The deformation and failure of sandstone samples are closely related to energy changes in the material. To explore the energy evolution during the process of sandstone sample damage, loading and unloading tests with different test paths were conducted. The results show that more energy is stored and consumed before the stress reaches its peak, while after the peak stress, more energy is released and consumed. Energy dissipation increases internal cracking, leads to sample damage and lithologic deterioration, and reduces the bearing capacity of the sample. During triaxial unloading of the confining pressure, the higher the initial unloading confining pressure, the more the elastic energy stored, and the more the energy released when the sandstone sample fails, resulting in more severe damage. Therefore, during the excavation of high-stress rock masses, large amounts of elastic energy stored in sandstone can be rapidly released, leading to rock burst disasters. Additionally, during triaxial unloading confining pressure tests, the damage in sandstone when the sample is close to failure increases more rapidly than that during conventional triaxial compression tests because of the unloading effect of the confining pressure. This phenomenon also illustrates that the failure of sandstone induced by unloading is more sudden than that induced by loading.
Mining, tunnel excavation, and other projects in deep mines involve high-stress rock unloading, which often leads to serious accidents that threaten the security of mining [
By analyzing the evolution of energy in the material, valuable numerical results about the failure regularity of rock have been acquired. Lan et al. [
These results provide important references for the study of rock energy evolution and the relationship between rock energy evolution and rock deformation and failure. However, these studies mainly examined marble and granite under uniaxial compression, conventional triaxial compression, and triaxial unloading and confining pressure. In addition, a number of studies have investigated energy evolution in sandstone with certain conventional test paths, such as uniaxial compression and triaxial compression, but research on energy conversions during triaxial unloading confining pressure tests in sandstone is scarce.
To explore the essential characteristics of sandstone deformation and failure, we performed triaxial compression and triaxial unloading tests of sandstone under different confining pressures and studied the energy changes during the damage process under triaxial compression and triaxial unloading confining pressure. The results may be used as guidance for the prevention and control of mine disasters caused by excavation and stress unloading in deep mines.
Samples of test rock were taken from the roof sandstone of the No. 3 coal in Yangcun Coal Mine, Jining, Shandong, China. According to the engineering rock mass test standards, the coal block was cut into 50 × 100 mm (diameter × height) cylinders. To ensure that the specimens were similar and homogeneous, they were subjected to ultrasonic testing, and the specimens that showed high wave velocities were removed, leaving the specimens with similar velocities. The rock samples used in the test are shown in Figure
Processing of rock samples.
Triaxial loading and unloading tests were carried out on the MTS815.02 electrohydraulic servo rock mechanics test system of the China University of Mining and Technology. The test system could meet the test requirements under a variety of complex paths. The test plan in this study was as follows:
Therefore, the experiment adopted the loading path of increasing axial pressure and decreasing confining pressure. Under this path, the failure of the specimen is quickest and most dangerous. The experiment was divided into three stages: (1) increase the confining pressure (
Stress path of unloading confining pressure.
Test process and rock samples after testing.
When conventional triaxial compression tests or confining pressure triaxial unloading tests are carried out, the specimen is placed in the triaxial chamber of the testing machine, and the specimen and the testing machine are regarded as a closed system. Thus, during the deformation, damage, and destruction of the specimen in the triaxial chamber, no energy is exchanged with the outside environment, and energy exchange occurs solely within the testing apparatus.
In conventional triaxial compression tests, in addition to the axial energy
Furthermore, the total energy
Combining formulas (
The loading path of the triaxial unloading confining pressure tests lies between those of the uniaxial compression tests and the conventional triaxial compression tests. When the unloading rate of the confining pressure is relatively fast, the confining pressure decreases to 0 rapidly, which is similar to a uniaxial stress state. When the unloading rate of the confining pressure is relatively slow, the confining pressure decreases by a very small value for a period of time, which is similar to a conventional triaxial test. Therefore, the principle of energy calculation in the conventional triaxial test is also applicable to triaxial unloading confining pressure tests.
The calculation methods of the above energy [ The energy of the hydrostatic stress state ( The axial energy ( In these formulas, At a certain time where Formula (
According to formulas (
According to the stress-strain curve obtained from the experimental results, the energy of the sandstone under different test conditions is calculated using the energy calculation methods described in this paper. The results are plotted in certain curves to analyze the characteristics of energy evolution.
Figure
Energy curves of the rock sample under a confining pressure of 10 MPa.
At the beginning of the test,
Then, the sample is in the elastic stage (between dashed lines A and B), and as the axial force increases,
After the sandstone enters the plastic deformation stage (between points B and C),
When the sandstone reaches the peak stress (point C), the elastic energy
After point D, the stress-strain curve indicates a rapid release of
After point E,
During the triaxial unloading confining pressure, the hydrostatic energy also accumulates at the hydrostatic loading stage of the sandstone. Therefore, all the curves originate at the hydrostatic pressure point, and the analysis focuses on the energy evolution after the hydrostatic loading stage. Figure
Energy evolution curves of sandstone samples under unloading confining pressure. (a) Energy-time curve (C-7-0.05). (b) Energy-axial-strain curve (C-7-0.05). (c) Energy-radial-strain curve (C-7-0.05). (d) Energy-volume-strain curve (C-7-0.05).
At the beginning of the test, both
Before the confining pressure is applied, as the axial force increases,
When the confining pressure is applied,
When the peak stress value is reached, a large amount of elastic energy is rapidly released. The cracks in the sandstone converge and form a macroscopic fracture surface. The dissipation energy
After the plastic flow stage, the axial strain continues to increase with time, whereas both energy indexes tend to be stable.
According to the energy-radial-strain curve and the energy-volume-strain curve during the experiment, at the beginning of the test, the rules of energy varying with radial strain and volume strain are similar to the rules of energy varying with time and axial strain, which increases with strain. When the confining pressure is applied, the radial strain increases, and the value of volume strain is negative, indicating dilatancy in the sandstone. At the same time,
Overall, under the different test paths, such as the conventional triaxial compression test or the triaxial unloading confining pressure test, the energy in sandstone is mainly in the form of storage and dissipation before the peak stress, whereas the energy is mainly in the form of release and dissipation after the peak stress. In addition, energy dissipation expands the internal cracks in the sandstone, which results in damage to the rock and a decrease in the bearing capacity, while energy release causes instability and failure in the sandstone, as shown in Figure
Figures
Comparison of elastic energy curves under different confining pressures.
Comparison of dissipated energy curves under different confining pressures. (a) Dissipative energy-radial strain curve. (b) Dissipated energy-deviation stress curve.
At the axial loading stage, the higher the initial unloading confining pressure is, the greater the elastic energy of the sandstone is. The growth rate of elastic energy at high confining pressure is higher than that at low confining pressure. Then, when the confining pressure is unloaded, the elastic energy still increases, but the increase rate decreases with the unloading of the confining pressure. After the peak stress, a large amount of elastic energy is released, and the value of
During the early stages of the test, the dissipation energy increases gradually; however, under different confining pressures, the growth curve changes from linear to nonlinear. Moreover, the higher the confining pressure is, the faster the growth rate is. After the start of the unloading, the dissipation energy increases further. When the stress is near the peak value, the sandstone specimen fails, and dissipation increases sharply. The higher the initial confining pressure is, the greater the dissipation energy is, as shown in Figure
The energy indexes of the sandstone at the peak stress under different initial unloading confining pressures are shown in Table
Energy levels at peak stress under different initial unloading confining pressures.
Lithology | Initial unloading confining pressure (MPa) | Unloading pressure rate (MPa/s) |
|
Energy at peak stress (kJ/m3) | ||||
---|---|---|---|---|---|---|---|---|
|
|
|
|
|
||||
Sandstone | 4 | 0.05 | 129.08 | 351.01 | −16.96 | 298.70 | 35.35 | 334.06 |
7 | 0.05 | 132.39 | 452.57 | −22.23 | 368.06 | 62.28 | 430.34 | |
10 | 0.05 | 136.94 | 556.68 | −55.36 | 416.70 | 84.62 | 501.32 | |
13 | 0.05 | 145.56 | 581.67 | −61.86 | 430.23 | 89.58 | 519.81 | |
16 | 0.05 | 154.78 | 599.73 | −63.37 | 440.85 | 95.50 | 536.36 | |
19 | 0.05 | 175.7 | 722.39 | −88.05 | 523.95 | 110.40 | 634.35 |
To accurately reflect the energy characteristics of the sandstone specimens during this stage with different initial unloading pressures, we eliminate the energy of the specimen obtained from the axial loading stage, and the energy difference Δ
Energy transformation during the unloading of the confining pressure under different initial confining pressures.
During the unloading of the confining pressure stage, all the axial energy increments Δ
Figure
Energy transformation of loading and unloading confining pressure periods under different initial unloading confining pressures.
The accumulations of Δ
In practical engineering, the release of elastic energy in rock with high-stress conditions is significantly higher than that in rock under low-stress conditions when the rock fails. Therefore, when a high-stress rock mass is excavated, much elastic energy can be released rapidly, resulting in rock burst and other geological disasters.
Unlike Δ
In this paper, we assume that any external work performed on the sandstone specimens is converted into elastic energy and dissipative energy and that all the dissipation energy contributes to the damage to the sandstone. Hence, during the experiment, the damage caused by different stress paths, such as axial loading and radial unloading, can be regarded as the work of dissipation energy. The more the dissipation energy used, the greater the damage to the sandstone. Therefore, based on the definition of damage by Kachanov [
According to the elastic mechanics and related references, the constitutive stress-strain relation in the rock considering the damage characteristics can be expressed as
Substituting formula (
Based on the test results and formula (
In the conventional triaxial compression test (Figure
Damage curves of rock samples under conventional triaxial compression based on the dissipation energy.
Comparing the damage curves of sandstone under different confining pressures in conventional triaxial loading tests, Figure
According to the damage evolution of the triaxial unloading of the confining pressure (Figure
Damage curves of rock samples under unloading of the confining pressure based on the dissipation energy.
Comparing the damage to sandstone during conventional triaxial compression tests and triaxial unloading confining pressure tests, the sudden increase in the damage to the sandstone near the peak stress is more obvious in the latter tests because of the unloading effect of the confining pressure.
The energy evolution characteristics of sandstone samples under different confining pressures are studied in this paper, and some differences in the scale and mechanical properties are found between rock samples and actual rock masses. Studying the energy evolution law of rock samples during unloading confining pressure failure can further clarify the energy characteristics of the unloading confining pressure failure of rock samples. The greater the burial depth, the more the energy stored in the rock mass. Inducing a sudden release of a large amount of energy during excavation is easy. Therefore, during the process of deep rock mass excavation, some engineering measures should be taken to release internal energy in a timely manner. Although the research results cannot be directly used in field engineering, the principles revealed by the results are similar to those of unloading failure during the actual rock excavation process. Further applying these principles in engineering is the direction of follow-up research.
In this study, we conducted conventional triaxial compressional loading tests and triaxial unloading confining pressure tests under different loading paths for sandstone specimens. The results show that more energy is stored and consumed before the stress reaches its peak, while after the peak stress, more energy is released and consumed. The energy dissipation causes the internal cracks in the sandstone to expand, resulting in damage and deterioration, and the released energy causes the failure of the sandstone.
In the triaxial unloading confining pressure tests, the higher the initial confining pressure, the more the elastic energy stored in the sandstone; thus, under higher confining pressures, more energy is released when the failure of sandstone occurs, resulting in more severe disasters. In actual engineering, the release of elastic energy in rock under high-stress conditions is significantly higher than that in rock under low-stress conditions when the rock fails. Therefore, when a high-stress rock mass is excavated, much elastic energy can be released rapidly, resulting in rock burst and other geological disasters.
The higher the initial confining pressure, the more the energy consumed when the radial deformation does work to overcome the confining pressure during the axial loading stage. The dissipation energy also increases with the initial confining pressure caused by the appearance and development of cracks. This result demonstrates that the initial confining pressure has a great influence on the radial deformation of sandstone samples.
Compared to conventional triaxial compression tests, triaxial unloading confining pressure tests cause more obvious changes in the rock damage when the failure of sandstone occurs, and the unloading effect of the confining pressure contributes to this phenomenon. The rock failure caused by unloading is also concluded to be more serious than that caused by loading.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest related to the publication of this paper.
This study was supported by the National Key R&D Program of China (2018YFC0604705), the National Natural Science Foundation of China (51574156), and the Shandong Province Higher Educational Science and Technology Program (J18KA195).