## 3.1 Evolution stage division of multivariate parameters

The multivariate parameters include shear force, displacement, ringing count rate, accumulative ringing count, energy rate, accumulative energy, natural frequency and time variation.

The rock mass can be considered as a mechanical system composed of physical parameters such as stiffness, mass, and damping. When the rock mass is damaged under the stress state of compression and shear, it will cause changes in the physical system of the rock mass, resulting in changes in the dynamic characteristic parameters. As a Rayleigh quotient of generalized stiffness and generalized mass, natural frequency reflects the dynamic characteristics of rock mass. As one of the dynamic parameters, natural frequency is easy to measure and has high precision and stability32.

The calculation formula of undamped natural frequency is as follows:

$${f}_{n}=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$$

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The calculation formula of damped natural frequency is as follows:

$${f}_{nz}=\frac{1}{2\pi }\bullet \sqrt{\frac{k}{m}}\bullet \sqrt{（1-{\xi }^{2}）}$$

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where \({f}_{n}\) is the undamped natural frequency; \({f}_{nz}\) is the damped natural frequency; *k* is the stiffness of the material; *m* is the mass of specimen; and \(\xi\) is the damping ratio.

Except for structures with active damping mechanisms, such as shock absorbers, the damping ratio \(\xi\) of structures is general much less than 10%. For a damping ratio of 10%, \({f}_{nz}=0.99499{f}_{n}\); therefore, the damping ratio has little effect on the natural frequency of the structure. The natural frequency measured in this paper can be regarded as the undamped natural frequency. When analyzing the change of natural frequency, only the effects of stiffness and quality are considered, and the influence of the damping ratio is ignored.

When the rock is subjected to load, microcracks gradually occur, then expand and merge, accompanied by the release of elastic strain energy, namely acoustic emission (AE)33,34. The basic parameters of acoustic emission ringing count rate, accumulative ringing count, energy rate and accumulative energy are often used to analyze the evolution process of rock damage and fracture35–37. Through the displacement-time curve, the evolution process of compression-shear fracture of specimens with different brittleness degree is divided into three stages: compaction stage and elastic deformation stage, constant deformation stage and accelerated deformation stage. According to the multivariate parameters response, three mechanical characteristic points are found: damage point, expansion point and critical instability point. The multivariate parameters response laws are shown in Fig. 7, Fig. 8 and Fig. 9.

Compaction stage and elastic deformation stage: At the starting point A of time 0, the initial natural frequency of the specimens with brittleness index of 0.917, 0.910 and 0.906 are 849 Hz, 630 Hz and 835 Hz, respectively. At this time, the axial compaction has been completed under the action of axial force of 10 kN. At the initial stage of the test, the axial force made the natural frequency of the specimen increase rapidly. With the increase of shear force, the specimen was pressed tangentially. Under the action of axial force and shear force, the original crack of the specimen was closed, and a small amount of acoustic emission events could be observed at this stage.

Constant deformation stage: The displacement-time curve at this stage is approximately linear, and the specimen deforms at constant velocity at this stage. The natural frequency decreased greatly at point B, indicating that the crack growth damage occurred in the process of compression-shear test, and the natural frequency decreased. The point B with a steep drop in natural frequency is defined as the damage point, which corresponds to the starting point of constant deformation stage. From Fig. 8 and Fig. 9, it can be seen that the ringing count rate and energy rate of acoustic emission increase near the damage point, indicating that the acoustic emission activity is enhanced. Figure 7 shows that the brittleness index of the specimen is large, the stress drop caused by peak failure is large, and the ringing count rate and energy rate of acoustic emission are very large, in contrast, the acoustic emission phenomenon of the specimen with the brittleness index of 0.917 at the damage point B is not obvious. In the whole process of constant deformation stage, the natural frequency was relatively stable, and a small amount of acoustic emission events occurred.

Accelerated deformation stage: In this stage, the displacement-time curve is nonlinear, and the slope increases, the specimen enters the accelerated deformation stage. At the starting point C of accelerated deformation, the natural frequency changed obviously, showing a substantial rise. With the increase of shear force, the specimen was compressed in the shear direction, and the specimen expanded in the axial direction. Under the action of axial compaction, the specimen was compacted again, resulting in an increase in the stiffness of the specimen at point C, resulting in a substantial increase in the natural frequency. The point C with a substantial increase in the natural frequency is defined as the expansion point, and the expansion point corresponds to the starting point of accelerated deformation. The critical instability point D is determined by the change of acoustic emission parameters and natural frequency. The ringing count rate and energy rate increased obviously near the point D, and the acoustic emission activity in CD section was obviously enhanced. The natural frequency showed rise, steep drop and stable jitter in CD section. The response of natural frequency in CD section represents the precursor information of critical instability for three specimens with different brittleness degree. The acoustic emission activity was obviously enhanced in the accelerated deformation stage, and the ringing count rate and energy rate were obviously increased in this stage, which indicated that this stage changes from the process dominated by microcracks to the process dominated by a small amount of macro cracks. After the shear force reached the peak, it dropped sharply. The slope of the shear displacement curve increased rapidly at the peak, acoustic emission activity increased rapidly, and ringing count rate and energy rate reached maximum at this time.

## 3.2 Acoustic emission characteristics of RA and AF

Acoustic emission waveform characteristics are usually considered as an effective way to reflect fracture failure mode. The rise time/amplitude (RA) and the average frequency (AF) are often used for qualitative analyses of rock fracture mechanisms. Many studies shown that acoustic emission signals with low AF and high RA values usually represent the generation or development of shear cracks, and those with high AF and low RA values are the generation or development of tensile cracks38–41. The basic waveform figure and crack classification of Acoustic Emission are shown in Fig. 10.

The RA-AF parameters distribution of specimens with different brittleness degree are shown in Fig. 11, Fig. 12, Fig. 13. The original data distribution map such as Fig. 11(a), Fig. 12(a), Fig. 13(a), the actual acoustic emission data acquisition is too large to identify the crack type. In order to better express the distribution of acoustic emission parameters and reflect the crack type, the density of RA-AF parameter distribution is calculated by using the concept of probability density of random data in mathematics. The calculation results are shown in Fig. 11(b), Fig. 12(b) and Fig. 13(b). The density of data in red area is the highest and that in blue area is the lowest.

It can be seen from the above density nephogram that the cracks generated by the compression-shear tests of the three specimens with different brittleness degree are mainly tensile cracks. With the decrease of the brittleness degree of the specimen, the core density region (red region) gradually decreases, and the area sum of the AF region as a whole decreases except the blue region with the smallest density, indicating that the tensile failure weakens with the decrease of the brittleness degree. The greater the brittleness of the specimen, the greater the strain energy, the more complete the crack propagation, and the tensile failure is enhanced.

## 3.3 Characteristic of b-value

The concept of the b-value derives from seismology. The b-value is a parameter that characterizes the magnitude-frequency relationship of earthquakes. Gutenberg and Richter42 proposed the famous G–R relation, in the case of the AE technical, the G-R relationship between cumulative frequency and magnitude is expresses as follows:

$${{log}}_{10}N=a-b\left(\frac{{A}_{dB}}{20}\right)$$

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Where \({A}_{dB}\) is amplitude of AE in dB, and N is the number of AE hits or events with amplitude greater than \({A}_{dB}\)43.

The b-value of the three specimens with different brittleness degree are shown in Fig. 14, Fig. 15 and Fig. 16.

The b-value represents the ratio of weak to strong events. Microcracks will produce a large number of weak acoustic emission, which means that the proportion of weak events increases, so microcracks lead to relatively high b-value. In contrast, large-scale fracture lead to relatively lower b-value because of the increased proportion of strong events produced44,45. As shown in Figs. 14, 15 and 16, the b-value of the three specimens fluctuate at a relatively high level during the compaction stage and elastic deformation stage. Larger brittleness specimens (Fig. 14, Fig. 15) in the compaction stage and elastic deformation stage, constant deformation stage in the loading process of crack size is different, acoustic emission energy difference is large, resulting in large fluctuations. After entering the accelerated deformation stage, the b-value of the three specimens with different brittleness degree show a downward trend as a whole. The b-value of the three specimens with different brittleness degree show a continuous decline at low level before failure. This is consistent with the conclusions drawn by many scholars through research41,46−48. Before the failure of the specimen, with the increase of stress, the crack size increased continuously, and the amplitude of the acoustic emission signal continued to increase. The microcrack ran through the macrocracks, and the instability of the specimen released more energy, resulting in more major events, and the b-value decreased.