# Discontinuous Yielding of Fe E420 under High Strain Rate Loading
## 3 Results  
In Figure 3, the quasi-static test results are reported in the form of tensile stress to strain. Independent of the treatment, the result of a material Young modulus (\(E\)) was equal to \(193\pm 3\)  GPa. It can be seen that Fe E420 under the described conditions does not show a noticeable DY behaviour. Once the yielding point (\(\sigma_{y}\)) reached a stress value of 425 MPa, interestingly, we obtained a Luders plateau with a YPE of about the 2.6% of strain. Then, the stress increased slightly toward a plastic plateau until the failure occurred at a strain to failure (\(\varepsilon_{f}\)) equal to 28%. Figure 2: DT-Hopkinson bar system schematic representation. Interestingly, the maximum reached stress (\(\sigma_{max}\)) resulted in a wide range of strain that was equal to 519 MPa. The stress to stain curves, resulting from the quenched material, produced a substantially different behaviour. The treatment produced a much higher yielding point (630 MPa) associated to a more pronounced hardening behaviour. A reduced Luders plastic plateau was visible and the maximum stress, equal to 702 MPa, was obtained for a strain-rate in the range of \(3.5\pm 4.5\%\). Furthermore, the strain to failure was strongly reduced by the quenching, resulting in measurements halved compared to the as-is material. The discussed stress and strain characteristic values are here collected in Table 2 for both tested conditions. The dynamic test output data processing started from the Hopkinson bar wave analysis. In Figure 4a, the incident (\(\epsilon_{i}\)) the reflected (\(\epsilon_{f}\)) and the transmitted (\(\epsilon_{t}\)) waves were reported over the time for the test performed at 500 s\({}^{-1}\). The test execution time can be divided in two periods, the incident wave propagation time and the transmitted and reflected wave propagation time. The incident wave is measured on the incident bar as well as the reflected wave, while the transmitted wave, representing the specimen loading signals, was acquired on the transmitting bar. The green dashed line in Figure 4a sets the time at which the wave signals, coming from the specimen, were detected. The flat shape of the incident wave proves the successful execution of the test. The actual applied strain-rate can be calculated, starting from the reflected wave signal, according Equation (1). The strain-rate over the test time for the considered test, is reported in Figure 4b. As expected, it is not constant over the test: it reaches a first peak value of 500 s\({}^{-1}\); then, because of the specimen elongation, it decreases to a minimum value of 412 s\({}^{-1}\) (green dotted line); and, after that, due to the specimen necking, it rises again to 554 s\({}^{-1}\) (red dotted line) just before the specimen failure. The mean value of strain-rate over the test was so calculated from the curve, between the exposed maximum and minimum values, and is equal to 470 s\({}^{-1}\). Similar considerations were made for all of the conducted tests. The tests at a higher strain-rate also resulted in an average value close to the desired nominal of 1000 s\({}^{-1}\). The stress over strain curve for the 1000 s\({}^{-1}\) test, is here reported in Figure 5a, as an example. A DY behaviour can be clearly appreciated. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **E (GPa)** & \(\sigma_{y}\) (MPa) & \(\sigma_{max}\) (MPa) & \(\epsilon_{f}\) & \(\epsilon_{\sigma_{max}}\) \\ \hline As is & 191.95 & 425 & 519 & 0.279 & 0.197 \\ WQ & 195.27 & 630 & 702 & 0.118 & 0.048 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimentally measured stress and strain properties for the as-is and WQ material under quasi-static testing conditions. Figure 3: Quasi-static tensile stress over strain curves. In Figure 5b, two images of the failed specimen are reported. The deformation field as well as the necking phenomenon evolution during the tests were monitored through the high frame rate camera acquisition. In Figure 5c, a collection of four frames from the tests are presented.