# SHPB Experiment and MPM Simulation on Dynamic Properties of Unsaturated Clay Under Impact Loading
## 2 SHPB Test on Unsaturated Clay
### SHPB Setup and Testing Method  
The dynamic impact experiment on unsaturated clay employed the SHPB apparatus, namely ALT1000. Figure 1 shows the photograph and schematic diagram of the SHPB setup. The incident and transmitted bars are made of aluminum, each measuring 2500 mm in length and 50 mm in diameter. The gas pressures (0.05-0.25 MPa) can be adjusted using an air compressor and pneumatic value, thereby controlling the striking velocity of the bullet (400 mm in length) and achieving various loading rate conditions. Moreover, a cylindrical tube made of 4030 steel was used to apply passive confinement on the soil specimen. The inner and outer diameters of the cylindrical tube are 50.1 mm and 55.1 mm, respectively. The modulus and Poisson's ratio of the steel are 195 GPa and 0.28, respectively. A rubber disk with a thickness of 2.0 mm and a diameter of 45 mm  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\rho\) (g/cm\({}^{3}\)) & \(\omega\) (\%) & \(G_{s}\) & \(e\) & \(\omega_{L}\) (\%) & \(\omega_{P}\) (\%) & \(c\) (kPa) & \(\varphi\) (\({}^{\circ}\)) \\ \hline
1.74 & 40.3 & 2.69 & 1.27 & 83.9 & 48.6 & 202.47 & 5.94 \\ \hline \end{tabular} \(\rho\)—density, \(\omega\)—moisture content, \(G_{s}\)—specific gravity, \(e\)—porosity, \(\omega_{L}\)—liquid limit, \(\omega_{P}\)—plastic limit, \(c\)—cohesive force, \(\varphi\)—internal friction angle.  
\end{table}
Table 1: Properties of clay sample.  
was employed as a pulse shaper to mitigate high-frequency oscillations and dispersion effects. To reduce friction effects, a thin layer of petroleum jelly was applied between the soil specimen and the inner wall of the steel tube.  
As illustrated in Figure 1b, strain gauge 1 was positioned at the center of the incident bar to record dynamic signals, including the incident stress wave propagating from left to right and the reflected stress wave propagating from right to left along the incident bar. Additionally, strain gauge 2 was positioned to record the transmitted stress wave propagating from left to right along the transmission bar. According to the computing theory of a one-dimensional stress wave, the stresses at the front and back ends of the specimen are considered to be equal, as defined in Equation (1). Equation (2) is used for calculating the strain rate, with the corresponding strain obtained through integration.  
\[\sigma_{\text{front}}=\frac{A}{A_{s}}E(\epsilon_{i}+\epsilon_{r})\cong\sigma_{ \text{back}}=\frac{A}{A_{s}}E(\epsilon_{t}) \tag{1}\]  
\[\dot{\epsilon}(t)=-\frac{2v}{l_{s}}\epsilon_{r},\;\epsilon(t)=-\frac{2v}{l_{s} }\int_{0}^{t}\dot{\epsilon}(t)\text{d}t \tag{2}\]  
where \(\sigma_{\text{front}}\) denotes the calculated stress of the specimen at the front-end surface (i.e., the contact surface of the incident bar and the soil); \(\sigma_{\text{back}}\) denotes the calculated stress of the specimen at the back-end surface (i.e., the contact surface of the transmission bar and the soil); \(\epsilon_{i}\), \(\epsilon_{r}\), and \(\epsilon_{t}\) denote the incident, reflected, and transmitted signals, respectively; \(E\) denotes the elastic modulus of the bar; \(A\) and \(A_{s}\) denote the cross-sectional areas of the bar and specimen, respectively; \(l_{s}\) denotes the specimen thickness; and \(v\) denotes the elastic wave velocity of the bar.  
Figure 1: Split Hopkinson Pressure Bar equipment: (**a**) photograph and (**b**) schematic diagram.