## 4 Deformation Characteristics  
### Elastic Modulus  
The elastic modulus \(E\) is a crucial parameter that reflects the deformation characteristics of rock. It is important to consider that the coal sample is influenced by gas pressure during the testing process, the axial effective stress \(\sigma_{1}^{\prime}\) and radial effective stress \(\sigma_{2}^{\prime}=\sigma_{3}^{\prime}\) are calculated according to the principle of effective stress, and the equations are, respectively [35],  
\[\sigma_{1}^{\prime}=\sigma_{1}-p_{1} \tag{1}\]  
Figure 4: Bias stress–volume–strain curves. \[\sigma_{2}^{\prime}=\sigma_{3}^{\prime}=\sigma_{3}-\frac{1}{2}(p_{1}+p_{2}) \tag{2}\]  
In the formula, \(\sigma_{1}\) and \(\sigma_{3}\) are the axial and radial stresses, respectively, and \(p_{1}\) and \(p_{2}\) are the inlet gas pressure and outlet gas pressure, respectively. The elastic modulus of rock during triaxial cyclic loading and unloading is calculated as follows [19]:  
\[E=\frac{(\Delta\sigma_{1}^{\prime}+2\Delta\sigma_{3}^{\prime})(\Delta\sigma_ {1}^{\prime}-\Delta\sigma_{3}^{\prime})}{\Delta\sigma_{3}^{\prime}(\Delta \varepsilon_{1}-2\Delta\varepsilon_{3})+\Delta\sigma_{1}^{\prime}\Delta \varepsilon_{1}} \tag{3}\]  
In the formula, \(\Delta\sigma_{1}^{\prime}\) and \(\Delta\varepsilon_{1}\) are the axial effective stress increment and axial strain increment in the elastic interval of bias stress loading/unloading section, and \(\Delta\sigma_{3}^{\prime}\) and \(\Delta\varepsilon_{3}\) are the radial effective stress increment and radial strain increment in the elastic interval of bias stress loading/unloading section, respectively. The variation pattern of the elastic modulus is illustrated in Figure 5. Generally, the elastic modulus during the bias stress unloading phase is higher than that during the loading phase. With regard to the overall stress level, the elastic modulus shows a decreasing trend as the stress level increases. For instance, when the cyclic period is 120 s, the average loaded elastic modulus at each stress level is 2.07 GPa, 1.80 GPa, and 1.51 GPa, respectively. This is due to the combined influence of increased shear stress, the development and propagation of internal cracks are intensified, leading to a deterioration in the coal samples' resistance to deformation. The pattern of elastic modulus at the same stress level can be attributed to the internal hardening and softening reactions of the coal samples. From Figure 5, it can be found that at stress level 1, the elastic modulus increases with the number of cycles. This is mainly due to the compaction and closure of the pores and cracks within the coal sample, as well as the filling and compaction of the detrital particles which act as a hardening effect. At stress level 2, the elastic modulus curve takes on a groove shape. This likely occurs because the new formed fissure structure has not yet stabilized as to the next stress level, over time, the particles and cracks gradually adjust and stabilize, the elastic modulus stabilizes or even increases slightly. At stress level 3, the bias stress is further increased, leading to the extension and connection of internal cracks. The weakening effect becomes predominant, resulting in a gradual decrease in specimen stiffness and elastic modulus as the number of cycles increases. Figure 5 demonstrates a clear upward trend in the elastic modulus curve as the cyclic period increases. Figure 6 illustrates that the average elastic modulus increases as the cyclic period slows down at each stress level. As observed at stress level 1, the average loading elastic modulus during cyclic periods of 120 s, 300 s, and 480 s were 2.07 GPa, 2.17 GPa, and 2.30 GPa, reflecting increases of 4.83% and 10.60%, respectively. The results indicate that an increase in the acceleration of the cyclic period reduces the stiffness of the coal samples, diminishes their ability to withstand external deformation, and renders the samples more vulnerable to destabilizing damage.