## 4 Verification of the Brittleness Indexes  
### For Red-Bed Sandstone under Saturated Condition  
This study validates the brittleness evaluation using red-bed sandstone. The red-bed sandstone underwent triaxial compression tests under different confining pressures in its natural state and in its saturated state. The experimental process for red-bed sandstone under saturated conditions was as follows: the samples were divided into two groups, totaling eight samples. The first group underwent saturation testing, while no further operations were required for the second group. In the saturation test, the samples were placed in a water tank for natural absorption. Initially, water was injected to a height of 1/4 of the specimen and, every 2 h, water was injected to heights of 1/2 and 3/4 of the specimen. After 6 h, the samples were fully submerged, and after 48 h of water absorption, they were dried in a drying oven before testing. Each group of samples underwent triaxial compression tests under confining pressures of 2, 4, 8, and 16 MPa, resulting in stress-strain curves, as shown in Figure 3.  
Figure 3: The stress-strain curves of red-bed sandstones under different confining pressures: (a) 2 MPa; (b) 4 MPa; (c) 8 MPa; (d) 16 MPa.  
From the graph, it is evident that under confining pressures of 2, 4, 8, and 16 MPa, the post-peak curve is roughly similar, especially showing a multi-step downward trend under saturated conditions, while the peak strength of red-bed sandstone in its natural state is significantly higher than that of red-bed sandstone under saturated condition. After saturation, the strength of the sandstone deteriorates, and under low confining pressure, the degree of deterioration is higher. Furthermore, based on the axial strain-axial stress curve, it can be observed that the total strain of red-bed sandstone in its natural state is significantly lower than that of red-bed sandstone under saturated condition. This indicates that the brittleness is higher in its natural state compared to that under saturated conditions.  
Using the brittleness index formula proposed in this study, the brittleness values of red-bed sandstone under different confining pressures in its natural state and in its saturated condition can be calculated based on the stress-strain curves mentioned above. The results are shown in Figure 4. It can be observed that under natural conditions, the brittleness values of red-bed sandstone increased by 18.52% from 2 MPa to 4 MPa, 62.59% from 4 MPa to 8 MPa, and 48.65% from 8 MPa to 16 MPa. Under saturated conditions, the brittleness value increased by 50.22% from 2 MPa to 4 MPa, decreased by 78.93% from 4 MPa to 8 MPa, and decreased by 52.50% from 8 MPa to 16 MPa. Under saturated conditions, compared to the natural state, the brittleness values decreased by 21.55%, 75.17%, and 88.21% for 2 MPa, 8 MPa, and 16 MPa, respectively. Under saturated conditions at 4 MPa confining pressure, the brittleness value increased by 17.85% compared to the natural state. Therefore, both in the natural state and under saturated conditions, the brittleness index of red-bed sandstone decreases with increasing confining pressure, which is consistent with the trend of decreasing brittleness with increasing confining pressure in rock masses. However, an anomalous phenomenon is observed in the brittleness under saturated conditions at 4 MPa, which is believed to be attributed to experimental errors caused by sample collection. Overall, the brittleness of red-bed sandstone under saturated conditions is lower compared to its natural state.