Damage and Deterioration Characteristics of Sandstone Under Multi-Stage Equal-Amplitude Intermittent Cyclic Loading and Unloading

Abstract

The surrounding rocks of roadways are typically subjected to cyclic loading–unloading stress states in underground engineering. In addition, cyclic loads are discontinuous under real working conditions, usually while loading rock mass in a cycle–intermission–cycle manner. Based on the XTDIC 3D (XTOP Three-dimensional Digital Image Correlation) full-field strain measurement system and AE (Acoustic Emission) system, the work performed uniaxial cyclic loading–unloading tests with constant-pressure durations of 0, 0.5, 2, and 6 h. The purpose was to investigate the damage degradation mechanism of sandstone under multi-stage equal-amplitude intermittent cyclic loading and unloading. The results are as follows. (1) As the constant-pressure duration increased, the uniaxial compressive strength of sandstone samples decreased, along with a decline in elastic modulus and a deterioration in stiffness and deformation recovery capacity. (2) The evolution of deformation localization zones became more intense in sandstone samples during cyclic loading and unloading with the increased constant-pressure duration. The maximum principal strain field became more active at failure. Sandstone samples exhibited shear failure accompanied by spalling failure and an increased failure degree. (3) As the constant-pressure duration increased, the damage variable of sandstone samples increased, indicating that the constant-pressure stage promoted the damage degradation of sandstone samples. The above results reveal the damage degradation mechanism of sandstone under multi-stage equal-amplitude intermittent cyclic loading and unloading, which is of significant importance for maintaining the safety of underground engineering.

1. Introduction

The instability of rock mass caused by excavation is a key disaster-inducing factor leading to safety production accidents in coal resource development [1,2,3,4]. The deformation and failure processes of rocks become highly complex under geological structures or engineering disturbances, presenting a dynamic evolution of damage [5,6,7]. When the intensity of construction gradually increases, cyclic loading and unloading disturbances (e.g., construction blasting, mechanical vibrations, and working face mining) disrupt the original natural equilibrium state of surrounding rocks during the construction of underground chambers and roadways. These disturbances lead to frequent occurrences of surrounding rock deformation disasters [8]. The deformation response of rocks changes under cyclic loading due to the complexity of rock microstructures. Cyclic loading can induce fatigue damage and deformation in rocks, which leads to structural damage and fracture [9].

Energy evolution law and damage degradation mechanisms of surrounding rocks under cyclic loading are crucial for studying the stability of rock mass under repeated engineering disturbances [10]. Song et al. [11] conducted fatigue mechanical tests and microscopic scanning electron microscopy (SEM) experiments on hard-bedded sandstone to address this issue. Bedding effects on fatigue damage are revealed based on macro- and micro-scale characteristics. Yin et al. [12] developed a self-designed pressurized water immersion test device for coal samples. Uniaxial cyclic loading–unloading tests are performed on both dry and pressurized water-immersed coal samples by integrating full-field strain measurement systems, AE monitoring, X-ray diffraction (XRD), and SEM. The strength, deformation and failure characteristics, AE signals, and energy evolution of coal samples are investigated to uncover the degradation mechanism of mechanical properties under pressurized water immersion and cyclic loading–unloading process. Peng et al. [13] utilized a triaxial rock testing system for cyclic loading–unloading experiments under different confining pressures. The damage evolution process of coal rocks under confining pressure is analyzed to define a damage variable based on energy analysis along with its evolution law. Meng et al. [14] performed triaxial cyclic loading–unloading tests on rock samples under six different confining pressures, revealing the evolution and distribution patterns of elastic and dissipative energy before and after the peak stress of loaded rock samples. Ding et al. [15] employed an RMT-150B rock mechanics testing system for uniaxial compression and cyclic loading–unloading tests on natural coal samples from the 3^# coal seam of a mine in Shaanxi Province under different stress levels. They studied the damage deformation and energy evolution characteristics of coal samples under cyclic loading–unloading conditions by comparing strength and deformation parameters in various loading modes.

Researchers have studied rock mechanical properties and deformation–failure characteristics under cyclic loading and unloading. Zhao et al. [16] conducted cyclic loading–unloading tests on different coal-rock monoliths and composites, along with PFC numerical simulations under identical conditions. They explored the mesoscopic crack propagation and energy evolution processes of coal-rock mass with different lithology during cyclic loading–unloading deformation. Chen et al. [17] investigated the influence of alternating cyclic loading–unloading tests on the creep behavior of sandstone, discussing the mechanism of cyclic stress variations in rock stability. Su et al. [18] performed triaxial cyclic loading–unloading tests on coal samples using an RMT-150B rock mechanics testing machine to analyze the deformation and strength characteristics of coal samples under triaxial cyclic loading and unloading. Chen et al. [19] conducted multi-stage constant-amplitude cyclic loading–unloading triaxial tests on sandstone samples under different confining pressures. Permeability and acoustic emission (AE) signals are monitored in real time during the experiments to analyze the effects of confining pressure and cyclic loading on characteristic stress, permeability, and AE parameters (b and RA/AF ratio). Xu et al. [20] performed triaxial cyclic loading–unloading creep tests to separate viscoelastic and viscoplastic strains during creep deformation. Corresponding viscoelastic and viscoplastic creep models are established, with methods proposed for determining model parameters. Gao et al. [21] used an MTS815 testing system to monitor the deformation behavior of rocks under cyclic loading at different confining pressures and loading rates. Mechanical and energy performance is analyzed based on experimental data. Yuan et al. [22] conducted graded cyclic loading–unloading tests to analyze stress–strain hysteresis curves, deformation parameters, and macroscopic fracture characteristics during the cyclic process. In addition, they revealed the deformation and failure mechanisms of water-saturated sandstone under graded cyclic loading and unloading.

The aforementioned research focuses on the impact of continuous cyclic loading on the mechanical properties and damage degradation mechanisms of rocks, which is of great significance for studying the degradation characteristics and mechanisms of coal and rocks under cyclic loading. However, there are few studies considering intermittent cyclic loading. Influenced by roadway excavation, coalface mining, and overlying strata movement, cyclic loading under actual working conditions is discontinuous. Rock mass is loaded in a cyclic–intermittent–cyclic manner. Therefore, the work considered the non-continuous nature of cyclic loading under real working conditions. Uniaxial multi-stage equal-amplitude intermittent cyclic loading–unloading tests on sandstone were performed by combining the XTDIC full-field strain measurement system and the AE system. The strength, deformation and failure characteristics, AE signals, and energy evolution of sandstone were investigated to reveal the damage degradation mechanisms of sandstone under the cyclic loading–unloading process and constant stress. This research will provide a theoretical basis and data reference for the differentiated support design in underground mines, the optimization of mining and excavation rhythms, and the early warning of roof stability based on time effect.

2. Test Materials, Equipment, and Methods

2.1. Test Materials

The rock used in this experiment is gray–white sandstone, a typical rock type commonly found in the coal seam roof strata of coal mines in Northwest China. It was collected from a mine in Ordos City, Inner Mongolia Autonomous Region. As shown in Figure 1, X-ray diffraction analysis indicates that the sandstone is composed of cristobalite, kaolinite, and microcline. All samples were extracted from the same large sandstone block to minimize the unavoidable inconsistencies in initial characteristics among individual samples. The rocks had an intact structure with no apparent joints or fractures. Following the testing standards of the International Society for Rock Mechanics (ISRM), cylindrical rock samples with a diameter of 50 mm were first drilled from the large sandstone block using a core drilling machine. A rock saw was then used to cut the samples into standard sandstone ones with a height of 100 mm. Finally, a grinding machine was employed to polish both ends of the samples, ensuring that the flatness deviation of end surfaces was less than 0.05 mm and the diameter error was less than 0.2 mm.

Five randomly selected samples were subjected to conventional uniaxial compression tests to determine the average uniaxial compressive strength (UCS) of this batch of sandstone samples. The loading rate was set at 0.5 kN/s, and the measured average UCS was 30.573 MPa. From the remaining sandstone samples, 12 samples with the most consistent parameters were selected and divided into groups A, B, C, and D.

Table 1. Pre-test sample parameters.

Sample Number	Loading Method	Diameter/cm	Height/cm	Mass/g	Density g/cm3	Uniaxial Compressive Strength/MPa
DZ-1	Convention Uniaxial Compression	4.86	10.12	472.6	2.5173	30.373
DZ-2		4.84	10.14	472.3	2.5316	30.473
DZ-3		4.82	10.08	472.6	2.5695	29.973
DZ-4		4.84	10.08	472.8	2.5493	31.173
DZ-5		4.84	10.12	472.2	2.5571	30.473

and

Table 2. Parameters of positive test samples.

Sample Number	Loading Method	Diameter/cm	Height/cm	Mass/g	Density g/cm3	Constant Pressure Duration/h
A-1	Multi-stage equal amplitude intermittent cyclic loading and unloading	4.82	10.2	472.4	2.5382	0
A-2		4.84	10.12	472.8	2.5393
A-3		4.84	10.08	472.3	2.5467
B-1		4.82	10.12	470.5	2.5480	0.5
B-2		4.82	10.22	469.6	2.5182
B-3		4.82	10.36	474.3	2.5091
C-1		4.82	10.1	471.5	2.5584	2
C-2		4.86	10.18	472.5	2.5020
C-3		4.8	10.12	470.9	2.5714
D-1		4.82	10.04	471.2	2.5721	6
D-2		4.9	10.04	471.9	2.4925
D-3		4.86	10.26	473.2	2.4862

list the parameters of each group.

2.2. Test Scheme and Setup

The main control and monitoring systems for this test included the loading system, XTDIC 3D full-field strain measurement system, and the MISTRAS series PCI-2 AE system. The loading system consisted of a Shimadzu AG-X250 electronic universal testing machine with a maximum load capacity of 250 kN. The XTDIC 3D full-field strain measurement system mainly comprised CCD cameras, an image acquisition card, a monitor, and a computer. The MISTRAS series PCI-2 AE monitoring system used R3α AE sensors with a frequency response range of 20–100 kHz. The main amplifier gain was set to 40 dB during the test, with a threshold of 45 dB, a floating threshold of 6 dB, and a sampling frequency of 1 MHz. Medical vaseline was used as couplant and evenly applied to the contact area between the AE probe and the sample. Transparent tapes were used to secure the AE probe to the backside of the artificial speckle field. Figure 2 presents the test system.

Groups A–D were designed under intermittent cyclic loading and unloading, with each group comprising three parallel samples. Based on the average uniaxial compressive strength of sandstone, the stress gradients for cyclic loading and unloading in groups A–D were set at 10, 30, 50, and 70%, respectively. Specifically, the cyclic loading and unloading stress gradients were defined as 10–30%, 30–50%, and 50–70% of σₘ (the maximum average compressive strength of sandstone), with a loading rate of 0.5 kN/s. Each cycle consisted of loading and unloading, and 20 cycles were performed for each group. Multi-stage constant-amplitude intermittent cyclic loading and unloading tests were conducted with constant-pressure durations of 0, 0.5, 2, and 6 h, respectively, at the end of the previous cyclic loading–unloading stage. This test design takes into account the actual mining conditions in the mine: 0 h represents extremely frequent disturbances with almost no stress relaxation time; 0.5 h simulates short operation intervals (such as the passage cycle of a coal cutting cycle, local support operations, or brief breaks during shift changes); 2 h simulates medium-length operation intervals (such as major production breaks within a work shift, or certain periods when operations need to be paused for equipment maintenance, moving supports, etc.); 6 h simulates longer periods of stress application (such as the duration of a work shift, night shift maintenance periods, or the relatively stable stage of surrounding rock before it reaches stress equilibrium during the advancement of the working face). The design of the cyclic loading stress levels aims to allow the rock samples to undergo a complete loading process from low stress to medium stress and then to high stress, simulating the entire stress state changes in the surrounding rock of the roadway from the initial stage affected by mining until it approaches failure. Table 2 and Figure 3 show the test scheme and loading path.

3. Test Results and Analysis

3.1. Characteristics of Stress–Strain Curves

Figure 4 shows the axial stress–strain curves of sandstone samples under multi-stage constant-amplitude cyclic loading and unloading, as well as multi-stage constant-amplitude intermittent cyclic loading and unloading (0.5, 2, and 6 h). Figure 5 illustrates the variation in strain during the constant-pressure phase for each group. Distinct patterns can be observed in comparative test groups A–D (Figure 4). The peak stress of sandstone samples gradually decreases at a diminishing rate under intermittent cyclic loading and unloading. Corresponding peak strain progressively increases, though the increase rate slows. The most significant change occurs between groups A and B. Internal microcracks in the sandstone samples continue to develop during the constant-pressure phase. Original voids are further compacted, and particle contact points undergo slippage, which leads to strain accumulation.

In the first stage, differences among the four curves are minor due to identical test parameters, with slight variations attributed to the natural heterogeneity of sandstone samples. In the second stage, the test parameters are adjusted. Group A proceeds to the next constant-amplitude cyclic loading–unloading phase, while groups B, C, and D enter constant-pressure phases of 0.5, 2, and 6 h, respectively. As the constant-pressure duration increases, strain correspondingly increases. In the third stage, following a similar trend as the second stage, strain further increases with longer constant-pressure durations. It results in more pronounced separation of the stress–strain curves in this phase. Additionally, the axial stress–strain curves transition from a loose to a dense pattern within the same loading–unloading cycle. Sandstone samples experience both elastic deformation (partially recoverable upon unloading) and plastic deformation (permanently retained).

Strain during the constant-pressure phase in groups B, C, and D increases with prolonged stress-holding duration (Figure 5). However, compared to the first constant-pressure stage, the strain increment in the second stage decreases across all test groups. The internal structure of sandstone gradually stabilizes after two cycles of loading–unloading at different stress levels and one constant-pressure loading. Consequently, the plastic deformation induced by each subsequent loading–unloading cycle diminishes, which reduces strain accumulation.

Figure 6 presents the variation in uniaxial compressive strength σ of sandstone after multi-stage constant-amplitude intermittent cyclic loading and unloading. σ declines with increasing constant-pressure duration. Compared to group A (no holding time), the compressive strength of groups B, C, and D decreases by 5.28, 11.5, and 14.68%, respectively. The constant-pressure phase promotes continuous microcrack propagation within sandstone, exacerbating internal damage. Concurrently, cyclic loading and unloading further contribute to cumulative damage, which weakens the load-bearing structure and strength of the sample.

3.2. Elastic Modulus

The elastic modulus is a crucial parameter for characterizing the deformation properties of rocks and holds significant importance in elucidating the mechanical behavior of rock mass [15,23]. Stepped-amplitude cyclic loading and unloading are applied to sandstone samples at different stress levels in this experiment. It leads to continuous redistribution of the stress field within rock samples and ongoing changes in their internal structure. As a result, the elastic modulus E of sandstone samples in this test differs significantly from that obtained in uniaxial tests. Equations (1) and (2) are used to calculate the loading and unloading elastic moduli of sandstone samples during the multi-stage stepped-amplitude cyclic loading–unloading phases, respectively.

$$E_i^{+}={\displaystyle \frac{\Delta {\sigma }_i^{+}}{\Delta {\epsilon }_i^{+}}}={\displaystyle \frac{{\sigma }_i^{+\max }-{\sigma }_i^{+\min }}{{\epsilon }_i^{+\max }-{\epsilon }_i^{+\min }}}$$

(1)

$$E_i^{-}={\displaystyle \frac{\Delta {\sigma }_i^{-}}{\Delta {\epsilon }_i^{-}}}={\displaystyle \frac{{\sigma }_i^{-\min }-{\sigma }_i^{-\max }}{{\epsilon }_i^{-\min }-{\epsilon }_i^{-\max }}}$$

(2)

where E_i⁺ and E_i⁻ represent the elastic moduli in the ith loading–unloading cycle, respectively; Δσ_i⁺ and Δσ_i⁻ represent the stress variations in the ith elastic-stage loading–unloading cycle, respectively; Δε_i⁺ and Δε_i⁻ represent the axial strain variations in the ith elastic-stage loading–unloading cycle, respectively.

Figure 7 illustrates the evolution of loading–unloading elastic moduli of sandstone samples under different constant-pressure durations and cycle numbers. Variations in elastic moduli among different groups of sandstone samples are generally similar in the first stage due to consistent test parameters. Minor differences primarily result from the inherent discreteness of rocks. Upon entering the second and third stages, as the constant-pressure duration gradually increases, the overall elastic moduli of all other groups of sandstone samples decline compared to group A.

Sandstone samples undergo plastic deformation during the constant-pressure phase. Microcracks within samples and contact points between particles experience displacement in the process, which reduces the samples’ stiffness and deformation resistance. Sustained stress induces gradual permanent deformation in sandstone samples, which affects their recovery capacity during cyclic loading and unloading and decreases elastic moduli.

The elastic modulus is relatively small during the first loading–unloading cycle ($E_1^{+}$ < $E_i^{+}$ (i > 1)). However, it stabilizes as the number of loading–unloading cycles increases. According to the report in ref. [24] on the damage characteristics of rocks under multi-stage characteristic-stress cyclic loading and unloading, the damage behavior of rocks exhibits a memory effect under cyclic loading. That is, the damage path of rocks develops along the damage path generated under previous loading conditions. Consequently, the initial loading induces the greatest damage quantity within the same stress stage. As the number of loading–unloading cycles increases, the damage quantity in rocks gradually stabilizes.

3.3. Deformation and Failure Characteristics

In Figure 8, 22 characteristic points (a-v) are selected from the axial stress–strain curves of samples A-1, B-1, C-1, and D-1 to investigate the strength and deformation–failure characteristics of sandstone.

Figure 9 depicts the progression of the maximum principal strain field within samples A-1, B-1, C-1, and D-1 throughout multi-stage stepped-amplitude intermittent cyclic loading and unloading, followed by loading until failure. Using the initial state (point a) as the reference image, strain field nephograms are generated for subsequent characteristic points. Sandstone generally exhibits single-inclined-plane shear failure under multi-stage stepped-amplitude intermittent cyclic loading and unloading, with a visible main inclined crack. Failure is complex during cyclic loading and unloading due to the multi-stage cyclic disturbances, exhibiting intense failure. Stress concentration-induced failure occurs due to end-face effects. Noticeable spalling appears at the sample end, while fractures propagate from the central region toward both ends.

When cyclic loading and unloading reach characteristic point b in the first stage, internal microcracks, pores, and other defects in the sandstone sample are compressed and densified. Stress concentration becomes evident, and strain contour lines appear densely distributed. Upon unloading to point c, strain contours gradually become sparser, with stress concentration weakened. As the number of loading–unloading cycles increases (from points c to g), pre-existing fissures in rocks are further compacted during loading, with microcracks gradually propagated and new cracks initiated. Continuous damage accumulation inside the sample leads to renewed stress concentration, which results in increasingly dense strain contours.

The subsequent maximum principal strain fields of samples B-1, C-1, and D-1 exhibit more pronounced activity compared to sample A-1 after the first constant-pressure stage (characteristic point h). Continuous damage accumulation occurs within rocks during loading, indicating that the constant-pressure stage exacerbates internal damage in sandstone samples. Moreover, as the constant-pressure duration increases, new stress concentration phenomena emerge more rapidly in the samples. Taking sample C-1 as an example, a new localized deformation band first appears in the central region (characteristic point l). Subsequently, the strain contours gradually converge toward this localized deformation zone. Compared to sample A-1, samples B-1, C-1, and D-1 exhibit earlier formation of new localized deformation bands during cyclic loading and unloading. The expansion rate of these deformation bands accelerates with increased loading–unloading cycles, accompanied by a steeper gradient in strain contour variations. The damage accumulation rate in sandstone samples increases significantly under multi-stage stepped-amplitude intermittent cyclic loading and unloading. Samples subjected to longer constant-pressure durations are more prone to stress concentration at weak points, which leads to the earlier development of new localized deformation bands.

Following the multi-stage stepped-amplitude intermittent cyclic loading–unloading tests, sandstone samples are further loaded until instability and failure occur (from characteristic points t to v). Localized deformation zones in all sample groups expand, accompanied by increasingly pronounced stress concentration phenomena. When the stress reaches the peak strength of the samples (characteristic point v), the pre-existing and newly formed fractures, along with microcracks, interconnect within sandstone samples, which leads to shear failure. Significant stress concentration is observed at the dominant macroscopic fracture, accompanied by spalling damage. Ultimately, cracks in sandstone samples fully propagate and interconnect under axial stress, with instability failure.

The area of the plastic hysteresis loop continuously accumulates under multi-stage constant amplitude intermittent cyclic loading and unloading. The internal damage of sandstone samples intensifies with plastic deformation accumulated, macroscopically manifesting as reductions in strength and resistance to deformation. Simultaneously, stress concentration at the defects of the sample becomes increasingly pronounced, which leads to more severe deformation and failure. The internal damage of rocks continuously increases during the cyclic loading–unloading process, weakening their load-bearing structure. It causes a reduction in load-bearing capacity as well as failure.

3.4. Evolution Law of Energy and AE in Sandstone

When subjected to external force, rocks undergo deformation. Assuming there is no heat exchange with the external environment during this physical process, according to the first law of thermodynamics [25],

$$U_i=U_i^e+U_i^d$$

(3)

$$U_i^e={\displaystyle {\int }_{\epsilon ″}^{\epsilon ‴}\sigma d\epsilon }$$

(4)

$$U_i^d={\displaystyle {\int }_{\epsilon \prime }^{\epsilon ‴}\sigma d\epsilon }-{\displaystyle {\int }_{\epsilon ″}^{\epsilon ‴}\sigma d\epsilon }$$

(5)

where U_i represents the total energy input to the rock sample by external force; U^e_i represents the elastic energy of the loaded rock sample; U^d_i represents the dissipated energy of the loaded rock sample.

Based on reversible elastic energy stored in the loaded sandstone sample, elastic energy and dissipated energy can be calculated using the stress–strain curve of rock samples from the multi-stage constant-amplitude intermittent cyclic loading–unloading test [26]. Sandstone is assumed to be a viscoelastic material, with no heat exchange between the sandstone samples and the external environment during the test. Neglecting energy losses such as thermal radiation, Figure 10 presents the schematic diagram and calculation method for the elastic energy and dissipated energy of the sandstone sample in the ith loading–unloading cycle.

Input energy, elastic energy, and dissipated energy are calculated for each cycle within the same constant-amplitude cyclic loading–unloading stage (Figure 11). The generation of dissipated energy primarily stems from internal damage and plastic deformation of rocks. This process reflects progressively developed internal defects in the sandstone sample and weakened strength until its eventual loss [27,28].

Dissipated energy reaches its maximum during the first loading–unloading cycle at the same stress level. The first cycle is more prone to inducing plastic deformation and microcrack propagation, which significantly increases dissipated energy within the sandstone sample. As the loading–unloading cycle number increases, dissipated energy gradually decreases and eventually stabilizes. The slope of the curve noticeably increases at the next stress level. As damage accumulates in the sandstone sample, the damage rate intensifies significantly once the loading stress reaches a higher level. Accumulated elastic energy in the sandstone sample remains constant within the same cyclic loading–unloading stage. The energy storage mechanism does not change with the increased cycle number during the damage process.

The AE phenomenon involves the release of elastic waves with diverse frequencies and energy levels from a material’s internal structure when the material undergoes loading, deformation, or fracture. The characteristic parameters of AEs are monitored during the loading process of rocks, reflecting the development of internal cracks and damage in rock samples [29,30,31,32,33]. The ring-down count presents the frequency of microcrack activity within rocks during loading.

Rock failure typically undergoes elastic, plastic, and failure stages, with significant differences in AE activity. The stress level is low in the initial loading stage (elastic stage), and rocks only exhibit a small number of microcracks. The growth of AE ring-down counts in sandstone samples is slow, indicating minimal microcrack activity. Microcracks propagate at an accelerated rate during the plastic stage, which increases the growth rate of ring-down counts. Penetrated cracks in the failure stage cause severe deformation and large-scale failure in rocks. Consequently, AE ring-down counts surge dramatically until instability failure.

Figure 12 shows the AE ring-down count characteristics of sandstone samples under multi-stage constant-amplitude cyclic loading and unloading. The AE activity significantly intensifies at the beginning of each new constant-amplitude cyclic loading–unloading stage. As the constant amplitude cycle number increases, the AE activity gradually stabilizes. The initiation, propagation, and penetration of microcracks within the sample are relatively active in the early phase of the same loading–unloading cycle. AE activity progressively stabilizes with the increased cycle number. The damage characteristics of rocks under cyclic loading exhibit memory effects. The damage path develops along the trajectory formed by previous loading–induced damage, which aligns with the energy evolution pattern observed in the sandstone samples.

As the duration of constant pressure increases, the AE ring-down counts decrease during the instability failure of sandstone samples. Although the constant-pressure duration in this experiment is relatively short, the low strength of sandstone samples allows the creep process to occur under constant stress. It leads to the gradual propagation and redistribution of internal microcracks. Samples generate numerous new microcracks upon entering the multi-stage constant-amplitude cyclic loading–unloading phase, which accelerates the damage evolution process. Microcracks connect and penetrate to form dominant cracks with prolonged constant pressure time. Long-term loading causes damage to the concentration in the main crack region. Consequently, fewer micro-fracture events are required for the rapid propagation of the main crack during failure, which decreases ring-down counts.

3.5. Sandstone Damage Degradation Mechanism

The damage and failure of rocks originate from the initiation, propagation, and penetration of numerous internal microcracks, degrading macroscopic mechanical properties. In prior investigations of rock damage quantification, researchers have defined damage variables based on dissipated energy. However, they did not explicitly differentiate between energy dissipated as a result of plastic damage and energy expended to overcome viscous effects. According to the study in ref. [34] on the energy evolution of sandstone under cyclic loading and unloading as well as the viscoelastic deformation of sandstone, dissipated energy is further subdivided into plastic damage energy and damping energy (the area enclosed by the plastic hysteresis loop). Plastic damage energy is obtained by subtracting part of the damping energy from the total dissipated energy. Damage variable D is defined by

$$D={\displaystyle \sum _{k=1}^i(U_d^k-U_d^z)/({\displaystyle \sum _{k=1}^i{U}_d^k+U_e^k})}$$

(6)

where U^k_d represents dissipated energy generated in the kth cycle; U^k_e represents elastic energy generated in the kth cycle; U^z_d represents damping energy generated in the zth cycle.

According to Equation (6), the average D of the four groups of sandstone samples is calculated as 0.376, 0.389, 0.41, and 0.456, respectively. All values range between 0 and 1, consistent with the definition of the damage variable. Figure 13 illustrates the variation in the average damage variables across the different groups of sandstone samples. The average damage variables of the four groups are relatively close in the first stage, exhibiting a slow growth trend. As the duration of constant pressure increases, average damage variables increase. Compared to group A, the damage variables of groups B, C, and D increase by 3.46, 9.04, and 21.28%, respectively, after multi-stage constant-amplitude intermittent cyclic loading and unloading. The constant stress promotes the propagation of internal cracks in sandstone samples, which increases damage within rocks. Simultaneously, damage continuously accumulates inside the samples under cyclic loading and unloading, with the damage variable increasing.

Sandstone is not an ideal material due to its lack of continuity, isotropy, and homogeneity. Friction between fine particles and the viscosity of internal fluids cause sandstone to exhibit nonlinear hysteresis effects during loading. Sandstone samples exhibit decreased compressive strength, increased strain, and reduced elastic modulus as the constant-pressure duration extends under multi-stage constant-amplitude cyclic loading and unloading. The internal stress of sandstone samples undergoes periodic variations during cyclic loading and unloading, which leads to stress concentration and release. The internal damage effects gradually become apparent and intensified. New microcracks and pores form inside the samples, while pre-existing cracks propagate and penetrate, which results in irreversible damage. This process exacerbates internal damage and weakens the load-bearing structure of the sandstone samples.

4. Conclusions

(1)

As the duration of constant stress increased, the uniaxial compressive strength of sandstone samples decreased by 5.28, 11.5, and 14.68, respectively, under multi-stage equal-amplitude cyclic loading and unloading and constant stress. Internal micro-crack propagation and particle displacement in sandstone induced during the constant stress phase intensified with the extended constant stress duration. This decreased the stiffness and deformation recovery capacity of sandstone samples, as well as the elastic modulus.

(2)

As the duration of constant stress increased, the evolution of deformation localization zones became more pronounced in sandstone samples during multi-stage cyclic loading and unloading. Deformation localization zones covered a broader area during the loading phase, with the maximum principal strain field at failure becoming more active. Stress concentration phenomena at defects in the sandstone samples grew increasingly significant. The samples experienced tensile-shear failure accompanied by spalling damage and increased fragmentation degree.

(3)

As the stress level of multi-stage cyclic loading and unloading with equal amplitude increased, the damage rate of sandstone accelerated. Dissipated energy was the maximum during the first cycle and then stabilized in subsequent cycles. The variation in cumulative dissipated energy aligned with the decline of the elastic modulus, reflecting the progressive failure process from dynamic expansion to stabilization of internal defects. AE activity initially intensified with increasing stages before stabilizing during multi-stage cyclic loading and unloading. The damage path depended on prior cumulative effects. As the duration of constant stress increased, the damage variable of sandstone samples increased. The constant stress promoted the deterioration of sandstone samples.

The research in this paper provides important reference data for evaluating the long-term strength of in situ rock masses. The results show that the rhythm of mining activities significantly affects the stability of the roof and surrounding rock of the roadway. Currently, this study mainly focuses on the effect of a single constant stress duration, but there are still some deficiencies. In the future, in-depth research can be conducted on aspects such as multi-dimensional stress states and microscopic mechanisms, with the aim of providing data references for the support design of working faces in production sites, thereby ensuring mine safety.