# Scale Effects on Shear Strength of Rough Rock Joints Caused by Normal Stress Conditions

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## Abstract

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_{n}applied to rock samples under direct shear tests. In this research, a two-dimensional particle flow code (PFC2D) is adopted to build a synthetic sandstone rock model with a standard joint roughness coefficient (JRC) profile. The manufactured rock model, which is adjusted by the experiment data and tested by the empirical Barton’s shear strength criterion, is then used to research scale effects on the shear strength of rock joints caused by normal stresses. It is found that the failure type can be affected by JRC and σ

_{n}. Therefore, a scale effect index (SEI) that is equal to JRC plus two times σ

_{n}(MPa) is proposed to identify the types of shear failure. Overall, shearing off asperities is the main failure mechanism for rock samples with SEI > 14, which leads to negative scale effects. It is also found that the degree of scale effects on the shear strength of rock joints is more obvious at low normal stress conditions, where σ

_{n}< 2 MPa.

## 1. Introduction

**Table 1.**Review of scale effects on the shear strength of rock joints [21].

Authors | Rock Types | Sample Size | Normal Stress (MPa) | Scale Effect |
---|---|---|---|---|

Azinfar et al. [13] | Silicon rubber | 25–2500 cm^{2} | 0.3, 0.8, 1.4 | O, N, P |

Barton and Choubey [19] | Granite | 9.8 × 4.5, 45 × 50 cm | 0.1–2 | N |

Bandis et al. [22] | plaster | 6–36 cm | 1 | N |

Bahaaddini et al. [21] | Sandstone | 5–40 cm | 0.5 | N |

Castelli et al. [23] | Cement | 100–400 cm^{2} | 0.75, 1.5, 3 | N |

Fardin [24] | Concrete | 5 × 5–20 × 20 cm^{2} | 1, 2.5, 5, 10 | N |

Hencher et al. [25] | Limestone | 44–531 cm^{2} | 0.0245 | O |

Johansson [26] | Granite | 36, 400 cm^{2} | 1 | O |

Ohnishi et al. [12] | Concrete | 100–1000 cm^{2} | 0.26–2.04 | P |

Pratt et al. [14] | Quartz diorite | 60, 142–5130 cm^{2} | 3 | N |

Ueng et al. [15] | Cement | 7.5–30 cm^{2} | 0.3, 0.6, 0.9 | O, N |

Vallier et al. [27] | - | 10–200 cm | 2 | N |

Yoshinaka et al. [17] | Granite | 20–9600 cm^{2} | 0.26–2.04 | N |

## 2. Synthetic Rock Model for Numerical Tests

#### 2.1. Synthetic Rock Model Based on PFC2D

_{j}, shear stiffness k

_{sj}, and normal stiffness k

_{nj}when the SJM is put into the BPM [32]. The synthetic rock model constructed by the BPM and SJM has the ability to simulate various mechanical responses of jointed rock masses including peak strength [31], scale effect [33], anisotropy [34], and cracking processes [30,33] in rocks and rock-like materials.

#### 2.2. Calibration of Numerical Models

_{j}, shear stiffness k

_{sj}, and normal stiffness k

_{nj}. In this research, the values of k

_{nj}= 25 GPa, k

_{sj}= 13 GPa, and μ

_{j}= 0.75 were selected using the inverse-modeling calibration approach to ensure that the numerical rock model can give a similar response as that from laboratory tests with joint shear stiffness K

_{s}= 6.4 GPa/m, normal stiffness K

_{n}= 28.6 GPa/m, and joint friction angle φ

_{b}= 37.6°. The calibration procedure was as follows: (1) The normal deformability compression test was carried out to calibrate normal stiffness k

_{nj}. (2) The shear test was carried out to calibrate shear stiffness k

_{sj}under normal stress of 1 MPa condition. (3) Direct shear tests were undertaken and friction coefficient μ

_{j}was calibrated. Figure 2, Figure 3 and Figure 4 present the final mechanical responses of the synthetic rock models after the final calibration.

_{n}generated by the synthetic rock specimen is 28.6 GPa/m, which is close to the laboratory test results with K

_{n}= 28.8 GPa/m.

_{s}generated by the synthetic rock specimen is 6.4 GPa/m, which is the same as laboratory test results with K

_{s}= 6.4 GPa/m.

_{b}generated by the synthetic rock specimen is 36.1°, which is close to laboratory test results with φ

_{b}= 37.6°.

## 3. Validation of Synthetic Rock Models

#### 3.1. Barton’s Shear Strength Model

_{b}is the joint friction angle. JCS is the joint compression strength, which is equal to UCS of intact rock in this research. JRC stands for joint roughness coefficient and can be calculated using standard joint profiles.

#### 3.2. Numerical Simulation Results

_{b}= 36.1° and JCS = 27.4 MPa. Figure 7 compares the shear strength obtained from numerical simulations to Barton’s model, indicating that the usage of synthetic rock models is capable of generating adequate shear strength of rock joints.

## 4. Configuration of Rock Samples for Scale Effect Investigations

## 5. Results and Discussion

_{i}is the joint length of the rock sample, y

_{i}the shear stress of the rock sample, and N is the number of the testing sample. k > 0 means the rock joint has a positive scale effect and k < 0 means the rock joint has a negative scale effect. The value of k can be calculated using three groups of data. For example, for rock samples with JRC = 2 under the normal stress σ

_{n}= 5 MPa, the shear strength of rock samples with joint lengths l = 100, 200, and 300 mm are 4.2, 4.4, and 4.6 MPa, respectively. Therefore, data (100, 4.2), (200, 4.4), and (300, 4.6) were put into Equation (2) to calculate the value of k. The result shows k = 0.4, which means the scale effect is positive. Table 4 shows comprehensive scale effect results of rock samples with various JRC profiles under different normal stress conditions. In Table 4, P means positive scale effect and N means negative scale effect.

_{n}= 0.5 MPa, the number of shear cracks is 10 and the scale effect is positive. However, when a rock sample with JRC = 20 under the normal stress σ

_{n}= 5 MPa, the number of shear cracks is 280 and the scale effect is negative.

_{n}= 0.5 MPa, the value of SEI = 2 + 2 × 0.5 = 3. The number P3 in Table 4 means the rock sample with SEI = 3 has a positive scale effect. It was found that 20 out of 21 rock samples have negative scale effects when SEI > 14, and 29 out of 33 rock samples have positive scale effects when SEI < 14.

_{n}= 0.5 MPa). The relations between SEI values and shear crack numbers of rock samples are also plotted in Figure 11. We can find that the number of shear cracks is low when SEI < 14. However, the number of shear cracks dramatically increases when the value of SEI is over 14, where the controlling failure mechanism transforms sliding to shearing off asperities.

_{n}< 2 MPa.

_{n}= 1 MPa, then, it decreases sharply to 0.14 when σ

_{n}increases to 2 MPa. After that, there is a slight change in CV values with a further increase of σ

_{n}from 2 MPa to 10 MPa. Such change in CV values with normal stresses is similar to that of the numerical results in Figure 12.

## 6. Conclusions

_{n}< 2 MPa.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**The direct shear test on the synthetic rock specimen with a planar joint under normal stress of 1 MPa.

**Figure 7.**Comparison failure envelopes obtained from numerical simulations and the Barton’s empirical model.

**Figure 8.**Two types of configuration of samples for scale effect investigations (

**a**) division of the Barton’s JRC profile (

**b**) assembly of repeated Barton’s JRC profile.

**Figure 9.**Results of scale effects on the shear strength of rock samples under various normal stress conditions.

**Figure 10.**Failure pattern and crack number of rock samples (40 mm × 100 mm) at peak shear strength.

**Figure 13.**Laboratory data of scale effect on shear stress of rock joints under various normal stress conditions (data from Fardin [24]).

Parameters | Values |
---|---|

Minimum particle radius: R_{min} (mm) | 0.28 |

Maximum particle radius: R_{max} (mm) | 0.42 |

Stiffness ratio: k_{n}/ k_{s} | 2.1 |

Effective modulus: E_{c} (GPa) | 4.1 |

Bond tensile strength: Tb (MPa) | 11.2 |

Bond friction angle: Φb (°) | 35 |

Cohesion: cb (MPa) | 11.2 |

Friction coefficient: u | 0.2 |

Porosity ratio: e | 0.16 |

**Table 3.**Comparison of mechanical properties calculated from the numerical model and tested from laboratory.

Properties | Parameters | Laboratory Test | PFC Model |
---|---|---|---|

Intact rock properties | UCS (MPa) | 27.40 | 27.40 |

E (GPa) | 4.20 | 4.20 | |

ν | 0.20 | 0.21 | |

Joint properties | K_{n} (GPa/m) | 28.6 | 28.6 |

K_{s} (GPa/m) | 6.40 | 6.40 | |

φ_{b} (°) | 37.60 | 36.10 |

JRC | Normal Stress σ_{n} (MPa) | |||||
---|---|---|---|---|---|---|

0.5 | 1 | 2 | 3 | 4 | 5 | |

2 | P3 | P4 | P6 | P8 | P10 | P12 |

4 | P5 | P6 | P8 | P10 | N12 | N14 |

6 | P7 | P8 | P10 | P12 | P14 | N16 |

8 | P9 | P10 | P12 | N14 | N16 | N18 |

10 | P11 | P12 | P14 | N16 | N18 | N20 |

12 | P13 | P14 | N16 | N18 | N20 | N22 |

14 | N15 | N16 | N18 | N20 | N22 | N24 |

16 | P17 | P18 | P20 | P22 | N24 | N26 |

18 | N19 | P20 | N22 | N24 | N26 | N28 |

20 | N21 | N22 | N24 | N26 | N28 | N30 |

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**MDPI and ACS Style**

Shen, J.; Sun, C.; Huang, H.; Chen, J.; Wu, C.
Scale Effects on Shear Strength of Rough Rock Joints Caused by Normal Stress Conditions. *Sustainability* **2023**, *15*, 7520.
https://doi.org/10.3390/su15097520

**AMA Style**

Shen J, Sun C, Huang H, Chen J, Wu C.
Scale Effects on Shear Strength of Rough Rock Joints Caused by Normal Stress Conditions. *Sustainability*. 2023; 15(9):7520.
https://doi.org/10.3390/su15097520

**Chicago/Turabian Style**

Shen, Jiayi, Chenhao Sun, Huajie Huang, Jiawang Chen, and Chuangzhou Wu.
2023. "Scale Effects on Shear Strength of Rough Rock Joints Caused by Normal Stress Conditions" *Sustainability* 15, no. 9: 7520.
https://doi.org/10.3390/su15097520