# Energy Behaviour of Coal Failure under Uniaxial Cyclic Loading/Unloading
## 4 Results and Analysis
### Energy Evolution Model of Coal Failure Process  
The coal specimen was loaded with cyclic loading/unloading at the same stress value, which would have a significant impact on its damage and deformation, and it would further affect the energy state and stability. The damage was closely related to the number of cycles. For the cyclic loading/unloading, if the number of cycles was 0 (\(N=0\)), the coal specimen was intact (\(D=0\)). When the cycle numbers reached the maximum cycle number (\(N=N_{f}\)), the damage of coal specimen is more serious. Then, the damage variable reached the peak value, and it may be a failure. The relationship between the damage variation and the cycle numbers can be expressed as Equation (7):  
\[D=1-\left[1-(\frac{N}{N_{f}})^{\frac{\beta}{1-\alpha}}\right] \tag{7}\]  
where the \(D\) is the damage variable of coal specimen; the \(N\) is the cycle numbers of coal specimen under the same stress value; \(\beta\) is the stress-related constant of the loading process of the coal sample, the larger \(\beta\) the greater the damage to the coal sample during the destruction process and the \(\alpha\) is the stress index of coal specimen, which can be expressed as Equation (8):  
\[\alpha=1-\gamma(\frac{\sigma_{\max}-\sigma_{f}}{\sigma-\sigma_{\max}}) \tag{8}\]  
where the \(\gamma\) is the coefficient related to the properties of coal specimen and it is a constant; the \(\sigma_{\max}\) is the maximum axial stress; the \(\sigma_{f}\) is the threshold of yield stress and the \(\sigma\) is the stress value under any strain state of the coal specimen. The dissipated energy evolution curve of the coal specimen is shown in Figure 12. The dissipated energy state evolution equation is as below during the failure process. It is shown that the Equation (9) can well coincide with the evolution pattern of the energy state under axial cyclic loading. \[U_{d}=(1-v)E_{0}\frac{1}{1+e^{-(\varepsilon-\varepsilon_{c})^{2}+\zeta}} \tag{9}\]  
Figure 12: Trend diagram of coal specimen dissipation energy under axial cyclic loading. **(a)** The dissipated energy evolution curve of the coal sample S1. **(b)** The dissipated energy evolution curve of the coal sample S2. The dissipated energy behaviour of the coal specimen had obvious evolution characteristics under axial cyclic loading. The dissipative energy behaviour of coal specimens has a distinct evolutionary feature under axial cyclic loading. Therefore, the energy state evolution equation can represent the energy state evolution mode of coal specimens. An energy evolution model was used to describe the energy state and stability of the coal specimen. The cumulative micro-flaws can represent the failure state and stability degree of the coal specimen to a large extent. The calculation formula for the generation of micro-flaws is Equation (10). \[V(\varepsilon)=A\frac{\alpha}{\gamma}\mathrm{e}^{(-\frac{\mathrm{i}I_{0}}{ \alpha})}[\mathrm{e}^{(\frac{\mathrm{i}I_{0}(\varepsilon)}{\alpha})}-1] \tag{10}\]  
where the \(\
u_{\varepsilon}\) is the formation rate of the micro-crack in the coal specimen which related to strain parameters; the \(A\), \(\gamma\) and \(\alpha\) are parameters that related to the properties of coal specimens and they are constant; The \(U_{0}\) is the minimum activation energy generated by micro-flaws in the coal specimen and the \(U_{d}(\varepsilon)\) is the energy state of the dissipated energy. The coal failure is closely related to its energy state and strain characteristics. Therefore, when the coal failure stress (\(\sigma_{c}\)) reaches the peak value, the relationship between the coal's strength (\(\sigma_{c}\)) and the energy state is Equation (11). \[\sigma_{c}=\frac{1}{\gamma}[\alpha\ln(\frac{\gamma\varepsilon}{\alpha}U_{d}( \varepsilon)+e^{-\frac{\mathrm{i}I_{0}}{\alpha}})+U_{0}] \tag{11}\]  
This equation expresses an energy destruction criterion for coal fracture evolution characterized by the dissipation energy.