## 3 Tensile Strength of Granite Samples After High-Temperature and Water-Cooling Treatment  
The tensile strength calculated by the Mohr-Coulomb model is unreasonably higher than the experimental result, and this discrepancy can usually be corrected by introducing the \"tension cut-off\" criterion [36]. Therefore, it is also required to study the tensile strength of granite samples after high-temperature and water cooling treatment.  
The pre-existing experimental results have shown that high temperature heating and water-cooling treatment has obvious effects on the tensile strength of granite [17; 23; 25; 32], however, it is still necessary to obtain the general evolution behaviors of tensile strength with different treatment situations. Table 5 lists the basic information of the samples and treatment conditions for studying the tensile strength variation characteristics.  
In order to systematically compare the tensile strength of different granite samples, the normalized tensile strength \(\sigma_{\text{IN}}\) is defined as shown in Equation (6).  
\[\sigma_{\text{IN}}=\sigma_{\text{LT}}/\sigma_{\text{IR}} \tag{6}\]  
where \(\sigma_{\text{LT}}\) is the tensile strength of granite after the heating treatment with temperature of \(T\)\({}^{\circ}\)C and water cooling, and \(\sigma_{\text{IR}}\) is the tensile strength of granite under room temperature. The normalized tensile strength \(\sigma_{\text{IN}}\) of different granite samples are plotted in Figure 8.  
Figure 8: Normalized tensile strength of different granite samples and the linear fitting curve.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Samples** & **Sample Size (mm)** & **Mineral Contents 1** & **Heating Rate** & **Cooling Water 2** & **References** \\ \hline Gonghe & & & Pl (40\%-50\%), & & 20 \({}^{\circ}\)C & \\ granite-1 & \(\varphi\)50 \(\times\) 25 & Qz (20\%-25\%), & & 3–5 \({}^{\circ}\)C/h & & \\ Gonghe & & Bt (5\%-10\%) & & 60 \({}^{\circ}\)C & \\ granite-2 & & & & & \\ \hline Gonghe & & & & 100 \({}^{\circ}\)C & \\ granite-3 & & & & & \\ \hline Songliao & \(\varphi\)50 \(\times\) 25 & Qz (26.1\%), Mi (8.9\%), & 30 \({}^{\circ}\)C/h & 20 \({}^{\circ}\)C & [17; 37] \\ granite & & & PF (36\%), Pf (25.2\%) & & \\ Rizhao & \(\varphi\)50 \(\times\) 25 & PI (35\%), PF (40–45\%), & & 3 \({}^{\circ}\)C/h & RM & [23] \\ granite & & & Qz (20–25\%), Bt (3–5\%) & & & \\ Pingyi & \(\varphi\)50 \(\times\) 25 &
\begin{tabular}{c} II (25\%), Qz (28\%), \\ FS (43\%) \\ \end{tabular} & N/A & RM & [25; 38] \\ granite & & & & & \\ \hline \hline \end{tabular} \({}^{\dagger}\)Qz— quartz; Pl— plagioclase; Mc—microcline; Mi—mica; PF—Potassium feldspar; Ab—albite; Bt—biotite; Il—little; Fs—feldspar. \({}^{2}\)RM—room temperature.  
\end{table}
Table 5: Samples and treatment conditions.  
It is observed that the normalized tensile strength \(\sigma_{\text{tN}}\) of these six groups of granite samples decrease generally linearly with the increasing heating temperature, with the similar slope, so a unified empirical Equation (7) can be fitted to describe the tensile strength behaviors of different granite samples. The normalized tensile strength and the linear fitting curve are shown in Figure 8.  
\[\sigma_{\text{tN}}=mT+n \tag{7}\]  
where \(m\) and \(n\) are both fitting coefficients. Figure 8 also presents the linear fitting curve of the normalized tensile strength \(\sigma_{\text{tN}}\) with increasing heating temperature \(T\). It can be obtained that \(m=-0.0013\), and \(n=1.03\), with \(R^{2}=0.94\) for these groups of experimental data.