# Investigation of Damage Evolution in Heterogeneous Rock Based on the Grain-Based Finite-Discrete Element Model

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## Abstract

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## 1. Introduction

## 2. Basic Principles of FDEM

## 3. FDEM Simulation of Acoustic Emission

## 4. Numerical Grain-Based Model

## 5. Mesoscopic Parameter Verification of Grain-based Model

^{−7}ms was adopted to ensure the numerical stability of the model in Irazu. A schematic diagram of the model under loading is presented in Figure 6. The force and displacement were obtained by monitoring the nodes of the upper loading platen.

_{max}is the maximum radial load, R is the radius of the disc and t is the thickness of the disc.

## 6. Analysis of the Effect of Meso-Heterogeneity on the Mechanical Properties

#### 6.1. Effect of the Grain Size

#### 6.2. Effect of the Preferred Grain Orientation

_{o}) was used to measure the heterogeneity of the numerical model, where the higher the value of S

_{o}is, the more heterogeneous the model. The models were generated in MATLAB and transferred to Gmsh via a C++ subroutine. The formula used to calculate the values of S

_{o}[34] shown in the table is expressed as

_{25%}and Q

_{75%}correspond to the diameters smaller than 25% and 75%, respectively, of the grains on the grain size cumulative frequency diagram.

_{max}is the ultimate load during the loading process and$\Delta S$ is the axial displacement corresponding to the ultimate load.

## 7. Conclusions

- The FDEM-GBM considers the complexity of the contacts between different mineral grains in the rock and the physical and mechanical properties of different mineral phases; therefore, the FDEM-GBM has the ability to simulate the effect of the grain size in heterogeneous rocks.
- Because the mineral grain has significant influence on the crack propagation paths, the FDEM-GBM can capture the location of fractures more accurately than the conventional models. Shear cracks occur near the loading area, while tensile and tensile-shear mixed cracks occur far from the loading area. The applied stress must overcome the tensile strength of the intergrain contact.
- The UTS and the ratio of the number of intergrain tensile cracks to the number of intragrain tensile cracks are negatively correlated with the grain size. During the whole process of splitting simulation, shear microcracks play the dominant role in energy release; particularly, they occur in later stage. Under different grain sizes conditions, the crack growth paths are significantly different. The larger the grain size is, the more complicated the crack propagation path. When the grain sizes are 1.5 and 2 mm, the fracture sections are smoother and straighter.
- With the increase of preferred grain orientation, the UTS presents a “V-shaped” characteristic distribution. During the whole process of splitting simulation, shear microcracks play the dominant role in energy release; particularly, they occur in later stage. Under different preferred grain orientation conditions, the crack growth paths are significantly different, especially 30°, 45° and 60°. When the preferred orientations are 30°, 45° and 60°, the fracture sections are more complicated.
- The preferred grain orientation considerably influences the energy storage capacity of Beishan granite. With the increase of grain orientation, the energy storage capacity decreases significantly. The greater the UTS is, the higher the energy storage capacity.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Failure models and failure criterion of CCEs in FDEM: (

**a**) schematic of Fracture Process Zone (FPZ) in brittle geomaterials; (

**b**) numerical representation of theoretical FPZ model in Irazu; (

**c**) exaggerated view of 4-noded CCEs located along edges of all adjoining triangular finite elements; (

**d**) tensile failure mode (mode I); (

**e**) shear failure mode (mode II); and (

**f**) mixed tension-shear failure criterion (mode III) (modified from Liu et al. [17]).

**Figure 2.**The evolution of normal bonding stress and kinetic energy (AE energy) of CCE nodes, ${E}_{k}$, as a function of time, T, for the tensile failing CCEs [17].

**Figure 3.**Micro-structure of Beishan granite (modified from [21] with permission from Elsevier).

**Figure 4.**Grain size distribution of minerals in the grain-based model (PDF in the figure stands for the probability density function).

**Figure 5.**(

**a**) typical Voronoi diagram. The red dots indicate the centers of the Voronoi polygons and the green dots indicate the seeds for generating Voronoi polygons; and (

**b**) a combination of multiscale grains.

**Figure 8.**Comparison of the failure phenomenon: (

**a**) fractured section of laboratory test (modified from [28] with permission from Elsevier); (

**b**) acoustic emission (AE) evolution; (

**c**) failure modes; and (

**d**) magnitudes of AEs, M

_{e}.

**Figure 11.**Characteristics of the samples and microcracks under different grain size conditions: (

**a**) the four numerical samples; (

**b**) the fracture paths (green, red and yellow segments represent tensile, shear and tensile-shear mixed cracks, respectively); and (

**c**) the crack angle distributions.

**Figure 12.**The variations of the fracture process under different grain size conditions: (

**a**) size = 1.5 mm; (

**b**) size = 2 mm; (

**c**) size = 2.5 mm; (

**d**) size = 3 mm (A represent the model; B, C and D represent the fracture process).

**Figure 13.**AE evolution and magnitudes of AEs under four different grain sizes conditions. (

**a**) the AE evolution, (

**b**) the magnitudes of AEs, M

_{e}, calculated from the kinetic energy of the sources using the technique illustrated in Section 3.

**Figure 15.**Microstructure of the quartz in granite [14] with permission from Elsevier.

**Figure 16.**The tensile stress-displacement curves under different preferred grain orientation conditions.

**Figure 18.**Characteristics of the sample and crack with five different preferred grain orientations: (

**a**) the numerical samples; (

**b**) the models with the preferred grain orientations (highlight grains represent grains with preferred orientations); (

**c**) the fracture paths (green red and yellow segments represent tensile, shear and tensile-shear mixed cracks, respectively); and (

**d**) the crack angle distributions.

**Figure 19.**The variation of the fracture process under different preferred grain orientation conditions: (

**a**) orientation = 0°; (

**b**) orientation = 30°; (

**c**) orientation = 45°; (

**d**) orientation = 60°; and (

**e**) orientation = 90° (A represents the model; B, C and D represent the fracture process).

**Figure 20.**AE evolution and magnitudes of AEs under different grain orientation conditions: (

**a**) the AE evolution; and (

**b**) the magnitudes of AEs, M

_{e}, calculated from the kinetic energy of the sources using the technique illustrated in Section 3.

Mineral | Density (kg·m^{−3}) | Tensile Strength (MPa) | Young’s Modulus (GPa) | Poisson’s Ratio (-) |
---|---|---|---|---|

K-feldspar | 2560 | 5–10 | 69.8 | 0.28 |

Plagioclase | 2630 | 5–10 | 88.1 | 0.26 |

Quartz | 2650 | 10–11 | 94.5 | 0.08 |

Biotite | 3050 | 4–7 | 33.8 | 0.36 |

Property | Qz | Fsp | Ma | |
---|---|---|---|---|

Intragrain | Density (kg·m^{−3}) | 2600 | 2600 | 3050 |

Young’s modulus (GPa) | 80 | 70 | 40 | |

Poisson’s ratio (-) | 0.07 | 0.26 | 0.27 | |

Friction coefficient (-) | 1.2 | 1.2 | 1.2 | |

Cohesion (MPa) | 25 | 25 | 25 | |

Tensile strength (MPa) | 20 | 15 | 10 | |

Mode I fracture energy (J·m^{−2}) | 900 | 300 | 600 | |

Mode II fracture energy (J·m^{−2}) | 1800 | 600 | 1200 | |

Fracture penalty (GPa) | 400 | 350 | 200 | |

Normal penalty (GPa·m) | 80 | 70 | 40 | |

Tangential penalty (GPa·m^{−1}) | 800 | 700 | 400 | |

Qz-Qz | Fsp-Fsp | Ma-Ma | ||

Homophase boundary | Friction coefficient (-) | 1.1 | 1.1 | 1.1 |

Cohesion (MPa) | 20 | 20 | 20 | |

Tensile strength (MPa) | 15 | 10 | 10 | |

Mode I fracture energy (J·m^{−2}) | 700 | 250 | 450 | |

Mode II fracture energy (J·m^{−2}) | 1400 | 500 | 900 | |

Fracture penalty (GPa) | 200 | 175 | 100 | |

Normal penalty (GPa·m) | 40 | 35 | 20 | |

Tangential penalty (GPa·m^{−1}) | 400 | 350 | 200 | |

Qz-Fsp | Qz-Ma | Fsp-Ma | ||

Heterophase boundaries | Friction coefficient (-) | 0.9 | 0.9 | 0.9 |

Cohesion (MPa) | 20 | 20 | 20 | |

Tensile strength (MPa) | 10 | 6 | 6 | |

Mode I fracture energy (J·m^{−2}) | 50 | 20 | 20 | |

Mode II fracture energy (J·m^{−2}) | 500 | 200 | 200 | |

Fracture penalty (GPa) | 350 | 200 | 275 | |

Normal penalty (GPa·m) | 70 | 40 | 55 | |

Tangential penalty (GPa·m^{−1}) | 700 | 400 | 550 |

Scheme | Grain Size (mm) | Grain Orientation (º) |
---|---|---|

1 | 1.5, 2.0, 2.5, 3.0 | - |

2 | - | 0, 30, 45, 60, 90 |

Sample | Number of Different Types of Microcracks | |||||
---|---|---|---|---|---|---|

Intergranular Cracks | Transgranular Cracks | |||||

Mode I | Mode III | Mode II | Mode I | Mode III | Mode II | |

A-1 | 119 | 23 | 2 | 4 | 68 | 1 |

A-2 | 121 | 43 | 2 | 7 | 48 | 3 |

A-3 | 150 | 56 | 3 | 10 | 50 | 6 |

A-4 | 130 | 70 | 3 | 19 | 52 | 10 |

Sample | Grain Number, N_{P} | Grain Size Coefficient, S_{o} | Preferred Orientation, φ (°) |
---|---|---|---|

B-1 | 303 | 1.06 | 0 |

B-2 | 319 | 1.07 | 30 |

B-3 | 329 | 1.06 | 45 |

B-4 | 328 | 1.07 | 60 |

B-5 | 321 | 1.06 | 90 |

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**MDPI and ACS Style**

Zhang, S.; Qiu, S.; Kou, P.; Li, S.; Li, P.; Yan, S.
Investigation of Damage Evolution in Heterogeneous Rock Based on the Grain-Based Finite-Discrete Element Model. *Materials* **2021**, *14*, 3969.
https://doi.org/10.3390/ma14143969

**AMA Style**

Zhang S, Qiu S, Kou P, Li S, Li P, Yan S.
Investigation of Damage Evolution in Heterogeneous Rock Based on the Grain-Based Finite-Discrete Element Model. *Materials*. 2021; 14(14):3969.
https://doi.org/10.3390/ma14143969

**Chicago/Turabian Style**

Zhang, Shirui, Shili Qiu, Pengfei Kou, Shaojun Li, Ping Li, and Siquan Yan.
2021. "Investigation of Damage Evolution in Heterogeneous Rock Based on the Grain-Based Finite-Discrete Element Model" *Materials* 14, no. 14: 3969.
https://doi.org/10.3390/ma14143969