## 1 Introduction  
The mining process in deep mines encounters severe conditions characterized by \"high geo-stress, high temperature and high osmotic pressure\" [1, 2, 3]. The disruptive impact of repeated loads, such as construction blasting and face mining, disrupts the natural equilibrium of coal resources. This alteration affects the strength, deformation, and damage characteristics of the coal body, making it susceptible to dynamic disaster incidents related to the destabilization and destruction of coal rock. Consequently, this poses a significant risk to the safe mining operations in coal mines [4, 5, 6]. The mechanical behavior and energy characteristics of raw coal under cyclic loading conditions differ from those observed under conventional stress conditions. Therefore, it is essential to investigate the deformation damage and energy evolution patterns of the coal body subjected to cyclic loading with varying cyclic periods. Rocks exhibit diverse mechanical behaviors and damage patterns when subjected to various dynamic loads [7, 8, 9]. [PERSON] [10, 11, 12] studied the deformation characteristics and damage features of fracture-containing rocks under uniaxial and triaxial cyclic loading and pointed out that localized damage and deformation occur during the unloading of confining stress. Additionally, radial and volumetric strains increase significantly, the deformation modulus initially rises and then declines, and Poisson's ratio continuously increases. [PERSON] [13; 14; 15] conducted triaxial cyclic loading and unloading compression tests on sandstones subjected to varying upper limit axial stresses. This research revealed the rules of damage evolution, life prediction, and the damage mechanisms of sandstones. [PERSON] [16; 17] used conventional and cyclic triaxial dynamic load tests at various frequencies on coal rock under high stress to investigate the damage mechanisms and the laws governing strength deterioration of coal rock induced by dynamic loading. [PERSON] [18] analyzed the mechanical properties and crack extension behavior of sandstone through a series of triaxial experiments and proposed a damage variable based on crack evolution. [PERSON] and [PERSON] [19; 20] investigated the deformation characteristics of coal samples under various cyclic loading and unloading stress paths. They established the evolution law of damage variables based on the loading and unloading response ratio. The elastic modulus and Poisson's ratio of the rock during variable amplitude loading exhibit a nonlinear decrease and increase, respectively, in relation to the loading rate, and show a negative correlation with the stress amplitude [21]. The stress paths utilized in the aforementioned studies primarily involve cyclic variations in confining stress or axial stress alone. There is a scarcity of experimental research examining the cyclic loading and unloading of rocks while simultaneously altering both axial and confining stress. Consequently, there is a significant gap in the literature regarding the impact of cyclic periods on the deformation properties of rocks under conditions of simultaneous changes in axial and confining stress. The deformation damage of rock is a process that progresses from local rupture to partial destruction and ultimately to complete crushing and destabilization, this process is accompanied by energy input, accumulation, and dissipation. In essence, rock damage is fundamentally an energy-driven process of destabilization and destruction [22; 23; 24; 25]. Currently, many scholars have sought to understand the behavior of rock during cyclic loading and unloading from an energy perspective. However, most of their research has concentrated on uniaxial and conventional triaxial experimental methods [26; 27; 28]. [PERSON] [22] performed uniaxial cyclic loading and unloading tests on coal samples at various loading rates and analyzed the interaction mechanisms of input, elastic, and dissipated energies at each stage, as well as their correlations with loading rates.