## 1 Introduction  
As underground engineering has progressed to vast and burial depths and spans, the stress state of the rock mass has become more complex. Rock mass collapses are common, with tensile, compressive-shear, and tensile-shear failure being three common failure types [1]. Mechanical fatigue can be viewed as a process of gradual accumulation of damage in rock specimens under stress or strain, a process that leads to changes in the structure and properties of rock specimens and the creation and expansion of cracks and fractures [2]. When an external load is applied to a rock specimen, it generates stresses in different directions, and these stresses, when reaching their limit values, cause small cracks or deformations inside the specimen [3, 4], which results in the generation of AE signals. Laboratory testing is an essential tool for investigating rock failure processes. The use of AE monitoring during testing to assess the elastic wave signals of internal rupture of rock specimens or rock-like specimens can efficiently predict the evolutionary stage of specimen failure, providing a theoretical foundation for disaster monitoring and early warning [5]. There have been many valuable studies on the AE evolution of specimens during direct tensile and compressive-shear failure compared to tensile-shear failure [6, 7, 8, 9, 10, 11] but few reports on the AE evolution of rock specimens of the same shape and size under direct tensile, tensile-shear, and compressive-shear failure modes. Experimental investigations on tensile-shear failure have primarily focused on the analysis of specimen rupture characteristicsand the formulation of strength requirements [12; 13], but the depth is insufficient in studies on the evolution of AE signals during failure. At present, there is no research on quantitative frequency non-band classification methods for different rock samples. Previous researchers mainly determined the frequency characteristics of rock acoustic emission signals by band distribution, and most of the previous studies showed that rock acoustic emission signals present multi-band characteristics [9; 11; 12], and the specific values should be divided into three or five bands according to the nature of the rock samples, which is a hierarchical distribution characteristic. Different rocks and rock samples have different frequency-band distributions [8; 14], which do not allow diverse frequency clustering analysis. This method is currently used more in the field of geotechnical testing, but the use of frequency-band classification is sufficient to reflect the frequency characteristics. AE signal analysis primarily consists of parametric and waveform analysis. The former describes the features of AE signals using a set of simplified waveform parameters, such as count, energy, amplitude, duration, etc. [15]. The latter refers to selecting appropriate signal-processing methods for AE signal time-domain waveforms to obtain AE signal characteristics, such as Fourier transform-based spectral analysis, time-domain analysis, etc. [16]. The amplitude is an essential characteristic feature of the AE waveform, reflecting the maximum amplitude in the signal decay waveform, which can reflect the magnitude of the AE signal and is frequently used to perform a preliminary crack scale assessment [17]. In many cases, the study of the key physical variables characterizing the AE signal from the frequency domain is more effective and intuitive in reflecting the information and occurrence mechanism of rock fracture [18] than the research of AE waveform features in the time domain. Many scholars have paid close attention to rock failure spectral features study recently. [PERSON] [19] discovered that the AE spectrum of marble failure tests was characterized by an apparent double peak frequency (PF) independent of the test method and that the double peak frequency could represent two separate microfracture forms of the rock.