## Abstract

The mechanical response characteristics and occurrence mechanism of coal and rock under unloading conditions are key to evaluating the stability and control of engineering rock excavation. Triaxial unloading confining pressure tests of coal and rock based on different unloading rates and different initial confining pressure conditions were conducted with fractal theory, research into the fractal characteristics of coal and rock acoustic emission time series under an unloading confining pressure, and the correlations with the unloading failure of coal and rock. The correlation dimension of the acoustic emission signal showed a variation law of a sudden increase, followed by a decrease, increase, and continuous decrease during the experiment. The average time differences between the time when the correlation dimension of coal and sandstone decreased and the actual fracture time of the rock sample were 13.6 s and 16.7 s, respectively. The change in correlation dimension showed the change in internal damage and fracture of the rock sample during the test. The HURST exponent of the acoustic emission time series was calculated at the beginning of the loading. The HURST index showed an overall stable trend and fluctuated around 0.5. A sudden drop in the HURST index resulted when the axial load reached 85% of the peak stress of the rock sample. When the HURST index dropped suddenly (minimum less than 0.2) and then increased rapidly (maximum above 0.8), the rock sample was close to rupture. A HURST index of 0.8 can be used as an index of rock sample unloading and fracture.

## 1. Introduction

Complex rock mass projects, such as high and steep slopes, underground caverns, and dam foundations, are often encountered in engineering construction. These rock mass projects often have characteristics of a large scale, high stress, complex geological environment, and strict requirements for engineering protection [1–5]. They often involve large-scale excavation, which leads to large-scale unloading of engineering rock mass and results in large deformation that affects the stability and safety of rock mass engineering. Traditional rock mechanics research is based on rock loading conditions. Elastic–plastic mechanics theory and methods result in many analytical results being different from actual engineering [6–8]. For underground coal mining, mining damage is the process of unloading damage of coal strata, which is often accompanied by unstable rock bursts and other disasters and results in high losses. Therefore, it is important to study the mechanical properties of coal during unloading [9, 10].

Different methods and technical approaches [11–21] have been used to study the mechanical properties and safety stability of rock during damage and failure. Agioutantis et al. [12] studied the acoustic emission characteristics in a marble three-point bending experiment and found that when the failure strength reached 90%, the acoustic emission events increased sharply, which can be used as a criterion for rock failure. Carpinteri et al. [13] improved the method to determine the start time of the acoustic emission signal and used acoustic emission technology to monitor the crack propagation pattern of concrete material. The method could be used to locate the crack initiation time. Chang et al. [14] used a moment tensor analysis method and rock acoustic emission data under triaxial compression to study the effect of microcrack accumulation on rock microscopic damage and analyze the mechanism of rock macroscopic damage. Many studies [22–30] have shown that fractal theory combined with the acoustic emission data of rocks can describe the dynamic evolution of cracks in rocks. Zhang et al. [25] designed granite uniaxial and triaxial experiments, used acoustic emission technology to monitor experimental progress, and found that the acoustic emission time series had fractal characteristics and that the correlation dimension showed an increasing and then decreasing pattern. Biancolini et al. [22] studied the acoustic emission events that were generated by the formation and expansion of cracks in rocks under fatigue test conditions and found that the fractal dimension can describe the dynamic evolution characteristics of cracks during loading. Kong et al. [23] studied the acoustic emission data of coal and rock mass based on fractal theory and divided the loading of coal and rock mass into three stages: initial loading, continuous loading and final failure. In the final failure stage, the fractal dimension of the acoustic emission data can be used as the criterion for the impending macroscopic damage of rock. Yin et al. [24] studied the fractal characteristics of acoustic emission at different stress levels during rock failure and showed that the continuous decrease in fractal dimensions of acoustic emission can be used to predict rock failure.

In 1951, British hydraulic scientist Hurst proposed HURST, which is an indicator that can describe fractal Brownian motion, to describe the long-term water storage capacity of reservoirs. This parameter can analyze whether the series is time-dependent and the R/S method to calculate the HURST index was provided. The HURST index has become an important parameter to analyze the fractal characteristics of the time series and has been applied widely. Meng [31] used the HURST index to analyze the changing characteristics of a country’s stock index futures market. He et al. [32] applied the R/S analysis method to calculate the HURST index of landslide displacement data and analyzed its application in landslide prediction. Niu [33] analyzed the relationship between the HURST index and fractal Brownian motion. Zhu et. al [34] analyzed the time series of ventricular fibrillation based on the HURST index. Tan et al. [35] studied the trend in earthquake magnitude based on the HURST index. The acoustic emission signal is a nonstationary random signal whose frequency and statistical characteristics change with time, which satisfies the characteristics of nonlinearity and randomness. Therefore, the acoustic emission signal satisfies the existence condition of a fractal dimension. Another fractal feature extraction method has been used to analyze the acoustic emission signal; that is, the self-similar feature of the acoustic emission signal has been extracted based on the HURST index, and the fracture and instability sound of the rock under a triaxial unloading confining pressure test has been analyzed.

Based on triaxial unloading confining pressure test research of coal and rock at different initial confining pressures and unloading rates, the acoustic emission data of coal and rock samples during unloading failure has been analyzed by using fractal theory. Compared with existing research results [32], this paper focuses on analyzing the variation law of the correlation dimension of the acoustic emission time series with loading time and stress level and discusses the internal relationship between the correlation dimension and the damage and fracture of rock samples. The HURST index calculation program was compiled to calculate the HURST index of the acoustic emission time series of rock samples under different test conditions, and the unloading failure precursors of coal and rock samples were analyzed through the variation characteristics of the HURST index of the acoustic emission time series. The research results have important engineering significance in the stability evaluation of rock mass engineering excavation and the analysis and prediction of failure characteristics under high stress and increase the theoretical research on the fracture response and failure mechanism of coal and rock loading and unloading under high stress conditions.

## 2. Materials and Methods

### 2.1. Sample Selection

Samples of test coal and rock were taken from No. 3 coal and its roof sandstone in Yangcun Coal Mine, Jining, Shandong, China. According to engineering rock mass test standards, the coal block was cut into $50\u2009mm\xd7100\u2009mm$ (diameter × height) cylinders. To ensure that the specimens were similar and homogeneous, they were subjected to ultrasonic testing, and the specimens that showed high wave velocities were removed, to leave specimens with similar velocities. The rock samples that were used in the test are shown in Figure 1.

### 2.2. Test Program

Triaxial loading and unloading tests were carried out on the MTS815.02 electrohydraulic servo rock mechanics test system of the China University of Mining and Technology. The system could meet the test requirements under a variety of complex paths. The test plan in this study was as follows.

*Loading tests:* triaxial tests were carried out on specimens at different confining pressures. Confining pressures (4, 7 or 10 MPa) were applied gradually under a hydrostatic pressure. Under a constant confining pressure, the axial pressure was increased by axial displacement control at 0.002 mm/s until the specimen failed [36, 37].

*Triaxial unloading confining pressure tests:* during mining, the lateral stress in coal bodies (equivalent to the confining pressure) is reduced gradually, and the advanced support pressure (equivalent to the axial stress) is increased gradually. In working face mining, the confining pressure is reduced, and the axial pressure of the rock mass and its surroundings is increased.

We used a loading path of increasing axial pressure and decreasing confining pressure where specimen failure was the quickest and the most dangerous. The experiment was divided into three stages: (1) increase the confining pressure (*σ*_{3}) gradually to a predetermined value (4, 7, or 10 MPa) according to the hydrostatic pressure condition. (2) Keep the value of *σ*_{3} unchanged and increase the axial pressure (*σ*_{1}) to eighty percent of the peak compressive stress from conventional triaxial tests though the stress control method. (3) Use the displacement control method to increase the value of *σ*_{1} and, at the same time, decrease the value of *σ*_{3} at different rates (0.02, 0.05, 0.08, 0.11, and 0.14 MPa/s) until the specimen is destroyed. Stop the decreasing confining pressure immediately after the specimen is damaged, and continue to load the axial pressure to the residual strength of the specimen using the displacement control method [37, 38]. The stress path of the unloading confining pressure is shown in Figure 2, and the test process and posttest samples are shown in Figure 3.

## 3. Analysis of Unloading Failure Characteristics of Coal and Rock

Tables 1 and 2 show test data of the conventional triaxial compression test of coal and rock and the triaxial unloading confining pressure test with different initial unloading confining pressures and different unloading confining pressure rates.

The parameter names that are represented by each symbol in Tables 2 and 3 are *ε*_{1}: axial strain; *ε*_{3}: lateral strain; *ε*_{v}: volume strain; *σ*_{3}: confining pressure, MPa; *v*_{σ3}: unloading confining pressure rate, MPa/s; *σ*_{1}-*σ*_{3}: peak principal stress difference, MPa; *σ*_{3}$\u2032$: confining pressure when rock sample fails, MPa; *σ*_{3}-*σ*_{3}$\u2032$: confining pressure reduction, MPa.

The stress-strain curve of coal and rock under triaxial compression and unloading confining pressure is shown in Figures 4–7. In the triaxial unloading confining pressure test, under different unloading pressure and different unloading rate, the brittleness characteristics of coal and sandstone are obvious compared with the conventional triaxial compression test, with brittle failure sound in the process of the test, and the axial strain stress curve of sandstone shows an obvious sudden decline trend after the peak value.

With the increase of axial pressure (axial loading stage), the stress-strain curve is nearly linear (except the compaction stage). Compared with the circumferential strain, the slope of the axial strain is smaller, and the increase rate of the circumferential strain of the sample is less than the axial strain, which is similar to the conventional triaxial compression. At this time, the change of the volumetric strain is mainly affected by the axial strain.

There are differences in the circumferential strain changes of rock samples of different lithology. The circumferential strain slope of coal is small and that of sandstone is large, which is mainly due to the development of joints and fissures in coal. Under the action of confining pressure, the circumferential deformation of coal is obvious.

After the beginning of unloading confining pressure, due to the axial loading by displacement control, the axial strain basically continues to increase at the original speed, while the increase speed of circumferential strain increases significantly. The change law of volumetric strain and circumferential strain is basically the same, and the volumetric strain also begins to turn left. Moreover, the increase speed of circumferential strain and volumetric strain is faster and faster, and the rock sample begins to expand, indicating that the circumferential strain plays a leading role in the stage of unloading confining pressure.

With the continuous reduction of confining pressure, the bearing capacity of rock sample begins to decline, obvious capacity expansion occurs, the rock sample breaks and loses stability, and a crisp sound is generated during failure. In the triaxial unloading confining pressure test, with the decrease of confining pressure, the restraint effect of confining pressure on the rock sample surface is weakened, which makes the failure degree of rock sample unloading confining pressure more severe than that of conventional triaxial compression test.

## 4. Fractal Features of Acoustic Emission Time Series

### 4.1. Determination of Delay Time

Delay time is an important phase space reconstruction parameter. If $\tau $ is too small, the difference between any $xi$ and $xi+\tau $ in $Xi=xi,xi\u2212\tau ,\cdots ,xi\u2212m\u22121\tau T$ is too small, which makes it difficult to distinguish, and does not result in independent coordinate components, which makes the reconstruction space messy. If $\tau $ is too big, then any quantity in $Xi=xi,xi\u2212\tau ,\cdots ,xi\u2212m\u22121\tau T$ is irrelevant, and the two quantities are independent, which makes it difficult to reflect the true law of the trajectory in the phase space.

The mutual information method is effective to calculate the time delay in phase space, and has a wide range of applications.

In practical application, the grid division method can be used to divide the sample space composed of $ai$ and $bj$ into a network, The probability value is obtained by calculating the number of points in the grid, the probability value is obtained by calculating the number of points in the grid, and the minimum time $\tau min$ corresponding to the first minimum point of the mutual information function is taken as the delay time.

The delay time curves of coal and sandstone from the mutual information method for the triaxial unloading confining pressure test are shown in Figure 8. 16-N-21 (C-10-0.02), S-2-3 (S-10-0.14), and other samples were used as examples in the analysis.

The calculated optimal delay time of the coal and sandstone acoustic emission time series is shown in Table 3.

### 4.2. Determination of Embedding Dimensions

In 1983, Grassberger and Procaccia jointly proposed G-P algorithm, or G-P correlation integral method. As a correlation index saturation method, it can not only calculate the correlation dimension of attractor, but also obtain the embedding dimension of reconstructed phase space. When G-P algorithm is used to calculate the optimal embedding dimension of rock acoustic emission time series $x1,x2,\cdots ,xn\u22121,xn,\cdots $, the calculation steps are as follows:

- (1)
The acoustic emission time series is normalized, and the smaller embedding dimension is selected to reconstruct the phase space according to the calculated time delay

- (2)
Calculate the correlation function $Cr=1/N2\u2211i,j=1N\theta r\u2212Xi\u2212Xj$. The distance between phase points $Xi$ and $Xj$ is $Xi\u2212Xj$, and $\theta z$ represents Heaviside function

- (3)
Take the appropriate value of $r$, calculate the value of $Dm=lnCr/lnr$, get the double logarithm curve, and fit the scale area with good linearity of the double logarithm curve to obtain the correlation dimension

- (4)
Increase the value of $m$, $m>m1$, and repeat the calculation of steps (2) and (3) until the calculated value of $D$ remains basically stable and does not increase with the increase of the value of $m$; at this time, the value of $m$ is the best embedding dimension, and the value of $D$ is the correlation dimension

The double logarithmic curve of $lnCr\u2212lnr$ was calculated by using the G–P algorithm, and the straight line segment of the curve was fitted to obtain the change curve of the correlation dimension $D$ with the embedded dimension $m$, as shown in Figure 9.

The fractal characteristics of the acoustic emission time series of the coal and sandstone samples for the triaxial confining pressure relief tests are obvious. From the slope of the straight-line segment of the double logarithmic curve, the smaller value corresponds to the smaller slope, and the distance between each curve is larger. When the embedding dimension $m$ increases, the slope of the straight-line segment of the double logarithmic curve increases, the curves gradually become denser, and the distance between them decreases. The calculated optimal embedding dimension and associated dimensions are shown in Table 4.

### 4.3. Fractal Chaos Characteristics of Rock Acoustic Emission Time Series

Many methods exist for the identification of time series fractal chaos, and some methods can be used for qualitative analysis, such as the principal component analysis method. Some methods can be used for quantitative analysis, such as the maximum Lyapunov index method and saturated correlation dimension method. The calculation methods of the largest Lyapunov index method include the definition method, Wolf method, P-norm method, Jacobian method, and small-data volume method. The small-quantity method is reliable for small data, less computationally intensive and easy to operate. In this work, the Lyapunov index was obtained by using a small amount of data to analyze quantitatively the chaotic characteristics of the acoustic emission time series of the rock sample in the triaxial confining pressure relief test, as shown in Figure 10. The Lyapunov index has a stable motion and a low sensitivity to initial conditions in the direction of $\lambda <0$. In the direction of $\lambda >0$, the motion trajectories are separated, form a long-term sensitivity to the initial conditions, and enter a chaotic motion state. The critical state of the system is $\lambda =0$.

The small data volume method was used to calculate the relationship between time and the separation factor by using the $\tau $ and $m$ values that were obtained earlier, and the curve was fitted with a straight line to obtain the slope of the fitted straight line, as shown in Figure 11. The calculated maximum Lyapunov indices are shown in Table 5.

The calculation results show that the maximum Lyapunov index of the acoustic emission time series of coal and sandstone exceeds 0, as shown in Table 6. The acoustic emission time series of coal and sandstone rock samples in the triaxial unloading confining pressure test have fractal and chaotic characteristics, and a nonlinear analysis of the acoustic emission time series can be carried out by using the relevant method of fractal chaotic time series analysis.

## 5. Variation Law of Correlation Dimension of Acoustic Emission Sequence

The acoustic emission process of the triaxial unloading confining pressure of rocks is self-similar, the time series of acoustic emission has obvious fractal characteristics, and the correlation dimension of fractal features that characterizes an acoustic emission time series reflects the internal evolution characteristics of rock sample damage and fracture during the test. Therefore, the acoustic emission time series of coal and sandstone under a triaxial unloading confining pressure were analyzed, the variation law of the correlation dimension of the acoustic emission time series with the loading time and stress level was obtained by calculation, and the internal relationship between the correlation dimension and the damage and fracture of the rock sample was analyzed.

According to the associative dimension algorithm, MATLAB software was used to calculate the correlation dimension of the acoustic emission time series of coal and sandstone for the triaxial unloading confining pressure, and the acoustic emission ringing count rate was selected when the correlation dimension was calculated.

The variation curves of the correlation dimension of the acoustic emission ringing count rate of coal and sandstone in the triaxial unloading confining pressure test are shown in Figures 12 and 13 (some test results are listed). During the test, from the initial stage of loading to the damage and fracture of the rock sample, the correlation dimension of the acoustic emission ringing count rate changed, and the variation law of the correlation dimension of the acoustic emission ringing count rate of coal and sandstone was similar.

In the initial stage of loading, the correlation dimension of the acoustic emission signal of the rock sample was small. As the loading progressed, the acoustic emission ring count rate began to increase, the value of the associated dimension increased with the increase in axial stress, and its growth mode was “wave-shaped.” When the rock sample approached the peak stress, the correlation dimension of the acoustic emission signal peaked, and the correlation dimension changed abruptly and increased to an extreme value again.

The change curve of the correlation dimension showed that the time interval in which the correlation dimension suddenly dropped was in the time interval of the “relatively low period” of the acoustic emission ringing count rate. After the correlation dimension reached the extreme point again, the rock sample showed macroscopic fracture, and the correlation dimension showed a downward trend. The correlation dimension of the acoustic emission signals of different rock samples differed in numerical and local variation characteristics, but overall, the correlation dimension of the acoustic emission signal of the rock sample showed an increase, sudden decrease, increase, and continuous decrease during the experiment.

Literature shows that a larger correlation dimension yields greater chaos of damage and fracture activities inside the rock sample. At the beginning of the test, with the increase in loading time, the correlation dimension increased gradually, and the chaotic nature of the internal motion state of the rock sample became more obvious. When the rupture approached, the correlation dimension dropped suddenly, and the chaos of the internal motion state of the rock sample decreased. When the peak stress was reached, the rock sample was fractured macroscopically, and the correlation dimension value continued to decrease. The above analysis shows that the change in fractal dimension (correlation dimension) of the acoustic emission parameters reveals changes in the internal damage and fracture of the rock sample during the test. Therefore, the variation law of the correlation dimension can be used as precursor information of rock sample rupture and instability during the test.

The difference between the time when the correlation dimension of the rock sample suddenly dropped and the actual fracture time is shown in Table 7. Table 7 shows that the time intervals of the difference between the time when the correlation dimension of coal and sandstone suddenly dropped and the actual fracture times of the rock samples were [8.6 s, 19.6 s] and [13.9 s, 22.0 s], respectively.

## 6. Acoustic Emission Precursor of Rock Failure Based on HURST Index

The HURST index varies between 0 and 1, and it represents different properties in different intervals:

- (1)
When $0\u2264H<0.5$, the time series has a “mean reversion” property; increments at the front of the time series are opposite to those at the back, i.e., if the time series increases during the early period, it will decrease in the later period. The time series shows a strong volatility; for a HURST close to 0, the volatility is stronger

- (2)
When $H=0.5$, the time series conforms to Brownian motion, which is irrelevant and random walk

- (3)
When $0.5<H\u22641$, the time series has a long memory and persistence, and the increments before and after the time series tend to be consistent. For a HURST close to 1, the trend is stronger, and for a HURST closer to 0.5, the random walk is stronger

The HURST index calculation program was written by using a MATLAB program to calculate the HURST index of the acoustic emission time series of rock samples under different test conditions. In the calculation, the acoustic emission ringing count rate for the three-axis unloading confining pressure of coal and sandstone was used to calculate the time series.

The calculation results of the acoustic emission time series HURST index of some coal and sandstone rock samples for the triaxial unloading confining pressure test are shown in Figures 14 and 15 (partial test results are listed). The change curve of the HURST index of coal rock and sandstone for the triaxial unloading confining pressure test shows that the evolution characteristics of the HURST index for different lithology rock samples differ, but the overall law is similar.

Before the axial load reaches the peak stress of the rock sample, the HURST index for each test condition showed characteristics of a repeated fluctuation, overall stability, and slight increase. At the beginning of this stage, the internal crack of the rock sample changed little, and the acoustic emission activity was low. With an increase in load, the crack began to expand, and the acoustic emission activity increased. When the crack expanded, the acoustic emission signal increased. When the crack propagation ended, the acoustic emission signal weakened again until the next crack propagation occurred. The intensity of the acoustic emission signals continued to change, which resulted in the HURST index showing characteristics of repeated fluctuations. However, as the load increased, the crack propagation increased, new cracks were generated, the strength of the acoustic emission signal increased, and the difference between the ringing count rates that were generated by the strong and weak signals increased, which increased the HURST index slightly. However, because of the relatively low AE activity in general, the HURST index showed a stable trend. The index fluctuated around 0.5, with the overall fluctuation range being within [0.3, 0.7], and the dense fluctuation range was [0.4, 0.6].

After the axial load reached 85% of the peak stress of the rock sample, the HURST index began to drop suddenly. At that time, the acoustic emission ring count existed in a relatively silent period, and the acoustic emission activity was low. The intensity of the produced acoustic emission signal was relatively small compared with the active period, which led to a decrease in HURST index. The minimum HURST index of the different rock samples was less than 0.2. The HURST index then increased rapidly, the rock sample entered a macroscopic fracture state, and the acoustic emission activity increased sharply. The maximum HURST index of the coal and sandstone exceeded the fluctuation range at 0.8. After a period in the higher range, the rock sample entered the residual plastic stage, the acoustic emission activity began to decrease again, and the HURST index decreased gradually.

The variation law of the HURST index was related closely to the strength of the acoustic emission activity and the degree of contrast of the acoustic emission signal during the test. Therefore, the sudden drop in HURST index occurred before the fracture of the rock sample. The HURST index increased rapidly and exceeded the normal fluctuation range before the rock sample failure and instability. The minimum of the sudden drop was less than 0.2. The maximum that was reached by the rapid rise exceeded 0.8, after which the rock sample ruptured and became unstable. Therefore, when the HURST index dropped suddenly (minimum less than 0.2) and then increased rapidly (maximum exceeds 0.8), the rock sample was about to rupture and could be used as a precursor to rock sample failure. The initial time when the HURST index of the rock sample dropped suddenly and the time when the maximum exceeded 0.8 is show in Table 8.

Table 8 shows that the initial time of the sudden drop occurs in the quiet period of the ringing count rate of rock sample acoustic emission. The difference intervals between the initial occurrence time of coal and sandstone suddenly collapse and the actual rupture times were [10.8 s, 21.0 s] and [16.2 s, 23.4 s], respectively, and the intervals between the time when the maximum HURST index exceeded 0.8 and the actual rupture time were [0.7 s, 1.3 s] and [0.8 s, 1.4 s], respectively. The difference between the initial time of the sudden drop of the HURST index and the actual rupture time greatly exceeded the difference between the time when its maximum exceeded 0.8 and the actual rupture time. Therefore, the time when the HURST index of the rock sample reached 0.8 was the same as the time when the macroscopic fracture of the rock sample was very short. When the HURST index reached 0.8, the rock samples broke immediately; therefore, the HURST index of 0.8 can be used as a judgment index for the fracture time of the rock samples.

## 7. Conclusions

The time interval in which the correlation dimension dropped suddenly was within the time interval of the “relative quiet period” of the acoustic emission ringing count rate. The correlation dimension of the acoustic emission signal of the rock sample showed a changing law of an increase, sudden decrease, increase, and continuous decrease during the experiment.

According to the test results, the average time difference between the time when the correlation dimension of coal and sandstone dropped suddenly and the actual fracture time of the rock sample was 13.6 s and 16.7 s, respectively. The change in internal damage and rupture of the rock sample during the test can be revealed from the change law of the correlation dimension.

During the initial period of loading, the HURST index showed an overall stable trend and fluctuated around 0.5. When the axial load reached 85% of the peak stress of the rock sample, the HURST index dropped suddenly.

When the HURST index dropped suddenly (minimum less than 0.2) and then increased rapidly (maximum above 0.8), the rock sample was about to rupture; therefore, the HURST index of 0.8 can be used as an indication of rock sample fracture.

## Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

## Conflicts of Interest

The authors declare that they have no conflicts of interest.

## Acknowledgments

This research has been supported by the National Natural Science Foundation of China (51574156). This research was also partially funded by the Shandong Province Higher Educational Science and Technology Program (J18KA195).