## 1 Introduction  
The failure mode of three-dimensional stress rock mass is a critical parameter for predicting the stability of geological structures, including rock slopes, dam foundations, deep tunnels, and roadways [1, 2]. Deep mining causes intense readjustment of mining stress, leading to concentrated stress redistribution on the fault plane. Understanding the failure modes and energy evolution mechanisms in rock masses under high-stress conditions is crucial for reducing accident rates. Consequently, scholars, including [PERSON], have extensively researched the mechanical properties of deep rock masses based on triaxial compression tests [3]. [PERSON] et al. explored how fillers impact the deformation characteristics, mechanical properties, and failure modes of rock masses [4]. [PERSON] et al. [5] assessed how the loading rate affects the peak strength and post-failure resilience of rock masses under confining pressure, providing insights into the long-term deformation and stability of underground structures. [PERSON] et al. [6] found that confining pressure significantly influences the shear expansion of rock masses, noting that under low pressures, rock masses compress before expanding, whereas under high pressures, they predominantly undergo shear compression. [PERSON] et al. [7] performed triaxial compression tests on rock specimens at various temperatures and confining pressures, focusing on how these factors affect rock mechanics. However, these theories and experiments rely on conventional triaxial stress (\(\sigma_{1}>\sigma_{2}=\sigma_{3}\)) and do not accurately represent the actual three-dimensional stress state, neglecting the significant influence of intermediate principal stress \(\sigma_{2}\) on rock deformation and strength. Numerous in situ stress measurements demonstrate significant anisotropy in stress from shallow to mid-depth levels (\(\sigma_{1}>\sigma_{2}>\sigma_{3}\)) [8; 9; 10; 11]. The confirmed influence of \(\sigma_{2}\) on the compressive strength of jointed rock mass underscores the importance of conducting physical and mechanical property tests under true triaxial stress conditions. Since Mogi developed the first genuine rock triaxial testing machine in 1980 [12], researchers including [PERSON] et al. and [PERSON] et al. have extensively studied marble [13; 14], limestone, shale, sandstone, and granite using triaxial testing equipment [15]. Research indicates that the splitting or shear failure of marble is influenced by the intermediate and minimum principal stresses [16]. True triaxial experiments on gray sandstone specimens showed that adjusting the \(\sigma_{2}/\sigma_{3}\) ratio can increase strength by 310%. [PERSON] et al. [17] examined how intermediate principal stress affects crack morphology, stress-strain responses, and acoustic emission (AE) activity. They proposed a damage model that accounts for both micro- and macro-damages--micro-cracks and pre-existing defects, respectively--to elucidate the failure characteristics of rock specimens with a single defect under true triaxial compression. Building on the linear and Coulomb strength criteria for marble, [PERSON] et al. [15] developed a refined true triaxial strength criterion that considers the effects of joint orientation and the intermediate and minimum principal stresses, establishing the relationship between failure mode, joint inclination, and intermediate principal stress. During the coal mining process, changes in the three-dimensional stress within the rock mass can easily trigger disasters such as rock bursts, activation of floor structures, and water intrushes. Deep mining in high-stress environments more readily induces fault slips, leading to the formation and expansion of water-conducting fissures and subsequent water intrush channels at the working face, ultimately causing disasters.