# Percolation Threshold of Red-Bed Soft Rock during Damage and Destruction
## 2 Research Contents and Methods
### Constitutive Model of Percolation in Rock Damage
\[\sigma_{1}=\frac{\lambda}{(\lambda+G)}\sigma_{3}+\bigg{(}\lambda+2G-\frac{\lambda^{2 }}{\lambda+G}\bigg{)}\varepsilon_{1} \tag{8}\]  
Substituting \(\lambda\) and \(G\) into the above equation, we obtain:  
\[\sigma_{1}=2\
u\sigma_{3}+E\varepsilon_{1} \tag{9}\]  
Because \(\sigma_{1}=\sigma_{3}+\Delta\sigma_{1}\), \(\varepsilon_{1}=\sigma_{3}^{0}+\Delta\varepsilon_{1}\) and \(\sigma_{2}^{0}=\sigma_{3}^{0}=\sigma_{2}=\sigma_{3}\), then \(\Delta\sigma_{1}=E\Delta\varepsilon_{1}\) or  
\[\sigma_{1}-\sigma_{3}=E\Big{(}\varepsilon_{1}-\varepsilon_{3}^{0}\Big{)} \tag{10}\]  
Following confined pressure loading, the axial strain is zeroed, so \(\varepsilon_{3}^{0}=\frac{\sigma_{3}^{0}(1-2\
u)}{E}\) is cleared and the triaxial compression is recorded as:  
\[\sigma_{1}-\sigma_{3}=E\varepsilon_{1} \tag{11}\]  
Then, according to the strain equivalence hypothesis, \(\varepsilon=\frac{\widetilde{\sigma}}{E}=\frac{\sigma}{(1-D)E}\), then  
\[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-\widetilde{\sigma}_{3}}{E}= \frac{\sigma_{1}-\sigma_{3}}{E(1-D)} \tag{12}\]  
That is,  
\[\sigma_{1}-\sigma_{3}=E\varepsilon_{1}(1-D) \tag{13}\]  
By substituting the amount of damage in the percolation process into the above equation, we obtain:  
\[\sigma_{1}-\sigma_{3}=E\varepsilon_{1}(1-D_{a}+D_{b}\frac{\Omega}{\Omega_{0}}) \tag{14}\]  
The elastic modulus in the above formula is the elastic modulus in the lossless state. However, the elastic modulus measured by the stress-strain curve obtained from the indoor compression test entailed initial damage. Therefore, let the elastic modulus containing the initial damage be \(\widetilde{E}\), so that  
\[\widetilde{E}=E(1-D_{a}), \tag{15}\]  
where \(D_{a}\) is the initial damage. Therefore, the constitutive model of percolation damage under triaxial compression is:  
\[\sigma_{1}-\sigma_{3}=E\varepsilon_{1}(1-D)=E\varepsilon_{1}\bigg{[}1-D_{a}-D _{b}\frac{\Omega}{\Omega_{0}}\bigg{]}=\widetilde{E}\varepsilon_{1}\bigg{[}1- \frac{D_{b}}{1-D_{a}}\frac{\Omega}{\Omega_{0}}\bigg{]} \tag{16}\]  
By applying the same method, we can find the constitutive model of percolation damage under uniaxial compression:  
\[\sigma_{1}=\frac{\lambda}{(\lambda+G)}\sigma_{3}+\bigg{(}\lambda+2G-\frac{\lambda^{2 }}{\lambda+G}\bigg{)}\varepsilon_{1} \tag{8}\]0  
Using the constitutive model of percolation damage, the percolation threshold of red-bed soft rock can be obtained experimentally. The method is as follows:  
(1) conduct compression test with rock samples of red-bed soft rock and record the stress-strain curve and AE data during the compression test; (2) determine the cumulative AE number \(\Omega_{0}\) at the time of red-bed soft rock percolation; (3) use porosity as the initial damage \(D_{a}\) of soft rock. Then, the amount damaged by the percolation process (\(D_{b}\)) is adjusted to make the compressive strength equal, and the constitutive model curve for percolation damage is fitted to the stress-strain curve to obtain the amount damaged by the percolation process (\(D_{b}\)). The percolation threshold of the red-bed soft rock sample is then \(D_{p}=D_{a}+D_{b}\).