Experimental study on creep mechanical properties and failure modes of anthracite in dry and saturated states

Abstract

Water is crucial to the durability and safety of underground engineering. To investigate the effect of long-term goaf water infiltration on creep characteristics and long-term stability of the No.3 anthracite pillar in the Qinshui coalfield in Shanxi Province, conventional uniaxial compression and uniaxial creep tests were conducted under dry and saturated conditions. In addition, based on the test results, a nonlinear damage viscoelastic–plastic creep model was developed to describe each creep stage in detail. The experimental results reveal the following. (1) When the loading stress exceeds the long-term strength, the anthracite creep in dry and saturated states undergoes decay, steady-state and accelerated creeps; (2) the total instantaneous elastic strain of the anthracite sample increased linearly with the stress level, and the growth rate of the saturated sample exceeded that of the dry sample; (3) the average of the long-term strength of anthracite under dry and saturated conditions is 7.604 MPa and 4.489 MPa respectively, and the corresponding long-term strength to instantaneous strength ratios are 0.73 and 0.68, respectively; (4) the developed nonlinear damage viscoelastic–plastic creep model can simultaneously describe the two-stage creep characteristics of anthracite under low stress and the three-stage creep characteristics under high stress; (5) the failure form of the dry coal sample after the creep test was a shear failure, whereas the failure form of the saturated coal sample was ductile owing to water softening. When anthracite was transformed from the dry to the saturated state, its instantaneous and long-term strengths decreased by ~ 37% and ~ 41%, respectively.

Introduction

The rheological properties of coal or rock as critical mechanical properties are closely related to the long-term stability of underground engineering^1,2,3. The water content of coal or rock mass is one of the most important factors influencing its creep instability⁴. In the long-term bearing process, the coal pillars left behind by previous mining in an underground coal mine exhibit prominent creep characteristics. Coal is considerably weakened by both the infiltration of groundwater and the accumulation of water in the goaf. Long-term immersion reduces the physical and mechanical properties of coal, affording internal degradation and damage that affects the creep properties of coal^5,6. Recently, many experiments on the creep properties of rocks in the water-bearing state have been conducted. Based on the test data, several creep constitutive models of rocks that account for water-cut weakening have been developed, and the effect of water on rock creep characteristics has been further investigated. Yu⁷ conducted uniaxial compression, triaxial compression and Brazilian splitting tests on red sandstone with various water contents and discovered that the instantaneous compressive and tensile strengths of red sandstone decreased exponentially with the increase in the water content. On this basis, a uniaxial creep experiment was conducted under the combined action of water and force. The results indicate that the instantaneous strain and steady creep rates of the sample increased exponentially with increasing moisture content, while the creep strain and failure time reduced. Based on the mechanical properties and creep principles of carbonaceous slate under dry and saturated states, Song⁸ investigated the effect of water on the strength and deformation of carbonaceous slate. Jin⁹ conducted a graded loading creep test of quartz sandstone in dry and saturated states and concluded that water considerably influenced the creep failure mode of sandstone and that the failure mode of the specimen under the influence of water was primarily splitting failure. To investigate the creep properties of mudstone in Tianshui, Gansu Province, Zhang¹⁰ conducted triaxial creep tests with varying moisture contents and loading stresses on mudstone, determined the relationship between the creep strain and time, and then developed a suitable creep model based on the test characteristics. Li¹¹ et al. used sandstone samples in the Wuhan River-crossing Tunnel to conduct shear creep tests under dry and saturated states and analysed the effect of water on the shear creep characteristics of sandstone and its influencing mechanism. Wan¹² et al. conducted triaxial creep tests of red sandstone samples with different moisture contents after freezing and thawing and discovered that the creep amount and creep rate of red sandstone samples increased with moisture content while the stress grade and long-term strength decreased considerably. Liu¹³ et al. analysed the effect of water saturation on the creep characteristics and long-term strength of underground gangue cemented backfill using a triaxial graded loading creep test and established a fractional constitutive model that can describe each creep stage of the gangue cemented backfill.

As the primary method for describing the creep characteristics of coal or rock, the creep model has always been the focus of rock rheology research. Sun¹⁴ et al. conducted uniaxial compression and uniaxial creep tests on sandstone under different moisture conditions and proposed an improved Nishihara model based on the damage principle. Liu¹⁵ et al. developed a four-element fractional creep model capable of describing the accelerated creep stage of the weakly cemented soft rock; the model parameters were obtained by inverting the experimental results. By connecting the Nishihara creep damage constitutive model with variable parameters in series with the expansion model, Zhang¹⁶ developed an expansion creep damage model that can describe each creep stage of red mudstone under hydration. Shi¹⁷ established the SBN-burgers model, representing each creep stage of the rock samples based on the creep curve characteristics of water-saturated tuffaceous siltstone. The model parameters were regressed, and the fitting correlation exceeded 0.90. Scholars also conducted creep tests on carbonaceous slate¹⁸, argillaceous siltstone¹⁹, granite²⁰ and shale²¹, discussed the influence of water on rock creep deformation characteristics and long-term strength and established creep models of various rocks considering water-bearing damage based on the test results. Wu²² et al. carried out microscopic tests and triaxial creep tests of coal samples under different water content conditions, and concluded that with the increase of water content, the content of kaolinite in coal decreased, and the axial load required for coal samples to enter steady-state creep and accelerated creep decreased. Based on the multi-stage creep experiment of submerged coal samples, Li²³ constructed the fractional creep constitutive model of coal and rock under the condition of water immersion, verified the reliability of the creep model and parameters, and put forward the stability evaluation method of submerged strip coal pillar based on TFAHP- cloud model. Han²⁴ conducted a simulation test and applied research on the influence of goaf water on the stability of coal pillars, expounded the failure mechanism and microstructure change characteristics of coal samples under different soaking conditions, and analyzed the influence of soaking time on the stability of coal pillars. Li²⁵ studied the creep damage characteristics and permeability evolution law of coal pillar dam samples of underground water storage structure in coal mine under the coupling effect of multiple fields, and obtained that the creep failure mode of coal samples with different water contents changed from tensile failure to shear failure, constructed the seepage-creep model of coal body with damage, and obtained the optimization formula for calculating the reasonable width of coal pillar considering the coupling effect of seepage and creep.

However, based on the results of previous research, it is apparent that scholars focus on the strength tests and creeps of all types of rocks in a water-bearing state, whereas coal in a water-bearing state has received little attention. Owing to the vast difference in the water content between coal and rocks, the impact degree of the “water-time” effect on them is also different^26,27, and the existing creep results of the water-bearing rocks cannot fully characterise the creep characteristics of coal in the water-bearing state. Thus, the No.3 anthracite in the Qinshui coalfield of Shanxi Province was selected as the research sample of this study. The conventional uniaxial compression test and uniaxial creep tests were conducted under dry and saturated conditions, and the influence of water on the instantaneous and creep mechanical properties of the anthracite samples was analysed. The results of this study will provide some theoretical basis and support for the mining design of the coal pillar left by previous mining activities under long-term goaf water soaking, the analysis of the long-term stability of the coal pillar and the study of the coal pillar instability mechanism.

Testing equipment and tested coal samples

Tested samples Preparation

These samples of No. 3 anthracite coal were obtained from the Guanlingshan Coal Mine in the Qinshui Coalfield in Shanxi Province. Following the International Institute of Rock Mechanics testing regulations, standard cylinder samples of Φ 50 mm × 100 mm along the vertical bedding direction were cut from coals using a yarn cutting machine. The upper and lower surfaces of the coal samples were then finely polished using an end grinder to ensure the parallelism of the two end faces and the verticality of the samples along their axial directions. To reduce the discreteness of the samples and ensure homogeneity, the longitudinal wave velocity of each processed coal sample was measured using an ultrasonic pulser-receiver (Model: HC-F600). The wave velocities of the initial batch of samples ranged from 1673.53 to 2658.14 m/s, indicating inherent variability. Samples with wave velocities within a strict threshold of ± 5% of the mean value were selected for this study. The Fig. 1 shows the wave velocity distribution of all initially prepared specimens. The samples selected for the formal tests (highlighted in blue) are those with velocities within the ± 5% tolerance band around the mean value, ensuring sample homogeneity. The processing steps and processed coal samples are depicted in Fig. 2(a)–Fig. 2(b), and related testing equipment in Fig. 2(c)–Fig. 2(h).

Fig. 1

Distribution of ultrasonic P-wave velocities for processed coal samples.

Fig. 2

Tested specimens preparation process and related testing equipment.

We dried the selected test samples by placing them in a drying box for 12 h at a temperature of 100 °C. They were then coated after drying and cooling to room temperature to keep their moisture content constant. The samples were divided into two groups, with eight specimens in each dry and saturated groups. Four specimens were used for uniaxial compression testing, and the remaining four were used for uniaxial creep testing. Weighed dry samples were soaked using a soaking technique. The treatment procedure was as follows. The coal samples were soaked in a water tank at a constant room temperature of 20 ± 2 °C. To maintain water quality and prevent any potential microbial activity, the distilled water was replaced every 24 h. Initially, distilled water was injected to one-fourth the height of the specimen, the specimen was removed after 1 h, the surface moisture was wiped off and the moisture content was calculated after weighing. After replacing the sample, it was injected with distilled water to a depth of one-half its original height, soaked for 2 h, weighed and its water content was calculated in the same manner as described above. According to this, three-fourth of the specimen’s height was injected with water for 4 h, and the specimen was submerged in water for 8 h. After that, the samples were weighed every 8 h, and their moisture content was calculated until there was no longer any change in the moisture content of the samples. Figure 3 depicts the variation law of the water content of the saturated specimen with time. From the graph, we can conclude that the relationship between moisture content and time is

$$\omega {\text{=-3}}{\text{.8806}}{{\text{e}}^{ - 8.2305t}}+3.21 ({{\text{R}}^2}={\text{0}}.{\text{996}})$$

(1)

where ω represents the moisture content of the anthracite sample. t denotes the soaking time of the coal sample and R² denotes the correlation coefficient of the fitted curve. According to the above equation, when time t exceeds 120 h, the mass of the coal samples no longer increases, and the moisture content stabilizes at a constant value of 3.21%. This state is defined as the saturated state, indicating that the sample has reached a water absorption equilibrium with its environment under atmospheric pressure. This saturation moisture content is intrinsically linked to the pore structure of the anthracite. The saturated water absorption closely correlates with the apparent porosity of the coal, suggesting that the 3.21% moisture content primarily occupies the interconnected pores and microfractures within the coal matrix. The presence of water in these pores plays a critical role in weakening the coal by softening the coal matrix and reducing the frictional resistance between mineral particles, which is the fundamental reason for the reduction in its mechanical strength.

Fig. 3

Variation of moisture content of the anthracite samples with time.

Testing equipment

Under dry and saturated conditions, anthracite samples were subjected to uniaxial compression tests at Key Laboratory of In-Situ Property Improving Mining of the Ministry of Education at Taiyuan University of Technology. As depicted in Fig. 2(g), the instrument dutilised was a 205D-TS microcomputer-controlled electronic universal testing machine. This instrument is primarily used for mechanical tests on coal and rock samples, such as compression, tension, bending and shearing. The maximum axial load is 250 kN; force measuring resolution is 0.5 N; axial displacement resolution is 0.04 μm and data acquisition frequency is 1000 Hz.

The HSW-200R rock creep testing machine was utilised to conduct uniaxial creep tests on anthracite samples under both dry and saturated conditions. As shown in Fig. 2(h), the test system primarily comprises an axial pressure loading system, a confining pressure loading device, a temperature control system, a kettle body, a displacement sensor and a data acquisition system. During the test, air pressure was used to load the axial pressure; the maximum axial stress was 50 MPa; the stress resolution was 0.01 MPa and the voltage stabilising effect was adequate. During the test, the specimen and the upper and lower plugs were sealed with a heat -shrinkable sleeve to prevent the moisture content from changing, and the axial deformation was measured in real time by an external extensometer. The axial strain was calculated by △L/L, where L is the height of the specimen and △L is the deformation of the sample. The extensometer accuracy is 0.001 mm. The temperature control range of the kettle containing the specimen was from 0 °C to 300 °C, and the temperature accuracy was controlled within ± 0.1 °C.

Uniaxial compression tests

Uniaxial compression test

The conventional uniaxial compression test was the simplest and the most fundamental method for determining the mechanical properties of coal or rock. On the one hand, the complete stress–strain curve obtained from uniaxial compression testing can reveal the strength, deformation and failure characteristics of coal or rock. In addition, the rock mechanical properties derived from the uniaxial compression test could serve as a foundation for the graded loading stress grade of the creep test. Considering the discreteness of mechanical parameter test results of anthracite samples during uniaxial compression herein, one group of dry and saturated anthracite samples was tested in parallel, with four coal samples in each group subjected to the uniaxial compression test. Finally, a representative sample for data analysis was chosen.

During the test, the prepared standard specimens were positioned in the centre of the bearing plate of the 205D-TS testing machine, and the spherical base was adjusted so that the testing machine, the upper and lower bearing plates and the specimen were horizontally aligned. After clearing the displacement and stress, the specimen was loaded at 0.2 MPa/s until it was destroyed, and the axial stress and axial displacement were recorded twice per second.

Mechanical parameters of anthracite in dry and saturated States

Fig. 4

Stress–strain curves of the anthracite samples during uniaxial compression tests under dry state.

Fig. 5

Stress–strain curves of the anthracite samples during uniaxial compression tests under saturated state.

The axial stress–axial strain curves of anthracite samples in the uniaxial compression test under dry and saturated states are shown in Figs. 4 and 5; it shows that water considerably affects the mechanical properties of anthracite specimens. The saturated specimen had lower peak stress than the dry specimen. The stress–strain curve was concave at the initial loading stage regardless of whether it was dry or saturated, indicating that the strain rate decreased with increasing stress primarily due to the closing of micro-cracks in coal samples under the action of external forces. The stress–-strain curve was approximately linear, which could be because the friction force between the fissures prevents the mutual dislocation of the micro-fissures after the fissures were closed, affording linear elastic deformation. As the axial stress increased, the curve changed from linear to nonlinear, and the sample entered the strain hardening stage. The stress in the 1#, 2# dry coal samples and 3# saturated coal sample suddenly decreased and then rapidly increased possibly caused by the compaction or destruction of the original internal cracks. The samples exhibited obvious strain softening after exceeding the peak stress.

Comparing the deformation curves of the dry and saturated samples reveals that the dry coal sample is brittle and would be destroyed as soon as it reaches its peak value. Compared to the dry coal sample, the saturated coal sample exhibited a “softening” trend and a flatter stress–strain curve, in this paper, softening coefficient (average peak intensity in saturated state/average peak intensity in dry state) was used to express the softening degree of coal sample after saturation. The action of water diminished the friction and cohesive forces between mineral particles in the coal sample, and the elastic activity space of the coal sample expanded. The compressive strength and elastic modulus decreased macroscopically, deformation and softening occurred under load and ductility increased. The peak points of the axial stress–axial strain curves represent the instantaneous compressive strength of the samples. The elastic modulus (Es) is the slope of the approximately linear portion of the axial stress–strain curve, obtained from the dotted line in Figs. 4 and 5. The conventional mechanical parameters of the anthracite samples in their dry and saturated states are shown in Table 1.

Table 1 Conventional mechanical parameters of anthracite under dry and saturated states.

Uniaxial creep test

Testing program

The uniaxial creep tests of the anthracite samples were conducted in a step-by-step loading mode, and samples under dry and saturated states were loaded step-by-step from low to high. The loading rate was 0.05 MPa/s, and each stress level was continuously loaded for 48 h, after which the next stress level was performed until the specimen failed. Table 2 lists the loading scheme of the axial stress of anthracite with dry and saturated states in the uniaxial creep test.

Table 2 Loading plan for the uniaxial creep test of dry and saturated coal samples.

Test results

As depicted in Figs. 6 and 7, axial strain–time curves of the coal samples under dry and saturated states were obtained through uniaxial creep tests of the anthracite samples under graded loading.

Fig. 6

Creep test curve of anthracite samples in dry state. (a) Dry-1#, (b) Dry-2#, (c) Dry-3#.

Fig. 7

Creep test curve of anthracite samples in saturated state. (a) Saturated-1#, (b) Saturated-2#, (c) Saturated-3#.

From the uniaxial compression creep test curves of the anthracite samples under dry and saturated conditions shown in Figs. 6 and 7, it is evident that the coal samples undergo instantaneous deformation at each stress loading, with the creep deformation of the specimen increasing with time under increasing stress levels. Below, typical 2# dry coal samples and 2# saturated coal samples are taken as examples for analysis.

When the stress level is low, the creep of the anthracite sample is very small, and as the stress level increases, the creep deformation increases. For example, when the first three stress levels were applied to a dry sample, there was almost no creep deformation. When the fourth stress level (6.4 MPa) was applied to the sample, the axial strain of the coal sample only increased by 0.33 × 10⁻³ after 47.8 h from the instantaneous time of 0.21 h. When the stress level increased to 9.5 MPa, the axial strain of the coal sample increased rapidly by 0.38 × 10⁻³ in 12.2 h.

At medium and low-stress levels, the creep deformation of anthracite samples mainly occurred in two stages: deceleration and steady creeps. However, when the stress level was high (the loading stress exceeded the long-term strength of the coal sample), the deformation rate increased rapidly with time, and the specimen was quickly destroyed; thus, accelerated creep occurred. For instance, using the creep test curve of a saturated anthracite sample, the sample underwent instantaneous elastic deformation when the first axial stress (1.2 MPa) was applied. After that, the axial deformation rate decreased gradually with time, reaching zero after ~ 6.7 h, after which the anthracite sample deformation remained constant.

When the second axial stress (2.4 MPa) was applied, the coal sample exhibited instantaneous deformation at the loading instant. After ~ 14.8 h, the axial deformation rate decreased to 0.091 × 10⁻⁵/h, indicating that it passed through the decay creep stage and then maintained the same rate, indicating that it passed through the steady creep stage. When the third (3.6 MPa) and fourth (4.8 MPa) axial stresses were applied, the sample initially underwent instantaneous strain and decay creep and then maintained the same rates of 0.170 × 10⁻⁵/h and 0.284 × 10⁻⁵/h, i.e., it underwent the steady creep stage. The anthracite sample entered the steady creep stage after 9.94 h when the final axial stress (6.2 MPa) was applied, and the creep rate of 0.478 × 10⁻⁵/h was maintained for ~ 21.12 h. The creep rate increased abruptly and accelerated until the specimen was destroyed.

As shown in Figs. 6 and 7, the steady-state creep rate of anthracite samples increased with increasing stress levels in both the dry and saturated states because anthracite contains relatively well-developed primary cracks. In proportion to the increase in axial stress, the extension rate of primary cracks in the samples and the creep rate of coal samples increased.

Comparing Figs. 6 and 7, it can be concluded that under the same stress level, the creep of saturated coal samples was considerably greater than that of dry coal samples; under the loading stress of 4.8 MPa, the creep curves of anthracite samples were stable under both conditions, and the dry anthracite samples stabilised after loading for 5.3 h, and the creep amount under this stress was extremely small, only 0.012 × 10⁻³. However, the creep rate of the saturated anthracite sample tended to remain stable until 23.24 h later, and the creep amount was considerably greater than that of the dry coal sample, reaching 0.053 × 10⁻³, ~four times higher than the dry coal sample. In addition, the total strain of the saturated anthracite sample was ~ 6.050 × 10⁻³ when it reached steady creep under this stress, whereas the total strain of the dry anthracite sample was only 2.691 × 10⁻³, which was approximately half that of the saturated coal sample. The aforementioned phenomenon demonstrates that water has a pronounced softening effect on anthracite, which considerably affects the creep deformation of coal samples and improves the creep ageing characteristics of anthracite.

Instantaneous elastic strain

Figure 8 depicts the variation in the instantaneous elastic strain of anthracite samples with axial stress levels under dry and saturated states. The data points and fitting curves in Fig. 7 represent the mean values derived from three replicate samples. Use the t-distribution to calculate the confidence interval, at a 95% confidence level, with the formula being:

$$CI=\mu \pm {t_{\alpha /2,n - 1}} \cdot \frac{\sigma }{{\sqrt n }}$$

(2)

where: µ is the sample mean, σ is the sample standard deviation, n is the sample size, and t_{α/2, n−1} is the critical value of the t-distribution with n−1 degrees of freedom (α = 0.05).

The total instantaneous elastic strain of coal samples under various stress levels is nearly linearly related to the loading stress level. The correlation coefficients of a linear fitting equation of the anthracite samples under dry and saturated conditions are 0.993 and 0.987, respectively, indicating that the instantaneous elastic strain has an excellent linear relationship with the stress level. The slopes of the linear fitting equation for anthracite in its dry and saturated states are 0.4407 and 1.0806, respectively. In a saturated state, the instantaneous elastic strain of the anthracite sample increased more rapidly than in a dry state. This is because the softening effect of water weakens the bonding ability of the internal particles of coal, and the amplitude and speed of mutual dislocation between internal particles of the coal sample increased when it was stressed, yielding a higher instantaneous strain rate for the saturated coal sample.

Fig. 8

Relationship between instantaneous elastic strain and stress levels of anthracite samples under dry and saturated conditions.

Creep strain rate

By calculating the slope of the creep curves with time, it is possible to determine the relationship between strain rate and time in the creep test of the anthracite samples. Owing to space limitations, herein, the creep principle of the samples under the final axial stress under dry and saturated conditions was investigated, and the variation curves of the axial steady-state creep rates and creep time are shown in Figs. 9. and 10.

Fig. 9

Variation of the creep rate of anthracite with time in dry state (σ = 9.5 MPa).

Fig. 10

Variation of the creep rate of anthracite with time in saturated state (σ = 6.2 MPa).

Figure 9 depicts the variation of the axial strain rate of the anthracite sample in a dry state with time under the last axial stress (σ = 9.5 MPa). The axial strain rate decreases gradually from 1.85 × 10⁻³·h⁻¹ to 0.03 × 10⁻³·h⁻¹ and then becomes relatively stable in zone AB, which is defined as the decay creep stage. The steady-state creep stage occurs when the strain rate is approximately kept constant at 0.03 × 10⁻³·h⁻¹ in zone BC of the creep curve during which the axial strain increases slowly. The strain rate increases rapidly in zone CD until the anthracite sample experiences plastic failure, and this stage is known as the accelerated creep stage. Thus, the time-dependent creep properties of anthracite in dry state were evident, and it went through three typical creep stages: the decay creep stage (AB), the steady-state creep stage (BC) and the accelerated creep stage (CD). T₁ (2.2 h) is the end time of the decay creep and the start time of the steady creep in Fig. 7, while T₂ (12.2 h) and T₃ (14.2 h) are the start and end times of the accelerated creep, respectively. Decay creep, steady creep and accelerated creep took 2.2 h, 10.0 h and 2.0 h, respectively. The creep strain increased with time during the steady-state creep stage, and when the accumulated deformation was insufficient to maintain this load level, the anthracite sample underwent accelerated creep. The axial strain and strain rate increase dramatically in the accelerated creep stage, as shown in Fig. 8, indicating that the sample finally failed owing to continuous compaction, expansion and the extension of the primary micro-cracks in the coal sample.

Figure 10 depicts the variation of axial strain rate of the anthracite sample with time under the last axial stress (σ = 6.2 MPa) in the saturated state. Under the action of this axial stress, the sample, such as anthracite in the dry state, went through three stages: decay creep state (AB), steady-state creep state (BC) and accelerated creep state (CD). The decay creep, steady-state and accelerated creep durations are 10.9 h, 18.3 h and 6.9 h, respectively. According to Fig. 9, the time of the decay creep, steady creep, and accelerated creep of the anthracite sample in the saturated state lasted longer than that of the anthracite sample in the dry state, and the above three creep stages of anthracite samples in the saturated state take about 5.0 times, 1.8 times and 3.5 times longer than those in the dry state, respectively. This is because the influence of water softening, the time of compaction, extension, and expansion of primary micro-cracks in the coal sample became longer under axial stress, extending the time for the specimen to bear cumulative deformation damage.

Long-term strength

When the load carried by coal or rock is less than its instantaneous strength and lasts for an extended period, the coal or rock may be damaged due to creep. Therefore, the strength of coal or rock decreases as the external load action time increases. Typically, the long-term strength of coal or rock is defined as the lowest strength value of action time t^28,29,30. Therefore, the long-term strength of coal is an important time effect index for measuring the long-term stability of underground engineering.

Currently, methods for determining the long-term strength of coal include the isochronous stress–strain curve method, the excessive creep method, and the steady-state creep method, among others. The benefits and drawbacks of these three methods were examined and discussed in reference³¹. Finally, the tangent method of the steady-state creep rate curve was proposed as an improved method for accurately determining the long-term strength of rock based on the steady-state creep rate method. Consequently, the tangent method of the steady creep rate curve was used to calculate the long-term strength of anthracite.

Figure 11 depicts the relationship curve between axial steady creep rate and stress in the case of uniaxial creep of anthracite sample in the dry state. Figure 11 clearly shows that there is a critical point in the curve. The tangents of the curve are drawn before and after the critical point, and the stress corresponding to the intersection point O represents the long-term strength of anthracite.

As shown in Table 3, the long-term strengths of anthracite under dry and saturated conditions obtained using the above method are 7.418 MPa and 4.318 MPa, respectively. Under dry and saturated conditions, the ratios of long-term strength to uniaxial compressive strength are 0.71 and 0.66, respectively.

Fig. 11

The axial steady-state creep rate of anthracite under dry state.

Table 3 Long-term strength of anthracite sample under dry and saturated conditions.

In view of the long-term strength attenuation characteristics of coal samples after being saturated with water, taking the re-mining working face 3101 of No.3 coal seam in Guanlingshan Coal Industry as an example, according to the situation revealed on the spot, the strength of residual coal pillars is seriously reduced under the long-term immersion of goaf water, which is characterized by serious coal fragmentation and coal pillar spalling (as shown in Fig. 12(a)), the ratio of coal strength without goaf water immersion to coal strength under goaf water immersion is about 0.62, which is basically consistent with the experimental results in Table 3. Therefore, according to the underground detection results, pre-mining grouting reinforcement is carried out on the coal pillars in areas with long-term soaking of goaf water, and it can be found that the reinforced coal pillars are more stable during re-mining (as shown in Fig. 12(b)).

Fig. 12

Stability of coal body before and after grouting reinforcement of residual coal pillar under long-term goaf water immersion.

Creep model and parameter identification of anthracite

Nonlinear damage viscoelastic–plastic creep model of anthracite

According to the results of the graded loading creep test of anthracite in dry and saturated states, the creep curve of anthracite had the following characteristic.

• 1. The coal samples produced instantaneous strain after each level of axial stress loading, indicating the presence of an elastic element in the creep model.

• 2. Dynamically, the strain of the anthracite sample gradually increased. There were some differences between the creep curves of the anthracite samples under dry and saturated conditions when the stress level was low. The creep deformation of anthracite in the dry state gradually stabilised after the decay creep stage, whereas the creep deformation of anthracite in the saturated state occurred in two stages, decay creep and steady creep, implying that the creep model should include viscous elements. To make the creep models in different states consider the above differences, the viscosity coefficient of viscous elements in the creep model in the dry state could be adjusted.

• 3. According to the creep test curve of coal samples in dry and saturated state, it can be seen that the specimen experiences three stages of attenuation creep, stable creep and accelerated creep only at the last stress level, and the coal sample finally destroyed. Consequently, an element in the creep model that can describe the accelerated creep stage is required.

Among traditional linear creep models, the Burgers creep model could simultaneously describe instantaneous elastic deformation, decay creep, and the steady creep of rock¹⁴. Therefore, when the stress level was low (i.e., less than the long-term strength of coal), the traditional Burgers model was used to describe the creep characteristics of anthracite. However, when the stress level was high (the loading stress was greater than the long-term strength of the coal), anthracite samples exhibited accelerated creep characteristics under both dry and saturated states. Because the Burgers model has no yield limit, it cannot describe the creep characteristics of coal or rock when the applied stress exceeds the long-term strength, i.e., the accelerated creep characteristics. Therefore, a nonlinear viscoplastic element capable of describing the accelerated creep stage must be connected in series using the Burgers model.

Coal or rock mass creep failure is a process in which the primary micro-cracks and fissures in the coal and rock mass continuously penetrate and expand, and material damage gradually accumulates, affording a gradual reduction in the ability to resist deformation and damage, causing a gradual increase in deformation until damage¹⁷. The internal damage factor of coal or rock mass was defined as

$$D=\left\{ \begin{gathered} 1 - {e^{ - {{(t - {t^*})}^n}}} (t \geqslant {t^*}) \hfill \\ 0 (t<{t^*}) \hfill \\ \end{gathered} \right. n>0$$

(3)

where D denotes the damage factor, t denotes the creep time, n denotes a constant, and t* denotes the starting time at which the coal sample enters the accelerated creep stage. The coal sample was in the steady creep stage when t < t*, and the damage factor D was close to zero. When t ≥ t*, the coal sample moves from the steady creep stage to the accelerated creep stage. At this point, the damage factor D gradually increased with time t and approached 1.

Herein, a nonlinear damage viscoplastic element was connected in series based on the Burgers’ model to describe the change in deformation of anthracite in the accelerated creep stage with time. Figure 13 depicts the developed nonlinear damage viscoelastic–plastic creep model. In the model, E₁ and E₂ represent the instantaneous elastic modulus and viscous modulus, respectively, η₁, η₂ and η₃ are viscosity coefficients and σ_∞ is the long-term strength of the coal. The following is the corresponding creep equation.

Fig. 13

The schematic of nonlinear viscoelastic-plastic creep model for damage.

When the loading stress σ < σ_∞, the fourth part of the above model fails, and the model degenerates into the Burgers model. Burgers’ model compries elasticity, viscosity and a viscoelastic body, and its constitutive equation is as follows.

$$\sigma +\left[ {\frac{{{\eta _1}}}{{{E_1}}}+\frac{{\left( {{\eta _1}+{\eta _2}} \right)}}{{{E_1}{E_2}}}{E_1}} \right]\mathop \sigma \limits^{\cdot } +\frac{{{\eta _1}{\eta _2}}}{{{E_1}{E_2}}}\mathop \sigma \limits^{{\cdot \cdot }} ={\eta _1}\mathop \varepsilon \limits^{\cdot } +\frac{{{\eta _1}{\eta _2}}}{{{E_2}}}\mathop \varepsilon \limits^{{\cdot \cdot }}$$

(4)

The creep equation for the Burgers model is

$$\varepsilon {\text{=}}\frac{\sigma }{{{E_2}}}+\frac{\sigma }{{{\eta _1}}}t+\frac{\sigma }{{{E_2}}}\left[ {1 - \exp \left( { - \frac{{{E_2}}}{{{\eta _2}}}t} \right)} \right]$$

(5)

When the loading stress σ ≥ σ_∞, t < t*, the nonlinear damage variables in the above model have no effect, and the corresponding creep state equation is

$$\left\{ \begin{gathered} {\sigma _1}{\text{=}}{E_1}{\varepsilon _1} \hfill \\ {\sigma _2}={\eta _1}{\mathop \varepsilon \limits^{\cdot } _2} \hfill \\ {\sigma _3}={E_2}{\varepsilon _3}+{\eta _2}{\mathop \varepsilon \limits^{\cdot } _3} \hfill \\ {\sigma _4}={\sigma _\infty }+{\eta _3}{\mathop \varepsilon \limits^{\cdot } _4} \hfill \\ \sigma ={\sigma _1}={\sigma _2}={\sigma _3}={\sigma _4} \hfill \\ \varepsilon {\text{=}}{\varepsilon _1}+{\varepsilon _2}+{\varepsilon _3}+{\varepsilon _4} \hfill \\ \mathop \varepsilon \limits^{\cdot } {\text{=}}\mathop {\mathop {{\varepsilon _1}}\limits^{\cdot } +\mathop {{\varepsilon _2}}\limits^{\cdot } +\mathop {{\varepsilon _3}}\limits^{\cdot } +\mathop {{\varepsilon _4}}\limits^{\cdot } }\limits^{{\text{.}}} \hfill \\ \end{gathered} \right.$$

(6)

In the above equation, σ₁, σ₂, σ₃ and σ₄ respectively correspond to the stresses of Sects. 1, 2, 3 and 4 in the model; ε₁, ε₂, ε₃ and ε₄ correspond to the strains of parts 1, 2, 3 and 4 in the model, respectively, and σ and ε denote the total stress and strain of the model, respectively.

The Laplace transformation of Eq. (6) yielded the corresponding creep Eq.

$$\varepsilon =\frac{\sigma }{{{E_1}}}+\frac{\sigma }{{{\eta _1}}}t+\frac{\sigma }{{{E_2}}}\left[ {1 - \exp -\frac{{{E_2}}}{{{\eta _2}}}t } \right]+\frac{{\sigma - {\sigma _\infty }}}{{{\eta _3}}}t$$

(7)

When the loading stress σ ≥ σ_∞, t ≥ t*, all the above model and nonlinear damage variables are activated, and the corresponding creep state equation is

$$\left\{ \begin{gathered} {\sigma _1}{\text{=}}{E_1}{\varepsilon _1} \hfill \\ {\sigma _2}={\eta _1}{\mathop \varepsilon \limits^{\cdot } _2} \hfill \\ {\sigma _3}={E_2}{\varepsilon _3}+{\eta _2}{\mathop \varepsilon \limits^{\cdot } _3} \hfill \\ {\sigma _4}={\sigma _\infty }+{\eta _3}(1 - D){\mathop \varepsilon \limits^{\cdot } _4}={\sigma _\infty }+{\eta _3}{e^{ - {{(t - {t^*})}^n}}}{\mathop \varepsilon \limits^{\cdot } _4} \hfill \\ \sigma ={\sigma _1}={\sigma _2}={\sigma _3}={\sigma _4} \hfill \\ \varepsilon {\text{=}}{\varepsilon _1}+{\varepsilon _2}+{\varepsilon _3}+{\varepsilon _4} \hfill \\ \mathop \varepsilon \limits^{\cdot } {\text{=}}\mathop {\mathop {{\varepsilon _1}}\limits^{\cdot } +\mathop {{\varepsilon _2}}\limits^{\cdot } +\mathop {{\varepsilon _3}}\limits^{\cdot } +\mathop {{\varepsilon _4}}\limits^{\cdot } }\limits^{{\text{.}}} \hfill \\ \end{gathered} \right.$$

(8)

The Laplace transformation of Eq. (8) yielded the corresponding creep Eq.

$$\varepsilon =\frac{\sigma }{{{E_1}}}+\frac{\sigma }{{{\eta _1}}}t+\frac{\sigma }{{{E_2}}}\left[ {1 - \exp -\frac{{{E_2}}}{{{\eta _2}}}t} \right]+\frac{{\sigma - {\sigma _\infty }}}{{{\eta _3}}}{e^{{{(t - {t^*})}^n}}}t$$

(9)

In summary, the creep equation of anthracite in dry and saturated states is

$$\left\{ \begin{gathered} \varepsilon {\text{=}}\frac{\sigma }{{{E_2}}}+\frac{\sigma }{{{\eta _1}}}t+\frac{\sigma }{{{E_2}}}\left[ {1 - \exp \left( { - \frac{{{E_2}}}{{{\eta _2}}}t} \right)} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma <{\sigma _\infty } \hfill \\ \varepsilon =\frac{\sigma }{{{E_1}}}+\frac{\sigma }{{{\eta _1}}}t+\frac{\sigma }{{{E_2}}}\left[ {1 - \exp -\frac{{{E_2}}}{{{\eta _2}}}t } \right]+\frac{{\sigma - {\sigma _\infty }}}{{{\eta _3}}}t{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma \geqslant {\sigma _\infty } t<{t^ * } \hfill \\ \varepsilon =\frac{\sigma }{{{E_1}}}+\frac{\sigma }{{{\eta _1}}}t+\frac{\sigma }{{{E_2}}}\left[ {1 - \exp -\frac{{{E_2}}}{{{\eta _2}}}t } \right]+\frac{{\sigma - {\sigma _\infty }}}{{{\eta _3}}}{e^{{{(t - {t^*})}^n}}}t{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma \geqslant {\sigma _\infty } t \geqslant {t^ * } \hfill \\ \end{gathered} \right.$$

(10)

In order to better distinguish the accuracy and practicability of the established model, we compare the established model with the traditional creep model, as shown in Table 4.

Table 4 Comparison of the proposed model with existing creep models.

As illustrated in Table 4, a systematic comparison with existing models highlights the novelty and advantages of the proposed nonlinear damage viscoelastic-plastic model:

Unlike the Improved Nishihara model¹⁴, the proposed model can accurately capture the full three-stage creep behavior, including the critical accelerated creep stage leading to failure, through the introduction of a nonlinear damage viscous-plastic element.

Compared to the Expansion creep damage model¹⁶ and the SBN-Burgers model¹⁷, the proposed model offers superior physical interpretability. Each component (spring, dashpot, frictional slider) and parameter in the model corresponds to a specific physical mechanism (instantaneous elasticity, viscoelasticity, viscoplastic yielding, and damage), making the model parameters easier to determine and the model behavior easier to understand.

While the proposed model does not incorporate an explicit chemical water-damage variable like the model in¹⁶, it implicitly and effectively captures the water-induced weakening effect. The significant reduction in identified parameters (e.g., elastic moduli E₁, E₂ and viscosity coefficients η₁, η₂) for saturated samples (as shown later in Table 5) quantitatively reflects the mechanical degradation caused by water softening. This approach directly links the model parameters to the macroscopic mechanical response under different environmental conditions.

Therefore, the proposed model provides a more comprehensive and physically grounded framework for describing the nonlinear creep behavior of anthracite, especially under water-saturated conditions.

Parameter identification and model verification

According to the test results, the uniaxial creep test data of anthracite was iteratively calculated to determine the model parameters using the above nonlinear damage viscoelastic–plastic creep model based on the 1stOpt program and L–M optimisation algorithm. Table 5 displays the creep parameter identification results of anthracite samples under dry and saturated states under various stress levels.

Table 5 Creep model parameters of the anthracite samples under dry and saturated states.

Table 4 shows that when nonlinear damage viscoplastic elements are employed, the coal sample experiences accelerated creep at the final stress level. The established damage nonlinear viscoelastic–plastic creep model accurately describes the anthracite creep curve at all stress levels, and the parameter identification results were well fitted, with correlation coefficients exceeding 0.92. Simultaneously, it is discovered that the creep parameters of anthracite change with stress level and time.

Figures 14 and 15 compare the creep and theoretical model curves of anthracite samples under dry and saturated conditions. The fitting curve of the test data is consistent with the prediction curve of the model, and the nonlinear damage viscoelastic–plastic creep theoretical model can simultaneously replicate the two-stage creep characteristics of anthracite under low stress and the three-stage creep characteristics under high stress, indicating that the creep model established herein is reasonable and accurate. The developed model and its steps serve as a good reference for recovering high-quality anthracite resources left by long-term goaf water immersion, predicting nonlinear coal deformation and evaluating the long-term stability of the coal pillar.

Fig. 14

Comparison of experimental data and the model curve of anthracite samples under dry conditions.

Fig. 15

Comparison of experimental data and the model curve of anthracite samples under saturated conditions.

Evolution law of creep parameters of anthracite

To investigate the variation of creep parameters of anthracite with stress levels, the variation curves of creep parameters of anthracite samples with axial stress levels under dry and saturated conditions according to Table 4 are shown in Figs. 16, 17, 18 and 19.

Fig. 16

Changes of instantaneous elastic modulus E₁ of the anthracite samples with stress levels under dry and saturated conditions.

Fig. 17

Changes of viscous modulus E₂ of the anthracite samples with stress levels under dry and saturated conditions.

Fig. 18

Variation curve of viscosity coefficient η₁ with stress levels under dry and saturated conditions.

Fig. 19

Variation curve of viscosity coefficient η₂ with stress levels under dry and saturated conditions.

Figure 16 depicts the change in instantaneous elastic modulus with stress level. Figure 15 shows that the change law of instantaneous elastic modulus E₁ of the anthracite in dry state differed from that in saturated state. The instantaneous elastic modulus decreased exponentially with stress level under dry conditions but linearly with stress level under saturated conditions, indicating that as loading stress increased, the internal micropores of the anthracite were closed, cracks were compressed and the instantaneous strain value gradually increased, yielding a decrease in E₁.

The variation in viscosity modulus with stress levels is shown in Fig. 17. The principle of change of the viscosity modulus E₂ of anthracite samples under dry and saturated conditions was the same, and both increased exponentially with the increase of stress levels. It demonstrates that as the stress level increased, the anthracite underwent creep hardening, and the growth rate of saturated samples exceeded that of dried samples, ascribed to the effect of water softening, which improved the ability of the sample to resist deformation.

Figures 18 and 19 show the variation curves of viscosity coefficients η₁ and η₂ with stress levels. The viscosity coefficient η₁ represents the creep rate in the steady creep stage.

The stress-dependent evolution of the viscosity coefficients η₁ and η₂ is not merely a mathematical fitting outcome but reflects the underlying physical processes of microcrack activity within the coal sample.

The gradual linear decrease of η₂ with increasing stress level (Fig. 19) can be interpreted as a continuous accumulation of damage. The viscosity coefficient η₂, associated with the steady-state creep, represents the material’s resistance to sustained deformation. As the axial load increases, the primary microcracks and defects within the coal initiate stable propagation. This propagation reduces the effective load-bearing area and facilitates inter-particle sliding, leading to a progressive weakening of the coal’s structural integrity. Macroscopically, this weakening is manifested as a reduction in η₂, indicating that the same stress level causes a higher steady-state creep rate as damage accumulates.

The variation of η₁ (Fig. 18) reveals a more complex competition between different micro-mechanisms. The coefficient η₁ is primarily related to the decay creep stage. For the dry sample, the initial increase in η₁ at low stress levels may be attributed to the compaction and closure of initial microcracks. This process enhances the contact between crack surfaces, increasing frictional resistance and temporarily improving the material’s ability to resist transient deformation, hence an increase in η₁. As the stress level further increases beyond a certain threshold (around 4.8 MPa for the dry sample), the mechanism shifts towards damage accumulation through microcrack initiation and stable propagation. This dominant damage process overwhelms the initial compaction effect, leading to a subsequent decrease in η₁. For the saturated sample, the weakening effect of water significantly reduces the frictional resistance along crack surfaces. This means the compaction effect is less pronounced, and the damage process (crack propagation) initiates at a much lower stress level. Consequently, η₁ for the saturated sample exhibits a consistent decreasing trend from the beginning, governed primarily by water-accelerated damage.

The role of water is therefore clearly articulated through these parameters: Water saturation lubricates crack surfaces and softens the coal matrix, which (1) diminishes the initial strengthening effect from crack closure (lower and monotonically decreasing η₁), and (2) significantly promotes crack propagation under lower stresses, leading to a more rapid deterioration of the resistance to both transient and steady-state creep (lower values of both η₁ and η₂ across all stress levels).

Failure characteristics analysis of anthracite

Many factors influence the macro-failure modes of anthracite samples under dry and saturated conditions, including coal sample characteristics, stress level, and water content. The failure modes of the anthracite samples differ considerably owing to differences in water saturation and stress. Figures 20 and 21 show the failure modes of anthracite samples after the uniaxial compression and uniaxial creep tests with two different water contents. Figure 20 shows the macroscopic failure patterns of the dry and saturated samples in the uniaxial compression test. Because of the stress, the internal cracks of the dry coal sample began to compress and close during the test. The original cracks expanded or formed new cracks as the stress increased, and the cracks expanded and penetrated, making the coal sample crack further. The coal sample cracked during the loading process, accompanied by a splash of small coal pieces, and the entire spalling failure of the coal sample across the vertical section was a typical brittle shear failure. The coal sample in the saturated specimen was crushed at the upper end owing to the influence of water softening, and the damaged part expanded and extended several macro-cracks along the vertical direction. The extended cracks did not penetrate the entire specimen, ~ 70% of its length. The crushed coal pieces surrounding the severely crushed part fell off, revealing a clear case of weak plane shear failure.

Figure 21 depicts the macroscopic failure patterns of dry and saturated samples following a uniaxial creep test. The dry coal sample is sheared along the internal fracture surface under continuous loading at all stress levels, and the failure surface runs through the entire specimen in the diagonal direction, and the failure surface at the upper end of the coal sample expands and shifts locally. The water-saturated coal sample is subjected to extrusion outward expansion failure at the upper end, and the fracture runs from the upper to the lower parts of the specimen, forming several cracks that run through the entire specimen. The entire specimen is loose and collapsed on the macroscopic level, indicating typical ductile failure.

Fig. 20

Failure modes of the anthracite sample under dry and saturated conditions during a uniaxial compression test.

Fig. 21

Failure modes of the anthracite sample under dry and saturated conditions during the uniaxial creep test.

Conclusions

The instantaneous uniaxial compression and creep tests of anthracite samples under dry and saturated conditions were performed herein, along with the creep characteristics of anthracite samples in two types of water conditions, and the variation rules of instantaneous elastic strain, steady creep rate, and axial stress level of the anthracite samples were analysed. The relationship between long-term strength and instantaneous strength of anthracite samples under the dry and saturated conditions was compared, providing a decision-making reference for the safety evaluation and engineering application of the long-term stability of old coal pillars under long-term goaf water immersion.

(1) The stress curve of the anthracite samples under dry and saturated conditions is a “linear–nonlinear–linear” transformation relationship before the stress peak, according to a uniaxial compression test. The saturated coal sample shows a considerable softening trend after reaching the peak. The dried sample exhibits higher compressive strength and elastic modulus than the saturated sample, with a softening coefficient of 0.62.

(2) The instantaneous elastic strain of anthracite increased linearly with stress levels under both dry and saturated conditions, with the saturated sample growing faster than the dry sample. This was because the softening effect of water weakened the bonding ability of internal coal particles, and when the coal sample was stressed, the amplitude and speed of mutual dislocation among the internal coal particles increased.

(3) The long-term strength of anthracite samples under dry and saturated conditions was determined using the steady creep rate method to be 7.418 MPa and 4.318 MPa, respectively, and the ratio of long-term strength to instantaneous strength was 0.71 and 0.66, respectively. The long-term strength of anthracite under the two states decreased by ~ 30% compared to the instantaneous strength.

(4) When the loading stress was less than the long-term strength of the coal, the samples only underwent decay creep and steady creep. When the loading stress exceeded the long-term strength of the coal, the anthracite samples exhibited accelerated creep characteristics in dry and saturated states. The established nonlinear damage viscoelastic–plastic creep theoretical model can simultaneously reflect the two-stage creep characteristics of the anthracite under low stress as well as the three-stage creep characteristics under high stress.

(5) In the uniaxial compression test, the failure modes of the dry and saturated coal samples were brittle shear failure and weak plane shear failure, respectively. Following the creep test, the dry coal sample demonstrated shear failure, whereas the saturated coal sample demonstrated typical ductile failure owing to water softening.