## 1 Introduction
Owing to its simple mathematical form and ease of parameter determination, it has become one of the most widely used models in rock and soil mechanics. It is commonly applied in slope stability, tunneling, wellbore integrity, hydraulic fracturing, and mining excavation analyses. Despite its simplicity, the MC criterion provides valuable insights into the failure behavior of geomaterials across a broad range of geological and engineering contexts [20; 21; 22]. [PERSON] et al. employed the MC failure criterion to investigate the relationships between stress-strain curves, stress ratio, deviatoric stress, and the ratio of volumetric strain to axial strain. Based on numerical simulations, they successfully reproduced the rock failure mechanisms observed in uniaxial and triaxial compression tests, enabling the rapid evaluation of rock strength under different loading conditions. This approach effectively reduced both the time and economic costs associated with rock mechanical strength assessment in mining engineering [23]. [PERSON] et al. proposed a power law-based MC criterion for deep Longmaxi Formation shale and evaluated its reliability, sensitivity, and applicability. Their work provides important references and theoretical support for drilling, completion, and production enhancement in deep shale gas reservoirs [24]. However, as the depth of underground engineering continues to increase, rock mechanics test data indicate that the failure envelope of rocks under high confining pressure is not a linear line, and the MC failure criterion shows significant errors under high confining pressure conditions [25; 26; 27]. Additionally, some rock failure criteria are developed for the specific rocks taking the special properties of the rocks, such as shale considering the anisotropy and hydration of shale [28; 29] and carbonatite considering the hole and pore heterogeneous distribution [30]. Although there are many existing rock failure criteria, few of them can be used for the rock under high-temperature and high-pressure conditions. Under high-temperature conditions, the mismatch in thermal expansion between different minerals in the rock induces shear stresses, which, in turn, trigger the initiation and propagation of microcracks. In addition, cementing materials such as carbonates or clay minerals are prone to thermal decomposition or dehydration at elevated temperatures, leading to a reduction in intergranular bonding strength and compromising the structural integrity of the rock. Moreover, under high confining pressure, the lateral constraint alters the failure mode of the rock from brittle to ductile behavior, and its strength characteristics exhibit pronounced nonlinearity. Scholars have proposed various forms of nonlinear rock failure criteria, which can be categorized into hyperbolic [31], parabolic [32], elliptical [33], and exponential [34; 35] failure criteria. Although these nonlinear failure criteria can address the limitations of classical failure criteria under high confining pressure, most of them are empirical formulas that are only applicable to specific regions and rock types. Furthermore, the existing failure criteria rarely account for the effects of temperature, assuming that the nonlinear characteristics of rock strength envelopes in deep formations are solely due to high confining pressure while neglecting the impact of high temperature on rock mechanical properties. In summary, the current classical rock failure criteria are only applicable under low confining pressure conditions, while nonlinear failure criteria neglect the impact of temperature on rock mechanical properties, making it difficult to accurately assess rock strength under ultra-deep high-temperature and high-confining pressure conditions. As a result, they cannot be widely applied in the wellbore stability evaluation of ultra-deep reservoir oil and gas resources.