## 4 Relationship between \(K_{ic}\) and \(\sigma_{t}\)
### Theoretical Analysis  
Theoretical analysis was conducted to explore the correlation between the mode-I fracture toughness and tensile strength of rock further. The stress field near the crack tip when a rock is subjected to biaxial tensile stress can be considered in terms of fracture mechanics [32] as shown in Figure 8.  
According to elastic mechanics [33], the stress field near the crack tip can be approximated as  
\[\sigma_{x}=\frac{K_{I}}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\bigg{(}1-\sin\frac {\theta}{2}\sin\frac{3\theta}{2}\bigg{)} \tag{10}\]  
\[\sigma_{y}=\frac{K_{I}}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\bigg{(}1+\sin\frac {\theta}{2}\sin\frac{3\theta}{2}\bigg{)} \tag{11}\]  
Figure 8: Stress near the crack tip. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{T}\) (\({}^{\circ}\)C) & **Linear Formula** & \(\mathbf{\mathbf{\mathbf{R}}^{2}}\) & **Exponential Formula** & \(\mathbf{\mathbf{\mathbf{R}}^{2}}\) \\ \hline
20 & y = 0.1420x & 0.9448 & y = 0.4288 exp(0.1049x) & 0.9728 \\
100 & y = 0.1379x & 0.9821 & y = 0.4328 exp(0.1025x) & 0.9637 \\
200 & y = 0.1274x & 0.9035 & y = 0.2376 exp(0.1444x) & 0.8713 \\
300 & y = 0.1077x & 0.8457 & y = 0.2959 exp(0.1100x) & 0.7027 \\
400 & y = 0.1184x & 0.7721 & y = 0.0508 exp(0.3512x) & 0.9421 \\
500 & y = 0.1129x & 0.8163 & y = 0.0430 exp(0.3687x) & 0.9184 \\
600 & y = 0.1054x & 0.9478 & y = 0.0335 exp(0.4255x) & 0.8092 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitting formulas for different temperatures\[\tau_{xy}=\frac{K_{I}}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\sin\frac{\theta}{2}\cos \frac{3\theta}{2} \tag{12}\]  
If the area around the crack tip does not undergo significant plastic deformation, it is considered to be an inelastic stress region when micro-cracks occur. The principal stress of the area near the crack tip can be expressed by elastic mechanics as  
\[\sigma_{1}=\frac{\sigma_{x}+\sigma_{y}}{2}+\sqrt{\left(\frac{\sigma_{x}- \sigma_{y}}{2}\right)^{2}+\tau_{xy}{}^{2}} \tag{13}\]  
\[\sigma_{2}=\frac{\sigma_{x}+\sigma_{y}}{2}-\sqrt{\left(\frac{\sigma_{x}- \sigma_{y}}{2}\right)^{2}+\tau_{xy}{}^{2}} \tag{14}\]  
where \(\sigma_{1}\) is the maximum principal stress. Substituting Equations (10-12) into Equation (13)  
\[\sigma_{1}=\frac{K_{I}}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\bigg{(}1+\sqrt{2} \sin\frac{\theta}{2}\cos\frac{3\theta}{2}\bigg{)} \tag{15}\]  
Previous research has identified the following main material fracture criteria: the maximum circumferential tensile stress criterion, strain energy density criterion, maximum energy release rate criterion, and modified maximum tangential stress (MMTS) criterion [11; 34; 35; 36; 37]. However, the experimental results available show that, for rocks, fracture toughness is controlled by the maximum principal stress (i.e., tensile strength) [11] and the expansion of microcracks is one of the causes of rock damage [11; 26; 38; 39; 40; 41; 42]. Microcrack propagation is caused by the microcrack being subjected to tensile stress rather than compressive stress or shear stress. Moreover, rock exhibits brittle properties, and the tensile strength is much smaller than the compressive strength. When the tensile stress in the rock exceeds the maximum tensile stress, damage will occur. Therefore, the condition of the maximum tensile stress being equal to the tensile strength, \(\sigma_{1}=\sigma_{t}\), can be considered a critical case [21]. When the stress intensity factor at the crack tip reaches a critical value, the crack expands, that is, \(K_{I}=K_{IC}\). If crack tip strength failure and crack initiation are used as equivalent criteria, the crack propagation radius reaches the critical crack propagation radius, that is, \(r\)=\(r_{\text{IC}}\). Equation (15) can be then be rewritten as  
\[\sigma_{t}=\frac{K_{IC}}{\sqrt{2\pi r_{IC}}}\cos\frac{\theta}{2}\bigg{(}1+ \sqrt{2}\sin\frac{\theta}{2}\cos\frac{3\theta}{2}\bigg{)} \tag{16}\]  
The crack propagation angle (\(\theta\)) is the direction of crack propagation. For a pure mode-I crack, the crack propagation angle \(\theta\) = 0\({}^{\circ}\).