## 2 Statistical Thermal Damage Constitutive Modeling of Rocks
### Applicability of the Strength Criterion
The presence of soft rock such as sandstone and limestone is the main reason for large deformations in the tunnel. [PERSON] et al. [28] conducted a deformation study of soft rock based on the gray prediction method, and the control of different support schemes was analyzed by numerical simulation. Taking into account the engineering problems associated with the above typical rocks, granite [29], limestone [30], marble [31] and sandstone [32] were selected for this study. The corresponding test data and mechanical parameters are shown in Table 1. The friction angle can be obtained by fitting the perimeter pressure to the strength, the internal friction coefficient is the tangent of the internal friction angle, and \(\psi_{T}\) and \(m\sigma_{u}\) are obtained by fitting the experimental data in Table 1.  
As shown in Figure 2, the predicted strengths using the modified strength criterion in this paper have a good fit to experimental values of more than 0.95, and the strength criterion that takes into account the presence of initial cracks is more accurate in predicting rock strength compared with the original H-B strength criterion. Although the H-B strength criterion takes into account the effect of confining pressure to a certain extent, the predicted strengths are still higher to a certain extent than the experimental valuesunder high confining pressures, whereas the modified H-B strength criterion overcomes this shortcoming by taking into account the presence of initial cracks in the parameter acquisition process. The modified H-B strength criterion adopted in this paper solves the problem of the predicted stresses of rocks under high confining pressure being too large, which shows that the modified H-B strength criterion can better evaluate the strength properties of rocks. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Rock Type** & \begin{tabular}{c} **Confining** \\ **Pressure/MPa** \\ \end{tabular} & \
\begin{tabular}{c} **Peak** \\ **Strength/MPa** \\ \end{tabular} &
\begin{tabular}{c} **Internal Friction Angle/** \\ \end{tabular} & \(\varphi\) & \(\psi\tau\) & \(\mathbf{m\sigma\omega\)/MPa} & \(\mathbf{R^{2}}\) \\ \hline \multirow{4}{*}{Granite} & 0 & 116.852 & & & & & & \\  & 10 & 208.038 & & & & & & \\  & 20 & 301.143 & 47.36 & 1.08 & 8.493 & 2808.47 & 0.9571 \\  & 30 & 370.961 & & & & & & \\  & 40 & 364.571 & & & & & & \\ \hline \multirow{4}{*}{Limestone} & 0 & 65.525 & & & & & & & \\  & 2 & 77.088 & & & & & & \\  & 4 & 83.512 & & & & & & \\  & 5 & 88.653 & & & & & & \\  & 8 & 96.360 & 34.56 & 0.69 & 2.687 & 352.13 & 0.9738 \\  & 10 & 102.784 & & & & & & \\  & 15 & 112.206 & & & & & & \\  & 20 & 128.908 & & & & & & \\  & 30 & 144.754 & & & & & & \\ \hline \multirow{4}{*}{Marble} & 0 & 53.115 & & & & & & & \\  & 5 & 126.738 & & & & & & \\  & 10 & 173.766 & & & & & & \\  & 20 & 245.427 & & & & & & \\  & 30 & 303.524 & & & & & & \\  & 54 & 417.883 & & & & & & \\ \hline \multirow{4}{*}{Sandstone} & 0 & 55.69 & & & & & & & \\  & 5 & 84.57 & & & & & & \\ \cline{1-1}  & 10 & 107.37 & & & & & & \\ \cline{1-1}  & 15 & 125.43 & & & & & & \\ \cline{1-1}  & 25 & 156.80 & & & & & & \\ \cline{1-1}  & 35 & 186.43 & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of various rocks under different confining pressures. Figure 2: Relationship between strength criterion and test strength. The strength predictions of the H-B strength criterion and the modified H-B strength criterion under different confining pressures are shown in Figure 2.