## 3 Experimental Results and Discussions
### Variation Law of Characteristic Stress
According to the theoretical model of [PERSON] and [PERSON], the initial crack stress \(\sigma_{\text{ci}}\) under uniaxial compression can be expressed as follows:  
\[\sigma_{\text{ci}}=\frac{\sqrt{3}K_{\text{IC}}}{\sqrt{\pi a}\Big{(}\sqrt{1+ \mu^{2}}-\mu\Big{)}} \tag{11}\]  
where \(K_{\text{IC}}\) is the fracture toughness of the rock, \(a\) is the half-length of the inclined crack, and \(\mu\) is Poisson's ratio. All the parameters, except for the half-length of the crack \(a\), in Equation (11) are material constants. Therefore, the initial crack stress \(\sigma_{\text{ci}}\) is mainly controlled by the inclined crack length. As the number of freeze-thaw cycles increases and  
Figure 9.0 Schematic diagram of the freeze–thaw influence on microcrack propagation inside the rock: (**a**) initial crack inside the rock; (**b**) crack growth under the freeze–thaw influence. the freezing temperature decreases, \(a\) gradually increases. Therefore, the initial crack stress \(\sigma_{\text{ci}}\) decreases with the increasing value of \(D_{NT}\). Wing crack propagation process when multiple cracks interact each other can be seen as Figure 9.1  
In addition, from the point of numerical simulation, [PERSON] et al. [46] also found that multiple oblique cracks in the rock under uniaxial compression propagate through wing cracks. According to the models of [PERSON] and [PERSON] [47], the stress intensity factor at the tip of the wing crack under uniaxial compression is  
Figure 9.2  
where \(K_{\text{I}}\) is the stress intensity factor at the tip of the wing crack, \(L\) is the length of the wing crack, \(\theta_{0}\) is the angle of the oblique crack, and \(N_{A}\) is the number of cracks per unit area. When \(K_{\text{I}}\) equals the fracture toughness \(K_{\text{IC}}\), the wing crack continues to propagate. In Equation (11), the parameters are constants, except for \(L\) and \(\sigma_{\text{axial}}\), thus the wing crack length \(L\) can be determined via the axial stress \(\sigma_{\text{axial}}\). From the results of [PERSON] and [PERSON] and Equation (12), the general variation trend of the axial stress \(\sigma_{\text{axial}}\) along with wing crack length \(L\) is shown in Figure 9.3  
Figure 9.4 Wing crack initiation under uniaxial compression. Figure 9.5 Diagram of the interaction of multiple cracks. From Figure 14, it can be seen that before \(L\) reaches \(L_{f}\), the axial stress \(\sigma_{\rm axial}\) must be increased to maintain the wing crack propagation. Furthermore, the wing crack propagation can be continued even if the stress \(\sigma_{\rm axial}\) no longer increases after \(L\) reaches \(L_{f}\). Therefore, when the axial stress increases to \(\sigma_{\rm max}\), the crack is unstable and propagates and \(\sigma_{\rm max}\) can be seen as the peak strength \(\sigma_{\rm p}\) of the rock sample. From Equation (12), we can find that the \(\sigma_{\rm max}\) decreases with the increase of the initial inclined crack length \(a\). Combining the result stating that \(a\) will increase with the increase of \(D_{NT}\), it can be concluded that \(\sigma_{\rm p}\) will decrease with the increase of \(D_{NT}\). In addition, from the research results of [PERSON] et al. (2018), it was found that the crack damage stress is about 0.70-0.82 of the peak strength, indicating that crack damage stress \(\sigma_{\rm cd}\) will gradually decrease with the increase in \(D_{NT}\). Therefore, it can be concluded that the essential reason for the effect of freezing and thawing on the change of the characteristic stress of the rock is that the length of the microcrack in the rock increases, thereby the initial crack stress \(\sigma_{\rm ci}\), the crack damage stress \(\sigma_{\rm cd}\), and the peak strength \(\sigma_{\rm p}\) gradually decrease with the increase in the number of freeze-thaw cycles and the decrease in the freezing temperature.