Precursory Indicator for Mode I Fracture in Brittle Rock through Critical Slowing Down Analysis

The abrupt rock-related hazards, such as landslide, rock burst, and collapse, seriously threaten the safety and service life of engineering works. Precursory information on critical transitions preceding sudden fracture is of great signiﬁcance in rock mechanics and engineering. This study investigates the critical slowing down feature of acoustic emission (AE) signals and precursory indicators during the mode I fracture process of brittle rock. Cracked chevron notched Brazilian disc (CCNBD) specimens were utilized, accompanied by acoustic emission monitoring. The principle of critical slowing down was introduced to study AE count sequences, and the variance and autocorrelation coeﬃcient versus loading time curves were analyzed. The results show critical slowing down phenomenon exists during mode I rock fracture. The variance and autocorrelation coeﬃcient of AE counts grow signiﬁcantly prior to rock fracture, and thus, the signiﬁcant growth of variance and autocorrelation coeﬃcient of AE signals can act as the precursory indicator of rock fracture. Compared to the autocorrelation coeﬃcient, the precursors determined by the variance are more remarkable. The time interval between the precursory indicator using the critical slowing down theory and fracture moment ranges from 2% to 15% of the entire loading time. The ﬁndings in this study could facilitate better understandings on the rock fracture process and early-warning technique for rock fracture-related geological disasters.


Introduction
e stability of rock structures is important in extensive engineering fields, such as coal mining, tunnel excavation, water resources, and hydropower engineering. By contrast, rock failure and instability during the engineering construction and performance periods may cause a series of geological disasters, including landslide, rock burst, collapse, and large deformation of the tunnel, which seriously threatens the safety and service life of engineering works. erefore, it is of great significance to study the precursory information and early-warning technology of rock fracture.
A rock inevitably contains preexisting defects, e.g., microcracks, cavities, and pores. When the rock is subjected to external forces, the stress concentrations are generated around preexisting defects and the rapid release process of localized strain energy is accompanied by acoustic emission (AE) phenomena [1][2][3]. Since German scientist Kaiser [4] verified the acoustic emission phenomenon of common engineering materials, the AE technique has been widely used in rock mechanics and engineering [5][6][7]. Many scholars investigated the characteristics and precursory information of AE signals during the rock deformation and damage process, e.g., the remarkable increase in the number and release rate of AE counts or events [8,9], the change of seismic b-value [10][11][12], the variation of spatial correlation length [13,14], frequency spectrum characteristics [15,16], and fractal characteristics [17][18][19][20][21]. e critical slowing down phenomenon exists in a complicated dynamical system as the system shifts suddenly from one stable state to another contrasting one, which has attracted widespread attention in many fields [22][23][24][25]. Dakos et al. [22] and Wu et al. [26] assumed that slowing down could be viewed as a precursor for upcoming catastrophic climate change. e literature [27][28][29] investigated the critical slowing down phenomenon preceding a bifurcation from a stable equilibrium to a chaotic state in the ecosystem. Yan et al. [30] stated that the critical slowing down characteristics are a possible early sign for upcoming earthquakes. Ren and Watts [31] showed statistical measures occur in the energy supply infrastructure upon reaching abrupt large-scale disruptions in the electrical system. Obviously, the critical slowing down theory provides a new sight to identify whether a dynamical system is undergoing a critical transition.
Although the critical transition feature has been widely studied in extensive fields [23,25], spanning from the ecosystem to the climate and seismic systems, the critical slowing down characteristics of AE signals preceding rock failure are in their infancy. Kong et al. [32] and Wei et al. [33] analyzed the early signs of AE counts in coal and sandstone during uniaxial compression tests. Zhang et al. [34] investigated the critical slowing down characteristics of electric potential (EP) generated during the sandstone failure process subjected to compression. However, the critical slowing down feature of AE signals during the rock damage process under other types of loading, especially mode I fracture, is still absent. e purpose of this paper is to study the critical slowing down feature of acoustic emission (AE) signals and reveal the early-warning information during the mode I fracture process of brittle rock. e mode I fracture tests were carried out using cracked chevron notched Brazilian disc (CCNBD) specimens, which is a suggested method for determining mode I fracture toughness [35]. Real-time acoustic emission monitoring was conducted during the loading process. Variances and autocorrelation coefficients of AE count sequences with loading time were investigated through the critical slowing down theory. e average time intervals between the precursory indicator and rock fracture point for brittle rock were statistically analyzed. Finally, the critical transition characteristics of rock fracture were revealed.

Critical Slowing Down Theory
e critical transition can be characterized and predicted by the changes of some statistical parameters when a complex dynamical system undergoes sudden transition from a stable state to another contrasting one. As a concept in statistical physics, critical slowing down describes the fluctuation phenomenon when the system approaches the critical point. Specifically, critical slowing down refers to the phenomenon that the fluctuation duration is lengthened, the recovery rate decreases, and the ability to return to the previous state declines [23,27]. Critical slowing down phenomenon can be statistically characterized by the variance and autocorrelation coefficient. e variance (δ) is the deviation of sample data to its mathematical expectation, expressed as follows: where S and E(x i ) represent the standard deviation and mathematical expectation, respectively, and x i and N are the i th data and the number of data, respectively. Autocorrelation coefficient statistically describes the correlation between different periods of the same variable. e autocorrelation coefficient (β(k)) of the variable x i with lag length k can be denoted as follows: It is assumed that the state variable has a repeated disturbance after each period (Δt). e return to equilibrium is approximately exponential during the perturbation process with a certain recovery rate (α). is can be described in an autoregressive (AR) model as follows [23]: where u n denotes the deviation of the state variable from equilibrium and ω n is a random quantity from a normal distribution.
If the recovery speed (α) and period (Δt) are independent of the deviation (u n ), this model can be simplified into an AR (1) model: e autocorrelation coefficient β � e αΔt is approximately one for red noise and zero for white noise. e variance of the AR (1) model (equation (5)) is found to be e recovery rate of small amplitude disturbances is increasingly lower when the system approaches the critical point [22,23]. e variance will approximate infinity and the autocorrelation coefficient will approach 1 as the recovery rate approaches 0. erefore, the significant increase of variance and autocorrelation coefficient of one system parameter can be regarded as the precursor close to the critical point [30][31][32][33][34]. Here refers to the AE count of marble samples during the mode I fracture process.

Experimental Setup
e cracked chevron notched Brazilian disc (CCNBD) specimens are suggested for determining mode I fracture toughness by the International Society for Rock Mechanics (ISRM) [35]. e CCNBD marble specimens were prepared according to the requirements of the suggested method, as shown in Figure 1.
e radius (R) and thickness (B) of marble specimens are 50 mm and 32 mm. e dimensions of the final chevron notched crack length (a 1 ) and initial chevron notched crack length (a 0 ) are 38 mm and 13 mm on average. e density of marble specimens is 2.68 × 10 3 kg/m 3 on average. e average uniaxial compression strength (UCS) and tensile strength are 53.85 and 4.05 MPa. Mineral composition analysis showed that white marble is constituted by only a single mineral, namely, calcite. e forcecontrolled loading mode was adopted in this study. e loading rate was constant at 1 kN/min. In total, there are 7 CCNBD marble specimens for mode I fracture tests.
A rock mechanics testing system (model: MTS815 Flex Test GT) with a maximum normal load of 4600 kN was used to carry out mode I fracture tests. e force, displacement, and crack opening displacement were recorded during the tests. Force was measured by force sensors with a range of 0 to 1000 kN, while displacement and crack opening displacement were measured by linear variable differential transducers (LVDTs) with a range of ±2.5 mm and crack opening displacement meter (COD) with a range of −2 mm to 5 mm, respectively. is configuration used in this study is according to the suggested method by ISRM [35]. An acoustic emission monitoring system (model: PCI-2), produced by American Physical Acoustics Corporation, was utilized to automatically monitor AE signals during the whole loading process. e experimental setup of acoustic emission monitoring is shown in Figure 2. Eight Micro30 sensors, which have a good sensitivity to AE signals, were equally distributed on the surface of each CCNBD specimen symmetrically with respect to the loading direction. Vaseline was used to guarantee good connections between AE sensors and rock sample. e threshold level was set as 35 dB in the experiments. e sampling rate was 1 MHz and the preamplification was set as 40 dB. e Micro30 sensor has a wide frequency response [36].
To analyze the critical slowing down feature of AE signals, the variances and autocorrelation coefficients of AE counts with loading time were calculated through a batch program written by MATLAB software. Figure 3 shows the typical CCNBD sample after mode I fracture tests. According to the recommended method by ISRM, the mode I fracture toughness (K ΙC ) of specimen can be determined by the following equation [35]:

Mechanical Properties of Marble under Mode I Fracture.
where Y * min refers to the critical dimensionless stress intensity for the CCNBD specimen, which is calculated by geometry dimensions of specimen only. P max refers to the peak force recorded during the experiments. e symbols B and D are the thickness and diameter of the disc specimen, respectively. e mode I fracture toughness of marble specimens was calculated by equation (7), as listed in Table 1. e mode I fracture toughness is 0.636 MPa × m 1/2 on average, ranging from 0.447 to 0.797 MPa × m 1/2 . e number of AE hits released during the rock fracture process is on average 20479 with a range of 13659 to 27321.
According to the ISRM suggested method [35], the minimum valid diameter (D min ) of CCNBD specimens can be estimated as follows: where σ t denotes the tensile strength of rock specimens. e minimum valid diameter (D min ) for marbles in this study is approximately 68.6 mm calculated by equation (8).
e diameter of marble specimens is 100 mm, which is much greater than D min . erefore, the mode I fracture toughness in this study is valid and accurate. e typical force-crack opening displacement curve of the specimens is shown in Figure 4. Obviously, the crack opening displacement grows slowly with the increasing force at the initial loading stage. e crack opening displacement increases significantly after the applied force reaches the peak, while the applied force on the CCNBD specimens remains approximately constant at this stage.

Characteristics of AE Signals Released during Mode I Fracture Process.
e typical spatial distribution of AE signals is presented in Figure 5. Note that black line represents the boundary of rock specimen and red lines refer to macroscopic cracks distributed in rock specimens. It can be found that the spatial distribution of AE sources agrees well with macroscopic failure pattern of rock specimens subjected to mode I fracture loading. Figure 6 shows typical acoustic emission count curve versus loading time during the mode I fracture test. Note that red dotted line in Figure 6 refers to the force-loading time curve during mode I rock fracture. On the left vertical axis is AE count while on the right vertical axis is force in kN. ere are few AE signals produced by the specimen at the initial stage. is is because the external force at this loading stage is too small to cause microcrack initiation and propagation in rocks. At this stage, AE signals result from the closure of preexisting cracks. Subsequently, the AE counts increase gradually with increasing force, which reflects the onset and propagation of microcracking. Significant growth of AE counts occurs as the applied force approaches the peak. Formation and penetration of macroscopic cracks occur at this stage. e release of AE counts is consistent with the rock fracture process.

Critical Slowing Down Feature during Mode I Fracture
Process. To investigate the critical slowing down feature, appropriate window length and lag step length need to be adopted to determine the variance and autocorrelation coefficient of AE counts. Window length is the basic unit of sequence analysis and refers to the number of data in each   sequence selected from the total data. e lag step length refers to the lag length from the sequence of a specific window length to another new identical sequence. Hence, the effect of different window lengths and lag step lengths on the results was analyzed. Figure 7 shows the variance and autocorrelation coefficient curves of AE counts under the same lag step length of 50 with different window lengths of 80, 100, and 120. For different window lengths, the overall trend of variance and autocorrelation coefficient curves is almost the same although there are differences in their magnitude. e variance and autocorrelation coefficient curves of AE counts with different lag step lengths under the same window length are presented in Figure 8. e window length was set constant at 100, and the lag step length was set as 20, 50, and 80, respectively. When the window length remains constant, the variance curves are consistent for different lag step lengths, while the lag step length influences autocorrelation coefficient curves. Overall, the window length and lag step length have a little effect on the results, especially the trend of variance and autocorrelation coefficient curves. e window length of 100 with a lag step length of 50 was adopted in the following analysis. Figure 9 shows the variance and autocorrelation coefficient curves of AE counts with loading time. Obviously, there are sudden and significant increases in the variance and autocorrelation coefficient curves preceding rock fracture. is indicates the critical slowing down phenomenon exists during the rock fracture process. By comparing Figures 9(a) and 9(b), the precursory indicator is easier to identify in the variance curve compared to the autocorrelation coefficient curve. e critical slowing down feature characterized by variances is more remarkable than autocorrelation coefficients. is is due to differences between the mathematical meaning and range of the two parameters. e variance is the deviation of sample data to its mathematical expectation, while autocorrelation coefficient statistically describes the correlation between different periods of the same variable. e variance can be a very large value, whereas the range of autocorrelation coefficient is from −1 to 1. us, the change of variance is more remarkable compared to autocorrelation coefficient. According to the critical slowing down feature of AE counts, the sudden and significant growth of variance and autocorrelation coefficient, especially variance, can be viewed as the precursory indicator for imminent rock fracture. e precursory indicator is automatically identified by the magnitude and gradient of changes.
Time intervals between the precursory indicator and failure moment for different specimens are shown in Figure 10. e time intervals between the precursory indicator and failure moment range from 3.36 to 42.94 seconds with an average of 13.68 seconds.
To study the relationship between the precursory indicator and the loading process, ratios of precursory indicator to failure moment in time and force are shown in Table 2.
e average ratios of the precursory indicator to failure moment in time and load are 95.69% and 95.62%, respectively. e ratios of the precursory indicator to failure moment range from approximately 85% to 98% in terms of both time and force. In other words, the period between precursory signal and fracture time of marble specimens occupies 2%-15% of the fracture time after analysis of critical slowing down feature. erefore, the critical slowing down theory has great potential for predicting rock fracture.

Discussion
According to analysis results, the phenomenon of critical slowing down exists during the mode I rock fracture process, which is a new method to identify whether a rock is approaching fracture. is further verifies that the critical slowing down feature is ubiquitous for complex dynamical systems in nature, spanning from ecosystems to seismic systems [27,29,30]. e time intervals between the   Shock and Vibration 5 precursory indicator using critical slowing down theory and fracture moment (averagely 4% with a range of 2%-15%) are not very short or long compared to total loading time. is is of great significance for early-warning sudden fracture of rock because it is meaningless if the early-warning time is too short or too long. e findings in this study could facilitate better understandings on the rock fracture process and early-warning technique of rock fracture-related geological disasters.
As shown in Section 4.3, the range of time intervals between the precursory indicator and failure moment is from 3.36 to 42.94 seconds, and the period between precursory signal and fracture time of marble specimens occupies 2%-15% of the loading time. It can be found that there are differences between the results of different marble specimens. e differences may be attributed to the discrepancy between microstructure of specimens although white marble is constituted by only calcite. e spatial   distribution and quantity of preexisting defects, e.g., preexisting microcracks, pores, and cleavage, affect the rock fracture process subjected to external load.
A comparison was made between the precursory point obtained by the AE count curve directly and that of variance and autocorrelation coefficient, as shown in Figures 6 and 9. It is found that the precursory indicator through the variance and autocorrelation coefficient of AE counts approximates that obtained by the AE count curve directly. However, compared to the direct determination of the precursory indicator through AE response, i.e., the remarkable growth in AE counts [8,9], the precursory indicator through critical     slowing down analysis is more precise and recognizable. Furthermore, compared to seismic b-value, which is also applied to analyze the rock failure process [10][11][12], the change of the variance and autocorrelation coefficient preceding rock fracture is easier to identify. is advantage is crucial for engineers.
To discuss whether there is a certain relationship between the ratio of the precursory indicator to failure moment and mode I fracture toughness, a comparison was made among different marble specimens. Scatter diagram between the ratios of the precursory indicator to failure moment and mode I fracture toughness of marble specimens is shown in Figure 11. e ratios of the precursory indicator to failure moment in terms of time and force are relatively stable when the mode I fracture toughness of rock specimens varies.
Acoustic emission response can reflect the influence of preexisting defects on the rock failure process indirectly. However, direct measurement of preexisting defects through three-dimensional CT scanning or other methods would be much more convincing. Further study on the effect of preexisting defects on the critical slowing down characteristics is needed. Furthermore, the present study investigated the precursory information for mode I fracture of rock based on the critical slowing down theory. Further study on mode II and mixed-mode loading conditions is needed.

Conclusions
e study investigated the precursory indicator of AE signals during mode I rock fracture through critical slowing down analysis. e following conclusions are drawn: (1) e critical slowing down phenomenon exists during mode I rock fracture. e variance and autocorrelation coefficient of AE count sequences grow significantly preceding rock fracture. (2) e significant growth of variance and autocorrelation coefficient of AE signals can be viewed as the precursory indicator of rock fracture. Compared to autocorrelation coefficient, the precursors determined by variance are more remarkable. (3) e time interval between the precursory indicator using critical slowing down theory and fracture moment ranges from 2% to 15% of the loading time. (4) e difference between different specimens may be attributed to the discrepancy between microstructure of specimens, especially the spatial distribution and quantity of preexisting defects.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.