# A New Shear Strength Criterion for Rock Masses with Non-Persistent Discontinuities Considering the Nonlinear Progressive Failure Process

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{n}is the normal stress; k is the connectivity rate; c, c

_{d}, and c

_{r}denote the cohesion of a rock mass, discontinuities and rock bridges respectively; and φ, φ

_{d}, and φ

_{r}denote the friction angle of a rock mass, discontinuities and rock bridges respectively.

## 2. The Establishment of the New Shear Strength Criterion

#### 2.1. Data Collection

#### 2.2. The New Shear Strength Criterion

#### 2.3. The Fitting Curve of Correction Coefficients Based on Test Results

_{d}can be determined according to Equation (7) proposed by Barton (1973) [4]:

_{b}is the basic friction angle of discontinuities.

## 3. The Reliability of the New Shear Strength Criterion

_{1}and c

_{2}in fourth and sixth columns respectively. And then the ratios of the cohesion estimated by the Jennings and new criterions to the real cohesion obtained by laboratory tests are shown as R

_{c1}and R

_{c2}in fifth and seventh columns respectively. It is obvious that the ratio of the estimated cohesion to the real cohesion can reflect the reliability of the criterions. Thus, we show these ratios of samples with non-persistent discontinuities in Figure 3a. The red color denotes the results estimated by the new criterion while the blue color denotes the results estimated by the Jennings criterion. The various shapes of labels denote the data from different references. The estimated result is better if the ratio is closer to 1. It can be seen that the estimated values by the new criterion are superior to those by the Jennings criterion for samples with connectivity rates of 0.2, 0.5 and 0.8. It is difficult to judge which is better for connectivity rates of 0.4 and 0.6. Therefore, we adopted a quantitative factor, i.e., the variance of the ratios to 1 (V

^{2}) (Equation (8)).

^{2}denotes the variance, R

_{n}denotes the ratio and n denotes the number of data.

_{1}and tanφ

_{2}in fourth and sixth columns respectively in Table 5. And then the ratios of internal friction coefficients estimated by the criterions to real internal friction coefficients obtained by laboratory tests are shown as R

_{f1}and R

_{f2}in fifth and seventh columns respectively. These ratios of samples with non-persistent discontinuities are shown in Figure 4a. The red color denotes the results estimated by the new criterion while the purple color denotes the results estimated by the Jennings criterion. The various shapes of labels denote the data from different references. It indicates that the estimated values by the new criterion are superior to those by the Jennings criterion for samples with connectivity rates of 0.2 and 0.8. It is difficult to judge which is better for connectivity rates of 0.4, 0.5 and 0.6. The variance is also adopted to judge qualitatively. The variances of internal friction coefficients estimated by the Jennings and new criterions are 0.02 and 0.01 respectively, which means the latter one is superior to the former one obviously.

## 4. Discussion

#### 4.1. The Nonlinear Features of the Rock Bridge Strength during Progressive Failure

^{2}is not high. This is primarily because the data for such type of test are not sufficient enough for its difficulties of sample preparation and experiment implementation, and thus a further research is needed to incorporate many more data. The exponential function, which is not the best fitting one to characterize the correction coefficient A, is adopted mainly regarding the previous research of the variation trend of cohesion during the progressive failure process, i.e., an exponential form [33]. Despite of these, the new criterion has an obvious progress compared with the Jennings criterion.

#### 4.2. The Influence of Sample Types

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of artificial rock mass samples with various discontinuity positions and connectivity rates.

**Figure 2.**The correction coefficient A of the cohesion and the correction coefficient B of the internal friction coefficient for rock bridges with the connectivity rate.

**Figure 3.**Comparison of the ratios of the estimated cohesions by the Jennings and new criterions to the test results of the cohesions with various connectivity rates. (

**a**) The cohesion ratio with the connectivity rate; (

**b**) The average cohesion ratio with the connectivity rate; (

**c**) The error with the connectivity rate.

**Figure 4.**Comparison of the ratios of the estimated internal friction coefficients by the Jennings and new criterions to the test results of the internal friction coefficients with various connectivity rates. (

**a**) The ratio of the internal friction coefficient with the connectivity rate; (

**b**) The average ratio of the internal friction coefficient with the connectivity rate; (

**c**) The error with the connectivity rate.

**Table 1.**Strength parameters of artificial rock mass samples with various discontinuity positions and connectivity rates.

Sample Type | Connectivity Rate | Quality Ratio of Materials | Cohesion of the Rock Bridge (MPa) | Angle of the Internal Friction of the Rock Bridge (°) | Cohesion of the Discontinuity (MPa) | Angle of the Internal Friction of the Discontinuity (°) | Reference |
---|---|---|---|---|---|---|---|

T-type I-type | 0.6 | Sand: Gypsum: Water = 3:3:2 | 4.23 | 26.55 | 0 | 35.2 | [15] |

T-type I-type | 0.4 | Cement: Sand: Water = 5:5:2 | 5.2 | 56.31 | 0.19 | 39.69 | [17] |

T-type I-type C-type | 0.2, 0.4, 0.5, 0.6, 0.8 | Cement: Sand: Water = 5:5:2 | 4.7 | 59.24 | 0.63 | 37.95 | [19] |

T-type | 0.5 | Cement: Sand: Water = 2:3:1 | 3.93 | 39.5 | 0 | 32.3 | [25] |

F-type I-type B-type | 0.2, 0.4, 0.6 | Quartz sand: Cement = 1:1 | 8.3 | 37.16 | 1.63 | 32.8 | [21] |

**Table 2.**Direct shear test results of artificial rock mass samples with various discontinuity positions and connectivity rates.

Sample Type | Connectivity Rate | Shear Rate | Cohesion (MPa) | Coefficient of the Internal Friction | Reference |
---|---|---|---|---|---|

T-type | 0.6 | 0.003 mm/s | 1.186 | 0.559 | [15] |

I-type | 1.057 | 0.573 | |||

T-type | 0.4 | 0.005 mm/s | 3 | 1.4 | [17] |

I-type | 3.4 | 1.3 | |||

T-type | 0.2 | 0.005 mm/s | 3.205 | 1.836 | [19] |

T-type | 0.4 | 2.585 | 1.664 | ||

I-type | 0.6 | 1.9367 | 1.13 | ||

C-type | 0.5 | 2.259 | 1.238 | ||

C-type | 0.6 | 3.504 | 1.29 | ||

C-type | 0.8 | 2.687 | 0.759 | ||

T-type | 0.5 | 0.005 mm/s | 1.423 | 0.821 | [25] |

F-type | 0.2 | 1 kN/s | 5 | 0.985 | [21] |

F-type | 0.4 | 4.9 | 0.743 | ||

F-type | 0.6 | 3.37 | 0.787 | ||

I-type | 0.4 | 3.76 | 0.649 | ||

B-type | 0.4 | 8.2 | 0.61 |

**Table 3.**Correction coefficients A and B of the strength parameters for rock mass samples with various discontinuity positions and connectivity rates.

Sample Type | Connectivity Rate | A | B | Reference |
---|---|---|---|---|

T-type | 0.6 | 0.701 | 0.677 | [15] |

I-type | 0.6 | 0.625 | 0.749 | |

T-type | 0.4 | 0.937 | 1.187 | [17] |

I-type | 0.4 | 1.065 | 1.076 | |

T-type | 0.2 | 0.819 | 1.256 | [19] |

T-type | 0.4 | 0.827 | 1.358 | |

I-type | 0.6 | 0.829 | 0.985 | |

C-type | 0.5 | 0.827 | 1.035 | |

C-type | 0.6 | 1.663 | 1.262 | |

C-type | 0.8 | 2.322 | 0.504 | |

T-type | 0.5 | 0.724 | 1.225 | [25] |

F-type | 0.2 | 0.704 | 1.412 | [21] |

F-type | 0.4 | 0.853 | 1.066 | |

F-type | 0.6 | 0.720 | 1.321 | |

I-type | 0.4 | 0.624 | 0.861 | |

B-type | 0.4 | 1.516 | 0.775 |

Sample Type | Connectivity Rate | Test Result of the Cohesion c (MPa) | Estimated Cohesion by the Jennings Criterion c _{1} (MPa) | Ratio of c_{1} to cR _{c1} | Estimated Cohesion by the New Criterion c _{2} (MPa) | Ratio of c_{2} to cR _{c2} | Reference |
---|---|---|---|---|---|---|---|

T-type | 0.6 | 1.186 | 1.692 | 1.427 | 1.7316 | 1.46 | [15] |

I-type | 0.6 | 1.057 | 1.692 | 1.601 | 1.7316 | 1.6382 | |

T-type | 0.4 | 3.000 | 3.196 | 1.065 | 2.7347 | 0.9116 | [17] |

I-type | 0.4 | 3.400 | 3.196 | 0.940 | 2.7347 | 0.8043 | |

T-type | 0.2 | 3.205 | 3.886 | 1.212 | 2.7939 | 0.8717 | [19] |

T-type | 0.4 | 2.585 | 3.072 | 1.188 | 2.6551 | 1.0271 | |

I-type | 0.6 | 1.9367 | 2.258 | 1.166 | 2.302 | 1.1886 | |

C-type | 0.5 | 2.259 | 2.665 | 1.180 | 2.5096 | 1.1109 | |

C-type | 0.6 | 3.504 | 2.258 | 0.644 | 2.302 | 0.657 | |

C-type | 0.8 | 2.687 | 1.444 | 0.537 | 1.6594 | 0.6175 | |

T-type | 0.5 | 1.423 | 1.965 | 1.381 | 1.835 | 1.2896 | [25] |

F-type | 0.2 | 5.000 | 6.966 | 1.393 | 5.0373 | 1.0075 | [21] |

F-type | 0.4 | 4.900 | 5.632 | 1.149 | 4.8957 | 0.9991 | |

F-type | 0.6 | 3.370 | 4.298 | 1.275 | 4.3757 | 1.2984 | |

I-type | 0.4 | 3.760 | 5.632 | 1.498 | 4.8957 | 1.302 | |

B-type | 0.4 | 8.200 | 5.632 | 0.687 | 4.8957 | 0.597 |

**Table 5.**Comparison of the internal friction coefficients estimated by the Jennings and new criterions.

Sample Type | Connectivity Rate | Test Result of the Internal Friction Coefficient tanφ | Estimated Internal Friction Coefficient by the Jennings Criterion tanφ_{1} | Ratio of tanφ_{1} to tanφR _{f1} | Estimated Internal Friction Coefficient by the New Criterion tanφ_{2} | Ratio of tanφ_{2} to tanφR _{f2} | Reference |
---|---|---|---|---|---|---|---|

T-type | 0.6 | 0.559 | 0.623 | 1.1154 | 0.5988 | 1.0711 | [15] |

I-type | 0.6 | 0.573 | 0.623 | 1.0876 | 0.5988 | 1.045 | |

T-type | 0.4 | 1.400 | 1.232 | 0.88 | 1.3199 | 0.9428 | [17] |

I-type | 0.4 | 1.300 | 1.232 | 0.9477 | 1.3199 | 1.0153 | |

T-type | 0.2 | 1.836 | 1.491 | 0.8123 | 2.0005 | 1.0896 | [19] |

T-type | 0.4 | 1.664 | 1.303 | 0.7829 | 1.4186 | 0.8525 | |

I-type | 0.6 | 1.130 | 1.140 | 1.0088 | 1.0581 | 0.9364 | |

C-type | 0.5 | 1.238 | 1.208 | 0.9762 | 1.2148 | 0.9812 | |

C-type | 0.6 | 1.290 | 1.114 | 0.8637 | 1.0581 | 0.8202 | |

C-type | 0.8 | 0.759 | 0.926 | 1.2195 | 0.86 | 1.133 | |

T-type | 0.5 | 0.821 | 0.728 | 0.887 | 0.7208 | 0.8779 | [25] |

F-type | 0.2 | 0.985 | 0.735 | 0.7464 | 0.961 | 0.9756 | [21] |

F-type | 0.4 | 0.743 | 0.713 | 0.9594 | 0.757 | 1.0188 | |

F-type | 0.6 | 0.787 | 0.690 | 0.8763 | 0.6529 | 0.8296 | |

I-type | 0.4 | 0.649 | 0.713 | 1.0972 | 0.757 | 1.1664 | |

B-type | 0.4 | 0.610 | 0.713 | 1.1673 | 0.757 | 1.241 |

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**MDPI and ACS Style**

Zheng, B.; Qi, S.; Guo, S.; Huang, X.; Liang, N.; Zou, Y.; Luo, G.
A New Shear Strength Criterion for Rock Masses with Non-Persistent Discontinuities Considering the Nonlinear Progressive Failure Process. *Materials* **2020**, *13*, 4694.
https://doi.org/10.3390/ma13214694

**AMA Style**

Zheng B, Qi S, Guo S, Huang X, Liang N, Zou Y, Luo G.
A New Shear Strength Criterion for Rock Masses with Non-Persistent Discontinuities Considering the Nonlinear Progressive Failure Process. *Materials*. 2020; 13(21):4694.
https://doi.org/10.3390/ma13214694

**Chicago/Turabian Style**

Zheng, Bowen, Shengwen Qi, Songfeng Guo, Xiaolin Huang, Ning Liang, Yu Zou, and Guangming Luo.
2020. "A New Shear Strength Criterion for Rock Masses with Non-Persistent Discontinuities Considering the Nonlinear Progressive Failure Process" *Materials* 13, no. 21: 4694.
https://doi.org/10.3390/ma13214694