Introduction

Coal is a kind of organic rock formed from compaction and induration of plant remains. An extensive microscopical study (Cao and Xian 2001) indicated that coal is inhomogeneous and is composed of particles of many different shapes and sizes. Coal also has many weak planes and structures such as micro-holes, micro-cracks, bedding, and joints. Undoubtedly, coal is a heterogeneous substance containing many inherent defects. The existence of these defects not only reduces the coal’s strength but also leads to additional damage under even small cyclic stresses. If these stresses are imposed, then fatigue damage will occur (Xi et al. 2004; Du et al. 2016).

During underground coal mining, some coal pillars, such as sublevel pillars, fault protection pillars, strip coal pillars when strip-pillar mining is used, and coal pillars in room-and-pillar mines, are left in the mine. These pillars all have the function of supporting the roof or roofs and separating strata no matter what shape the pillars are or how they are distributed. The stability of these pillars plays a decisive role in mine safety as well as ensuring the integrity of the buildings on the ground above the mine and protecting the surface environment from ground subsidence. The pillars will in some cases be subjected to repeated stresses cause by multiple cycles of mining activity, especially when strip-pillar mining or room-and-pillar mining methods are used. The effect of repeated mining stresses on the coal pillars will be much more significant in mines exploiting multiple closely spaced coal seams (Chen et al. 2016a; Chen et al. 2016b). It goes without saying that coal pillars will be repeatedly damaged during multi-seam mining operations. In these operations, the deformation of the coal pillars is increased and the supporting capacity is reduced. For this reason, studying damage to coal subjected to cyclic loading is important for designing the size of coal pillars.

Many experiments have been carried out on the strength and deformation of relatively hard, dense, and homogeneous rocks under cyclic loading, lithologies like marble, granite, limestone, sandstone, and so on. Through numerous experimental studies, Ge and Lu (1992), Ge et al. (2003) proved that a rock fatigue “threshold value” exists under cyclic loading and that fatigue damage to the rock can be determined from the complete static stress–strain curve. The strain at the time of fatigue failure is the post-peak strain of the complete static stress–strain curve. This corresponds to the maximum cycling stress. In addition, according to deformation creep theory, the axial strain can be divided into three stages, namely the primary creep stage, the constant or steady-state stage, and the tertiary, exponentially increasing strain rate stage. Xu et al. (2000, 2001) studied the influence of the number of cyclic uniaxial loads on rock deformation and the rocks’ Young’s modulus. Xiao et al. (2010a, b) found through constant amplitude cyclic loading tests on granite that it is more reasonable to consider the lateral deformation or lateral deformation rate as the criterion for imminent rock fatigue failure. Wang et al. (2012) proposed an elastic and brittle constitutive model with damage, a model that included a linear shear yield function with tension-off and a nonlinear shear yield function, when taking the decrease of elastic modulus into account for the stress drop in brittle rocks. Experimental work by Feng et al. (2009) showed that the maximum cyclic stress and the amplitude of the cyclic loading were the main factors affecting the fatigue life of the samples studied. Lu and Li (2016) and Lu et al. (2015) found that the fatigue failure of a sandstone sample could be determined from the axial strain, lateral strain, and volumetric strain on the complete stress–strain curve. Compared to strength, deformation is a more suitable fatigue failure criterion for rock. Zou et al. (2016) conducted conventional cyclic loading tests and showed that the process that damages the rock can be divided into three stages, namely the initial damage stage, the micro-crack initiation and stable evolution stage, and the destruction stage.

In summary, because coal has micro-pores, is low strength, and it is difficult to prepare appropriate test samples by coring, the results of cyclic loading test on coal and the research on coal fatigue damage are still quite incomplete. More research on the strength, deformation, energy dissipation, and fatigue damage evolution of coal under cyclic loading needs to be carried out.

Experimental methods and sample descriptions

For this study, cyclic loading fatigue experiments on coal samples were carried out on an MTS815.02 servo-controlled rock mechanical test system (MTS Systems Corporation, Eden Prairie, MN, USA), which is shown in Fig. 1. Medium hard coal from the No. 3 seam in the Xinhe Colliery, Jining, Shandong Province, China, was used for this research.

Fig. 1
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MTS815.02 servo-controlled rock mechanical test system

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Compressive loading is used as a controlling variable in the cyclic loading experiments. A sine wave with a frequency of 0.5 Hz is taken as the waveform for the cyclic loading. The loading waveform is shown in Fig. 2. In Fig. 2, the parameters are as follows: σmax is the maximum value of cyclic loading; σmin is the minimum value of cyclic loading; Δσ = σmax − σmin is the amplitude of the loads; T is the period; \( f=\frac{1}{T} \) is the frequency.

Fig. 2
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Diagram of the waveform for the compressive cyclic loading experiments

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The diagram of the waveform for the cyclic loading illustrates the factors influencing the strength and deformation of the coal. Only one frequency (0.5 Hz) and the sine waveform are used in these experiments. The experiments mainly study how the different maximum stresses and stress scopes affect the strength, deformation, and fatigue damage of the samples.

Considering that the energy consumption of the MTS815.02 test system is quite high and the fatigue experiments are very time-consuming, the cyclic loading for each coal sample is set to 2000 cycles at the upper limit of each stress level. The test is stopped if fatigue breakage of the coal sample does not occur. Then, the stress level is increased to continue the test until fatigue breakage occurs at the specified stress level at fewer than 2000 cycles.

In order to minimize the adverse effects of the bedding planes and cleats of the coal samples on the test results (He et al. 2016; He et al. 2017), the structurally intact, jointless medium hard coal samples have been carefully selected from the No. 3 coal seam in the Xinhe Colliery, Jining, Shandong Province, China. Immediately afterwards, they were packaged, marked, and shipped to the laboratory of Shandong University of Science and Technology for processing. Coal sample processing equipment (coring machine (ZS-100B); cutting machine (DQ-4); grinding machine (AHM-200)) are shown in Fig. 3.

Fig. 3
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Coal sample processing equipment

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In order to reduce the dispersion of test results caused by the individual differences of natural coal samples, the core sampling was carried out on a large, intact, and jointless coal body, and the coring machine speed was kept as low as possible during the core drilling process to reduce artificial disturbance to the coal core. The coring process is shown in Fig. 4. Coal samples were cored and prepared according to the ISRM (International Society for Rock Mechanics) suggested method with approximate dimensions of 100 mm in height and 50 mm in diameter. When the samples were processed and formed, the two end surfaces of them were polished to make the surface roughness less than 0.02 mm. After air seasoning, they would be numbered and sealed with preservative films to prevent weathering. Nearly 100 standard coal samples were taken as shown in Fig. 5.

Fig. 4
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Coal sample coring process

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Fig. 5
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Standard coal samples

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Coal belongs to sedimentary rocks. Due to the complex internal structure of sedimentary rocks, various mineral minerals, and great differences in sedimentary environment, even the mechanical parameters of the same sedimentary rock in the same mining area may differ greatly, especially the coal with lower strength and more developed joints. At the same time, the influence of man-made disturbance caused by coal sample processing is large, which makes the mechanical parameters have a large dispersion. Therefore, before carrying out the cyclic loading test on coal samples, it is necessary to select coal samples with similar physical and mechanical properties so as to reduce the influence of the sample dispersion to the test results as much as possible.

Acoustic velocity is similar to rock mechanics parameters, and it is closely related to rock mechanical properties, structure, and stress state. Therefore, acoustic velocity can be used as an evaluation index to characterize rock mechanical properties. At the same time, in the process of testing the acoustic velocity, coal samples will not be damaged by external forces. The test will take a short time, be easy to operate, and can be tested repeatedly, thus minimizing the test error. Therefore, in order to reduce the influence of the discreteness of the samples on the test results, the acoustic velocity test was performed on all the coal samples that were processed, and the sample with the greater dispersion of the acoustic velocity was removed (Yang and Chen 2005). The four samples with the most similar p wave velocity were used for testing and numbered sequentially as XHC1, XHC2, XHC3, and XHC4. The acoustic velocity testing instrument is shown in Fig. 6, and the four coal samples used in the cyclic loading test are shown in Fig. 7. The sample sizes, P wave velocities, and predicted strengths are listed in Table 1.

Fig. 6
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Acoustic velocity testing instrument

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Fig. 7
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The four coal samples used in the cyclic loading test

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Table 1 Size, P wave velocities, and predicted strengths for coal samples used in the cyclic uniaxial compression tests
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Coal strength under cyclic loading

Table 2 lists the uniaxial cyclic loading test results for the coal samples and Fig. 8 presents the cyclic loading axial stress–strain curves for two of the samples. The strength of the coal samples under uniaxial cyclic loading is described below.

  1. 1)

    At the beginning of the cyclic loading, when the maximum stress is low, the coal sample will not suffer fatigue failure even after 2000 loading cycles (for example, coal sample XHC1). This shows that there is a cyclic stress threshold value for fatigue failure in coal. Fatigue tests on marble and sandstone show that the threshold value for fatigue failure strength in marble is about 85% of its uniaxial strength; the threshold value for the fatigue failure strength in sandstone is about 83% of its uniaxial strength (Lin and Wu 1987). Compared with hard and dense rocks such as marble or granite, coal has more inherent defects and is less homogeneous, less dense; and the threshold value for its fatigue failure strength is lower. The test results show that the threshold value for the fatigue failure strength for coal from the No. 3 seam in the Xinhe Colliery is 78% of its uniaxial compressive strength under cyclic loading.

  2. 2)

    When a cyclic loading and unloading experiment is carried out below the fatigue failure threshold value, there will be a certain amount of fatigue damage but the samples will not be broken. As for the coal samples from the Xinhe Colliery, for one sample, the elastic modulus decreased by 3.7% after 2000 loading/unloading cycles with the maximum value of cyclic loading at 67% of its uniaxial strength. For the other sample, the elastic modulus decreased by 6.4% after 2000 cycles with the maximum value of its cyclic loading is 62% of its uniaxial strength (Table 2).

Table 2 Test conditions and result for coal samples under uniaxial cyclic loading
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Fig. 8
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Axial stress–strain curves for two coal samples under cyclic loading

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These results suggest that when designing coal pillars, if stresses from multiple rounds of mining are to be repeated, which can occur during strip-pillar or room-and-pillar mining, care should be taken to increase the coal-pillar-strength safety factor by some percentage.

Coal deformation under cyclic loading

Axial deformation

Changes in plastic hysteresis

Coal contains a large number of micro-pores and cracks; its elastic behavior is significantly nonlinear and it presents some hysteresis. The area within the hysteresis loop represents the energy dissipation of the stress cycle, and there is a plastic deformation lag loop in each loading or unloading cycle during fatigue (Xi et al. 2003). As shown in Fig. 9a–d, the area of the hysteresis loop is very large at the start of the unloading. This is because the pre-existing cracks consume the energy of the coal as they are compacted and closed by confining pressure, extension, and the friction between the cracks. As the number of cycles increases, the hysteresis area is gradually reduced and tends to be stable. The explanation for this is that the greater number of cycles causes the original cracks to extend and the generation of new cracks to decrease. Thus, the system gradually becomes stable. As can be seen from Fig. 10a–d, for the coal samples with fatigue damage, after a certain number of cycles, when the coal samples are close to fatigue failure, the hysteresis area increases until the fatigue damage appears, meaning that the new cracks gradually increase in number and length and eventually connect with each other. This means that the energy consumed to overcome crack initiation, extension, and penetration will increase sharply. Each loading cycle will produce a finite amount of plastic deformation and thus plastic hysteresis will gradually transfer to the direction of strain increase, presenting three stages of “sparseness-denseness-sparseness.”

Fig. 9
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Four stress–strain curves showing lag and the hysteresis loops for the initial several loading–unloading cycles at the beginning of a cyclic loading test

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Fig. 10
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Four stress–strain curves showing lag and the hysteresis loops for the last several loading–unloading cycles before sample fatigue failure during a cyclic loading test

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Changes in axial deformation

The axial strain of every lag loop at the upper limit of stress can be considered as the axial strain of every cycle. Figures 11 and 12 show the changes in axial stress versus the number of test cycles for coal samples XHC2 and XHC1, respectively. Sample XHC2 underwent fatigue failure but sample XHC1 did not.

Fig. 11
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Graph of axial strain versus number of cycles for coal sample XHC2 under cyclic loading

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Fig. 12
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Graph of axial strain versus number of cycles for coal sample XHC1 under cyclic loading

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Figure 11 shows that, for coal samples experiencing fatigue failure, fatigue can be divided into three stages. At the beginning, deformation develops rapidly but after a few cycles, the increase in deformation slows and deformation tends to be steady. Deformation increases only slowly as the number of cycles increases. However, in the final stage, deformation increases sharply and the sample suddenly fails after only a few additional cycles. From Fig. 11, it is evident that strain develops quickly in the first and the third stages. More than 2/3 of all the deformation in sample XHC2 accumulates in those two short stages and most of that deformation is in the third stage. In the second stage, axial strain develops quite slowly, but the second stage occupies the vast majority of the sample’s fatigue life.

For sample XHC1, the coal samples with no fatigue failure, the axial deformation occurs in only two stages. At the beginning, the deformation develops quickly but, again, after a few cycles (16 cycles for XHC1), the deformation does not significantly increase and the increase in deformation is nearly zero.

Transversal deformation

Figure 13 shows the curves for axial stress versus the value for transversal strain at the upper limit of stress for each cycle for coal sample XHC2. The transversal deformation of the coal samples with fatigue failure can be divided into two stages. In the first stage, the sample is only damaged a little by transversal deformation and this deformation occurs over many cycles (that is, it takes a long time). As the number of cycles increases, the transversal strain accumulates only slowly. In the second stage, immediately before fatigue failure, the transversal strain increases sharply and reaches its maximum as the coal sample is failing.

Fig. 13
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Graphs showing transversal strain in coal sample XHC2

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As the number of cycles increases, the transversal strain at maximum cycle stress is obviously smaller than the axial strain and the rate at which it increases is also slower. Then the transversal strain increases rapidly just before fatigue failure. In the loading/unloading cycles, the transversal strain recovers less during unloading, showing that the coal sample’s volume increases at this time. The transversal deformation increasing sharply and the deformation relaxing only a little when the load is removed is an indication that the coal will fail from fatigue damage soon.

Irreversible deformation

Under cyclic loading, the elastic portion of deformation in coal or rock will recover during unloading, but the plastic deformation is irreversible and any plastic deformation will be preserved. The size, growth trend, and the total accumulation of irreversible deformation can reflect a rock’s mechanical properties which are directly related to fatigue. Therefore, it is instructive to analyze fatigue failure in terms of the irreversible deformation.

The strain of unloading at the lower stress limit on each hysteresis loop can be considered to approximate the irreversible deformation. This is not completely accurate for plastic strain because there is still a little elastic deformation included in this value. Figures 14 and 15 illustrate the relationships between irreversible strain and number of cycles for coal samples XHC2 and XHC1, respectively. Sample XHC2 experienced fatigue failure; sample XHC1 did not.

Fig. 14
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Graph showing irreversible strain versus number of cycles for sample XHC2 (with fatigue failure)

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Fig. 15
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Graph showing irreversible strain versus number of cycles for sample XHC1 (no fatigue failure)

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As can be seen in Fig. 14, for the coal samples with fatigue failure, the development of irreversible deformation during fatigue can be divided into three stages. (1) At the beginning, the irreversible deformation increases relatively slowly. (2) The irreversible deformation gradually becomes stable after a few cycles (17 cycles for coal sample XHC2) and the deformation rate then remains roughly constant. (3) The irreversible deformation then begins to accelerate rapidly until the sample fails.

In the coal samples with no fatigue failure, XHC1, Fig. 15 shows that after a short initial stage of rapid deformation, the irreversible deformation stabilizes. The deformation does not increase significantly as the number of loading cycles increases meaning that the damage does not increase; under cyclic loading, there is no further accumulation of damage after the initial damage.

Energy dissipation from coal under cyclic loading

Coal failure is essentially a process of energy dissipation. Coal’s deformation and failure modes are closely related to energy conversion; therefore, the energy evolution can reflect the coal’s deformation, damage, and failure. Under uniaxial cyclic loading, some of the energy absorbed by the material forms elastic strain energy. At the same time, energy is consumed in the form of heat energy and radiant energy, energy that has been called dissipation energy (He et al. 2015). In general, the area inside the hysteresis loop formed by the stress–strain curve of the material under cyclic loading can be used to describe the magnitude of this dissipation energy (Xiao et al. 2010a, b). Figures 16 and 17 show the curves for dissipated energy per unit volume (also called energy density, “U”) for coal samples XHC1 and XHC2 versus the number of cycles.

Fig. 16
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Graph of the dissipated energy per unit volume versus number of cycles for coal sample XHC1 under cyclic loading (no fatigue failure)

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Fig. 17
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Graph of the dissipated energy per unit volume versus number of cycles for coal sample XHC2 under cyclic loading (with fatigue failure)

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As can be seen in Fig. 16, the shape of the energy dissipation/volume curve for XHC1 is shaped like a horizontal “L.” Energy dissipates quickly at first but then as the number of cycles increases, dissipation per unit volume does not change. In contrast, for sample XHC2, the shape of the curve before failure is similar to that of XHC1, but the dissipation energy per unit volume increases rapidly just before fatigue damage failure. The recumbent L-shaped curve turns into a U. This shows that although the coal can store a significant quantity of deformation energy, once the coal sample is damaged and fails, the stored energy will be dissipated in a short time resulting in a sudden decrease in strength. In the end, the coal will become unstable.

Fatigue damage evolution of coal under cyclic loading

When a load is imposed on a material, micro-cracks influence the material’s strength before macro-cracks appear. Jean (1985) put forward the concept of continuous damage mechanics to model the damage to materials. For a one-dimensional problem, the damage stress–strain equation based on the strain equivalency hypothesis is:

$$ \sigma =E\left(1-D\right)\xi $$
(1)

where σ, ξ, and E denote the stress on material, the strain, and the elastic modulus, respectively; (1-D) denotes the proportion of effective load-bearing area in the whole area. D denotes a damage variable where D = 0 means the integral material, without damage, and D = 1 means the volume cell is completely damaged, so 0 < D < 1 reflects the state of the material’s damage.

Using the elastic modulus, Eq. 1 can be expressed as:

$$ D=1-\frac{E^{\hbox{'}}}{E} $$
(2)

where E denotes the elastic modulus of undamaged material and E′ denotes the elastic modulus of damaged material. During a period of continuously increasing damage, the current state of damage can be calculated from the material’s elastic modulus.

The most common method for determining an elastic modulus for calculation purposes it to use the unloading stiffness as the elastic modulus of the material. This is applied to calculations for elastic damage, but it is not accurate for elastic–plastic or viscous material damage. Xie et al. (1997) improved the elastic modulus method and provided an elastic–plastic material damage definition that is affected by irreversible plastic deformation. The equation for that definition is:

$$ D=1-\frac{\varepsilon -{\varepsilon}^{\prime }}{\varepsilon}\cdot \frac{E^{\prime }}{E} $$
(3)

where E′ and E denote the unloading stiffness of elastic–plastic material and the initial elastic modulus, respectively, and ε′ stands for the residual deformation after unloading.

Coal can be considered to approximate an elastic–plastic material. If the values of every strain after each cyclic unloading are deemed to be residual plastic deformation, then the damage variable D for every cycle of loading and unloading for coal sample XHC2 can be calculated using Eq. 3. Doing so results in the points plotted in Fig. 18a, a graph showing the damage variable versus number of cycles for a sample with fatigue failure. Identical assumptions and calculations produce Fig. 18b for coal sample XHC1, a plot showing the development of damage versus number of cycles for a sample with no fatigue failure.

Fig. 18
figure 18

Graphs of damage variable “D” versus number of cycles for coal samples

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Figure 18a clearly shows that for the coal samples with fatigue failure, the curve for damage versus cyclic fatiguing under cyclic loading can be divided into three stages. At the initial cyclic stage, damage increases as the number of cycles increases. After a certain number of cycles, damage continues to increase with the increase with the number of cycles, but only slowly, until just before fatigue failure. At this point, the damage increases sharply and the coal sample soon fails.

For the coal samples with no fatigue failure, the damage versus cyclic fatiguing curve can only be divided into two stages. In the initial stage, the damage increases sharply as the number of cycles increases. After those first few cycles, however (16 cycles for coal sample XHC1), the sample is basically not damaged further; damage does not increase any more. It means that when a cyclic loading and unloading experiment is carried out below the fatigue failure threshold value, a certain amount of damage will be produced that will eliminate the pre-existing cracks. These cracks are compacted and closed by confining pressure and the friction between them but only during the early cycles of the experiment. After that, no significant damage occurs.

As can be seen in a microscopical view, damage is characterized by the expanding of many initial cracks distributed in the coal samples, the appearance of new cracks, and then the joining of those micro-cracks into macro-cracks (Yang et al., 2014). Compared with other kinds of rocks, coal contains more micro-cracks. In the coal samples that underwent fatigue failure, the cyclic loading beyond the threshold value of fatigue failure causes the initial cracks to be compacted and to close and new cracks to form and expand. This causes damage to increase sharply in the initial cyclic loading and unloading stage and also causes some irreversible deformation. After a certain number of cycles, the rate of compaction, closure, and increase of micro-cracks in the coal becomes steady and the rate at which damage increases subsides. At this time, there are a certain number of new micro-cracks appearing and expanding and these consume some energy during every loading and unloading cycle. This causes irreversible deformation to increase slightly and the elastic modulus to decrease, so there will be a certain amount of damage. When the damage accumulates to the required level, many new cracks will appear in the coal and gradually connect. Then, the damage will increase sharply and a large number of cracks will join and connect to form macro-cracks. Finally, fatigue failure of the coal samples occurs and the damage has reached its maximum extent.

Conclusions

The experimental study of coal’s fatigue damage under cyclic loading is important for designing coal pillar size because pillar strength is affected by the stresses from repeated cycles of coal mining.

Compared with hard and dense rocks, coal is more inclined to fatigue breakage. Under uniaxial cyclic loading, the fatigue failure threshold value for samples from the No. 3 coal seam in the Xinhe Colliery is less than 78% of its uniaxial compressive strength. The experiments run on these samples show that there will also be a certain amount of fatigue damage to this coal when cyclic loading and unloading occurs at stress levels below the fatigue failure threshold value.

Under cyclic loading, coal’s axial deformation and irreversible deformation can be divided into three stages. They are the beginning stage, the stable deformation stage, and the accelerating deformation stage. The accumulation of deformation in these three stages will eventually cause fatigue failure. Transversal deformation in the coal can be divided into two stages: a steady deformation stage and an accelerating deformation stage. When the transversal deformation increases significantly but then does not recover or recovers only very little when the load is removed, the coal is going to break soon.

As the number of loading/unloading cycles increases, a graph of energy dissipation per unit volume versus number of cycles presents a horizontal L-shaped curve for the coal sample with no fatigue failure. However, for the coal sample with fatigue failure, the same graph shows a U-shaped curve.

Under cyclic loading, the damage the coal suffers can be divided into three stages. First the coal is damaged significantly in the initial stages of cyclic loading and unloading because there are many pre-existing cracks. Then, as the number of loading/unloading cycles continues to increase, the damage increases, but very slowly. If the cyclic stress threshold value has been exceeded and the damage accumulates to the required level, then just before failure, the damage will increase sharply. The coal sample will then suffer fatigue failure. The coal’s deformation and failure versus cyclic loading and unloading are the coal’s fatigue damage evolution.