## 4 Parameter Determination of the HJC Model for Sandstone and Its Application  
### Parameter Determination Method of the HJC Model for Sandstone  
The HJC constitutive model proposed by [PERSON] et al. contains 21 parameters. The value of the parameter has a great influence on the calculation accuracy. After the HJC model was proposed, scholars introduced the numerical calculation of rock dynamics but did not propose a set of test methods for the determination of the parameters for specific rocks. The HJC model includes three parts: the equation of the yield surface, the equation of damage evolution and the equation of the state, as shown in Figure 12. The equation of the yield surface is given as:  
\[\sigma^{*}=\left[\mathrm{A}(1-\mathrm{D})+\mathrm{B}\mathrm{p}^{*N}\right] \left[1+Cln\left(\dot{\epsilon}^{*}\right)\right] \tag{17}\]  
where \  
parameter; \(B\) is the normalized pressure-hardening coefficient; \(C\) is the strain rate coefficient; and \(N\) is the pressure-hardening exponent. The model incrementally accumulates damage, \(D\), from both the equivalent plastic strain and plastic volumetric strain, which is expressed as:  
\[D=\sum\frac{\Delta\varepsilon_{p}+\Delta\mu_{p}}{\varepsilon_{p}{}^{f}+\mu_{p}{ }^{f}}=\sum\frac{\Delta\varepsilon_{p}+\Delta\mu_{p}}{D_{1}(p^{*}+T^{*})^{D_{2}}} \tag{18}\]  
where \(\Delta\varepsilon_{p}\) and \(\Delta\mu_{p}\) are the equivalent plastic strain and plastic volumetric strain, respectively, during a cycle of integration; \(T^{*}=T/\sigma_{c}\) is the normalized maximum tensile hydrostatic pressure, where \(T\) is the maximum tensile hydrostatic pressure the material can withstand and \(D_{1}\) and \(D_{2}\) are material constants. The equation of the state is as follows:  
\[p=\left\{ \begin{array}{l}k_{1}\mu_{p}\leq P_{c}\\ \frac{P_{c}-P_{c}}{\mu_{c}-\mu_{l}}+P_{c},P_{c}\leq P\leq P_{l}\\ k_{1}\overline{\mu}+k_{2}\overline{\mu}^{2}+k_{3}\overline{\mu}^{3},P\geq P_{l }\end{array} \right. \tag{19}\]  
Figure 12: The HJC model. where \(\mu=\rho/\rho_{0}\) is the volumetric strain; \(\rho\) and \(\rho_{0}\) are the current density and initial density, respectively; \(k=P_{c}/\mu_{c}\) is the elastic bulk modulus; \(P_{c}\) and \(\mu_{c}\) are the pressure and volumetric strain, respectively, when the material begins to undergo plastic deformation; \(\mu_{l}\) and \(P_{l}\) are the volumetric strain and pressure, respectively, when the air voids are completely removed from the material; \(\overline{\mu}=\left(\mu-\mu_{l}\right)/\left(1+\mu_{l}\right)\) is the modified volumetric strain; \(\mu_{l}\) is the volumetric strain when the density \(\rho\) reaches the grain density \(\rho_{grain}\); \(k_{1}\), \(k_{2}\) and \(k_{3}\) are material constants. As can be seen from the above introduction, the HJC model parameters can be divided into five categories, namely the basic mechanical parameters, limit surface parameters, pressure parameters, damage parameters and strain-rate-dependent parameters, as shown in Table 1. The test methods for the value of each parameter include the uniaxial compression test, triaxial compression test, single impact test and repeated impact test. The basic mechanical parameters are RO, FC, G and T, which can be measured using uniaxial compression tests. The static stress-strain curves of sandstone specimens are shown in Figure 5. According to the basic theory of rock mechanics, the mechanical parameters can be obtained as shown in Table 2.  
The limit surface parameters include A, B, N and SFMAX. The influence of the damage and strain rate is not considered in the determination of the limit surface parameters; thus,  
\[\sigma^{*}=A+B\cdot p^{*N} \tag{20}\]  
The limit surface of the HJC model and the envelope of the Mohr-Coulomb (M-C) criterion start from the same point and have an intersection, as shown in Figure 13. In the M-C criterion, the intercept C of the envelope is the cohesion, which is obtained via a series of triaxial compression tests. Therefore, the triaxial compression tests of sandstone