## 4 The Influence of Damage Evolution on Electrical Resistivity
### The Relationship between AE Damage Variable and Electrical Resistivity  
According to the working principle of \(LCR\) tester, the electrical resistivity of the sample is calculated by the following formula [39].  
\[\rho=R\cdot S/L \tag{8}\]  
where \(\rho\) is the electrical resistivity of coal mass; \(R\) is the real-time resistance value of loading coal in \(\Omega_{2}\)\(L\) is the length of the coal sample in m and \(S\) is the cross-sectional area of sample in m\({}^{2}\).  
As can be seen from the foregoing description, the damage and failure of coal mass are a process of deformation and fracture, and the formation, expansion and fracture of microcracks in coal mass can have a great impact on electrical resistivity [40]. Moreover, it is believed that the conductivity of coal mass is mainly formed by the free movement of the charge on account of conductivity theory of solid electrolyte, while the effective bearing area of free charge changes during the damage evolution of coal mass [41]. Consequently, based on the damage variable in continuum damage mechanics and  
Figure 7: Damage variable curve of multi-stage loading coal sample with time.  
the AE damage variable under uniaxial compression, the relationship between AE damage variable and electrical resistivity is established by plugging Equations (1) and (7) into Equation (8) as follows:  
\[\rho=\frac{\mathrm{R}}{L}\Big{[}\big{(}C_{d}\cdot\big{(}\sigma_{\mathrm{c}}- \sigma_{p}\big{)}\big{/}\big{(}C_{0}\cdot\sigma_{p}\big{)}\big{)}+1\Big{]}S \tag{9}\]  
In order to verify the effectiveness and rationality of expression, the theoretical and experimental curves of electrical resistivity with time are calculated according to the above relational expression as shown in Figure 8. The above results show that the resistivity theoretical curve is basically consistent with the experimental curve, and it makes clear that the relationship between AE damage variable and electrical resistivity is accurate and reliable for forecasting the occurrence time of coal-rock mass. Also, it can avoid the influence of other factors such as temperature, frequency on results as well as it obtains the failure time of coal-rock mass without delaying. In addition, both of resistivity test results and theoretical calculation results decrease step by step with the gradual increasing of damage variable. When the damage variable increases to a peak abruptly, that is to say, the damage and failure of the coal sample reaches the maximum, and electrical resistivity suddenly increases to a peak in a short time as the main fracture occurring. The relationship expression between AE damage variable and electrical resistivity is compared with the previous methods of using a single test to describe resistivity variation regularities, it is more comprehensive and accurate to describe the resistivity variation regularities during the process of damage evolution of loading coal mass adopting the joint test method of resistivity and AE technique. Therefore, it is helpful to understand the evolution process of coal-rock dynamic disasters and promote the application and development of electrical prospecting technology.