## 3 Test Results and Theoretical Analysis
### Analysis of the Deformation Properties during Post-Peak Loading and Unloading  
As there are fewer cracks in the sample, the permeability of the sample is 0 during the pre-peak loading and unloading process, and the permeability of the sample increases rapidly after the sample is destroyed. The permeability formula reflects the relationship between gas permeability and gas pressure, gas flow rate, sample length and cross-sectional area of the sample. It does not directly reflect the relationship between stress and permeability, but simply studies the relationship between axial stress, confining pressure or pore pressure (provided by gas pressure) and permeability can only explain the one-way action relationship. The permeability is the result of joint action of axial stress, confining pressure and gas pressure, which needs comprehensive consideration. Therefore, the study of the relationship between effective stress and permeability has more practical significance than the study of the relationship between a certain force and permeability alone, and it has a certain positive significance for studying coal and gas outburst. According to the theory of geotechnical mechanics, the effective stress is the difference between the total stress acting on the coal seam and the fluid pressure existing in the pore or fissure, that is to say:  
\[\sigma_{\mathrm{e}}=(\sigma_{1}+2\sigma_{3})/3-(p_{1}+p_{2})/2. \tag{3}\]  
In the formula, \(\sigma_{\mathrm{e}}\) is the average effective stress of coal sample, MPa; \(\sigma_{1}\) and \(\sigma_{3}\) are the axial stress and confining pressure of coal samples, respectively, MPa; \(p_{1}\) and \(p_{2}\) are the gas pressure at the inlet and outlet of raw coal samples, respectively, MPa. The deformation parameters of each specimen under different stress states during the post-peak loading and unloading confining pressure are shown in Table 2.  
Table 2 shows that in the process of loading and unloading confining pressure post-peak, the axial direction of the sample is applied by the displacement control, which causes the axial strain to increase linearly during the process of loading and unloading confining pressure, that is, the axial direction of the sample is compressed deformation. During the process of loading and unloading confining pressure, the radial deformation of specimens is shrinkage with confining pressure loading, and expansion with confining pressure unloading. According to the test results in Table 2, the relationship curves of axial strain-effective stress and radial strain-effective stress under different stress states were plotted as shown in Figure 6.  
As can be seen from Figure 6, for the dry coal sample, the radial strain of the sample decreases from \(2.37\times 10^{-2}\) to \(1.86\times 10^{-2}\) after the first loading confining pressure, which decreases by \(21.54\%\). After the first unloading confining pressure, the radial strain increases to \(2.99\times 10^{-2}\), which increases by \(25.89\%\); the radial strain of the sample decreases to \(2.60\times 10^{-2}\) after the second loading confining pressure, which decreases by \(12.83\%\). After the second unloading confining pressure, the radial strain increases to \(3.43\times 10^{-2}\), which increases by \(15\%\); the radial strain of the sample decreases to  
Figure 5: The peak strength, peak axial strain of the sample with water content curves. after the third loading confining pressure, which decreases by 4.45%. After the third unloading confining pressure, the radial strain increases to \(3.72\times 10^{-2}\), which increases by 8.3%; the radial strain of the sample decreases to \(3.54\times 10^{-2}\) after the fourth loading confining pressure, which decreases by 4.48%. After the fourth unloading confining pressure, the radial strain increases to \(3.99\times 10^{-2}\), which increases by 7.29%.