## 4 The Influence of Damage Evolution on Electrical Resistivity  
### The Establishment of AE Damage Model  
In order to further analyze the damage evolution process of coal sample, AE accumulative ringing count is used as a characteristic parameter to describe the damage characteristics of the coal sample. It was used to describe the process from microscopic damage to macroscopic fracture of materials with the concept of continuous factor and define the damage variable by [PERSON] and [PERSON] [34; 35]. \[D=A_{d}/A \tag{1}\]  
where \(A_{d}\) is the damage cross-sectional area of loading coal sample and \(A\) is the cross section area in the initial undamaged state. The counts of AE rings per unit area when the coal sample is destroyed as follows:  
\[\mathrm{C_{w}}=\frac{C_{0}}{A} \tag{2}\]  
where \(C_{0}\) is ringing counts of the whole section of coal sample in complete failure. When the damage area reaches \(A_{d}\), the AE accumulative ringing count of the \(A_{d}\) area destruction is defined as follows:  
\[C_{d}=C_{w}A_{d}=\frac{C_{0}}{A}A_{d} \tag{3}\]  
Hence,  
\[D=\frac{C_{d}}{C_{0}} \tag{4}\]  
The damage variable can be modified to Equation (4) due to the different failure conditions of coal mass. \[D=D_{U}\frac{C_{d}}{C_{0}} \tag{5}\]  
where \(D_{u}\) is the critical damage value. For convenience of calculation, the critical damage value can be expresses as follows:  
\[D_{U}=1-\frac{\sigma_{c}}{\sigma_{p}} \tag{6}\]  
[PERSON] and [PERSON] conducted a detailed study of uniaxial and triaxial compression in coal and rock mass, and the analytical expressions of damage variable were deduced on this foundation of AE accumulative ringing count. In this paper, the theoretical analysis was carried out with the analytic formula of damage variable of coal-rock mass under uniaxial compression only as follows [10; 36; 37]:  
\[\mathrm{D_{c}}=D_{u}\frac{C_{d}}{C_{0}}=\left(1-\frac{\sigma_{c}}{\sigma_{p}} \right)\frac{C_{d}}{C_{0}} \tag{7}\]  
where \(D_{u}\) is the critical damage value of sample; \(C_{d}\) is the AE accumulative ringing count of coal and rock mass at any time under the loading; \(C_{0}\) is the AE accumulative ringing count of completely destroyed coal and rock mass; \(\sigma_{c}\) is the residual strength and \(\sigma_{p}\) is the ultimate strength. Consequently, the damage of coal and rock mass is analyzed basing on the AE parameters and the results of stress strain, and combining with the damage variable expression of AE accumulative ringing count, and the damage curve of uniaxial compression in coal sample is shown in Figure 7.  
It can be seen from the Figure 7 that there is a good correspondence relationship between damage variable and resistivity variation curve of coal mass, and the curve of damage variable has a stepladder upward trend, while the resistivity curve shows a stepladder downward trend with time. Besides, the curve of damage variable is consistent with the curve of cumulative AE ringing counts, and there are three different types of growth trends, i.e., smooth change, abrupt rise, muted growth. In addition, it is clear that the values of damage variable are relatively small in the initial stage of loading, which indicates the damage degree of loading coal sample increases slowly at the initial stage of loading. It is precisely because of the compaction of internal microfracture structures that new conductive channels are generated, resulting in a decrease in resistivity at the initial stage of loading [38]. When the stress increases gradually to 9 MPa, the curve of damage variable is still increasing constantly during the constant loading stage even, and it shows that new cracks constantly form in coal sample and the interaction, extension and penetration between internal microcracks with each other lead to the destruction of coal sample to the end.. Meanwhile, all of conductive channels are destroyed, and thus electrical resistivity increases to a peak suddenly along with the movement circuits of electrons are cut off or blocked.