Introduction

Rock creep indicates the deformation characteristics of rock under constant load, and it is different from rock characteristics under static loading and dynamic loading (Kong et al. 2020). Creep significantly affects the stability of engineering rock masses.

Many experimental studies investigated the compression-creep properties of rock, and most researchers used multi-level loading when conducting such tests. However, the excavation and support can either increase or decrease the stress in the surrounding rock during underground engineering. Therefore, in compression-creep tests, multi-level loading is inconsistent with the manner in which stress changes in the surrounding rock. Instead, multi-level loading-unloading is used to conduct rock compression-creep tests in this study, which is more consistent with how stress changes in the surrounding rock during underground engineering.

Compare to multi-level loading tests, relatively fewer researchers have used multi-level loading-unloading in compression-creep tests. Zhao et al. (2009) conducted viscoelastic/plastic-creep experiments on soft ore rock under circular incremental loading-unloading and found that soft rocks enhance ability to resist instantaneous elastic deformation and instantaneous plastic deformation during the multi-level loading-unloading in the cyclic process. Yang et al. (2014a) conducted large-scale in situ compressive-creep tests on the weak layer using a rigid bearing plate and loading-unloading cycles and expressed that the five-parameter generalized Kelvin model is a good representation of the creep behavior of the weak rock layer. Yang et al. (2015) performed triaxial compression experiments on deeply buried marble specimens to investigate their short-term and creep behavior under cyclic loading and pointed out that the steady-state creep rate of marble increases nonlinearly with the increasing axial deviatoric stress. Zhao et al. (2017a) carried out a series of triaxial creep tests on intact and cracked Maokou limestone specimens under multi-level loading-unloading cycles and found that the viscoelastic strain converges to a certain value with time, but the visco-plastic strain keeps increasing with time. Zhao et al. (2017b) presented an extensive laboratory investigation of the nonlinear rheological mechanical characteristics of hard rock under cyclic incremental loading-unloading and found that the proportion of instantaneous plastic strain to total instantaneous strain has an increasing tendency with higher deviatoric stress overall. Similarly, relatively few researchers have used unloading confining stress when conducting compression-creep tests. Huang et al. (2017) conducted triaxial unloading confining pressure creep tests on sandy mudstone specimens to study the dilatancy and fracturing behavior of soft rock. They revealed that when the deviatoric stress is larger and the experienced creep time is longer, the unloading effect and creep characteristics are more apparent and are accompanied by more obvious lateral dilatancy. Hu et al. (2019) performed cyclic loading-unloading creep experiments on the artificial layered cemented specimens with various layer angles. They showed that time-independent strains have linear correlations with the applied stress ratio, whereas time-dependent strains have nonlinear correlations with the stress ratio.

When rock material is subjected to an external load, expansion of internal micro-cracks and friction between particles both act as ready sources of acoustic emission (AE). Recently, many tests have been conducted on rock AE. Yang et al. (2014b) conducted multi-stage triaxial compression tests on cylindrical red sandstone specimens and displayed that the spatial AE events are not active and distribute stochastically in the specimen before peak strength, while after peak strength, the spatial AE events are very active and focus on a local region beyond final microscopic failure plane. Kim et al. (2015) found that the AE energy and damage parameter from the moment tensor of granite are very close not only on the stress thresholds of crack initiation and crack damage but also close on the patterns of progressive damage evolution. Zhang et al. (2015) pointed out that the AE activity and energy release of rock salt with coarse grains and non-uniform structures decrease with increasing stress level. Meng et al. (2016) found that the acoustic emission intensity of rock specimens during uniaxial cyclic loading-unloading compression changes repeatedly. Wang et al. (2019) showed that the peak AE activity of saturated karst limestone under uniaxial compression occurs near the peak stress of the rock. Meng et al. (2018) found that the AE characteristics are closely related to the stress-strain properties of rock materials and are affected by the developmental stage and degree of internal micro-cracks. Zhang et al. (2018) divided the AE energy curves of anisotropic coal under triaxial stress into three types: rapid facture type, stable fracture type, and plastic fracture type.

Because rock creep takes a relatively long time and involves a relatively complex failure mechanism, few studies about rock creep AE exist currently, especially regarding the analysis and investigation of AE characteristics during rock creep process under different loading modes.

Therefore, in the present study, we conduct creep-AE tests and theoretical analysis on red sandstone under multi-level loading-unloading. We experimentally obtain the creep-AE characteristics of the rock during (i) multi-level loading and (ii) multi-level loading-unloading; those characteristics include AE amplitude, cumulative AE event, and cumulative AE ring-down count.

Test equipment, specimen characteristics, and experimental groups

Test equipment

The creep-AE test equipment used in the present study comprises two main parts, namely, a triaxial rheological testing machine (HPTAS; GDS Instruments, UK) and an AE system as shown in Fig. 1. The uniaxial compression-creep tests were conducted on the triaxial rheological testing machine with a maximum axial load of 400 kN. The AE system is composed of an AE acquisition board (PCI-2; Physical Acoustics, USA) and control software (AEwin; Physical Acoustics).

Fig. 1
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Triaxial rheological testing machine and AE system

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Specimen characteristics and experimental groups

The red sandstone specimens used in present experiments were obtained from the south of Jiangxi Province in China. After processing, the rock specimens were dried naturally under the same conditions. In the meantime, in order to keep a high requirement, the specimens with obvious cracks on the surface were removed, and then the sonic test was used to screen out the rock specimens with uniform integrity and homogeneity. Finally, we selected 13 representative specimens for the uniaxial compression test and uniaxial compression-creep test in this study. These 13 representative cylindrical specimens are shown in Fig. 2.

Fig. 2
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Processed red sandstone specimens

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Uniaxial compression tests

We conducted uniaxial compression tests on the five specimens (HR-9#, HR-10#, HR-11#, HR-12#, and HR-13#). The uniaxial compressive strength of the present red sandstone is 27–35 MPa with an average of 30.48 MPa. This average uniaxial compressive strength provided the test basis for the loading control of the creep-AE tests under loading-unloading.

Tests of uniaxial compression creep and acoustic emission characteristics

We tested four specimens (HR-5#, HR-6#, HR-7#, and HR-8#) for their creep-AE characteristics under multi-level loading and other four specimens (HR-1#, HR-2#, HR-3#, and HR-4#) for their creep-AE characteristics under multi-level loading-unloading. The test parameters of all the specimens are listed in Table 1.

Table 1 Test parameters of all the specimens
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Loading methods of creep tests

The loading methods used in the present creep tests are multi-level loading and multi-level loading-unloading. Based on the average uniaxial compressive strength of the red sandstone specimens, the creep tests were divided into 10 different stress levels, and the corresponding load of each stage is 10 kN, 15 kN, 20 kN, 25 kN, 30 kN, 35 kN, 40 kN, 45 kN, 50 kN, and 55 kN. The first and tenth stresses are approximately 16% and 85%, respectively, of the average uniaxial compressive strength of the red sandstone specimens.

Creep tests: multi-level loading

During the creep tests, the axial stress was increased incrementally from low to high. Each stage of the load was increased to the next stage after completion of the predetermined time until the final level of stress was reached. During this process, the AE monitoring was synchronized with the compression-creep test.

Creep tests: multi-level loading-unloading

Different from multi-level loading process, the stress was unloaded to zero after maintained for the creep time, and then it was reloaded to the next stress level until the final stage of the set stress was applied. During this process, the AE monitoring was again synchronized with the compression-creep test. The two loading methods used in the creep tests are shown in Fig. 3.

Fig. 3
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Loading methods of creep test: (a) multi-level loading and (b) multi-level loading-unloading

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In the figures of this manuscript, the time of the creep test is defined as Tt; the axial strain of the specimen is defined as ε1.

Comparison of acoustic emission amplitude of creep tests with two loading paths

The maximum amplitude of the AE waveform signal is referred to as the acoustic emission amplitude (AE amplitude). The relationships among AE amplitude, axial strain, and time for specimen HR-7# are shown in Fig. 4, and those for specimen HR-3# are shown in Fig. 5.

Fig. 4
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Relationships among AE amplitude, axial strain, and time of multi-level loading creep test on specimen HR-7#

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Fig. 5
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Relationships among AE amplitude, axial strain, and time of multi-level loading-unloading creep test on specimen HR-3#

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During the loading process, new internal cracks emerge and expand in the specimen after applying higher level of stress, and AE also behaves significantly. When a red sandstone specimen is subjected to creep test involving multi-level loading only, the frequency at which the AE amplitude changes is relatively high at low stress and decreases with the stress increases. When the stress reaches the rock creep threshold, both the alteration frequency and the AE amplitude increase, and the AE amplitude reaches its maximum when the specimen undergoes accelerated creep damage. The variation in AE amplitude reveals the damage evolution of the internal rock microstructure to a certain extent because the rock interior is dominated by the compaction of micro-cracks under the low stress. As the stress increases, the intrinsic cracks inside the rock expand and connect, their number decreases, and their scale increases. When the stress reaches the creep threshold, new internal cracks emerge and expand until to the macroscopic instability damage.

When a red sandstone specimen is subjected to creep test involving multi-level loading-unloading, its internal structure undergoes the compaction of the original cracks and pores and the emergence of new cracks under the low several stress levels. The AE activity is enhanced during the loading-unloading process at each stage but tends to weaken when the load is maintained. The AE amplitude decreases as the stress increases. Therefore, under low stress, the rock interior is dominated by crack compaction, and the loading process has little effect on the rock stability. Under higher stress, the internal damage evolution of the rock is dominated by crack propagation.

The variations of AE amplitude during the creep tests of the red sandstone specimen under two loading paths have similar characteristics: The AE amplitude of the rock under each stress-level creep condition exhibits the trend of “elevation → stationary → elevation.” The AE amplitude decreases as the stress increases. However, the AE amplitude variation under these two loading paths is different: The alteration frequency and AE amplitude during the creep process involving multi-level loading-unloading are both significantly higher than those under multi-level loading.

Comparison of cumulative acoustic emission parameters of creep tests with two loading paths

Cumulative acoustic emission event

The number of AE event accumulated during the statistical time is called the cumulative AE event. The relationships among the cumulative AE event, axial strain, and time for specimen HR-7# are shown in Fig. 6, and those for specimen HR-3# are shown in Fig. 7.

Fig. 6
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Relationships among cumulative AE event, axial strain, and time of multi-level loading creep test on specimen HR-7#

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Fig. 7
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Relationships among cumulative AE event, axial strain, and time of multi-level loading-unloading creep test on specimen HR-3#

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When the red sandstone specimen is subjected to creep test under only multi-level loading, the cumulative AE event increases linearly with time during the first stress-level creep. The cumulative AE event exhibits an upwardly convex trend with increasing stress, and this trend becomes more obvious with a higher stress. While the curve appears to increase linearly once the stress reaches a certain value. This shows that under low stress, the AE activity inside the rock is more severe, and many rock cracks appear to be closed. As the stress increases, the AE activity weakens, and the original cracks inside the rock expand and connect. When the stress reaches a certain threshold, many new cracks in the rock are created and the AE activity is enhanced, which results in the cumulative AE event increasing.

When the red sandstone specimen is subjected to creep test under multi-level loading-unloading, the variation of the cumulative AE event with time in the creep process is similar to that under multi-level loading only, but the value is obviously higher.

Cumulative acoustic emission ring-down count

Acoustic emission ring-down count refers to the number of oscillations of the signal produced by it crossing the threshold voltage, and the number of acoustic emission ring-down count accumulated during the statistical time is called the cumulative AE ring-down count. The relationships among cumulative AE ring-down count, axial strain, and time for specimen HR-7# are shown in Fig. 8, and those for specimen HR-3# are shown in Fig. 9.

Fig. 8
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Relationships among cumulative AE ring-down count, axial strain, and time of multi-level loading creep test on specimen HR-7#

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Fig. 9
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Relationships among cumulative AE ring-down count, axial strain, and time of multi-level loading-unloading creep test on specimen HR-3#

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When the red sandstone specimen is subjected to creep test under only multi-level loading, the cumulative AE ring-down count increases linearly with time at low stress levels. Then the cumulative AE ring-down count exhibits an upwardly convex trend with increasing stress, and the higher the stress, the more obvious that trend. When the stress reaches a certain value, the cumulative AE ring-down count appears to increase linearly with time. At the final stress level, the rock exhibits accelerated creep damage, and the cumulative AE ring-down count increases dramatically with time.

When the red sandstone specimen is subjected to creep test under multi-level loading-unloading, the variation of the cumulative AE ring-down count with time in the creep process is similar to that under multi-level loading only, while the value is obviously higher.

Damage variable for creep-AE characteristics of rock

Evolution of rock damage

Heterogeneous statistical description

Because the mesostructure of rock materials is highly random in shape, size, direction, and distribution, it is feasible to use statistical analysis to obtain a discrete description of the rock damage evolution. We choose a micro-body v in a heterogeneous rock sample, where v is deemed large enough to contain many micro-cracks and micro-pores and can therefore be regarded as a continuous medium. Under this premise, we introduce the following Weibull statistical distribution (Weibull 1951) to analyze the mechanical properties of this micro-body:

$$ \varphi (a)=\frac{m}{a_0}{\left(\frac{a}{a_0}\right)}^{m\hbox{-} 1}{e}^{\hbox{-} {\left(\frac{a}{a_0}\right)}^m} $$
(1)

where a is a mechanical parameter of the rock material (strain, elastic modulus, etc.), a0 is the mean value of that mechanical parameter, m is the morphological parameter of the Weibull distribution function and reflects the homogeneity of the rock material, and φ(a) is a statistical probability density function.

Damage evolution equation

The damage of the rock material is related to its internal micro-fracture structure. When the microstructure changes, the strength of the internal micro-body is affected; this can be expressed by the damage variable D and the probability density function of the microstructure damage in the form

$$ \frac{d D}{d\varepsilon}=\varphi \left(\varepsilon \right) $$
(2)

where ε is the material strain of the rock.

From Eqs. (1) and (2), we obtain

$$ \frac{d D}{d\varepsilon}=\varphi \left(\varepsilon \right)=\frac{m}{\varepsilon_0}\times {\left(\frac{\varepsilon }{\varepsilon_0}\right)}^{m\hbox{-} 1}\times {e}^{\hbox{-} {\left(\frac{\varepsilon }{\varepsilon_0}\right)}^m} $$
(3)

And D is obtained by integrating Eq. (3), namely

$$ D={\int}_0^{\varepsilon}\varphi (x) dx={\int}_0^{\varepsilon}\left(\frac{m}{\varepsilon_0}{\left(\frac{x}{\varepsilon_0}\right)}^{m\hbox{-} 1}{e}^{\hbox{-} {\left(\frac{x}{\varepsilon_0}\right)}^m}\right) dx=1\hbox{-} {e}^{-{\left(\frac{\varepsilon }{\varepsilon_0}\right)}^m} $$
(4)

Creep acoustic emission damage-variable model

In the present study, the damage variable is defined by the damage area, and the damage of the rock material is caused mainly by a reduction in the unit effective area. When establishing a creep AE damage-variable model, the following assumptions must be made.

  1. 1)

    There are a certain number of micro-defective micro-elements in any cross section of the specimen, and there is a certain difference in the intensity of each micro-element.

  2. 2)

    The intensity characteristics of the micro-element satisfy a Weibull statistical distribution.

  3. 3)

    Under the action of an external load, the rock specimen is evenly stressed and the strain is evenly distributed.

During the creep process of rock materials, the strain increases with time. The microstructure damages when its strain exceeds a certain value. Therefore, the damage area of the rock cross section can be expressed as (Tang and Xu 1990; Chen et al. 1997)

$$ s={s}_0{\int}_0^{\varepsilon}\varphi (x) dx $$
(5)

where s0 is the cross-sectional area of the specimen in the initial (undamaged) state, φ(x) is the intensity probability density function of the micro-body, and ε is the strain of the rock material.

According to Eq. (5), the damage variable D of the rock can be expressed as

$$ D=s/{s}_0={\int}_0^{\varepsilon}\varphi (x) dx $$
(6)

Assuming that the cumulative AEs (AE ring-down count, AE event, etc.) generated by the damaged micro-element in unit cross-sectional area is n, the cumulative number ΔN of AEs generated within area ΔS can be expressed as

$$ \Delta N=\mathrm{n}\times \Delta S $$
(7)

The cumulative AE Nm arises when the cross-sectional area S0 of the rock specimen is completely destroyed, where upon Eq. (7) can be rewritten as

$$ \Delta N=\frac{N_m}{S_0}\times \Delta S $$
(8)

In the present study, we use strain as the variable of the statistical distribution. When the increment of axial strain of the specimen is Δε during the creep process, the increment of the corresponding damage area ΔS of the micro-element can be expressed as

$$ \Delta S={S}_0\times \varphi \left(\varepsilon \right)\times \Delta \varepsilon $$
(9)

Substituting Eq. (9) into Eq. (8), we obtain

$$ \Delta N/{N}_{\mathrm{m}}=\phi \left(\varepsilon \right)\cdot \Delta \varepsilon $$
(10)

During the creep deformation process, when the strain of the rock specimen increases to ε, the cumulative AE produced in the rock is

$$ \Delta N={N}_m\varphi \left(\varepsilon \right)\varepsilon ={N}_m{\int}_0^{\varepsilon}\varphi (x) dx $$
(11)

From Eqs. (6), (10), and (11), we obtain the following relationship between D and the number of AEs from the rock during the creep process:

$$ D=\Delta N/{N}_m $$
(12)

Because the AEs from rock have a certain relationship with the damage variable of the rock, the variation of AE parameters can be used to evaluate the damage evolution of the rock.

ΔNi (i = 1,2,….n) represents the cumulative AE generated during the creep process under stress level i, and Nm represents the sum of the cumulative AE during the creep process under each stress level.

Equations (5), (6), (7), (8), (9), (10), (11), and (12) are derived by Tang and Xu (1990) and Chen et al. (1997).

Verification and analysis of creep acoustic emission damage-variable model

We analyzed the test data of specimen HR-7# to study the applicability of the creep AE damage-variable model under the creep conditions of multi-level loading, which are presented in Tables 2, 3, and 4 and Fig. 10. And the applicability of the creep AE damage-variable model under the creep conditions of multi-level loading-unloading was also studied by using specimen HR-3#; those results are presented in Tables 5, 6, and 7 and Fig. 11.

Table 2 Results of theoretical model (Eq. (4)) characterizing the damage variable for specimen HR-7#
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Table 3 Results of cumulative AE event characterizing the damage variable for specimen HR-7#
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Table 4 Results of cumulative AE ring-down count characterizing the damage variable for specimen HR-7#
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Fig. 10
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Comparison of creep AE damage variables of creep test involving multi-level loading

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Table 5 Results of theoretical model (Eq. (4)) characterizing the damage variable for specimen HR-3#
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Table 6 Results of cumulative AE event characterizing the damage variable for specimen HR-3#
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Table 7 Results of cumulative AE ring-down count characterizing the damage variable for specimen HR-3#
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Fig. 11
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Comparison of creep AE damage variables of creep test involving multi-level loading-unloading

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According to analysis of the specimen subjected to multi-level loading, the results of the AE parameters characterizing the rock creep damage variable are consistent with those calculated by the theoretical model (Eq. (4)). Upon analyzing the specimen subjected to multi-level loading-unloading, although the rock creep damage variable characterized by AE parameters is different from that calculated by the theoretical model (Eq. (4)), the degree of disagreement is relatively small, and variation trend of the rock creep damage variable characterized by the theoretical model (Eq. (4)) is similar to that characterized by the creep AE model. Therefore, the creep AE damage-variable model could be used to characterize the evolution of rock damage during the creep process involving multi-level loading-unloading.

When the red sandstone specimen is subjected to creep test involving multi-level loading only, the damage variable exhibits a U-shaped distribution with stress. The damage variable is larger at low stress, smaller at medium stress, and larger once again at high stress. At low stress, the cracks and pores inside the rock are closed; thus, the damage variable is larger, but the damage in this stage has a limited effect on the stability of the rock. With increased stress, the major internal damage of rock is the expansion of the original crack. When the stress reaches the creep threshold, the damage evolution of the rock is characterized by the expansion of new cracks leading to macroscopic instability failure; thus, the damage variable reaches its maximum.

When the red sandstone specimen is subjected to creep test involving multi-level loading-unloading, the variation of the damage variable is similar to that observed involving only multi-level loading.

Conclusions

Creep-AE tests on red sandstone specimens under only multi-level loading and multi-level loading-unloading were carried out, and some main conclusions are as follows:

  1. (1)

    During the creep process involving multi-level loading-unloading, the temporal variation of AE amplitude is similar to that observed during the creep process involving only multi-level loading, and both exhibit a trend of “sudden increase ➝ stationary fluctuation ➝ sudden increase.” However, the frequency of AE is significantly higher during the creep process involving multi-level loading-unloading.

  2. (2)

    Under the previous stress levels, the temporal variations of the cumulative AE event and cumulative AE ring-down count during the creep process involving only multi-level loading are similar to those during the creep process involving multi-level loading-unloading.

  3. (3)

    The creep AE damage-variable model could be used to characterize the damage evolution of the internal rock micro-crack structure during the creep process involving either only multi-level loading or multi-level loading-unloading.