# Numerical Analysis of the Mechanical Behaviors of Various Jointed Rocks under Uniaxial Tension Loading

^{*}

## Abstract

**:**

_{t}), however, the changes in α and β have less influence on the Young’s modulus in tension (E

_{t}). With respect to joint spacing, the simulations show that the effects of joint spacing on σ

_{t}and E

_{t}are negligible. In relation to the joint density, the numerical results reveal that the joint intensity of rock mass has great effect on E

_{t}but insignificant effect on σ

_{t}.

## 1. Introduction

_{t}) of rocks is much lower than their compressive strength, and (2) joints and fractures of rock mass can offer little resistance to tensile stress [2]. After an excavation is taken underground, tension failure and its induced fractures will occur in surrounding rock masses near the opening [3,4,5,6,7,8]. The initiated fractures will develop/propagate and bring weakening effects on the rock masses under the effect of time and constant stress disturbance (development of entry or shaft, mining activities, etc.). As it is well established that the Young’s modulus of a rock mass is directly related to the fracture or joint intensity [9,10,11,12,13], the aforementioned fracture development can be described as a Young’s modulus degradation process in the view of an equivalent material method in continuum mechanics.

_{t}) was always no larger than the compressive Young’s modulus (E

_{c}) [17,18,19,20,21]. Hawkes et al. [17] conducted direct tension tests on different kinds of rocks, and the results showed that the ratio of E

_{t}/E

_{c}was 1/9 for Barre sandstone and 0.5 for Barre granite. Stimpson and Chen [20] acquired the ratios to be 0.5, 1.0, 0.7, and 0.3–0.4 for four different rocks from cyclic loading uniaxial tension and compression tests. Similar results were obtained by Yu et al. [22] with an originally developed loading frame for direct tension. It has been addressed by researchers that the improper assumption that E

_{t}equals to E

_{c}may lead to errors in calculating stress distributions around underground openings by means of analytical or numerical analysis, as well as in determining the tensile strength of rocks with Brazilian tests [18,20,22,23].

## 2. Numerical Modelling

_{t}and E

_{t}under different joint conditions are calculated, and the effects of joint condition on the tensile strength (σ

_{t}) and E

_{t}are hereby investigated.

## 3. Effects of Joints on the Tensile Properties

#### 3.1. Effect of the Dip Angle of A Single Joint

_{t}) of the pre-peak curves vary monotonically increases with the increase of α.

_{t}and E

_{t}of the specimen are 0.78 MPa and 0.80 GPa, respectively. When α reaches 50°, the σ

_{t}and E

_{t}increase to 1.05 MPa and 0.81 GPa, which are approximately 1.35 and 1.01 times the values in the case of α = 0°, respectively. In comparison, the changes in α have a greater influence on σ

_{t}than E

_{t}. In reality, joint surfaces are usually rough, and the asperities provide resistance to shear stress, and the joint roughness is positively correlated to the shear strength [35]. The joint roughness is described with normal stiffness, shear stiffness, cohesion, etc. in 3DEC. When α is not 0°, the shear strength of the joint will provide extra resistance to the tensile load on rock failure and deformability, which leads to the high σ

_{t}and E

_{t}when is α higher. However, when α = 0°, the joint is perpendicular to the tensile load, so no extra resistance can be provided from the shear behavior of the joint.

#### 3.2. Effects of Parallel Joints on the Tensile Properties

#### 3.2.1. Effects of Joint Spacing and Dip Angle of Two Parallel Joints

_{t}and E

_{t}, four sets of models with different dip angles (α = 20, 30, 40, and 50°) were built (as illustrated in Figure 4) under the condition of n = 2, and for each set of the model, four subsets with different joint spacing (d = 10, 20, 30, and 40 mm) were analyzed. The test program specifics are shown in Table 4, and the stress–strain curves under uniaxial tensile load are shown in Figure 5. The results are shown in Table 5.

_{t}and E

_{t}are negligible. When d decreases from 40 to 10 mm, σ

_{t}and E

_{t}merely decrease by 0.003 MPa and 0.002 GPa, respectively. This result is obtained under the premise that no joint development is considered. In practice, joints will develop, and adjacent joints will connect and merge under mechanical loading. However, this study focuses on the effects of existing joints on the tensile properties. Similar to the results from Section 3.1, σ

_{t}and E

_{t}are affected by the dip angle of the two parallel joints, but the effects vary. The dip angle has a greater effect on σ

_{t}than E

_{t}. As the dip angle increases from 20 to 50°, σ

_{t}increases from 0.88 to 1.05 MPa, while E

_{t}lightly increases from 0.71 to 0.73 GPa. When comparing with Table 3, it can be noticed that the value of σ

_{t}is identical in the cases of single-jointed and double-jointed models with the same dip angle.

#### 3.2.2. Effect of Joint Density of Parallel Joints

_{t}and E

_{t}is investigated by means of the number of joints (n) inside the specimen. Four sets of models with different dip angles (α = 20, 30, 40, 50°) were built (as illustrated in Figure 6) under the condition of d = 5 mm, and for each set of the model, six subsets with different numbers of joints (n = 1, 2, 3, 4, 5, 6) were analyzed. The test program specifics are shown in Table 6 and the stress–strain curves under uniaxial tensile load are shown in Figure 7. The results are shown in Table 7.

_{t}significantly drops from 0.81 to 0.52 GPa, while σ

_{t}remains constant. The influencing pattern of n on E

_{t}is identical for different values of α. These results indicate that the fracture intensity of a rock mass has a great effect on E

_{t}but a negligible effect on σ

_{t}.

_{t}with different values of α after fitting is shown in Figure 8. As can be seen, E

_{t}is negatively correlated to n, and the relationship varies with different values of α. The change in E

_{t}with n will be less significant in cases of higher values of α.

#### 3.3. Effects of Intersecting Joints on the Tensile Properties

#### 3.3.1. Effect of the Intersection Angle of an Intersecting Joints Set

_{t}) and the slopes (E

_{t}) of the pre-peak curves vary and

_{t}monotonically increase with the increase of β.

_{t}and E

_{t}of the specimen are 0.80 MPa and 0.71 GPa, respectively. When β reaches 100°, the σ

_{t}and E

_{t}increase to 1.05 MPa and 0.73 GPa, which are approximately 1.3 and 1.03 times the values in the case of β = 20°, respectively. In contrast, the changes in β have a greater influence on σ

_{t}than E

_{t}.

#### 3.3.2. Effects of Joint Spacing and Intersection Angle of a Joints Set

_{t}and E

_{t}, four sets of models with different dip angels (β = 20, 40, 60, and 100°) were built (as illustrated in Figure 12), and for each set of the model, four subsets with different joint spacing (d = 10, 20, 30, and 40 mm) were analyzed. The test program specifics are shown in Table 9. The stress–strain curves under uniaxial tensile load are shown in Figure 13 and the results are shown in Table 10.

_{t}and E

_{t}are trivial. When d increases from 10 to 40 mm, σ

_{t}and E

_{t}merely increase by 0.0004 MPa and 0.0008 GPa, respectively. Similar to the results from Section 3.3.1, σ

_{t}and E

_{t}are affected by the joint spacing of the intersection joints, but the effect varies. The intersection angle has a greater effect on σ

_{t}than E

_{t}. As the intersection angle increases from 40 to 100°, σ

_{t}doubles from 0.88 to 1.05 MPa, while E

_{t}lightly increases from 0.58 to 0.61 GPa. It can be noticed that the value of σ

_{t}is identical in the cases of single-jointed and double-jointed models with identical β.

#### 3.3.3. Effect of Joint Density of Intersection Joints

_{t}and E

_{t}are investigated by means of the number of joints (n) inside the specimen. Four sets of models with different dip angles (β = 40, 60, 80, and 100°) were built (as illustrated in Figure 14) under the condition of d = 5 mm, and for each set of the model, six subsets with different numbers of joints (n = 4, 6, 8, 10, and 12) were analyzed. The test program specifics are shown in Table 11 and the stress–strain curves under uniaxial tensile load are shown in Figure 15. The results are shown in Table 12

_{t}significantly drops from 0.73 to 0.34 GPa, while σ

_{t}remains constant. The influencing pattern of n on E

_{t}is identical for different values of α. These results indicate that the fracture intensity of a rock mass has a great effect on E

_{t}but a negligible effect on σ

_{t}.

_{t}with different values of β after fitting is shown in Figure 16. As can be seen, E

_{t}is negatively correlated to n, and the relationship varies with different values of β. The change in E

_{t}with n will be less significant in case of a higher value of β.

## 4. Discussion

_{ci}) < 100 MPa, the elastic modulus of a rock mass (E

_{t}) is estimated from the following equation [39]:

_{ci}is the uniaxial compressive strength of intact rock, and D is a factor that depends on the degree of disturbance to which the rock mass has been subjected by blast damage and stress relaxation. When D = 0, the relationships between E

_{t}and GSI under the conditions of different values of σ

_{ci}are shown in Figure 18a. With the increase of the development of rock mass fissures, the Young’s moduli decline notably. The degradation patterns with the number of parallel and intersection joints are illustrated in Figure 19a and Figure 20a, respectively, which agree with the numerical results presented in the previous sections.

_{b}is the reduced value of the material constant m

_{i}for the rock mass, and is given by:

_{t}and GSI under the conditions of different values of σ

_{ci}are shown in Figure 18b. In a certain interval, the influence on the tensile strength is negligible with the increase of the development of rock mass fissures. Then, the rationalities of the conclusion in this paper were proved, as illustrated in Figure 19b and Figure 20b.

## 5. Conclusions

- (1)
- For rock specimens with a single joint, the tensile strength (σ
_{t}) is positively related to the joint angle (α). However, the Young’s modulus in tension (E_{t}) is barely influenced by α. - (2)
- For rock specimens with parallel joints, the perpendicular distance (d) between the joints has negligible effects on σ
_{t}and E_{t}. The influencing pattern is similar to the tests with single joints by changing α while keeping d constant. The number of parallel joints, or joint density, has notable effects on E_{t}but few effects on σ_{t}. - (3)
- For rock specimens with intersecting joints, the number of intersecting joints has notable effects on E
_{t}but few effects on σ_{t}. For a given number of intersecting joints, σ_{t}is positively correlated to the angle between two interesting joints (β).

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 13.**Tensile behaviors of intersection-jointed rock specimens with different intersection angles.

**Figure 15.**Tensile behaviors of intersection-jointed rock specimens with different numbers of intersection joints.

**Figure 18.**The effect of GSI onE

_{t}, and σ

_{t}of rock mass (

**a**) Relationship between GSI and E

_{t,}(

**b**) Relationship between GSI and σ

_{t.}

**Figure 19.**Effect of the number of parallel fractures on E

_{t}and σ

_{t}. (

**a**) Effect of the number of fractures on E

_{t},(

**b**) Effect of the number of fractures on σ

_{t.}

**Figure 20.**Effect of the number of intersecting parallel fractures on E

_{t}and σ

_{t}. (

**a**) Effect of the number of fractures on E

_{t,}(

**b**) Effect of the number of fractures on σ

_{t.}

Lithology | Density (kg/m^{3}) | Bulk modulus K (GPa) | Shear modulus G (GPa) | Cohesion C (MPa) | Internal friction angle φ (°) | Tensile strength (MPa) |
---|---|---|---|---|---|---|

Sandstone | 2630 | 26.49 | 19.05 | 3.75 | 25.9 | 2.25 |

Lithology | Cohesion C (MPa) | JKN (GPa/M) | JKS (GPa/M) | Joint friction angle (°) | Tensile strength (MPa) |
---|---|---|---|---|---|

Sandstone | 3.67 | 26.49 | 19.05 | 25.9 | 1.88 |

**Table 3.**Tensile strengths (σ

_{t}) and Young’s moduli in tension (E

_{t}) of single-jointed rock specimens with different dip angles.

α | σ_{t} (MPa) | E_{t} (GPa) |
---|---|---|

α = 0° | 0.78 | 0.80 |

α = 10° | 0.80 | 0.80 |

α = 20° | 0.88 | 0.80 |

α = 30° | 0.91 | 0.80 |

α = 40° | 0.93 | 0.81 |

α = 50° | 1.05 | 0.81 |

Sets of models | α | Subsets with respect to d |
---|---|---|

1 | 20° | 10 mm |

20 mm | ||

30 mm | ||

40 mm | ||

2 | 30° | 10 mm |

20 mm | ||

30 mm | ||

40 mm | ||

3 | 40° | 10 mm |

20 mm | ||

30 mm | ||

40 mm | ||

4 | 50° | 10 mm |

20 mm | ||

30 mm | ||

40 mm |

α | σ_{t} (MPa) | E_{t} (GPa) |
---|---|---|

20° | 0.88 | 0.71 |

30° | 0.91 | 0.71 |

40° | 0.93 | 0.73 |

50° | 1.05 | 0.73 |

Sets of models | α | Subsets with respect to n |
---|---|---|

1 | 20° | n = 2 |

n = 3 | ||

n = 4 | ||

n = 5 | ||

2 | 30° | n = 2 |

n = 3 | ||

n = 4 | ||

n = 5 | ||

3 | 40° | n = 2 |

n = 3 | ||

n = 4 | ||

n = 5 | ||

4 | 50° | n = 2 |

n = 3 | ||

n = 4 | ||

n = 5 |

n | σ_{t} (MPa) | E_{t} (GPa) |
---|---|---|

n = 1 | 0.93 | 0.81 |

n = 2 | 0.93 | 0.73 |

n = 3 | 0.93 | 0.66 |

n = 4 | 0.93 | 0.61 |

n = 5 | 0.93 | 0.56 |

n = 6 | 0.93 | 0.52 |

α | σ_{t} (MPa) | E_{t} (GPa) |
---|---|---|

β = 20° | 0.80 | 0.71 |

β = 40° | 0.88 | 0.71 |

β = 60° | 0.91 | 0.71 |

β = 80° | 0.93 | 0.73 |

β = 100° | 1.05 | 0.73 |

Sets of Models | d | Subsets with Respect to β |
---|---|---|

1 | d = 10 mm | β = 40° |

β = 60° | ||

β = 80° | ||

β = 100° | ||

2 | d = 20 mm | β = 40° |

β = 60° | ||

β = 80° | ||

β = 100° | ||

3 | d = 30 mm | β = 40° |

β = 60° | ||

β = 80° | ||

β = 100° | ||

4 | d = 40 mm | β = 40° |

β = 60° | ||

β = 80° | ||

β = 100° |

**Table 10.**σ

_{t}and E

_{t}of intersection-jointed rock specimens with different intersection angles.

β. | σ_{t} (MPa) | E_{t} (GPa) |
---|---|---|

β = 40° | 0.88 | 0.58 |

β = 60° | 0.91 | 0.58 |

β = 80° | 0.93 | 0.58 |

β = 100° | 1.05 | 0.61 |

Sets of Models | β | Subsets with Respect to n |
---|---|---|

1 | 40° | n = 4 |

n = 6 | ||

n = 8 | ||

n = 10 | ||

n = 12 | ||

2 | 60° | n = 4 |

n = 6 | ||

n = 8 | ||

n = 10 | ||

n = 12 | ||

3 | 80° | n = 4 |

n = 6 | ||

n = 8 | ||

n = 10 | ||

n = 12 | ||

4 | 100° | n = 4 |

n = 6 | ||

n = 8 | ||

n = 10 | ||

n = 12 |

n | σ_{t} (MPa) | E_{t} (GPa) |
---|---|---|

n = 2 | 0.93 | 0.73 |

n = 4 | 0.93 | 0.58 |

n = 6 | 0.93 | 0.50 |

n = 8 | 0.93 | 0.44 |

n = 10 | 0.93 | 0.38 |

n = 12 | 0.93 | 0.34 |

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**MDPI and ACS Style**

Shu, J.; Jiang, L.; Kong, P.; Wang, Q.
Numerical Analysis of the Mechanical Behaviors of Various Jointed Rocks under Uniaxial Tension Loading. *Appl. Sci.* **2019**, *9*, 1824.
https://doi.org/10.3390/app9091824

**AMA Style**

Shu J, Jiang L, Kong P, Wang Q.
Numerical Analysis of the Mechanical Behaviors of Various Jointed Rocks under Uniaxial Tension Loading. *Applied Sciences*. 2019; 9(9):1824.
https://doi.org/10.3390/app9091824

**Chicago/Turabian Style**

Shu, Jiaming, Lishuai Jiang, Peng Kong, and Qingbiao Wang.
2019. "Numerical Analysis of the Mechanical Behaviors of Various Jointed Rocks under Uniaxial Tension Loading" *Applied Sciences* 9, no. 9: 1824.
https://doi.org/10.3390/app9091824