## 4 Results and Discussions
### Mechanical Response Considering Softening Effect  
#### 4.2.1 Strength Characteristics  
The softening effect considered in numerical simulation is essentially to attenuate the contact cohesion and contact tensile strength according to Equations (7) and (8). In this paper, soft rock specimens were immersed in water for 1 day, 4 days, 8 days, and 15 days, respectively. Since the attenuation degree of internal friction angles in the test is small, the contact internal friction angles of the heterogeneous models with different immersion time are consistent with the values in Table 4. The parameter calibration process described in Section 3 is used.  
Figure 9 shows the stress-strain (displacement) simulation curves for heterogeneous models with different immersion time. The simulation results indicate that the post-peak characteristics of the stress-strain curves of uniaxial compression are quite evident, which is due to the presence of quartz and feldspar with very good mechanical properties in the heterogeneous model. During the loading process, the mineral composition of the model increases its resistance to external loads, resulting in a gradual decrease in the stress-strain curve after reaching the peak stress. According to Figure 10, the uniaxial compressive strengths for different immersion times of 0 days, 1 day, 4 days, 8 days, and 15 days are 29.65 MPa, 4.40 MPa, 2.18 MPa, 1.88 MPa, and 1.21 MPa, respectively, with errors compared to the average experimental values of 1.61%, \(-\)5.38%, \(-\)8.40%, \(-\)4.44%, and \(-\)5.47%, respectively. The tensile strengths for the same immersion time are 2.656 MPa, 0.332 MPa, 0.152 MPa, 0.114 MPa, and 0.085 MPa, respectively, with errors compared to the average experimental values of \(-\)0.11%, \(-\)7.78%, \(-\)5.59%, 3.64%, and 4.94%, respectively.  
In summary, the small errors observed in both uniaxial compressive strengths and tensile strengths support the validity of the meso-scale softening damage factor to some extent. These errors suggest that the meso-scale softening damage factor has the potential to capture the softening behavior of soft rocks reasonably well. However, the presence of discrepancies between simulated and experimental results indicates that there may be areas where the model can be further refined or enhanced to improve its accuracy in representing the mechanical response of materials. It is crucial to consider the limitations and assumptions associated with the meso-scale softening damage factor and validate its applicability across different materials, loading conditions, and degradation mechanisms. Overall, while the errors are relatively small, they serve as indicators for potential areas of improvement in the meso-scale softening damage factor. Further calibration, validation against additional experimental data, and refinement of the modeling approach could help enhance the prediction accuracy and reliability of the model, leading to a better  
Figure 9: Stress–strain (displacement) curves of heterogeneous models with different immersion time: (**a**) uniaxial compression condition; and (**b**) Brazilian splitting condition.  
understanding of the softening behavior of materials and improved applicability in practical engineering scenarios.