Abstract
The study of the deformation characteristics and damage evolution law of the underground water-bearing rock mass under reciprocating loads such as mine earthquake and mechanical vibration is a very crucial aspect of underground engineering. In this pursuit, the present study was envisaged to assess the deformation characteristics and damage evolution law of sandstone with different water contents under various cycles. Specifically, the uniaxial and cyclic loading and unloading tests, X-ray diffraction (XRD), and scanning electron microscope (SEM) tests of the sandstone under dry, unsaturated, and saturated conditions were carried out under laboratory conditions. Subsequently, the change laws of elastic modulus, cyclic Poisson’s ratio, and irreversible strain in the loading section of sandstone under different water content conditions were analyzed. Based on the two-parameter Weibull distribution, the coupled damage evolution equations of sandstone under water content and load were established. The results showed that with an increase in the water content in the sandstone, the loading elastic modulus of the corresponding cycles exhibited a gradual decrease. Microscopic analysis revealed that kaolinite was present in the water-bearing sandstone in a lamellar structure, with flat edges and many superimposed layers, and the proportion of kaolinite gradually increased with an increase in the water content. The poor hydrophilicity and strong expansibility of kaolinite are the key factors in reducing the elastic modulus of sandstone. With the increase of the number of cycles, the cyclic Poisson’s ratio of sandstone experienced three stages: an initial decrease, followed by a slow increase, and finally a rapid increase. The decrease was mainly observed in the compaction stage; the slow increase existed in the elastic deformation stage; and the rapid increase was seen in the plastic deformation stage. Furthermore, with the increase of water content, there was a gradual increase in the cyclic Poisson’s ratio. The concentration degree of the distribution of the rock microelement strength (the parameter m) under the corresponding cycle of sandstone with different water content states exhibited an initial increase followed by a subsequent decrease. With the increase in the water content, the parameter m under the same cycle gradually increased, and the change rule of parameter m corresponded to the development of internal fractures in the sample. With an increase in the number of cycles, the internal damage of the rock sample gradually accumulated, and the total damage increases gradually but the growth rate decreases gradually.
1. Introduction
Under the influence of dynamic disturbances such as roadway excavation and coal seam mining, the underground aquifer of a coal mine inevitably leads to water migration,1,2 and water seeps into the rock mass. External cyclic forces such as mine earthquake,3−5 blasting,6 and mechanical vibration will show different degrees of deformation and damage characteristics on the rock mass.7−9 Therefore, performing uniaxial cyclic loading and unloading tests on the sandstone with different water contents under laboratory conditions and studying changes in the deformation and damage characteristics of sandstone are important prerequisites for maintaining the stability of rock mass.
Domestic scholars have conducted extensive research on the macroscopic deformation,10 damage constitutive models,11−15 and macroscopic and microscopic damage and failure characteristics16,17 of water-bearing coal and rock masses under uniaxial and triaxial compression conditions. Xu et al.10 studied the corresponding relationship between the water content state and rock complete stress–strain curve, surface deformation field, deformation localization, and damage evolution characteristics using 3D digital image correlation technology. Based on the elastic damage mechanics, Wang et al. and Li et al.11,12 characterized the overall damage variables of coal and rock under different water contents and established the segmented damage constitutive model of coal and rock under hydraulic coupling. Li et al.13 established the mechanical properties and damage constitutive model of the limestone tailing foundation filled with different water saturation levels. Zhang et al.14 established a statistical damage constitutive model that could reflect multiple test curves. Bian15 studied the parameter degradation mechanism of shale under immersion conditions from a microscopic perspective. Based on the generalized strain equivalence principle and statistical mesodamage mechanics theory, they established the damage constitutive model of rock under water weakening and uniaxial loading and verified the correctness of the model. Lai16 analyzed the mechanical properties and failure modes of coal and rock samples from the macro–microscale and retrieved the damage evolution process of the coal and rock samples from the mesoscale based on acoustic emission parameters. They obtained the macro-mesoscale damage evolution characteristics of coal and rock samples under the hydraulic coupling effect. Nie et al.17 believed that filling bodies with different water contents have different damage characteristics, and water can inhibit the damage development of dry filling bodies, but the increase of water content of filling bodies will promote the damage development. The above analysis does not involve cyclic loading but it has important reference significance for studying and analyzing cyclic loading and unloading.
Some scholars have carried out relevant research on the rock masses in different water-bearing states under uniaxial cyclic loading and unloading conditions. Feng et al.18 carried out uniaxial cyclic loading and unloading tests on dry, natural, and saturated sandstone and showed that the unloading modulus initially increased and later decreased with an increase in the number of cycles. Under the same cycle, with an increase in the water content, there was a reduction in the unloading modulus. The damage definition based on the energy dissipation shows that the initial damage value of sandstone was not zero, and it experienced a damage evolution process of an initial decrease and a subsequent increase. Wang et al.19 performed the uniaxial cyclic loading and unloading tests and acoustic emission monitoring on dry and saturated sandstone specimens and studied the strength, deformation, acoustic emission characteristics, and the change of loading and unloading response ratios of sandstone in different water content states. The peak characteristic time of loading and unloading response ratios of the saturated specimens was 12.5% earlier than that of dry specimens. The above research has not thoroughly analyzed the damage and deformation characteristics of sandstone samples at different cycles, and the description of deformation parameters has not been analyzed from the perspective of microminerals. The damage evolution equation of water-bearing sandstone during the cycling process has not been established, and the description of the entire process of sandstone under load failure during water-bearing cycling is not deep and rigorous enough. Therefore, based on the existing research,20,21 the mechanical properties, macroscopic crack evolution, microscopic fracture morphology, and damage evolution of different water-bearing sandstones under uniaxial cyclic loading and unloading were analyzed in the present study. Combined with the X-ray diffraction analysis and scanning electron microscopy, the microaction mechanism of the water-bearing sandstone was analyzed. Based on the generalized form of the principle of strain equivalence and the two-parameter Weibull distribution characteristics, the coupled damage evolution equation of water-bearing sandstone was established, and the damage evolution law of sandstone in different water-bearing states in each cycle was obtained. The research results can provide guidance in the study of rock mass damage under the conditions of underground aquifer water movement, tunnel excavation, and water erosion of roadway rock mass.
2. Materials and Methods
2.1. Sample Preparation
In order to reduce the dispersion of the test data, the material used in the test was selected from the same roof and floor sandstone of Pansidong Coal Mine in Huainan, and the surface was free of macroscopic visible joints, cracks, and other defects. According to the national standard,22 the sample prepared under laboratory conditions was of a 50 mm × 100 mm standard cylindrical sandstone piece. In order to reduce the end effect of sandstone specimen during the test, the end face parallelism of sandstone specimen was controlled within ±0.02 mm, and the end face was perpendicular to the axis of rock sample, as shown in Figure 1. Sandstones with different water contents were prepared by natural immersion as follows:
-
(1)
The sandstone samples were placed in a drying oven at a constant temperature of 106 °C for drying for 24 h. When the weight of the samples weighed for two consecutive times was approximately equal, the drying was assumed to be completed, and the mass at this time was recorded as “m0”.
-
(2)
A group of sandstone samples were randomly placed in a water tank for the preparation of saturated samples. When the weight of the samples weighed for two consecutive times was approximately equal, the preparation of saturated samples was considered to be complete, and the measured mass was recorded as “m1”.
-
(3)
According to the specific water content obtained in step (2), the intermediate water content for preparation was taken, and the mass of the corresponding sandstone sample was obtained. The remaining group of sandstone samples was placed in the water tank to prepare the unsaturated sample. When the sandstone sample reached the set mass, it was considered that the preparation of the unsaturated sample was completed and recorded as “m2”.
Figure 1.
Preparation of the sandstone samples.
The calculation of the water content was obtained as follows
 |
1 |
where i is 0, 1, and 2.
2.2. Test Scheme Design
Using the RMT-150B rock mechanics testing machine, two tests of uniaxial compression and uniaxial cyclic loading and unloading were performed. Each test was divided into three groups, viz., dry group, unsaturated group, and saturated group (Table 1). The average uniaxial compressive strength of sandstone samples in the dry state, unsaturated state, and saturated state under uniaxial compression was determined to be 101.37, 55.16, and 44.07 MPa, respectively, as the reference value for the setting of uniaxial cyclic loading and unloading parameters, and the experiment was reasonably designed. The cyclic loading and unloading test processes adopted the stress control. In order to ensure that under cyclic loading and unloading conditions, the saturated specimen could reach enough cycles before being damaged, and fully reflect the test law, each loading section employed 5 MPa as a cyclic gradient to uniformly increase the axial stress. The cyclic loading and unloading rates were set at 0.5 MPa/s, and the specific path of cyclic loading and unloading was 0 → 5 → 1.25 → 10 → 1.25 → 15 → 1.25 → 20 → 1.25 → 25 → 1.25 MPa. The test was carried out according to this rule until the test piece was damaged, and the test was stopped at this point.
Table 1. Design of the Test Scheme.
| test method | sample status | sample number | water content (%) | average water |
|---|---|---|---|---|
| uniaxial compression test | drying (0) | D-1 | 0 | 0 |
| D-2 | 0 | |||
| unsaturated (50%) | D-3 | 1.01 | 1.03 | |
| D-4 | 1.05 | |||
| saturation (100%) | D-5 | 2.13 | 2.14 | |
| D-6 | 2.15 | |||
| uniaxial cyclic loading and unloading tests | drying (0) | X-1 | 0 | 0 |
| X-2 | 0 | |||
| X-3 | 0 | |||
| unsaturated (50%) | X-4 | 0.98 | 1.01 | |
| X-5 | 1.00 | |||
| X-6 | 1.05 | |||
| saturation (100%) | X-7 | 2.09 | 2.07 | |
| X-8 | 2.04 | |||
| X-9 | 2.08 |
3. Results and Discussion
3.1. Stress–Strain Curve
The cyclic loading and unloading stress–strain curves of the sandstone with different water contents are shown in Figure 2. It can be seen from the figure that the peak stress, peak axial strain, and number of cycles of sandstone were of the order saturated state < unsaturated state < dry state. In the process of cyclic loading and unloading, the area of hysteretic loop increased suddenly at a certain cycle, indicating that the dissipated energy increased suddenly at this cycle stage.23 In such a case, when the stress increases in a uniform manner, the strain changes abruptly and a relatively severe failure can occur in the sandstone, the porosity of old fractures increases, accompanied by the generation of new fractures. However, the number and structure of fractures are not enough to make the sandstone unstable, so the load gets increased. As a result, after the crack is compacted, it can still bear greater external force.
Figure 2.
Cyclic loading and unloading stress–strain curves of sandstone in different water-bearing states.
The stress–strain curves of sandstones with different water content states all undergo a stage of presparse to medium dense to postsparse, and as the water content increases, the presparse section of the curve weakens, while the postsparse section strengthens, especially in the postsparse section, which is caused by the secondary damage of water to the sandstone. This is because the water film adsorption force and pore water pressure formed by micropores and fractures in water-bearing sandstone can inhibit the initial compaction deformation (presparse section) of sandstone. In the unstable crack propagation stage (post-thinning section), due to cyclic loading, the internal micropores of water-bearing sandstone are fully developed, resulting in new fractures that intersect with old fractures, leading to the outflow of internal pore water, causing secondary damage to the interior of the sandstone, making the strain enhancement effect significant, and ultimately leading to a significant pattern of weakened prethinning and enhanced post-thinning on the curve.
3.2. Loading Elastic Modulus
Zhou et al.24 proposed two methods to calculate the elastic modulus of rock materials. (1) The approximate equivalent method was used to calculate the elastic constants of rocks in each loading and unloading cycle. It states that each loading and unloading process can cause an irreversible deformation with the increase in the loading and unloading times. (2) For each loading and unloading cycle, the linear section of the loading stress–strain curve is used to calculate the elastic modulus. Since the characteristics of the mesomechanical changes in the interior of sandstone should be considered, method (2) was selected. Briefly, 30–70% of the loading curve was considered to calculate the loading elastic modulus, but in the actual calculation, the data was selected according to the straight line section of the specific curve for calculation. The average value of the calculation results was taken, and the obtained curve is shown in Figure 3. Since the unloading curve is a free unloading process after removing the external force of the testing machine, it is of little significance.
Figure 3.
Relationship between the loading elastic modulus and cycle times of sandstone.
The loaded elastic modulus of sandstone in different water content states increased with an increase in the number of cycles, but the growth rate decreased gradually when it was close to failure and decreased at the last cycle. With an increase in the water content, the loaded elastic modulus of sandstone showed a gradual decrease. The change process of each curve is divided into three stages: (1) stage of uniform increase (elastic stage); (2) stage of deceleration and increase (stable plastic deformation stage); and (3) reduction stage (unstable plastic deformation stage), as shown in Table 2. During the compression process of cyclic external force, the internal microstructure of sandstone in the elastic stage (some microstructures with higher closed stress except for micropores and cracks in the compaction stage) is compacted, the antideformation ability of the overall structure is enhanced, and the elastic modulus is also increased. When the sandstone is in the plastic deformation stage, new microcracks appear inside, and the growth rate of the antideformation ability of the overall structure gradually decreases. When the new cracks appear to a certain extent, the antideformation ability of the overall structure decreases and the elastic modulus decreases.
Table 2. Corresponding Cycle Times of Sandstone with Different Water Contents at Each Stage.
| sandstone state | stage (1) corresponding cycle | stage (2) corresponding cycle | stage (3) corresponding cycle stage |
|---|---|---|---|
| dry state | 7–15 | 15–19 | 19–20 |
| unsaturated state | 5–10 | 10–12 | 12–13 |
| saturation state | 4–8 | 8–10 | 10–11 |
Apart from this phenomenon, free water is present in the micropores and fractures in the water-bearing sandstone, and there is an adsorption of the water film on the contact surface between the micropores, the inner wall of the fractures, and the water film. At the same time, the pressure of the pore water also exists in the micropores and fractures during the compression process of rock samples, which weakens the rigidity of the overall structure. With the increase in the water content, there is an increase in the free water content and a corresponding increase in the water film adsorption force and pore water pressure. This further weakens the overall stiffness of the sandstone, resulting in the gradual reduction of the sandstone loading elastic modulus.
3.3. Microscopic Analysis of Change Law of Elastic Modulus under Loading
The elastic modulus of the rock refers to the ratio of stress to strain in the elastic range of rock, which is a mechanical parameter describing the deformation characteristics of the rock. Macroscopically, the elastic modulus is a measure of the ability of rock materials to resist the elastic deformation. Microscopically, it is a reflection of the bonding strength between the atoms, ions, or molecules. Therefore, it is related to the bonding mode, crystal structure, chemical composition, microstructure, temperature, internal micropore size, pore number, and water content.
Feng et al.18 and Wang et al.19 calculated and analyzed the cyclic loading and unloading elastic moduli of sandstone containing different water contents under a uniaxial cyclic loading and unloading and found that the change trend of the elastic modulus of dry and saturated specimens generally increased with an increase in the number of cycles. It was considered that the elastic modulus of the specimen weakened after immersion due to the effect of water. Feng et al.18 carried out X-ray diffraction analysis of the sandstone and found that the sandstone contained montmorillonite and other mineral components. Wang et al.19 used the vacuum-saturated method to prepare the test piece, which was different from the natural saturated method, and did not analyze the microscopic mechanism. The microminerals montmorillonite, illite, and kaolinite have great differences in hydrophilicity. The montmorillonite has the strongest hydrophilicity, large water absorption and expansion, highest surface area, and is most unstable. Illite has hydrophilicity between montmorillonite and kaolinite. Kaolinite has poor hydrophilicity, the smallest surface area, and exhibits high stability. The instability of the microminerals is the internal factor that causes the elastic modulus to decrease with an increase in the water content. Based on the above analysis, we speculate that the influence of water on the elastic modulus of sandstone should consider not only the water film force and pore water pressure of the internal micropores and fractures of sandstone under axial load but also the difference of the micromineral composition in water-bearing sandstone that has a certain influence on the change of elastic modulus of sandstone. The mineral composition and mineral structure of sandstone were analyzed as follows.
Figure 4 shows the schematic diagram of the X-ray diffraction analysis of sandstones in different water-bearing states. The main mineral composition in different water-bearing states was as follows: in the dry state, quartz and albite; in the unsaturated state and saturated state, quartz, albite, and kaolinite. Sandstone changes from the dry state to water-bearing state, and kaolinite changes from none to presence, which indicates that kaolinite is generated in sandstone under water-bearing condition.
Figure 4.
X-ray diffraction analysis of sandstone samples in different water-bearing states.
Figure 5 shows the scanning electron microscope image analysis of the water-bearing sandstone. It was observed that kaolinite exhibited a lamellar structure, with flat edges and many overlapping layers. At the same time, it has poor hydrophilicity and strong stability. Zhang et al.25 found that albite forms kaolinite under the action of weakly acidic water, and the corresponding reaction equation is shown in formula 2. Under water-bearing conditions, albite and the weakly acidic water in sandstone undergo a hydration reaction to form kaolinite, and the mineral content of kaolinite gradually increases with the increase in the water content (soaking time). The crystal cell structure of kaolinite is connected by O2– and OH– with strong adhesion. It is extremely difficult for water molecules to enter into the crystal cell. Due to the poor hydrophilicity and high stability of kaolinite, the crystal of kaolinite clay will expand slightly when it is soaked in water,26 and its expansion mainly comes from the intergranular expansion caused by the thickening of the combined water film on the particle surface.27 Therefore, even if albite hydration reaction generates softened mineral kaolinite under water content conditions, due to the special expansibility of kaolinite, the elastic modulus of sandstone will gradually decrease with an increase in the water content under the action of axial force at each stage of the cycle
 |
2 |
To sum up, water exhibited a weakening effect on the elastic modulus of sandstone. Based on the above results, the following three perspectives were observed: (1) the physical weakening effect of water on the internal structure of sandstone mainly manifests in the erosion, water wedge, and lubrication of water on the internal structure of sandstone, which reduces the strength of sandstone samples and the elastic modulus. (2) The hydration effect of water on the internal material structure of sandstone is mainly represented by the reaction equation in Formula 2. With the increase of water content and the deepening of reaction degree, the content of kaolinite gradually increases. On the other hand, the poor hydrophilicity and strong expansibility of kaolinite cause the decrease of elastic modulus of sandstone. (3) There is water film adsorption force between the inner wall of micropores, cracks, and water film, and there is pore water pressure between the free water in micropores and cracks, which reduces the rigidity of the whole structure. With an increase in the water content, the pore water pressure formed gradually increases, and the overall structural rigidity gets reduced, resulting in the reduction of elastic modulus of the sandstone.
Figure 5.
SEM analysis of water-bearing sandstone samples.
3.4. Cyclic Poisson’s Ratio
The curve of the change of Poisson’s ratio of sandstone in different water-bearing states with cycles is shown in Figure 6. Poisson’s ratio of each cycle under cyclic loading and unloading conditions is defined as cyclic Poisson’s ratio (μ0). Under different water content conditions, the cyclic Poisson’s ratio experienced the following three stages: an initial decrease, followed by a slow increase, and finally manifested a rapid increase. With the increase in the water content, there was a gradual increase in the cyclic Poisson’s ratio.
Figure 6.
Relation of Poisson’s ratio and cycle time.
Most of the microcracks randomly distributed in the sandstone have a certain angle with the axial load. Figure 7 shows the loading and deformation diagram of the microcracks in sandstone. Let us consider an internal microcrack of sandstone for stress analysis. With the microcrack tip as the reference line, the axial load can be decomposed into Fx in the direction of the parallel reference line and Fy in the direction of the vertical reference line. Here, Fx is used for crack tip development and propagation, and Fy is used for crack compaction. At the compaction stage with small load, the size of Fx is not enough to make the crack grow. At this time, the compaction effect of Fy is mainly considered. Therefore, in the initial compaction stage, the pore compaction leads to an increase in the axial strain and a decrease in the cyclic Poisson’s ratio. In the elastic stage, there is an increase in the transverse strain and the axial strain, and the cyclic Poisson’s ratio increases slowly. After entering the plastic stage, internal cracks develop in the sample and the transverse strain increases rapidly, resulting in a rapid increase in the cyclic Poisson’s ratio. Based on the analysis of experimental laws, it is found that as the water content increases, Poisson’s ratio of sandstone samples increases at all stages of cycling, and the water content increases Poisson’s ratio of sandstone samples. The harder the sandstone material, the smaller Poisson’s ratio; the softer the material, the higher Poisson’s ratio, which also indicates that water has a softening effect on sandstone. Under water-bearing conditions, taking the saturated state as an example, during the initial compaction stage, there is pore water pressure inside the sandstone, and the axial strain is relatively small.
Figure 7.
Schematic diagram of load and deformation of microcracks in sandstone.
3.5. Cumulative Irreversible Strain
The deformation of rock consists of reversible elastic deformation and irreversible plastic deformation. The deformation produced by the loading section exhibits both of them. The elastic deformation recovery in an unloading section leaves plastic deformation, which is the process of cumulative damage of rock deformation. The formula for calculating cumulative irreversible strain is as follows
 |
3 |
where εax is the axial cumulative irreversible strain, εaxper(n) is the nth loading and unloading axial irreversible strain, and εax(1) is the first loading and unloading axial irreversible strain.
Figure 8 shows the relationship between the cumulative irreversible strain and the number of cycles of sandstone under different water-bearing states. With an increase in the cycle, the cumulative irreversible strain gradually increased. With an increase in the water content, the cumulative irreversible strain exhibited a gradual decrease. This is due to the existence of water, which leads to the generation of pore water pressure and membrane adsorption in the internal microfracture of sandstone. Further, the existence of a large amount of bound water in the internal skeleton of sandstone causes the expansion of the sandstone and inhibits axial deformation.
Figure 8.
Relation of the cumulative irreversible strain and cycle time.
4. Coupling Damage Characteristics of Water-Bearing and Loaded Sandstone
4.1. Coupling Damage Relationship between Water-Bearing and Loaded Sandstone
Water has a significant impact on the physical and mechanical parameters of sandstone, and it is important to consider the damage characteristics of sandstone under different water conditions. In this study, the damage evolution law of sandstone under uniaxial cyclic loading is expressed based on the difference in the elastic modulus of sandstone under different water conditions. Let the damage variable Dωin represent the damage value of unsaturated or saturated sandstone under the nth cycle, which changes with the increase of cycle n. Then, the damage variable can be expressed as
 |
4 |
where Eω0n and Eωi refer to the loading elastic modulus of the sandstone in the dry, unsaturated, or saturated state under different cycles and are defined as the elastic modulus of benchmark damage state and water damage state; i is 1 and 2, respectively.
Under the influence of external factors (such as water, force, etc.), the rock material gets damaged to a certain extent internally. From the mesoscopic perspective, due to the natural heterogeneity of the internal structure of rock materials, each element of rock materials is randomly distributed, and the damage parameter is the macroscopic description of the irreversible mesostructural changes in the material. Therefore, the statistical damage density of the element failure can be introduced to obtain the following relationship between the water-loaded damage parameter D and the statistical distribution density of the element failure
 |
5 |
where φ(ε) is a measure of the damage rate of the rock element in the process of loading, which reflects the damage degree of rock from a macroperspective. The two-parameter Weibull distribution is introduced to characterize the statistical distribution of the rock element strength, and the description of rock damage evolution equation under load can be obtained as
 |
6 |
where parameter m represents
the concentration degree of rock microelement strength distribution,
which changes with the increase of cycle number n, and is a function of cycle number; 
and is calculated from test data.
According to the extension form of the strain equivalence principle proposed by Zhang,28 that is, “when a material is affected by force F, and the damage is expanded, if any of the two damage states are selected, then the effective stress of the material in the first damage state acting on the strain caused by the second damage state is equivalent to the effective stress of the material in the second damage state acting on the strain caused by the first damage state”. Most rocks in the natural state have some degree of original damage, and there are very few truly nondestructive rock materials. Therefore, the dry state of the sample is defined as the baseline damage state. First, the dry state is defined as the first damage state, and the initial immersion softening damage of the sandstone is defined as the second damage state; then
 |
7 |
By analogy, the water-softening damage of sandstone is defined as the first damage state, and the water-loaded damage is defined as the second damage state; then
 |
8 |
where Dn is the damage variable of the loaded sandstone.
The dry state of sandstone is defined as the first damage state, and the water-loaded total damage is defined as the second damage state; then
 |
9 |
It can be obtained from simultaneous eqs 4, 7, and 9
 |
10 |
where Dn is the total damage variable of water-loaded sandstone.
The elastic modulus E of the reference damage state (dry state) is defined in formula 7Eω0n. It is easy to obtain the test data, thus avoiding the solution of the elastic modulus of the nondestructive rock required in the traditional constitutive equation. In Formula 10, the damage of sandstone increases under the combined action of water content and load and shows obvious nonlinear characteristics. The essence of water-bearing damage is the lubrication, water wedge, and corrosion between the mineral particles and the rock micropores. The effect of load causes the rock grains to slip and stagger. The resulting damage and water-bearing damage are coupled and interact with each other, which will inevitably lead to changes in the mechanical properties of the rock. The coupling effect of water and load weakens the total damage, and DnDωi represents the coupling term. Figure 9 shows the coupled damage process of the water-bearing and loaded sandstone.
Figure 9.
Schematic diagram of the coupling damage process of water-bearing loaded sandstone.
Substitute eqs 4 and 6 into eq 10, the total damage evolution equation of water-bearing and loaded sandstone is
 |
11 |
4.2. Evolution Law of Parameter “m”
The parameter m represents the concentration degree of distribution of the rock microelement strength, which changes with an increase in the cycle number n, and is a function of cycle number (Table 3). Based on the analysis of the test data, the variation rule of parameter m of sandstone sample in different water-bearing states with cycle times is shown in Figure 10. The value of parameter m in different water-bearing states initially increased followed by a subsequent decrease. This is because, before the sandstone failure (compaction stage and elastic stage), with an increase in the upper limit stress of each cycle, the sample was gradually compacted, the stiffness increased, and there was an increase in the intensity concentration. After the plastic deformation of sandstone, microfractures appeared in its interior, resulting in the decrease in the intensity concentration. Compared with Figure 3, it can be found that the cycle number of the point where the area of the hysteresis loop suddenly increased, and the m value of the corresponding sample showed a decrease to a certain extent, specifically in the 14th and 19th cycles during drying, the 12th and 13th cycles in the unsaturated condition, and cycles 7, 10, and 11 at saturation. Depending on the damage of sandstone, there was a difference in the reduction of the strength dispersion degree. As a result, the reduction degree of parameter m also exhibited a variation.
Table 3. Variation Law of the Loaded Elastic Modulus and Parameter m.
| loading
elastic modulus, GPa |
parameter m |
|||||
|---|---|---|---|---|---|---|
| cyclic loading | dry | unsaturated | saturated | dry | unsaturated | saturated |
| 2 | 1.207 | 1.920 | 2.662 | |||
| 3 | 1.201 | 2.291 | 2.680 | |||
| 4 | 5.92 | 1.221 | 2.411 | 2.704 | ||
| 5 | 7.29 | 6.33 | 1.253 | 2.481 | 2.860 | |
| 6 | 7.74 | 6.77 | 1.286 | 2.578 | 3.007 | |
| 7 | 8.52 | 7.94 | 7.11 | 1.317 | 2.641 | 2.700 |
| 8 | 9.09 | 8.41 | 7.79 | 1.358 | 2.682 | 2.961 |
| 9 | 9.73 | 9.04 | 8.26 | 1.390 | 2.762 | 2.995 |
| 10 | 10.26 | 9.57 | 8.47 | 1.416 | 2.802 | 2.363 |
| 11 | 10.88 | 10.11 | 7.95 | 1.446 | 2.804 | 1.839 |
| 12 | 11.27 | 10.26 | 1.471 | 2.268 | ||
| 13 | 11.73 | 9.28 | 1.497 | 1.541 | ||
| 14 | 12.12 | 1.554 | ||||
| 15 | 12.51 | 1.546 | ||||
| 16 | 12.54 | 1.514 | ||||
| 17 | 12.98 | 1.522 | ||||
| 18 | 13.31 | 1.533 | ||||
| 19 | 13.45 | 1.555 | ||||
| 20 | 12.39 | 1.380 | ||||
Figure 10.
Variation law of parameter m with cycle times.
4.3. Analysis of the Damage Evolution Law for Water-Bearing and Loaded Sandstone
The damage evolution law for the sandstone with different water contents in different cyclic loading sections is shown in Figure 11. The damage rules of sandstone in the two water-bearing states are similar. In the initial cycle stage, the total damage of water-bearing rock samples increased gradually from 0 under the coupling effect of the axial load and water content. Compared with the initial cyclic damage, the damage value of the rock sample at the beginning of each cycle gradually showed an increase with an increase in the cycle number. The irreversible damage gradually increased, which indicates that with an increase in the cycle number, the internal damage of the rock sample gradually accumulated, thereby leading to an increase in the overall damage. It can be seen that the loading section under each cycle of cyclic loading and unloading was different from the loading section of the stress–strain curve.
Figure 11.
Evolution law of the total damage of sandstone with different water contents in (a) unsaturated state and (b) saturated state.
Further analysis of the total damage evolution law of water-bearing sandstone under loading at each cycle showed that the damage growth of the curve of a single cyclic loading section was relatively slow at the initial stage of loading. The damage growth gradually accelerated as the loading continued to the middle stage, and the highest growth in the damage growth was observed in the last stage, thereby leading to a gradual increase in the overall damage growth rate. From the comparative analysis of the total damage of each cycle, it was found that with an increase in the number of cycle, the damage growth rate in the early, middle, and late stages gradually decreased. It can be seen that the growth rate of the loaded damage of the water-bearing sandstone showed increases gradually, but the growth rate decreases gradually.
5. Conclusions
In the present study, the mechanical properties, macroscopic crack evolution, microscopic fracture morphology, and damage evolution of different water-bearing sandstones under uniaxial cyclic loading and unloading were analyzed. Additionally, using X-ray diffraction and scanning electron microscopy, the microaction mechanism of the water-bearing sandstone was analyzed. The conclusions of the study are as follows:
-
(1)
In general, the cyclic loading and unloading elastic moduli of the sandstone with different water contents gradually increased with an increase in the number of cycles, but the growth rate decreased gradually when it was close to failure, and decreased at the last cycle. The change could be divided into three stages: (1) stage of uniform increase (elastic stage); (2) stage of deceleration and increase (stable plastic deformation stage); and (3) reduction stage (unstable plastic deformation stage). With an increase in the water content, the corresponding cyclic loading and unloading elastic moduli gradually decreased. The water film adsorption and pore water pressure formed by micropores in sandstone samples; free water in fractures and pore walls could be the possible reasons for the decrease in the elastic modulus of sandstone samples.
-
(2)
Based on the analysis of XRD and SEM studies, it was found that the mineral particles albite in sandstone samples react with water to form kaolinite, and the higher the water content, the higher was the content of kaolinite. The poor hydrophilicity and strong stability of kaolinite are other reasons for the gradual decrease in the cyclic loading and unloading elastic moduli of water-bearing sandstone samples with an increase of the water content.
-
(3)
With an increase in the number of cycles, the cyclic Poisson’s ratio of the sandstone experienced three stages: an initial decrease, followed by a slow increase, and finally a rapid increase. The decrease existed in the compaction stage, the slow increase existed in the elastic deformation stage, and the rapid increase existed in the plastic deformation stage. With the increase in the water content, the cyclic Poisson’s ratio gradually increased. The cumulative irreversible strain increased gradually with an increase in the number of cycles. Further, with an increase in the water content, the cumulative irreversible strain gradually decreased.
-
(4)
Based on the generalized form of the strain equivalence principle and the two-parameter Weibull distribution characteristics, the coupled damage evolution equation of water-loaded sandstone was established. The parameter m of sandstone in different water-bearing states under corresponding cycles followed the law of an initial increase with subsequent decrease. The change law of parameter m corresponded to the development of internal fractures of the sample, and with an increase in the water content, the parameter m increased at the same cycle. With an increase in the number of cycles, the internal damage of the rock sample gradually accumulated, and the total damage increases gradually but the growth rate decreases gradually.
Acknowledgments
The authors are very grateful to the reviewers and editors for their valuable comments and suggestions.
This research was funded by the Key Projects of the Joint Fund of the National Natural Science Foundation of China (U21A20110), the Major Special Projects of Science and Technology in Shanxi Province (20191101016), and the Institute of Energy, Hefei Comprehensive National Science Center (Anhui Energy Laboratory).
The authors declare no competing financial interest.
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