Using discrete element software, namely, particle flow code as two-dimensional program (PFC2D), two types of models were established: vertical fissure hole combination and horizontal fissure hole combination with ratios of major and minor axis of ellipse being 1, 1.2, 1.5, 2, and 3, which corresponded to a total of ten samples. The failure mode, mechanical behavior, and stress state before and after crack generation in elliptical hole crack combination models with different ratios of major and minor axis were analyzed. The crack development, stress field evolution, and acoustic emission characteristics of the vertical fissure model and horizontal fissure model were studied at the optimized ratio of major and minor axis of ellipse being 1.5. The results showed that elliptical hole fissure with different ratios of major and minor axis resulted in the decrease in the strength and elastic modulus of rock and increase in the peak strain of rock. The effect of the horizontal fissure model on the peak strength, peak strain, and elastic modulus of rock was found to be greater than that of the vertical fissure hole model. Ellipses with different ratios of major and minor axis in various models slightly influenced the rock failure modes, and their failure modes corresponded to tensile shear failure and tensile failure. Before crack formation, the tensile stress concentration areas of each model were, respectively, distributed at the upper and lower ends of the vertical fissure and the major axis of ellipse, and the compressive stress concentration areas were distributed at both ends of the major axis of ellipse and the fissure in the horizontal direction. After the model failed, the compressive stress concentration areas of the vertical fissure model and the horizontal fissure model transferred to the left upper part and the right upper part of the model along the left end of the hole and the right end of the fissure, respectively. When the ratio of major and minor axis of ellipse was 1.5, cracks in the vertical model and the horizontal model of fissure developed along the axial direction at the ends of cracks and holes, respectively, and then secondary cracks were generated at the ends of left and right sides. The maximum compressive stress in each stage of the vertical fissure model was greater than that of the horizontal fissure model, and when the model was damaged, its stress release was more.

Numerous defects such as cracks, holes, and weak structural planes are present in natural rock mass. The existence of these defects significantly influences the structural characteristics and mechanical properties of rock, and the instability and failure of rock engineering are closely related to the crack initiation and extension characteristics of fissure and holes in rock [

For a long time, extensive research efforts have been devoted to the study of the defects in rocks such as fissure and holes [

Moreover, with respect to the combination defects of ellipse and fissure hole, scholars have also carried out relevant related research. For example, Yang et al. [

The above-mentioned studies mainly focused on fracture, hole defect, and other defects. However, in actual engineering, the interior of rock material is very complex, usually containing various defects such as fissure and holes of different sizes [

Rock mass containing preexisting oval-like flaws [

Cundall and Strack proposed the DEM in 1971 [

Parallel bond model.

In general, after the rock numerical model is established, the normal and tangential bond strength of the model can directly reflect the macro strength of the rock. Under the influence of external load, when the stress transmitted between the particles exceeds the bond strength between the particles, the bond between the particles breaks, causing the generation of microcracks in the rock sample [

When PFC2D is used for numerical simulation test, the microscopic parameters of particles are very important. However, these microscopic parameters cannot be directly obtained from laboratory tests. Therefore, it is necessary to check the microscopic parameters in order to ensure that the numerical simulation results are consistent with the laboratory test results [

Calibration of numerical parameters.

In order to study the influence of ellipses and fissure defects with different ratios of major and minor axis on rock failure and mechanical behavior, two types of models were established: vertical fissures and horizontal fissures vertically intersecting the major axis of the ellipse, which were respectively called vertical fissure hole model and horizontal fissure hole model, as shown in Figure ^{2}. In this study, different ratios of major and minor axis were used for investigation. The height of the model was 100 mm, the width was 50 mm, the major axis 2a of the ellipse was 12 mm, the minor axis of the ellipse was 2b, the longest axis of the ellipse was 12 mm and the shortest axis was 4 mm, and the length of the fissure was determined according to the length of the minor axis of the ellipse. The sum of the total length of the two fissures was at least 10 mm and at most 18 mm, and the sum of the total length of the two fissures and the minor axis of the ellipse was always 22 mm. The details about the specific size of the model are presented in Table

Schematic illustration and numerical model illustration of hole fissure defects. (a) Schematic illustration. (b) Numerical model illustration.

Size parameters of elliptic fissure hole model with different ratios of major and minor axis.

Type | Number | Height (mm) | Length (mm) | 2a (mm) | 2b (mm) | |
---|---|---|---|---|---|---|

Vertical fissure model | ① | 100 | 50 | 12 | 12 | 5 |

② | 100 | 50 | 12 | 10 | 6 | |

③ | 100 | 50 | 12 | 8 | 7 | |

④ | 100 | 50 | 12 | 6 | 8 | |

⑤ | 100 | 50 | 12 | 4 | 9 | |

Horizontal fissure model | ⑥ | 100 | 50 | 12 | 12 | 5 |

⑦ | 100 | 50 | 12 | 10 | 6 | |

⑧ | 100 | 50 | 12 | 8 | 7 | |

⑨ | 100 | 50 | 12 | 6 | 8 | |

⑩ | 100 | 50 | 12 | 4 | 9 |

A Shimadzu AG-X250 precision universal testing machine was used as the loading system, and it was used for performing regular compression and tensile tests. Displacement-based loading control method was used to perform the uniaxial compression test until the sample fractured. The loading rate was set to 0.01 mm/s. All the tests were conducted according to ISRM standards (Brown 1981). The numerical model was loaded based on displacement loading, the loading rate was 0.01/s, the bottom of the sample was fixed, and the left and right sides were released.

The numerical simulation and laboratory testing of the stress-strain curve and failure mode of combination defects of ellipse and fissure hole are shown in Figure

Comparison between numerical simulation results and test results. (a) Vertical fissure model. (b) Horizontal fissure model.

Table

Comparison of mechanical properties between laboratory test results and numerical simulation results.

Category | Vertical fissure model (a/b = 3) | Horizontal fissure model (a/b = 3) | ||||
---|---|---|---|---|---|---|

Peak stress (MPa) | Elastic modulus (GPa) | Peak strain | Peak stress (MPa) | Elastic modulus (GPa) | Peak strain | |

Experiment result | 58.37 | 1.567 | 0.0231 | 36.19 | 1.233 | 0.0176 |

Numerical result | 56.98 | 1.316 | 0.0196 | 36.93 | 1.035 | 0.0157 |

Error (%) | 2.38 | 16.02 | 15.15 | 2.0 | 16.05 | 10.8 |

Figure

The stress-strain curve and crack number-strain curve. (a) Vertical fissure stress-strain curve and crack number-strain curve. (b) Horizontal fissure stress-strain curve and crack number-strain curve.

Figure

Mechanical parameters of fissure holes under uniaxial compression.

Mechanical parameters of sandstone model based on PFC.

Parameter | Value |
---|---|

Minimum particle diameter (mm) | 0.3 |

Particle diameter ratio | 1.5 |

Density (kg/m^{3}) | 2490 |

Contact modulus of the particle (GPa) | 3.19 |

Contact bond gap (mm) | 0.05 |

Porosity | 0.1 |

Parallel bond friction angle (°) | 38 |

Parallel bond tensile strength (MPa) | 22.3 |

Normal critical damping ratio | 0.5 |

Parallel bond cohesive force (MPa) | 33 |

The peak strength, peak strain, and elastic modulus of the vertical fissure hole model were found to be larger than those of the horizontal fissure hole model on the whole. Moreover, the peak strength, peak strain, and elastic modulus of the vertical fissure hole model with the same ellipse major and minor axis ratio are also larger than those of the horizontal fissure hole model; nonetheless, the trends of the peak strength, peak strain, and elastic modulus of the two models are basically the same. The peak strength and peak strain of the two models show an increasing trend first and then a decreasing one followed by again increasing. The maximum is when major and minor axis ratio is 1.2, and the minimum is when major and minor axis ratio is 1.5; however, the elastic modulus is basically the same. It shows that ellipses with different major and minor axis ratios have different effects on the strength and peak strain of rocks but have little effect on elastic modulus.

Figure

Tensile shear mixed failure: the failure modes of all specimens in the vertical fissure model with different major and minor axis ratios of ellipse and the horizontal fissure model specimens with ellipse major and minor axis ratios of 1.2 and 2.0 correspond to tensile shear mixed failure. Cracks causing instability of specimens in the vertical fissure model and the horizontal fissure model mainly start at the end points of the major axis of ellipses and at the left and right end points of horizontal fissure, respectively. The shear fissure surface penetrates the rock sample along the diagonal direction; however, in the fissure vertical model and the fissure horizontal model, cracks from the two tips of vertical fissure and the upper and lower ends of ellipses penetrate to the end points of the rock sample approximately axially. The tensile shear failure of the vertical fissure model was more obvious than that of the horizontal fissure model; however, the specimen failure of the vertical fissure model was more obvious.

Tensile failure: when the ratio of major and minor axis of ellipse in the horizontal fissure model is 1, 1.5, and 3.0, the rock sample eventually undergoes tensile failure, and the cracks that eventually cause instability of the rock sample mainly start at the left and right ends of the fissure and at the upper and lower ends of the hole, which are approximately distributed in a “

Final failure illustration of model. (a) Fissure vertical model with different ratios of major and minor axis of ellipse. (b) Horizontal fissure model with different ratios of major and minor axis of ellipse.

Figure

Final crack number after model failure.

Table

Force chain diagram before crack generation and after failure.

Vertical fissure model | Horizontal fissure model | ||
---|---|---|---|

(a) | (b) | (c) | (d) |

In Table

Based on the results of the large number of models studied, one of the five types of ratios of major and minor axis was selected as the optimized research value (i.e., the ratio of major and minor axis of 1.5). Figures ^{3} (point a in Figure ^{3} (point b in Figure ^{3} (point c in Figure ^{3} (point d in Figure

Crack propagation and maximum principal stress illustration when the major and minor axis ratio a/b = 1.5 of vertical fissure ellipse.

Stress-strain curve and acoustic emission-strain curve when the major and minor axis ratio a/b = 1.5 of vertical fissure ellipse.

Figures ^{3} (point a in Figure ^{3} (point b in Figure ^{3} (point c in Figure ^{3} (point d in Figure

Crack propagation and maximum principal stress illustration when the major and minor axis ratio a/b = 1.5 of horizontal fissure ellipse.

Stress-strain curve and acoustic emission-strain curve when the major and minor axis ratio a/b = 1.5 of horizontal fissure ellipse.

In this study, the horizontal fracture model can be regarded as the vertical fracture model by clockwise rotation of 90°. The above-mentioned analysis indicates that the mechanical properties, failure modes, and stress concentration areas before the crack and after the failure of the vertical crack model and the horizontal model are different. Even the mechanical properties and failure of the two models with the same ratio of major and minor axis of ellipse are different. This shows that the arrangement of ellipses and fissure defects significantly influences the mechanical properties and failure characteristics of rocks. The influence of horizontal fissure defect combination on rock mechanical properties is greater than the combination of vertical fissures on rock mechanical properties. This study is based on the analysis of a 2D uniaxial compression test. The stress of the sample is the force in the vertical direction. Not only does the arrangement of the combination of fissure and defects affect the mechanical properties of the rock, but the direction of the applied force also significantly influences the mechanical properties of the rock.

The influence of elliptical fissure holes with different ratios of major and minor axis on the mechanical properties of rock was found to be different. The peak strength and elastic modulus of the vertical fissure hole model and horizontal fissure hole model were less than those of the intact rock sample, and the peak strain was greater than that of the intact rock sample, among which the influence of horizontal fissure hole on the rock strength was greater than that of vertical fissure hole on the rock strength. The crack formation of the horizontal fissure hole model preceded the crack formation of the crack fissure hole model.

The failure modes of elliptical fissure holes with different ratios of major and minor axis were divided into the following two types: tensile shear mixed failure and tensile failure. The failure modes of elliptical fissure vertical model with different ratios of major and minor axis and elliptical fissure horizontal model with ratios of major and minor axis of 1.2 and 2.0 were tensile shear mixed failure. Tensile failure occurred when the ratios of major and minor axis of ellipse in the horizontal fissure model were 1, 1.5, and 3.0.

Before crack formation, the tensile stress concentration area was distributed at the upper and lower ends of the axial fissure and the axial elliptical major axis, and the compressive stress concentration area was distributed at both ends of the elliptical major axis and the fissure in the horizontal direction. When the model finally failed, the compressive stress concentration area of the vertical fissure model was distributed on the left side of the fissure hole, and the compressive stress concentration area of the horizontal fissure model was distributed on the right side of the fissure hole.

The cracks in the vertical fissure model and the horizontal fissure model when the ratio of major and minor axis of ellipse was 1.5, were first developed, respectively, along the axial direction at the ends of the fissure and hole, and then secondary cracks occurred at the left and right ends. The maximum compressive stress of each stage of the vertical fissure model was greater than that of the horizontal fissure model. When the model was destroyed, the stress release was also greater. The stress released on the right side of the vertical fissure model and on the left side of the horizontal fissure model was more obvious.

All data are available within the article or from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by the National Natural Science Foundation of China (51904167, 51474134, and 51774194), Taishan Scholars Project, Taishan Scholar Talent Team Support Plan for Advantaged & Unique Discipline Areas, Shandong Provincial Natural Science Fund for Distinguished Young Scholars (JQ201612), Shandong Provincial Key Research and Development Plan (2017GSF17112), and Project of Open Research Fund for Key Laboratories of Ministry of Education for safe and efficient coal mining (JYBSYS2019201).