{s}\) is the specimen gauge length and \(\varepsilon_{r}(t)\) is the reflected wave strain. \[\dot{\varepsilon}(t)=-\frac{2\cdot C_{0}}{L_{s}}\cdot\varepsilon_{r}(t) \tag{1}\]  
Since the parameters in (1), as well as the reflected wave signal, are known, the expression can be calculated and also integrated to obtain the specimen engineering strain \(\varepsilon(t)\). Then, the engineering stress \(\sigma(t)\) can be calculated according to Equation (2), where \(E_{b}\) is the bar material Young modulus, \(A_{s}\) and \(A_{b}\) are, respectively, the specimen and bars cross section net area, and \(\varepsilon_{t}(t)\) is the transmitted wave strain.