## 4 Relationship between \(K_{ic}\) and \(\sigma_{t}\)  
### Empirical Relationship  
Underground engineering is carried out in a high-temperature environment, and temperature should therefore be a key consideration in such contexts. Exposure to high temperature will change the physical and mechanical properties of the rock [5; 26; 27; 28]. Table 2 combines the results obtained in this study and experimental data obtained by other studies [18; 29; 30; 31] regarding the effects of temperature on \(K_{IC}\) and \(\sigma_{t}\). The \(K_{IC}\) test methods used in these other studies were SR for FS marble and FS gabbro, SCR3 PB for Stripa granite, TPBSEN for BS granite, and SCB for Sichuan sandstone. Tensile strength was obtained via BD testing for FS marble, FS gabbro, Stripa granite, and Sichuan sandstone. In this paper, 25 \({}^{\circ}\)C is approximated as 20 \({}^{\circ}\)C.  
Table 2 shows that there is a large amount of variation in the mode-I fracture toughness and tensile strength of rocks treated to the same temperature in the different studies. This variation is due to differences in the rocks tested and test methods employed. In tests of the tensile strength of rock, the test method, rock heterogeneity, contact of the iron plate of the machine with the rock, etc., will all affect the test results. Results regarding the mode-I fracture toughness of rock will also be affected by such factors [22]. It can also be seen from Table 2 that both the mode-I fracture toughness  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \multicolumn{2}{c}{**FS Marble**} & \multicolumn{2}{c}{**FS Gabbro**} & \multicolumn{2}{c}{**Stripa Granite**} & \multicolumn{2}{c}{**BS Granite**} & \multicolumn{2}{c}{**SC**} \\  & & & & & & & & & & **Sandstone** \\ \hline \(\mathbf{T}\) & \  
and the tensile strength gradually decrease with heating to greater temperature. This indicates that temperature exposure has an effect on mode-I fracture toughness and tensile strength. Straight-line relationships between the tensile strength and mode-I fracture toughness of rocks exposed to specific temperatures can be plotted on the basis of the data in Table 2 (Figure 7), and the expressions for these lines can be obtained by fitting. Figure 7: _Cont._  
It can be seen from Figure 7 that, for rocks exposed to the same temperature, \(K_{IC}\) gradually increases with an increase in \(\sigma_{t}\), making \(K_{IC}\) and \(\sigma_{t}\) broadly positively correlated. Table 3 gives the empirical formulas relating tensile strength and mode-I fracture toughness, where \(y\) represents mode-I fracture toughness and \(x\) represents tensile strength. The manner that \(K_{IC}\) varies with \(\sigma_{t}\) at each processing temperature is reflected by the slope of the fit line, termed the proportionality factor. In the linear fits, the proportionality factor for 20 \({}^{\circ}\)C, 0.1420, is the largest. That is, \(K_{IC}\) changes most with \(\sigma_{t}\) in rock exposed to this temperature. The proportionality factor for 600 \({}^{\circ}\)C is the smallest, 0.1054; that is, \(K_{IC}\) changes least with \(\sigma_{t}\). Therefore, the positive correlation between \(K_{IC}\) and \(\sigma_{t}\) is different for rock that has been exposed to different temperatures. There are also differences in the fitting coefficient R\({}^{2}\). For processing to 20 \({}^{\circ}\)C, 100 \({}^{\circ}\)C, 200 \({}^{\circ}\)C, and 600 \({}^{\circ}\)C, the fitting coefficients R\({}^{2}\) were 0.94, 0.98, 0.90, and 0.95, respectively, for all of which R\({}^{2}\)\(\geq\) 0.90. \(K_{IC}\) and \(\sigma_{t}\) thus show a very obvious linear relationship under these processing conditions. For processing temperatures ranging from 300 \({}^{\circ}\)C to 500 \({}^{\circ}\)C, the fitting coefficient ranged from 0.77 to 0.85. These coefficients reflect a less pronounced linear relationship between \(K_{IC}\) and \(\sigma_{t}\).