## 5 Conclusions  
The deformation and failure law of sandstone after confining pressure unloading damage was studied using laboratory tests and numerical simulation. This can be used to effectively analyze the structural damage changes of sandstone in the process of reloading after confining pressure unloading from a microscopic perspective such as the crack hot spot, force chain structure, and moment magnitude. The main conclusions drawn based on the findings of the study are summarized below.  
1. There was a positive correlation between the confining pressure unloading and the peak strain, with the circumferential peak strain proving to be more sensitive to confining pressure unloading damage effects. A negative correlation was observed between the strength of a specimen and its confining pressure unloading. The decrease in strength was more apparent as the unloading failure confining pressure approached. Confining pressure had a certain inhibitory effect on the attenuation of the bearing capacity of the specimens.
2. The initiation stress (\(\sigma_{ci}\)) decreased more than the damage stress (\(\sigma_{cd}\)) with increasing confining pressure unloading. The microdefects in the specimens further propagated and developed under confining pressure unloading conditions, shortening the stable crack propagation stage; the differences between the initiation stress (\(\sigma_{ci}\)), damage stress (\(\sigma_{cd}\)), and triaxial compressive initiation stress level (\(K\)) also decreased.
3. With a gradual increase in the confining pressure unloading, the penetration fracture zone was more evident and expansive in the model, and the distribution of dense crack areas was more concentrated in the fracture zone and fracture area. The total number of cracks increased, with tensile cracks being dominant and accompanied by tension-shear and compression-shear cracks as a result of deformation and failure caused by pressure-induced tensile fractures.
4. The anisotropic mechanical behavior of the model was evident under deformation and failure, with increases in bond breaking, weakening of the normal contact force, and enhancement of the tangential contact force, all of which led to serious damage. The average interval of the tangential contact force was the largest in the direction of crack expansion and propagation. The strong force chains were shown to primarily bear external loads, whereas the weak force chains played a key auxiliary role in maintaining stability. Macroscopic fractures occurred easily in the vacuum areas of the force chains.
5. The number of cracks developing in the confining pressure unloading damage process indicated that the loading process did not cause damage to the specimens. The  
Figure 16: Statistical frequency and proportional distribution of moment magnitude values. (**a**) Confining pressure unloading damage, (**b**) Loading after confining pressure unloading damage.  
moment magnitude distributions changed from randomly scattered distributions to regular accumulations, with dominant fractures accumulating to form potential fractures. Finally, fracture zones further propagated and formed on the dominant fractures based on the damage caused by confining pressure unloading disturbance. Increasing levels of acoustic emissions caused the distributional range of the moment magnitude to expand and the average moment magnitude to decrease, indicating that the overall degree of damage caused by bonding fractures in the model gradually intensified.  
Conceptualization, B.M. and X.C.; methodology, B.M.; software, B.M.; validation, X.D.; formal analysis, B.M.; data curation, X.D.; writing--original draft preparation, B.M.; writing--review and editing, X.D., X.C.; supervision, X.D.; funding acquisition, X.C. All authors have read and agreed to the published version of the manuscript.  
This work was funded by the National Natural Science Foundation of China, grant numbers 51979218 and U1965107.  
Data Availability Statement: Data will be made available on request.  
The authors declare no conflict of interest.