## 1 Introduction  
The intrinsic mechanical mechanism of the continuous deformation of the roadway surrounding rock after excavation is the process of stress adjustment of the perimeter rock from a higher triaxial stress state until it reaches equilibrium [1; 2; 3]. This process causes the weakening of the surrounding rock, causing power disasters such as rock bursts, seriously affecting the construction of the project and the personal safety of workers. Therefore, it is of great significance to study the mechanical properties of the unloaded rock body to analyze the stability of the surrounding rock and to determine a reasonable support method for the surrounding rock.  
After the roadway is excavated, the surrounding rock stresses are adjusted again, forming crushed zones, fissure zones, and original rock stress zones. Many scholars have carried out a series of studies on the phenomenon of decreasing confining pressure and increasing axial pressure within the rift zone, focusing on different loading and unloading paths [4; 5; 6; 7], fractal characteristics [8], acoustic emission signals [9; 10; 11; 12], and energy evolution [13]. For example, [PERSON] [5] designed an incremental cyclic loading and unloading pressure test (ICLUP text) and proposed a new model to characterize the evolution of pre-peak unloading damage; [PERSON] [8] found an exponential relationship between the energy consumption density and the average size of rock chips through impact tests and analyzed the impacted fragments. In engineering, acoustic emission (AE) monitoring systems are commonly used to monitor the signals generated by the internal rupture of the material during the whole process from loading to damage and are used to analyze the damage and destruction characteristics of the material [14; 15; 16; 17; 18]. For example, [PERSON] [19] proposed a correction method for the \(b\)-value of acoustic emission and used it to analyze local damage in reinforced concrete structures subjected to dynamic loading; [PERSON] [20] quantified the extent of damage to concrete structures under fatigue loading by analyzing the acoustic emission signals generated from three-point bending tests with four different concrete volumes in terms of \(b\)-value, improved \(b\)-value (Ib-value), severity, and strength. However, during the stress adjustment process, the stress on the radial direction of the roadway is distributed in a gradient, which causes structural level damage to the surrounding rock, so the consideration of gradient stress more accurately presents the surrounding roadway rock stress situation. In this regard, [PERSON] [21] analyzed the stress distribution pattern around the roadway and provided an analytical solution for the stress distribution in the plastic zone of the circular roadway; [PERSON] [22] proposed the theory of gradient damage, which suggests that the stress gradient at depth is an important factor in the occurrence of damage in the surrounding rock.  
After the roadway is excavated, under the action of gradient stress, the surrounding pressure forms different unloading spans, causing different degrees of damage to the surrounding rock. Numerous scholars have focused on the study of the mechanical properties of the rock body after stress redistribution and different damages to the surrounding rock, but the study of what kind of bearing capacity the critically damaged surrounding rock still has is still relatively weak. Therefore, this paper designs 13 groups of plus and minus circumferential pressure schemes to explore the critical damage range of coal samples. Secondly, a uniaxial loading test was carried out on the critically damaged coal samples, and finally, AE acoustic emission signal, fractal theory, and energy theory were utilized to investigate the characteristics of the critical damage of coal samples under the action of different radial gradient stresses, so as to provide theoretical guidance for the analysis of the stability of surrounding rock and the support of the roadway.