## 4 Deformation Characteristics
### Poisson's Ratio  
Poisson's ratio \(\mu\) is a parameter reflecting the relative deformation of radial and axial directions of the rock in the progressive destruction process of cyclic loading, and the  
Figure 5: Relationship of elastic modulus with number of cycles.  
Figure 6: Elastic modulus at different cyclic periods.  
[PERSON]'s ratio \(\mu_{l}\) for the bias stress loading section and the Poisson's ratio \(\mu_{u}\) for the unloading section are calculated by the formula [28]:  
\[\mu_{l}=(\varepsilon_{3,l}-\varepsilon^{\prime}_{3,l})/(\varepsilon_{1,l}- \varepsilon^{\prime}_{1,l}) \tag{4}\]  
\[\mu_{u}=(\varepsilon_{3,u}-\varepsilon^{\prime}_{3,u})/(\varepsilon_{1,u}- \varepsilon^{\prime}_{1,u}) \tag{5}\]  
where \(\varepsilon_{1,l}\), \(\varepsilon^{\prime}_{1,l}\), \(\varepsilon_{3,l}\), and \(\varepsilon^{\prime}_{3,l}\) are the axial and radial strains in the elastic interval of the biased stress loading section, respectively, and \(\varepsilon_{1,u}\), \(\varepsilon^{\prime}_{1,u}\), \(\varepsilon_{3,u}\), and \(\varepsilon^{\prime}_{3,u}\) are the axial and radial strains in the elastic interval of the biased stress unloading section, respectively.  
From Figure 7, it can be noticed that the Poisson's ratio of the loaded section is greater than that of the unloaded section. Poisson's ratio increases in a generally stepwise manner as stress levels rise. This behavior indicates that radial deformation becomes more sensitive compared to axial deformation. Under the same stress level, the ductility of the specimen is somewhat limited in the first two levels. This limitation occurs because most of the pores and defects tend to be compressed and filled. In stress level 3, the stress is approaching the yield threshold of the coal samples. The previous cycles of loading and unloading have caused some degree of damage to the specimens, leading to the expansion and development of both new and old cracks. As a result, the radial deformation becomes more pronounced, and the decreasing trend of Poisson's ratio is effectively mitigated. A sustained increase in the Poisson's ratio over three to four cycles, occurring near the time of damage, can serve as a precursor to the rupture of the specimen.  
From Figure 7, it can be found that deceleration of the cycle period significantly constrains the Poisson's ratio and mitigates the radial expansion deformation of the specimen. The average Poisson's ratio for each stress level is shown to decrease with the increased cyclic period in Figure 8. In the full process cycle, the average Poisson's ratio of the loading section at T = 120 s, 300 s, and 480 s is 0.45, 0.38, and 0.29, respectively, which decreases by 15.56% and 23.68% in that order. This may be related to the fact that higher loading rates generate greater kinetic energy, leading to increased slip misalignment between the contact surfaces of the coal sample fissures and thus the more pronounced radial deformation.  
Figure 7: Cont.  
The linear fitting equations (in Table 1) allow us to express the relationship between cumulative irreversible deformation and the number of cycles at each stress level. It is observed that the rate of accumulation of irreversible deformation increases with higher stress levels. The rate of increase in cumulative irreversible deformation gradually accelerates as the cyclic period shortens under the same stress conditions. For stress level 1, the values of \(a\) at T = 120 s, 300 s, and 480 s are 0.012, 0.010, and 0.008, respectively. The results illustrate that a higher sensitivity to plastic deformation in coal samples, under conditions of high strain rate and high stress, is more likely to lead to structural deterioration and induce instability damage.