## 2 A Nonlinear MC Failure Criterion Considering High-Temperature and High-Pressure Conditions  
### Nonlinear MC Failure Criterion Under High Stress  
The mechanical properties of rock masses and the stress environment are the key factors leading to rock deformation and failure in underground space engineering, such as mining, drilling, and tunnel excavation [12]. To accurately determine rocks' strength in different formations, various failure criteria have been proposed by scholars. Among them, the MC failure criterion, due to its simplicity and practicality, has become one of the most widely used rock strength theories in geotechnical engineering [17]. This criterion characterizes rock shear strength through cohesion and the internal friction angle, effectively describing failure modes under different stress paths, particularly demonstrating significant advantages in the strength analysis of various rock masses. The MC failure criterion is expressed as follows [36]:  
\[\sigma_{1}=\frac{2c\cos\varphi}{1-\sin\varphi}+\frac{1+\sin\varphi}{1-\sin \varphi}\sigma_{3} \tag{1}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the major and minor principal stresses applying on the rock, respectively, MPA; \(c\) is the rock cohesion, MPa; \(\varphi\) is the rock internal friction angle, \({}^{\circ}\).  
Many research results have shown that, as the confining pressure increases, the failure mode of rocks gradually transitions from brittle failure to ductile failure. With the increase in confining pressure, the slope of the Mohr strength envelope decreases gradually, indicating that the rate of increase in shear strength diminishes and eventually approaches zero (as shown in Figure 1) [37; 38]. However, the MC criterion assumes that cohesion (\(c\)) and the internal friction angle (\(\varphi\)) are constants, which only allows for a linear representation of rock strength variations with confining pressure. It fails to capture the dynamic process of the transition from brittle to ductile failure as the confining pressure increases. To overcome the limitations of the linear MC criterion in predicting rock strength under high confining pressure conditions, many scholars have proposed improvements to the MC criterion. A representative study by [PERSON] et al. [38] introduced a quadratic function of confining pressure to develop a modified MC criterion model that describes the nonlinear coupling relationship between confining pressure and strength. [PERSON] et al. [38] conducted triaxial compression tests on 158 rock samples of various types. Through the experimental data analysis, they found that, as the confining pressure increased, the rock strength gradually increased, but the rate of increase diminished. When the confining pressure reached the uniaxial compressive strength, the rate of increase in the rock strength approached zero. Therefore, the mathematical expression of the modified MC failure criterion is as follows:  
\[\left\{\begin{array}{l}(\sigma_{1}-\sigma_{3})=\frac{2\cos\varphi}{1-\sin \varphi}+\frac{2\sin\varphi}{1-\sin\varphi}\sigma_{3}-\frac{1}{\sigma_{c}} \cdot\frac{\sin\varphi}{1-\sin\varphi}\sigma_{3}^{2},\ 0\leq\sigma_{3}\leq\sigma_{c}\\ (\sigma_{1}-\sigma_{3})_{\max}=\frac{2c\cos\varphi}{(1-\sin\varphi)^{2}},\ \sigma_{3}\geq \sigma_{c}\end{array}\right. \tag{2}\]  
where \(\sigma_{c}\) is the uniaxial compressive strength of the rock, MPa.  
Figure 1: Rock strength envelope. (**a**) Indiana limestone. (**b**) Daye marble [36; 37].