# Evolution Model of Coal Failure Using Energy Dissipation under Cyclic Loading/Unloading
## 4 Analysis and Discussion
### Energy Evolution Model of Coal Fracture  
#### 4.4.1 State Equation of Energy Damage  
It can be seen from the stress-strain curve that the cyclic loading/unloading process had a non-negligible influence on the damage and deformation characteristics of the coal, which then affected the energy state and stability. In recent years, the damage threshold value and damage evolution law under different confining pressures have been obtained based on the damage model [36]. Considering the initial damage and introducing the concept of equivalent modulus to make the model more suitable to the actual damage [17; 37] expressed the damage model as a power exponential function of dissipated energy and provided an effective method for evaluating fatigue damage. The concept of the damage variable has been introduced into the existing research, but the relationship between it and cyclic loading/unloading times has been rarely studied.  
Based on the nonlinear characteristics of coal, this paper tried to analyze the influence of constant amplitude cyclic loading/unloading times on coal damage from the perspective of energy dissipation, and established the corresponding equation of the energy damage state. However, under the equal amplitude cyclic loading and unloading, the coal sample was subjected to repeated cyclic loading and unloading at the same stress level, which had a non-negligible effect on the damage characteristics and deformation characteristics of the coal sample and then affected the energy state and stability degree of the coal sample. From the previous analysis, there was a closer relationship between the dissipative energy state inside the coal sample and the stability degree of the coal sample. Therefore, we explored the relationship between the damage change patterns of the coal samples under the same stress levels and the number of cycles based only on the dissipative energy change characteristics of the coal samples under repeated cyclic loading conditions. By referring to the research results of the injury variable theory [19; 36; 37], the damage variable of coal at the same stress level is determined as follows:  
\[D=\frac{U_{d_{i}}-U_{d_{j}}}{U_{\text{max}}-U_{d_{0}}} \tag{7}\]  
where \(D\) is the damage variable of coal under a constant amplitude cyclic loading; \(U_{d_{i}}\) and \(U_{d_{j}}\) are the dissipated energy of cycle \(i\) and cycle \(j\) at the same stress level, respectively; and \(U_{\text{max}}\) and \(U_{d_{0}}\) are, respectively, the maximum dissipated energy and the initial dissipated energy at the same stress level.  
Figure 11: Evolution diagram of energy tangent factor in coal specimen failure process.  
Data fitting was performed between the damage variables and cycle times in the process of constant amplitude cyclic loading to obtain the following variation relationship:  
\[D=1-[1-(\frac{N}{N_{f}})^{\frac{1}{1-\alpha}}]^{\frac{1}{1+\beta}} \tag{8}\]  
where \(N\) and \(N_{f}\) are, respectively, the cycle times and maximum cycle times at the same stress level; \(\beta\) is a constant; and \(\alpha\) is the stress index of axial load, which can be calculated by  
\[\alpha=1-\gamma(\frac{\sigma_{\max}-\sigma_{f}}{\sigma-\sigma_{\max}}) \tag{9}\]  
where \(\sigma_{\max}\) is the maximum axial stress; \(\sigma_{f}\) is the yield stress threshold; and \(\sigma\) is the stress under any strain state of coal.  
As shown in Figure 12, the fitting results show that this model had a good correlation with the experimental data, which could accurately describe the decreasing trend of the damage variable with cycle times.  
Figure 12: Relationship between damage variable and times of cyclic loading/unloading. (**a**,**b**) are the fitted curves under cyclic loading/unloading with equal amplitude at different (**I**–**III**) stages.