## 5 The Law of Energy Evolution  
During loading of the specimen, as shown in Figure 13, energy is transferred from the RMT-150C test machine into the specimen (assuming no energy exchange between the specimen and the environment), which is called the input energy \(U\) (area enclosed by ABD in the figure). After the energy is injected into the specimen, part of the energy is used for plastic deformation, crack extension, and crack gestation (area enclosed by ABC in the diagram) and part is used for elastic deformation (area enclosed by BCD in the diagram); the former is called dissipative energy, \(U_{d}\), and the latter is called \(U_{e}\). \[U=U_{d}+U_{e} \tag{7}\]  
The experiment is a uniaxial compression test, without considering the surrounding pressure, so the energy expression for each unit during the loading of the coal rock assemblage is:  
\[U=\int_{0}^{\varepsilon}\sigma\mathrm{d}\varepsilon \tag{8}\]  
\[U_{\mathrm{e}}=\frac{\sigma^{2}}{2E} \tag{9}\]  
where \(\sigma\) is the axial stress and \(\varepsilon\) is the strain corresponding to the axial stress. \(E\) is the modulus of elasticity. Based on the stress-strain curve in Figure 8, the pre-peak energy curve of the coal sample was plotted using Origin with Equations (7)-(9) to analyze the energy evolution of the critically damaged coal sample under different spans of unloading of the surrounding pressure. This is shown in Figure 14.  
The input energy, elastic energy, and dissipation energy curves of the critically damaged coal samples at different peritectic pressure unloading spans can be seen (Figure 14). The overall regularity of all the curves is strong, and they can be divided into two stages based on the variation in the curves: the stabilization stage and the growth stage. When looking at the elastic and dissipative energy curves in the four plots; the elastic energy  
Figure 13: Energy relation diagram of coal rock mass during the loading process. curve increases in duration as the unloading span decreases, while the dissipative energy curve is the opposite, with the stability zone decreasing in duration. This indicates that plastic damage occurs earlier within the coal sample from the undamaged control group to the time when the circumferential pressure is unloaded from 20 MPa to 60% of the 15 MPa triaxial compressive strength at 15 MPa axial pressure loading. In the no-damage control group, the elastic energy curve was above the dissipative energy curve from the beginning to the end, and the elastic energy was dominant during the loading process. At 15 MPa-60%, the dissipative energy curve is completely above the elastic energy curve. Coal samples unloaded at higher stress states have a greater degree of damage, and the resulting damaged coal samples have more fully developed internal fissures. Therefore, the threshold value required for crack expansion is lower during uniaxial reloading of damaged coal samples, so the energy is more easily used for crack expansion, resulting in the phenomenon that the dissipated energy increases with the increase of the unloading span of the confining pressure. In Figure 15, the dissipation energy ratio is calculated by converting each group of axial strains into a standardized axial strain and plotting the dissipation energy ratio. This section of the curve can be divided into the compression phase, the elastic phase, and the  
Figure 14: Energy curves under different paths. (**a**) No-damage control group. (**b**) 5 MPa-60% group. (**c**) 10 MPa-60% group. (**d**) 15 MPa-60% group. damage phase. In the compression phase, the dissipation energy is higher when the axial strain is at a low level because the natural fractures in the rock or the damage fractures caused by unloading are reduced by compression and plastic deformation of the rock sample occurs, and the dissipation energy percentage starts to decrease as the loading progresses. In the elastic phase, the sample begins to deform elastically, and the majority of the energy is elastic, so the dissipated energy ratio decreases further.