# Numerical Study of the Mechanical and Acoustic Emissions Characteristics of Red Sandstone under Different Double Fracture Conditions

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## Abstract

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## 1. Introduction

## 2. Numerical Model of Joint Red Sandstone

#### 2.1. Particle Flow Introduction

_{i}, which is generated by the overlap of particles and particles. The contact force vector, F

_{i}(which represents the action of two contact particles), can be resolved into normal and shear components with respect to the contact plane as [19]

^{n}and F

^{s}denote the normal and shear force components, respectively, and n

_{i}and t

_{i}are the unit vectors that define the contact plane. The normal force is calculated by

_{n}is the contact normal stiffness, U

^{n}is the overlap.

^{s}is initialized to zero. Each subsequent relative shear–displacement increment, ΔU

^{s}; produces an increment of elastic shear force, ΔF

^{s}; the increment of elastic shear force is given by

_{s}is the contact shear stiffness.

_{n}) and shear (k

_{s}) stiffness values, and the model is a point contact, representing a force. The parallel-bond model consists of a parallel bond of finite size (circular or rectangular section), characterized by a force and a torque, as shown in Figure 1. In this experiment, the parallel-bond model was used to reconstruct the red sandstone. The size of the particle flow model was in accordance with the size of the laboratory test specimen, and the selected size was 50 mm × 100 mm. The microscopic parameters can be solved by applying Equation (1) [21].

_{c}is the elastic modulus of particle contact; ${\overline{E}}_{\mathrm{c}}$ is the elastic modulus of particle parallel bonding; k

_{n}and k

_{s}are the particle contact stiffness; ${\overline{k}}_{\mathrm{n}}$ and ${\overline{k}}_{\mathrm{s}}$ are the parallel bonding stiffness; and $\overline{R}$ is the average radius of the two contact particles.

#### 2.2. Parameters of the Microscopic Physical and Mechanical Properties of Red Sandstone

^{−2}. The numerical model used the same size and loading speed as the laboratory experiment. After repeated adjustments and comparisons, the finalized parameters of the microscopic physical mechanical properties were determined, as shown in Table 1. Under these parameters, the numerical simulation gave the basic mechanical parameters as 7.24 GPa, a peak stress of 124.77 MPa and a peak strain of 1.74 × 10

^{−2}. The difference between the experimental data and the laboratory experiment was less than 5%. At the same time, the numerical simulation results were consistent with the stress–strain curve of the laboratory rock specimen and the failure mode (Figure 2 and Figure 3). The obtained parameters can better reflect the physical and mechanical properties of red sandstone. Therefore, it can be seen that the simulated failure mode and stress–strain curve are in good agreement with the laboratory test results, which indicates that the selection is reasonable and can be used in subsequent calculation and analysis.

#### 2.3. PFC Simulation of AE

#### 2.4. Construction of Particle Flow Model and Simulation Working Conditions

## 3. Analysis of the Test Results

#### 3.1. Analysis of the Mechanical Characteristics

#### 3.2. Analysis of the Failure Mode

#### 3.3. Analysis of the AE Characteristics of Rock Mass Damage

^{3}times. When α is 75°, the AE cumulative ringing count of rock samples reaches the minimum value of 9.67 × 10

^{2}times. That is, the peak stress of the rock mass is directly proportional to the AE cumulative ringing count.

## 4. Influence of Joint Geometric Properties on the Damage Evolution Characteristics of the Rock Mass

#### 4.1. Rock Mass Damage Model Based on AE Characteristics

_{d}is the cross-sectional area of material damage in a certain period and A is the cross-sectional material area without initial damage.

_{w}:

_{d}, the cumulative AE ringing count C

_{d}is

_{u}is the critical value of damage.

_{p}is the peak intensity, and σ

_{c}is the residual strength.

#### 4.2. Damage Evolution Law of Double-Jointed Rock Mass Based on AE Characteristics

## 5. Conclusions

- The peak stress and elastic modulus of samples increase with the increase in α, and their change trends are similar. The postpeak stress decreases with the increase in α. The fracture angle α has a nonlinear distribution, and the amplitude of the change is small. The peak stress and elastic modulus of rock samples first decrease and then increase with the increase in the rock-bridge angle, exhibiting nonlinear distributions. The variation trends of the elastic modulus and peak stress of specimens with different rock-bridge angles are similar. When β is less than or equal to 45°, the rock peak strain differs little and the peak strain decreases gradually as the rock-bridge angle increases.
- With the increase in the fracture angle, the final fracture mode of the sample can be divided into three categories: A positive Y shape, an inverted Y type and inclined failure. The failure modes of specimens with different rock-bridge angles are roughly classified as follows: When β is less than or equal to 45°: No coalescence failure; when β is greater than or equal to 45°: Coalescence failure. Under noncoalescence failure, the failure shapes of rock samples include V type, positive Y type, X type and inclined failure.
- The evolution process of AE in fractured rock mass includes three stages: The initial stage of AE, the increasing stage of AE and the stage of AE decrement. The fracture angle has little influence on the characteristics of AE from rock samples. The intensity of the AE signal and the count of the cumulative ringing of AE first increase and then decrease with the fracture angle, but the amplitudes of the changes are not significant. The strain range of rock mass that produces obvious AE events has a gradually decreasing trend under different fracture angles. The influence of the rock-bridge angle on the AE characteristics of the rock is mainly reflected in the intensity of AE signals and the corresponding strain values. With the increase in the rock-bridge angle, the intensity of the AE ringing counts first decreases and then increases.
- The damage evolution process of sandstone specimens with different joints can be divided into four stages: The initial damage stage, the stable increase stage, the accelerated development stage, and the stable damage stage. The effect of the fracture angle on the initial damage, steady increase and accelerated development stages is not great; this parameter mainly affects the stable damage stage. The smaller the fracture angle, the greater the strain value reached in the stable damage stage. The rock-bridge angle mainly influences the evolution of the damage variables of the specimen in the damage stable increase, accelerated damage development and stable damage stages. In the stage of stable damage increase, when β is less than or equal to 45°, the failure variation trends are similar. The occurrence time of damage at this stage is later than that for samples with β greater than or equal to 45°. In the stage of accelerated damage development, the greater the angle of the rock bridge is, the slower the change in the damage variable and the higher the degree of strain relaxation, which is related to the failure mode of the jointed rock mass. In the stage of damage stabilization, different rock-bridge angles also show different stable damage values.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Comparison between the numerical and experimental stress–strain curves of intact sandstone specimens.

**Figure 3.**Failure modes of intact red sandstone specimens obtained experimentally and through simulation.

**Figure 10.**Stress–strain–AE (acoustic emission) characteristic curves of double-fractured rock samples with different fracture angles.

**Figure 11.**AE cumulative ringing count curves of double-fractured rock samples with different fracture angles.

**Figure 12.**Stress–strain–AE characteristic curves of double-fractured rock samples under different rock-bridge angles.

**Figure 13.**Strain-AE cumulative ringing count curves of double-fractured rock samples under different rock-bridge angles.

**Table 1.**Microscopic physical mechanical parameters of the particle flow code (PFC) red sandstone model.

Smallest Diameter (mm) | Radius Ratio |

0.26 | 1.5 |

Density (kg/m^{3}) | Particle Contact Model Amount (GPa) |

2400 | 6.9 |

Particle Contact Stiffness Ratio | Model Amount of Parallel Bonding (GPa) |

3.0 | 6.9 |

Stiffness Ratio of Parallel Bonding | Normal Strength of Parallel Bonding (MPa) |

3.0 | 110 ± 10 |

Tangential Strength of Parallel Bonding (MPa) | Radius Factor of Parallel Bonding |

225 ± 10 | 1 |

Working Conditions | α (°) | β (°) | Working Conditions | α (°) | β (°) |
---|---|---|---|---|---|

Condition 1 | 0 | 45 | Condition 7 | 45 | 15 |

Condition 2 | 30 | 45 | Condition 8 | 45 | 30 |

Condition 3 | 45 | 45 | Condition 9 | 45 | 45 |

Condition 4 | 60 | 45 | Condition 10 | 45 | 60 |

Condition 5 | 90 | 45 | Condition 11 | 45 | 75 |

Condition 6 | 45 | 0 | Condition 12 | 45 | 90 |

Working Conditions | Peak Stress (MPa) | Peak Strain | Elastic Modulus (GPa) |
---|---|---|---|

Condition 1 | 97.78 | 1.65 | 6.44 |

Condition 2 | 102.74 | 1.57 | 6.56 |

Condition 3 | 110.32 | 1.68 | 6.78 |

Condition 4 | 110.48 | 1.59 | 7.00 |

Condition 5 | 121.44 | 1.76 | 7.16 |

Condition 6 | 113.79 | 1.69 | 6.82 |

Condition 7 | 111.82 | 1.64 | 6.76 |

Condition 8 | 114.72 | 1.66 | 6.78 |

Condition 9 | 110.32 | 1.68 | 6.78 |

Condition 10 | 97.85 | 1.49 | 6.72 |

Condition 11 | 78.21 | 1.24 | 6.49 |

Condition 12 | 84.96 | 1.25 | 6.81 |

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**MDPI and ACS Style**

Huang, D.; Chang, X.; Tan, Y.; Zhou, J.; Yin, Y.
Numerical Study of the Mechanical and Acoustic Emissions Characteristics of Red Sandstone under Different Double Fracture Conditions. *Symmetry* **2019**, *11*, 772.
https://doi.org/10.3390/sym11060772

**AMA Style**

Huang D, Chang X, Tan Y, Zhou J, Yin Y.
Numerical Study of the Mechanical and Acoustic Emissions Characteristics of Red Sandstone under Different Double Fracture Conditions. *Symmetry*. 2019; 11(6):772.
https://doi.org/10.3390/sym11060772

**Chicago/Turabian Style**

Huang, Dongmei, Xikun Chang, Yunliang Tan, Junhua Zhou, and Yanchun Yin.
2019. "Numerical Study of the Mechanical and Acoustic Emissions Characteristics of Red Sandstone under Different Double Fracture Conditions" *Symmetry* 11, no. 6: 772.
https://doi.org/10.3390/sym11060772