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To reveal the influence of the number and location of joints on rock failure mechanism, using Particle Flow Code (PFC) to simulate the calculation of a large amount of acoustic emission data generated during breeding, development, and penetration of rock cracks, the fracture parameters such as the spatial location, rupture azimuth, rupture type, stress state, and moment magnitude of acoustic emission events in various fracture stages of multijoint rock were studied based on the moment tensor theory, the P-T diagram method, and the T-

Acoustic emission (AE), which is a phenomenon of rapid release of strain energy causing the produce of transient elastic waves, is generated spontaneously by plastic deformation and microfailures initiation and growth when a rock is influenced by external force, internal force, or temperature [

Kaiser conducted a study of materials acoustic emission characteristics in 1950, which was the origin of modern acoustic emission techniques [

Ji analyzed frequency characteristics of AE at the stage of rock failure by using uniaxial compression tests of granite [

Moment tensor has been applied in geotechnical engineering for 20 years; however, the number of researches using moment tensor in analysis of rock failure types is still very small, especially in the field of rock fracture mechanism and evolvement rule; there is scarcely any related research work. Therefore, this paper utilizes Particle Flow Code (PFC) and moment tensor inversion method to obtain some parameters of rock failure events, such as spatial position, moment magnitude, moment tensor, T-

In PFC simulative approach, the displacement triggered by the total contact forces, which act on the surfaces of particles, can be equivalent to the same effect, namely, the moment tensor, triggered by volumetric forces.

The two particles on either side of the failure can be defined as source particles. After the bond between them breaks, the source particles will move, and contacts surrounding the source particles will suffer some deformation. The contact force will, therefore, change due to the formation of the failure. The action zone of the microfailure is a circle with the centre of the circle being the microfailure centroid, and the radius is the maximum particle diameter. Then an summation operation around the contacts surrounding the failure can be performed to calculate components of the moment tensor as

Combined with equation (

Combined with equation (

Moment tensor of rock rupture is a second-order symmetric tensor, and all of its three principal eigenvalues are real, and there exist three orthogonal principal axes, namely, three eigenvectors. Assume that the three principal eigenvalues of moment tensor are

One of the simplest decomposition methods is to determine the isotropic part and deviatoric part of moment tensor. If the isotropic part is positive, the rupture type is tension failure (uniform explosion), while if it is negative, the rupture type will be pressure failure (uniform implosion). And the proportions of two parts, respectively, take the form as [

If the proportion of isotropic part is greater than 30%, it can be considered that the event has obvious isotropic part. But one thing that should be noted in the two-dimensional coordinate department is that the maximum value of isotropic part is 66% because one of the eigenvalues is equal to zero.

Hudson (1989) has come up with two parameters T and

As shown in Figure

The T-

Through calculating the moment tensor of acoustic emission and using the equation (

A single source fracture azimuth can be expressed as the beach ball generally, as shown in Figure

The interpret Figures of beach ball. (a) Three-dimensional beach ball. (b) Two-dimensional beach ball.

The stress field distribution of beach ball.

The P-T distribution diagram.

In order to research the change law of fracture azimuth of a large number of acoustic emission events during rock triaxial compression, the P-axis and T-axis of fracture surface are projected to the equatorial plane (W-N-E-S plane) in this paper. Then, the stereographic projection of P-axis (shown as +) and T-axis (shown as •) is obtained, which is called P-T distribution diagram for short [

To reveal the effect of joint number and position to the mechanism of rock fracture, this paper simulated and computed a large amount of acoustic emission data of the process of the generation, growth, and coalescence of rock cracks using PFC (Particle Flow Code) software and then researched the failure parameters of acoustic emission incidents and the their evolution law such as spatial position, failure orientation, failure mode, stress state, and moment magnitude in each stage of multijoint rock failure based on moment tensor theory, the methods of P-T diagram and T-

This paper uses moment tensor inversion theory to calculate the failure process of the sample simulated on the condition of different joint numbers and positions. In order to research the failure mechanism of rock under uniaxial compression under the condition of different joint numbers and positions, we assume that the medium parameters of imitation specimen are isotropic.

The dimension and loading direction of imitation specimen in this paper are the same as the actual rock samples (

The microparameters (see Table

Mechanical parameters of parallel bond model.

Radius of minimum particle | Radius ratio of max-min particle | Density ^{−3}) | Interparticle friction coefficient | Elastic modulus | Normal-tangential stiffness ratio | ||
---|---|---|---|---|---|---|---|

0.50 | 1.66 | 2630 | 0.50 | 67000 | 2.5 | ||

Radius coefficient | Elasticity modulus | Normal-tangential stiffness ratio | Normal intensity | Tangential intensity | |||

Average value | Standard deviation | Average value | Standard deviation | ||||

1.0 | 67000 | 2.5 | 166 | ±38 | 166 | ±38 |

Mechanical parameters of Lac du Bonnet granite.

Elastic modulus | Uniaxial compressive strength (MPa) | Poisson’s ratio | P-wave velocity (m·s^{−1}) | S-wave velocity (m·s^{−1}) |
---|---|---|---|---|

67031 | 48～210 | 0.25 | 5820 | 3360 |

To quantitatively describe the location, orientation, and location of each joint of each rock sample, making the following definition for now (as shown in Figure

Interpretation diagram of joint orientation.

More than 10 groups of rock failure tests with different multijoints are simulated by PFC in this paper. The joint parameters of rock specimens are shown in Table

Joint parameters of rock specimens.

Sample number | |||||
---|---|---|---|---|---|

Joint parameters | 30-2 | 45-2-1 | 45-2-2 | 60-2 | 45-3-1 |

Number of joints (number) | 2 | 2 | 2 | 2 | 3 |

Joint orientation | 30 | 45 | 45 | 60 | 45 |

Joint spacing | 10 | 10 | 30 | 10 | 20 |

Joint mode | Mode 1 | Mode 1 | Mode 3 | Mode 1 | Mode 2 |

Sample number | |||||

Joint parameters | 45-3-2 | 45-4-1 | 45-4-2 | 45-5-1 | 45-5-2 |

Number of joints (number) | 3 | 4 | 4 | 5 | 5 |

Joint orientation | 45 | 45 | 45 | 45 | 45 |

Joint spacing | 15 | 13.3 | 10 | 10 | 7.5 |

Joint spacing | Mode 3 | Mode 2 | Mode 3 | Mode 2 | Mode 3 |

Joint orientation of rock specimens: (a) 30-2, (b) 45-2-1, (c) 45-2-2, (d) 60-2, (e) 45-3-1, (f) 45-3-2, (g) 45-4-1, (h) 45-4-2, (i) 45-5-1, and (j) 45-5-2.

Simulated result by PFC software: (a) 30-2, (b) 45-2-1, (c) 45-2-2, (d) 60-2, (e) 45-3-1, (f) 45-3-2, (g) 45-4-1, (h) 45-4-2, (i) 45-5-1, and (j) 45-5-2.

The whole simulated stress-strain curve results in the condition of different numbers and positions of joints are shown in Figure ^{−3}, 2.916 × 10^{−3}, 2.925 × 10^{−3}, 2.703 × 10^{−3}, 2.806 × 10^{−3}, 2.902 × 10^{−3}, 2.774 × 10^{−3}, 2.945 × 10^{−3}, 2.723 × 10^{−3}, and 2.980 × 10^{−3}, respectively.

Whole stress-strain curve of rock failure process: (a) 30-2, (b) 45-2-1, (c) 45-2-2, (d) 60-2, (e) 45-3-1, (f) 45-3-2, (g) 45-4-1, (h) 45-4-2, (i) 45-5-1, and (j) 45-5-2.

By analyzing the axial stress and strain difference values of the rock specimens at peak stress, some rules can be summarized as follows:

As the green arrows shown in Figures

As the blue arrows shown in Figure

As the yellow arrows shown in Figures

As the purple arrows shown in Figure

As the cyan arrows shown in Figure

Before the peak stress, the variation rate of elastic modulus is very small; the curve shows obvious fluctuation at the peak stress and a sharp decline after the peak stress; it means that the rock specimens have strong brittle features.

Axial stress difference values.

Axial strain difference values.

In the process of rock failure, the simulated results of parameters such as spatial position, failure magnitude, and fracture type of acoustic emission are shown in Figure

Spatial positions of acoustic emission events and calculated results of fracture parameters: (a) 30-2, (b) 45-2-1, (c) 45-2-2, (d) 60-2, (e) 45-3-1, (f) 45-3-2, (g) 45-4-1, (h) 45-4-2, (i) 45-5-1, and (j) 45-5-2.

By analyzing the parameters such as spatial location, moment magnitude, and fracture type of acoustic emission events in the process of rock failure, some rules can be obtained as follows:

Crack closure stage (O-A) and linear elastic stage (A-B): acoustic emission events have not yet begun to generate

Crack generation stage (B-C): acoustic emission events begin to appear in the area near the surface of joints and axial load; every fracture types have appeared; these moment magnitudes are very small

Crack increase stage (C-D): acoustic emission events gradually appear in all over the rock specimen and begin to concentrate in the area near the joints and two axial loading surfaces; these moment magnitudes are still very small; but linear tensile fracture events increase obviously

Peak stress (point D): acoustic emission events continuously increase, and linear shear fracture events and mixed fracture events with larger moment magnitude begin to appear at the end of each joint and the area between joints and extension direction of joints

Large deformation and cumulative damage stage (D-E): linear shear failure events and fixed failure events with larger moment magnitude mainly distribute at the end of each joint, along the direction of joints and along the direction of joint sets; when the rock specimens contain two joints, linear shear failure events and fixed failure events mainly distribute the area near joints and along the direction of joints; when the rock specimens contain more joints, if the rock specimens contain Joint Type 2, linear shear failure events and fixed failure events mainly distribute along the direction of two joints, which are located near two axial loading surfaces and the direction of joint sets; if the rock specimen contains Joint Type 3, linear shear failure events and fixed failure events mainly distribute along the direction of joint sets; when the number of joints is the same, linear shear failure events in the rock specimens containing Joint Type 3 are more than those in the rock specimens containing Joint Type 2

In the simulation results of rock stress-strain curve as shown in Figure

T-

As shown in Figure

Crack closure stage (O-A) and linear elastic stage (A-B): T-

Crack generation stage (B-C): T-

Crack increase stage (C-D): T-

Peak stress (point D): T-

Large deformation and cumulative damage stage (D-E): T-

In the simulation results of rock stress-strain curve, the rock failure mechanism and evolution law in each stage can be analyzed by the locations and moment magnitudes of acoustic emission as shown in Figure

P-T distribution diagram of moment tensor: (a) 30-2, (b) 45-2-1, (c) 45-2-2, (d) 60-2, (e) 45-3-1, (f) 45-3-2, (g) 45-4-1, (h) 45-4-2, (i) 45-5-1, and (j) 45-5-2.

As shown in Figure

Linear elastic stage (O-A), P-T value points are not yet appearing.

Crack generation stage (B-C), the principle compression stress component(P-axis) has appeared near the A-axis(Point W or Point E), and the poles in point E are arranged regularly pointing upper left 45°, as Box 1 shown in Figure

Crack growth stage (C-D), the principle compression stress components (P-axis) gather gradually near

Peak stress (point D), the principle compression stress components (P-axis) gather near the area of plus or minus 30 degrees by the direction of

Large deformation and cumulative damage stage (D-E), the principle compression stress components (P axis) expand from the area of plus or minus 30 degrees by the direction of

By analyzing the complete stress-strain curve, acoustic emission location, failure orientation, failure type, and moment magnitude of multijoint rock compress test in the condition of different numbers and orientation of joints, the following rules are obtained in this paper:

The axial stress and strain difference values of each multijoint sample decrease with the increase of the angle of the joint and

When the number of joints is the same, the axial stress and strain difference values of the sample containing type 2 joints are less than those of the rock specimen containing type 3 joints.

When the joints of rock specimens are of type 2, the axial stress and strain difference values decrease with the increase of the number of joints.

When the mode of rock specimens joints is of type 3, the axial stress difference values first decrease and then slightly increase with the increase of number of joints, which illustrates that stress fluctuation caused by the large-scale failure before and after peak stress has effect on the difference value, and the axial strain difference values decrease with the increase of number of joints.

Before the sample reaches the peak stress, the variation rate of the elastic modulus is small; the curve shows obvious fluctuation before the peak stress and a sharp decline after the peak stress, which shows that the rock specimens have strong brittle features.

Acoustic emission events first appear near the area containing joints and axial load plane, and all types of failures have appeared, but tension failure is more, and all the moment magnitudes of acoustic emission events are small.

In peak stress and large deformation stage, the linear shear failure events and fixed failure events with larger moment magnitude are mainly distributed in the end of each joint, along the direction of joints or the joint sets.

When the sample contains two pieces of joint, the linear shear failure events and fixed failure events are mainly distributed near the joints or in the direction of joint orientation.

When the rock specimens contain more joints, if the samples contain type two joints, the linear shear failure events and fixed failure events are mainly distributed in the direction of two joints close to the axial loading plane; if the samples contain type three joints, the linear shear failure events and fixed failure events are mainly distributed on the orientation of joint sets.

When the number of joints is the same, the failures in rock specimen containing type three joints are more than those in the samples containing type two joints.

First, the points of T-

The principal compressive stress component (P-axis) is first distributed near the direction of the axial stress, and part of the compressive stress components (P-axis) is deflected gradually to 45 degrees by the direction of axial stress; the principal tensile stress components (T-axis) start to appear in the area near the direction of plane

In conclusion, we can obtain the large number of mesoscopic fracture characteristics of acoustic emission events such as dimensional orientation, failure status, failure type, and moment magnitude in the progress of rock failure to study the failure mechanism of joint rock and macroevolution law of joint rock by utilizing the method of moment tensor inversion P-T diagram and T-

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

The author declares that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Major Systematic Project of Science and Technology Research and Development Plan of China National Railway Group Co., Ltd. (no. P2019G001) and the Scientific Research Foundation of China Academy of Railway Sciences Group Co., Ltd. (no. 2018YJ030).