## 4 Verification of the Brittleness Indexes
### For Cyan Sandstone under Freezing-Thawing Condition  
To validate the feasibility of the proposed method, another type of rock--cyan sandstone--collected from different regions, was used in this study. Triaxial compression tests were conducted under natural conditions and freezing-thawing conditions, with confining pressures set at 0, 3, 6, 9, and 12 MPa. The stress-strain curves of cyan sandstone under natural conditions and freezing-thawing conditions at these confining pressures are shown in Figure 5. From the graph, it can be observed that, under natural conditions, both the peak strength and the residual strength of cyan sandstone gradually increase with increasing confining pressure. This trend is also observed under freezing-thawing conditions. At the same confining pressure, the peak strength of the rock after freezing-thawing is significantly lower than that under natural conditions, while the difference in residual strength is not significant. For 0 and 3 MPa, the peak strain of the rock after freezing-thawing is significantly higher than that under natural conditions, while for high confining pressure, the peak strain difference between the two is smaller than that  
Figure 4: The brittleness index of red-bed sandstones: (**a**) Under natural conditions; (**b**) Under saturated conditions.  
under low confining pressure. This may be because the confining pressure suppresses the deformation of the rock. Clearly, during the initial compaction stage, the stress-strain curve shows a distinct upward concavity. In this stage, the initial cracks inside the rock gradually close under pressure until compaction is no longer possible, accumulating energy gradually. In the post-peak stage, the rock exhibits significant brittleness, with a noticeable stress drop, indicating that characterizing rock brittleness through stress-strain curves before and after the peak stress is feasible.  
The brittleness indexes calculated using the proposed formulas are shown in Figure 6. Under natural conditions, the pre-peak brittleness index, post-peak brittleness index, and overall brittleness index generally increase with increasing confining pressure, except for the pre-peak brittleness index at 3 MPa confining pressure, which is higher than the pre-peak brittleness index at 0 MPa confining pressure. This is because under a confining pressure of 3 MPa, the pre-peak curve has a weak corrective effect on the rock brittleness. Overall, the pre-peak brittleness index is significantly higher than the post-peak brittleness index. Furthermore, the trend of Bnew is basically consistent with Bpost, and its value is not significantly different, indicating that the post-peak brittleness index contributes more to the rock's brittleness, while the pre-peak brittleness index plays a supporting role in adjusting the size of Bnew. Similar trends are observed under freezing-thawing conditions. Additionally, at the same confining pressure, the brittleness indexes under natural conditions are higher than those under freezing-thawing conditions. Specifically, under freezing-thawing conditions at 0 MPa, 3 MPa, 6 MPa, 9 MPa, and 12 MPa, the brittleness indexes of cyan sandstone decrease by 8.69%, 31.39%, 17.00%, 13.57%, and 15.69%, respectively, compared to the brittleness indexes under natural conditions. In  
Figure 5: The stress-strain curves of cyan sandstones under different confining pressures: (**a**) 0 MPa; (b) 3 MPa; (c) 6 MPa; (d) 9 MPa; (e) 12 MPa.  
conclusion, the brittleness of cyan sandstone decreases with increasing confining pressure, and freezing-thawing action can reduce the brittleness of the rock.