# Shear Damage Simulations of Rock Masses Containing Fissure-Holes Using an Improved SPH Method

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## Abstract

**:**

## 1. Introduction

## 2. Basic Principles of SPH Method

#### 2.1. SPH Discrete Strategy

**x**is the coordinate vector;

**x**’ is the coordinate vector of a particular point; f(

**x**) is the field function, which stands for particle mass, density, energy, and velocity, etc.; Ω is the integral domain; and δ(

**x**−

**x**’) is the Dirac functions.

#### 2.2. Particle Approximation

#### 2.3. Governing Equations

_{ij}means v

_{i}–v

_{j}; x represents the particle position; σ

_{αβ}stands for the stress components; W is the so-called smoothing kernel function; and T represents the artificial viscosity.

## 3. Damage Model in SPH Method

#### 3.1. Damage Criterion

_{n}and σ

_{t}represent the normal stress and tensile strength, respectively; τ

_{f}stands for the shear stress on the failure surface; c represents the particle cohesion, while φ stands for the internal friction angle.

#### 3.2. Damage Treatments in SPH Method

## 4. SPH Model and Calculation Conditions

#### 4.1. Parameter Calibrations

_{0}stands for the mean value of the basic particle parameters.

#### 4.2. SPH Model of Rock Specimen Containing Fissure-Holes

_{N}is applied at the top of the model, and the tangential force τ

_{S}is applied at the upper left side of the model. The lower right side and the bottom side are the fix boundaries, as shown in Figure 4a. The whole model is divided into 200 × 200 = particles. The model mechanical parameters are listed below: elastic modulus E = 17 GPa, Poisson’s ratio μ = 0.2, m = 5. What should be stressed is that all the simulations in the current study use 2D plane stress conditions.

#### 4.3. SPH Calculation Conditions

## 5. SPH Simulation Results

#### 5.1. Failure Mode Analysis of Fissure-Hole Interactions

#### 5.2. Analysis of Initiation and Failure Pressures

## 6. Discussion

#### 6.1. Rock Crack Propagation Morphology

#### 6.2. Initiation Laws of Different Hole Shapes

## 7. Conclusions

- (1)
- The rock fracture properties can be realized with the SPH method by adding a fracture mark, ξ, to multiply it with the traditional kernel function.
- (2)
- Different hole–fissure numerical models have been established and simulated. “Wing cracks” initiate from the fissure tips, and the interaction locations between the holes and the fissures are at the hole corners. The numerical results are verified by comparisons with the previous experimental results.
- (3)
- The circular hole has the most reduction on the specimen strength, while the trapezoid hole has the least. The failure strength increases with an increase in the fissure angle.
- (4)
- The fissure lengths and numbers are the two key factors that influence the peak strength of rock masses. Meanwhile, an increase in the confining pressure also increases the shear strength of the specimen.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Comparisons between numerical results and experimental results [35].

**Figure 4.**Calculation model and particle divisions. (

**a**) Calculation model and boundaries; and (

**b**) particle divisions.

**Figure 5.**The interactions of holes and fissures under different hole shapes. (

**a**) Rectangle hole; (

**b**) circular hole; (

**c**) triangle hole; and (

**d**) trapezoid hole.

**Figure 6.**The interactions of holes and fissures under different fissure angles. (

**a**) θ = 15°; (

**b**) θ = 45°; (

**c**) θ = 60°; and (

**d**) θ = 75°.

**Figure 7.**The interactions of holes and fissures under different fissure numbers. (

**a**) N = 2; (

**b**) N = 4; (

**c**) N = 6; and (

**d**) N = 8.

**Figure 8.**The interactions of holes and fissures under different fissure lengths. (

**a**) l = 6 mm; (

**b**) l = 12 mm; (

**c**) l = 18 mm; and (

**d**) l = 24 mm.

**Figure 9.**The interactions of holes and fissures under different vertical stress. (

**a**) σ

_{N}= 0.5 MPa; (

**b**) σ

_{N}= 1 MPa; (

**c**) σ

_{N}= 1.5 MPa; and (

**d**) σ

_{N}= 2 MPa.

**Figure 11.**“Wing crack” morphology. (

**a**) “Wing crack” propagation calculated by SPH method; and (

**b**) previous experimental results [35].

**Figure 13.**Distributions of maximum principal stress under different hole shapes. (

**a**) Maximum principal stress of condition A1; (

**b**) maximum principal stress of condition A2; (

**c**) maximum principal stress of condition A3; and (

**d**) maximum principal stress of condition A4.

Model | Condition | Details | Model | Condition | Details |
---|---|---|---|---|---|

A1 | Rectangle Hole | B1 | θ = 15° | ||

A2 | Circular Hole | B2 | θ = 45° | ||

A3 | Triangle Hole | B3 | θ = 60° | ||

A4 | Trapezoid Hole | B4 | θ = 75° | ||

C1 | N = 2 | D1 | l = 6 mm | ||

C2 | N = 4 | D2 | l = 12 mm | ||

C3 | N = 6 | D3 | l = 18 mm | ||

C4 | N = 8 | D4 | l = 24 mm | ||

E1 | σ_{N} = 0.5 MPa | ||||

E2 | σ_{N} = 1 MPa | ||||

E3 | σ_{N} = 1.5 MPa | ||||

E4 | σ_{N} = 2 MPa |

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**MDPI and ACS Style**

Yu, S.; Yang, X.; Ren, X.; Zhang, J.; Gao, Y.; Zhang, T. Shear Damage Simulations of Rock Masses Containing Fissure-Holes Using an Improved SPH Method. *Materials* **2023**, *16*, 2640.
https://doi.org/10.3390/ma16072640

**AMA Style**

Yu S, Yang X, Ren X, Zhang J, Gao Y, Zhang T. Shear Damage Simulations of Rock Masses Containing Fissure-Holes Using an Improved SPH Method. *Materials*. 2023; 16(7):2640.
https://doi.org/10.3390/ma16072640

**Chicago/Turabian Style**

Yu, Shuyang, Xuekai Yang, Xuhua Ren, Jixun Zhang, Yuan Gao, and Tao Zhang. 2023. "Shear Damage Simulations of Rock Masses Containing Fissure-Holes Using an Improved SPH Method" *Materials* 16, no. 7: 2640.
https://doi.org/10.3390/ma16072640