## 3 Results
### Characterization of Damage and Its Evolution Characteristics Based on Infrasonic Energy  
The deformation and destruction of rock is the result of the internal evolution of rock damage, the internal damage of rock will release energy, which not only produces high-frequency signals but also produces low-frequency signals, and these signals reflect and contain information about the damage to the rock material, so infrasonic energy was chosen to characterize the rock damage. It was assumed that the microcell strength of the rock obeys the Weibull distribution [22]. The Weibull distribution of microelement strength is given as:  
\[F(\sigma)=1-\exp\left[-(\frac{\sigma}{\sigma_{0}})^{m}\right] \tag{2}\]  
where  
\(\sigma\) is the microelement strength, \(\sigma_{0}\) is the scale parameter, and \(m\) is the shape parameter. The damage area of the rock cross-section is as follows:  
\[S=S_{m}\int_{0}^{\varepsilon}\varphi(x) \tag{3}\]  
where  
\(S_{m}\) is the cross-sectional area of the material in its undamaged state;  
\(\varphi(x)\) is the probability distribution of the strength of the microelement. When the strain increment of the specimen is \(\Delta\varepsilon\), the damage area of the specimen is obtained from Equation (3). When the damage area of the specimen is \(\Delta S\),  
\[\Delta S=S_{m}\int_{0}^{\Delta\varepsilon}\varphi(x)dx \tag{4}\]  
The damage variable \(D\) is defined as the ratio of the damage area \(\Delta S\) to the initial cross-sectional area \(S_{m}\):  
\[D=\Delta S/S_{m}=\int_{0}^{\Delta\varepsilon}\varphi(x)dx \tag{5}\]  
Assuming that the infrasonic waves energy produced by damaging the microcell area is \(n\), then the infrasonic waves energy \(N\) will be produced by damaging the \(\Delta S\) area:  
\[N=n\Delta S=N_{m}\Delta S/S_{m} \tag{6}\]  
Figure 15: Cumulative infrasonic energy characteristics under the cyclic loading and unloading of rock specimen A-4. The ratio of infrasonic energy \(N\) to the cumulative infrasonic energy \(N_{m}\) at full damage:  
\[N/N_{m}=\Delta S/S_{m}=\int_{0}^{\Delta t}\varphi(x)dx \tag{7}\]  
The relationship between the damage variable \(D\) and the infrasonic cumulative infrasonic energy is obtained by couplings (4) and (6):  
\[D=N/N_{m} \tag{8}\]  
It is assumed that the infrasonic cumulative infrasonic energy produced by the rock at all levels of loading and unloading phases are \(N_{i+}\) and \(N_{i-}\), respectively; the damage variables are \(D_{i+}\) and \(D_{i-}\); and the cumulative damage variables are \(D_{+}\) and \(D_{-}\). The cumulative infrasonic energy of the cyclic infrasonic waves per stage is \(N_{i}\). The cumulative infrasonic energy \(\Sigma N_{i}\) is obtained by accumulating \(N_{i}\), as shown in Tables 5-10 and Figures 16 and 17. \[N_{i}=N_{i+}+N_{i-} \tag{9}\]  
\[D_{i\pm}=N_{i\pm}/N_{m} \tag{10}\]  
\[S=S_{m}\int_{0}^{\varepsilon}\varphi(x) \tag{3}\]0  
As shown in Figures 17 and 18, During the low-stress phase of the first and second cycles, the cumulative damage variable averaged 0.1072 by the end of the second cycle, and the rock specimens were less damaged. As the stress continues to increase, the damage variable and cumulative damage variable increase more significantly in the third and fourth cycles, where the cumulative damage variable averages 0.4397 after the fourth cycle; in the third and second cycle of the unloading phase of rock specimen A-4, the damage variable grows from 0.0204 to 0.0709, and the cumulative damage variable grows from 0.0642 to 0.1351. Entering the high-stress stage, in the fifth and sixth cycles, the maximum value of the damage variable occurs and the cumulative damage variable increases sharply. In the fifth cycle of the loading stage of rock specimen A-6, the damage variable grows from 0.1439 to 0.2046, and in the fourth cycle of the unloading stage, the cumulative damage variable grows from 0.5224 to 0.7269. After the fifth cycle of loading, reaching peak strength, the damage variable decreases and the rock specimen damage, although increasing, increases at a reduced rate.