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In order to investigate the feasibility and reliability of the three-dimensional particle flow method in simulating the type I fracture toughness test, four types of numerical samples were established by particle flow code PFC^{3D}: straight crack three-point bending (SC3PB), edge cracked flattened semicircular disc (ECFSD), cracked chevron notched Brazilian disc (CCNBD), and edge cracked flattened ring (ECFR). Three models with different strength parameters (group A, group B, and group C) were established for each type, in which group A parameters are obtained from the concrete model, group B parameters are applied for simulating marble, and group C parameters are for granite. The type I fracture toughness and the failure form of each model are obtained by conducting the numerical test, and the curves of load versus displacement of loading point are recorded. The numerical test results show that, with the same strength parameter, the maximum difference in test results of each specimen type is 0.39 MPa·m^{1/2}. The _{IC} of ECFR specimen is 0.13–0.28 MPa·m^{1/2} smaller than that of CCNBD specimen, and the _{IC} of ECFSD specimen is slightly higher than that of CCNBD sample. The _{IC} of SC3PB specimen is 0.06–0.21 MPa·m^{1/2} smaller than that of the CCNBD sample. When the loading rate is less than 0.01 m/s, the effect of loading rate on fracture toughness can be reduced to less than 0.1 MPa·m^{1/2}.

The research of theoretical and laboratory test for type I fracture toughness K_{IC} of rock materials is relatively mature [

Zhang et al. [^{2D}. The results show that the generation of secondary cracks is mainly affected by particle sizes. The size effect in particle flow software simulation has been studied by Wong et al. [_{IC} of SR and CCNBD specimen becomes smaller. The facture surface of SR specimen is rougher than that of CCNBD. It is also found that fracture toughness test results can be more consistent by the specimen with larger diameter than the ISRM suggested “minimum effective diameter 75 mm.” Based on the boundary effect theory, the fracture toughness of rock is studied by Guan et al. [

Experimental study on the fracture toughness of CCNBD specimen and the size effect was conducted by Wu et al. It is proposed that the results can be modified by geometric shape function, and then the real fracture toughness of rock can be obtained. In semicircular bending (SCB) test, the support type influence on rock fracture toughness is researched by Bahrami et al. [_{IC} of lapilli-ash tuff is researched by Wong et al. [_{IC} measured by semicircular bending (SCB) method is found to be lower than that using cracked chevron notched semicircular bending (CCNSCB) method. The CCNBD method produces more scattered results.

In laboratory tests of rock fracture toughness _{IC}, the sample processing is relatively difficult, and the test results are scattered. Therefore, numerical simulation method is widely used to test rock fracture toughness. Most of the numerical tests are simulated by two-dimensional software, and three-dimensional numerical calculation method has not been widely discussed. In this study, four types of sample are selected, and three-dimensional particle flow numerical simulation is used to test the type I fracture toughness of the samples. The results are compared and analyzed, which provides a reference for numerical test of fracture toughness.

Particle flow software PFC is widely used to simulate the deformation and failure process of elastic-plastic materials, such as rock, soil, and concrete, which can show the mechanical properties and failure mechanism from a micro perspective. By adjusting the parameters of particle and bond model, the mechanical characteristics of numerical model can be similar to actual materials. Failure process can be obtained by monitoring the number and location of microcracks and the stress in model. In order to study the applicability of numerical simulation in rock materials with different strength, three groups of micro parameters (group A, B, and C) are selected for calculation and analysis (see Tables

Particles microscopic parameters of group A.

Particle parameters | Parallel bond parameters | ||||||||
---|---|---|---|---|---|---|---|---|---|

Density (kg × m^{−3}) | Ratio of particle size | Contact modulus (GPa) | Contact stiffness (_{n} × _{s}^{−1}) | Friction coefficient | Bond modulus (GPa) | Ratio of bond stiffness (_{n} × _{s}^{−1}) | Normal bond strength (GPa) | Shear bond strength (MPa) | Multiplier of radius |

1850 | 1.6 | 2.3 | 2.0 | 0.45 | 2.3 | 2.0 | 11 | 15 | 1.0 |

Particles microscopic parameters of group B.

Particle parameters | Parallel bond parameters | ||||||||
---|---|---|---|---|---|---|---|---|---|

Density (kg × m^{−3}) | Ratio of particle size | Contact modulus (GPa) | Contact stiffness (_{n} × _{s}^{−1}) | Friction coefficient | Bond modulus (GPa) | Ratio of bond stiffness (_{n}×_{s}^{−1}) | Normal bond strength (MPa) | Shear bond strength (MPa) | Multiplier of radius |

2700 | 1.66 | 55 | 2.2 | 0.5 | 55 | 2.2 | 80 | 80 | 1.0 |

Particles microscopic parameters of group C.

Particle parameters | Parallel bond parameters | ||||||||
---|---|---|---|---|---|---|---|---|---|

Density (kg × m^{−3}) | Ratio of particle size | Contact modulus (GPa) | Contact stiffness (_{n} × _{s}^{−1)} | Friction coefficient | Bond modulus (GPa) | Ratio of bond stiffness (_{n} × _{s}^{−1}) | Normal bond strength (MPa) | Shear bond strength (MPa) | Multiplier of radius |

2800 | 1.66 | 5.0 | 3.0 | 0.8 | 35 | 3.0 | 70 | 140 | 1.0 |

The parameters in group A are calibrated according to the results of direct shear test in the laboratory. The parameters in group B are the micro parameters for marble taken by Huang et al. [

As PFC^{2D} is used in reference [

Mix cement, sand, and water evenly by the mass ratio of 1 : 1 : 0.4, put them into a cubic mold sized 15 cm × 15 cm × 15 cm (as shown in Figure

Carry out uniaxial compress test and direct shear test on samples under different normal stress (as shown in Figure

Establish PFC^{3D} numerical models of the same size, and carry out numerical test with the micro parameters of Huang et al. [

Similar model sample mold.

Direct shear test of similar model sample.

Shear stress-normal stress curve of sample.

The calibration and modification of parameters in group C are as follows:

Cylindrical specimens sized

Establish the numerical model of particle flow and carry out the numerical test according to step (1), and then adjust the micro parameters of the model repeatedly, until the load-displacement curves of numerical test agree well with those of laboratory tests (as shown in Figure

Brazil disk split test with granite specimens.

Load-displacement curve of Brazil test (parameter of group C).

Based on the above micro parameters, 12 numerical models of SC3PB, ECFSD, CCNBD, and ECFR were established, respectively, among which SC3PB and CCNBD models have been widely used, and ECFSD and ECFR are new type models studied by Zhang [

Three-point bending beam model. (a) Model size. (b) PFC^{3D} model.

Edge cracked flattened semicircular disc model. (a) Model size. (b) PFC^{3D} model.

Cracked chevron notched Brazilian disc model. (a) Model size. (b) PFC^{3D} model.

Edge cracked flattened ring model. (a) Model size. (b) PFC^{3D} model.

Displacement loading is used in the test. In order to select an optimal loading rate, the SC3PB and CCNBD sample models are established with parameters in group A, and the test is executed under three loading rates of 0.05 m/s [

Elastic stages of the curves, under different loading rates, are almost the same. The slower the loading rate, the lower the peak strength of the curve. For SC3PB sample, _{IC} is calculated by substituting the first load platform in the curve into formulas (

For CCNBD samples, _{IC} is calculated by substituting the maximum load [_{0} = _{0}/_{1} = _{1}/_{B} = _{B}/_{max}, the local maximum load, kN.

The fracture toughness of the samples under various loading rates is listed in Table ^{1/2}, are greatly affected by loading rate. The results of CCNBD samples are almost close to the error of 0.08 MPa·m^{1/2}. With the decrease of loading rate, the influence of loading rate becomes smaller. The slower the loading rate is, the closer the test is to the quasi-static loading. With the same displacement, there is less damage in the sample, and the results are more accurate. Therefore, theoretically, the loading rate should be controlled relatively low, but considering the calculation efficiency of numerical simulation, the loading rate should be increased properly under the condition that the results are accurate enough. According to the above test, the loading rate of this test is controlled as 0.01 m/s.

Test results of fracture toughness under different loading rates.

Loading rate | 0.05 m/s | 0.01 m/s | 0.002 m/s |
---|---|---|---|

SC3PB | 1.04 | 0.92 | 0.88 |

CCNBD | 1.19 | 1.13 | 1.11 |

For numerical test, first of all, the rationality and validity of the test results should be preliminarily judged. Through monitoring the generation and distribution of microcracks in the model, the failure form of the sample can be observed (as shown in Figures

Microcracks distribution in SC3PB specimen (A1). (a) Crack initiation. (b) Crack growth. (c) Specimen failure.

Microcracks distribution in SC3PB specimen (B1). (a) Crack initiation. (b) Crack growth. (c) Specimen failure.

Microcracks distribution in SC3PB specimen (C1). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

Then, record the curve of load _{A} = 1.11 kN, _{B} = 2.99 kN, _{C} = 3.27 kN; substituting these values into formula (_{IC} of model A1 is calculated as 0.92 MPa·m^{1/2}, _{IC} of model B1 is 2.49 MPa·m^{1/2}, and _{IC} of model C1 is 2.72 MPa·m^{1/2}.

During the test, the distribution of microcracks in the model is shown in Figures

Microcracks distribution in ECFSD specimen (A2). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

Microcracks distribution in ECFSD specimen (B2). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

Microcracks distribution in ECFSD specimen (C2). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

In B2 model, the crack occurs at the loading points firstly and then occurs at the end of precrack (as shown in Figure

Record the curve of load _{max} is the dimensionless stress intensity factor [_{max} = 1.0756; _{min} is the local minimum load, kN; and the other symbols are the same as before.

According to the test results, _{Amin} = 6.22 kN, _{Bmin} = 14.9 kN, _{Cmin} = 16.3 kN; then, _{IC} of models can be calculated. The results show that _{IC} = 1.11 MPa·m^{1/2} for A2 model, _{IC} = 2.66 MPa·m^{1/2} for B2 model, and _{IC} = 2.91 MPa·m^{1/2} for C2 model. The observation of the loading process shows that the local minimum value _{min} of the load curve corresponds to the crack initiation and propagation stage of the prefabricated crack. At this time, the specimen is not completely damaged. As the loading continues, the load curve increases, and the second peak value may exceed the first.

The distribution of microcracks in the models is shown in Figures

Microcracks distribution in CCNBD specimen (A3). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

Microcracks distribution in CCNBD specimen (B3). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

Microcracks distribution in CCNBD specimen (C3). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

The curve of load

According to the test results, _{Amax} = 12.2 kN, _{Bmax} = 18.4 kN, _{Cmax} = 20.4 kN; then, _{IC} of models are calculated. The results show that _{IC} = 1.13 MPa·m^{1/2} for A3 model, _{IC} = 2.55 MPa·m^{1/2} for B3 model, and _{IC} = 2.83 MPa·m^{1/2} for C3 model.

The distribution of microcracks in the model is shown in Figures

Microcracks distribution in ECFR specimen (A4). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

Microcracks distribution in ECFR specimen (B4). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

Microcracks distribution in ECFR specimen (C4). (a) Crack initiation. (b) Crack growth. (c) Sample failure.

The curve of load _{min} is substituted into formula (_{max} = 1.0468, _{Amin} = 5.71 kN, _{Bmin} = 13.0 kN, _{Cmin} = 15.2 kN, and the results are as follows: _{IC} = 1.00 MPa·m^{1/2} for A4 model, _{IC} = 2.27 MPa·m^{1/2} for B4 model, and _{IC} = 2.65 MPa·m^{1/2} for C4 model.

In each group of samples, SC3PB and CCNBD samples are damaged by the propagation of precut cracks, while ECFSD and ECFR samples are damaged by not only the propagation of precut cracks, but also the coalescence of vertical cracks. The load-displacement curve of each specimen will decrease slightly and then increase again, which is caused by stress release after crack initiation. By comparing the fracture toughness of each model (see Table ^{1/2}. In group B, the difference between the four sample types is 0.39 MPa·m^{1/2}. In group C, the difference between the four sample types is 0.26 MPa·m^{1/2}. On the whole, the fracture toughness of the ECFR samples is smaller than that of the ECFSD and CCNBD samples (Tables

Test results of fracture toughness (MPa·m^{1/2}).

SC3PB sample | ECFSD sample | CCNBD sample | ECFR sample | |
---|---|---|---|---|

A model | 0.92 | 1.11 | 1.13 | 1.00 |

B model | 2.49 | 2.66 | 2.55 | 2.27 |

C model | 2.72 | 2.91 | 2.83 | 2.65 |

By comparing the distribution of microcracks in each model, it can be found that, in ECFSD and CCNBD samples, both the crack growth and the fracture of the loading point can be obtained. The short crack propagation distance in the ECFSD specimen is not the main reason for specimen failure. It is because the crack tip is within the width covered by specimen loading platform as precut crack propagating. In the contact diagram (as shown in Figure

Contact force between particles in the ECFSD model.

The ECFR specimen can be regarded as a platform Brazilian disk specimen with edge crack and central circular hole. Compared with the disk or half-disk specimen, the ring-like specimen is more prone to compression deformation.

The calculated results are closely related to the ratio of internal and external radius

The fracture toughness test of rock with 4 types of specimens is researched by the numerical simulation method of particle flow. The failure forms and load-displacement curves are analyzed and compared. The conclusions in this research are listed as follows:

The results of the numerical test are reasonable and effective. The maximum difference between the test results of different samples with the same strength parameter is 0.39 MPa·m^{1/2}.

When the loading rate is reduced to 0.01 m/s, the effect of loading rate on fracture toughness can be reduced to less than 0.1 MPa·m^{1/2}. So, the loading rate of 0.01 m/s is reasonable.

During the loading process, the microcracks occur in multiple areas in ECFR specimens. The test results of ECFR specimens are 6%–11% smaller than those of CCNBD specimens. For ECFSD specimens, there are many microcracks generated along the loading direction as the propagation of prefabricated crack. The test load is larger than the other specimens, so the fracture toughness of ECFSD specimens is 0.08–0.11 MPa·m^{1/2} larger than that of CCNBD specimens.

The SC3PB and CCNBD specimens are spilt along the loading direction because of the propagation of prefabricated crack, and the test results of SC3PB are 0.06–0.21 MPa·m^{1/2} less than those of CCNBD specimens.

The article data used to support the findings of this study are included within the article.

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

This work was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant nos. 19KJD410001 and 18KJB440002), the Science and Technology Project of Housing and Construction in Jiangsu Province (Grant No. 2018ZD199), and the Science and Technology Project of Changzhou (Grant No. CJ20190020).

_{IC}using cracked chevron notched Brazilian disc specimen