## 4 Discussion
### Sensitivity Analysis of the Parameters on the Predicted Results  
To evaluate the contribution of each parameter in the proposed failure criterion to the prediction results, a sensitivity analysis was conducted. The proposed criterion involves four parameters. Among them, the cohesion \(c\) and internal friction angle \(\varphi\) are defined in the same way as in the classical MC criterion and are therefore not discussed in detail. The focus of the analysis is on the sensitivity of parameters \(n\) and \(B\). For the sensitivity analysis, all other parameters were held constant, as shown in Table 2. Starting from their initial values, parameters \(n\) and \(B\) were incrementally increased by 20% at each step. The corresponding predicted \(\sigma_{1}\) and the rate of change in \(\sigma_{1}\) were recorded and used as indicators for the comparative sensitivity analysis. Figure 4 presents the sensitivity analysis of parameter \(n\) on the strength prediction results for three types of rock. In this analysis, the confining pressures were set to 10 MPa, 30 MPa, and 60 MPa, respectively. As shown in Equation (9), all predicted \(\sigma_{1}\) increase with the increasing \(n\), indicating a positive correlation between \(n\) and \(\sigma_{1}\). Parameter \(n\) is associated with the critical confining pressure of the rock: a higher \(n\) implies a higher critical confining pressure at which the rate of strength increases with respect to the confining pressure, which tends to zero. This also reflects a greater critical compressive strength of the rock. Although all three rock types exhibit an increasing trend in the rate of change of predicted \(\sigma_{1}\) with the increasing confining pressure, the rate of increase differs significantly among them. The most pronounced difference in sensitivity to \(n\) is observed between granite and shale. When \(n\) increases by 2.0, the predicted \(\sigma_{1}\) for granite increases by 0.31%, 2.18%, and 6.07% under confining pressures of 10 MPa, 30 MPa, and 60 MPa, respectively. In contrast, for shale, the corresponding increases are 5.13%, 14.82%, and 25.01%. Figure 5 illustrates the sensitivity analysis of parameter \(B\) on the strength prediction results for the three types of rock. Negative values indicate that the predicted \(\sigma_{1}\) decreases with the increasing \(B\), which is consistent with Equation (9), showing a negative correlation between \(B\) and \(\sigma_{1}\). In contrast to parameter \(n\), all three rock types exhibit a decreasing trend in the rate of change of predicted \(\sigma_{1}\) with the increasing confining pressure. This is because the model assumes that the strength reduction caused by temperature is independent of the confining pressure, and rocks tend to have higher initial strength under higher confining pressure. Furthermore, the sensitivity of predicted \(\sigma_{1}\) to \(B\) varies significantly among different rock types. Granite shows considerably lower sensitivity to \(B\) compared to shale and carbonate rock. For example, when \(B\) increases by 2.0 under confining pressures of 10 MPa, 30 MPa, and 60 MPa, the predicted \(\sigma_{1}\) for granite decreases by 8.12%, 5.37%, and 3.74%, respectively. In contrast, for shale, the decreases are 29.80%, 16.04%, and 13.54%, and for carbonate rock, they are 28.54%, 18.13%, and 11.17%, respectively. Figure 4: Sensitivity analysis of \(n\) on the predicted \(\sigma_{1}\) for different rock types. Figure 5: Sensitivity analysis of \(B\) on the predicted \(\sigma_{1}\) for different rock types. In summary, the sensitivity of predicted \(\sigma_{1}\) to parameters \(n\) and \(B\) varies significantly across different rock types. Among them, granite exhibits the lowest sensitivity to both parameters, indicating that the proposed HTP-MC criterion has better adaptability for predicting the strength of granite. This observation is consistent with the results shown in Figure 3c. Moreover, the predicted \(\sigma_{1}\) is generally more sensitive to parameter \(B\) than to \(n\).