## 5 Conclusions  
In this study, the key parametric characteristics of red sandstone, including mechanical properties, infrasonic amplitude variations, energy rates, and cumulative energy, were analyzed under both uniaxial and cyclic loading-unloading conditions. By utilizing infrasonic energy as a quantitative indicator of rock damage, the evolutionary mechanisms governing damage progression in red sandstone were comprehensively investigated, the main conclusions are as follows:  
(1) The infrasonic signal activity generated by rock under loading can be characterized by three distinct stages: relative stability period, active period, and pre-failure precursor period. During the low-stress stage (relative stability period), both infrasonic amplitude and cumulative energy increment remain minimal and stable, reflecting limited microcrack activity as the rock primarily undergoes elastic deformation. As stress progresses to intermediate levels (active period), significant fluctuations emerge in these parameters due to intensified microcrack interactions and localized damage accumulation and the infrasonic amplitude and cumulative energy increment exhibit significant variations, indicating intensified infrasonic activity. At the high-stress stage (pre-failure precursor period), the characteristics of infrasonic signals undergo the most significant changes, with the infrasonic amplitude reaching its maximum value and the cumulative energy increment attaining its highest level, signaling the accelerated microcrack coalescence preceding the macroscopic failure.  
(2) The significant variation in infrasonic energy rate typically manifests prior to peak strength, the infrasonic energy rate generally exhibits a gradually decreasing trend after its initial emergence. The infrasonic energy rate remains notably low prior to peak strength, and this characteristic becomes more pronounced under cyclic loading-unloading conditions. Therefore, the emergence of substantial infrasonic signals accompanied by a sustained increase in energy rate can serve as a precursor characteristic for rock failure.  
(3) Rock damage characterization based on infrasonic energy shows that both loading and unloading phases generate damage. Under external loading, rock damage exhibits a monotonically increasing and irreversible trend, with the curve demonstrating an upward-curving nonlinear growth pattern. Based on infrasonic energy, it is possible to characterize damage not only during the loading phase but also during the unloading phase. Infrasonic energy can effectively reflect the evolutionary process of internal crack damage in rocks.  
Conceptualization, K.Z. and P.Z.; methodology, M.Z. and X.X.; software, M.Z. and Z.L. ([PERSON]); validation, K.Z. and M.Z.; formal analysis, M.Z.; investigation, M.Z., K.Z. and Y.Y.; resources, M.Z., K.Z. and Z.L. ([PERSON]); data curation, M.Z.; writing--original draft preparation, M.Z.; writing--review and editing, M.Z. and Z.L. ([PERSON]); visualization, M.Z.; supervision, K.Z. and P.Z.; project administration, M.Z.; funding acquisition, K.Z. All authors have read and agreed to the published version of the manuscript.  
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (52164004, 52104086); the key research and development program of Jiangxi Province, China (20212 BBG71009); and the Science and technology project of Jiangxi Provincial Department of Education, China (GJJ209413).  
Not applicable.  
The original contributions presented in the study are included in the article. Further inquiries can be directed to the corresponding author.  
The authors gratefully acknowledge the generous support of the Jiangxi University of Science and Technology, which facilitated the conduct of this research.