## 3 Parameter Determination Method for the JH-2 Model
### Determination of EOS Parameters  
As shown in Figure 2b and Equation (8), \(K2\) and \(K3\) should be calculated from the relationship between hydrostatic pressure, \(P\), and volumetric strain (\(\mu=\rho/\rho_{0}-1\)) under dynamic loading with high pressure levels. Therefore, the Hugoniot relation is introduced, which is expressed as the curve of shock wave velocity, \(u_{S}\), and particle velocity, \(u_{P}\), by  
\[u_{S}=S\cdot u_{P}+C\;(\mathrm{m/s}) \tag{17}\]  
where \(S\) is the fitting slope of the straight line and \(C\) is the bulk sound velocity.  
The Hugoniot equations for the conservation of mass and momentum [43; 44] can be expressed as Conservation of mass  
\[\rho_{0}u_{S}=\rho(u_{S}-u_{P}) \tag{18}\]  
Conservation of momentum  
\[P=\rho u_{P}(u_{S}-u_{P})=\rho_{0}u_{S}u_{P} \tag{19}\]  
Combining Equation (17) with the Hugoniot equations for the conservation of mass (Equation (18)) and momentum (Equation (19)) to determine the shock Hugoniot in the \(P-\mu\) plane as  
\[P=\frac{\rho_{0}C^{2}\mu(1+\mu)}{\left(1-(S-1)\mu\right)^{2}} \tag{20}\]  
The bulk sound velocity \(C\) is given by  
\[C=\sqrt{V_{P}^{2}-\frac{4}{3}V_{S}^{2}}=3.088\;\mathrm{km/s} \tag{21}\]  
Substituting the previously obtained values of \(\rho_{0}\), \(C\), \(P_{HEL}\), and \(\mu_{HEL}\) of Barre granite into Equation (20), gives \(S=5.93\). Therefore, for Barre granite in this example, Equation (17) can be transformed as  
\[u_{S}^{\prime}=S\cdot u_{P}^{\prime}+C=5.93u_{P}^{\prime}+3088\;(\mathrm{m/s}) \tag{22}\]  
Ideally, it would be better to have shock velocity and particle velocity data from the flyer plate impact tests for Barre granite so that they could be used to fit the linear relationship of Equation (22). However, if there is insufficient data for depicting the relationship, the new extrapolated method above can be chosen in most cases.  
Consequently, Equation (20) can be transformed as  
\[P=\frac{2.66\times 3.088^{2}\times\mu(1+\mu)}{\left(1-(5.93-1)\mu\right)^{2}} \tag{23}\]  
By assuming several sets of \(P\) and then calculating the corresponding \(\mu\) according to Equation (23), the values shown in Figure 3 are used to fit the values of \(K2\) and \(K3\) in Equation (8) when \(D=0\). The obtained fitting values are \(K2=-386\) GPa and \(K3=12800\) GPa, and the fitting curve is presented in Figure 3.  
[PERSON] [29] proposed 2D numerical simulation adjustment to determine the constant values of \(K2\) and \(K3\) (\(K1\) is the bulk modulus previously presented), in which 16 sets of constants in different values of \(K2\) and \(K3\) were assumed for the polynomial EOS. Through the comparison between the results of pressure-distance curves from lab-scale experiments and simulations, the best-fitting values of the constants are obtained as \(K2=-4500\) GPa and \(K3=300000\) GPa. However, it is highly demanding to complete these sets of numerical simulations, and the values of \(K2\) and \(K3\) fitted by the results from 2D numerical simulations may have a large deviation, which is not appropriate for use in 3D numerical simulations. Due to different from static problems, the propagation characteristics of stress waves in rock dynamics are complicated and involve some boundary conditions such as the transmission and reflection at boundaries or discontinuities. Therefore, unlike practical experiments, 2D models are prone to generate extra reflection and overlapping of stress waves at artificial boundaries.