## 7 Conclusions  
(1) The uniaxial reload deformation characteristics of sandstone are controlled by the degree of unloading before the peak. Sandstone with 40% to 70% of the unloading point is similar to intact sandstone, showing plastic-elastic deformation characteristics. The sandstone at 75% of the unloading point goes through the pore compacting stage, the linear elastic stage, and the crack stable extension stage, significantly shorter than those at 40% to 70% of the unloading point, and enters the crack unstable extension stage more quickly, showing plastic-elastic-plastic body deformation characteristics. With the increase in the unloading point to 90%, the sandstone directly enters the crack unstable extension stage, showing the elastic-viscous body deformation characteristics.  
(2) The reload sandstone acoustic emission characteristics before peak unloading coincide with the deformation characteristics. The cumulative energy characteristics of the sandstone at 40% to 60% of the unloading point are similar to those of the intact sandstone and can be roughly divided into three stages: smooth, stable growth, and secondary smooth. The damage intensity of the sandstone at 65% and 70% of the unloading point decreases, and the acoustic emission energy characteristics are mainly concentrated in the crack expansion stage. The crack initiation strength of the sandstone at 75% of the unloading point is similar to the damage strength, and the acoustic emission energy characteristics reveal a \"double peak\" phenomenon. The crack initiation intensity and damage intensity of the sandstone at 80-90% of the unloading point overlap, the acoustic emission energy characteristics are active, and the cumulative energy characteristics show a \"stepped\" growth.  
(3) The pre-peak unloading rupture pattern of the sandstone determines the reload damage characteristics. In uniaxial reloading, 40% to 70% of the unloading point specimens are consistent with the damage characteristics of intact sandstone, and the new primary cracks show \"Y\"-type double-slope shear damage characteristics. The secondary primary crack and a secondary secondary crack developed based on the tensile rupture pattern before the peak in 75% of the specimens at the unloading point, and the secondary primary  
Figure 14: Single-axis re-load crack closure expansion law.  
crack was the main cause of the damage to the specimens. The damage path of 80% to 90% of the specimens at the unloading point occurs in shear damage along the pre-peak unloading rupture morphology.  
(4) A closed crack mechanics analysis model under uniaxial reload was established, while the basic solution of pseudo-force for fine microcracks subjected to far-field stress and the stress intensity factor at the crack tip and the crack fracture angle were theoretically derived. Further, the relationship between the fracture angle \(\theta\) of rock compression-shear cracks, the crack angle \(\beta\), and the friction coefficient f at the crack surface was clarified.  
Conceptualization, Y.Z. and P.L.; methodology, Y.Z. and P.W.; software, C.M.; validation, Y.Z. and P.L.; formal analysis, H.R.; investigation, H.R. and X.W.; writing--original draft preparation, P.L.; writing--review and editing, P.L. and C.M.; supervision, Y.Z. and P.W.; funding acquisition, Y.Z. All authors have read and agreed to the published version of the manuscript.  
The work presented in this paper was financially supported by the National Natural Science Foundation of China (Grant No. 52274119, 52174110), the Natural Science Foundation of Hunan Province (No. 2021 JJ30273), and the Open Research Fund Program of Hunan Provincial Key Laboratory of Safe Mining Techniques of Coal Mines, Hunan University of Science and Technology (No. 969-E22227).  
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
Data Availability Statement: Data available on request due to privacy restrictions.  
The authors declare no conflict of interest.