# Experimental Investigation of Energy Evolution in Sandstone Failure during Triaxial Unloading Confining Pressure Tests

Authors: Rui Yang; Depeng Ma; Yongjie Yang
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1]. From a thermodynamic point of view, the failure process of a rock mass is seen as a state of energy-driven instability, and the deformation and failure of the rock mass are directly related to its energy conversion. Therefore, by focusing on the energy conversions during the processes of rock deformation and damage, the essential characteristics of rock failure can be further revealed.By analyzing the evolution of energy in the material, valuable numerical results about the failure regularity of rock have been acquired. Lan et al. [2] studied the distribution of elastic energy within a rock mass by analyzing the tectonic stress field distribution and obtained the stress and energy conditions at the time of rock burst. Cong et al. [3] and Yang et al. [4] performed conventional compression tests on marble specimens to analyze the changes in energy during the failure of the specimens. Based on experimental results, Liu et al. [5] analyzed the relationship between the energy and the strain in marble under loading and unloading. Based on conventional triaxial compression tests and unloading confining pressure tests of brittle granite under different experimental conditions, Chen et al. [6] obtained a criterion for rock burst based on energy principles. Chen et al. [7] revealed the characteristics of energy changes in marble under high confining pressure triaxial unloading tests. Zhao et al. [8] built the connection between the acoustic emission energy and the efficiency of rock failure under the combined action of dynamic and static loads on granite. Huang et al. [9] analyzed the energy evolution mechanism of coarse-grained marble under uniaxial compression. Pan et al. [10] presented comprehensive stiffness and comprehensive energy criteria for rock failure. Song and Yang [11] performed uniaxial compression tests of coal rock to examine the energy evolution of coal pillar failure. Zhang et al. [12] studied energy dissipation during tensile failure of saturated water samples. Based on uniaxial compression and triaxial compression tests, Ma et al. [13] studied the influence of the loading rate and confining pressure on the energy dissipation characteristics of coal samples. In addition, many scholars have studied the mechanical and acoustic emission characteristics of sandstone under different stress paths [14–19].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.


## 2. Materials and Methods

### 2.1. Sample Selection

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 Figure1.Figure 1
Processing of rock samples.
(a)
(b)


### 2.2. Test Program

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:Loading tests: triaxial tests of specimens under different confining pressures were carried out; the confining pressure (4 MPa, 7 MPa, or 10 MPa) was gradually applied under hydrostatic pressure. Under constant confining pressure, the axial pressure was increased by axial displacement control at a loading rate of 0.002 mm/s until the specimen failed.Triaxial unloading confining pressure tests: during the process of mining, the lateral stress in coal bodies (equivalent to the confining pressure) is gradually reduced, and the advanced support pressure (equivalent to the axial stress) is gradually increased. Working face mining is actually the process of reducing the confining pressure and increasing the axial pressure of the rock mass and its surroundings.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 (σ3) gradually to a predetermined value (4 MPa, 7 MPa, or 10 MPa) according to the hydrostatic pressure condition; (2) keep the value of σ3 unchanged and increase the axial pressure (σ1) to eighty percent of the peak compressive stress from conventional triaxial tests through the stress control method; and (3) use the displacement control method to increase the value of σ1, and at the same time, decrease the value of σ3 at a rate of 0.05 MPa/s until the specimen is destroyed. Then, the decreasing of the confining pressure is stopped immediately after the specimen is damaged, while the axial pressure continues to load to the residual strength of the specimen using the displacement control method. The stress path of the unloading confining pressure is shown in Figure 2, and the test process and posttest sample are shown in Figure 3.Figure 2
Stress path of unloading confining pressure.Figure 3
Test process and rock samples after testing.
(a)
(b)
(c)
(d)


## 2.1. Sample Selection

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 Figure1.Figure 1
Processing of rock samples.
(a)
(b)


## 2.2. Test Program

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:Loading tests: triaxial tests of specimens under different confining pressures were carried out; the confining pressure (4 MPa, 7 MPa, or 10 MPa) was gradually applied under hydrostatic pressure. Under constant confining pressure, the axial pressure was increased by axial displacement control at a loading rate of 0.002 mm/s until the specimen failed.Triaxial unloading confining pressure tests: during the process of mining, the lateral stress in coal bodies (equivalent to the confining pressure) is gradually reduced, and the advanced support pressure (equivalent to the axial stress) is gradually increased. Working face mining is actually the process of reducing the confining pressure and increasing the axial pressure of the rock mass and its surroundings.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 (σ3) gradually to a predetermined value (4 MPa, 7 MPa, or 10 MPa) according to the hydrostatic pressure condition; (2) keep the value of σ3 unchanged and increase the axial pressure (σ1) to eighty percent of the peak compressive stress from conventional triaxial tests through the stress control method; and (3) use the displacement control method to increase the value of σ1, and at the same time, decrease the value of σ3 at a rate of 0.05 MPa/s until the specimen is destroyed. Then, the decreasing of the confining pressure is stopped immediately after the specimen is damaged, while the axial pressure continues to load to the residual strength of the specimen using the displacement control method. The stress path of the unloading confining pressure is shown in Figure 2, and the test process and posttest sample are shown in Figure 3.Figure 2
Stress path of unloading confining pressure.Figure 3
Test process and rock samples after testing.
(a)
(b)
(c)
(d)


## 3. Energy Calculation Principles

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 energyU1 produced by the testing machine, the machine also produces and maintains the hydrostatic pressure, which also does positive work on the specimen (U0). After the hydrostatic pressure stage, the axial pressure continues to increase, and the axial force applied to the specimen is constantly positive. As the axial force increases, the radial deformation of the cylindrical specimen increases, which causes the confining pressure to perform negative work [20], which is denoted as U3. The total energy of the rock sample during the triaxial compression test is as follows:(1)U=U1+U3+U0.Furthermore, the total energyU is converted into two energies: the elastic energy Ue stored in sandstone that can be released when the specimen fails and the dissipation energy Ud that is consumed by internal crack expansion, friction, and other processes during the damage of the specimen. This relation can be expressed as follows:(2)U=Ud+Ue.Combining formulas (1) and (2), we obtain(3)U1+U3+U0=Ud+Ue.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 [21–23] are as follows:(1)
The energy of the hydrostatic stress state (U0) can be calculated directly from the formulas related to elasticity:(4)U0=321−2μEσ32.(2)
The axial energy (U1) and the energy consumed by negative work under the confining pressure (U3) can be calculated by the integral method according to the stress-strain curve:(5)U1=∫σ1dε1=∑i=0n12ε1i+1−ε1iσ1i+σ1i+1,(6)U3=2∫σ3dε3=∑i=0nε3i+1−ε3iσ3i+σ3i+1.In these formulas,σ1i and σ3i are the axial stress and confining pressure corresponding to a certain point, and ε1i and ε3i are the axial strain and circumferential strain corresponding to a certain point, respectively.(3)
At a certain timet, the elastic energy (Ue) stored in a rock sample can be calculated by the following formula:(7)Ue=12Eiσ12+2σ32−2μi2σ1σ3+σ32,whereEi and μi are the modulus of elasticity and Poisson’s ratio at a certain point, respectively.(4)
Formula (2) can be rewritten as follows, which can be adopted to define the dissipation energy (Ud):(8)Ud=U−Ue.According to formulas (1)–(8), we compiled a corresponding program to calculate the energy of each sample, and the calculation results were analyzed. In addition, we should explain that the energy calculated by formulas (1)–(8) is essentially the energy density of each energy index.


## 4. Energy Evolution Characteristics of Sandstone under Different Experimental Paths

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.


### 4.1. Energy Evolution Characteristics of the Loading Test

Figure4 shows the stress-strain curve (black line) and the evolution of the various energies (colored lines) in the specimen as the axial strain ε1 increases under a confining pressure of 10 MPa. The hydrostatic energy in the triaxial compression test is accumulated in the specimen during hydrostatic loading. Therefore, all the energy curves in the figure originate from the same point (the hydrostatic pressure), and the curves describe the energy evolution after the hydrostatic loading stage.Figure 4
Energy curves of the rock sample under a confining pressure of 10 MPa.At the beginning of the test,U3 shows little change, and Ud increases slightly. U1, Ue, and U increase with the axial strain, and they are almost indistinguishable. The results show that although energy dissipation occurs during the compaction stage when the sandstone displays the behavior of crack closure, the consumption is very low.Then, the sample is in the elastic stage (between dashed lines A and B), and as the axial force increases,U1 rises at an increasing rate. The concave curve of U1 indicates that although the natural cracks in the sandstone are narrow and closed in the compaction stage, new cracks appear in the sandstone as the axial stress increases, and the appearance and extension of the new cracks consume some energy. Therefore, Ud increases at this stage. At the same time, U3 increases with the axial force, indicating that the radial strain in the sandstone sample also increases. At this stage, the trends of Ue and U are still similar to that of U1, but the difference between U1 and Ue increases as a result of the increase in Ud and U3.After the sandstone enters the plastic deformation stage (between points B and C),U1 increases at a high rate. As the axial stress continues to increase, certain cracks are generated and developed, resulting in significant deformation by radial dilatancy of the sandstone. In this respect, both Ud and U3 began to increase rapidly. Near the peak stress, Ue and U continue to grow but at slower rates, while Ud and U3 increase at higher rates.When the sandstone reaches the peak stress (point C), the elastic energyUe is released, while the increase rates of Ud and U3 show further improvement. In addition, the total energy curve reaches the turning point and begins to decrease, and the differences between U1, Ue, and U increase gradually.After point D, the stress-strain curve indicates a rapid release ofUe. The strength of the specimen decreases rapidly, and the sample loses its bearing capacity because of the formation of a macroscopic fracture surface in the sandstone. The increase rate of U1 also decreases; however, Ud and U3 increase exponentially within a very short time, whereas U decreases suddenly as the sandstone enters the residual plastic phase.After point E,Ud and U3 continue to increase at slower rates. Ue continues to decrease at a slower rate, and U1 continues to increase at a slower rate. The total energy of the system increases again but also at a slower rate. After a period of intense energy conversion, the energy of the system gradually stabilizes, maintaining a relatively balanced state.


### 4.2. Energy Evolution during Triaxial Unloading Confining Pressure

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. Figure5 shows the energy evolution of sandstone under a confining pressure of 7 MPa and an unloading pressure rate of 0.05 MPa/s.Figure 5
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).
(a)
(b)
(c)
(d)At the beginning of the test, bothU3 and Ud show linear increases within short periods of time. U1, Ue, and U also increase slightly with the loading time and axial strain, and they follow very similar paths. These results illustrate that, during the compaction stage of axial loading, the energy dissipation in the sample is relatively low.Before the confining pressure is applied, as the axial force increases,U1 begins to increase gradually at an increasing rate, which produces a concave curve. At this stage, Ud also begins to increase gradually, indicating the occurrence and development of cracks in the sandstone, which results in a certain consumption of energy. As the radial strain of the sandstone specimen increases, U3 also shows obvious growth. At this stage, the trends of Ue and U are still similar to that of U1, but the differences between U1, Ue, and U increase gradually due to the increases in Ud and U3.When the confining pressure is applied,U1 and Ue continue to increase, and U3 and Ud also increase but at higher rates. However, the supply of energy is still higher than the consumption of energy. Hence, the total energy U increases with a higher growth rate than in the previous stage.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 energyUd and the radial strain increase rapidly, resulting in an increase in U3. Although the axial energy continues to increase at this time, it increases at a slower rate due to the decrease in axial force as a result of the damage to the sandstone. In addition, with the rapid decrease in U1, the total energy U also decreases sharply, and the energy conversion in this stage is more obvious than that in the previous stages.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,U1 continues to grow, while Ud and U3 show similar linear increases. Since Ud and U3 reflect the damage and the volume dilation of the samples, respectively, the two processes during the test are indicated to occur simultaneously.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 Figure5.


## 4.1. Energy Evolution Characteristics of the Loading Test

Figure4 shows the stress-strain curve (black line) and the evolution of the various energies (colored lines) in the specimen as the axial strain ε1 increases under a confining pressure of 10 MPa. The hydrostatic energy in the triaxial compression test is accumulated in the specimen during hydrostatic loading. Therefore, all the energy curves in the figure originate from the same point (the hydrostatic pressure), and the curves describe the energy evolution after the hydrostatic loading stage.Figure 4
Energy curves of the rock sample under a confining pressure of 10 MPa.At the beginning of the test,U3 shows little change, and Ud increases slightly. U1, Ue, and U increase with the axial strain, and they are almost indistinguishable. The results show that although energy dissipation occurs during the compaction stage when the sandstone displays the behavior of crack closure, the consumption is very low.Then, the sample is in the elastic stage (between dashed lines A and B), and as the axial force increases,U1 rises at an increasing rate. The concave curve of U1 indicates that although the natural cracks in the sandstone are narrow and closed in the compaction stage, new cracks appear in the sandstone as the axial stress increases, and the appearance and extension of the new cracks consume some energy. Therefore, Ud increases at this stage. At the same time, U3 increases with the axial force, indicating that the radial strain in the sandstone sample also increases. At this stage, the trends of Ue and U are still similar to that of U1, but the difference between U1 and Ue increases as a result of the increase in Ud and U3.After the sandstone enters the plastic deformation stage (between points B and C),U1 increases at a high rate. As the axial stress continues to increase, certain cracks are generated and developed, resulting in significant deformation by radial dilatancy of the sandstone. In this respect, both Ud and U3 began to increase rapidly. Near the peak stress, Ue and U continue to grow but at slower rates, while Ud and U3 increase at higher rates.When the sandstone reaches the peak stress (point C), the elastic energyUe is released, while the increase rates of Ud and U3 show further improvement. In addition, the total energy curve reaches the turning point and begins to decrease, and the differences between U1, Ue, and U increase gradually.After point D, the stress-strain curve indicates a rapid release ofUe. The strength of the specimen decreases rapidly, and the sample loses its bearing capacity because of the formation of a macroscopic fracture surface in the sandstone. The increase rate of U1 also decreases; however, Ud and U3 increase exponentially within a very short time, whereas U decreases suddenly as the sandstone enters the residual plastic phase.After point E,Ud and U3 continue to increase at slower rates. Ue continues to decrease at a slower rate, and U1 continues to increase at a slower rate. The total energy of the system increases again but also at a slower rate. After a period of intense energy conversion, the energy of the system gradually stabilizes, maintaining a relatively balanced state.


## 4.2. Energy Evolution during Triaxial Unloading Confining Pressure

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. Figure5 shows the energy evolution of sandstone under a confining pressure of 7 MPa and an unloading pressure rate of 0.05 MPa/s.Figure 5
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).
(a)
(b)
(c)
(d)At the beginning of the test, bothU3 and Ud show linear increases within short periods of time. U1, Ue, and U also increase slightly with the loading time and axial strain, and they follow very similar paths. These results illustrate that, during the compaction stage of axial loading, the energy dissipation in the sample is relatively low.Before the confining pressure is applied, as the axial force increases,U1 begins to increase gradually at an increasing rate, which produces a concave curve. At this stage, Ud also begins to increase gradually, indicating the occurrence and development of cracks in the sandstone, which results in a certain consumption of energy. As the radial strain of the sandstone specimen increases, U3 also shows obvious growth. At this stage, the trends of Ue and U are still similar to that of U1, but the differences between U1, Ue, and U increase gradually due to the increases in Ud and U3.When the confining pressure is applied,U1 and Ue continue to increase, and U3 and Ud also increase but at higher rates. However, the supply of energy is still higher than the consumption of energy. Hence, the total energy U increases with a higher growth rate than in the previous stage.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 energyUd and the radial strain increase rapidly, resulting in an increase in U3. Although the axial energy continues to increase at this time, it increases at a slower rate due to the decrease in axial force as a result of the damage to the sandstone. In addition, with the rapid decrease in U1, the total energy U also decreases sharply, and the energy conversion in this stage is more obvious than that in the previous stages.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,U1 continues to grow, while Ud and U3 show similar linear increases. Since Ud and U3 reflect the damage and the volume dilation of the samples, respectively, the two processes during the test are indicated to occur simultaneously.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 Figure5.


## 5. Influence of the Unloading Confining Pressure on Energy Evolution

### 5.1. Relationship between Energy Evolution and Unloading of the Initial Confining Pressure

Figures6 and 7 present a comparison of the elastic energy curves under different confining pressures and a comparison of the dissipated energy curves under different confining pressures, respectively. The evolution of the elastic energy and dissipated energy under different unloading of the initial confining pressures are as follows.Figure 6
Comparison of elastic energy curves under different confining pressures.Figure 7
Comparison of dissipated energy curves under different confining pressures. (a) Dissipative energy-radial strain curve. (b) Dissipated energy-deviation stress curve.
(a)
(b)


#### 5.1.1. Evolution of Elastic Energy with Initial Unloading Confining Pressure

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 ofU1 decreases sharply. Furthermore, the higher the initial confining pressure is, the greater the decrease in U1, causing a greater release of the elastic energies when the sandstone fails, as shown in Figure 6.


#### 5.1.2. Evolution of Dissipation Energy with Initial Confining Pressure

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 Figure7.


### 5.2. Relationship between the Energy at Peak Stress and the Initial Confining Pressure

The energy indexes of the sandstone at the peak stress under different initial unloading confining pressures are shown in Table1. Both energy indexes at peak stress are found to increase with the initial unloading confining pressure. During the axial loading stage, the ultimate strength and axial strain of the sandstone increase with initial confining pressure. The time of energy accumulation before the peak stress is also positively correlated with the initial confining pressure; thus, the axial energy, elastic energy, and total energy at the peak stress have similar tendencies. At initial confining pressures of 7, 10, 13, 16, and 19 MPa, the U1 values are 1.29, 1.59, 1.66, 1.71, and 2.06 times the value when the initial confining pressure is 4 MPa, respectively; the Ue values are 1.31, 1.40, 1.44, 1.48, and 1.75 times, respectively; and the U values are 1.29, 1.50, 1.56, 1.61, and 1.90 times the value when the initial confining pressure is 4 MPa, respectively.Table 1
Energy levels at peak stress under different initial unloading confining pressures.
Lithology
Initial unloading confining pressure (MPa)
Unloading pressure rate (MPa/s)
σ
1
−
σ
3 at peak stress (MPa)
Energy at peak stress (kJ/m3)
U
1
U
3
U
e
U
d
U
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
634.35U
3 is the cumulative energy used to overcome the work performed by the confining pressure during the increase in radial deformation. The higher the initial confining pressure, the higher the U3 increase during the axial loading stage. Similarly, the higher the initial confining pressure is, the greater the limitation on the sandstone specimen at the axial loading stage, and the more energy is consumed during the appearance and development of cracks in the sandstone; thus, Ud increases with the confining pressure. At confining pressures of 7, 10, 13, 16, and 19 MPa, the U3 values are 4.31, 3.26, 3.65, 3.74, and 5.19 times the value when the confining pressure is 4 MPa, respectively; and the Ud values are 1.76, 2.39, 2.53, 2.70, and 3.12 times the value when the confining pressure is 4 MPa, respectively.


### 5.3. Characteristics of Energy Conversion during the Unloading Confining Pressure Stage

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 ΔU between the point where the unloading of the confining pressure starts and the point where the peak stress is defined. At the same time, the conversion rate VΔU (VΔU = ΔU/Δt) is calculated, in which Δt is the time corresponding to ΔU. The energy increment (left axis) and the energy conversion rate (right axis) of the sandstone specimens during the unloading of the different confining pressures are shown in Figure 8.Figure 8
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 ΔU1 of the sandstone specimens, the energy dissipation increments ΔU3, the elastic energy increments ΔUe, the depletion energy increments ΔUd, and the total energy increments ΔU increase with initial confining pressure, and the conversion rate of each energy index shows the same behavior. Under an initial confining pressure of 19 MPa, the energy increments ΔU1, ΔU3, ΔUe, ΔUd, and ΔU of the specimens are 2.07, 4.82, 1.38, 3.32, and 1.49 times their values when the initial confining pressure is 4 MPa, respectively, and the conversion rates are 1.40, 3.26, 0.93, 2.25, and 1.01 times their values when the initial confining pressure is 4 MPa, respectively. These results indicate that the effect of the initial confining pressure on the five energy indexes is different. ΔU3 shows the largest change among these energy indexes, indicating that the initial unloading confining pressure has a great influence on the radial deformation of the sandstone samples.


### 5.4. Energy Comparison between the Axial Loading Stage and the Pressure Unloading Stage

Figure9 shows the energy differences in the sandstone specimens between the axial loading stage (before unloading of the confining pressure) and the unloading stage (the point when the unloading of the confining pressure starts to the point of peak stress) for the different initial unloading confining pressures.Figure 9
Energy transformation of loading and unloading confining pressure periods under different initial unloading confining pressures.The accumulations of ΔUe and ΔU in the sandstone samples during the axial loading stage are higher than those during the unloading stage. For example, under an initial confining pressure of 7 MPa, ΔUe and ΔU during the axial loading stage are 47.47 kJ/m3 and 53.08 kJ/m3, respectively, whereas they are 20.35 kJ/m3 and 25.31 kJ/m3 during the unloading stage, respectively. Thus, the axial loading stage contributes more energy storage in the sandstone samples. In particular, the driving energy (Ue) of sandstone failure reaches 70% during the loading stage, suggesting that the energy accumulation in the axial loading stage plays a major role in sandstone failure. Additionally, the higher the initial confining pressure is, the greater the elastic energy that accumulates during the axial loading stage, which induces the rapid release of Ue, causing more serious disasters.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 ΔUe and ΔU, ΔU3 shows the opposite behavior. During the axial loading stage, the accumulation of U3 in the sandstone is less than that during the unloading stage. This result indicates that the increase in radial deformation occurs mainly during the unloading stage, and the higher the initial confining pressure is, the higher the U3 that accumulates during the unloading stage.


## 5.1. Relationship between Energy Evolution and Unloading of the Initial Confining Pressure

Figures6 and 7 present a comparison of the elastic energy curves under different confining pressures and a comparison of the dissipated energy curves under different confining pressures, respectively. The evolution of the elastic energy and dissipated energy under different unloading of the initial confining pressures are as follows.Figure 6
Comparison of elastic energy curves under different confining pressures.Figure 7
Comparison of dissipated energy curves under different confining pressures. (a) Dissipative energy-radial strain curve. (b) Dissipated energy-deviation stress curve.
(a)
(b)


### 5.1.1. Evolution of Elastic Energy with Initial Unloading Confining Pressure

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 ofU1 decreases sharply. Furthermore, the higher the initial confining pressure is, the greater the decrease in U1, causing a greater release of the elastic energies when the sandstone fails, as shown in Figure 6.


### 5.1.2. Evolution of Dissipation Energy with Initial Confining Pressure

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 Figure7.


## 5.1.1. Evolution of Elastic Energy with Initial Unloading Confining Pressure

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 ofU1 decreases sharply. Furthermore, the higher the initial confining pressure is, the greater the decrease in U1, causing a greater release of the elastic energies when the sandstone fails, as shown in Figure 6.


## 5.1.2. Evolution of Dissipation Energy with Initial Confining Pressure

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 Figure7.


## 5.2. Relationship between the Energy at Peak Stress and the Initial Confining Pressure

The energy indexes of the sandstone at the peak stress under different initial unloading confining pressures are shown in Table1. Both energy indexes at peak stress are found to increase with the initial unloading confining pressure. During the axial loading stage, the ultimate strength and axial strain of the sandstone increase with initial confining pressure. The time of energy accumulation before the peak stress is also positively correlated with the initial confining pressure; thus, the axial energy, elastic energy, and total energy at the peak stress have similar tendencies. At initial confining pressures of 7, 10, 13, 16, and 19 MPa, the U1 values are 1.29, 1.59, 1.66, 1.71, and 2.06 times the value when the initial confining pressure is 4 MPa, respectively; the Ue values are 1.31, 1.40, 1.44, 1.48, and 1.75 times, respectively; and the U values are 1.29, 1.50, 1.56, 1.61, and 1.90 times the value when the initial confining pressure is 4 MPa, respectively.Table 1
Energy levels at peak stress under different initial unloading confining pressures.
Lithology
Initial unloading confining pressure (MPa)
Unloading pressure rate (MPa/s)
σ
1
−
σ
3 at peak stress (MPa)
Energy at peak stress (kJ/m3)
U
1
U
3
U
e
U
d
U
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
634.35U
3 is the cumulative energy used to overcome the work performed by the confining pressure during the increase in radial deformation. The higher the initial confining pressure, the higher the U3 increase during the axial loading stage. Similarly, the higher the initial confining pressure is, the greater the limitation on the sandstone specimen at the axial loading stage, and the more energy is consumed during the appearance and development of cracks in the sandstone; thus, Ud increases with the confining pressure. At confining pressures of 7, 10, 13, 16, and 19 MPa, the U3 values are 4.31, 3.26, 3.65, 3.74, and 5.19 times the value when the confining pressure is 4 MPa, respectively; and the Ud values are 1.76, 2.39, 2.53, 2.70, and 3.12 times the value when the confining pressure is 4 MPa, respectively.


## 5.3. Characteristics of Energy Conversion during the Unloading Confining Pressure Stage

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 ΔU between the point where the unloading of the confining pressure starts and the point where the peak stress is defined. At the same time, the conversion rate VΔU (VΔU = ΔU/Δt) is calculated, in which Δt is the time corresponding to ΔU. The energy increment (left axis) and the energy conversion rate (right axis) of the sandstone specimens during the unloading of the different confining pressures are shown in Figure 8.Figure 8
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 ΔU1 of the sandstone specimens, the energy dissipation increments ΔU3, the elastic energy increments ΔUe, the depletion energy increments ΔUd, and the total energy increments ΔU increase with initial confining pressure, and the conversion rate of each energy index shows the same behavior. Under an initial confining pressure of 19 MPa, the energy increments ΔU1, ΔU3, ΔUe, ΔUd, and ΔU of the specimens are 2.07, 4.82, 1.38, 3.32, and 1.49 times their values when the initial confining pressure is 4 MPa, respectively, and the conversion rates are 1.40, 3.26, 0.93, 2.25, and 1.01 times their values when the initial confining pressure is 4 MPa, respectively. These results indicate that the effect of the initial confining pressure on the five energy indexes is different. ΔU3 shows the largest change among these energy indexes, indicating that the initial unloading confining pressure has a great influence on the radial deformation of the sandstone samples.


## 5.4. Energy Comparison between the Axial Loading Stage and the Pressure Unloading Stage

Figure9 shows the energy differences in the sandstone specimens between the axial loading stage (before unloading of the confining pressure) and the unloading stage (the point when the unloading of the confining pressure starts to the point of peak stress) for the different initial unloading confining pressures.Figure 9
Energy transformation of loading and unloading confining pressure periods under different initial unloading confining pressures.The accumulations of ΔUe and ΔU in the sandstone samples during the axial loading stage are higher than those during the unloading stage. For example, under an initial confining pressure of 7 MPa, ΔUe and ΔU during the axial loading stage are 47.47 kJ/m3 and 53.08 kJ/m3, respectively, whereas they are 20.35 kJ/m3 and 25.31 kJ/m3 during the unloading stage, respectively. Thus, the axial loading stage contributes more energy storage in the sandstone samples. In particular, the driving energy (Ue) of sandstone failure reaches 70% during the loading stage, suggesting that the energy accumulation in the axial loading stage plays a major role in sandstone failure. Additionally, the higher the initial confining pressure is, the greater the elastic energy that accumulates during the axial loading stage, which induces the rapid release of Ue, causing more serious disasters.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 ΔUe and ΔU, ΔU3 shows the opposite behavior. During the axial loading stage, the accumulation of U3 in the sandstone is less than that during the unloading stage. This result indicates that the increase in radial deformation occurs mainly during the unloading stage, and the higher the initial confining pressure is, the higher the U3 that accumulates during the unloading stage.


## 6. Analysis of Sandstone Damage Based on Dissipation Energy

### 6.1. Damage Evolution Model

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 [24] and the research of Liu et al. [25] and other studies, the model of rock damage evolution based on the energy transformation principle is obtained as follows:(9)D=1−σCσPUdUdmax,where Udmax is the value of the dissipation energy when the rock is completely destroyed, Ud is the dissipation energy when the damage area of the section reaches a certain value Ad, σP is the peak stress, and σC is the residual stress.According to the elastic mechanics and related references, the constitutive stress-strain relation in the rock considering the damage characteristics can be expressed as(10)σ1=2μσ3+E1−Dε1.Substituting formula (9) into formula (10), we obtain(11)σ1=2μσ3+E1−1−σCσPUdUdmaxε1.


### 6.2. Energy-Based Damage Characteristics of Sandstone under Triaxial Compression

Based on the test results and formula (11), we calculate the damage curves of sandstone under conventional triaxial loading and triaxial unloading confining pressure tests, as shown in Figures 7 and 8, respectively.In the conventional triaxial compression test (Figure10), cracks and joints develop in the sandstone, and damage occurs at the beginning of loading, but the damage is limited because of the low loading stress. Then, as the strain increases, the damage in the specimens grows at a gradually increasing rate. The growth curve changes from linear to nonlinear. When the rock sample fails and becomes unstable, the damage curve increases rapidly.Figure 10
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, Figure10 shows that, for low confining pressures, the damage increases steadily during the test, and although the growth rate becomes higher as the peak stress is approached, no sudden increase phenomenon occurs when the confining pressure is relatively high. This result indicates that the confining pressure prevents the fractures in sandstone from developing, which induces a low damage value until the specimen nears the peak stress, when increasingly more cracks appear, develop, gather, and form a macroscopic fracture surface, causing the damage to increase immediately.According to the damage evolution of the triaxial unloading of the confining pressure (Figure11), during the early stages of the test, the damage to sandstone is limited, and the dissipation energy is very low because of the large number of primary microcracks. With increasing axial stress, the microcracks in the sandstone begin to expand, and new cracks appear, which induces an increase in dissipation energy and damage. When the unloading stage starts, a large number of cracks in the sandstone begin to expand and converge. Moreover, a large number of new larger cracks appear continuously, and the dissipation energy of and the damage to the specimen increase rapidly. When the specimen fails, the dissipation energy increases suddenly, and the damage shows the same tendency. The sandstone then enters the plastic flow stage in which the dissipation energy is reduced and the damage tends to be relatively stable.Figure 11
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.


## 6.1. Damage Evolution Model

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 [24] and the research of Liu et al. [25] and other studies, the model of rock damage evolution based on the energy transformation principle is obtained as follows:(9)D=1−σCσPUdUdmax,where Udmax is the value of the dissipation energy when the rock is completely destroyed, Ud is the dissipation energy when the damage area of the section reaches a certain value Ad, σP is the peak stress, and σC is the residual stress.According to the elastic mechanics and related references, the constitutive stress-strain relation in the rock considering the damage characteristics can be expressed as(10)σ1=2μσ3+E1−Dε1.Substituting formula (9) into formula (10), we obtain(11)σ1=2μσ3+E1−1−σCσPUdUdmaxε1.


## 6.2. Energy-Based Damage Characteristics of Sandstone under Triaxial Compression

Based on the test results and formula (11), we calculate the damage curves of sandstone under conventional triaxial loading and triaxial unloading confining pressure tests, as shown in Figures 7 and 8, respectively.In the conventional triaxial compression test (Figure10), cracks and joints develop in the sandstone, and damage occurs at the beginning of loading, but the damage is limited because of the low loading stress. Then, as the strain increases, the damage in the specimens grows at a gradually increasing rate. The growth curve changes from linear to nonlinear. When the rock sample fails and becomes unstable, the damage curve increases rapidly.Figure 10
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, Figure10 shows that, for low confining pressures, the damage increases steadily during the test, and although the growth rate becomes higher as the peak stress is approached, no sudden increase phenomenon occurs when the confining pressure is relatively high. This result indicates that the confining pressure prevents the fractures in sandstone from developing, which induces a low damage value until the specimen nears the peak stress, when increasingly more cracks appear, develop, gather, and form a macroscopic fracture surface, causing the damage to increase immediately.According to the damage evolution of the triaxial unloading of the confining pressure (Figure11), during the early stages of the test, the damage to sandstone is limited, and the dissipation energy is very low because of the large number of primary microcracks. With increasing axial stress, the microcracks in the sandstone begin to expand, and new cracks appear, which induces an increase in dissipation energy and damage. When the unloading stage starts, a large number of cracks in the sandstone begin to expand and converge. Moreover, a large number of new larger cracks appear continuously, and the dissipation energy of and the damage to the specimen increase rapidly. When the specimen fails, the dissipation energy increases suddenly, and the damage shows the same tendency. The sandstone then enters the plastic flow stage in which the dissipation energy is reduced and the damage tends to be relatively stable.Figure 11
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.


## 7. Discussion

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.


## 8. Conclusions

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.



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*Source: 7419752-2019-05-02.xml*