Dynamic Response and Energy Characterisation of High-Strength Sandstone under Progressive Cyclic Loading Based on Sustainable Mining

Abstract

In the research on sustainable mining and environmental preservation, understanding the dynamic behaviour of rock formations in deep, high-stress mining environments is essential. In order to acquire the laws of rock dynamic disaster generation from mining in deep, high-stress environments, this research adopts a multistage and multi-cycle triaxial cyclic loading test to obtain the stress–strain curves and macroscopic deformation characteristics of hard sandstone under different surrounding pressures. The results show that the cumulative damage displacement of hard sandstone under cyclic loading at a certain stress level for the first 3–4 cycles is half of the total damage displacement at that cycle stage, and its peak volumetric strain will increase with the increase. The elastic energy density ratio and dissipation energy density ratio of hard sandstone under cyclic loading show a sinusoidal fluctuation trend, and the fluctuation gradually decreases with the increase in the number of cycles and the increase in the cyclic stress level. Under the cyclic loading of different surrounding pressures, the hard sandstone shows brittle damage characteristics, where the damage form is mainly shear damage with a small amount of tensile damage in low surrounding pressure and the damage form is mainly shear damage, tensile damage, and local compression damage in high surrounding pressure. The study reveals the deformation and damage law, energy evolution, and dissipation characteristics of high-strength hard sandstone. It is essential for the development of mining strategies that minimize the impact on the environment, reduce the dynamic hazards generated by mining, and maximize the efficiency of resource extraction

1. Introduction

With the rapid development of the world’s economy, the demand for energy is increasing in various countries. With the energy extraction, surface and shallow resources are gradually depleted. The mining depth of mines and oil and gas wells further increased and gradually developed into deep mining [1,2]. However, deep underground rock has a stronger engineering mechanical response than shallow rock due to its complex storage environment and higher stress conditions. The greater depth of the rock body will result in the existence of a large amount of energy within the rock. When the energy is disturbed and suddenly released, it will cause greater harm to underground geotechnical engineering and cause greater losses to people and property [3,4,5]. The rock burst and impact phenomena produced by stress release in sandstone with higher strength are more serious. At the same time, the change in external load is the main factor that induces the stress release of rock.

In mining, when the deep surrounding rocks and coal pillars in a high-stress state are affected by the induced stresses generated by mining activities such as the excavation of roadways or chambers and rock support in the neighbouring areas, it will certainly affect the stress state of the surrounding rocks themselves [6]. At the same time, the superposition of the original rock stresses and the induced stresses due to the excavation of the geotechnical mass of the surrounding rock will lead to the disturbance of the total stresses on the surrounding rock, and therefore the total stresses on the surrounding rock in deep underground engineering will always be maintained within a certain range. During excavation, support, or mining, the stress state in the surrounding rock may be cyclically fluctuating [7,8]. Under this cyclic effect of stress, certain cracks and damages are more likely to occur inside the surrounding rock [9,10]. When the induced stress caused by the surrounding rock’s internal cracks reaches a certain level, a series of deep geotechnical engineering disasters such as rock explosions, roof falls, and so on will occur. The surrounding rock of a deeply buried roadway is usually in a high-stress environment, mostly in a state of high stress in three directions. The excavation of the roadway will have an impact on this stress state, and researchers found that through on-site monitoring, the tangential stress of the surrounding rock closer to the roadway edge gang has a tendency to increase and then decrease with the increase in the distance, and the radial stress is the unloading rebound state that decreases and then increases [11,12]. In this case, with the excavation of the adjacent roadway or refuge, the stress influence on the coal pillar between the two roadways easily leads to the occurrence of instability damage. The reason for this is that the coal column subjected to higher rock stress fluctuates in the total stress due to the change in the surrounding stress state, and its internal structure produces a certain stress adjustment, which affects the stability of the coal column between the roadways in the adjustment of internal stress under higher ground stress conditions [13]. Sandstone is a common rock in underground engineering, and the roof slabs of some coal mines are mostly sandstone. The sandstone with high strength tends to gather large energy internally under the action of a cyclic load of mining and excavation, and the sudden release of energy in a short period of time will produce a large threat to people and property. Therefore, it is of great significance to research the mechanical properties, internal energy evolution characteristics, and damage destruction laws of high-strength sandstone under cyclic loading to maintain the stability of the surrounding rock in deep mining roadway and ensure the safe operation of deep geotechnical engineering.

Many scholars have carried out a lot of research on the mechanical strength properties of deep rock under complex stress conditions such as cyclic loading and unloading by means of laboratory tests, theoretical analysis, field observations, numerical simulation, etc. Meng et al. [14] investigated the acoustic emission properties of sandstone samples under uniaxial cyclic unloading and investigated the deformation and damage process of rock materials under cyclic loading and the effect of the loading and unloading rate on the strength of rock by exploring the acoustic emission properties of sandstone specimens under uniaxial cyclic unloading. The effect of the loading and unloading rate on the strength of the rock was investigated. Wang et al. [15] explored the damage characteristics and the form of instability of rocks in the static loading and fatigue loading phases through triaxial cyclic loading and unloading tests and also investigated the internal fracture expansion law of the rock. Peng et al. [16] found that the volumetric strain of the rock has a certain effect on the susceptibility of the rock to fracture extension by cyclic loading tests at different surrounding pressures. By increasing the surrounding pressure, the irreversible strain in axial and radial directions can be changed so that the change process of rock strain increases. In the research on the energy evolution characteristics of loaded rocks, H. Munoz et al. [17] analysed the energy characteristics of the pre-peak phase of rocks based on conventional uniaxial and triaxial rock compression tests. Yang et al. [18] analysed the law of influence of surrounding pressure on the energy dissipation and energy release characteristics of rock damage crack extension and revealed the energy mechanism of the crack extension during the damage of rock samples under different triaxial stresses. Huang et al. [19] analysed the characteristics of the energy transformation of the absorbed strain energy, dissipated strain energy, and elastic strain energy storage and release of the rock samples based on the marble peak-front unloading surrounding pressure test and the energy principle, revealing the strain energy transformation mechanism of the damage rupture evolution of the rock samples. Zhang et al. [20] analysed the energy evolution characteristics of marble under triaxial loading and unloading conditions and revealed the influence laws of the surrounding pressure and unloading rate on the energy accumulation and dissipation of rock samples. In the research on rock energy damage criteria, E. Gaziev et al. [21] proposed that when the shape alteration energy of a material reaches its limit value, then the unit begins to break down and the continuous accumulation of the total strain energy is a necessary condition that leads to the destruction of the material.

In summary, although there is much research on the mechanical properties of cyclic loading and unloading of different rocks, most of it focuses on lower-strength rocks. There is still some space for targeted research on deep, high-strength sandstone. Therefore, the research on the mechanical properties and energy evolution characteristics of high-strength hard sandstone under different peripheral pressure triaxial cyclic loading is an important demand for the current deep geotechnical engineering construction. In this research, conventional triaxial loading and triaxial multistage cyclic loading and unloading tests were carried out on high-strength sandstones under different surrounding pressures by a multi-field coupled triaxial tester. The stress–strain response characteristics of high-strength sandstones under different stress conditions, the strength properties, and the strain distribution law in cyclic loading and unloading were investigated. The energy evolution law and damage characteristics of high-strength sandstone under triaxial cyclic loading and unloading conditions were also analysed. It provides a reference for the development of technologies for the excavation of roadways and chambers, surrounding rock support, and resource extraction in underground engineering.

2. Experimental Conditions and Experimental Programme Design

2.1. Sample Preparation

Deep sandstone is characterised by high strength and low permeability due to the high temperature and high pressure during the diagenetic process. However, when it is disturbed by external stresses, it is prone to stress-release phenomena such as rockbursts and impacts. In order to improve the engineering application of high-strength sandstone and better prevent the rock explosion and impact phenomenon in high-stress environments, this research selects high-strength sandstone as the research object (the sandstone is buried at a depth of approximately 500 m). The uniaxial compressive strength of the sandstone samples selected for the research is more than 90 MPa, which belongs to high-strength hard sandstone. Sample cores were taken from deeply embedded rock in their natural state, and the samples were screened by rock wave velocity tests to ensure that the experimental samples had good homogeneity and low dispersion. The standard cylindrical sample size of ϕ = 50 mm and H = 100 mm recommended by the International Society for Rock Mechanics (ISRM) was chosen [22]. The samples are sanded so that they have a flat appearance and no visible natural cracks. In this case, the end flatness of the sample should be within no more than 0.05 percent of the diameter of the sample. The surface roughness of the sample is chosen from the standards recommended by ISRM (surface roughness should not exceed 0.02% of the sample diameter, and there should be no visible cracks and holes). Both end faces of the sample need to be perpendicular to the axis. The deviation of the end faces from the axis should be controlled within ±0.5°.

2.2. Experimental Equipment

The triaxial rock multi-field coupling test platform was selected for the experiment, as shown in Figure 1. The test platform is mainly composed of a control system, an axial pressure system, a surrounding pressure system and triaxial pressure chamber, and high-precision sensors. The test platform loading system can apply a maximum of 500 MPa of axial stress and 60 MPa of surrounding pressure. The pressure accuracy is ±0.01 MPa. The effect of surrounding pressure is mechanically counteracted by a self-balancing piston head with a triaxial pressure chamber. The axial stress loaded on the sample is directly the stress difference. The triaxial pressure chamber is equipped with high-precision axial displacement transducers and circumferential strain gauges. The displacement sensor has a range of 12 mm and an accuracy of ±0.001 mm. The test platform can be applied to the conventional triaxial mechanical test and external multi-field coupling test of rocks. The principle of the experimental equipment is shown in Figure 2.

2.3. Experimental Programme

We have taken deep mining as an example. When the work face advances along the strike direction, the distribution of the upper roof rock support pressure near the work face is shown in Figure 3.

In the vicinity of the work face, stress-raising and stress-reducing zones are formed within a certain range. As the work face advances and the hydraulic support of the roadway top plate is “supporting and shifting”, the rock layer of the roadway top plate will be subjected to the cyclic alternating action of “loading and unloading”. The interior of the top rock layer at a point then exhibits repeated changes in stress in the vertical direction. In order to analyse the change pattern of the mechanical properties of the top rock layer at this point of the process, high-strength sandstone samples were investigated using triaxial graded cyclic loading and unloading. In order to investigate the mechanical and energetic characteristics of sandstone after thorough disturbance, the number of loading cycles per stage is 12. According to the experimental purpose and experimental equipment specifications in the triaxial multistage cyclic loading and unloading test, the stress-controlled loading method is adopted, in which the surrounding pressure is first loaded to a predetermined value and then the axial load is applied. The surrounding pressures were 0 MPa, 5 MPa, 10 MPa, 15 MPa, and 20 MPa with a loading rate of 0.1 MPa/s. The axial cyclic loading is divided into three stages of 0.45σ to 0.55σ, 0.55σ to 0.65σ, and 0.65σ to 0.75σ, where σ is the average of the peak stresses of the samples obtained from the conventional triaxial compression test at the corresponding surrounding pressure. The axial cyclic loading and unloading rates were both 0.1 MPa/s. After completing the corresponding cycle, loading and unloading continue to load at a rate of 0.1 MPa/s until the sample is destroyed, and the stress loading path is shown in Figure 4. In order to obtain the peak strength of the sample at different surrounding pressures, a conventional triaxial compression test at the corresponding surrounding pressure should also be carried out. After the sample reaches the peak strength, the residual strength of the sample is obtained by displacement loading in a controlled manner at a rate of 0.02 mm/min. The details of the experimental programme are shown in

Table 1. Experimental programme and parameters.

Test Loading Method	Surrounding Pressure σ2, σ3/MPa	Surrounding Pressure Loading Rate/MPa/s	Cyclic Loading and Unloading-Axial Pressure σ1/MPa	Axial Pressure Loading and Unloading Rate/MPa/s	Samples No.
Conventional triaxial loading	0,5,10,15,20	0.1	/	/	S1~S15
Triaxial cyclic loading and unloading			0.45σ~0.55σ, 0.55σ~0.65σ, 0.65σ~0.75σ	0.1	SC1~SC15

.

3. Triaxial Cyclic Loading and Unloading Experiment on High-Strength Sandstone

3.1. Characteristics of Stress–Strain Response of Triaxial Cyclic Loading and Unloading under Different Surrounding Pressures

Sandstone is a common brittle rock. Its stress–strain relationship has certain response characteristics from the deformation to the destruction of the sample under triaxial cyclic loading and unloading conditions. C. D. Martin [23] previously suggested that the volumetric strain of rocks could be used as a research method to analyse the mechanical properties of rocks. The volumetric strain ${\epsilon }_v$ of a rock sample can be expressed in terms of axial strain ${\epsilon }_1$ and lateral strain ${\epsilon }_3$ as shown in Equation (1).

$${\epsilon }_v={\epsilon }_1+2{\epsilon }_3$$

(1)

According to the experimental results, it can be seen that the difference between the peak values of the stress–strain curves obtained from the three samples under the same stress condition is small, which is less than 6%. High-strength sandstone samples are more homogeneous and less discrete. In order to simplify the analysis while ensuring scientific and reasonable experimental conclusions, one sample was selected from each group of three samples to be analysed. The conventional triaxial compressive stress–strain curves of the samples under different surrounding pressures are shown in Figure 5.

When the sample is in a certain constant surrounding pressure condition, the stress–strain curve exists in the pore-compacting stage, elastic stage, yield stage, and destructive stage. In the pore-compacting stage, the internal pore of the rock is continuously reduced by external stress, which is manifested by the rapid increase in axial strain and radial strain under low-stress conditions within a short period of time. With the increase in the surrounding pressure, the axial strain in the pore-compacting stage increases. The increase in surrounding pressure in this process has a hindering effect on the deformation of the rock produced by axial pressure. However, hard sandstone is internally homogeneous and has a small pore volume, so the pore-compacting stage of the sample is shorter. The stress–strain curve will rapidly enter the elastic stage. In the elastic stage, the stress and strain of sandstone samples are linearly proportional to each other. As the axial stress on the sample increases during the yield stage when the sample is approaching its peak strength, small cracks grow internally and the strain in the sample increases rapidly. The sample will enter the destructive stage after reaching the peak strength. The sample will produce a large number of cracks and the strain will increase rapidly.

The stress–strain curves of triaxial cyclic loading and unloading under different surrounding pressures are shown in Figure 6. A significant hysteresis effect was observed in the stress–strain curve of the rock samples during cyclic loading experiments, forming a closed stress–strain hysteresis loop between the unloading curve and the loading curve of the next cycle [24,25]. Defects such as microfractures, pores, and joints contained within natural rock cause non-ideal elasticity in rock deformation. When unloading after loading to a certain stress level, the unloading curve does not match the original loading curve and is lower than the original loading curve. The unloading curve forms a closed annular region with the loading curve of the next cycle. The presence of hysteresis loops in the stress–strain curve also represents energy dissipation. The area of the hysteresis loop can represent the amount of energy dissipated in the loaded rock sample as a result of crack closure, expansion, connection penetration, and destruction (dissipation energy). The larger the area of the hysteresis loop, the more energy is dissipated and the greater the damage to the rock sample.

The hysteresis loop curves for different cyclic loading stages at various surrounding pressures are shown in Figure 7. Under cyclic loading, there is a stiffness degradation in high-strength sandstone samples where the peak point displacement per cycle increases with the number of cycles. The stiffness degradation is progressively more pronounced with the increase in the surrounding pressure. The hysteresis loops are more dispersed in the first 3–4 cycles and the distribution of hysteresis loops generated by the rest of the loading cycles is more concentrated, with some hysteresis effect of hysteresis loops. From the variation in the hysteresis loop area of sandstone samples under the same surrounding pressure, it can be seen that the energy dissipation in the same level of cyclic loading decreases slightly with the increase in the number of cycles. This indicates that the first stage of cyclic loading caused some damage to the sandstone samples and micro cracks formed in the samples. The later stages of cyclic loading caused the samples to consume less energy to produce the same strain. Sandstone samples are subjected to strain accumulation under cyclic loading, as evidenced by the stress–strain curve where the end point of a single cycle is higher than the start point. The cyclic loading creates a permanent deformation of the samples. This strain accumulation increases with the number of cyclic loads applied.

3.2. Strength Characteristics of Sandstones under Triaxial Cyclic Loading at Different Surrounding Pressures

The peak strength, peak strain, and modulus of elasticity of the samples under conventional triaxial compression with different surrounding pressures and triaxial cyclic loading with different surrounding pressures are shown in

Table 2. Strength parameters of sandstone samples under conventional triaxial compression and triaxial cyclic loading at different surrounding pressures.

No.	Loading Method	Surrounding Pressure/MPa	Average Peak Strength/MPa	Average Peak Strain/10−3	Average Modulus of Elasticity/GPa
S1–S3	Conventional triaxial compression	0	94.2	5.9	18.5
S4–S6		5	133.5	8.2	22.4
S7–S9		10	158.1	8.9	22.7
S10–S12		15	183.7	10.2	23.5
S13–S15		20	204.9	11.3	23.4
SC1–SC3	Triaxial cyclic loading and unloading	0	81.6	6.4	15.0
SC4–SC6		5	125.3	8.5	20.7
SC7–SC9		10	151.1	9.4	22.8
SC10–SC12		15	177.5	10.0	23.6
SC13–SC15		20	198.1	11.3	23.7

. The sample undergoes predominantly elastic deformation in unloading and can be considered as having no damage rupture generated within the rock. Its deformation is the elastic recovery caused by the release of elastic energy saved by the rock and the unloaded stress–strain curve is approximated as a straight line. The slope of the unloading curve is taken as the modulus of elasticity of the rock at this time.

As shown in Figure 8, the peak strength of sandstone samples in both the conventional triaxial compression test and the triaxial cyclic loading test increased with the increase in the surrounding pressure. The increase in surrounding pressure restricts and closes the tiny cracks and pores within the sandstone, thus increasing its load-bearing capacity. When the cycle reaches a certain number of cycles or a certain intensity, it will lead to internal structural damage to the sample and cause its peak strength to be reduced compared to the peak strength in conventional triaxial compression. This deteriorating effect on peak strength is more pronounced at lower surrounding pressures. The peak strain of sandstone samples under both conventional triaxial compression and triaxial cyclic loading increases with increasing surrounding pressure. When the surrounding pressure reaches a certain value, the peak strain of the sample has a tendency to reach a certain value and remain stable. The modulus of elasticity of the sample also increases with increasing surrounding pressure. The closure of micro cracks under surrounding pressure resulted in a denser interior of the sample and a consequent increase in the modulus of elasticity. The modulus of elasticity of sandstone samples under cyclic loading increases with the increase in surrounding pressure. After the surrounding pressure reaches a certain value, the modulus of elasticity of the sandstone sample is gradually stabilised by continuing to increase the surrounding pressure. This is due to the fact that the internal structure of the sample may gradually deteriorate after undergoing repeated stresses. This includes phenomena such as crack extension and penetration, which lead to the elastic decay of the sample and a decrease in the changes in the elastic modulus.

3.3. Strain Distribution Pattern of Sandstone under Triaxial Cyclic Loading with Different Surrounding Pressures

The strain curves of sandstone samples under triaxial cyclic loading and unloading with different surrounding pressures are shown in Figure 9. The volumetric strain of the sample is taken to characterise the overall deformation of the sample. When the surrounding pressure is low (0 MPa or 5 MPa), the sample volumetric strain before cyclic loading and unloading is mainly concentrated in the tiny cracks and pore structures of the sample. At this time, there is obvious plastic deformation of the sample, which is manifested in the beginning of the volumetric strain curve as an upward concave and downward convex. With cyclic loading, some of the micro cracks and pores close and the volumetric strain of the sample is distributed uniformly throughout the post-peak stage. When the surrounding pressure is high (15 MPa or 20 MPa), most of the micro cracks have closed. Under the action of cyclic loading micro cracks slide, the sample volumetric strain is concentrated in the peak after the stage of the occurrence of obvious macroscopic damage. It is manifested by the rapid change in the volumetric strain curve along the horizontal direction. From the figure, it is also found that the stress–volume–strain curve can be divided into a volume-decreasing phase and a volume-increasing phase. When the external load is small, the gradual increase in the load will lead to the gradual closure of the internal micro cracks in the sandstone sample, resulting in a gradual decrease in the volume of the sample. When the external load reaches the critical load, the external load will cause the internal micro cracks of the sample to turn from closed to open, which will lead to an increase in the volume of the sample and destabilisation. The point of maximum triaxial cyclic loading and unloading volumetric strain is taken for analysis. The maximum values of volumetric strain were 0.37%, 0.47%, 0.56%, 0.60%, and 0.65% when the surrounding pressure was 0 MPa, 5 MPa, 10 MPa, 15 MPa, and 20 MPa, respectively. According to the conventional triaxial loading volumetric strain maxima in Figure 5, the volumetric strain maxima are 0.09%, 0.40%, 0.46%, 0.51%, and 0.57% when the surrounding pressures are 0 MPa, 5 MPa, 10 MPa, 15 MPa, and 20 MPa, respectively. The peak volumetric strain of the sandstone samples increased significantly with increasing surrounding pressure, and cyclic loading produces greater cumulative damage within the sandstone sample at the same volumetric strain, making the sample unable to withstand larger load increments after the peak volumetric strain is reached. It has also been shown that hard rock that has undergone more stress adjustments under the same conditions has a much smaller bearing potential than those that have undergone fewer stress adjustments after the external load exceeds the value of the stress that caused the damage.

4. Characteristics of Energy Evolution of Triaxial Cyclic Loading and Unloading in High-Strength Sandstone

4.1. Energy Properties of Sandstone under Triaxial Cyclic Loading and Unloading

According to the law of energy conservation, it is assumed that the action of external loads does not cause changes in the internal thermal energy of the sandstone sample. Therefore, the energy input from outside is mainly mechanical energy. In this process, mechanical energy is converted into elastic energy stored in the elastic deformation of the sample, plastic energy corresponding to plastic deformation, and energy released by dissipation of the destruction of the sample. During the pre-peak phase of the stress–strain curve, most of the energy input into the testing machine is stored in the sample in the form of releasable elastic energy. The elastic energy density $U^e$ can be expressed as Equation (2) and the elastic strain ${\epsilon }_{}^e$ can be expressed as Equation (3).

$$U^e=\frac{1}{2}{\sigma }_1{\epsilon }_1^e+\frac{1}{2}{\sigma }_2{\epsilon }_2^e+\frac{1}{2}{\sigma }_3{\epsilon }_3^e$$

(2)

$$\begin{array}{l}{\epsilon }_1^e=\frac{1}{E_1}\left[{\sigma }_1-{\mu }_1\left({\sigma }_2+{\sigma }_3\right)\right]\\ {\epsilon }_2^e=\frac{1}{E_2}\left[{\sigma }_2-{\mu }_2\left({\sigma }_1+{\sigma }_3\right)\right]\\ {\epsilon }_3^e=\frac{1}{E_3}\left[{\sigma }_3-{\mu }_3\left({\sigma }_1+{\sigma }_2\right)\right]\end{array}\}$$

(3)

where ${\sigma }_1$ is the principal stress, ${\epsilon }_1^e$ is the elastic strain in the direction of the principal stress, $E_i$ is the elastic modulus, and ${\mu }_i$ is Poisson’s ratio. It is assumed that the sandstone sample is homogeneous and isotropic, and the testing machine is loaded with equal surrounding pressure (${\sigma }_2={\sigma }_3$). Therefore, it can be simplified to obtain Equation (4).

$$U^e=\frac{1}{2E}\left({\sigma }_1{}^2+2{\sigma }_3{}^2-4\mu {\sigma }_1{\sigma }_3-2\mu {\sigma }_3{}^2\right)$$

(4)

The surrounding pressure ${\sigma }_3$ is kept constant during the test loading. Equation (5) is obtained by integrating Equation (4).

$$d{U}^e=\frac{{\sigma }_1-2\mu {\sigma }_3}{E}d{\sigma }_1-\frac{1}{2E^2}\left({\sigma }_1{}^2+2{\sigma }_3{}^2-4\mu {\sigma }_1{\sigma }_3-2\mu {\sigma }_3{}^2\right)dE$$

(5)

During unloading, the rock undergoes mainly elastic deformation, so dE≈0. $d{\epsilon }_3=-\mu d{\epsilon }_1=-\frac{\mu }{E}d{\sigma }_1$, and Equation (5) can be reduced to

$$d{U}^e=\frac{{\sigma }_1}{E}d{\sigma }_1-2\frac{\mu {\sigma }_3}{E}d{\sigma }_1={\sigma }_{1}d{\epsilon }_1+2{\sigma }_{3}d{\epsilon }_3$$

(6)

The elastic energy released by unloading can be expressed as

$$U^e={\displaystyle {\int }_{{\epsilon }_1^d}^{{\epsilon }_1}{\sigma }_{1}d}\epsilon +2{\sigma }_3\left({\epsilon }_3-{\epsilon }_3^d\right)$$

(7)

where ${\epsilon }_1$ and ${\epsilon }_3$ are the axial and circumferential strains at unloading, respectively. ${\epsilon }_1^d$, ${\epsilon }_3^d$ are the axial and circumferential strains after unloading, respectively.

The energy consumed for crack extension, development, internal damage, etc., in a sample is called dissipation energy $U^d$ and can be expressed as

$$U^d=U-U^e={\displaystyle {\int }_{{\epsilon }_1^0}^{{\epsilon }_1}{\sigma }_{1}d}\epsilon -{\displaystyle {\int }_{{\epsilon }_1^d}^{{\epsilon }_1}{\sigma }_{1}d}\epsilon +2{\sigma }_3\left({\epsilon }_3^d-{\epsilon }_3^0\right)$$

(8)

where ${\epsilon }_1^0$, ${\epsilon }_3^0$ are the axial and circumferential strains at the start of loading, respectively.

In general, the energy evolution of loaded samples goes through the processes of energy input, accumulation, dissipation, and release. Considering the reversibility of the elastic energy and the irreversibility of the dissipation energy, only the axial elastic energy and dissipation energy are systematically investigated. The elastic and dissipation energies of the samples were calculated using triaxial cyclic loading and unloading test stress–strain curves. The calculation of the energy density of the sample in cyclic loading and unloading at a certain stress level is shown in Figure 10. The elastic energy density $U^e$ and dissipation energy density $U^d$ are shown in Equation (9).

$$\begin{array}{l}{U}_i^e={\displaystyle {\int }_{{\epsilon }_i^d}^{{\epsilon }_i}{\sigma }_{i}d}{\epsilon }_i\\ U_i^d={\displaystyle {\int }_0^{{\epsilon }_i}{\sigma }_{i}d}{\epsilon }_i-{\displaystyle {\int }_{{\epsilon }_i}^{{\epsilon }_i^d}{\sigma }_{i}d}{\epsilon }_i\end{array}\}$$

(9)

The cyclic loading count–energy density curve of the sample is shown in Figure 11. When the surrounding pressure of sandstone samples was kept constant, the total energy density U, elastic energy density $U^e$, and dissipation energy density $U^d$ of the samples at the same level of cyclic loading and unloading phases showed an overall tendency to decrease with the increase in cycle count. From the dissipation energy density curve and the hysteresis loop area of the stress–strain curve, it can be concluded that there is energy loss in the sample during cyclic loading and unloading. The energy dissipation in a single cycle decreases with increasing cycle count. The total energy of the sample decreases significantly in the first four cycles. The energy density curves of the loaded sandstone samples are characterised by more pronounced stages. The dissipated energy in a single cycle of the sample increases as it enters the next higher stress level of the stress cycle. When analysed in conjunction with Figure 9 and Figure 11, the energy density of the sample exhibits an aggregation of energy due to the incomplete unloading and small amplitude of the applied unloading stresses. Elastic energy occupies the main part. With the release of elastic energy and the increase in dissipation energy, the energy storage property of the sandstone sample decreases. The deformation of the loaded sandstone sample continues to increase until instability damage occurs.

The total energy density, elastic energy density, and dissipated energy density curves of cyclically loaded samples at different surrounding pressures are shown in Figure 12. According to Figure 12a, the total energy density increases slowly with increasing cyclic stress levels at a surrounding pressure of 0 MPa. The increase in total energy density is more significant at a surrounding pressure of 20 MPa. In the first stage of the cycle, the total energy density was at a relatively low level for all surrounding pressure conditions. The total energy density at 0 MPa surrounding pressure is maintained at approximately 75 kJ/m³, while the total energy density at 20 MPa surrounding pressure is close to approximately 262 kJ/m³. During the third stage of cyclic loading, a significant increase in the total energy density for each surrounding pressure condition was observed. The sandstone samples showed the most significant increase in total energy density at 20 MPa surrounding pressure, with the total energy density increasing from approximately 300 kJ/m³ to approximately 375 kJ/m³. This shows that the energy absorption capacity of hard sandstone under higher surrounding pressure is significantly improved. According to Figure 12c, the dissipation energy densities of the hard sandstone samples in each surrounding pressure condition in the first cyclic loading and unloading stage at low stress levels fluctuate with the increase in cycle count. At 0 MPa surrounding pressure, the dissipation energy density of the samples fluctuated around 4.2 kJ/m³, and at 20 MPa, the dissipation energy density fluctuated in a larger range (6.1 kJ/m³~16.5 kJ/m³). Significant peaks in dissipation energy density can be observed during the loading and unloading phases in the second stage of the cycle, especially at higher surrounding pressures (15 MPa~20 MPa). In this stage, the dissipation energy density at 20 MPa reaches a maximum value of approximately 27.8 kJ/m³. The peak value of the dissipated energy density of the sandstone samples in the loading and unloading phases in the third stage of the cycle decreased at most surrounding pressures. Especially at 20 MPa, it is reduced from the peak value to approximately 6.8 kJ/m³. This may indicate that the energy dissipation capacity of the sandstone samples recovered a little or the samples adapted to the cyclic loading as the cycling progressed. This also indicates that the dissipation energy of hard sandstone samples increases dramatically under cyclic loading at high surrounding pressures and high stress levels.

The energy absorption capacity of hard sandstones increases with increasing surrounding pressure and increasing cyclic loading stress levels. The variation in dissipated energy density indicates the damage and plastic deformation characteristics of sandstone samples under triaxial cyclic loading. At low surrounding pressures, the dissipation energy density is relatively low. The dissipation energy density of the sandstone samples increased with increasing surrounding pressure. It may be that hard sandstone samples produce more damage or micro cracks at higher surrounding pressures, causing them to have more energy dissipation. The energy evolution characteristics exhibited by hard sandstones under triaxial cyclic loading and unloading may be related to the processes of micro crack expansion, closure, and penetration in the sandstones, which in turn influence their energy evolution behaviour.

4.2. Patterns of Energy Distribution in Sandstone

The energy absorbed by a loaded sandstone sample from the exterior is mainly converted into elastic and dissipation energy. The ratios of elastic energy and dissipation energy may affect the damage and destruction characteristics of sandstone samples. According to Figure 13, the overall fluctuation trend of the ratio of elastic energy and the ratio of dissipation energy with the increase in cyclic loading and unloading times are both close to a sinusoidal function. In Figure 13a, the elastic energy percentage increases and peaks as the cycle count increases in the same level of cyclic loading. From the different cyclic loading stages, the elastic energy percentage in the first cyclic loading stage reaches the highest. Afterwards, the peak value of the elastic energy percentage decreases as the number and level of loading cycles increase. The fluctuation of the elastic energy percentage of the samples gradually decreases. This also indicates that a decrease in the modulus of elasticity of the sandstone samples may have occurred with the increase in the cycle count of loading. The accumulation of micro cracks may have occurred within the rock samples. The elastic energy percentage will show an overall weakly increasing trend with increasing surrounding pressure. As an example, the average value of the elastic energy percentage at a surrounding pressure of 20 MPa is approximately 1.5% higher than that at a surrounding pressure of 0 MPa. This indicates that higher surrounding pressure can promote the elastic recovery of sandstone samples. It is possible that the effect of surrounding pressure reduces the expansion and slip of micro cracks within the rock. The trend of dissipation energy percentage changes in sandstone samples at different surrounding pressures in Figure 13b is similar to the trend of elastic energy percentage changes. The dissipation energy percentage of sandstone samples showed a fluctuating decrease with the increase in cycle count. The percentage of dissipation energy decreases as the cycle count increases for sandstone samples under the same cyclic loading. This indicates that cracks and defects within the rock stabilise after many cycles of loading, resulting in less energy being dissipated for each cycle thereafter. Regarding the effect of different surrounding pressures on the percentage of dissipation energy of hard sandstone samples, it can be found that at lower surrounding pressures (0 MPa and 5 MPa), the percentage of dissipation energy is relatively high. This occurs due to the greater susceptibility of sandstone samples to plastic deformation and internal damage at low surrounding pressures. As the surrounding pressure increased to 15 MPa or 20 MPa, the proportion of dissipated energy decreased. It may be due to the high surrounding pressure, which enhances the bearing capacity of sandstone samples and inhibits crack extension.

4.3. Damage Characteristics of Hard Sandstone under Triaxial Cyclic Loading and Unloading

The macroscopic damage characteristics of rock depend on the distribution of micro cracks and the expression of detailed mechanical properties [26]. The main forms of macroscopic damage in loaded rock are shear damage, tensile damage, compression damage, and laminar spalling. The macroscopic crack morphology and crack distribution of hard sandstone samples after conventional triaxial compression and triaxial cyclic loading experiments can be observed to obtain their macroscopic damage characteristics and patterns. The damage characteristics of the loaded sandstone samples are shown in Figure 14. The red part indicates the main macroscopic cracks that lead to the strength loss and destabilisation of the sample, while the green part indicates the micro cracks or tension cracks due to the loading of the rock.

As shown in the figure, conventional triaxial compression conditions, sandstone samples in the surrounding pressure of 0 MPa (without the constraints of the surrounding pressure), and the load damage after the rock of the obvious macro cracks are more developed. The form of damage is mainly dominated by shear damage and the samples are more broken locally. As the surrounding pressure increases, the main cracks that cause the sample to lose strength become more and more obvious. The form of damage is dominated by shear damage. Shear misalignment of a sample along one or more planes inclined in the direction of the maximum principal stress is observed. These planes are usually located at the interface of the weaker mineral grains in sandstone samples. Smooth friction surfaces and shear marks can be observed on the damaged surface. Under triaxial cyclic loading and unloading conditions, the damage characteristics of the samples are more obviously different from those of conventional triaxial compression due to repeated loading and unloading. It is also closer to the damage form under actual natural conditions. Under cyclic loading, the sandstone samples were always subjected to alternating compression–expansion cycles. This also promotes the alteration of primary cracks within the sample and the generation of new cracks, which in turn leads to an increase in the level of rock damage. As the loading and unloading frequency increases, the internal particles of the sample may be damaged and collapse. Deformation damage of rock is the whole process of energy dissipation and energy release. Destabilisation damage of rock is the result of an instantaneous release of energy in the rock [27]. When there is no constraint of surrounding pressure (the surrounding pressure is 0 MPa), the macroscopic cracks distributed throughout the sandstone samples are more developed. The form of damage is mainly shear damage and damage through the sample, accompanied by tensile damage secondary cracks approximately perpendicular to the direction of the maximum principal stress. At lower surrounding pressures (5 MPa and 10 MPa) at which the samples were exposed, shear damage penetrated the samples. Some of the samples showed crack extension and connection. Micro cracks within the sandstone samples also show stress concentration and produce fewer tension cracks as the cracks slip and expand. When the surrounding pressure is higher (15 MPa and 20 MPa), the sandstone samples will also develop more secondary cracks perpendicular to the penetrating cracks. The sample is more broken locally. The form of damage mainly consists of shear damage and tensile damage inclined to the direction of the principal stress. There is also localised compression damage manifested under high surrounding pressure due to the contraction of the internal pore structure of the sandstone sample under high stress. The damage form of the sample is more complex. The phenomenon of rock damage accumulation is obvious. The crack network on the sample surface and inside is also more developed.

5. Conclusions

In this research, conventional triaxial compression tests and triaxial cyclic loading and unloading tests were carried out using a triaxial rock multi-field coupled test platform for high-strength hard sandstones. In the research programme, multiple stress cycles were set up and repeated loading and unloading was carried out in each stress cycle. The mechanical properties and energy evolution characteristics of high-strength hard sandstone under triaxial cyclic loading and unloading conditions were investigated from the stress–strain response characteristics, strength characteristics, energy characteristics, and macroscopic damage characteristics of high-strength hard sandstone under different surrounding pressures. The main conclusions are as follows:

1. The stress–strain curve of high-strength hard sandstone under cyclic loading can be divided into the stages of pore compaction, elasticity, yielding, and destruction. As the cycle count increases and the surrounding pressure increases, the sandstone shows significant stiffness degradation. The hysteresis loops in the same level of stress cycling show a significant sparse distribution with the increase in cycle count. The first three to four cycles produce a cumulative cyclic damage displacement that is approximately half of the cumulative damage displacement at that stage. The effect of cyclic loading and unloading under low surrounding pressure conditions on the peak strength of sandstone is more pronounced. The peak volumetric strain of hard sandstone samples under cyclic loading and unloading conditions increases with increasing surrounding pressure.

2. The total energy density, elastic energy density, and dissipated energy density of high-strength sandstone at the same stage of cyclic loading show an overall decrease with the increase in cyclic times. The cyclic load dissipation energy decreases as the cyclic loading times increase. The dissipation energy is larger at the beginning of the cycle (first to fourth) of each cycle stage. This part of the cyclic loading plays a major role in damaging rock. There is a rapid increase in the dissipation energy of hard sandstone samples under cyclic loading with high surrounding pressure and high stress levels.

3. The elastic energy density ratio and dissipation energy density ratio of hard sandstone under cyclic loading tend to fluctuate as a sinusoidal function. As the cycle count increases, the elastic energy density ratio gradually rises to reach the peak value of this cycle stage. The fluctuations of the ratio of elastic energy and dissipation energy decreased gradually with the increase in cyclic stress level and cycle count. The accumulation of micro cracks within the sandstone also reduces the elastic energy percentage. Higher surrounding pressure promotes the recovery of elasticity in hard sandstone samples. It can also lead to a faster stabilisation of the percentage of elastic energy and dissipation energy. Cracks and defects within the rock stabilise after many cycles of loading. This causes the percentage of dissipation energy within the same cyclic phase to decrease with increasing cycle count.

4. Hard sandstones subjected to cyclic loading have more pronounced brittle damage characteristics. The form of damage at low surrounding pressure is mainly shear damage with a small amount of tension cracking. The generation of through cracks when surrounding pressure is high is accompanied by secondary cracks perpendicular to the direction of the maximum principal stress. The sandstone samples are partially damaged badly. Its damage form is also mainly shear damage, tensile damage, and partial compression damage.