## 2 A Nonlinear MC Failure Criterion Considering High-Temperature and High-Pressure Conditions
### The Nonlinear MC Failure Criterion  
In the 158 triaxial compression test database used for statistical analysis by [PERSON] et al. [38], 53 datasets from sandstone accounted for 33.54% of the total test data. Therefore, the conclusion that the critical confining pressure equals the uniaxial compressive strength may not apply to all rock types [38]. For example, [PERSON] et al. [39] found that Solnhofen limestone enters the critical state when the confining pressure is 0.34\(\sigma_{\rm c}\), while Bent sandstone does not enter the critical state until the confining pressure reaches 1.5\(\sigma_{\rm c}\). Considering that the MC criterion still provides relatively accurate predictions for various rock types under low confining pressure, this study follows the approach of [PERSON] et al. [38] and introduces a correction term based on the MC criterion. However, it assumes that the critical confining pressure under constant compressive strength is \(n\sigma_{\rm c}\) rather than \(\sigma_{\rm c}\). Its mathematical form is as follows:  
\[(\sigma_{1}-\sigma_{3})=\frac{2c\cos\varphi}{1-\sin\varphi}+\frac{2\sin \varphi}{1-\sin\varphi}\sigma_{3}-A\sigma_{3}{}^{2}\quad 0\leq\sigma_{3} \leq n\sigma_{\rm c} \tag{3}\]  
where \(A\) and \(n\) are parameters related to rock type, while the meanings of the other parameters are the same as those in the MC criterion.  
When the confining pressure reaches the critical condition, i.e., \(\sigma_{3}=n\sigma_{\rm c}\), Equation (3) should satisfy the following conditions:  
\[\frac{\partial(\sigma_{1}-\sigma_{3})}{\partial\sigma_{3}}=0 \tag{4}\]  
By solving Equation (4), the following can be obtained:  
\[A=\frac{2\sin\varphi}{2n\sigma_{\rm c}(1-\sin\varphi)} \tag{5}\]  
In addition, the failure criterion proposed by [PERSON] et al. [38] neglects the impact of temperature on rock mechanical properties and thus does not perform well in predicting rock strength in deep high-temperature and high-pressure environments. [PERSON] et al. [40] analyzed the shear characteristics of saw-cut smooth granite surfaces under various thermal-hydro-mechanical conditions based on the Mohr-Coulomb failure criterion and found that the shear strength of granite gradually decreases with the increasing temperature. The study by [PERSON] et al. [41] on the mechanical properties of sandstone at elevated temperatures also indicated a linear relationship between compressive strength and temperature, with compressive strength decreasing as the temperature increases. Accordingly, a linear model describing the relationship between temperature and rock strength is established as follows:  
\[\sigma^{T}=B\Delta T+C \tag{6}\]  
where \(\sigma^{T}\) is the temperature-dependent rock strength, MPa; \(\Delta T\) is the temperature difference between the actual temperature and room temperature, \({}^{\circ}\)C.  
We assume that the effect of temperature on the rock mechanical properties is independent of the confining pressure. Therefore, a linear relationship model between temperature and the rock uniaxial compressive strength can be established to represent the temperature-dependent reduction in rock strength. The mathematical form is as follows:  
\[\Delta\sigma_{\rm c}^{T}=\sigma_{\rm c}-\sigma^{T}=\sigma_{\rm c}-B\Delta T-C \tag{7}\]  
Additionally, from Equation (6), it is known that, when \(\Delta T=0\), \(\Delta\sigma_{\rm c}^{T}=0\). Therefore, it can be concluded that C = \(\sigma_{\rm c}\). By substituting this into Equation (7), the following can be obtained:  
\[\Delta\sigma_{\rm c}^{T}=B\Delta T \tag{8}\]Therefore, the nonlinear MC failure criterion (abbreviated as HTP-MC criterion) can be obtained by combining Equations (3), (5) and (8), as shown in Equation (9):  
\[(\sigma_{1}-\sigma_{3})=\frac{2c\cos\varphi}{1-\sin\varphi}+\frac{2\sin\varphi}{ 1-\sin\varphi}\sigma_{3}-\frac{2\sin\varphi}{2n\sigma_{c}(1-\sin\varphi)}{ \sigma_{3}}^{2}-B\Delta T\quad 0\leq\sigma_{3}\leq n\sigma_{c} \tag{9}\]