Introduction

The mechanical property of coal is loading-rate and loading-direction dependent. Coal specimens generally exhibit different mechanical responses as the loading direction and the loading rate change1,2,3,4,5,6,7. Meanwhile, the phenomenon that mechanical property varies with the loading direction is described as the mechanical anisotropy of coal, including mechanical parameters and failure characteristics8,9,10. During the coal mining process, coal mass around the roadway and the workingface spaces is frequently subjected to various degrees of cyclic disturbances that come from different directions11,12,13,14,15, which complicates the mechanical anisotropy feature of coal and makes the anisotropic coal mass exhibit diverse failure characteristics16,17,18. Thus, exploring the anisotropic mechanical behavior of coal affected by the cyclic loading-unloading rate is meaningful for coal pillar designing, roadway support assessment, and coal mass failure mechanism analysis.

Primary structures are broadly distributed in coal mass, which are generated during the sedimentary and geotectonic movement process of the coal formation, including mineral inclusions, pre-existing discontinuities, and weakness interfaces9,19. These primary structures contribute to the mechanical anisotropy of coal via affecting the generation, propagation, and coalescence of micro and macro cracks10,20,21. Meanwhile, these primary structures behave spatial directional distribution characteristic, and the bedding plane is generally chosen as the reference structure in the mechanical anisotropy investigation22,23,24, considering the visually recognizable feature of it. In the past decades, explorations and descriptions of the mechanical anisotropy of coal were broadly carried out based on the inclination variation of the bedding plane to the loading direction10,25,26, viz., the anisotropy angle.

The influence of cyclically loading-unloading process on the mechanical parameters of coal and rock material is complicated27,28,29,30. Mechanical parameters and failure characteristics of coal and rock exhibit different variation features as they are loaded under different loading rates31,32, loading path33,34, fatigue stress level35,36, and the amount of loading-unloading cycles11,37. In recent years, the mechanical anisotropy of coal was also proved strengthened when it was subjected to the cyclical loading-unloading process38. However, the mechanical response of anisotropic coal affected by the cyclical loading-unloading rate remains unknown, corresponding investigations or experiments are rarely mentioned. Thus, extensive works are needed to implement in this area, and clarify the evolution characteristics of mechanical parameters and failure modes of coal as being cyclically loaded-unloaded under different rates.

In this study, a detailed exploration on the mechanical anisotropy of coal affected by the cyclic loading-unloading rate was systematically carried out under the uniaxial compressive condition. Three stress cyclical loading-unloading rates (0.025 MPa/s, 0.050 MPa/s, 0.100 MPa/s) and five anisotropy angles (from 0º to 90º, with an interval of 22.5º) were chosen. The microscopic or macroscopic failure characteristics of coal specimens are respectively monitored by the acoustic emission (AE) and digital image correlation (DIC) monitoring instruments. After that, the mechanical anisotropy of coal affected by the cyclic loading-unloading rate was thoroughly analyzed.

Experimental scheme

Coal specimen Preparation

The block coal was excavated from the 4# coal seam of Manlailiang Coal Mine, located in Inner Mongolia, China. To facilitate the observation of strain and fracture propagation on the coal surface, coal specimens were processed with the quadrilateral prism form. It was prepared with a height of 100 mm and a side-to-height length ratio of 1:2. Meanwhile, five anisotropy angles, namely angles between the loading-unloading direction and the inclination of the bedding plane, were chosen from 0º to 90º, with an angle interval of 22.5º, as shown in Fig. 1(a).

Fig. 1
Fig. 1The alternative text for this image may have been generated using AI.
Full size image

The anisotropy angle and part of the specimens used in this study.

Sixty specimens, respectively, with five anisotropy angles mentioned above, were acquired, twelve specimens for each anisotropy angle. For specimens with the same anisotropy angle, they were randomly divided into 4 groups, namely, three specimens for each anisotropy angle per group. Part of the specimens used in this study are shown in Fig. 1(b).

Experimental instruments

Uniaxial testament facility

The servo-controlled stiff uniaxial cyclic loading-unloading equipment was adopted in this experiment, which was produced by the Mechanical Testing & Simulation (MTS) Corporation. It has both the cyclic and the monotonic loading-unloading capabilities. Meanwhile, it also provides two loading-unloading models, respectively, the strain and the stress control model. The maximum loading capability of this instrument is 300 kN, with an accuracy of ± 0.3%. The experimental equipment and its general layout during the experiment are exhibited in Fig. 2.

Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
Full size image

Layout of the experimental equipment.

Acoustic emission monitoring equipment

A broadly used acoustic emission monitoring device was adopted to detect the AE activity generated during the cyclic loading-unloading process, which was produced by Physical Acoustics Corporation (PAC), as shown in Fig. 2. This system includes a PIC-2 device, a 6-type preamplifier, and four micro 30 s sensors. It has a bandwidth frequency of 1 kHz ~ 3 MHz, and the maximum signal amplitude is 100 dB. Here, the amplification gain was chosen as 40 dB during the monitoring process.

Digital speckle monitoring system

A non-contact displacement/strain video measuring instrument was adopted in this experiment (IMETRUM, model ISM-CONTAR-VG5-2D-10G), as shown in Fig. 2. It consists of a system controller, a dedicated digital camera − 10GigE, with an acquisition of 68 fps and a resolution ratio of 4096 × 3000, six-core cables, a calibration tool, a digital image correlation (DIC) full-field measurement module, and an LED light source.

Test methodology

To acquire a reasonable cyclic loading-unloading stress increment interval, specimens with different anisotropy angles were firstly tested under the monotonic uniaxial compressive condition, with a loading rate of 0.025 MPa/s. The maximum and the minimum UCS were obtained with value of 9.24 MPa (anisotropy angle 90º) and 1.26 MPa (anisotropy angle 22.5º), with a ratio of 7.33. Thus, a reasonable stress increment interval is needed to ensure most of the specimens experience relatively close cyclic loading-unloading cycles.

Due to the mean of the minimum and maximum UCS value is 5.20 MPa. As the stress increment is set as 0.4 MPa, he average cyclic loading-unloading times specimens experimented with is 13. Which means most of the specimens can experience a relatively enough loading-unloading process before failure. Meanwhile, half of the stress increment (0.2 MPa) was chosen as the residual stress after per unloading process. As a result, the initial unloading stress is set as 0.6 MPa (0.4 MPa for the stress increment, 0.2 MPa for the residual stress). Three loading/unloading rates were chosen, separately 0.025 MPa/s, 0.050 MPa/s, and 0.100 MPa/s. The specimens were cyclically loaded and unloaded until failure, the loading-unloading path is shown in Fig. 3.

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
Full size image

The cyclic loading-unloading paths used in this research.

Strength and deformation anisotropy

Stress-strain curves

The stress-strain curves of specimens with different anisotropy angles under three cyclic loading-unloading rates were summarized, as shown in Fig. 4. It indicates that the stress and strain vary with the anisotropy angles and cyclic loading-unloading rates. Specimens usually have greater UCS and experience more axial deformation before the specimen failure at anisotropy angles of 90º and 67.5º than at 22.5º and 45º. Instead, specimens were observed with a greater UCS value, while less axial strain at the anisotropy angle 0º.

Meanwhile, specimens experience more cyclic loading-unloading cycles in anisotropy angles of 0º, 67.5º, and 90º than those of 22.5º and 45º. The increasing cyclic loading-unloading rate has a more complicated influence on the stress, the deformation, and the failure characteristic of coal. Thus, calculation and statistical analysis should be made to reveal the mechanical variation affected by the cyclic loading-unloading rate.

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
Full size image

The stress-strain curves of specimens with different anisotropy angles under various loading-unloading rates: (a) 0º, (b) 22.5º, (c) 45º, (d) 67.5º, and (e) 90º, (f) schematic diagram of loading-unloading stress-strain curves.

Uniaxial compressive strength

The mean values of UCS in specimens with different anisotropy angles were summarized in Table 1. It behaves as a U-shaped characteristic at the anisotropy angle range of 0º ~ 90º. The UCS usually obtained the maximum value at the anisotropy angle 90º, and with the minimum value at anisotropy angles 22.5º or 45º, as shown in Fig. 5. This feature accords with the UCS anisotropy characteristics in coal and stratified rocks reported before39,40, which illustrates that the internal structure, especially the bedding plane, still notably impacts the mechanical failure of coal.

Table 1 The average UCS value in specimens with different anisotropy angles under various Cyclic loading-unloading rates.
Full size table

The UCS exhibits an overall reducing trend as the cyclic loading-unloading rate increases from 0.025 MPa/s to 0.100 MPa/s. It reduces from 5.18 MPa to 4.71 MPa, with a reduction of 0.47 MPa, as shown in Table 1. Meanwhile, this variation is more complicated in coal specimens with different anisotropy angles. An overall reducing feature was observed at anisotropy angles 0º, 22.5º, 67.5º, and 90º, respectively, with reductions of 0.02 MPa, 0.60 MPa, 0.62 MPa, and 1.34 MPa. While it increases from 1.62 MPa to 1.86 MPa at the anisotropy angle of 45º, with an increment of 0.24 MPa. The anisotropy feature of the UCS reduces with the increasing cyclic loading-unloading rate; the standard deviation of specimens with different anisotropy angles reduces from 3.45 MPa to 3.13 MPa, as it increases from 0.025 to 0.100 MPa/s. In addition, the error bar represents the discreteness of the experimental data, as shown in Fig. 5. Which indicates the UCS usually has greater discreteness as the specimen has a greater UCS value, namely, at anisotropy angles of 0º and 90º.

Fig. 5
Fig. 5The alternative text for this image may have been generated using AI.
Full size image

The UCS of specimens with different anisotropy angles under varies loading-unloading rates.

Based on the similarly U-shaped anisotropy feature of the observed in the coal and layered rock39,41, the UCS anisotropy under the cyclic loading-unloading condition, which is usually represented in forms as

$${\sigma _\theta }=M - N\cos 2(\theta - {\theta _{\hbox{min} }})$$
(1)

where M and N are material-related constants; θ is the anisotropy angle, º; θmin is the anisotropy angle with the minimum UCS, º; σθ is the UCS with the anisotropy angle θ, MPa.

For specimens with different anisotropic angles, the standard deviation of the UCS in specimens has a firstly increasing and then decreasing trend, as the loading-unloading rates increase. Which represents the shape of the UCS-anisotropy angle curve also exhibits a similar variation feature. Based on the changing characteristic of these curves, it can be characterized via adding a parabolic control function to Eq. (1). After numbers attempts, a unified correlation was developed and can be represented in forms as

$${\sigma _{\theta ,v}}=(A{v^2}+Bv+C)\left[ {M+N\cos 2(\theta - {\theta _{\hbox{min} }})} \right]$$
(2)

where σθ,v denotes the UCS of specimen with the anisotropy angle of θ and the cyclically loading-unloading rate of v. The A, B, and C are material property and loading-unloading rates related parameters, which are obtained by regression analysis based on the experimental data, to maintain dimensional consistency on both sides of the formula Eq. (2). The dimension of A and B, respectively, s2/MPa, s/MPa. Parameter C is a dimensionless constant. The fitting curve of the experimental data and Eq. (2) exhibits a good correlation, with a correlation coefficient of 0.89, as shown in Fig. 6. Which also indicates the applicability of this correlation in describing the correlation of the UCS, the cyclic loading-unloading rate, and the anisotropic angle, at the cyclic loading-unloading range of 0.025 ~ 0.100 MPa/s.

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
Full size image

The regression analysis result of Eq. (2) of the UCS under different cyclic loading-unloading rates.

Meanwhile, difficulties still exist in precisely describing all the experimental data variation features in a formula, due to the experimental error and relatively greater UCS value range caused by the heterogeneity of coal specimens. In addition, the strength threshold may also exist in the UCS of coal42, which decides the strength increases or decreases with the increasing loading-unloading rate as the UCS is greater or lower than the strength thresholds. Considering the experimental data obtained in this research is relatively limited, the strength threshold is difficult to determine from these data or revealed by Eq. (2). Thus, more investigation can be implemented in future research.

Axial strain

The peak strain of coal specimens under the cyclic loading-unloading conditions was summarized in Table 2. Variation features of these specimens were exhibited in Fig. 7.

Table 2 The axial strain of specimens with different anisotropy angles under various cyclic loading-unloading rates.
Full size table

In general, the axial strain exhibits a firstly reducing and then increasing feature at the anisotropy angle range of 0º ~ 90º, the minimum value is usually observed at the anisotropy angle 22.5º or 45º. This accords with the anisotropy character of UCS, namely, specimens with greater UCS generally have a greater value of peak strain, apart from specimens with the anisotropy angle 0º. Where specimens have lower axial deformation capability, since the loading-unloading direction is perpendicular to the bedding plane, and the axial deformation mainly depends on the axial deformation of the layered coal matrix9. The discreteness axial strain in specimens with different anisotropy angles is significantly greater than that of the UCS, as shown by the error bar in Fig. 7. Specimens with greater UCS values still have greater discreteness, at anisotropy angles 0º and 90º. Meanwhile, notable variability of axial strain was also observed, at anisotropy angle 45º, under a lower loading-unloading rate (0.025 MPa/s).

Fig. 7
Fig. 7The alternative text for this image may have been generated using AI.
Full size image

The axial strain of specimens with different anisotropy angles under various loading-unloading rates.

The anisotropy feature of the peak strain changes minimally as specimens have been cyclically loaded-unloaded, besides that at the loading-unloading rate of 0.050 MPa/s, where the axial strain increases with the anisotropy angle at 0º ~ 22.5º, while it reduces at the anisotropy angle range of 22.5º ~ 45º, and then it increases to the maximum at 90º.

Unlike the axial strain reduction with the increasing loading-unloading rates in sandstone43, the increasing cyclic loading-unloading rate also gains the axial deformation capability, the average value of peak strain increases from 3.01% to 3.22%, as the loading rate increases from 0.025 MPa/s to 0.100 MPa/s. Meanwhile, the axial strain variation in different anisotropy angles also behaves inconsistently, as the cyclic loading-unloading rate increases from 0.025 MPa/s to 0.100 MPa/s. It increases with the loading-unloading rate at the anisotropy angles 0º and 90º, while having an overall increasing behavior at 22.5º and an overall reduced trend at 67.5º or 45º. The anisotropy feature of axial strain decreases with the increasing cyclic loading-unloading rate, and the standard deviation reduces from 1.43% to 1.34%, as it increases from 0.025 to 0.100 MPa/s.

Elastic modulus

To investigate the effect of the cyclic loading-unloading rate on the stress and deformation anisotropy of coal, the tangent elasticity modulus was calculated. The elastic modulus exhibits a U-shaped anisotropy feature, as the anisotropy angle increases from 0º to 90º, as shown in Fig. 8. However, it behaves as a symmetric anisotropy characteristic with the UCS. The elastic modulus generally has the maximum value at the anisotropy angle 0º, while the minimum value at 45º or 22.5º, and the second largest value at 90º. The discreteness of the elastic modulus is consistent with that of the UCS, the error bar indicates a greater variation range at anisotropy angles 0º and 90º, as shown in Fig. 8.

Fig. 8
Fig. 8The alternative text for this image may have been generated using AI.
Full size image

The elastic modulus variation with the anisotropy angle under various cyclical loading-unloading rates.

The value of elastic modulus has an overall reducing feature with the increasing cyclic loading-unloading rate, the average elastic modulus increases from 0.39 GPa to 0.41 GPa, and then reduces from 0.41 GPa to 0.37 GPa, as the loading-unloading rate increases from 0.025 MPa/s to 0.100 MPa/s, as shown in Table 3.

Table 3 The elastic modulus of specimens with different anisotropy angles under various Cyclic loading-unloading rates.
Full size table

Meanwhile, the elastic modulus has a similar anisotropy feature to the UCS as the cyclic loading-unloading rate gain. It exhibits firstly decreasing and then increasing features at anisotropy angles of 0º and 67.5º, while a firstly decreasing and then reducing feature at 90º. Meanwhile, it respectively illustrates an overall increasing and decreasing character at 22.5º and 45º, as the cyclic loading-unloading rate increases from 0.025 MPa/s to 0.100 MPa/s.

The anisotropy of elastic modulus also increases as the cyclic loading-unloading rate gain, the standard deviation increases from 0.22 GPa to 0.24 GPa, as the cyclic loading-unloading rate increases from 0.025 MPa/s to 0.100 MPa/s. Based on the previous research, the strength thresholds may exist in coal. The strength increases or decreases with the increasing loading-unloading rate as the UCS is greater or lower than the strength thresholds42. These may be the reasons that the UCS, the peak strain, and the elastic modulus behave differently as the cyclic loading-unloading rate increases in specimens with different anisotropy angles, as shown in Tables 1 and 2, and 3.

Anisotropic failure characteristics

Failure anisotropy

Typical failure characteristics of anisotropic coal specimens under different loading-unloading rates are exhibited in Fig. 9. The failure pattern of the coal specimen changes with the anisotropy angle. The effect of the bedding plane is pronounced at anisotropy angles of 22.5º and 45º, where the major failure crack developed along the bedding plane of the coal specimen, as shown in Fig. 9. Meanwhile, it also leads to a relatively lower bearing and deformation capacity of specimens at these two anisotropy angles.

Fig. 9
Fig. 9The alternative text for this image may have been generated using AI.
Full size image

Typical failure characteristics of coal specimens with different anisotropy angles and loading-unloading rates: (a) monotonic loading; (b) 0.025 MPa/s; (c) 0.050 MPa/s; (d) 0.100 MPa/s.

The influence of the bedding plane gradually reduces at the anisotropy angle of 67.5º. The major failure cracks cross and have a greater angle with the inclination of the bedding plane, though coalescence of it with the crack developed along the bedding plane also partially exists, as shown in Fig. 9. Which makes the mechanical parameters mentioned above has relatively greater values.

The layered rupture feature is observed, and failure cracks developed along the bedding plane at the anisotropy angle of 0º, due to the lower cohesion perpendicular to the bedding plane. This may be the reason that specimens have a relatively lower axial peak strain, greater UCS, and the highest elastic modulus, since the bearing and axial deformation capacity of coal are determined by the layered coal matrix.

The influence of the bedding plane is less obvious at the anisotropy angle of 90º. The major failure cracks are developed across the bedding plane layers, the deformation and the bearing capacity of coal are mainly subject to the coal matrix, as shown in Fig. 9. Thus, the greater UCS and axial strain, while a lower elastic modulus, are observed.

In addition, compared with other types of sedimentary and stratified rocks44, including shale, schist, and gneiss. Specimens exhibit similar anisotropy failure characteristics, while coal specimens behave more fragmented than other sedimentary rocks. The effect of bedding plane is also less obvious, since more cracks that penetrate through the bedding planes are observed, though the macroscopic damage cracks developed along the bedding plane at anisotropy angles 22.5º and 45º. More damaged small coal blocks were observed at coal specimens with different anisotropy angles, due to the coal matrix is weaker than the rock matrix.

Failure anisotropy affected by loading-unloading rates

The influence of cyclic loading-unloading rate on the anisotropic failure characteristics of coal is significant at anisotropy angles of 0º and 90º. As the cyclic loading-unloading rate increases from 0.025 MPa/s to 0.10 MPa/s, the failure pattern changes from a layered failure pattern to an X-shaped, and the amount of cracks developed along the bedding plane decreases at the anisotropy angle 0º, as shown in Fig. 10. Meanwhile, the after-failure specimen becomes more intact, and the fragments are not cast far away from the damaged specimen at the anisotropy angle of 90º, as shown in Fig. 9 (d).

Penetrating cracks become mainly propagated along the bedding plane, and the fragmentation behaves more integrated with greater size, at the anisotropy angles of 22.5º and 45º, though the fragment degree at 45º is greater than that of 22.5º, as shown in Figs. 9 (b), (c), and (d). Wider crack bands, more dusty characteristics, and fewer cracks generated parallel to the loading-unloading direction around the crack band were observed at the anisotropy angle 67.5º, as shown in Figs. 9 (b), (c), and (d).

The DIC photographs, approximate 1 s before the coal specimen failure, were chosen and applied for the strain field analysis. The processed image of specimens with different anisotropy angles under various loading-unloading rates is shown in Fig. 10.

Fig. 10
Fig. 10The alternative text for this image may have been generated using AI.
Full size image

The horizontal strain variation in specimens with different anisotropy angles: (a) monotonic loading; (b) 0.025 MPa/s; (c) 0.050 MPa/s; (d) 0.100 MPa/s.

Before the failure of coal specimens, the horizontal strain concentration zones are more intensively distributed in the main through-crack area, as shown in Fig. 10. The area of horizontal strain concentration reduces with the increasing loading-unloading rate, especially at that of 0.050 MPa/s. This also may illustrate the reason that a relatively less fractured development after specimen failure, which contribute to the increasing integrity of the damaged coal specimen under greater loading-unloading rates condition.

Anisotropy revealed by AE activities

AE count

Acoustic emission is a fracture generation and propagation-related activity, which also microscopically reflects the failure intensity and characteristics during the cyclic loading-unloading process. Thus, the cumulative AE counts in specimens with different anisotropy angles under various cyclic loading-unloading conditions were summarized in Table 4; Fig. 11.

Table 4 The variation of cumulative AE counts with the anisotropy angle of coal specimens under various Cyclic loading-unloading rates.
Full size table

The cumulative AE counts present a less obvious U-shaped anisotropy feature under the cyclic loading-unloading condition. It generally has minimum values at the anisotropy 22.5º, while the maximum values at 90º. Meanwhile, specimens also have greater cumulative AE counts discreteness in specimens with greater UCS value, namely at anisotropy angles 0º and 90º, where the error bar is more significantly observed, as shown in Fig. 11. The cumulative AE counts have similar anisotropic characteristics to those of axial strain, as shown in Figs. 7 and 11. The positive correlation is due to the axial strain represents the irreversible axial deformation generated during the specimen failure process. A greater value of axial strain indicates that more axial deformation occurred, which may contribute to more cracks, namely, a greater amount of AE events, generated during the cyclic loading-unloading process. In addition, greater axial deformation usually indicates a stronger carrying capacity of coal, which usually represents a greater value of the UCS and the elastic modulus. The crack propagation generally dissipated more energy under this condition, thereby resulting in more high-energy acoustic emission events being generated and monitored.

The increasing cyclic loading-unloading rate strengthens the AE activity, the average cumulative AE counts of specimens with different anisotropy angles gains from 4.03 × 105 to 4.83 × 105, as the cyclic loading-unloading rate increases from 0.025 MPa/s to 0.100 MPa/s. However, the variation characteristics of the cumulative AE counts with the loading-unloading rates also change with anisotropy angles. It exhibits an overall increasing feature at the anisotropy angles 0º, 22.5º, 45º, and 90º, while exhibiting an overall reducing feature at the anisotropy angle 67.5º.

Fig. 11
Fig. 11The alternative text for this image may have been generated using AI.
Full size image

The cumulative AE counts variation with the anisotropy angle and the cyclic loading-unloading rates.

AE energy

The cumulative absolute AE energy was also calculated and summarized in specimens with different anisotropy angles, as shown in Table 5; Fig. 12. It behaves similarly anisotropic changing characteristics with the cumulative AE counts, meanwhile, the data discrepancy revealed by the error bar also exhibits consistency in specimens with different anisotropy angles, as shown in Figs. 11 and 12. Specimens with a greater value of the UCS and the elastic modulus generally monitored more AE energy, particularly in specimens with the anisotropy angle of 0º. Where the coal specimen exhibits layered failure characteristics, leading to the coal specimen being damaged with lower axial deformation, greater UCS value, and thus greater elastic modulus.

Table 5 The cumulative absolute AE energy of specimens under various cyclic loading-unloading rates.
Full size table

The cumulative absolute AE energy exhibits a U-shaped variation feature at the anisotropy angle 0º~ 90º, under the cyclic loading-unloading condition. The value of the cumulative absolute AE energy is minimum at the anisotropy angle of 22.5º, while maximum at 90º. This indicates that a greater amount of cracks were generated and AE energy dissipated in specimens with an anisotropy angle of 90º, where the impact of the bedding plane is less obvious. While the crack generation and energy dissipation are less obvious, as specimen failure has been significantly affected by the bedding plane, namely, anisotropy angles 22.5º and 45º.

Fig. 12
Fig. 12The alternative text for this image may have been generated using AI.
Full size image

The cumulative absolute AE energy variation with the anisotropy angle and loading approaches.

The cumulative absolute AE energy exhibits an overall increasing characteristic as the cyclic loading-unloading rate increases from 0.025 MPa/s to 0.100 MPa/s. It has an overall increasing feature at the anisotropy angles 0º, 22.5º, 45º, and 90º, respectively, with increments of 3.65 × 105 aJ, 0.48 × 105 aJ, 0.88 × 105 aJ, and 1.12 × 105 aJ, while an overall reduction at the anisotropy angle 67.5º, with a reduction of 0.58 × 105 aJ. The increasing cumulative AE counts and absolute AE energy may be correlated to the greater amount of cracks with longer length generated as the cyclic loading-unloading rate increases. Which accords with the increasing fragmented block size in specimens with greater cyclic loading-unloading rate.

Meanwhile, the anisotropy of failure characteristic revealed by the AE activity increases with the cyclic loading-unloading rate, the standard deviation of cumulative AE counts and absolute AE energy separately gains from 3.18 × 105 to 3.29 × 105, from 3.18 × 105 to 3.48 × 105, as the cyclic loading-unloading rate increases from 0.025 MPa/s to 0.100 MPa/s.

Micro-failure affected by Cyclic Loading-unloading rates

Damage variable

In general, micro-cracks are usually generated in coal and rock material before the emergence of macro-fractures, and the initiation and growth of micro-cracks are defined as damage27. The tiny irreversible damage that accumulates in each cyclic loading-unloading cycle affects the final failure feature of coal. Thus, the impact of the cyclic loading-unloading rate on the failure process of coal is investigated in this part. In general, the physical quantity used to characterize irreversible damage described above is defined as a damage variable. In this research, the residual strain at the last cyclic loading-unloading cycle and peak strain of coal were chosen to evaluate the damage degree as impacted by the cyclical loading-unloading rate. The calculation equation was developed in forms as Eq. (3)

$$D=\frac{{{\varepsilon _L}}}{{{\varepsilon _P}}}$$
(3)

Where εL and εp are respectively the residual strain in the last cyclic loading-unloading cycle and the peak strain, %, D represents the damage variable of coal. The damage variable in specimens with different anisotropy angles and cyclic loading-unloading rates was exhibited in Fig. 13.

Fig. 13
Fig. 13The alternative text for this image may have been generated using AI.
Full size image

The damage variable of the coal specimen under different cyclic loading-unloading rates.

The damage variable behaves as a parabolic type variation feature versus the anisotropy angle at the loading-loading rate of 0.025 MPa/s and 0.050 MPa/s. It has the minimum value at the anisotropy angle 0º, the maximum value at anisotropy angle 45º (0.025 MPa/s) or 22.5º (0.100 MPa/s), as shown in Fig. 13. This anisotropy feature indicates specimens have a greater degree of damage at the anisotropy angles 45º and 22.5º during the cyclic loading-unloading process. This accords with the failure feature that the major failure crack mainly developed along the bedding plane, where cracks propagate more easily and contribute to a lower value of UCS, peak strain, and elastic modulus. This characteristic may correlate with the response inconsistencies of specimens with different UCS values under the effect of the same loading-unloading rates42. The strengthening effect and weakening effect of loading rate on coal may occur in specimens with different UCS values.

The anisotropy feature of the damage variable also changes at the cyclic loading-unloading rate of 0.100 MPa/s. It changes from a parabolic type to a wave-type variation. The damage variable firstly increases to maximum (0º ~22.5º), then reduces to the minimum (22.5º ~45º), finally gains to the second largest (45º ~90º), as shown in Fig. 13. This indicates more damage generated by the cyclic loading-unloading process in specimens with greater UCS under greater rates, conversely, less damage occurred in specimens with lower UCS values.

In addition, the damage variable has an overall reducing feature with the increasing loading-unloading rate. It respectively with values of 0.537, 0.507, and 0.528, under the loading-unloading rate of 0.025 MPa/s, 0.050 MPa/s, and 0.100 MPa/s.

Dissipated energy density

In this study, the cumulative dissipated energy density (CDED) before the last loading process was adopted to illustrate the damage feature of the coal specimen during the cyclic loading-unloading process, as shown in Fig. 14. Specimens with greater UCS value also observed with greater discreteness in specimens with different anisotropy angles, namely, anisotropy angle of 0º to 90º. The CDED is an unrecoverable deformation and strength increment-related energy index. It behaves a U-shaped variation character as the anisotropy angle increases from 0º to 90º. Which indicates plastic deformation and strength increment is maximum at the anisotropy angle of 90º, while it is less obvious at that of 22.5º and 45º.

Fig. 14
Fig. 14The alternative text for this image may have been generated using AI.
Full size image

Cumulative dissipated energy density in specimens with different anisotropy angles and loading-unloading rates.

In combination with the parabolic type variation feature of damage variables, a greater percentage of plastic strain generated during the cyclic loading-unloading process at anisotropy angles of 22.5ºand 45º. This may be the reason that the relatively integrated failure pattern of coal specimens observed at anisotropy angles of 22.5ºand 45º, since cracks easily developed along the bedding plane, the specimen was destructed with fewer cracks generated, as shown in Fig. 9. This is consistent with the lower AE activity parameter observed at these two anisotropy angles.

The CDED has an overall reducing feature with the increasing loading-unloading rate, it respectively has the average value of 4.88 × 104 J/m3, 5.50 × 104 J/m3, and 4.70 × 104 J/m3, at the loading-unloading rate of 0.025 MPa/s, 0.050 MPa/s, and 0.100 MPa/s. A greater value of CDED represents more deformation, greater stress increments, and more loading-unloading cycles exist in these specimens. Meanwhile, a greater value of the UCS was observed in specimens with the greater CDED energy density, namely, the UCS and the CDED are positively correlated, as shown in Figs. 5 and 14. Thus, a power function correlated relationship was proposed after multiple attempts. This correlation can be described by Eq. (4)

$${U_d}=\sum\limits_{{i=1}}^{n} {{U_{di}}} =\sum\limits_{{i=1}}^{n} {\left( {\int\limits_{{{\varepsilon _i}}}^{{{\varepsilon _j}}} {{\sigma _l}d\varepsilon - \int\limits_{{{\varepsilon _k}}}^{{{\varepsilon _j}}} {{\sigma _{ul}}d\varepsilon } } } \right)} =m{\sigma ^n}$$
(4)

Where Ud represents the cumulative dissipated energy density, J/m3, m is the material-related parameter, J/(m3•MPa), n is the loading rate correlated parameter, and σ is the UCS of specimens with different anisotropy angles, MPa. The regressive analysis result was summarized in Table 6.

Obviously, the experimental data have a good correlation with Eq. (4), as shown in Fig. 15, and respectively with correlation coefficients of 0.85, 0.97, and 0.79 at the loading-unloading rate of 0.025 MPa/s, 0.050 MPa/s and 0.100 MPa/s. Parameters of Eq. (4) are exhibited in Table 6. Based on Eq. (4) and Fig. 15, this verifies that the CDED positively correlated with the UCS in per loading-unloading rates.

Fig. 15
Fig. 15The alternative text for this image may have been generated using AI.
Full size image

Correlations between UCS and the CDED in specimens with different loading-unloading rates.

Table 6 Parameters of Eq. (4) obtained by regression analysis.
Full size table

Conclusion

In this study, the impact of the cyclic loading-unloading rate on the anisotropic mechanical response of coal was experimentally investigated. Research findings and conclusions were summarized as below.

  1. a.

    The UCS and elastic modulus maintain a U-shaped anisotropic characteristic under different cyclic loading-unloading rates. The anisotropy of UCS and axial strain decreases, while the elastic modulus ascends with the increasing loading-unloading rate. The correlation of the UCS, the loading-unloading rate, and the anisotropic angle of coal specimens can be described by a cosine function.

  2. b.

    The failure characteristics of coal are bedding plane related at anisotropy angles of 0°, 22.5°, and 45°, where the penetrating crack mainly propagates along the bedding plane. The after-failure coal specimen is more intact, and the anisotropic failure feature of the coal specimen becomes more notable, as the loading-unloading rates increase.

  3. c.

    The anisotropy of micro-failure characteristics revealed by the AE activity rises with the cyclic loading-unloading rate. The cyclic loading-unloading generates more damage as specimens with lower UCS, the damage variable caused by it exhibits a parabolic anisotropy feature at 0º~90º, and the greater value was generally observed at 22.5º or 45º. Meanwhile, this anisotropy feature changes to wave-type at the greater loading-unloading rate.

  4. d.

    The dissipated energy density generated during the cyclic loading-unloading process exhibits a U-shaped variation feature at anisotropy angle 0º~90º, with relatively lower values at 22.5º and 45º. Which shows a positive exponential correlation with UCS, namely, specimens with greater uniaxial compressive strength have more energy dissipated during the cyclic loading-unloading process.