## 4 Parameter Determination of the HJC Model for Sandstone and Its Application
### Parameter Determination Method of the HJC Model for Sandstone
In addition, the influence of the hydrostatic pressure on \(\sigma_{d}^{*}\) and \(\dot{\epsilon}\) should be eliminated in the calculation of the parameter C. The specific method is as follows: \(\sigma_{d}^{*}\) and the corresponding \(\dot{\epsilon}\) are taken when \(\sigma_{d}^{*}\) is approximately \(1/3\sigma_{d}^{*}\); then, the relationship between the obtained series of \(\sigma_{d}^{*}\) and \(\dot{\epsilon}\) is fit with a linear equation, the slope of which is the value of C. Here, the normalized dynamic compressive strength is introduced:  
\[\sigma_{d}^{*}=\frac{\sigma}{\sigma_{c}} \tag{28}\]  
where \(\sigma_{d}^{*}\) is the normalized dynamic compressive strength. The corresponding points of \(\sigma_{d}^{*}\) and \(\dot{\epsilon}\) for sandstone are shown in Table 6, and the fitting relationship is shown in Figure 17. The strain rate coefficient C is 0.0127. The reference strain rate EPSO is a fixed value, i.e., 1.0. \[PC=\frac{\sigma_{c}}{3} \tag{29}\]  
\[UC=\frac{PC}{K} \tag{30}\]  
\[UL=\frac{\rho_{grain}}{\rho}-1 \tag{31}\]  
where \(PC\) and \(UC\) are the hydrostatic pressure and volume strain corresponding to the elastic limit, respectively, and \(UL\) is the volumetric strain at plastic deformation. \(PC\), \(UC\) and \(UL\) are calculated to be 25.38, 0.0016, and 0.08, respectively. The value of \(PL\) is determined by \(\mu_{l}\) in Figure 11c, and \(K1\), \(K2\) and \(K3\) are determined by the stress amplitude and corresponding equivalent volumetric strain under impact loading. Therefore, it is necessary to carry out impact tests to obtain the fitting relationship by using Equation (32):  
\[p=k_{1}\overline{\mu}+k_{2}\overline{\mu}^{2}+k_{3}\overline{\mu}^{3} \tag{32}\]  
Figure 16: The relationship between \(p^{*}\) and \(\sigma^{*}\). \begin{table}
\begin{tabular}{c c c c} \hline \(A\) & \(B\) & \(N\) & \(SFMAX\) \\ \hline
0.3 & 1.95 & 0.54 & 1.2 \\ \hline \end{tabular}
\end{table} 
Table 5: The limit surface parameters of sandstone. The relationship between the stress amplitude and equivalent volumetric strain under impact loading is shown in Figure 18. The parameters related to the state equation are calculated as shown in Table 7. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Strain Rate \(\dot{\varepsilon}\) (s\({}^{-1}\))** & **Dynamic Compressive Strength (MPa)** & **Normalized Dynamic Compressive Strength \(\
u_{d}^{*}\)** \\ \hline
85.14 & 78.71 & 1.0 \\
90.43 & 80.32 & 1.1 \\
95.87 & 87.82 & 1.2 \\
87.96 & 95.98 & 1.3 \\
100.74 & 102.38 & 1.3 \\
102.74 & 96.79 & 1.2 \\
120.18 & 116.8 & 1.5 \\
130.06 & 125.14 & 1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Dynamic compressive strengths of sandstone under different strain rates. Figure 17: The relationship between \(\sigma_{d}^{*}\) and \(\dot{\varepsilon}\) for sandstone. Figure 18: The fitting relationship between \(p^{*}\) and \(\overline{\mu}\). The damage parameters include D1, D2 and EFMIN. The damage in the HJC model is determined by the equivalent plastic strain and volumetric strain. When determining the parameter values of D1 and D2, it is assumed that the material is only damaged under repeated impacts, and that the corresponding damage value is 1, so the following equation is obtained:  
\[\varepsilon_{p}^{f}+\mu_{p}^{f}=D_{1}(p^{*}+T^{*})^{D_{2}} \tag{33}\]  
where \(e_{p}^{f}+\mu_{p}^{f}\) is the fracture plastic strain under repeated impacts. EFMIN is the failure strain under repeated impacts. The specific method is to draw a common tangent of the peak strength of the constitutive curve cluster obtained under repeated impacts, and the strain value corresponding to the intersection of the common tangent and the abscissa axis is the failure strain under repeated impacts, as shown in Figure 19. The failure strain of sandstone is 0.00465. The relationship between the strain and damage is determined by Equation (32).