The Influence of Cyclic Load Amplitude on Mechanical Response and Acoustic Emission Characteristics of Granite

Abstract

Cyclic loading and unloading tests with varying stress amplitudes were carried out to study the evolution trends of the elastic modulus, plastic strain, dissipated energy density, and acoustic emission events for granite under cyclic loading with accidental extreme stress. The results were as follows: (1) The loading deformation modulus before and after extreme stress is different. In addition, at extreme stress levels, the loading deformation modulus of granite specimens decreases by approximately 5~13%, but the unloading deformation modulus does not change significantly. (2) When extreme stress causes rock damage, most of the plastic deformation potential of the rock at this stress level is released in advance. The stress constantly varies in the subsequent low-amplitude cycle, and the plastic strain caused by the extreme stress is partly recovered. (3) As the extreme stress increases, the cumulative dissipated energy density of granite increases significantly. Compared with the control group without a stress extremum, the cumulative dissipated energy density of samples in two groups with stress extrema of 20 kN and 40 kN increased by 48% and 153%, respectively. (4) A significant acoustic emission event occurs only when the rock is subjected to a load exceeding the maximum historical stress for the first time, and the acoustic emission intensity is positively correlated with the difference between this stress and the historical maximum value. (5) Extreme stress values below the crack damage threshold reduce the crack growth potential of the rock in advance, and extreme stress above the crack damage threshold aggravates rock damage.

1. Introduction

After a long and complex geological process, rocks contain many scattered primary discontinuities. Disturbances caused by external forces cause these primary discontinuities to continuously develop, gradually expand to larger ranges, and finally penetrate, leading to rock instability and failure. In practical engineering, many rock masses are subject to cyclic stress disturbances [1], and the corresponding mechanical properties of rock greatly differ from those under static loads [2,3]. Under this condition, to obtain a more accurate and in-depth understanding of the mechanical properties of rocks under cyclic loading, many scholars have studied their mechanical properties [4,5,6], deformation characteristics [7,8,9,10], and damage evolution [11,12,13] under cyclic loading and unloading conditions. The results demonstrated that the material properties of rocks are related to the maximum stress and the stress amplitude, loading waveform, and frequency [14,15,16].

To date, scholars have mainly performed cyclic loading and unloading at constant stress amplitudes and explored the mechanical response and fatigue characteristics of rock under cyclic loading. However, in practical engineering, the forms of cyclic load may be more complex and diverse. For example, the future sustainable energy system needs to introduce integrated energy storage technology. Compressed air energy storage (CAES) system is an electric energy storage system that can realize large capacity and long-term electric energy storage [17], which can solve the intermittence and instability of current clean energy. CAES systems typically use underground caverns (salt or hard rock caves) as gas storage facilities. When the electricity consumption is low, the excess electricity is used to drive the compressor to generate high-pressure air and inject it into the underground cavern. When the electricity consumption is high, the compressed air obtains heat energy through the combustion chamber and then enters the turbine to generate electricity. Therefore, the surrounding rock of the gas reservoir bears cyclic load, and different power requirements will cause the load imposed on the surrounding rock to change constantly [18]. At present, the global energy pattern is shifting from non-renewable fossil energy to renewable energy (e.g., solar energy, hydropower and wind energy), among which hydropower was rated as the best renewable energy in 2019 [19]. In the past two decades, pumped storage power stations (PSPS) in China have developed rapidly [20]. According to the prediction of the National Energy Administration of China, the total production scale of pumped storage power plants in China may reach approximately 120 million kilowatts by 2030. At a pumped storage power station, the process of water flow through the diversion tunnel imposes cyclic loads on the neighbouring rock mass [21,22]. In certain specific situations, such as the use of a water hammer in a diversion tunnel or blasting for tunnel excavation [23,24,25], the rock mass is subjected to instantaneous loading and unloading, with a high peak load and high loading rate; subsequently, stress waves with low amplitudes are generated. In these engineering cases, the cyclic loads experienced by the rock have similar characteristics; the peak value of the cyclic load is not fixed and is always less than the peak strength of the rock.

To sum up, the widely built integrated energy storage technologies (such as CAES and PSPS) make the form of cyclic load on the surrounding rock more complex. The inhomogeneity of the form and amplitude of these cyclic loads has an important impact on the fatigue properties and damage evolution of rocks. Therefore, if we want to study the fatigue characteristics of rock more comprehensively and deeply, we need to carry out more complex and diverse forms of cyclic load experiments. In this study, in order to simulate the instantaneous loading and unloading of surrounding rock caused by water hammer and tunnel excavation blasting, we improved the conventional cyclic load experiment. By adding the form of stress extremum to the low amplitude cyclic load, the surrounding rock under cyclic load in the actual engineering environment is more truly simulated, and based on this, the deformation response, energy characteristics and acoustic emission characteristics of rock under this stress condition are studied.

2. Experimental Materials and Procedures

2.1. Experimental Samples

The granite used in the experiment was obtained in Shaanxi. According to the International Society for Rock Mechanics and Rock Engineering (ISRM), the collected rock blocks were processed into $\varnothing$ 50 mm × 100 mm cylindrical standard specimens, and specimens with uniform texture, no apparent microcracks, and similar densities and wave velocities were selected for the experiment.

2.2. Experimental Scheme

The experiment was carried out using the MTS815 Flex Test GT (MTS Industrial Systems Co., Ltd., Shanghai, China) electrohydraulic servo rock mechanics experimental system at North China University of Water Resources and Electric Power in China, as shown in Figure 1, which can measure the axial and radial deformation of a specimen in real time, record the load, stress, displacement and strain values, and simultaneously plot the load-displacement and stress-strain curves. Acoustic emission data were acquired by using the PCI-II acoustic emission (AE) monitoring system (Physical Acoustics Company Beijing Office, Beijing, China) from the American Physical Acoustic Corporation (PAC). The MTS815 Flex Test GT electrohydraulic servo rock mechanics experimental system used in the experiment is regularly maintained and calibrated by a qualified professional organization to ensure that it is in good working condition. Prior to the start of the experiment, the AE monitoring system was tested in a broken-lead experiment, and the monitoring results were accurate.

In the experiment, axial force control was adopted, and the loading waveform was a cosine wave with a period of 2 s. The uniaxial compressive strength of the granite sample used in the experiment was measured to be approximately 116~143 MPa through uniaxial compression experiments. To study the mechanical response and acoustic emission characteristics of rocks at different stress levels, the axial load was applied in a step-by-step manner based on the upper and lower limits of the load, and the axial force level in each stage was increased by 40 kN. To simulate the occasional large instantaneous loads that are applied in actual engineering, the stress extremes under three working conditions were set to 40 kN, 20 kN, and 10 kN. Two to three sets of experiments were conducted under each working condition, and the loading paths are shown in

Table 1. Experimental stress path (kN).

Stage	Low-Amplitude Cycle			High-Amplitude Cycle
	Con1	Con2	Con3	Con1	Con2
1	0–10	0–10	0–10	0–40	0–20
2	40–50	40–50	40–50	40–80	40–60
3	80–90	80–90	80–90	80–120	80–100
4	120–130	120–130	120–130	120–160	120–140
5	160–170	160–170	160–170	160–200	160–180

and Figure 2.

3. Results and Discussion

3.1. Deformation Features

Due to space limitations, the experimental data for only one typical rock sample for each of the three loading conditions explored in the experiment are analysed in detail. A typical stress-strain curve is shown in Figure 3.

The microcracks inside the rock are repeatedly closed, opened, developed, and penetrated under the actions of cyclic loading and unloading, resulting in irreversible nonlinear deformation of the rock. Macroscopically, the stress-strain curve forms a hysteresis loop, as shown in Figure 4.

In the plot of the stress-axial strain curve, if the slope of line OP between the starting point O and the loading end point P of a cycle is defined as the loading deformation modulus $E_1$, then it includes the axial elastic strain and plastic strain. Similarly, the slope of the line PN between the loading endpoint P and the unloading endpoint N of a cycle is defined as the unloading deformation modulus $E_2$, which includes only the axial elastic strain $E_1$. These values can be calculated by using the following formulas:

$$E_1=\frac{\sigma }{{\epsilon }_{1e}+{\epsilon }_{1p}}=\frac{PM}{OM}$$

(1)

$$E_2=\frac{\sigma }{{\epsilon }_{1e}}=\frac{PM}{NM}$$

(2)

where $E_1$ is the loading deformation modulus, $E_2$ is the unloading deformation modulus, ${\epsilon }_{1e}$ is the axial elastic strain, and ${\epsilon }_{1p}$ is the axial plastic strain.

Figure 5 shows the deformation modulus of the granite specimens in Case 1. Notably, point O represents the starting point of each stress stage, and points A and B are the two stress extremes in the stage. During the rock loading-unloading cycle, the loading deformation modulus $E_1$ is slightly smaller than the unloading deformation modulus $E_2$ due to the influence of plastic deformation. As the cyclic load gradient increases, both the loading deformation modulus $E_1$ and the unloading deformation modulus $E_2$ of the granite specimens increase continuously in the form of a gradual slope. This trend is similar to the results obtained in cyclic loading tests of general rock [26].

An interesting phenomenon is also observed in Figure 5: under extreme stress, the loading deformation modulus $E_1$ of the granite sample decreased by approximately 5–13% compared to that under a low-amplitude cyclic load at the same stress level, and the unloaded deformation modulus $E_2$ did not change significantly. This phenomenon occurred because $E_2$ is only related to elastic strain, and $E_1$ is affected not only by elastic strain but also by plastic strain. Under extreme stress, the microcracks expand, intersect, or even penetrate through the rock sample, resulting in a large plastic strain, which leads to a significant decrease in $E_1$ under extreme stress.

In addition, we found that during the low-amplitude cycle outside the extreme stress point, the loading deformation modulus $E_1$ notably varied. Compared with that near point B, the change in amplitude of $E_1$ before and after point A is larger, possibly because when the rock sample was subjected to a load exceeding the maximum historical stress for the first time, the internal structure of the rock underwent a significant variation. At the same stress level, the adjustment potential of the internal structure of the rock was largely reduced at point A; thus, the $E_1$ changes before and after point B are small. This phenomenon provides us with clues to understand the rock deformation mechanism.

Figure 6 shows the plastic strain generated by a single loading and unloading cycle with different stages and at a low amplitude. The results of each stage are divided into three clusters of curves according to the progress of the experiment, corresponding to the OA segment (without extreme stress), AB segment (experiencing one extreme stress), and BO segment (experiencing two extreme stresses).

First, it should be noted that the magnitude of macroscopic deformation increases when cracks in the rock develop; thus, macroscopic deformation can be regarded as an intuitive reflection of rock damage. Additionally, the opening of some cracks and the closing of other cracks occur at the same time. Since axial deformation is not only a measure of the development of internal cracks but also a measure of the closure of internal cracks, annular deformation can better reflect the development process of rock damage than can the axial deformation.

The figure illustrates that the extreme stress points influence the deformation characteristics of rocks, and this effect is manifested differently in terms of axial strain and circumferential strain. After the rock is subjected to the first extreme stress (point A), the compressive effect of subsequent cyclic loads in the axial direction of the rock is significantly reduced. When the load level is low, this phenomenon even manifests as negative axial plastic strains in the AB and BO sections, reflecting the internal adjustment of the rock during subsequent cyclic loads, restoring some axial compression caused by extreme stress.

In the first three stages with comparatively lower stress levels, in the low-amplitude cyclic loading (segment AB and segment BO) after the extreme stress, the specimen’s circumferential plastic strain even exhibits positive values, which contradicts the conventional test results that uniaxial compression specimens are stretched in the circumferential direction. One possible explanation for this phenomenon is that extreme stress causes the strong expansion of internal cracks in the rock, leading to excessive plastic deformation of the sample in the circumferential direction. Subsequent low-amplitude cyclic loads have certain effects on the internal cracks in the rock, partially restoring the excessive tensile deformation caused by extreme stress. The above characteristics are transformed in the last cycle—the hoop plastic strain of the sample becomes negative and the absolute value increases after the stress extreme; that is, the extreme stress value significantly increases the damage inside the rock, resulting in an increase in the Poisson ratio of the sample.

Overall, the influence of extreme stresses (points A and B at each stress level) on rock deformation characteristics can be summarized based on two phenomena. First, the extreme stress release associated with the plastic deformation potential of the rock occurs. Second, when the stress is large, the extreme stress causes damage to the rock. These two effects exist at the same time during the loading process, but the corresponding weights change with the level of the axial load. When the axial load is small, the extreme stress does not cause significant damage inside the rock. Therefore, the former effect plays a dominant role, and the weight of the latter increases gradually with increasing axial load.

3.2. Energy Characteristics

Rocks are deformed under the actions of unequal amplitude loading and unloading cycles. According to the first law of thermodynamics, it is assumed that there is no heat exchange between these physical processes and the outside world; that is, a closed system is assumed (the total energy of the rock materials in an isolated system remains unchanged). If the plastic properties of the sample during the deformation and failure process are not considered, the energy input into the sample by the work carried out by the external force in each cycle is $U_i$. The releasable elastic strain energy $U_{ei}$ is stored inside the rock material during each cycle of loading, as shown in Figure 7, and this variable is numerically equal to the area enclosed by the stress-strain curve during unloading. The difference between $U_i$ and $U_{ei}$ is the energy lost $U_{di}$ during damage or plastic deformation inside the rock material per cycle [27]. The total energy, elastic energy, and dissipated energy described in this paper all refer to the energy of the rock unit, that is, the energy density. The formula for $U_{di}$ is as follows:

$$U_{di}=U_i-U_{ei}={\displaystyle {\int }_0^{\epsilon +}{\sigma }_i\mathrm{d}{\epsilon }_i}-{\displaystyle {\int }_{\epsilon +}^{\epsilon -}{\sigma }_i\mathrm{d}{\epsilon }_i}$$

(3)

where $\epsilon +$ is the axial strain corresponding to the loading end point of a certain cycle, $\epsilon -$ is the axial strain corresponding to the endpoint of unloading in a certain cycle, and $i$ is the number of cycles, where $i=1,2,3,\cdots ,\mathrm{n}$.

According to the uniaxial cyclic loading and unloading stress-strain curve, the variations in the elastic energy density and dissipated energy density of granite can be obtained as shown in Figure 8. The dissipated energy density of the rock in the first few cycles of each stress level (the OA segment) is highest because when the rock is subjected to a load exceeding the maximum historical stress for the first time, the local stress leads to changes in the microcracks in the rock, causing energy dissipation. In addition, although the amplitude of the cyclic loads applied in the OA section is 10 kN, the magnitude of the dissipation work and the absolute value of the axial force are significantly correlated, suggesting that the stress level of the cyclic load significantly affects the evolution and expansion of the microcracks in the rock.

Overall, as the extreme stress value in the experiment increased, the cumulative dissipated energy density under the three sets of loading conditions significantly increased. At the end of the 5th stress level, the cumulative dissipated energy densities of the three groups of specimens with extreme stress values of 10 kN, 20 kN, and 40 kN were 11.1 kJ/mm³, 16.4 kJ/mm³, and 28.1 kJ/mm³, respectively. The laws of thermodynamics dictate that energy dissipation is an essential property of rock deformation and failure, and it reflects a process of continuous development, weakening, and, finally, loss of microdefects inside the rock. Energy dissipation is directly related to damage, and the experimental results show that the extreme stress value significantly accelerates the accumulation of damage inside the rock. In addition, by comparing the results for the three sets of loading conditions, after the rock sample is subjected to extreme stress, the dissipated energy density generated by the subsequent cyclic loading (section AB and section BC) is significantly reduced, possibly due to the drastic internal effects of the extreme stress on the rock. Additionally, the crack propagation potential of the rock under subsequent cyclic loads is released in advance.

3.3. Characteristics of Acoustic Emission Events

When rock is subjected to stress, microscopic damage and cracks occur inside, and these cracks continue to expand as the damage continues, releasing energy and resulting in acoustic emission events. The number of acoustic emission events and the absolute energy can be used to identify and track damage and failure processes in rocks. The absolute energy of an acoustic emission event is obtained by the square integration of the signal voltage, and the corresponding formula is as follows:

$$E={\displaystyle {\int }_{t_1}^{t_2}{V}^2}(t)dt$$

(4)

where $E$ is the absolute energy, $V(t)$ is the variation in the voltage of the acoustic emission signal with time, and $t_1$ and $t_2$ are the start time and end time of the acoustic emission signal, respectively.

Figure 9 shows the number of acoustic emission events under three sets of working conditions with different extreme stress levels. Figure 10 shows the corresponding acoustic emission event energy. Since the acoustic emission energy varies greatly and spans multiple orders of magnitude, for clarity, the distribution range of the absolute energy is provided in logarithmic form.

In the rock stress process, the number of acoustic emission events obeys a certain law. The experimental results show that significant acoustic emission events only occur at the initial loading, unloading, and extreme stress points along each stress gradient. This phenomenon is consistent with the Kaiser effect of rock [28]; that is, when the load does not exceed the maximum load in the previous loading cycle, less acoustic emission occurs because the cracks and damage generated inside the rock expanded during the previous loading process and now require higher stress levels to continue to expand, resulting in new acoustic emission events. Interestingly, although the extreme stresses in the experiment are smaller than the initial loading and unloading amplitudes of the next-level stress gradient, the extremum stress still seems to have an effect on the acoustic emission events related to the initial loading and unloading in the next stage. For example, for the three sets of working conditions, the ratio of the number of acoustic emission events at the initial loading and unloading points of the next stage to the extreme stress level in the previous stage is 0.49–1.38, 1.10–3.13, and 12.00–41.14, respectively. These values demonstrate that in one loading cycle, the acoustic emission intensity is positively correlated with the extent to which the stress exceeds the maximum historical stress. This conclusion can be regarded as a supplement to the Kaiser effect to a certain extent.

At the same time, the extreme stress significantly affects the rock’s damage evolution pattern. When no extreme stress is considered (Working Condition 3), high-energy acoustic emission events often occur during the initial loading and unloading of each stress level. In the cases in which extreme stress is considered (working condition 1 and working condition 2), high-energy acoustic emission events occur not only during the initial loading and unloading of each stress level but also at the extreme stress level. Since the energy of an acoustic emission event is closely related to the extent of microcrack propagation, the results indicate that extreme stresses aggravate rock damage. In particular, in the first three stress levels under working condition 1, there are almost no acoustic emission events in the low-amplitude cyclic loading period after the extreme stress is reached, indicating that the extreme stress causes the rock to undergo dramatic internal changes and that the internal microdamage and crack propagation potential of the rock are reduced in advance. At the last two stress levels under working condition 1, the extreme stress seems to aggravate the acoustic emission associated with the subsequent low-amplitude cyclic load, suggesting that the impact of the extreme stress on the rock is different from that at lower stress levels. Extreme stress exacerbates rock damage. Since the crack damage threshold is the starting point of unstable crack propagation in the rock, it is reasonable to assume that the boundary point of these two diametrically opposite effects is related to the crack damage threshold. Relevant studies have shown that the crack damage threshold is only related to the characteristics of the rock itself and has nothing to do with the loading conditions [29]. Therefore, the crack damage threshold of granite is 70.7 MPa according to the turning point of the volume strain in stress-strain curves for the same group of rock samples obtained based on uniaxial compression tests. The threshold value is at the fourth stress level in the cyclic loading experiment, which is consistent with the above assumption.

4. Conclusions

(1)

The loading deformation modulus is different before and after the extreme stress level is reached. Notably, at the stress extreme point, the loading deformation modulus of the granite sample decreased by approximately 5–13% compared to that under a low-amplitude cyclic load at the same stress level, and the unloaded deformation modulus did not change significantly.

(2)

When rock is subjected to cyclic loads with varying amplitudes, accidental extreme stress causes rock damage and releases most of the rock’s plastic deformation potential in advance of subsequent continued loading. Changes continuously occur within the rock due to stress, and some of the plastic strain loss at the extreme stress level is recovered.

(3)

The energy dissipation characteristics of rock and the magnitude of an extreme stress are correlated, and the cumulative dissipated energy density increases significantly as the extreme stress increases. At the end of the cyclic load experiment, the cumulative dissipated energy density of samples with extreme stress values of 20 kN and 40 kN increased by 48% and 153%, respectively, compared with those for the control group without extreme stress.

(4)

An acoustic emission event occurs only when the rock is subjected to a load exceeding the maximum historical stress for the first time, and the acoustic emission intensity is positively correlated with the difference between the stress and the historical maximum value. Extreme stress below the crack damage threshold reduces the crack growth potential of the rock in advance, and extreme stress above the crack damage threshold aggravates rock damage.