## 2 Split Hopkinson Pressure Bar Theory  
The SHPB set-up usually consists of a striker and two long bars, called an input bar and an output bar, with the same diameter and material (typically a high-strength material such as maraging steel) (Figure 3). A material sample is placed between the bars, and it is loaded by a stress impulse generated by the striker impact on the front surface of the input bar. This generates a trapezoidal stress impulse (incident wave), which travels through the impacted bar. When the elastic wave reaches the specimen, due to the mismatch of mechanical impedances between the bar and the specimen material, part of the incident wave is reflected back (reflected wave), and the rest of the incident wave is transmitted through the specimen. It compresses the specimen at high rates, and the rest of the wave travels to the output bar as a transmitted wave. The incident and the reflected signals are recorded by the strain gauges, which are fastened on the input bar, and the transmitted signals are recorded by the strain gauges located on the output bar. The information about the dynamic behavior of the sample is included in the incident, reflected, and transmitted signals. Time-resolved engineering stress, strain rate, and strain in the specimen are defined by the following equations [1]:  
\[\sigma(t) = \frac{A}{2A_{s}}E[\varepsilon_{l}(t)+\varepsilon_{R}(t)+ \varepsilon_{T}(t)] \tag{1}\] \[\dot{\varepsilon}(t) = \frac{c_{0}}{L_{s}}[\varepsilon_{l}(t)-\varepsilon_{R}(t)- \varepsilon_{T}(t)]\] (2) \[\varepsilon(t) = \frac{c_{0}}{L_{s}}\int\limits_{0}^{t}[\varepsilon_{l}(\tau)- \varepsilon_{R}(\tau)-\varepsilon_{T}(\tau)]d\tau \tag{3}\]  
where \(A_{s}\) and \(L_{s}\) are the initial cross-section area and length of the specimen, respectively; \(c_{0}\), \(E\), and \(A\) are the sound velocity, Young's modulus, and bar initial cross-section, respectively; \(\varepsilon(t)\) is the strain signal sensed by strain gauges; and the \(I\), \(R\), and \(T\) subscripts denote incident, reflected, and transmitted pulses, respectively.  
The strain acceleration, which is the effect of a non-constant strain rate, can produce additional axial stress and radial stress in the specimen [16; 17; 18; 19]. The inertia and friction effects in the form of overstress are given by [14]:  
\[\Delta\sigma(t) = \frac{\mu\sigma_{exp}}{3(l_{s}/d_{s})}+\frac{pd_{s}^{2}}{12} \left[\left(\frac{l_{s}}{l_{s}}\right)^{2}-\frac{3}{16}\right](\dot{ \varepsilon}^{2}+\ddot{\varepsilon})+\frac{3\phi d_{s}^{2}}{64}d_{s}^{2}, \tag{4}\]  
where \(d_{s}\) and \(l_{s}\) are the current diameter and length of the specimen, respectively, \(\upmu\) is the friction coefficient, and \(\sigma_{exp}\) is the measured specimen stress. According to Equation (4), an assurance of a constant strain rate in the measuring range, the choice of the initial sample length and diameter leading to \(l_{s}/d_{s}\approx\sqrt{3}/4\), and lubrication contacts between the specimen and bars cause the overstress to reach a minimum. [PERSON] and [PERSON]'s study shows that a sample length-to-diameter ratio should be \(l_{s}/d_{s}\approx\sqrt{3v}/4\)[20].  
Figure 3: Split Hopkinson pressure bar (SHPB) set-up with strain gauges position and recorded signals.