## 3 Experimental Results and Discussions
### Analysis of the AE Source Mechanism  
The crack model identification is determined based on the moment tensor analysis theory proposed by [PERSON] (2018). When the seismic source is a synchronous source, the body force in each direction of the unit body can be expressed using the moment tensor, which is expressed in Equation (13):  
\[M_{\rm ij}=\left(\begin{array}{cc}m_{11}&m_{12}&m_{13}\\ m_{21}&m_{21}&m_{22}\\ m_{31}&m_{32}&m_{33}\end{array}\right) \tag{13}\]  
The various components of the \(M_{\rm ij}\) can be obtained using Equation (14):  
\[A(x)=\frac{C_{S}Re(t,r)}{R}(r_{1}\,r_{2}\,r_{3})\times\left(\begin{array}{cc }m_{11}&m_{12}&m_{13}\\ m_{21}&m_{21}&m_{22}\\ m_{31}&m_{32}&m_{33}\end{array}\right)\left(\begin{array}{c}r_{1}\\ r_{2}\\ r_{3}\end{array}\right) \tag{14}\]  
where \(C_{S}\) is the sensor coupling coefficient (determined before the test); \(R_{\rm e}\)(\(t\),\(r\)) is the reflection coefficient of the seismic source wave; \(R\) is the distance vector between the seismic source and the sensor; \(r_{1}\), \(r_{2}\), and \(r_{3}\) are the components in the directional cosine matrix. Due to the symmetry, there are six unknown quantities in Equation (13) to be determined; therefore, at least six sensors are required during the test. According to the method suggested by [PERSON], the classification of AE sources can be determined using  
\[\left.\begin{array}{c}\lambda_{1}/\lambda_{1}=X+Y+Z\\ \lambda_{2}/\lambda_{1}=-0.5Y+Z\\ \lambda_{3}/\lambda_{1}=-X-0.5Y+Z\end{array}\right\} \tag{15}\]  
Figure 14: Diagram of the axial stress \(\sigma_{\rm axial}\) as a function of the wing crack length \(L\).  
\(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are the three standardized eigenvalues of the moment tensor. The AE source is shear fracture when \(A\geq 60\%\), the AE source is tensile fracture when \(A\leq 40\%\), and it is mixed type when \(40\%<A<60\%\). Taking samples S-T1-0 and S-T1-20 as examples, the results can be seen in Figures 15 and 16.  
From Figures 15 and 16, the tensile, shear and mixed AE percentages of sandstone samples at different crack propagation can be obtained, as shown in Tables 4 and 5.  
Figure 16: AE spatial location and crack source identification of sample S-T1-20. A blue AE denotes a tensile crack, a red AE denotes a mixed type, and a green AE denotes a shear crack. (**a**) The results when \(\sigma_{\text{cc}}\) was reached; (**b**) the results when \(\sigma_{\text{ci}}\) was reached; (**c**) the results when \(\sigma_{\text{cd}}\) was reached; (**d**) the results when \(\sigma_{\text{p}}\) was reached.  
Figure 15: AE spatial location and crack source identification of sample S-T1-0. A blue AE denotes a tensile crack, a red AE denotes a mixed type, and a green AE denotes a shear crack. (**a**) The results when \(\sigma_{\text{cc}}\) was reached; (**b**) the results when \(\sigma_{\text{ci}}\) was reached; (**c**) the results when \(\sigma_{\text{cd}}\) was reached; (**d**) the results when \(\sigma_{\text{p}}\) was reached.  
From the Tables 4 and 5, it can be seen that the proportion of tensile cracks in the rock during the loading process gradually increased. In addition, the tension-type AE gradually increased with the increase in the freeze-thaw damage during the stable and unstable growth stages of the cracks, which indicated that the rock that experienced freeze-thaw damage for a long time was more prone to tensile failure.