## 4 Parameter Determination of the HJC Model for Sandstone and Its Application
### Application of the Parameter Determination Method in LS-DYNA  
The model of the SHPB system with a diameter of 50 mm was established in LS-DYNA, as shown in Figure 20. The lengths of the impact bar, incident bar and transmitted bar are  
Figure 19: The failure strain under repeated impacts. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline RO (kg/m\({}^{3}\)) & 2610 & PC (MPa) & 25.38 \\ FC (MPa) & 76.13 & UC & 0.0016 \\ G (GPa) & 10.49 & PL (MPa) & 5.22 \\ T (MPa) & 7.63 & UL & 0.08 \\ A & 0.297 & K1 (MPa) & 10.67 \\ B & 1.947 & K2 (MPa) & \(-\)62.56 \\ N & 0.537 & K3 (MPa) & 157.8 \\ SFMAX & 1.2 & D1 & 0.005 \\ C & 0.0127 & D2 & 1.0 \\ EPSO (s\({}^{-1}\)) & 1.0 & EFMIN & 0.00465 \\ \hline \end{tabular}
\end{table} 
Table 8: The parameters of the HJC model for sandstone. 400 mm, 2000 mm and 2000 mm, respectively. The diameter of the specimen is 50 mm, and the length is 40 mm. The material of the bar is defined as an elastic material, and the properties of the aluminum-magnesium alloy are a density of 2800 kg/m\({}^{3}\), elastic modulus of 77 GPa and Poisson's ratio of 0.27. The dynamic stress-strain curves of the sandstone under different impact velocities are extracted to calculate the elastic limit, peak strength, maximum strain and other mechanical indicators. The impact velocities are taken as 7.5 m/s, 9.5 m/s, 11.5 m/s and 13.5 m/s, and the incident wave at each impact velocity is shown in Figure 21. The stress waves generated by the collision of elastic bars conform to the one-dimensional wave theory, and their amplitudes conform to the calculation results of \(\sigma=\rho\)Cv/2; the durations are all 158.4 us, consistent with the theoretical calculations. The stress and failure state of the specimen after the stress wave loading are intercepted, as shown in Figure 22. Figure 22a shows that the stress wave acts on the contact surface of the specimen and the input bar relatively uniformly at this time, and the maximum compressive stress is consistent with the amplitude of the incident wave; with the passage of time, the stress wave gradually moves towards the specimen. Internal propagation and action of the force occur, and stress reflection occurs on the end face of the specimen, with the stress cloud diagram shown in Figure 22b. Then, the internal compressive stress of the specimen gradually increases, individual elements meet the failure criteria and the element is deleted, indicating that the specimen begins to crack at the macroscopic scale. Finally, more elements are damaged and deleted, which shows the fracture of the specimen at the macroscopic level. Figure 21: The incident stress wave under different impact velocities in LS-DYNA. Figure 20: The SHPB model. In LS-DYNA, the stress-time history curve and strain-time history curve of the object element can be directly extracted from the calculation results, and the stress-strain curve can be obtained after corresponding the two data points. A summary of the stress-strain curves is shown in Figure 23, which shows that with increasing impact velocity, the dynamic peak strength also increases, which conforms to the characteristic that the strength increases with the strain rate; at the same time, with increasing impact velocity, the dynamic elastic modulus of the curve also increases, which is also in line with the relevant situation for the previous SHPB trial. The dynamic mechanical properties of sandstone at each impact speed and a comparison with the test conditions are shown in Table 9.  
Figure 22: The simulation results of the specimen failure process. (**a**) Stress wave acting on the specimen. (**b**) Propagation of the stress wave to the specimen. (**c**) Sporadic unit failure. (**d**) Overall failure of the specimen. Table 9 shows that the simulated strain rate, dynamic compressive strength and maximum strain of the sandstone specimen are highly similar to the experimental results under the same impact velocity, and the error does not exceed 10%.