# Energy Behaviour of Coal Failure under Uniaxial Cyclic Loading/Unloading
## 4 Results and Analysis
### Energy Evolution Characteristics of the Coal Specimen
#### 4.2.2 Evolution Characteristics of Energy Tangent Factor  
The change rate of the energy is a very important parameter for judging the state of the coal specimen. It is also of great significance for judging the energy evolution stage and energy drive mechanism. To show the change of energy growth rate of the coal specimen under axial load, the energy growth tangent factor was proposed. The calculation formula can be expressed as Equation (5):  
\[k_{TF}=\frac{dE_{i}}{d\sigma_{i}}=\frac{\Delta E}{\Delta\sigma} \tag{5}\]  
where the \(k_{TF}\) represents the energy tangent factor; the energy factor increment \(\Delta E=E_{i+1}-E_{i}\), and the stress increment \(\Delta\sigma=\sigma_{i+1}-\sigma_{i}\).  
Energy Growth Pattern under Graded Cyclic Load  
It can be known that the elastic energy growth rate keeps increasing, while the dissipative energy tangent factor fluctuates at a low value at stage I, as shown in Figure 9. It mainly shows crack closure and cracking behaviour in this stage, and there is basically no plastic deformation. Therefore, the elastic energy tangent factor is continuously increasing. The tangent factor of elastic energy fluctuates at a higher value, and the dissipative energy tangent factor also fluctuates after a certain jump at stage II. The reason for the jump of the dissipative energy tangent factor was that the deformation gradually reached the yield stage with the increase of crack growth, and irreversible plastic deformation occurred in the coal specimen. Then, it would lead to a step-like jump of dissipative energy tangent factor.  
Figure 9: The energy tangent factor during grading cyclic loading/unloading. (**a**) Elastic energy tangent factor and dissipative energy tangent factor of coal sample S1. (**b**) Elastic energy tangent factor and dissipative energy tangent factor of coal sample S2.  
It can be known that the tangent factor curve of elastic energy and dissipative energy shows a large value of continuous decline at stages III and IV. This was mainly caused by how the high energy storage state inhibited the energy storage capacity of the coal specimen. When the coal specimen is in a high energy state, the stable state decreases, and the energy storage rate decreases after being restrained. Therefore, the variation of the energy tangent factor exists in the state of \"promoting storage with low energy and inhibiting storage with high energy\" under different stress states.  
Energy Growth Pattern under Equal Amplitude Cyclic Load  
It can be known that the tangent factor of elastic energy and dissipative energy are both in stable fluctuation during the equal amplitude loading/unloading. The change of the energy growth rate will only fluctuate to a certain extent, as shown in Figure 10. However, the energy tangent coefficients are different for different deformation or stress states. The elastic energy is at a lower value in the first stage and at a higher value in the second stage. Furthermore, it is also at a lower value in the third stage, but still higher than the first stage. The change of dissipative energy tangent factor also shows a change trend of low value rise-high value maintain-second high value decline. It indicates that there also exists a \"low facilitation, high inhibition\" model of change.  
Figure 10: The energy tangent factor under equal amplitude cyclic loading/unloading. (**a**) Elastic energy tangent factor and dissipative energy tangent factor of coal sample S3 in stage I. (**b**) Result of coal sample S4 in stage I. (**c**) Result of coal sample S3 in stage II. (**d**) Result of coal sample S4 in stage II. (**e**) Result of coal sample S3 in stage III. (**f**) Result of coal sample S4 in stage III.  
indicated that the coal specimen was completely destroyed. Therefore, the energy damage variable displays a pattern of the small rapid growth-slow growth-jump mutation state.