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In order to reveal the influence of prophase stress levels on the fatigue damage characteristics of granite, uniaxial fatigue tests of granite with different prophase stress levels were carried out on the basis of the MTS 815.04 rock mechanics test system. The results show that, under the same number of cycles, the failure degree increases with the increase of the prophase stress level. Under the low upper limit of cyclic stress, the tangent modulus and dissipated energy increase significantly with the increase of prophase stress level at the early stage of the cycle loading, while the increasing trend is not obvious with the increase of prophase stress level at the late stage. Under the high upper limit of cyclic stress, the tangent modulus and dissipated energy are less affected by the prophase stress level. The development trend of elastic release energy is not obvious with the increase of prophase stress level, which is less affected by the number of cycles. From the damage parameters defined by dissipative energy, under the low upper limit of cyclic stress, the initial damage is less affected by the prophase stress level. With the increase of the number of cycles, the influence of the prophase stress level on the development trend of the damage variable increases gradually. And the development trend of damage variables shows “C-shaped” damage.

Fatigue damage occurs inside the rock under cyclic loading, accompanied by the initiation and expansion process of cracks, which further affects the engineering stability [

In recent years, a large number of experimental studies have been carried out on the fatigue characteristics of rocks with different types of initial damage. Mokhfi et al. [

In addition, some scholars have studied the fatigue damage characteristics of rocks under different freeze-thaw conditions. Ling et al. [

As mentioned above, most of the researches focused on the influence of high temperature and freeze-thaw action on the fatigue damage characteristics of rock, and few researches on the fatigue damage and deformation characteristics of rock under different prophase stress states for now. In this paper, uniaxial fatigue tests of granite with different prophase stress levels were carried out on granite samples. The deformation characteristics, mechanical characteristics, and fatigue characteristics of granite are analyzed to study the influence of prophase stress level on the fatigue damage characteristics of rock.

The granite samples used in this experiment were taken from Tuanshan Village, Huangbai Town, Miluo City, Yueyang City, Hunan Province. The samples were drilled from the same block and cut into cylinders with a diameter of 50 mm and a length of 100 mm. They were prepared in strict accordance with the standards of the International Society for Rock Mechanics (ISRM). In addition, the deviation is controlled within 0.5 mm, and the nonparallelism of the end face is less than 0.03 mm, as shown in Figure ^{3} and longitudinal wave velocity is between 3240 m/s and 3320 m/s, which can be used to ensure the uniformity of the samples and the reliability of the test results.

Granite samples.

X-ray pattern (a) and chemical analysis of the granite (b).

The MTS 815.04 rock mechanics test system was used to conduct the fatigue test, which is composed of hydraulic pump, host, servo control unit, and data acquisition unit, as shown in Figure ^{9} N/m, and the maximum axial force is 4600 kN. Three sets of uniaxial cyclic load tests under different prophase stress levels were carried out on the system. In addition, three different upper cyclic stress limits were set under the same prophase stress level to analyze whether the fatigue damage characteristics of granite under different prophase stress levels show similar laws under different upper limit of cyclic stress. The scheme is shown in Table

MTS 815.04 rock mechanics test system.

The sample loading scheme.

Groups | Specimens | Prophase stress level (MPa) | Upper limit of cyclic stress (MPa) | Cycles |
---|---|---|---|---|

Group 1 | G1-C1 | 125 | 105 | 300 |

G1-C2 | 115 | |||

G1-C3 | 125 | |||

Group 2 | G2-C1 | 130 | 105 | 300 |

G2-C2 | 115 | |||

G2-C3 | 125 | |||

Group 3 | G3-C1 | 135 | 105 | 300 |

G3-C2 | 115 | |||

G3-C3 | 125 |

Schematic diagram of stress path.

Under the same upper limit of cyclic stress, the stress-strain curve shape is affected by the prophase stress level. The upper limit of cyclic stress 125 MPa is taken as an example, as shown in Figure

Stress-strain curves of rock samples under different prophase stress levels with the same upper limit cyclic stress. (a) Prophase stress level 125 MPa. (b) Prophase stress level 130 MPa. (c) Prophase stress level 135 MPa.

Elastic modulus is one of the important mechanical properties of rock, which reflects the ability of rock to resist deformation. With the increase of elastic modulus, the stiffness increases gradually, and the rock is harder to deform. Because of the damage caused by the loading of the rock mass, the loading and unloading curves do not coincide, which form a hysteresis loop. The elastic modulus of the loading section is defined by the slope of the OA section on the stress-strain curve at the loading stage, which is the tangent modulus, as shown in Figure

Schematic diagram of tangent modulus of loading section.

In order to explore the influence of the prophase stress level on the tangent modulus, the relation diagram between tangent modulus and the prophase stress level under different cycle times in the loading process was drawn, as shown in Figure

The relationship between tangent modulus and prophase stress level in the loading process. (a) Upper limit of cyclic stress 105 MPa. (b) Upper limit of cyclic stress 115 MPa. (c) Upper limit of cyclic stress 125 MPa.

On the one hand, tangent modulus generally presents an upward trend with the increase of the prophase stress level. That is because, before the cyclic loading, the closure degree of internal microcracks is different under different prophase stress levels. With the increase of prophase stress level, the degree of microcracks closing increases, and the deformation trend decreases. As a result, the tangent modulus increases with the increase of the prophase stress level.

On the other hand, tangent modulus shows different development trends with the increase of the prophase stress level under different upper limit of cyclic stress. Under the low upper limit of cyclic stress, the tangent modulus in the first cycle increases significantly with the increase of prophase stress levels. With the increase of the number of cycles, the development trend of tangent modulus is gradually stable. On the contrary, the tangent modulus tends to be stable in the whole cycle under the high upper limit of cyclic stress. The samples generally experience the initial fatigue stage, the constant velocity stage, and the acceleration stage in the whole cycle process, and the tangent modulus shows the same trend. Under the low upper limit of cyclic stress, because the work done by external force on samples is small, the rock samples go through the initial stage of fatigue, resulting in a long process from compaction to stability of the microcrack inside the samples. It can be conducted that different prophase stress levels in the early stage of cyclic load test have a great influence on the tangent modulus. With the increase of the number of cycles, the crack of rock under cyclic load gradually compresses and tends to a stable state into the constant velocity stage, and the elastic modulus also enters a stable state, while, under the high upper limit of cyclic stress, the external force does more work on the samples, which leads to the short compaction stage of the microcracks inside the samples and quickly enters the stable state. The results show that the prophase stress level has little effect on the tangent modulus under the high upper limit of cyclic stress.

It can be drawn that, with the increase of the number of cycles, the tangent modulus at the same prophase stress level gradually increases. The tangent modulus under the first two cyclic loads increases greatly, but the increase is not obvious after that. That is, the newly generated damage from the third cycle is extremely small.

According to the variation trend of tangent modulus with prophase stress level in the same number of cycles as shown in Figure

The relationship between

It can be clearly observed that the development trend presents an inversely proportional relationship under different upper limit of cyclic stress. The development trend is gradually stable with the increase of upper limit of cyclic stress. Moreover, the inverse function is used to fit the development trend

Therefore, it can be inferred that the influence of prophase stress level on tangent modulus decreases with the increase of the upper limit of cyclic stress. Under the low upper limit of cyclic stress, the tangent modulus is greatly affected by the prophase stress level at the early stage of the cycle. On the contrary, the prophase stress level has little influence on the elastic modulus under the high upper limit of cyclic stress.

Energy dissipation of rock is the essential property of rock deformation and failure. It reflects the continuous closure of microscopic defects in rock itself, such as microcracks and holes, the development and evolution of new cracks, and the continuous weakening and final loss of material strength.

In the process of rock deformation and failure, material and energy are always exchanged with the outside world, and the deformation and failure can be regarded as the damage evolution process of energy dissipation [^{d} is expressed as dissipated energy; ^{e} is expressed as elastic release energy.

Combined with the test conditions of uniaxial cyclic load, the energy distribution under cyclic load is shown in Figure ^{d} is represented by the area enclosed by the loading curve, unloading curve, and strain axis. Elastic release energy ^{e} is represented by the unloading curve, the area enclosed by the vertical line of maximum strain

Distribution of elastic energy and dissipated energy of loaded rock units.

Therefore, the dissipated energy and elastic release can be calculated by integral:

Figure

The relationship between elastic energy and prophase stress level. (a) Upper limit of cyclic stress 105 MPa. (b) Upper limit of cyclic stress 115 MPa. (c) Upper limit of cyclic stress 125 MPa.

The relative data for the elastic release energy.

Number of cycles | Elastic release energy (J·m^{−3}) | ||||||||
---|---|---|---|---|---|---|---|---|---|

Upper limit of cyclic stress 105 MPa | Upper limit of cyclic stress 115 MPa | Upper limit of cyclic stress 125 MPa | |||||||

Prophase stress level (MPa) | Prophase stress level (MPa) | Prophase stress level (MPa) | |||||||

125 | 130 | 135 | 125 | 130 | 135 | 125 | 130 | 135 | |

1st | 0.084 | 0.100 | 0.118 | 0.082 | 0.102 | 0.120 | 0.085 | 0.101 | 0.119 |

10th | 0.083 | 0.100 | 0.119 | 0.082 | 0.102 | 0.120 | 0.085 | 0.100 | 0.119 |

30th | 0.083 | 0.100 | 0.119 | 0.082 | 0.102 | 0.121 | 0.084 | 0.100 | 0.119 |

50th | 0.083 | 0.100 | 0.119 | 0.081 | 0.102 | 0.120 | 0.084 | 0.100 | 0.119 |

100th | 0.093 | 0.100 | 0.119 | 0.081 | 0.102 | 0.121 | 0.084 | 0.100 | 0.119 |

200th | 0.083 | 0.100 | 0.119 | 0.081 | 0.102 | 0.121 | 0.085 | 0.100 | 0.119 |

30th | 0.081 | 0.098 | 0.119 | 0.080 | 0.102 | 0.121 | 0.084 | 0.100 | 0.119 |

Figure

The relationship between dissipated energy and prophase stress level. (a) Upper limit of cyclic stress 105 MPa. (b) Upper limit of cyclic stress 115 MPa. (c) Upper limit of cyclic stress 125 MPa.

Moreover, dissipation energy shows different trends under different upper limits of cyclic stress. Under the low upper limit of cyclic stress (105 MPa), the trend of dissipation energy in the first cycle is the most obvious with the increase of the prophase stress level. And then, with the increase of cycles, the trend of dissipation energy tends to be stable. On the contrary, under the high upper limit of cyclic stress (125 MPa), the dissipated energy basically keeps a stable development trend with the increase of the prophase stress level. That is because, under the low upper limit of cyclic stress, the external force does less work on the rock samples. And the rock goes through the initial fatigue stage, during which the microcracks are constantly compacted. It means that different prophase stress levels in the early stage of cyclic load test have a great influence on the dissipated energy. With the increase of the number of cycles, the development trend of microcracks under cyclic load gradually tends to be stable, while the influence of the prophase stress level is also gradually weakened, while, under the high upper limit of cyclic stress, the external force does more work on the samples, so that the microcracks close quickly and tend to a stable state. It indicates that the dissipated energy is less affected by the prophase stress level.

Therefore, it can be inferred that the influence of the prophase stress level on the dissipated energy decreases with the increase of the upper limit of cyclic stress. Under cyclic loading with a low-stress upper limit, the dissipated energy is greatly affected by the prophase stress level in the early period of cycle. On the contrary, under cyclic loading with a high-stress upper limit, the influence of prophase stress level on dissipated energy is small.

The damage of rock and other materials under external load requires energy consumption, which means that the damage of materials and energy consumption are synchronous, and both are irreversible. On this basis, Jin et al. [

Calculation of constitutive energy and dissipation energy.

Figure

The relationship between the damage variable and cycles under different prophase stress levels. (a) Upper limit of cyclic stress 105 MPa. (b) Upper limit of cyclic stress 115 MPa. (c) Upper limit of cyclic stress 125 MPa.

The difference of the initial damage variable is small at different prophase stress levels, indicating that the prophase stress level has little influence on the initial damage of rock. Under the low upper limit of cyclic stress, the development trend of damage variable gradually increases with the increase of the number of cycles. And it tends to be stable after 150 cycles. On the contrary, under the high upper limit of cyclic stress, the damage variable basically stays stable with the increase of cyclic numbers. It can be inferred that, with the increase of prophase stress levels, the variation trend of the damage variable decreases gradually.

As a kind of heterogeneous and complex geological material, different parts of rock mass are in different initial stress states. In this paper, uniaxial cyclic loading tests of granite under different prophase stress levels were carried out. The impact of prophase stress level on the fatigue damage characteristics of granite samples is investigated from the perspectives of stress-strain curve characteristics, tangent modulus of loading section, energy characteristics, and damage parameters. The following conclusions can be drawn based on the test results.

By studying the stress-strain curves of the same upper limit of cyclic stress under different prophase stress levels, it is found that the strain is greatly affected by the prophase stress levels. Before the cyclic loading, the rock samples deformation increases with the increase of the prophase stress level. In addition, during the cycle, the damage deformation gradually increases with the increase of the prophase stress level.

Through the study on the influence of different prophase stress levels on tangent modulus, it is shown that the influence of prophase stress levels on tangent modulus is more obvious. Especially, under the cyclic loading with a low-stress upper limit, the elastic modulus is greatly affected by the prophase stress level at the early stage of the cycle. On the contrary, under the cyclic loading of a high-stress upper limit, the influence of the prophase stress level is small. In addition, this paper introduces the development trend

Through the study of energy characteristics, it is found that the influence of the prophase stress level on dissipated energy is smaller with the increase of the upper limit of cyclic stress. Under a low upper limit of cyclic stress, the dissipation energy is markedly affected by the prophase stress level in the early stage of the cycle, whereas it is less affected by prophase stress level in the high upper limit of cyclic stress.

Through the damage parameters defined based on the energy dissipated energy, it is found that the initial damage variable is less affected by the prophase stress level. But, with the increase of the number of cycles, the prophase stress level has a more obvious influence on the changing trend of the damage variable. That is, with the increase of the prophase stress level, the damage variable tends to be stable with the increase of the number of cycles.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The authors would like to acknowledge the financial support for this study supported by the Second Tibetan Plateau Scientific Expedition and Research Program (grant no. 2019QZKK0904), the National Natural Science Foundation of China (grant no. 42007254 and 41831290), the Natural Science Foundation of Zhejiang Province (grant no. LQ20E080006), and the General Research Project of Zhejiang Education Department (grant no. Y202043108).