# A Numerical Study on the Crack Development Behavior of Rock-Like Material Containing Two Intersecting Flaws

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation Procedure

## 3. Results and Discussion

#### 3.1. Mechanical Properties of Intact Models

#### 3.2. Mechanical Behavior of Specimens Containing Flaws

#### 3.2.1. Effect of the Flaw Geometries on Peak Strength

- (1)
- For low values of α (below app. 50°), the peak strength first increases up to about β = 60° and then decreases.
- (2)
- For higher values of α (above app. 50°), the peak strength first increased up to about β = 45° and then decreased.
- (3)
- For the values of β (above app. 15°), the peak strength decreased with increasing α.

#### 3.2.2. Crack Initiation and Coalescence Behavior

#### 3.3. Crack Initiation Stress

- The ration ${\sigma}_{ci}/{\sigma}_{cn}$ varies between 0.2 and 0.4.
- For the same values of α, the minimum values of ${\sigma}_{ci}/{\sigma}_{cn}$ are observed at β = 30°, which means that the first crack initiates more easily (except α = 60°, β = 45°). For four cases of β = 90° and one case of β = 0°, α = 90° is the maximum value of ${\sigma}_{ci}/{\sigma}_{cn}$ observed.

#### 3.4. Crack Coalescence

- A macroscopic tensile crack initiated from the tip of one of the flaws extends upwards or downwards to the other flaw but does not reach the tip and some microscopic shear or mixed tensile-shear cracks participate at coalescence. This is classified as one-tip-linkage. According to the coalescence position, the one-tip-linkage mechanism contains three sub-types: Coalescence position near the tip (Figure 11(1),(2)), coalescence position at the flaw, but far away from the tip (Figure 11(4)), or coalescence position outside the flaw with a certain distance from the tip (Figure 11(3)).
- A macroscopic tensile crack initiated from the tip of one flaw extends upwards or downwards to the tip of the other flaw. The coalescence occurred by the linkage of tensile cracks and is classified as two-tips-linkage. The two tips linkage has two sub-types: Straight linkage (Figure 11(5)) and arc linkage (Figure 11(6)).

#### 3.5. Crack Coalescence Stress

## 4. Conclusions

- For any intersection angle α, the strength is increases for flaw inclination angle β, ranging from 0° to about 45°. For intersection angle α up to about 45°, the strength is further slightly increased for inclination angle β, bigger than 45°. For intersection angle α, it is bigger than about 45° and the strength is strongly decreased for inclination angle β, bigger than 45°.
- The macro-crack initiated from the tips of two intersect flaws that did not occur simultaneously. The crack initial stress σ_ci of the first crack, obtained at the tip of flaw (A flaw or B flaw), normalized by the respective specimen peak strength ${\sigma}_{cn}$, which changed between 0.2 and 0.4 with different flaw geometries.
- Two major crack coalescence patterns were observed: (a) One-tip-linkage (either near the crack tip, at the flaw but far away from the crack tip or outside the flaw at a certain distance from the tip) and (b) two-tips-linkage (straight or arc linkage). The geometry of flaws governed the coalescence type.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Constructing a macro-crack based on connecting centroids of micro-cracks [18].

**Figure 3.**Geometries of the model containing two intersecting flaws: (

**a**) Flaws are numbered A and B; (

**b**) flaw inclination angle β, intersection angle α and flaw length L; and (

**c**) four bridging zones.

**Figure 6.**The peak strengths of the model containing a single flaw against β for different flaw initial inclination angles. ① ② ③ ④ ⑤ curves corresponding to five flaw initial inclination angles in Figure 7.

**Figure 7.**Five flaw initial inclination angles of the model containing a single flaw. The red dotted line represents the final state of rotation for the flaw. The blue arrow represents the direction of rotation.

**Figure 8.**Axial stress (black line) and total crack number (blue line) versus axial strain with different values of α and β. Five points, A, B, C, D, and E, on the stress–strain curve are monitored to observe the crack initiation, propagation, and coalescence.

**Figure 11.**Observed crack coalescence patterns for two intersecting flaws (left: Simulation results, right: Simplified sketch). White and red segments represent microscopic tensile and shear cracks, respectively. The arrows represent the crack propagation direction.

**Figure 12.**Ratio between coalescence stress ${\sigma}_{co}$ and peak strength ${\sigma}_{cn}$ for different values of α and β.

Particle Parameters | Value | Parallel Bond Parameters | Value |
---|---|---|---|

Ec Young’s modulus of the particle | 2.5 GPa | $\overline{\mathrm{E}}\mathrm{c}$ Young’s modulus of the parallel bond | 2.5 GPa |

kn/ks ratio of normal to shear stiffness of the particle | 1 | $\overline{\mathrm{k}}\mathrm{n}/\overline{\mathrm{k}}\mathrm{s}$ ratio of normal to shear stiffness of the parallel bond | 1 |

Grain friction coefficient, | 0.01 | $\overline{\mathsf{\sigma}}\mathrm{c}$ tensile strength of the parallel bond | 6 MPa |

Rmax/Rmin ratio of max particle radius to min particle radius | 1.66 | $\overline{\mathrm{c}}$ cohesion of the parallel bond | 15 MPa |

Rmin Lower bound of particle radius | 0.21 mm | $\overline{\mathsf{\lambda}}$ radius multiplier | 1 |

ρ particle density | 1.83 g/m^{3} | $\overline{\mathsf{\Phi}}$ Friction angle of the parallel bond | 0 |

**Table 2.**Lab test results [43] vs. numerical simulation results.

Parameter | Experimental Tests | BPM |
---|---|---|

Density (g/cm^{3}) | 1.54 | 1.54 |

Young’s modulus (GPa) | 5.96 | 6.09 |

Poisson’s ratio | 0.15 | 0.15 |

Uniaxial compressive strength (MPa) | 33.85 | 34.2 |

Tensile strength (MPa) | 3.2 | 4.8 |

**Table 3.**The crack evolution pattern at different stages of loading with different flaw geometry. The cracks corresponding to five points on the stress–strain curve in Figure 8. Microscopic tensile and shear cracks are shown in white and red, respectively. Original flaws are in yellow.

Flaw Geometry | Crack Evolution Pattern at Different Stages of Loading | ||||
---|---|---|---|---|---|

(A) | (B) | I | (D) | I | |

α = 90° β = 30° | |||||

α = 60° β = 0° | |||||

α = 45° β = 45° | |||||

α = 30° β = 0° |

**Table 4.**Crack coalescence patterns for two intersecting flaws with different values of α and β. Microscopic tensile and shear cracks are shown in white and red, respectively. Original flaws are in yellow.

Flaw Geometry | The Crack Coalescence Patterns of Loading | ||||
---|---|---|---|---|---|

β = 0° | β = 30° | β = 45° | β = 60° | β = 90° | |

α = 90° | |||||

α = 60° | |||||

α = 45° | |||||

α = 30° |

Angle/° | Coalescence Types of Different Geometries | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

β = 0° | β = 30° | β = 45° | β = 60° | β = 90° | ||||||

Type | Zone | Type | Zone | Type | Zone | Type | Zone | Type | Zone | |

α = 90° | (3) | II | (6) | II IV | (6) | II IV | (3) (6) | II III | (3) | II |

α = 60° | (6) | III | (2) (3) | I III | (1) (2) (3) (4) (5) | I II IV | (1) (5) (6) | I Ⅱ IV | (3) | I III |

α = 45° | (3) | II | (2) (6) | I III | (2) (3) (4) (6) | I II | (1) (4) (6) | II IV | (3) (6) | I II |

α = 30° | (5) (6) | I III | (5) (6) | I III | (2) (3) | I II | (2) (3) (4) | I III IV | (3) (4) (6) | I III |

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**MDPI and ACS Style**

Dai, B.; Chen, Y.; Zhao, G.; Liang, W.; Wu, H. A Numerical Study on the Crack Development Behavior of Rock-Like Material Containing Two Intersecting Flaws. *Mathematics* **2019**, *7*, 1223.
https://doi.org/10.3390/math7121223

**AMA Style**

Dai B, Chen Y, Zhao G, Liang W, Wu H. A Numerical Study on the Crack Development Behavior of Rock-Like Material Containing Two Intersecting Flaws. *Mathematics*. 2019; 7(12):1223.
https://doi.org/10.3390/math7121223

**Chicago/Turabian Style**

Dai, Bing, Ying Chen, Guoyan Zhao, Weizhang Liang, and Hao Wu. 2019. "A Numerical Study on the Crack Development Behavior of Rock-Like Material Containing Two Intersecting Flaws" *Mathematics* 7, no. 12: 1223.
https://doi.org/10.3390/math7121223