Non-linear Response of Acoustic Emission and Electric Potential During Creep Failure of Coal under Stepwise Increasing Loads: Insights from Multifractal Theory

Abstract

The combination of acoustic emission and electrical potential monitoring methods holds promise for monitoring and warning of rock bursts due to its comprehensive reflection of the damage process. However, the response features during the creep failure process remain unclear. In this paper, a coal creep test was conducted using a combination of electric potential and acoustic emission monitoring. The response characteristics were analyzed, their multifractal characteristics were analyzed, and the joint response mechanism was explored. This research demonstrated a significant correlation among acoustic emission and electrical potential signals and creep deformation and failure. At the start of loading, a brief increase in both signals was observed. As deformation progressed, the signals became steady, and their intensity and fluctuation notably increased during accelerated creep failure. Quantitative analysis of acoustic emission count rates and electric potential intensity during creep processes revealed a quadratic relationship of acoustic emission count rates with stress and strain variations, in addition to an exponential correlation with mean electric potential intensity. Additionally, the statistical analysis of the multifractal characteristics before coal sample instability and failure revealed consistent trends in the characteristic values of Δα and Δf(α), with initial decrease followed by slight fluctuations, culminating in a sudden abnormal change preceding failure. Finally, leveraging the mechanisms of acoustic emission and electrification under load, this study discusses the multifractal characteristics of acoustic-electric signals and verifies their complementary roles in accurately predicting coal rock creep failure. These studies provide essential theoretical groundwork and references for improving dynamic disaster monitoring in coal mines.

Introduction

As the depth of coal mining increases, rock burst disasters intensify. These disasters result from the rapid rheological failure process of rock masses under high strength loads (He et al. 2005; Mishra and Verma 2015; Li et al. 2017; Zhou et al. 2019; Bian et al. 2022; Hu et al. 2023). In this process, the energy stored in rock is released in the forms of elastic energy (Kong et al. 2019; Feng et al. 2022a; Feng et al. 2022b; Kong et al. 2022; Li et al. 2023a, 2023b), acoustic and electromagnetic energy (Wang and He 2000; Aydan and Akagi 2004), which require numerous geophysical techniques for monitoring and advance warning of rock burst disasters (Wang et al. 2011; Wang et al. 2012; Qiu et al. 2020; Li et al. 2021a, 2021b, 2021c, 2021d; Xie et al. 2022; Yin et al. 2024). The experimental study of the electrical and acoustic response signal laws during the creep failure of rock and coal is the basis for predicting rock burst disasters using acoustic and electrical methods.

As early as the late 1930s, rock rheological tests were preliminarily explored. (Griggs 1939) conducted rheological experiments on weak rocks and first proposed rock rheology as an independent topic (Tjongkie 1982). Creep, a significant aspect of rheology, is recognized as a major contributor to substantial deformations in the surrounding rock during deep mining operations (Yang et al. 2014). Previous rock creep experiments have shown that typical creep deformation exhibits three stages of behavior: (1) deceleration creep, (2) stable creep, and (3) accelerated creep, which are particularly evident in soft rock creep experiments. However, with the deepening of creep experiments and theoretical research, it was found that hard rocks also exhibit strong time effects at higher stress levels and can produce significant creep deformation over a longer period of time (Malan 2002). Graded loading and unloading creep experiments on different types of rock samples showed that the occurrence of each stage is closely related to the duration of the creep experiment and the applied stress level (Li and Xia 2000). The yield stress is the critical value for the occurrence of material creep failure. When the applied stress level is below this critical value, the creep rate continues to decrease and remains constant. Conversely, as the creep rate continues to increase, the sample ultimately fails. Subsequently, some scholars characterized the rock creep process quantitatively from a mechanical perspective and established constitutive models for rock creep under different experimental conditions, achieving fruitful results (Yang et al. 1999; Gasc-Barbier et al. 2004; Fabre and Pellet 2006; Wang et al. 2022a). However, in addition to describing the creep stress and deformation state of coal and rock using theoretical mechanical methods, the use of geophysical methods to monitor coal and rock creep has also been studied by scholars.

Acoustic emission (AE), the phenomenon of rapid release of energy during material deformation to generate transient elastic waves, has been utilized by several scholars to investigate the progressive failure process of rock and coal, leading to significant achievements. Rock bursts are nonlinear creep failure processes that occur in coal and rock mass systems under high ground stress during mining activities. These processes involve the steady accumulation of energy and the release of unstable energy. Thus, using AE technology to monitor rock bursts in coal mines is a sound engineering practice. Existing research has shown that AE parameters such as counting and energy can reflect the rock creep process effectively (Baud and Meredith 1997; Wang et al. 2022b, 2022c; Li et al. 2023a, 2023b, 2023c; 2023d), determine the damage evolution (Yang et al. 2024) and long-term strength of rocks (Shi et al. 2019; Zha et al. 2021; Ding et al. 2023a, 2023b), and investigate the influence of loading history on rock creep behavior (Dong et al. 2022). Moreover, AE characteristics can help elucidate how rocks deform when subjected to the combined effects of creep and fatigue (Zhao et al. 2022). In addition, some scholars combined AE and other monitoring methods to study the microscopic mechanism of creep (Liu et al. 2022a, 2022b, 2023). (Chu et al. 2022) explored the brittle creep mechanism of rock samples under a microscope through AE and nuclear magnetic resonance measurements and studied the AE characteristics during creep. (Shi et al. 2022) analyzed and studied the damage evolution during creep using AE and numerical imaging DIC, and studied the AE characteristics. The indoor experiments and on-site applications demonstrate that AE is an effective indicator of the stress state and damage evolution during creep. To date, AE technology has been used widely for monitoring and warning of dynamic disasters in underground mines.

The characteristic response of AEs is highly sensitive at the instance of fracture in coal rock materials, allowing for precise spatial localization of microcrack events. However, due to the discontinuous nature of rock fracturing, the response of AE signals during the stage from the onset of rock loading to fracture is suboptimal, posing challenges in comprehensively characterizing the evolution of rock damage. In contrast, electric potential techniques, owing to their high sensitivity and ability to facilitate continuous monitoring, offer complementary advantages to AE monitoring methods.

Electrical potential (EP), the potential difference generated by charge separation and movement, initially originated from research on the surface electrification phenomenon of coal fractures caused by the Earth's electric field anomalies during volcanic eruptions and earthquake preparation stages (Johnston 1989; Varotsos et al. 1993; Vallianatos et al. 2004). Scholars have investigated the phenomenon of rock fracture electrification (Eccles et al. 2005; Takeuchi et al. 2006; Li et al. 2021a, 2021b; Zhang et al. 2022). The research results show that the instantaneous charge of marble, granite, and other rocks sharply increases when loaded and unloaded quickly, and then decreases rapidly. The variation in natural electric field is atypical, and a clear correlation exists between the alteration in EP and the rate of stress change when stress undergoes sudden changes (Hao et al. 2004; Haas et al. 2013). Coal, a unique rock mass material, can also produce surface electrification anomalies when subjected to load and fracture. Research has revealed that EP is closely related to the damage and fracture state of coal, and the real time variation in EP amplitude can reflect the degree of deformation under loading (Li et al. 2009; Niu et al. 2018, Niu et al. 2022). Scholars have extensively investigated the mechanism of EP generation by examining the impact of its effects and response laws. (Li et al. 2009) suggested that friction electrification and crack expansion during the coal loading process lead to charge separation, providing a moving charge source for EP. (Niu et al. 2018) found through testing and analysis of EP signals at coal mine sites that there exists clear similarity in the distribution of EP and stress field, and the abnormal distribution area of EP space is consistent with the localization of coal body damage. (Liu et al. 2018) revealed the microscopic mechanism of natural electric field anomalies during coal rock failure based on the theory of crack tip discharge. Predecessors have extensively studied the EP effect, response law, and mechanism of rock fracture and have preliminarily explored applications of EP monitoring and warning against dynamic disasters. However, due to the characteristics of deep coal mines, such as high crustal stress and strong rheology, EP test research under the creep conditions of coal will be more closely related to the development processes of coal mine dynamic disasters. Hence, studying the response pattern of EP signals generated under creep conditions is highly important.

Considerable progress has been made in studying the response patterns of AE or EP signals during conventional loading and fracturing processes in coal rocks. However, there is a paucity of research on the synchronized response patterns of AE and EP during creep processes. Therefore, this study conducted triaxial creep mechanical tests on coal samples, simultaneously capturing deformation, AE and EP signals. It investigated the synchronized response patterns of acoustic–electric signals during coal mass creep failure, identified the anomalous characteristics of acoustic–electric signals before coal mass instability, and quantitatively characterized the nonlinear evolution features of AE and EP signals using multifractal theory. By integrating both monitoring techniques and employing multifractal analysis, subtle changes in the mechanical behavior of coal rock could be discerned and the variations in both were analyzed comprehensively, providing a comprehensive understanding of the evolving damage process. This allows more accurate identification of key precursors to creep failure, thereby enhancing the early warning capability of the monitoring system.

Experimental Details

Experimental System

A self-developed conventional triaxial loading control experimental system was used for simulating the three-dimensional stress state of coal bodies in coal mines. The axial stress was loaded by hydraulic pressure pushing the plunger, and the confining pressure was loaded by hydraulic pressure on the rubber sleeve. Two independent hydraulic pumps controlled the three-dimensional loading stress, with a dual pump (continuous flow) flow rate range of 0.001–107 ml/min and accuracy of 0.5% of the indicated value. The loading system provided maximum axial and confining pressures of 50 and 40 MPa, respectively. The axial deformation of the specimen was measured using an external linear variable displacement sensor (LVDT). The COBWEB-DAU multichannel data acquisition system collected EP signals with a DC coupling and multichannel parallel acquisition method and a sampling frequency of 195 Hz. The EP signal was processed by an analogue-to-digital converter and recorded, stored, and displayed in real time by the host. The primary monitoring indicator of EP was the potential intensity (in mV). The AE signals were collected and stored using the Express-8 model 24-channel AE system from the American company Physical Acoustics. Four NANO-30 model AE sensors were employed in the experiment, which were tightly adhered to the surface of the test cylinder. Their operating frequencies ranged mainly between 150 and 400 kHz, with resonant frequency of 300 kHz. The acquisition threshold for the AE signals was set at 32 dB, with sampling frequency of 1 MSPS and a corresponding preamplifier gain of 40 dB. The key parameters monitored for AE were signal intensity and ring-down counts. This study primarily analyzed the ring-down counts of AE signals during the creep process of coal samples. The experimental setup is shown in Figure 1.

Figure 1

Schematic diagram and device photos of the experimental system: (a) strain data collection system; (b) loading and control system; (c) AE data acquisition system; (d) EP data acquisition system. 1 = AE probe; 2 = axial plunger pressure head; 3 = LVDT displacement measurement sensor; 4 = confining pressure valve; 5 = high precision cylindrical piston pump; 6 = displacement and pressure testing system; 7 = stress–strain data display screen; 8 = AE signal preamplifier; 9 = AE data acquisition instrument; 10 = AE data display screen; 11 = sample and electrode copper sheet; 12 = EP data acquisition instrument

Sample Preparation

The sample was a cylindrical standard coal sample with height of 100 mm and diameter of 50 mm. The surface was polished with fine sandpaper, and the unevenness of the surfaces at both ends was less than 0.02 mm. After wiping the surface with alcohol, the sample was placed indoors and allowed to dry naturally. Prior to conducting the experiments, the basic mechanical parameters of the specimens were tested, and the results are shown in Table 1. The electrode piece used to measure the EP signal was coupled with the coal sample surface through a highly sensitive conductive paste and fixed with hot melt adhesive to ensure fine electrical contact between the electrode piece and the coal, avoiding signal loss during the experiment. The wire connecting the electrode piece passed through the upper wire connection of the clamp chamber and was connected to the potential meter. The lead port was tightly matched with a PVC conical sealing ring and bolts to ensure sealing of the loading system. The AE probe was secured to the sample holding chamber's surface with hot melt adhesive, and Vaseline was applied to improve the conductivity of the signal.

Table 1 Basic mechanical parameters of coal sample

Test Steps

The coal sample, with electrodes and wires properly arranged, was placed in the loading chamber. Stainless steel pads with diameter of 50 mm were added at both ends of the specimen, and plunger heads were installed simultaneously on both ends until the pads were compacted, ensuring the specimen's fixation at the center of the chamber. Once the experimental system was set up and debugged, the stress loading system was initiated. An axial stress of 1–2 MPa was applied to stabilize the coal sample, followed by applying confining pressure to a preset value of 2–4 MPa. The confining pressure was maintained constant, and axial pressure was applied to the present value. Each level of axial pressure was maintained for approximately 20 hours until the deformation of the sample stabilized, after which the next level of axial stress was applied. This process continued in a cyclic manner until the sample experienced creep failure. The loading path is shown in Figure 2. Simultaneously, the stress–strain data acquisition system, multichannel AE data acquisition system, and EP data acquisition system were activated to collect stress, strain, AE, and EP data during the coal sample loading and creep experiments. After the coal sample underwent creep failure, the experimental loading and data acquisition systems were stopped, marking the end of the experiment. The triaxial creep test scheme is shown in Table 2.

Figure 2

Test steps and loading path: (a) test coal sample; (b) prepared coal sample; (c) initial loading; (d) AE, EP, and strain data collection; (e) loading path; (f) damaged sample

Table 2 Triaxial creep test scheme

Test Results

In this study, creep tests were conducted on seven sets of coal samples under graded loading conditions, and strain, AE, and EP signals were synchronously collected during the creep process. Two sets of specimens, in which the entire process of creep failure was observed, were selected, and the evolution laws of AE and EP with creep deformation were studied. The sample numbers were C1 and C2.

Time-Varying Response Characteristics of AE during Creep Failure

The entire creep failure process of two groups of specimens, C1 and C2, under graded loading conditions is shown in Figure 3. AE count signals during the creep process were collected simultaneously. The figure illustrates a notable correlation between the coal sample creep deformation and AE count curves at varying stress levels. At each stage of stress loading, the instantaneous deformation increases, and the AE counts significantly increased abnormally, exhibiting a discontinuous sudden increase corresponding to the stress loading gradient and then stabilizing. Before the failure stress, the deformation rate decreased gradually to a stable level over time, and the AE intensity then decreased to a lower level. In the failure stress level stage, the deformation changed from stable growth to accelerated growth until the sample underwent unstable failure, at which point the AE count significantly increased.

Figure 3

AE response during the creep process

Analysis was conducted on C2, and only deceleration and steady-state creep were observed at a graded stress level of 6–10 MPa. At every stress level, the microcracks and pores in the sample were compressed and closed, resulting in instantaneous elastic deformation and a small increase in the AE count, but with relatively high amplitude. As the time of constant load application increased, the deformation tended to stabilize, and the AE count and strength decreased significantly. When the axial stress increased to 11.5 MPa, the specimen underwent accelerated creep failure after 1 h of creep stress. The AE counts and intensity generated by the coal sample under pressure during this stage increased significantly compared to those at low-stress levels. According to the graded loading deformation curve, under constant load, the deformation of coal samples will exhibit two situations: maintaining stability with time and accelerating growth, leading to instability and failure. The accelerated creep stage is the final stage of specimen creep deformation and failure, and the initiation of accelerated creep marks the beginning of specimen failure. For engineering practice, studying the accelerating creep deformation stage is important for preventing the failure of coal mine tunnels and stress concentration areas. Therefore, conducting an in-depth analysis of the AE response characteristics during the accelerated creep failure stage is necessary.

In the creep test, the second-level creep of C1 and the fifth-level creep of C2 caused accelerated creep failure. From the creep failure curve, it was observed that the creep failure process had gone through three complete stages of creep deformation: deceleration, steady-state, and acceleration. To delineate the three stages of creep deformation more accurately and systematically, a more in-depth analysis of the creep rate during the creep deformation process under the given stress conditions is necessary. We introduced the autocorrelation function to analyze the time series. Autocorrelation is a crucial concept in time series analysis, indicating a certain degree of correlation between the values at one moment and those at another moment in the time series. Its calculation formula is:

$${\rho }_{l}=\frac{\text{Cov}\left({r}_{t},{r}_{t-l}\right)}{\sqrt{\text{Var}\left({r}_{t}\right)\text{Var}\left({r}_{t-l}\right)}}=\frac{\text{Cov}\left({r}_{t},{r}_{t-l}\right)}{\text{Var}\left({r}_{t}\right)}$$

(1)

where ${\text{cov}} \left( {r_{t} ,r_{t - l} } \right)$ is the covariance of the creep rate at time interval l with r_t; $\text{Var}\left({r}_{t},{r}_{t-l}\right)$ is the variance of the creep rate at time interval l with r_t; ρ_l is autocorrelation. The autocorrelation of creep rate was calculated and the results are shown in Figure 4. Based on this, the creep failure stages of two sets of experimental data were divided into stages. Figure 5 shows the characteristic curves of deformation and AE response during the creep failure stage.

Figure 4

Strain rate and its autocorrelation

Figure 5

Characteristics of strain and AE response during creep failure stage: I = deceleration creep; II = steady-state creep; III = accelerated creep

Figure 5 demonstrates a significant correlation between AE counts and creep, which can be categorized into three stages according to the creep deformation law: (1) The initial active period. After applying a constant load for 0.47 h and 0.14 h, respectively, the strain continuously increased, but the rate of strain increase decreased gradually and tended to stabilize. Due to the compression of the instantaneous axial stress, the initial stress equilibrium in the coal formation was disrupted, and the skeleton and particles in the coal body slipped and moved, resulting in further compaction and closure of cracks. The strain energy was discharged through the AE, and the AE signal was relatively active. (2) The short calm period. The creep entered a stable increasing stage, and the weak crystal units inside the sample were completely destroyed. The cracks were further compacted and closed, leading to an increase in the intensity of the coal. Correspondingly, the AE activity entered a short-term stable and quiet period. (3) The active period. With the steady increase in crack damage, the cumulative strain increased, and internal damage dominated the coal. The accumulated cracks in the steady creep process began to extend and develop, and the dislocation and sliding of crystal particles formed more macrocracks, which ultimately resulted in macro-instability and fracture. The coal underwent an accelerated creep failure process, and the strain energy it contained was released fully, leading to a simultaneous maximum value of the AE signal.

Figure 6 shows the variation in the creep rate of specimen C2 under different creep stress conditions. Under the noncreep failure stress level of 6–10 MPa, the steady-state creep rate of the sample remained between 7.54 × 10^-9 and 1.12 × 10^-8, and an approximately linear positive correlation with the change in stress was observed. However, compared to the steady-state creep rate of 5.83 × 10^-5 during the creep failure stage, this rate of change was very low. Moreover, a statistical analysis was conducted on the variation in the AE count rate during the steady-state creep stage of C2 during the graded loading process with respect to the strain rate and loading stress level. Figure 7a displays the results indicating a quadratic relationship between the AE count rates and loading stress. During primary loading, the AE count rate tended to increase because of the sliding of weaker structures and the closure of cracks. The AE count rate reached the lowest level at σ = 7.5 MPa and dɛ/dt = 1.08 × 10^-8, and subsequently increased with escalating stress. Figure 7b reveals the relationship between the AE counting rate and creep rate. Due to the rapid increase in strain during the catastrophic failure stage, the magnitude of creep rate increased from 10^-8 to 10^-5. Therefore, the stage before the stress level of the creep failure was subjected to amplification analysis. The relationship between the strain rate and the AE count rate was also a quadratic function. The AE count rate under the initial loading stress gradient was significantly greater than that under the subsequent creep stress gradient, and reached its maximum during the creep failure stage.

Figure 6

Relationship between strain rate and axial stress

Figure 7

Variation patterns of AE count rate with strain rate and loading stress: (a) relationship between stress and AE count rate; (b) relationship between strain rate and AE count rate

Time-Varying Response Characteristics of EP during Creep Failure

Figure 8 shows the EP response during the creep process. Compared with the strain and EP strength curves during the creep test process of the graded loading of the sample, the trend of the two changes also showed significant synchronicity. Using Figure 8b as example, there was a notable surge in the EP intensity at the initial moment of stress loading, which then stabilized gradually with slight decrease. In the steady-state deformation stage, the immediate EP strength increased slightly under loading. As stress levels increased, the overall EP strength also improved, albeit to a lesser extent. In the creep failure stage, the EP intensity experienced a significant enhancement. As creep time increased, it substantially increased, and unstable failure triggered a sudden increase. There is a certain correlation between the EP signal and the deformation process of the coal samples.

Figure 8

EP response during the creep process

Statistical analysis was conducted on the variation pattern of the average EP during the steady-state creep stage of C2 during the graded loading process between the strain rate and the loading stress level (Fig. 9). From Figure 9, it can be determined that the EP value showed an exponential growth trend with the magnitude of stress, while there was a slight difference in the relationship with the creep deformation rate. Before the instability failure stress (11.5 MPa), the steady-state creep deformation rate of the sample was small and displayed little variation, with strain rate range of $7.54\times 1{0}^{-9}\sim 1.42\times 1{0}^{-8}$ mV. However, at the failure stress level (11.5 MPa), due to the sudden rheological failure of the sample, the strain rate increased sharply, reaching $5.83 \times 10^{ - 5}$ mV, with maximum difference of 10⁴ orders of magnitude from the strain rate level in the nonfailure stage. Therefore, to reflect the variation in EP strength comprehensively with the strain rate of the specimen, the strain rate and EP curve of the undamaged stage were amplified locally and combined with the failure stage data to reflect the variation in EP comprehensively with respect to the strain rate of different creep stages.

Figure 9

Variations of average EP with loading stress level and strain rate: (a) relationship between stress and EP change; (b) relationship between strain rate and EP change

The second-level creep of sample C1 and the fifth-level creep of C2 caused accelerated creep. Figure 10a, b depicts the strain and EP response characteristics. Under the applied stress conditions, the specimens experienced unstable failure for 3.57 h and 1.04 h, respectively. Compared with C2, sample C1 exhibited a much shorter time from stable deformation to unstable failure, indicating a relatively slow failure process. In contrast, sample C2 experienced a faster and more severe process of transition from stable deformation to unstable failure at this stress level. For the corresponding EP, it can also be observed from the graph that during the steady-state stage before the failure of C1, the EP increased slightly with no significant change, and only a significant sudden increase occurred in the final acceleration stage. The EP fluctuation amplitude of C2 before instability failure was relatively large, there was a significant steady upwards trend, and there was a significant sudden increase before accelerated creep failure.

Figure 10

Strain and EP response characteristics during creep failure stage: I = deceleration creep; II = steady-state creep; III = accelerated creep

Multifractal Characteristics of AE-EP Signals during Creep Failure

Through the analysis of AE and EP signals generated during the creep failure process of coal samples in section Time-varying response characteristics of AE during creep failure and section Time-varying response characteristics of EP during creep failure, respectively, it was found that both exhibit good synchronicity with creep deformation at different creep stages, which can better respond to the creep process. Previous researches indicate that the signals of AE and EP can be employed effectively in studying the evolution of damage and the process of instability failure in materials (Wang et al. 2018a, 2018b; Li et al. 2019a, 2019b). However, during long-term creep experiments, the AE and EP signals generated during creep fluctuated considerably due to the heterogeneity of the material structure and the presence of crack defects. Qualitative analysis of sudden changes in EP and AE signals as precursor information is insufficient when predicting specimen instability and failure. To enhance the precision of identifying the precursor points of creep instability failure in coal samples, additional examination and refinement of EP and AE signals are necessary.

The multifractal analysis method is an important means for revealing potential patterns and structures in complex time series data. The deformation and fracture process of coal and rock under loading is a nonlinear process, and the closure, initiation, propagation, and aggregation of internal cracks are not continuous. The generated AE and EP signals are also dispersed and nonlinear (Liu et al. 2019, 2022a, 2022b, 2024; Zhang et al. 2021; Li et al. 2022; Zheng et al. 2022). The use of multifractal calculation methods to analyze the processes of AE and EP changes in coal rock samples is helpful in revealing the deformation and failure process.

Multifractal Theory

A multifractal structure is an infinite set composed of fractal measures with multiple scaling exponents on the fractal structure. Singularity reflects the degree of unevenness in a sequence, and existing studies have shown that the fractures of faults and joints in rocks (Xie et al. 2001), AE spatial positioning results (Li et al. 2019a, 2019b, 2023a, 2023b, 2023c, 2023d), and EP response characteristics (Hu et al. 2014) all exhibit fractal characteristics. Because the essence of creep failure of coal is the development of cracks and the continuous accumulation of damage and deformation, the deformation process can be considered to have fractal characteristics. Therefore, multifractal analysis of AE and EP signals during the creep process can offer fresh insights into monitoring and predicting the creep process and instability failure of coal.

This study employed the box dimension method to investigate the singularity evolution characteristics of the time series response data of EP and AE and focused on multiple fractal dimensions during creep damage and sudden failure of coal. Let a time series be {x(i)} and divide it into N subsets, each with a length of n, and with the probability distribution function {P_i(n)} for each subgroup. {P_i(n)} and n are related as follows:

$$\left\{ {P_{i} (n)} \right\} \propto \mathop {\lim }\limits_{n \to 0} n^{\alpha }$$

(2)

where α is the subset singularity index, which is a measure of the variability in the subset of the probability distribution within the time series {x(i)} at different scales n. As the length of the general subset sequence decreases, the number of subsets obtained (denoted as N_α(n)) increases. This increase is inversely proportional to n, and can be expressed mathematically as:

$$N_{\alpha } (n) \propto n^{ - f(\alpha )}$$

(3)

where f(α) reflects the frequency of the subset sequence corresponding to the constant value α appearing in all subsets, and it can also be considered as the fractal dimension of α; f(α) can be determined as:

$$f(\alpha ) = \mathop {\lim }\limits_{n \to 0} \frac{{\ln N_{\alpha } (n)}}{\ln n}$$

(4)

Introducing methods of statistical physics to solve multifractal spectra indirectly, the partition function (statistical moment) is defined as:

$$X_{q} (n) \equiv \sum P_{i} (n)^{q} \sim n^{\tau (q)}$$

(5)

where τ(q) is quality index. At a certain point in the calculation process, when |q| attains a particular value, the characteristic values of the multifractal spectrum almost do not change. To reduce computational workload, the value of q is typically constrained to a certain range. We can simplify the above equation by taking the logarithm of both sides:

$$\tau \left( q \right) = \mathop {\lim }\limits_{n \to 0} \frac{{\ln X_{q} (n)}}{\ln n}$$

(6)

When the time series {x(i)} exhibits multifractal characteristics, the lnX_q(n) – lnn curve shows a strong linear correlation. The generalized fractal dimension is defined as:

$$D = \frac{\tau \left( q \right)}{{q - 1}} = \frac{{\ln X_{q} (n)}}{{\left( {q - 1} \right)\ln n}}$$

(7)

After Legendre transformation, we obtain:

$$\alpha = \frac{{{\text{d}}\left( {\tau \left( q \right)} \right)}}{{{\text{d}}q}} = \frac{d}{dq}\left( {\mathop {\lim }\limits_{n \to 0} \frac{{\ln X_{q} \left( n \right)}}{\ln n}} \right)$$

(8)

$$f\left( \alpha \right) = \alpha q - \tau \left( q \right)$$

(9)

Analysis of Multifractal Characteristics of AE–EP Signals

Multifractal theory suggests that multifractal spectral features can indicate the distribution roughness of event time series. The AE and EP signals during sudden and unstable creep damage were analyzed using multifractal theory; α represents AE and EP events of different intensities. Figure 11 shows a schematic diagram of the multifractal calculations for the AE data time series. The α_min denotes the large signal in the subset sequence, while the α_max represents the small signal in the same subset. The multifractal spectral width, Δα = α_max-α_min, indicates the extent of variation in signal changes within the subset. As Δα increases, the signal strength fluctuates more intensely and exhibits more pronounced fractal characteristics. The value of f(α) represents the frequency of the AE counts and EP intensity time series with a singularity value of α, f(α_min) represents the frequency of occurrence of large signals, and f(α_max) represents the frequency of occurrence of small signals. Let Δf = f(α_max) – f(α_min), where Δf represents the proportion of AE and EP intensity signal values. A positive Δf value represents weak signal dominance, while a negative value represents strong signal dominance. The multifractal parameters, which were collected synchronously during the creep failure stage, were analyzed statistically. The statistical results are shown in Figure 12 for AE and Figure 13 for EP.

Figure 11

Schematic diagram of multifractal calculation process for AE data time series

Figure 12

Multifractal characteristic values of AE during creep failure stage of coal samples

Figure 13

Multifractal characteristic values of EP during creep failure stage of coal samples

Figure 12 shows the calculation results of the characteristic values of the AE multifractals during the creep failure stage for the two sets of samples. As shown in Figure 12, the trends of Δα and Δf showed a specific correlation. During the initial stage of stress loading, the two characteristic values experience varying degrees of fluctuation and reduction. As the loading continued, the two exhibited a three-stage pattern characterized by rapid reduction, stable maintenance, and sustained rapid fluctuation before the sample underwent creep failure. For a more in-depth analysis of the fractal characteristics of AE during creep, we focused on the creep failure stage of the C2 sample. The coal sample did not instantaneously fail under this stress level but underwent creep failure under the sustained action of stress, indicating that the stress level lies between the long-term and peak strengths of the sample. Under a high static load, the coal sample underwent immediate plastic deformation, and internal cracks expanded, leading to large-scale short-term fractures. The frequency of large AE signals increased relatively. Although the sample did not reach its peak strength, continuous loading caused the internal cracks to expand gradually, causing damage to accumulate over time. As a result, the intensity of the AE signal decreased compared to that in the initial loading stage. Therefore, the eigenvalues Δα and Δf of the multifractal exhibited a decreasing trend from high to low. This suggests a reduction in the intensity difference of the AE signals, as well as a significant decrease in the proportion of large signals. At the point at which the creep time reached 136.04 h, there was a significant decrease in Δf, while Δα underwent a significant increase. The large signal increases, and the difference between the large and small signals was magnified, indicating intensification of coal body fracture.

Figure 13 shows the calculation results for the fractal characteristic values obtained from the synchronously collected EP signals during the creep test of the C1 and C2 samples. Obviously, the multifractal characteristic values Δα and Δf of the EP signal before coal sample instability and failure exhibited a reverse synchronous trend. The Δα initially decreased and then exhibited small fluctuations and stability. As failure approached, the Δα rapidly increased in value, surpassing its maximum value during the initial decrease. Moreover, the Δf was negative during the sample's initial loading stage and remained mostly unchanged during the steady-state changes in the sample. However, it decreased rapidly to a negative value in the early stage of sample failure. This negative value of Δf signifies an increase in large cracks inside the coal and a strengthening of crack propagation. These characteristics aligned with the deformation behavior of the creep failure stage. By analyzing the fractal characteristics, it can be concluded that these signals underwent significant mutations prior to creep failure. However, the change trends of their fractal characteristics were not identical. The fractal characteristic parameter values of AE exhibited small, stable, or even decreasing trends in the initial upwards trends of Δα and Δf before failure. However, they continued to increase as the sample was damaged. In contrast, the fractal characteristic values of the EP signals collected synchronously exhibited sudden and more significant changes from stable state directly before failure.

Discussion

Mechanism of AE and EP in Coal Fracture Process

AE and EP are homologous and dissimilar phenomena. The former depends on the transient elastic waves induced by the multifrequency vibration of the material. Therefore, the question arises: is there a specific synchronous relationship between EP and AE signals during the coal and rock creep loading process? Understanding the interrelationship between the two signals has crucial theoretical value for establishing a joint acoustic and electrical warning method.

AE Mechanism

Generally, the physical and mechanical properties of coal rock materials, including strength, homogeneity, and grain size, affect the AE phenomenon decisively. Previous studies have shown that there are numerous micropores and microcracks inside the coal body. The essence of compressive failure results from the expansion, convergence, and rupture of inherent defects under tensile action. When the stress surpasses the ultimate failure strength of the defect unit, its activation leads to the failure of defects, resulting in the generation of AE.

The defect distribution of coal rock materials follows the Weibull distribution (Zhou 2007), thus:

$$n(\varepsilon ) = k\varepsilon^{m}$$

(10)

where n(ɛ) represents the number of defects activated by strain ɛ; k and m are constant values related to the distribution of material defects. When the strain increases by increment dɛ, the number of new defects generated is:

$${\text{d}}n = n^{\prime}(\varepsilon ){\text{d}}\varepsilon$$

(11)

Assuming that the defect units of the coal rock material correspond one-to-one with the AE counts, considering that the coal rock material itself contains initial defects; therefore, under the condition that the initial damage of the material is D, the AE count corresponding to dɛ is:

$${\text{d}}N = (1 - D)n^{\prime}(\varepsilon ){\text{d}}\varepsilon$$

(12)

Therefore, under the condition of strain ɛ, the cumulative AE count generated is:

$$N = \int_{{\varepsilon_{0} }}^{\varepsilon } {{\text{km}}(1 - D)\varepsilon^{m - 1} } {\text{d}}\varepsilon$$

(13)

where ɛ₀ is the initial damage strain of the material. Subsequently, the change rate of AE count is:

$$N^{\prime}\left( t \right) = {\text{km}}(1 - D)\varepsilon^{m - 1} \frac{{{\text{d}}\varepsilon }}{{{\text{d}}t}}$$

(14)

The Eq. (14) suggests that the change rate of the AE was affected primarily by strain and strain rate, and it correlated with coal materials' physical and mechanical characteristics, including inherent defects, homogeneity, and material scale, etc.

Electrification Reason

The electrification phenomenon during the deformation of coal and rock under loading remains a subject of diverse interpretations. The key aspects include the piezoelectric effect, friction-induced electrification, and potential dislocation effect. Consequently, to pinpoint the primary contributors to electromagnetic radiation during coal and rock deformation and failure, a comprehensive examination and investigation of these electrical phenomena is imperative.

In the realm of the piezoelectric effect, the charge density correlates directly with the external force. Assuming that the piezoelectric effect is the source of electrification in loaded coal, the potential increases linearly with stress. However, during initial loading, the electrical amplitude initially surges with increasing stress. Once stress stabilizes, the potential continues to increase, albeit not linearly. A sudden stress drop, if attributed to the piezoelectric effect as the charge source, would lead to an abrupt decrease in sample charge, given the proportional relationship between charge density and external force. However, the sharp stress decrease at the point of final rupture, coupled with a sudden potential increase, contradicts the piezoelectric effect hypothesis. This suggests that the piezoelectric effect may not be the primary driver of potential generation. This viewpoint aligns with those of other researchers (Cress et al. 1987), who similarly contend that the piezoelectric effect is not the main contributor to electrical signal generation in coal rock materials.

Microscopic imperfections, such as pores and microcracks, are inherently present within coal structures, and particle interactions during deformation and failure lead to friction. Consequently, frictional electrification pervades and persists throughout the entirety of this process, resulting in the generation of free charges through frictional forces during coal damage. During the compaction stage, in which new cracks do not form, friction enters the center stage. In contrast, in the plastic deformation stage, the primary factors are crack propagation and tip effects. Thus, the frictional electrification effect cannot be the principal electrification mechanism in the creep failure process.

The moving charge district (MCD) model, initially proposed by (Slifkin 1993) and subsequently refined by (Vallianatos and Tzanis 1998), posits that charge generation in brittle materials arises from the motion of dislocations or localized polarized defects. This phenomenon is inherently associated with brittle failure, as stress concentrations and crack openings occur once edge dislocations propagate, migrate, and accumulate, surpassing a critical stress threshold. The MCD model has gained increasing acceptance among researchers and has been experimentally validated (Stavrakas et al. 2003; Anastasiadis et al. 2007; Stergiopoulos et al. 2015). In this study, when lower stress levels were applied, the EP signal change exhibited a relatively minor amplitude. However, during the creep failure stage, there was a significant amplification in the overall EP growth. Therefore, it was rational for this study to employ the MCD model to elucidate the primary factors influencing EP changes during the coal rock fracture process. Research has shown that EP is formed by the superposition of the polarization electric field of the free charge coal rock medium generated by sample fracture. There are also many dislocations in coal. Under stress, these dislocations will shift relative to each side of their slip surface, resulting in the diffusion of charged cutting steps under nonequilibrium stress, leading to charge separation.

According to the MCD model: assuming that $\Lambda^{ + }$ represents the density of mechanical flavor edge dislocations necessary to accommodate uniaxial compression or tension, and $\Lambda^{ - }$ represents the density of dislocations of the opposite type, the movement of charged dislocations generates a transverse polarization that can be determined as:

$$P = (\Lambda^{ + } - \Lambda^{ - } ) \cdot q_{1} \cdot \frac{\delta x}{{\sqrt 2 }} = \delta \Lambda \cdot q_{1} \cdot \frac{\delta x}{{\sqrt 2 }}$$

(15)

where $q_{1}$ represents the charge per unit length, $\delta x$ denotes the distance traveled by dislocations, and $\delta \Lambda = (\Lambda^{ + } - \Lambda^{ - } )$ indicates the variation in edge dislocation density.

A crystal lattice employs Burger's vector ${\text{b}}$ to determine the distance, disregarding screw dislocations. When Burger's vector $b$ of these dislocations moves through a distance $\delta x$, the plastic strain contribution can be described as (Vallianatos et al. 2004):

$$\varepsilon = (\Lambda^{ + } - \Lambda^{ - } ) \cdot b \cdot \frac{\delta x}{2}$$

(16)

Defining electric current density J as the polarization change rate, combining Eq. (15) and (16) leads to a straightforward demonstration that (Vallianatos et al. 2004):

$$J = \frac{\partial P}{{\partial t}} \Rightarrow J = \frac{\sqrt 2 }{\beta } \cdot \frac{{q_{1} }}{b} \cdot \frac{{{\text{d}}\varepsilon }}{{{\text{d}}t}}$$

(17)

where $\beta = (\Lambda^{ + } + \Lambda^{ - } )/(\Lambda^{ + } - \Lambda^{ - } )$. By employing Eq. (17), one can explain the connection between the nonstationary deformation accumulation and the detected electric signal. It has been established that anticipated values are highly consistent with laboratory test measurements. Due to the fact that EP is formed by the superposition of free charge polarization electric fields generated by rupture, the polarization rate defined by current density can also reflect EP.

Precursor Characteristics of Coal Creep Failure Based on Combined Acoustic and Electrical Responses

In the process of deformation and failure of coal rock under loading, the internal cracks propagate and release deformation energy in the form of elastic waves. AE technology characterizes qualitatively the rock's stress and deformation state by monitoring the elastic waves released during this process. Despite advantages, such as high sensitivity, real time signal transmission, and strong localization capabilities, AE faces limitations in coal rock deformation monitoring. Its signals are susceptible to interference from material acoustic properties and environmental noise, and the selection of signal threshold values may lead to false positives or negatives. Moreover, the various characteristic parameters of AE make it challenging to accurately quantify the extent of coal rock deformation damage, rendering AE more suitable as an indicator of damage occurrence rather than a tool for quantitative measurement. Therefore, the necessity of seeking a multifaceted, collaborative monitoring approach becomes apparent. During the deformation and damage of materials, internal structural deformations and fractures generate EP signals. These signals, which are not constrained by the propagation paths of stress waves, provide relatively accurate continuous EP change information, facilitating the quantitative assessment of damage severity. Hence, by integrating AE and EP monitoring, a more comprehensive, sensitive, and reliable monitoring system can be established, enhancing our understanding and ability to predict material deformation and damage processes.

According to the characteristics of both signals under load, it can be inferred that the two signals are closely related to the stress and deformation. Figures 3 and 8 reveal a significant long-range correlation between the changes in AE and EP responses and the stress levels and creep damage evolution. During the steady-state creep stage, the signal tends to stabilize and the strength is relatively small, with significant enhancement occurring when there is instantaneous deformation or near failure. To further analyze the relationships between deformation, AE, and EP, they were summed and averaged at intervals of 0.1 h. Then, the AE counts were accumulated and processed to obtain the cumulative AE counts, average EP values, and deformation comparison curves (Fig. 14). From these curves, it can be determined that the cumulative counts of AE, EP, and deformation trends were essentially synchronized, showing a steady upwards trend. However, there were certain differences in the trends of the AE and EP changes. To analyze quantitatively the relationship between the AE and EP signals during creep failure, correlation analysis was conducted on the acoustic and electrical signals.

Figure 14

Joint response characteristics of AE and EP during the instability and failure stage

Let two sequences α(α₁, α₂,.., α_n) and β(β₁, β₂, ····, β_n) with a length of n. The correlation coefficient r_αβ between the two can be expressed as:

$$r_{\alpha \beta } = \frac{{\sum\limits_{n}^{i = 1} {\left( {\alpha_{i} - \overline{\alpha }} \right)} \left( {\beta_{i} - \overline{\beta }} \right)}}{{\sqrt {\sum\limits_{n}^{i = 1} {\left( {\alpha_{i} - \overline{\alpha }} \right)^{2} } } \sqrt {\sum\limits_{n}^{i = 1} {\left( {\beta_{i} - \overline{\beta }} \right)^{2} } } }}$$

(18)

The statistical correlation between the cumulative counts of AE and EP synchronously collected during the creep process showed a correlation coefficient of 0.97, indicating highly positive correlation. The changes in AE and EP during the creep deformation process are highly consistent. Therefore, the identification of the precursor characteristics of creep failure can be transformed into the precise identification of precursor characteristic points in the joint response of acoustic and electrical signals.

The characteristic values of multifractals can reflect the non-uniform characteristics of set branches. Previous studies have shown that crack propagation during rock failure has fractal characteristics. Therefore, studying the fractal characteristics of AE and EP signals generated synchronously during deformation provides an essential reference value for quantitatively describing the process of rock deformation and fracture. Figures 12 and 13 show the multifractal eigenvalues Δα and Δf of the AE counts and EP signals. The trends of the changes in Δα and Δf were similar. Overall, there was a significant change trend with respect to the three stages of creep deformation. During the deceleration creep stage, the acoustic and electrical multifractal characteristic values decreased rapidly, and the steady-state creep process tended to stabilize. This suddenly changed when the sample was approaching instability and failure.

Therefore, analyzing the trends of changes in the multifractal eigenvalues of AE or EP revealed that, while their overall trends were similar, there were minor differences. Figure 12 displays the fractal characteristic values Δα and Δf of AE. These values showed more pronounced fluctuations during the steady-state creep stage in comparison to the potential, suggesting a greater variation in the size of the AE counting data column, because although the coal body did not reach the peak stress level under high static load, new cracks emerged in coal due to exceeding its long-term strength. Under continuous high stress, the cracks intermittently accumulated and connected, and large, discontinuous and sporadic AE signals also appeared, resulting in significant differences in the AE values. The production of EP signals was linked to crack propagation, but it was an ongoing process that was influenced by the sample's deformation and deformation rate. While, cracks formed intermittently during the steady-state creep stage, there was no significant increase in the overall deformation or the deformation rate. Analysis of the EP fractal data revealed that Δα and Δf had smaller fluctuations before the onset of creep instability, and their magnitudes were significantly smaller than those of the AE fractal characteristic values. From this, it was determined that the AE signal changed significantly but fluctuated greatly before the failure of the coal, making it difficult to identify the precursor points of the failure. Although, the EP signal had a low recognition level for small fractures, the fractal characteristic values can identify more accurately the precursors of instability and failure. Therefore, we can considered combining AE and EP technology to improve the accuracy of predicting the creep failure of rocks.

Conclusions

In this study, joint monitoring and testing of AE and EP signals during creep failure was conducted. The response characteristics of the AE and EP signals during different creep deformation stages were analyzed, and both signals during the creep failure stage were characterized quantitatively using multifractal methods. The joint response mechanism of AE and EP was explored. The research results were as follows:

• (1) The AE and EP signals generated under triaxial creep conditions exhibited noticeable phase changes with creep deformation. At the moment of stress loading, the AE underwent a brief active period, followed by a significant decrease in strength during the steady-state creep stage and a relatively calm period, and reached its maximum value during the accelerated creep failure stage. Overall, the EP increased gradually with increasing deformation and multistage stress. During the accelerated creep failure stage, the fluctuation amplitude of the EP increased significantly, and there was a sudden increase before instability failure.

• (2) A statistical analysis of the multifractal characteristics of the AE and EP signals generated under creep failure stress levels was performed, and the multifractal characteristic values Δα and Δf of the two signals were obtained. The variation patterns of Δα and Δf were relatively synchronous. The fractal characteristic values of the three stages of coal sample deceleration creep, steady-state creep, and accelerated creep failure showed a rapid decrease, slight fluctuation, and abnormal mutation trend, respectively, indicating the similarity of the corresponding laws of AE and EP signals.

• (3) Both AE and EP indicate precursor responses to the creep failure of coal, and the two offer complementary advantages. The AE signal responds significantly but with high volatility, and it is challenging accurately to identify precursor points of damage. The EP signal does not have a high degree of recognition for small ruptures, but its fractal feature values significantly show the precursor feature of the damage points. Therefore, the precursor characteristics of creep failure based on the combined response of AE and EP can improve the accuracy of predicting creep failure.