## 4 Parameter Determination of the HJC Model for Sandstone and Its Application
### Parameter Determination Method of the HJC Model for Sandstone
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Implication** & **Category** & **Parameter** & **Implication** & **Category** \\ \hline RO & Mass density & & PC & Crushing pressure \\ \hline FC & Quasi-static uniaxial & & UC & Crushing volumetric strain \\  & compressive strength & Basic mechanical & & PL & Locking pressure \\ \hline G & Shear modulus & & UL & Locking volumetric strain \\ \hline T & Maximum tensile & & UL & Locking volumetric strain & Pressure \\  & hydrostatic pressure & & K1 & Pressure constant & parameters \\ \hline A & Normalized cohesive & & K2 & Pressure constant & \\  & strength & Limit surface & & K3 & Pressure constant \\ \hline B & Normalized & & D1 & Damage constant & \\  & pressure-hardening & & D2 & Damage constant & Damage \\ \hline N & Pressure-hardening & & EMIN & Amount of plastic strain & \\  & exponent & & & before fracture & \\ \hline SFMAX & Normalized maximum & & & & \\  & strength & & & & \\ \hline C & Strain rate coefficient & Strain rate & D2 & Damage constant & Damage \\ \hline EPSO & Quasi-static threshold & dependent & & & \\  & strain rate & parameters & EMIN & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classification of HJC model parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**RO (kg/m3)** & **FC (MPa)** & **G (GPa)** & **T (MPa)** & \(\mu\) & **E (GPa)** & **K (GPa)** \\ \hline
2610 & 76.13 & 10.49 & 7.63 & 0.22 & 25.60 & 15.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The basic mechanical parameters of sandstone. The limit surface of the HJC model and the envelope of the M-C criterion pass through the same point (0, \(C\)):  
\[\frac{C}{\sigma_{c}}=A \tag{21}\]  
Thus, the value of \(A\) is 0.297. In the triaxial compression test, the failure strength increases with increasing confining pressure. After a series of triaxial tests under different confining pressures, a set of \(p^{*}\) and \(\sigma^{*}\) values can be obtained using Equations (22)-(25). \[p=\frac{\sigma_{n}+2\sigma_{r}}{3} \tag{22}\]  
\[\Delta\sigma=\sigma_{n}-\sigma_{r} \tag{23}\]  
\[p^{*}=\frac{p}{f_{c}} \tag{24}\]  
\[\sigma^{*}=\frac{\Delta\sigma}{f_{c}} \tag{25}\]  
where \(\sigma_{n}\) is the axial pressure corresponding to sandstone failure in the triaxial test; \(\sigma_{r}\) is the confining pressure; and \(p^{*}\) and \(\sigma^{*}\) are the normalized hydrostatic pressure and normalized stress difference, respectively, as shown in Table 4. Figure 16 shows the relationship between \(p^{*}\) and \(\sigma^{*}\) by using Equation (26):  
\[\sigma^{*}=B\cdot p^{*}N \tag{26}\]  
Figure 15 shows that the values of \(B\) and \(N\) are 1.947 and 0.537, respectively. \(SFMAX\) is the ratio of the stress difference when the axial compressive strength of the specimen does not increase with increasing confining pressure from the triaxial test to the static compressive strength:  
\[SFMAX=\frac{\Delta\sigma_{\text{max}}}{\sigma_{c}} \tag{27}\]  
According to the triaxial compression test results, the axial pressure remains stable when the confining pressure is approximately 90 MPa. The limit surface parameters of sandstone are shown in Table 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(p\) & \(\Delta\sigma\) & \(p^{*}\) & \(\sigma^{*}\) \\ \hline
33.8 & 93.87 & 0.44 & 1.23 \\
40.1 & 105.40 & 0.53 & 1.38 \\
43.0 & 110.86 & 0.56 & 1.46 \\
46.0 & 115.51 & 0.60 & 1.52 \\
49.0 & 117.26 & 0.65 & 1.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The values of \(p\), \(\Delta\sigma\), \(p^{*}\) and \(\sigma^{*}\) under different confining pressures. \begin{table}
\begin{tabular}{c c} \hline \hline
**Confining Pressure** & **Axial Pressure** \\ \hline
2.5 & 96 \\
5.0 & 110 \\
6.0 & 116 \\
7.5 & 123 \\
10.0 & 127 \\
12.5 & 129 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The axial compressive strength under different confining pressures. The strain-rate-dependent parameters include C and EPSO. The parameter C is obtained by using a single impact test under different strain rates.