## 4 Deformation Characteristics
### Damage Variable \(D_{s}\)  
In order to characterize the degree of damage evolution in coal, [PERSON] [36] defined damage variables based on the decay of elastic modulus:  
\[D=1-\frac{\overline{E}}{E_{0}} \tag{6}\]  
where \(\overline{E}\) is the elastic modulus of the damaged rock, and \(E_{0}\) is the elastic modulus of the rock when it is non-damaged. However, the above equation is only suitable for describing  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**T/s**} & \multirow{2}{*}{**Stress Level**} & \multicolumn{2}{c}{**Fitting Formula**} & \multirow{2}{*}{\(\mathbf{R^{2}}\)} \\  & & (\(y\) = \(ax+b\), \(1\leq x\leq 6\)) & & \\ \hline \multirow{4}{*}{120} & 1 & y = 0.069 + 0.012x & 0.99 \\  & 2 & y = 0.231 + 0.015x & 0.99 \\  & 3 & y = 0.439 + 0.019x & 0.98 \\ \hline \multirow{4}{*}{300} & 1 & y = 0.051 + 0.010x & 0.99 \\  & 2 & y = 0.187 + 0.014x & 0.98 \\  & 3 & y = 0.369 + 0.018x & 0.99 \\ \hline \multirow{4}{*}{480} & 1 & y = 0.051 + 0.008x & 0.99 \\  & 2 & y = 0.171 + 0.012x & 0.99 \\ \cline{1-1}  & 3 & y = 0.346 + 0.017x & 0.99 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Fitting equations for cumulative irreversible deformation versus number of cycles for each stress level.  
Figure 9: Curve of irreversible deformation with the number of cycles.  
the elastic damage behavior, and for the loading and unloading process of an elastic-plastic material like coal, the damage variables need to be corrected [32]:  
\[D_{\mathrm{s}}=1-(1-\frac{\varepsilon_{p}}{\varepsilon})\frac{E_{u}}{E_{0}} \tag{7}\]  
where \(\varepsilon\) and \(\varepsilon_{p}\) are the total strain and total plastic strain, respectively, and \(E_{u}\) and \(E_{0}\) are the unloaded modulus and initial elastic modulus, respectively.  
Figure 10 shows the evolution curve of the damage variable. Based on the rate of change of the damage variable \(D_{S}\) at each stress level, coal damage can be categorized into three stages: deceleration, stabilization, and acceleration. During the deceleration damage stage, the original defects are gradually compacted, the density of the internal structure is enhanced, and the damage caused by subsequent cycling continues to diminish. During the stabilization damage stage, the specimen is further compacted to enhance its mechanical properties. No new cracks or rupture surfaces have emerged, as the extent of plastic deformation is negligible, and the cumulative damage exhibits a relatively stable pattern of change. In the acceleration damage stage, the specimen is subjected to higher stress disturbances, causing cracks to continue closing, developing, and expanding under cyclic loading. The plastic deformation zone expands. Consequently, the damage within the specimen continues to increase, and the rate of increase accelerates.  
To compare the variation rules of \(D_{\mathrm{S}}\) curves for coal samples with different cyclic periods, it can be observed that during the accelerated damage stage, the growth rates of \(D_{\mathrm{S}}\) are 2.21%, 1.68%, and 1.23% at T = 120 s, 300 s, and 480 s, respectively. It indicates that under identical stress conditions, a shorter cyclic period results in a more rapid accumulation of damage in the specimen prior to imminent failure, thereby increasing the intensity of internal degradation.  
Figure 10: Curve of the damage variable \(D_{\mathrm{S}}\).