## 3 Testing Results
### Damage Evolution Characteristics  
Damage reflects the deterioration degree of specimen mechanical properties under different loadings [22]. The elastic modulus, residual strain, dissipated energy, ultrasonic  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Cycle Number**} & \multicolumn{5}{c}{**Cumulative AE Hit Count in Several Key Stress Points**} \\ \cline{2-7}  & **3.5 MPa** & **4.0 MPa** & **4.5 MPa** & **5.0 MPa** & **5.5 MPa** & **5.826 MPa** \\ \hline
1 & 104 & — & — & — & — & — \\
2 & 9 & 164 & — & — & — & — \\
3 & 2 & 49 & 423 & — & — & — \\
4 & 0 & 0 & 267 & 774 & — & — \\
5 & 0 & 0 & 1 & 35 & 785 & — \\
6 & 0 & 0 & 0 & 2 & 27 & 1277 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The cumulative AE hit count of coal specimens at key stress points in different loading-unloading cycles.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cycle Number** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Cyclic loading maximum stress/MPa & 3.5 & 4 & 4.5 & 5 & 5.5 & 5.826 \\ Kaiser point/MPa & — & 3.886 & 4.166 & 4.708 & 5.501 & 5.819 \\ F\({}_{\text{r}}\) & — & 1.110 & 1.042 & 1.046 & 1.100 & 1.058 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The Kaiser effect point and the corresponding Felicity ratio.  
Figure 9: Felicity ratio versus cycle number curve.  
velocity, and cumulative AE counting can all be defined as damage variables for quantification calculation. The disadvantage of the elastic modulus method is that it can not determine the initial elastic modulus of the specimen. The residual strain method and the dissipated energy method can only calculate the damage according to the cycle number, thus resulting in the discontinuity of the damage process. The disadvantage of the ultrasonic velocity method is that the damage increases in fluctuation in the process of cyclic loading and unloading, which is inconsistent with the fact that the damage has been increasing. Instead, the cumulative AE counting method does not have the above defects. So, this method is adopted for damage calculation. The calculation equations are as follows:  
\[D_{1}=\frac{N_{1i}}{N_{1E}} \tag{2}\]  
where \(D_{1}\) is the damage calculated by the AE ring counting method, \(N_{1i}\) is the cumulative AE ring count at some time point, and \(N_{1E}\) is the cumulative AE ring count at failure.  
\[D_{2}=\frac{N_{2i}}{N_{2E}} \tag{3}\]  
where \(D_{2}\) is the damage calculated by the AE hit counting method, \(N_{2i}\) is the cumulative AE hit count at some time point, and \(N_{2E}\) is the cumulative AE hit count at failure.  
The damage evolution curves are shown in Figure 10. The damage calculated by the cumulative AE ring count basically increases according to the linear law, which can not reflect the fact that the damage is basically unchanged in the unloading phase. Instead, the damage calculated by the cumulative AE hit count can better represent the fact that the damage has been increasing in the loading phase and kept basically unchanged in the unloading phase. Therefore, the AE hit count as a damage variable can better describe the damage development of coal specimens subjected to cyclic loading and unloading.