## 5 Discussion of Failure Criteria of the Rock Based on Abrupt Changes in Energy  
From a mechanical standpoint, rocks are comprised of natural geomaterials riddled with imperfections, including pre-existing micro-cracks. The process of altering and breaking down under applied stress involves the growth of local fractures from these micro-cracks into fracture planes, eventually merging to trigger the widespread instability of the surrounding rock mass. This failure mechanism of rocks is intrinsically linked to the dynamics of energy conversion and represents an energy-induced instability. When subjected to external loads, energy is expended in the progression and merging of cracks, as well as in the creation of macroscopic fracture planes, particularly near the peak stress phase, encompassing the stages of pore or micro-crack compaction and both elastic and plastic deformations. To measure the degree of damage within the rock, the ratio of energy, Ke (characterized by the proportion of energy expended to the energy conserved during the loading phase of the rock), is utilized. This metric serves as an indirect indicator of the rock's stability under the applied load [49; 50; 51]. Ke is calculated as follows:  
\[K_{e}=\frac{u_{d}}{u_{e}} \tag{5}\]  
where \(u_{d}\) represents the energy lost through the beginning and the spread of cracks through the rock, and \(u_{e}\) signifies the elastic deformation energy contained within the rock, encompassing both axial and radial deformations. Illustrated by Figures 13 and 14, during the CLU trials at varying confining pressures, the energy consumption ratio Ke initially experiences a sharp decrease (stage I), followed by a gradual slowdown in reduction (stage II), a slight rise (stage III), a significant increase at the peak stress inflection point (stage IV) and, ultimately, a slow decline towards stabilization (stage V) as the cycle count rises. Stage I is defined by the densification of pores and the closure of microscopic fractures; stage II involves the process of elastic deformation; stage III involves the extension and initiation of new cracks; stage IV is marked by the coalescence of cracks into a macroscopic fracture plane; and stage V shows that despite fracturing, the rock specimens maintain a level of load-bearing capacity under the confining pressure. The analysis of the rock specimens' energy evolution during the CLU in Section 4.2 suggests that rock failure under load is defined by the rapid conversion of stored elastic strain energy into energy that is released and dissipated as cracks propagate, merge, and form fracture planes. Thus, the phase where the Ke ratio incrementally rises indicates the progression of the accrual of damage and plastic deformation within the rock. When Ke reaches a threshold and shows an inflection point, cracks propagate and coalesce, the main crack appears, and fracture planes are formed. Hence, this inflection point serves as a precursor signal for the impending failure of the rock when subjected to loading. The ratio of ud to we was used as the amplified energy signal for failure of the rock, and its abrupt change can be used as a failure criterion for the rock. Figure 13: _Cont._Figure 14: Variation in energy dissipation ratios in rock samples with strain under various confining pressures. Figure 13: Stress–strain–energy dissipation ratio curves for rock samples subjected to triaxial CLU tests. Ke \(<1\) indicates that the rock, during its loading phase, retains more accumulated elastic strain energy than the energy it releases and dissipates, signifying a state of stability. This condition is typically associated with the elastic and plastic deformation phases that occur prior to the peak, predominantly during the initial stages of deformation and the residual deformation phase. A Ke value of 1 signals a critical equilibrium state for the rock, usually observed around the peak stress level or within the residual deformation stage.