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Prediction of rock fracture is essential to understand the rock failure mechanism. The three-point bending test has been one of the most popular experiments for the determination of rock fracture parameters. However, the crack initiation and propagation of rock beam with the center notch and offset notch have not been fully understood. This paper develops a numerical method for modelling the notched beam cracking based on nonlocal extended finite element method (i.e., XFEM) and mixed mode rock fracture model. An example is worked out to demonstrate the application of the numerical method and verified with experimental results. The crack length development, crack pattern, crack opening and slipping displacements, and the load-crack mouth of displacement (P-CMOD) curve are obtained. The effects of offset notch location and mechanical properties on the crack length development, P-CMOD curve, and crack pattern are investigated and discussed. It has been found that the peak load of the notched beam nearly linearly increases with the increase of the notch offset ratio. The cracking of rock beam with offset notch is dominated by mode I fracture, but mode II fracture contributes more when crack deflection occurs. The fracture energy significantly affects the peak load, while it has little effect on the prepeak and postpeak slopes in the P-CMOD curve.

The failure of rock or rock mass in deep rock engineering is closely related to the crack initiation, propagation, coalescences, and nucleation from micro- to macroscales [

As a standard or suggested method for determining the strength, fracture energy, and fracture toughness of rock and concrete from RILEM (International Union of Laboratories and Experts in Construction Materials, Systems, and Structures) [

Computational simulations of rock fracture have been an active research field, which enhanced the understanding of failure mechanism of rock [

This paper attempts to develop a numerical method based on nonlocal XFEM for modelling notched rock beam cracking under the three-point bending load. Cohesive crack behaviour is used to describe the mixed mode fracture of rock. An example of rock beams with the notch at different locations is worked out to demonstrate the numerical method. The crack length development, crack pattern, crack opening and slipping displacements, and load-CMOD (P-CMOD) curve are obtained. The numerical results are compared with those from experiments. Finally, the effects of offset notch location and mechanical parameters on the crack initiation, prorogation, crack pattern, and P-CMOD curves are investigated and discussed.

Rock exhibits the tensile strain-softening behaviour due to an inelastic zone being developed ahead of the crack tip, often referred to as fracture process zone (FPZ) [_{n} and _{s} are the normal and shear stresses, respectively; _{n} and _{s} are corresponding normal and shear strains, respectively; _{n} and _{s} are the normal and tangential stiffness, respectively.

When the maximum principal stress reaches the criterion value (i.e., cohesive strength), crack initiation will occur and a damage value is introduced to reduce the stiffness for stress softening, i.e.,

Constitutive model for rock fracture.

The mixed mode fracture is considered and the mixed mode fracture energy is defined as follows [_{C} is the mixed mode fracture energy; _{I} and _{II} are the pure mode I and pure mode II fracture energies, respectively. _{I} and _{II} are the work ratios done by the normal and shear forces, respectively.

The relationship between the fracture energy and the fracture toughness can be established by Irwin’s formula [_{C} is the fracture toughness;

Furthermore, the damage value can be calculated by the following equation:

Once the damage value is determined, the residual stresses can be obtained. By the above equations, the stress-strain relationship of rock under mixed mode fracture is established.

To overcome the problem associated with matching the geometry of the discontinuity as the crack propagation, the extended finite element method was first introduced by Belytschko and Black [_{I} is the nodal enriched degree of freedom;

Illustration of normal and tangential coordinates for the crack.

The Heaviside function is expressed as follows:

In a polar coordinate system (

Once the crack initiation criteria are satisfied, a new crack will be created in the enriched element. The newly introduced crack is always orthogonal to the maximum principal stress direction. However, the direction will be affected by the local element in the mesh. To reduce the mesh dependency and improve the accuracy of crack direction, a nonlocal calculation technique is used as illustrated in Figure _{c} is the radius around the crack tip for averaging and

Illustration of nonlocal averaging of the stress around the crack tip.

A rock beam model with single notch at different locations is used to demonstrate the application of the developed numerical method. As shown in Figure

Geometry of the worked example.

Figure

Numerical model for the beam with the notch of 20% offset ratio: (a) the whole model and (b) enlarged part around the notch.

The basic parameters for the numerical simulations.

Symbols | Description | Value |
---|---|---|

Young’s modulus | 30 GPa [ | |

Poisson’s ratio | 0.25 [ | |

_{max} | Cohesive strength | 9 MPa [ |

_{I} | Mode I fracture energy | 100 N/m [ |

_{II} | Mode II fracture energy | 100 N/m [ |

_{c} | Radius of nonlocal averaging | 0.5 mm |

Figure

The crack processes and maximum principal stress under the loading displacement: (a) 0.018 mm; (b) 0.036 mm; (c) 0.073 mm; (d) 0.38 mm.

The displacements of the final crack: (a) opening displacement; (b) slipping displacement.

To investigate the effect of offset notch location on the cracking, numerical simulations for the beams with offset notch ratios 0, 10%, 20%, 30%, and 40% are carried out. Figure

The crack length developments with the loading displacement increasing for different notch offset ratios.

Crack patterns for the rock beams with different notch offset ratios.

The load-CMOD curves from the numerical simulations are obtained as Figure

P-CMOD curves for the rock beams with different notch offset ratios.

Figure

Experimental verification of the peak load.

The effects of Young’s modulus on the crack length development and P-CMOD curve are investigated. All the other parameters keep the same as listed in Table

Effect of Young’s modulus on the crack length development.

Effect of Young’s modulus on the P-CMOD curve.

Figure

Effect of cohesive strength on the crack length development.

Effect of cohesive strength on the P-CMOD curve.

Figure

Effect of fracture energy on the crack length development.

Effect of fracture energy on the P-CMOD curve.

Figure

Crack patterns for the rock beams with the notch of 20% offset ratio for different mechanical parameters: (a) _{max} = 9 MPa; _{f} = 100 N/m; (b) _{max} = 9 MPa; _{f} = 100 N/m; (c) _{max} = 9 MPa; _{f} = 100 N/m; (d) _{max} = 6 MPa; _{f} = 100 N/m; (e) _{max} = 12 MPa; _{f} = 100 N/m; (f) _{max} = 9 MPa; _{f} = 80 N/m; (g) _{max} = 9 MPa; _{f} = 120 N/m.

In this paper, a numerical method for notched rock beam cracking under the three-point bending load has been developed based on the nonlocal XFEM and mixed mode fracture model. The cohesive crack behaviour in the fracture process zone was employed to describe the rock cracking. A worked example for rock beams with notch offset ratio of 0, 10%, 20%, 30%, and 40% has been presented to demonstrate the application of the derived method and then verified with experimental results. The effects of the offset ratio, Young’s modulus, strength, and fracture energy on the crack length development, crack pattern, and P-CMOD curves were investigated and discussed. Conclusions can be given as follows:

The derived method based on nonlocal XFEM and rock mixed mode fracture is suitable for modelling the crack initiation and propagation of notched rock beam under the three-point bending load. The arbitrary crack is produced without the limitations of the mesh.

The peak load of the notched rock beam is close to linearly increase with the increase of notch offset ratio. The numerical results have a good agreement with those from experiments.

The cracking of rock beam with offset notch is dominated by mode I fracture but mode II fracture contributes more when crack deflection occurs. The cracking of rock beam with offset notch transfers from mixed mode fracture to mode I fracture.

The material mechanical parameters, that is, Young’s modulus, strength, and fracture energy, have no effect on the crack pattern. The peak load of notched rock beam increases with the increase of Young’s modulus, strength, and fracture energy. The fracture energy has little effect on the prepeak and postpeak slopes in the P-CMOD curves.

The data used to support the findings of this study are available from the authors upon request.

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

This work was supported by Fundamental Research Funds for the Central Universities (no. FRF-TP-18-015A3).