Excavation and earth surface processes (e.g., river incision) always induce the unloading of stress, which can cause the failure of rocks. To study the shear mechanical behavior of a rock sample under unloading normal stress conditions, a new stress path for direct shear tests was proposed to model the unloading of stress caused by excavation and other processes. The effects of the initial stresses (i.e., the normal stress and shear stress before unloading) on the shear behavior and energy conversion were investigated using laboratory tests and numerical simulations. The shear strength of a rock under constant stress or under unloading normal stress conforms to the Mohr Coulomb criterion. As the initial normal stress increases, the cohesion decreases linearly and the tangent of the internal friction angle increases linearly. Compared with the results of the tests under constant normal stress, the cohesions of the rock samples under unloading normal stress are smaller and their internal friction angles are larger. A strength envelope surface can be used to describe the relationship between the initial stresses and the failure normal stress. Shear dilatancy can decrease the total energy of the direct shear test under constant normal stress or unloading normal stress, particularly when the stress levels (the initial stresses in the test under unloading normal stress or the normal stress in the test under constant normal stress) are high. The ratio of the dissipated energy to the total energy at the moment failure occurs decreases exponentially with increasing initial stresses. The direct shear test under constant normal stress can be considered to be a special case of a direct shear test under unloading normal stress with an unloading amount of zero.

Shear failure is one of the fundamental forms of rock failure [

In the rock failure process, energy is converted, including energy accumulation, dissipation, and release [

In this study, a new loading path for direct shear tests is proposed to model the shear failure process of a rock under unloading normal stress caused by excavation or other processes. In the direct shear test under unloading normal stress (DSTUNS), the normal stress is gradually unloaded, while the shear stress remains unchanged. In addition, a direct shear test under constant normal stress (DSTCNS) was conducted for comparison and to determine the parameters. Laboratory tests and numerical simulations were conducted to compare the strength, failure, and energy evolution of the rock in the two types of direct shear tests.

The sandstone used in this study was collected from the Three Gorges Reservoir area of China, where there are a lot of landslides due to the rise and fall of reservoir water. Its average bulk density is 2390 kg/m^{3}. According to the X-ray diffraction results, the main minerals of the sandstone mainly include quartz, illite, feldspar, and a small amount of magnetite and hematite. An intact rock block without observable bedding planes was cut and ground into cubic test samples with the side lengths of 60 mm. The upper and lower surface parallelism of the sample is controlled within

The direct shear tests were carried out using a compression-shear test machine with a loading capacity of 600 kN in both the vertical and horizontal directions.

Step 1: Load the normal stress to the target level at a constant loading rate of 0.1 kN/s. Name the target normal stress as initial normal stress (INS)

Step 2: Load the shear stress to the desired level at a constant displacement rate of 0.1 mm/min while keeping the normal stress constant. Name the desired shear stress level as initial shear stress (ISS)

Step 3: Keep the shear stress constant and unload the normal stress at an unloading rate of 0.1 kN/s until the rock sample fails.

During unloading (Step 3), the shear failure occurring in the middle of the rock sample is caused by both the decrease in the normal compressive stress and the constant shear stress. As is well known, there is a positive correlation between the shear resistance and the normal compressive stress of rock materials. Thus, the shear resistance of the sample gradually decreases with decreasing normal stress. When the shear resistance is less than the ISS, failure occurs.

Prior to the beginning of unloading, the normal and shear stresses were applied. The shear resistance of the rock is related to the normal stress, cohesion, and friction angle. If the ISS is large enough or the INS is small enough, the sample may fail before unloading. Whereas it will not fail when the normal stress is unloaded to zero if the ISS is smaller than a certain value. Consequently, the DSTCNS was carried out to determine the values of the INS and ISS.

Specimen | INS (MPa) | ISS (MPa) | Specimen | INS (MPa) | ISS (MPa) | |
---|---|---|---|---|---|---|

1 | 20 | 15 | 16 | 35 | 15 | |

2 | 17 | 17 | 20 | |||

3 | 19 | 18 | 25 | |||

4 | 21 | 19 | 30 | |||

5 | 23 | 20 | 35 | |||

6 | 25 | 15 | 21 | 40 | 15 | |

7 | 18 | 22 | 20 | |||

8 | 21 | 23 | 25 | |||

9 | 24 | 24 | 30 | |||

10 | 27 | 25 | 35 | |||

11 | 30 | 15 | The cohesion and internal friction angle of the rock sample tested by DSTCNS are 11.22 MPa and |
|||

12 | 18 | |||||

13 | 21 | |||||

14 | 24 | |||||

15 | 27 |

The two-dimension particle flow code (PFC^{2D}) was used to perform the simulations of the DSTCNS and DSTUNS. The PFC^{2D} simplifies the rock material into discs and bonds. Discs are assumed to be rigid bodies, but they can overlap each other. The adjacent discs are connected by a bond. The discs are used to simulate the rock grains, and the bonds are used to simulate the cement between the rock grains. The mechanical relationship between particles obeys the Newton’s second law:

The linear parallel bond model, which can transmit both force and moment between the pieces, is commonly used in rock mechanic simulations [^{2D} can simulate rock fracture and fracture propagation. In this respect, the finite element method and the finite difference method have obvious defects. In the PFC, the energy consists of boundary work _{g}_{s}_{f}_{d}_{b}_{k}_{e}_{d}

where the elastic strain energy _{e}_{s}_{b}_{d}_{f}_{d}_{k}_{k}_{g}

Strain energy, slip energy, dashpot energy, and bond strain energy can be calculated as follows:

In this study, the planar dimensions of the numerically simulated sample are the same as those of the sandstone sample used in the laboratory tests. As shown in ^{2D} code applied the stress by moving walls or particles, and the stress-controlled mode was difficult to implement. Thus, the normal and shear stresses were applied in the displacement-controlled mode with a constant displacement loading and unloading rate of 0.01 m/s. This loading rate is low enough for a rock static loading simulation using the PFC^{2D} code [

There are no macro-parameters that reflect the mechanical characteristics of the rock material. It is essential to establish a correlation between the micro-parameters in the PFC^{2D} code and the experimental macro-parameters of the tested sandstone sample. A parameter calibration should be performed for the micro-parameter selection. According to Huang et al. [

Micro parameter | Value |
---|---|

Parallel bond tensile strength (MPa) | 6.4 |

Parallel bond cohesion (MPa) | 7.1 |

Parallel bond friction angle ( |
46 |

Ratio of normal to shear stiffness of the parallel bond | 1.5 |

Particle friction coefficient | 0.75 |

In addition, the normal displacement during the unloading process decreases faster than during the loading of the shear stress. The growth of the shear displacement leads to shear dilatancy deformation in the direct shear test due to crack propagation and frictional sliding along the sloping micro-failure surfaces inside the rock. Thus, it is guaranteed that shear dilatancy deformation will occur during the unloading of the normal stress due to the gradual increase in the shear displacement. Furthermore, the release of the elastic deformation in the normal direction occurs because of the decrease in the normal stress. Therefore, the decrease in the normal displacement consists of shear dilatancy deformation and elastic deformation release.

The M–C and Hoek–Brown (H–B) criteria are commonly used to describe the change in shear strength of rocks. The tensile strength of a rock is necessary in the H–B criterion. In addition, according to the experimental results from Huang et al. [

The root-mean-square error (RMSE) of which is 0.83 MPa. The internal friction angle and the cohesion are

In the DSTUNS, at the moment when the sample fails, the failure shear stress (

where

Test | INS (MPa) | Cohesion (MPa) | Internal friction angle ( |
---|---|---|---|

DSTUNS | 20 | 8.43 | 36.46 |

25 | 7.87 | 37.06 | |

30 | 6.94 | 37.48 | |

35 | 6.25 | 38.88 | |

40 | 4.42 | 38.90 | |

DSTCNS | 11.22 | 35.90 |

According to the above analysis, the failure normal stress

where

Since the RMSE of 0.58 MPa is small enough,

In the laboratory tests under unloading normal stress, surface spalling occurred around the shear plane on all of the samples (

_{d}_{d}

In the DSTUNS, the dissipated energy _{d}_{d}_{e}

The total energy changes in two different manners in the DSTCNS and DSTUNS. This is mainly due to the different failure patterns. As shown in

As shown in _{d}

As shown in

Thus, the dissipated energy ratio in the DSTUNS can be rewritten as

In both the DSTCNS and the DSTUNS, the normal stress and the shear stress are applied successively. However, there is no unloading process in the DSTCNS. The DSTCNS can be considered to be a special case of the DSTUNS, with an INS of

As shown in

In the process of deep tunnel excavation, shear slip fractures will occur in the surrounding rock accompanied by unloading. In this process, the normal stress on the potential shear fracture surface gradually decreases. Under the action of slope excavation or groundwater rising, the normal stress on the potential shear fracture surface of a slope decreases gradually. Based on the fact that the normal stress of the potential shear fracture surface gradually decreases in engineering rock mass, the DSTUNS was proposed to study the shear mechanical behavior and energy evolution characteristics of rock mass under unloading normal stress. The DSTUNS under different initial stress levels was carried out on the common sandstone in the Three Gorges Reservoir Area, and the numerical simulation is carried out by using PFC^{2D}. The characteristics of strength, deformation, fracture and energy evolution of rock in the DSTUNS were analyzed. The research results have certain guiding significance for the stability evaluation of rock mass in slope and deep tunnel.

In this study, the normal stress on the shear fracture surface gradually decreases and the shear stress remains unchanged. However, in the process of tunnel, slope excavation and groundwater rising, the shear stress on the potential shear fracture surface of rock will also change and will not remain constant. Therefore, the results of this study have certain limitations, and further research is needed.

In this study, a new stress path was proposed for simulating the shear failure of rocks under unloading normal stress. Then, laboratory direct shear tests and simulations were carried out under constant normal stress and unloading normal stress to investigate the shear deformation, the shear strength, and the energy conversion within the rock. The conclusions of this study are summarized as follows:

The shear strengths in the DSTCNS and DSTUNS conform to the M–C criterion. The cohesion and internal friction angle are related to the initial normal stress and the initial shear stress. As the initial normal stress increases, the cohesion decreases linearly and the tangent of the internal friction angle increases linearly. In addition, in the DSTUNS, the cohesion is smaller and the internal friction angle is larger than that in the DSTCNS.

When the sample is subjected to a high initial stress level, there are many micro-cracks evenly distributed inside the sample before the cracking begins at the ends of the shear plane. This results in a decrease in the shear resistance. Surface spalling occurred on all of the samples in the DSTUNS, while spalling rarely occurred in the DSTCNS.

Shear dilatancy caused a decrease in the total energy during the direct shear tests. A high initial stress is more likely to result in a reduction in the total energy. The dissipated energy ratio decreases exponentially with increasing initial shear stress. The DSTCNS is a special case of the DSTUNS, with an unloading amount equal to zero.