# Evolution Model of Coal Failure Using Energy Dissipation under Cyclic Loading/Unloading
## 4 Analysis and Discussion
### Energy Evolution Model of Coal Fracture
#### 4.4.3 State Equation of Coal Failure Strength  
According to the law of thermodynamics, energy conversion is the essential characteristic of the physical process of matter [8]. Based on the analysis of damage evolution, the relationships of the dissipated strain energy to the stress-strength ratio, loading rate, and temperature were studied [18; 41], which better predicted the fatigue life of the material. Based on the theoretical analysis of energy dissipation, we defined the concepts of unit dissipated energy, releasable strain energy, strength loss, and overall damage, and we proposed a rock strength criterion [7]. However, coal is a heterogeneous material with high voidage. Under the action of load, the change in external factors directly affects the formation rate of internal micro-defects and further causes the instability failure of coal. Therefore, from the perspective of internal micro-defects to explore the relationship between the strength characteristics and energy state, the energy failure criterion of coal failure can be obtained more directly.  
Under the action of equal axial load, the generation rate of micro-defects in coal can be expressed as  
\[v_{e}=Ae\biggl{(}-\frac{U_{0}-\gamma U_{d}(\varepsilon)}{\alpha}\biggr{)} \tag{13}\]where \(A\), \(\gamma\), and \(\alpha\) are the parameters related to the nature of coal; \(U_{0}\) is the lowest activation energy required for the generation of internal micro-defects; and \(U_{d}(\varepsilon)\) is the dissipated energy in the coal.  
By integrating Equation (13), the cumulative total amount of internal micro-defects is  
\[V(\varepsilon)=A\frac{\alpha}{\gamma}e^{(-\frac{U_{0}}{\alpha})}[e^{(\frac{ \varepsilon U_{d}(\varepsilon)}{\alpha})}-1] \tag{14}\]  
Therefore, when the coal reaches its failure load \(\sigma_{c}\), the relationship between \(\sigma_{c}\) and internal energy of coal is as follows:  
\[\sigma_{c}=\frac{1}{\gamma}[\alpha\ln(\frac{\gamma\varepsilon}{\alpha}U_{d}( \varepsilon)+e^{-\frac{U_{0}}{\alpha}}+U_{0})] \tag{15}\]  
In fact, when the coal reaches the yield stress threshold \(\sigma_{f}\), it has already entered the critical failure state. Therefore, the dissipated energy \(U_{fd}(\varepsilon)\) is taken as the key warning point of coal failure when the stress reaches \(\sigma_{f}\), and, similarly, the dissipated energy \(U_{cd}(\varepsilon)\) is taken as the fracture point of coal when the stress reaches \(\sigma_{c}\).  
Based on the relationships among the equations of the energy state, damage state, and strength state, an early warning model of coal failure based on energy dissipation was established:  
\[\left\{\begin{array}{l}D=\frac{U_{d_{i}}-U_{d_{j}}}{U_{\max}-U_{d_{0}}}\\ U_{d}=(1-v)E_{0}\frac{1}{1+e^{-(v-\varepsilon_{c})^{2}}+\frac{\gamma}{\varepsilon }}\\ \sigma_{c}=\frac{1}{\gamma}[\alpha\ln(\frac{\gamma\varepsilon}{\alpha}U_{d}( \varepsilon)+e^{-\frac{U_{0}}{\alpha}})+U_{0}]\end{array}\right. \tag{16}\]  
Additionally, some problems will require further in-depth investigation: (1) The test results in this paper are only at the laboratory scale, so the applicability of dynamic disasters on site such as rock burst should be further verified. (2) According to the compressive strength of the sample, it was necessary to further refine the stress loading gradient and increase the cycle number to improve the accuracy of the tests.