# A Quantitative Method to Predict the Shear Yield Stress of Rock Joints

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- (1)
- Empirical methods

- (2)
- Shear stiffness method

_{i}is the shear displacement at a certain shear stress, m represents the reciprocal of the initial shear stiffness (K

_{st}), and n is the reciprocal of the horizontal asymptote to the hyperbolic curve. By using Equation (1) to fit the experimental data, it is possible to readily derive the initial shear stiffness. Subsequently, a line originating from the origin O, with a slope of K

_{st}, intersects with the shear stress-shear displacement curve at point F, which represents the yield point [67], as shown in Figure 2. This method is on the basis of shear test results. However, it is worth noting that determining the initial shear stiffness may be imprecise in situations where the shear stress–shear displacement curve does not exhibit a hyperbolic shape.

- (3)
- Inflection point method

_{st}), as shown in Figure 3a. The shear stress difference (△τ) can then be obtained, and its relation versus shear displacement is illustrated in Figure 3b. The inflection point clearly corresponds to the yield point. Nonetheless, accurately pinpointing the inflection point remains a challenging task, as it is susceptible to subjective errors.

## 3. A New Displacement Reduction Method

#### 3.1. Modeling Process

_{i}/u

_{p}). The specific steps are as follows:

_{i}= 1 − u

_{i}/u

_{p}

_{i}is the displacement data in the pre-peak stage of the shear stress–shear displacement curve, and u

_{p}is the peak shear displacement. It is evident that w is a scalar quantity.

_{p}) and B (1, 0) can be identified, which correspond to the origin (0,0) and the peak point (u

_{p}, τ

_{p}) on the shear–stress displacement curve, respectively. It is obvious that points A (0, τ

_{p}) and B (1, 0) always lie on the shear stress-w curve. Connecting these two points results in a reference line denoted as AB in Figure 4.

#### 3.2. Verification and Applications

^{3}and 45°, respectively. The uniaxial compressive strength, indirect tensile strength, Young’s modulus, Poisson′s ratio, cohesion, and internal friction angle of the cement mortar material are 18.97 MPa, 1.637 MPa, 2.202 GPa, 0.2, 1.84 MPa, and 58.47°, respectively. During the test, prior to the application of shear stress, the specimen was subjected to a normal stress of 0.4 MPa. The test was completed once the shear displacement reached or exceeded 10 mm.

_{p}and the peak shear displacement u

_{p}were 2.93 MPa and 1.39 mm, respectively. Then, w

_{i}can be obtained based on Equation (2), and its relation versus shear stress τ, and the reference line are shown in Figure 8. The w variation between the experimental curve and the reference line exhibited a significant alteration with the increase in w, which is further illustrated in Figure 9. The yield shear stress corresponds to the maximum value of Δw, as shown in Figure 9, where it is determined as 2.42 MPa, accounting for 82.6% of the maximum shear stress.

#### 3.3. Comparative Analysis

- (1)
- Empirical methods

- (2)
- Shear stiffness method

- (3)
- Inflection point method

## 4. Discussion

#### 4.1. Effect of the Temperature

_{y}

**τ**

_{/}_{p}is larger than 90%.

#### 4.2. Effect of the Normal Stress

#### 4.3. Effect of the Shear Velocity

#### 4.4. Effect of the JRC

#### 4.5. Effect of the Shear Direction

## 5. Conclusions

_{p}) and B (1, 0) is introduced. The yield stress can then be determined as the displacement reduction coefficient difference between the experimental shear–stress displacement curve and the reference line maximizes. The effectiveness and precision are validated by many experimental data from published papers. Compared with previous methods, this method can enhance objectivity and effectively reduce human interference. Furthermore, the effects of external environmental factors except JRC on the yield stress are limited, which demonstrates that the yield stress of rock materials is dependent on in situ lithology to a large extent, and can be significantly affected by the joint surface roughness.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the shear stress–shear displacement curve (u

_{y}and u

_{p}are yield displacement and peak shear displacement, respectively; τ

_{y}and τ

_{p}are yield stress and peak shear stress, respectively).

**Figure 6.**Schematic diagrams of saw-tooth triangular joint cement mortar specimens (Modified after Xie et al. [70]).

**Figure 7.**Shear stress–shear displacement curve from the direct shear test [70].

**Figure 14.**Yield stresses under different normal stresses (Note: τ

_{p}data are from Zhou et al. [78]).

**Figure 16.**Yield stresses under the high velocity impact (Note: τ

_{p}data are from Fan et al. [80]).

**Figure 17.**Schematic diagram of the selected rock joints [81].

**Figure 18.**Yield stresses under different shear directions (Note: τ

_{p}data are from Bao et al. [81]).

References | τ_{y}_{/}τ_{p/}% | ||||||
---|---|---|---|---|---|---|---|

New Proposed Method | Empirical Methods | Shear Stiffness Method [65] | Inflection Point Method [68] | ||||

Goodman [60] | Xiao et al. [61] | Sun et al. [62] | Oh, Cording and Moon [53] | ||||

Bandis et al. [67] | 83.70 | 70–90 | 70 | 85 | 76.83 | 2.48 | 90.96 |

Papaliangas et al. [71] | 77.98 | 70–90 | 70 | 85 | 58.72 | 6.80 | 70.12 |

Grasselli [72] | 92.80 | 70–90 | 70 | 85 | 39.11 | 9.88 | 97.20 |

Nasir and Fall [73] | 62.69 | 70–90 | 70 | 85 | 17.27 | − | 57.57 |

68.42 | 70–90 | 70 | 85 | 72.93 | − | 61.50 | |

Bahaaddini [74] | 90.67 | 70–90 | 70 | 85 | 68.00 | − | 82.67 |

80.00 | 70–90 | 70 | 85 | 63.43 | − | 69.14 | |

Ge et al. [75] | 82.23 | 70–90 | 70 | 85 | 49.31 | − | 68.21 |

77.11 | 70–90 | 70 | 85 | 49.37 | − | 82.10 | |

53.84 | 70–90 | 70 | 85 | 40.82 | − | 45.19 | |

71.95 | 70–90 | 70 | 85 | 43.88 | − | 54.82 | |

Ong and Choo [76] | 78.77 | 70–90 | 70 | 85 | 74.82 | − | 65.63 |

74.71 | 70–90 | 70 | 85 | 71.73 | − | 66.73 | |

Xie et al. [70] | 82.6 | 70–90 | 70 | 85 | 38.2 | − | 78.16 |

Temperature/°C | τ_{y}/MPa | τ_{p}/MPa | τ_{y}/τ_{p}/% |
---|---|---|---|

20 | 10.49 | 10.69 | 98.13 |

100 | 9.00 | 9.50 | 94.74 |

200 | 9.00 | 9.61 | 93.65 |

300 | 8.94 | 9.03 | 99.00 |

400 | 7.78 | 8.36 | 93.06 |

Normal Stress/MPa | τ_{y}/MPa | τ_{p}/MPa | τ_{y}/τ_{p}/% |
---|---|---|---|

10 | 14.96 | 15.40 | 97.20 |

20 | 24.90 | 25.19 | 98.86 |

30 | 35.02 | 35.89 | 97.60 |

40 | 40.64 | 42.37 | 95.92 |

50 | 34.69 | 48.34 | 71.76 |

60 | 51.43 | 55.14 | 93.26 |

Shear Velocity/m/s | τ_{y}/MPa | τ_{p}/MPa | τ_{y}/τ_{p}/% |
---|---|---|---|

4.15 | 5.80 | 7.50 | 77.40 |

5.29 | 6.39 | 9.34 | 68.34 |

7.07 | 10.79 | 13.30 | 81.13 |

JRC | τ_{y}/MPa | τ_{p}/MPa | τ_{y}/τ_{p}/% |
---|---|---|---|

0.4 | 0.27 | 0.51 | 52.93 |

5.8 | 0.51 | 0.68 | 75.36 |

12.8 | 0.92 | 1.11 | 82.15 |

16.7 | 1.02 | 1.22 | 83.77 |

Shear Direction/° | τ_{y}/MPa | τ_{p}/MPa | τ_{y}/τ_{p}/% |
---|---|---|---|

0 | 0.36 | 0.45 | 81.81 |

45 | 0.39 | 0.49 | 80.39 |

90 | 0.40 | 0.61 | 65.50 |

135 | 0.36 | 0.47 | 76.14 |

180 | 0.42 | 0.56 | 74.78 |

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**MDPI and ACS Style**

Han, Z.; Xie, S.; Lin, H.; Duan, H.; Li, D.
A Quantitative Method to Predict the Shear Yield Stress of Rock Joints. *Minerals* **2023**, *13*, 500.
https://doi.org/10.3390/min13040500

**AMA Style**

Han Z, Xie S, Lin H, Duan H, Li D.
A Quantitative Method to Predict the Shear Yield Stress of Rock Joints. *Minerals*. 2023; 13(4):500.
https://doi.org/10.3390/min13040500

**Chicago/Turabian Style**

Han, Zhenyu, Shijie Xie, Hang Lin, Hongyu Duan, and Diyuan Li.
2023. "A Quantitative Method to Predict the Shear Yield Stress of Rock Joints" *Minerals* 13, no. 4: 500.
https://doi.org/10.3390/min13040500