## 4 Analysis of Micro-Scale Failure Evolution of Deep Granite Based on IET Function  
A large number of microcracks are randomly distributed in natural granite. When an external load is applied to the granite sample, the primary microcracks cause local stress concentration inside the sample, which further promotes the occurrence and propagation of internal microcracks. The expansion and aggregation of microcracks lead to the production of macroscopic cracks and local failure surfaces until the failure produces a large number of AE signals. These signals contain the evolution information of internal failure and fracture in the process of rock deformation and failure [20; 30]. As all the samples were obtained from the same core, their stress-strain curves indicate that the axial stress-strain relationship and the AE event curve of all samples under different confining pressures shows similar trends. This reflects that the distribution of microcracks in the deep granite samples is basically similar, with no remarkable difference between the samples. According to the deformation characteristics, the loading process of rock can be classified into five stages: (1) crack closure, (2) linear elastic deformation, (3) crack initiation and stable crack growth, (4) critical energy release and unstable crack growth, and (5) failure and post-peak behavior [31; 32]. During rock loading, the AE event rate of each fracture level differs considerably. The AE event monitoring method has been widely used to classify and verify the degree of rock cracking [33; 34]. To further explore the evolutionary relationship between the AE event number and the fracture degree, this study adopts the IET function \(F(\tau)\) proposed by [PERSON] and [PERSON] [35; 36]. The calculation method is as follows:  
(1) Define a sliding time window with \(N\) impact event intervals  
\[(t_{i-1},t_{i+N-1}) \tag{1}\]  
(2) Calculate the moving average of \(N\) event intervals in the specified time window:  
\[\tau_{i}=\frac{t_{i+N-1}-t_{i-1}}{N}(i=2,3,\ldots) \tag{2}\]  
Figure 3: SEM scanning image. (**a**) A1, (**b**) B1, (**c**) C1. special:  
\[\tau_{1}=\frac{t_{N}-t_{1}}{N}(i=1) \tag{3}\]  
(3) For a given time window:  
\[F(\tau)=\tau_{\rm i}^{-1}(i=1,2,3,\ldots) \tag{4}\]  
In this test analysis, we adopted \(n\) = 50, as explained by [PERSON] and [PERSON]. Here, \(N\) represents the resolution of AE characteristics and does not majorly affect the analysis results [36]. Figure 4 shows the stress-strain-\(F(\tau)\) evolution curves for the three granite samples. For sample A1, the value of the event interval function is extremely low and the curve is close to the horizontal axis within 0-2087 s, implying less AE activity. According to [PERSON] [31], a small amount of AE activity is mainly caused by the compaction of primary pores or shear slip on the primary failure surface. Within 784-1869 s, the stress curve of the rock sample is approximately a straight line, indicating that the rock is in the elastic loading stage. The microcracks close and destroy the cementation points on the rough grain surface of some fracture surfaces, and the \(F(\tau)\) value increases slightly. Within 1869-2087 s, the stress curve of the rock sample still shows a straight-line form, but with slightly decreasing slope. This is because the loading rate is reduced to 0.008 mm/s to obtain a clear post-peak curve in the loading stage. At the same time, the later value of \(F(\tau)\) in this stage increases significantly, which indicates that the rock sample undergoes the formation and development of microcracks and the phenomenon of microcrack nucleation. This is the fracture generation and development stage. Within 2087-3455 s, the stress curve no longer shows linear development, the stress value gradually increases to the maximum value, and a series of U-shaped evolution paths of the function \(F(\tau)\) appears. The U-shaped evolution path of \(F(\tau)\) is the path in which \(F(\tau)\) decreases, then stays at a relatively low level, and finally increases with time.