## 2 The Proposed Rock Brittleness Index  
According to the theory of elasticity, the axial strain of a lossless material under the pseudo-triaxial condition can be expressed as follows:  
\[\varepsilon_{1}=\frac{\sigma_{1}-2\
u\sigma_{3}}{E} \tag{1}\]  
where \(\sigma\) and \(\varepsilon\) represent stress and strain, respectively, and the subscripts denote the stress direction; \(\
u\) and \(E\) symbolize the Poisson's ratio and elastic modulus, respectively. Similarly, the axial strain of a damaged material is obtained from Equation (1) using [PERSON]'s strain equivalence assumption [21], i.e.,  
\[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-2\
u\widetilde{\sigma}_{3}}{E} \tag{2}\]  
where the superscript denotes the effective stress. Let the damage variable be \(D\); Equation (2) is rephrased as  
\[\varepsilon_{1}=\frac{\sigma_{1}-2\
u\sigma_{3}}{E(1-D)} \tag{3}\]  
Transforming Equation (3) yields  
\[\sigma_{1}=E\varepsilon_{1}(1-D)+2\
u\sigma_{3} \tag{4}\]  
According to the statistical damage theory [22], the damage variable can be determined by the following:  
\[D=1-\exp\left[-\left(\frac{\varepsilon_{1}}{F}\right)^{m}\right] \tag{5}\]  
where \(m\) and \(F\) denote two fitting parameters. Substituting Equation (5) into Equation (4) leads to  
\[\sigma_{1}=E\varepsilon_{1}\exp\left[-\left(\frac{\varepsilon_{1}}{F}\right) ^{m}\right]+2\
u\sigma_{3} \tag{6}\]It is clear that the following relationship is satisfied when the stress reaches the peak point:  
\[\sigma_{1}|_{\varepsilon_{1}=\varepsilon_{\rm 1p}}=\sigma_{\rm 1p} \tag{7}\]  
\[\left.\frac{d\sigma_{1}}{d\varepsilon_{1}}\right|_{\varepsilon_{1}=\varepsilon _{\rm 1p}}=0 \tag{8}\]  
where \(\sigma_{\rm 1p}\) and \(\varepsilon_{\rm 1p}\) denote the peak stress and peak strain, respectively. Substituting Equation (6) into Equations (7) and (8) yields the display expressions for the fitting parameters m and F as follows:  
\[m=\frac{1}{-\ln(1-D_{\rm P})} \tag{9}\]  
\[F=\varepsilon_{\rm 1p}\left[-\ln(1-D_{\rm P})\right]^{-1/m} \tag{10}\]  
where \(\rm D_{\rm P}\) represents the damage value at the peak point. It can be obtained by transforming Equation (4). \[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-2\
u\widetilde{\sigma}_{3}}{E} \tag{2}\]0  
According to the damage mechanics, it is found that the faster the damage develops, the more brittle the rock [23]. Changes in damage variables can be used to represent the rock's brittleness. Therefore, we can define a brittleness index as follows:  
\[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-2\
u\widetilde{\sigma}_{3}}{E} \tag{2}\]1  
Generally, the damage variable (D) ranges from 0 to 1, but it cannot reach 1 due to residual strength and other factors. Previous studies have shown that the damage variable reaches approximately 0.9 when the rock is destroyed [24]. D is taken as 0.9 in this paper. At a damage variable of 0.9, the physical interpretation of Equation (12) is shown in \[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-2\
u\widetilde{\sigma}_{3}}{E} \tag{2}\]2 This indicates that the faster damage development corresponds to a more brittle rock. Incorporating Equations (9) and (10) into Equation (12), the following formula is derived:  
\[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-2\
u\widetilde{\sigma}_{3}}{E} \tag{2}\]3  
\[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-2\
u\widetilde{\sigma}_{3}}{E} \tag{2}\]4 A schematic diagram for the definition of the proposed brittleness index. Additionally, substituting Equation (11) into Equation (13), the brittleness index, which depends on the peak strain, peak strength, Poisson's ratio and elastic modulus, can be expressed as  
\[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-2\
u\widetilde{\sigma}_{3}}{E} \tag{2}\]5  
By making \(\ln\left(\frac{\sigma_{1p}-2\
u\sigma_{3}}{E\varepsilon_{1p}}\right)=A\), the brittleness index (\(B\)) can be abbreviated as follows:  
\[\varepsilon_{1}=\frac{\widetilde{\sigma}_{1}-2\
u\widetilde{\sigma}_{3}}{E} \tag{2}\]6  
The brittleness index B characterizes the brittleness of the rock or rock mass.