## 2 Materials and Methods
### Calibration of RHT Model Parameters
#### 2.2.4 Material Damage Model Parameters  
The model damage curve is defined as:  
\[\sigma_{f}^{*}(P_{0}^{*},F_{r})= \begin{cases}A\left(P_{0}^{*}-F_{r}/+\left(A /F_{r}\right)^{-1/N}\right)^{N}&3P_{0}^{*}\geq F_{r}\\ F_{r}F_{s}^{*}/Q_{1}+3P_{0}^{*}(1-F_{s}^{*}/Q_{1})&F_{r}>3P_{0}^{*}\geq 0\\ F_{r}F_{s}^{*}/Q_{1}-3P_{0}^{*}(1/Q_{2}-F_{s}^{*}/(F_{t}^{*}Q_{1}))&0>3P_{0}^{ *}\geq 3P_{t}^{*}\\ 0&3P_{t}^{*}>3P_{0}^{*}\end{cases}  \tag{10}\]  
When the material is under quasi-static loading conditions, \(F_{r}\) takes the value of 1. Parameters A and N can be obtained from the expression for the damage surface at \(3P_{0}^{*}\geq F_{r}\). Thus, the failure surface expression when \(3P_{0}^{*}\geq F_{r}\) is as follows:  
\[\sigma_{f}^{*}(P_{0}^{*})=A\left(P_{0}^{*}-1/3+\left(A\right)^{-1/N}\right)^{N} 3P_{0}^{*}\geq F_{r}, \tag{11}\]  
where \(\sigma_{\text{f}}^{*}\) is the normalized strength, \(\sigma_{\text{f}}^{*}=\sigma_{\text{f}}/\sigma_{\text{c}}\), and A and N are the failure surface parameters. The results of normalized and Y are obtained from the experimental data of triaxial rock mechanics in Table 3, with P as \(x\)-axis and Y as \(y\)-axis, and the curve of mode 3 is obtained by linear fitting to obtain A = 2.4, N = 0.82, as shown in Figure 3.  
Figure 3: Linear fit to calibrated A and N parameters. The p and y in Table 3 are obtained from the following Equations (12) and (13) [30]:  
\[P_{0}^{*}=(\sigma_{1}+\sigma_{3})/3f_{c}, \tag{12}\]  
\[P_{0}^{*}=(\sigma_{1}-\sigma_{3})/f_{c}, \tag{13}\]  
In the RHT model, the damage variable is defined as the accumulation with plastic strain \(\varepsilon_{p}\):  
\[D=\sum\frac{d\varepsilon_{p}}{\varepsilon_{p}^{f}} \tag{14}\]  
where \(\varepsilon_{p}^{f}\) is the plastic strain at failure. When the stress state reaches the material's ultimate strength on the damage surface, damage accumulates gradually during inelastic deformation or plastic strain. In this model, the plastic strain at the point of damage is defined as [31]:  
\[\varepsilon_{p}^{f}=\left\{ \begin{matrix}D_{1}[P_{0}^{*}-(1-D)P_{t}^{*}]^{D_{2} }&P_{0}^{*}\geq(1-D)P_{t}^{*}+\left(\varepsilon_{p}^{m}/D_{1}\right)^{1/D_{2} }\\ \varepsilon_{p}^{m}&P_{0}^{*}<(1-D)P_{t}^{*}+\left(\varepsilon_{p}^{m}/D_{1} \right)^{1/D_{2}},\end{matrix} \right. \tag{15}\]  
where \(\varepsilon_{p}^{m}\) is the minimum damaged residual strain and D\({}_{1}\) and D\({}_{2}\) are the damage constants for the RHT model. The variables are based on the findings in the literature [32], D\({}_{1}\) = 0.04, D\({}_{2}\) = 1.0. In the RHT model, the maximum reduction in strength at the stretching meridian is described by the factor of the following equation:  
\[\text{Q}=\text{Q}_{0}+\text{B}_{\text{Q}}\text{P}_{0}^{*}, \tag{16}\]  
where Q\({}_{0}\) and B\({}_{\text{Q}}\) are introduced as lode angle-dependence factors to describe the maximum reduction in strength on the tensile meridian. Thus, the tensile strength can be obtained by multiplying Equations (1) and (2). Q\({}_{0}\) and B\({}_{\text{Q}}\) values can be obtained from the results of biaxial compression or triaxial tension. In this study, Q\({}_{0}\) = 0.68 and B\({}_{\text{Q}}\) = 0.0105 were determined using the method of Li [33]. In summary, the mineral rock RHT parameters are calibrated as shown in Table 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline Mass density RO (kg/m\({}^{3}\)) & 3100 & Elastic shear modulus SHEAR (GPa) & 21.76 \\ \hline Compressive strength FC (MPa) & 116.3 & Erosion plastic strain EPSF & 2.0 \\ \hline Compressive strain rate dependence exponent BETAC & 0.011 & Relative tensile strength FT\({}^{*}\) & 0.052 \\ \hline Compressive yield surface parameter GC\({}^{*}\) & 0.53 & Relative shear strength FS\({}^{*}\) & 0.187 \\ \hline Crush pressure PEL (MPa) & 77.53 & Residual surface parameter AF & 1.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RHT model parameters.