Research on Multijoint Rock Failure Mechanism Based on Moment Tensor Theory

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-Tdiagrammethod, and the T-k diagrammethod. It will be of great importance in the geotechnical engineering field.


Introduction
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 [1]. rough the computation analysis of rupture parameters and rupture types during rock failure, the procedure can reveal the mechanisms of rock failure and judge the evolution law. erefore, analyzing the mechanisms of rock failure has important theoretical significance and application value.
Kaiser conducted a study of materials acoustic emission characteristics in 1950, which was the origin of modern acoustic emission techniques [2]. Recently, many domestic and international scholars have carried out lots of research studies in rock engineering, mostly concentrating on studying the theory and application of experiments about Kaiser Effect and the law of AEs when a rock is subjected to sufficient load until damage. Liu analytically investigated the friction effect on the buckling behaviors of cellular structures under axial load [3,4]. Cheng introduced a meshless method based on the nonsingular weight functions for elastoplastic large deformation problems [5,6]. Ishida researched the acoustic emission monitoring of hydraulic fracturing in laboratory and field [7]. Landis researched the micro-macro-fracture relationships and acoustic emissions in concrete [8]. Ohtsu researched the crack classification and moment tensor based on acoustic emission [9,10]. Muralidhara researched the fracture process zone size and true fracture energy of concrete using acoustic emission [11]. On the basis of studying the uniaxial compression test of granite, He et al. analyzed the frequency characteristics of acoustic emission signals during different stages of rock failure process [12]. In the application field, Li utilized Kaiser Effect of rock acoustic emissions to determine the values and directions of the three principal stresses of surrounding rock in deep mining [13]. Gilbert first introduced the concept of moment tensor, which he defined as the first moment of equivalent volume force, to represent a point source [14].
Ji analyzed frequency characteristics of AE at the stage of rock failure by using uniaxial compression tests of granite [15]. Under the utilization of AE system, the regulation and characteristics of AE with stress and time were researched by Zhao [16][17][18]. e source was represented by the moment tensor. And displacement of far field expressed by moment tensor was linear without the assumption of source mechanism [19]. Initially, moment tensor was applied to rock engineering and rock mechanical test. And types of rock failures were researched using moment tensor in Foreign Countries. At URL, moment tensor was introduced to analyze the types of fracture at the source by Feignier, to explain the tensile fracture located in the spandrel during the process of drilling with machine. Moment tensor was divided into isotropic part (M ISO ) and deviator part (M dev ). Types of fracture at the source were quantified by the ratio of isotropic part (M ISO ) to the entire moment tensor [20]. Based on the ratio shear component of eigenvalue of moment value to the entire, fracture types of AE were judged, and location of fracture plane of AE was determined. Using PFC and PFC3D, failure of rock was simulated, and moment tensor was calculated by Hazzard [21,22]. And Hazzard described fracture source mechanism of microseism by using types of fractures introduced by Feignier. Based on the results of moment decomposition method in the same maximum axial direction, Cao explained the application of moment tensor in the judgment of types of rock failure and simulated reliability and applicability of moment tensor in the analysis of types of mining induced rock and coal fracture by using theory method [23]. Chai judged the types of fracture in the mining microseism by moment tensor, analyzed the applicability of moment tensor in engineering [24], and studied the internal fracture mechanism of simulated rock acoustic emission events using T-k diagram method and P-T graph method [25][26][27]. Cheng researched the element-free Galerkin method based on the nonsingular weight functions [28][29][30][31].
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. erefore, 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-k value, and rupture orientations, and conducts a quantitative research on the rock fracture mechanism and evolvement rule. It will be of great importance in geotechnical engineering field.

PFC Simulative Approach.
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. e 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. e contact force will, therefore, change due to the formation of the failure. e 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.
en an summation operation around the contacts surrounding the failure can be performed to calculate components of the moment tensor as where ΔF i is the i th component change in contact force, and R j is j th component of the distance between the contact point and the event centroid. Combined with equation (2), the moment magnitude M w can be calculated from the peak scalar moment by Combined with equation (2), the moment magnitude M w can be calculated from the peak scalar moment by

Moment Tensor Decomposition.
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 M x ≥ M z ≥ M y , and the corresponding three eigenvectors are t, b, p. In principal axial system, moment tensor can be diagonalized and decomposed as [14] M � where M i ′ are deviatoric moments, i � x, y, z. 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 [32] %DEV � 100 − |%ISO|, 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 2 Mathematical Problems in Engineering isotropic part is 66% because one of the eigenvalues is equal to zero.

T-k Plot
Method. Hudson (1989) has come up with two parameters T and k of moment tensor to analyze the source rupture type. e parameter T is used to measure deviatoric part of source with its range from pure positive CLVD, which is located at − 1, to pure negative CLVD, located at +1, crossing the pure DC located at original point. e parameter k is used to measure isotropic part with its range from uniform explosive type located at +1 to uniform implosive type located at − 1.
As shown in Figure 1, these two parameters can be drawn into an equal area source type plot. Assuming M x ≥ M z ≥ M y are the three eigenvalues of moment tensor, then T and k can be presented as

P-T Plot Method.
rough calculating the moment tensor of acoustic emission and using the equation (10) A single source fracture azimuth can be expressed as the beach ball generally, as shown in Figure 2. Figure 2(a) visually shows the rock fracture azimuth information in the 3D coordinate system (Strike φ s , Dip δ, Slide λ). Figure 2(b) is the stereographic projection shadow from the lower hemisphere to the equatorial plane of 3D beach ball. In Figure 3, P-axis and T-axis, respectively, represent compressive stress axis and tensile stress axis. A1 is shear fracture surface, and A2 is its orthogonal auxiliary plane. eir intersecting line is zero-axis (B axis). Figure 4 is the twodimensional beach ball of shear fracture type, tensile rupture type, mixed type, and its stress field distribution.
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. en, the stereographic projection of P-axis (shown as +) and T-axis (shown as •) is obtained, which is called P-T distribution diagram for short [33].

Uniaxial Compression Test of Multijoint Rock.
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-k diagram.

Calculation Model
is 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. e dimension and loading direction of imitation specimen in this paper are the same as the actual rock samples (R (X-Y plane) � 50 mm, H (Z-axis) � 100 mm, the vertical of Z-axis is the loading direction), and the bonding mode between particles is the parallel bond model. e microparameters (see Table 1) basically agree with the mechanical parameters (see Table 2) of Lac du Bonnet granite sample.
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 5), joint orientation angle for the angle of joint orientation and Z-axis (the direction of compression), joint length b � 20 mm, joint spacing can be expressed as symbol c.
More than 10 groups of rock failure tests with different multijoints are simulated by PFC in this paper. e joint parameters of rock specimens are shown in Table 3 and  By analyzing the axial stress and strain difference values of the rock specimens at peak stress, some rules can be summarized as follows:  Rock fracture surface       . is phenomenon is related to stress fluctuation caused by the large-scale failure at the peak stress. (5) As the cyan arrows shown in Figure 9, the axial strain difference values decrease with the increase in the number of joints when the joints of rock specimens are Joint Type 3 (Sample 45-3-2, Sample 45-4-2, and Sample 45-5-2). (6) 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.

Mathematical Problems in Engineering
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 11. e shape of acoustic emission events represents its fracture type (the circulars represent the linear tension fracture; the rhombuses represent the linear shear fracture; the squares represent the double couple shear fracture, the hexagons represent the middle mixed fracture with large principal stress). e space size of acoustic emission events represents the scale of moment magnitude.
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: 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    Mathematical Problems in Engineering (4) 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 (5) 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

T-k Distribution of Moment Tensor.
In the simulation results of rock stress-strain curve as shown in Figure 12, the rock failure mechanism and evolution law in each stage can be analyzed by the distribution of T-k parameters. As shown in Figure 12, the distributions and evolution laws of T-k value points in each rock specimens are very similar:

P-T Distribution of Moment Tensor.
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 10 and the distribution of T-k parameters as shown in Figure 13. As shown in Figure 13, P-T value points of moment tensor and the evolution law of stress field of each joint sample are similar:  Figure 13; it shows that when acoustic emission events happen, compression stress components are mainly distributed near the direction of axial stress and a part of them gradually points to 45 degrees by the direction of axial stress; the principle tension stress components (T axis) start to appear near the direction of plane X-Y (near line N-S). (3) Crack growth stage (C-D), the principle compression stress components (P-axis) gather gradually near Z- Mathematical Problems in Engineering 13

Conclusions
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: (1) e axial stress and strain difference values of each multijoint sample decrease with the increase of the angle of the joint and Z-axis. (2) 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. (3) 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. (4) 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. (5) 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. (6) 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. (7) 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. (8) 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. (9) 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. (10) 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. (11) First, the points of T-k values first appear and continue to increase near the line y � 1/2x; next, the points start to aggregate near the linear vector pair pole (− 1, 1/3); the points near the negative linear vector pair pole (1, − 1/3) also increase but are always dispersive; the points away from the line y � − 1/2x increase gradually but are still few. e above shows that the main type of failure is linear tension failure; linear shear failure events and fixed failure events have increased gradually with large energy, but the their number is small; double couple shear failure events are few, and homogenic expansion events and homogenic compression failure events have not appeared. (12) e principal compressive stress component (Paxis) 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 X-Y (line N-S); the principle compression stress components (Paxis) expand from the area of plus or minus 30 degrees by the direction of Z-axis (point W or point E) to the direction of plane X-Y (line N-S) further and finally form an obvious phenomenon of regular arrangement in some local area; inside each multijoint sample, there are obvious crescent regular arrangements near 30 to 60 degrees by the direction of Z-axis (point W or point E) , which illustrates that the multijoint structure can affect the fracture orientation of acoustic emission events to a certain extent; the principle stress components (T-axis) expand from the direction of plane X-Y (line N-S) to Z-axis (point W or point E).
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-k diagram. It can provide a new means to analyze the stability and the development tendency of rock and is an effective supplement to traditional methods.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.