## 2 The Existing Methods for Determining Rock Brittleness
The higher the degree of plastic deformation, the lower the brittleness of the rock mass, while the brittleness is independent of the amount of elastic deformation during the failure process. Therefore, if the amount of plastic strain occurring prior to failure is minimal, the rock mass is considered more brittle. As shown in Figure 2,  
Figure 1: Simplified stress-strain curves. the brown area represents the stored elastic strain energy prior to failure, and the slope of the line dividing the yellow area and the brown area corresponds to the elastic modulus. Hence, \(2W_{\text{e}}/E\) can be used to obtain the elastic strain occurring prior to failure, while the deformation prior to failure consists of plastic deformation and elastic deformation. Therefore, subtracting the elastic deformation from the total strain prior to failure provides the plastic deformation prior to failure, thus evaluating the brittleness value of the rock mass. The formula for the pre-peak brittleness index is given as follows:  
\[B_{pre}=log\left(\frac{1}{\varepsilon_{p}-\frac{2W_{e}}{E}}\right), \tag{1}\]  
where \(\varepsilon_{p}\) is the strain before peak strength, \(W_{e}\) is the elastic strain energy, and \(E\) is the elastic modulus. Herein, the log symbol represents the log for the decadic logarithm. Analyzing the brittleness of rock masses solely from an energy perspective prior to the peak does not allow for distinguishing between the energy required to overcome cohesive forces and the energy needed for friction and heat generation. Therefore, in this study, the brittleness of the rock mass after the peak is assessed based on plastic strain. This approach provides a comprehensive evaluation considering the specific energy consumed during crack propagation and accounts for the rock mass's post-peak behavior. The post-peak brittleness index represents the total strain energy required for failure after reaching the peak stress. The difference in crack propagation between brittle and non-brittle rocks during the failure process using a schematic diagram is demonstrated. It can be observed that brittle materials require less energy for crack propagation, while non-brittle materials require energy to sustain crack growth. Therefore, the lower the energy required for crack propagation in the post-peak stage, the higher the brittleness of the rock mass. As shown in Figure 2, the energy absorbed by the rock mass prior to the peak is divided into dissipation energy (yellow area) used for crack propagation, overcoming cohesive forces, friction, and heat dissipation, and the stored elastic strain energy (brown area) of the rock mass. Some of the stored elastic energy is utilized for crack propagation after the peak, while the remaining portion is residual elastic energy (green area). The energy for rock mass failure after the peak includes not only the stored elastic energy prior to the peak but also the external energy input after the peak (wall shape). Therefore, the total energy for post-peak rock mass failure is given by (brown area + green area \(-\) green area), which can be expressed as \(W_{\text{e}}+W_{\text{out}}-W_{\text{re}}\). However, \(W_{\text{e}}+W_{\text{out}}-W_{\text{re}}\) is inversely  
Figure 2: Schematic diagram of energy evolution for different stages. proportional to the brittleness of the rock mass. Therefore, the reciprocal form is used to express the brittleness of the rock mass. \[B_{post}=log\frac{1}{W_{e}+W_{out}-W_{re}}, \tag{2}\]  
where \(W_{e}\) is the stored elastic strain energy before the peak, \(W_{out}\) is the external energy input after the peak, and \(W_{re}\) is the residual elastic strain energy. In this study, a logarithmic function is employed to reduce the numerical disparity between the two components of the brittleness index, thus avoiding the influence of one component's magnitude on the magnitude of the other component. For the coupling of the two brittleness index components, a multiplication form is selected, following the same principle as mentioned earlier.