In order to study the bedding effect of coal and rock deformation, the surface deformation fields of coal and rock at different bedding angles were obtained by means of digital image correlation (DIC). By optimizing the statistical index describing the nonuniformity of strain field, the initiation and evolution characteristics of deformation localization were analyzed quantitatively. The concepts of equivalent cohesive force and equivalent tensile strength were put forward, and a transverse isotropic constitutive model of coal and rock was established. The results show that the maximum shear deformation field of uniaxial compressed coal and rock in parallel bedding is more likely to show the three-stage characteristics of uniform stage, localization stage, and failure stage. In the vertical and parallel bedding directions, the modified statistical index curve of coal and rock in the process of uniaxial compression shows typical two-stage characteristics. When loading perpendicular to bedding, the starting stress of deformation localization of coal and rock is closer to the peak strength, and when loading parallel to bedding, coal and rock are easier to show localization characteristics.

Bedding is the main structural symbol of coal and rocks, which can be divided into horizontal, current, and oblique bedding according to different bedding morphology. In coal resource mining, different types of bedding are often encountered, such as near-horizontal, oblique, and near-vertical bedding. The difference in the direction of coal and rock bedding will lead to significant differences in the evolution law of the stress field, the law of gas migration, the deformation and failure characteristics of coal seam and overburden, etc. in the front of the working face during mining. Studying the deformation and failure characteristics of coal and rocks in different bedding directions under stress is of great significance for disaster prevention and safe production of coal mines [13].

Scholars around the world have carried out lots of studies on the characteristics of bedding effects of coal deformation and failure. For example, Zhang et al. [4] revealed the effects of bedding plane on coal and rock energy release law, AE timing parameters, AE amplitude distribution, and spatial evolution characteristics under uniaxial loading by means of acoustic emission system; Liang et al. [5] studied the basic mechanical parameters of rock salt in different bedding directions through uniaxial and triaxial compression tests; Liu et al. [6] studied the influence of bedding on coal deformation and permeability and quantified the effects of bedding plane of coal rocks under true triaxial stress; Gatelier et al. [7] and Kim et al. [8] found out that the uniaxial compressive strength of sandstone decreases gradually with the increase of bedding dip; Louis et al. [9] described the mechanical anisotropy and failure characteristics of Rothbach sandstone with bedded structure by means of X-ray and CT and believed that the brittle strength of porous sandstone decreases gradually with the increase of bedding dip. Tien et al. [10] and Cho et al. [11] studied the elastic mechanical parameters, fracture modes, and failure mechanism of bedded rocks from different angles as well as the failure mechanism of transversely isotropic rocks. Liu et al. [12] used cubic anthracite coal samples to perform a series of permeability measurements and obtained the influence of bedding on the permeability of coal seam. Lei et al. [13] studied the damage and failure law of jointed rock during freeze-thaw cycle and established a constitutive model reflecting its failure characteristics. Li et al. [14] established a new constitutive model by considering the bedding characteristics of salt rocks, embedded the constitutive model into FLAC3D, and carried out engineering calculations. Most of the researches on the bedding effect of coal and rock mainly focus on the mechanical parameters, damage evolution law, and failure mechanism of bedding direction on coal and rock [1517]. There are few studies on the influence of bedding direction on coal rock deformation characteristics, and the deformation law of the whole sample is mostly replaced by the deformation value of a small area on the surface of the sample.

Many disasters in geotechnical engineering can be attributed to rock deformation. Deformation localization is a problem that cannot be ignored in the process of deformation and failure of coal and rock [1821]. Digital image correlation technology has the advantages of full field, noncontact, and high degree of automation, and it has been widely used in the research of rock deformation location [22]. Li et al. [23] measured the broken rock zone thickness through an approach based on DSCM. Liu et al. [24] investigated the strength and fracture mechanism of flawed rock by DIC. Shirole et al. [25] analyzed the stress-induced multiscale spatial heterogeneity in the strain-field of three types of rocks by DIC.

With the help of DIC, this paper studied the formation and evolution law of coal deformation localization and macroscopic failure mode in different bedding directions and established a transverse isotropic constitutive model considering the bedding direction. The research results provide a theoretical basis for safe mining of coal resources.

2.1. Sample Preparation

The coal samples used in the test were taken from Tazigou Coal Mine in Liaoning Province. Coal and rocks in this mine are characterized by obvious bedding characteristics and high brittle. The large pieces of raw coal were taken out from the working face, sealed with wax, and then transported to the laboratory. The coal samples were cut, respectively, according to the axial parallel bedding direction and the axial vertical bedding direction; and then, the standard square (70mm70mm70mm) coal samples meeting the requirements of ISRM were prepared through the grinding process. The coal samples prepared are shown in Figure 1. Coal samples selected for the test should be free of macroscopic structural inhomogeneity and defects, so as to reduce the dispersion as far as possible. Before the test, ultrasonic testing equipment was used to select coal samples, and samples with abnormal acoustic values were eliminated, so as to ensure that the acoustic value difference of samples in each group was within 100 m/s. The samples are divided into two groups: the loading direction is perpendicular to the bedding coal and rock, and the loading direction is parallel to the bedding coal and rock, with three samples in each group. The samples with loading direction perpendicular to the bedding are named V1-1, V1-2, and V1-3. The samples with loading direction parallel to the bedding are named P2-1, P2-2, and P2-3.

In this test, white light speckle method was employed to make artificial speckle field. Before the test, the surfaces of the coal samples to be observed were sprayed with black paint to avoid the influence of mineral particles in the coal samples on the speckle effect. After the black paint dried, random white spots were sprayed.

2.2. Test System

The test system mainly included electrohydraulic servo universal testing machine, CCD camera, LED light source, and computers. The electrohydraulic servo universal testing machine was used for the uniaxial compression test of coal samples, and the loading modes included loading axially perpendicular to the bedding plane and axially parallel to the bedding plane. The CCD camera was for continuous acquisition of images of the sample surfaces during the loading process, and the LED lamp provided white light source. The computer connected with the universal testing machine controlled the loading process and recorded the force and displacement information, and the computer connected with the CCD camera controls the image acquisition process and automatically stores the images. In the test, the image acquisition frequency was 15 frames/s, the resolution of the CCD camera was 1280 pixel 1024 pixel, and the loading rate of the universal testing machine was 0.1 mm/min. The test system is shown in Figure 2.

2.3. Digital Image Correlation Method

Digital image correlation (DIC), also known as digital speckle correlation method (DSCM), is to obtain deformation information of the region of interest through correlation calculation of two digital images before and after deformation of the specimen. The basic principle is that the regions of interest in the image before deformation are meshed, and each subregion is treated as rigid motion. For each subregion, correlation calculation is carried out according to the predefined correlation function through certain search methods. Find the region with the maximum number of correlation with the subregion in the deformed image, which is the position of the deformed subregion, and then get the displacement of the subregion. By calculating all subregions, the deformation information of the whole field can be obtained [26]. The calculation principle of digital speckle correlation method is shown in Figure 3.

3.1. Effects of Bedding Plane on Stress-Strain Characteristics

The coal samples with axis parallel to and perpendicular to the bedding plane basically experienced four stages: compaction, elasticity, yield, and failure, as shown in Figure 4. The coal samples under the two loading modes showed slightly different characteristics in each stage. The axial deformation in the initial compaction stage of coal samples with axis perpendicular to bedding plane was slightly larger than that parallel to bedding plane, which was possibly because most of the initial microfractures are parallel to the bedding plane direction, resulting in significant closure of the initial microfractures during the loading perpendicular to bedding plane. For the elastic stage, the stress-strain curves under the two loading modes were inclined straight lines, indicating that the deformation of coal and rocks increases linearly with the increase of stress. No obvious differences were found between the coal samples under the two loading modes. At the yield stage, fluctuation of stress was found in both loading modes, and during the test, small particles left the samples and catapult, accompanied by sound. The stress curve dropped rapidly after the failure of the coal sample, and the raw coal samples showed the characteristics of hard brittleness. The peak strength of coal samples with axis perpendicular to bedding plane was 42.689 ~44.902 MPa, with the average value of 43.638 MPa; the peak strength of coal samples with axis parallel to bedding plane was 21.316 ~25.675 MPa, with the average value of 23.952 MPa. The peak strength of coal samples with axis perpendicular to bedding plane was significantly higher than that parallel to bedding plane, and the former was about 1.82 times of the latter. The peak strain of coal samples with axis perpendicular to bedding plane was 2.914 ~4.909%, with the average value of 4.022%; the peak strain of coal samples with axis parallel to bedding plane was 1.390 ~2.431%, with the average value of 1.872%. Obviously, the peak strain of coal samples with axis perpendicular to bedding plane was greater than that parallel to bedding plane, and the former was about 2.15 times that of the latter.

3.2. Effects of Bedding Plane on Failure Mode

Figure 5 shows the residual block after the failure of the coal sample under uniaxial compression and the transverse strain nephogram obtained by processing the numerical speckle image near the failure time. The coal samples with axis perpendicular to the bedding plane appeared particle ejection during the loading process, accompanied by sound. When the axial stress reached the peak strength, along with the sudden release of energy, the coal sample was broken into pieces with a huge sound. In the process of loading with axis parallel to the bedding plane, particle ejection on the surface of the coal sample was very little. The fluctuation of stress curve is mainly caused by the crack propagation along the bedding plane. Moreover, the coal sample only cracked along the bedding plane without any fragmental ejection when the stress reached the peak strength, and the sound was also less than that of the loading perpendicular to bedding plane. Figure 5(a) shows that shear failure is the main failure mode of coal samples with axis perpendicular to the bedding plane. Figure 5(b) shows that the failure mode of coal samples with axis parallel to the bedding plane is mainly splitting failure.

In terms of failure mode, the failure mechanism and strength anisotropy of bedding coal and rock are mainly controlled by weak cementation between bedding planes. When parallel bedding is loaded, the fracture surface of coal sample is compressional fracture through the bedding plane. The flat tensile cracks initiated and extended along the weak plane and gradually formed penetrating cracks, which were in essence the instability of the pressure bar caused by Poisson ratio effect, and there were several penetrating cracks.

3.3. Effects of Bedding Plane on Evolution Characteristics of Deformation Field

Taking the deformation image obtained during the initial loading time as the reference image, the displacement field and strain field nephogram at the corresponding time in the whole process of loading can be obtained through calculation with the digital speckle correlation method. According to the principle of digital image correlation method, based on the measured displacement values of each pixel, the image processing software can directly calculate and give the horizontal strain, vertical strain, shear strain, and maximum shear strain of the sample at different loading times. The maximum shear strain field is calculated as follows:
(1)γmax=εxεy2+γxy2,
where εx is horizontal linear strain, εy is vertical linear strain, and γxy is shear strain.

To compare the similarities and differences in the surface deformation field evolution law of coal samples under the two loading modes, four point positions A, B, C, and D (Figure 6, representing the four typical stages of coal and rock failure) were selected on the stress-strain curve, and the nephogram of the maximum shear strain field at each point was obtained based on the digital speckle principle, as shown in Figure 7.

Figure 7 shows the strain values of coal samples at different loading times. The magnitude of the strain values is expressed by the color depth, so the coordinates and units are not given in the figure. It can be seen from Figure 7 that the bedding plane has a significant effect on the deformation field of coal and rocks. Figures 7(a) and 7(b) are two coal samples. The different initial microfissures of coal samples lead to different deformation concentration positions at the initial stage of compression, showing different positions of maximum deformation. For the coal and rock with axis perpendicular to the bedding plane, its maximum shear strain is mainly distributed transversely along the bedding plane direction, while for the coal and rock with axis parallel to the bedding plane, its maximum shear strain is mainly distributed vertically along the bedding plane direction. This may be because under axial vertical bedding loading, the deformation of coal and rock is mainly closed and staggered along the bedding plane, and the maximum shear strain is transverse distribution along the bedding plane. The coal samples with the axis are parallel to the bedding plane, the deformation of coal is mainly opening along the bedding plane, and the deformation localization zone shows vertical distribution along the bedding plane. It can be seen from Figure 7 that under the two loading modes, the maximum shear strain values of coal and rocks gradually increase with the increase of axial stress. The coal samples with the axis are parallel to the bedding plane, and the maximum shear deformation field of coal and rocks goes through three stages, namely, uniform deformation stage (Figure 7(b) A), localization stage (Figure 7(b) B and C), and failure stage (Figure 7(b) D); and the stage characteristics are similar to the research results of Wang et al. [27]. However, the coal samples with the axis are perpendicular to the bedding plane, and the stage characteristics are not obvious, which may be caused by the hard brittleness characteristics of coal and rocks. No obvious microfractures after the uniform deformation stage are found on the coal and rock surface before the sudden failure occurs; thus, no obvious localization zones formed. At the compaction stage of coal and rocks with the axis perpendicular to the bedding plane (Figure 7(a) A), due to the presence of initial microcracks; although the maximum shear strain value is very small, there is still a concentration area of deformation field; at the compaction stage of coal and rocks with the axis parallel to the bedding plane (Figure 7(b) A), there is no obvious microfracture closure, so the deformation field is relatively uniform.

3.4. Effects of Bedding Plane on the Characteristics of Deformation Localization

It can be seen from the evolution process of strain field localization that the main differences between localized deformation field and uniform deformation field include two aspects. First, the deformation of a few points (in the localization band) in the deformation field is much greater than that of other points (outside the localization band), which is the numerical feature of deformation field localization. Second, the spatial feature of deformation field localization is that a few points with large deformation are concentrated in one or several connected bands. According to the above characteristics, the deformation field statistical index can be used to quantitatively analyze the localization evolution process of the deformation field, and the deformation localization appears suddenly compared with the homogeneous deformation stage. When the deformation localization appears, the slope of the evolution curve of the statistical index of the deformation field will be abrupt, which can be used as the judgment of the starting position of the deformation localization. Accordingly, Sun et al. proposed the calculation method of statistical index [28]. The statistical indexes of strain field at different loading time of coal samples were solved by using this method. The relationship curve between statistical index values and loading time was drawn, as shown in Figure 8.

As shown in Figure 8, the Sw curve under the two loading modes shows a two-stage characteristic. In the initial stage, namely, the uniform deformation, the Sw curve is relatively gentle and increases slowly, while in the later stage, namely, the nonuniform deformation, the Sw curve shows an accelerated growth trend. Under the loading perpendicular to the bedding plane, the stress value is 40.27 MPa, that is, at 84.5% σc before the peak (σc is the peak strength of the coal and rock), the Sw curve turns, showing the start of the localization of coal and rock deformation. Under the loading parallel to the bedding plane, the stress value is 17.18 MPa, that is, at 57.3% σc before the peak (σc is the peak strength of the coal and rock), the Sw curve turns, showing the start of the localization of coal and rock deformation. Based on the above analysis, the initial stress of deformation localization of coal and rocks under loading axial perpendicular to the bedding plane is closer to the peak strength, and coal and rock under loading axial parallel to the bedding plane are more prone to show the characteristics of localization.

Coal and rocks, which show obvious different mechanical effects in parallel bedding and vertical bedding direction, are usually regarded as transversely isotropic body in engineering. In isotropic rock mass, Mohr-Coulomb criterion with tensile cutoff is used to represent the failure of rock mass, and the expression is as follows [29]:
(2)fs=σ1σ3Nϕ+2cNϕft=σ3σth=σ3σt+aPσ1σp,
where
(3)Nϕ=1+sinϕ1sinϕaP=1+Nϕ2+NϕσP=σtNϕ2cNϕ,
where fs=0 means Mohr-Coulomb shear yield criterion, ft=0 represents tensile yield criterion, h=0 indicates shear-tensile yield border, σ1 and σ3 are the first and the third principal stress, respectively, and c, ϕ, and σt are cohesive force, internal friction angle, and tensile strength, respectively. The maximum tensile strength shall be less than the value at the intersection of fs=0 and σ1σ3=0, that is,
(4)σmaxt=ctanϕ.
Inside the transverse isotropic rocks, the strength parameters of rock mass are related to direction. The strength parameters (cohesive force, tensile strength) of rock mass have the minimums cmin and σmint along the bedding plane and have the maximums cmax and σmaxt in the direction of perpendicular to the bedding plane, and with the change of the angle with the bedding plane, it may take the form of linear, sine, cosine, and quadratic sine. This is because when the angle between the potential failure surface and the bedding surface increases from 0° to 90°, the equivalent cohesion and the equivalent tensile strength must be monotonically increasing. The elementary functions describing monotonically increasing include linear, sine, cosine, and other forms. The maximum cohesion is measured by the direct shear experiment. When the shear direction is parallel to the bedding plane, the minimum cohesion can be measured. When the shear direction is perpendicular to the bedding plane, the maximum cohesion can be measured. The tensile strength is measured by the tensile test of rock samples. When the tensile direction is perpendicular to the bedding plane, the minimum tensile strength can be measured. When the tensile direction is parallel to the bedding plane, the maximum tensile strength can be measured. Based on this, the concepts of equivalent cohesive force and equivalent tensile strength are proposed.
(5)c=ccmin,cmax,θσt=σtσmint,σmaxt,θ,
where θ is the angle between the potential failure surface and the transverse isotropic plane. When the potential failure surface is parallel to the transverse isotropic plane, θ=0; the cohesive force c is the function of θ with the minimum cohesive force cmin and the maximum cohesive force cmax; tensile strength σt is the function of θ with the minimum tensile strength and the maximum tensile strength.
Assuming that the tensile stress is a negative and the compressive stress is a positive, the Mohr-Coulomb yield criterion of transversely isotropic rock mass with angle parameters is as follows:
(6)fs=τ+σtanϕcθft=σσtθh=ττPaPσσtθ,
where
(7)τP=cθσtθtanϕap=1+tanϕ2tanϕ.
The cosine relationship is used to describe the relationship among c, σt, and θ, as follows:
(8)cθ=cmin+1cosθcmaxcminσtθ=σmint+1cosθσmaxtσmint.

When there is stress inside the rock mass, there is a certain potential failure angle θ1 for each element to make the yield function fs obtain the maximum value, and there is a certain potential failure angle θ2 to make the yield function ft obtain the maximum value. When fs or ft of a certain element is greater than 0, the element produces plastic yield. Therefore, it is necessary to determine the potential failure angle θ1 and θ2 in the calculation process, so that the maximum value of fs or ft can be obtained to determine the potential failure surface.

In actual numerical calculation, it is difficult to calculate θ1 and θ2 accurately, so a simplified method is used to determine the failure mode. (1) When θ1=θ2=0, if plastic yield occurs in the element, it is shear or tensile failure along the bedding plane, and the failure marks are “zjshear” and “zjtension.” (2) When there is no bedding, the bedrock is subject to shear or tensile failure, and the plastic zone is marked as “shear” and “tension.”

Establish a local coordinate system on the potential failure plane (z-axis is perpendicular to the potential failure plane, x-axis is downward along the vertical potential failure plane, y-axis is perpendicular to the x-axis, and the coordinate system is right-handed). The principal stress is transformed from the global coordinate system to the local coordinate system where the potential failure surface is located by the transformation matrix Q, and the transformation formula is
(9)σ=QTσQ.
Assume that the normal vector of the potential failure plane is α, and the normal vector of the bedding plane is β; then, the expression of the potential failure angle θ is shown as follows
(10)cosθ=αβαβ.
The stress state in any direction at a point of bedding plane is calculated first, and then the normal stress and shear stress are calculated by the coordinate transformation formula. The stress tensor is converted to the stress component of the local coordinate system (that is, the x-axis and y-axis are parallel to the potential failure surface, and the z-axis is perpendicular to the coordinate system of the potential failure surface) where the potential failure surface is located. On the potential failure surface, normal stress and shear stress are defined as
(11)σ=σ33,τ=σ132+σ232.
When judged as shear or tensile failure along the bedding plane, the yield criterion is
(12)fs=σ132+σ232+σ33tanϕcθft=σ33σtθh=σ132+σ232τPaPσ33σtθ.

The determination of potential failure surface only needs to determine the potential failure surface which maximizes the yield function. When the yield function is greater than 0, plastic yield occurs on the corresponding potential failure surface, and the material enters the plastic stage. However, the plastic yield test cannot be carried out for every angle at every moment in the constitutive model, so two potential failure planes (the shear failure plane of solid rock and the failure plane along the bedding plane are included) are checked to simplify the calculation.

In plastic mechanics, the constitutive equation is usually expressed by incremental theory. The strain increment can be decomposed into elastic part and plastic part (Figure 9).
(13)Δεi=Δεie+Δεip.
The stress increment is calculated from the elastic stress increment and the plastic stress increment:
(14)Δσi=ΔσieΔσip.
In accordance with the incremental decomposition theory, the relationship between the stress increment and the strain increment is as follows.
(15)Δσie=FiΔεn,Δσip=FiΔεnp,Δσi=FiΔεne.
In the equation, F is a linear function of elastic strain increment. Therefore, by substituting the stress increment into the Equation (14), it can be obtained:
(16)Δσi=FiΔεnFiΔεnp.
Flow rule is used to express the relationship between plastic strain increment and yield function or plastic potential function, which can be expressed as
(17)Δεip=λgσi,
where Δεip is the increment of plastic strain; λ is a constant; g is the plastic potential function.
A flow rule which replaces the plastic potential function with yield function is called the associated flow rule, that is, fg. A flow rule characterized by a plastic potential function is called the disassociated flow rule. In the constitutive model for transverse isotropy of parallel multibedded coal and rock, the nonassociated flow rule is adopted for shear failure and the associated flow rule for tensile failure. The potential function is composed of the function gs defining the shear plastic flow and the function gt defining the tensile plastic flow, which is
(18)gs=σ132+σ232+σ33tanψcθgt=σ33σtθ,
where ψ is dilatancy angle.
Substitute the flow rule into the Equation (16), the following can be obtained:
(19)Δσi=FiΔεnλFigσn.
When plastic yield occurs in the model element, its stress state always meets the yield function, as follows:
(20)fσn=0fσn+Δσn=0.
Taking the shear stress as a stress component alone, the stress component of the potential failure surface element is as follows:
(21)σn=σ11,σ22,σ33,τT.
f is a linear function about σn. By decomposing fσn+Δσn, it can be obtained:
(22)fσn+fΔσn=0,
where f is a linear function obtained by subtracting the constant term from the yield function, that is,
(23)fΔσn=fΔσnf0n.
From Equations (20) and (22), it can be obtained:
(24)fΔσn=0.
Substitute Equation (19) into Equation (24), it can be obtained:
(25)fFnΔεnλfFngσn=0.
From the above, it can be obtained:
(26)λ=fFnΔεnfFng/σnf0n.
At the same time, the relationship between stress increment and total stress is expressed as
(27)σiN=σi+Δσi,σiNE=σi+FiΔεn,
where σiN is the total stress at t+Δt moment; σiNE is the total elastic stress at t+Δt moment. By substituting Equation (27) into Equation (26), it can be obtained:
(28)λ=fσnNEfFng/σnf0n.
By substituting Equation (19) into Equation (27), it can be obtained:
(29)σiN=σiNEλFigσn.

The constitutive model can be defined by the user’s secondary development of FLAC 3D using C++ language, and the custom constitutive model can be invoked after the “model configure plugin” command is configured. According to the secondary development platform provided by FLAC 3D, the constitutive model for parallel multibedded coal body is implemented using the coding language of C++. The rationality of the constitutive model is verified by comparing the numerical simulation results with the indoor test results (Figure 10).

It can be seen from the Figure 10 that the simulation curve coincides with the test curve at the elastic stage. The prediction of peak strength is relatively accurate, and the anisotropy of raw coal samples can be accurately characterized. However, due to the fact that there is a compaction stage in the actual test and homogeneous samples are used in the simulation, the simulation curve fits poorly with the test curve before the elastic stage. After entering the plastic stage, because of a high brittleness of raw coal, the breakage of raw coal samples in the test process leads to stress mutation, while numerical simulation cannot predict the sample breakage, so the curve cannot fit the sudden stress change.

  • (1)

    Coal samples under uniaxial compression perpendicular and parallel to the direction of bedding plane experienced four stages: compaction, elasticity, yield, and failure, and showed typical characteristics of hard brittleness; the deformation, peak strength, and axial peak strain at the initial compaction stage of coal and rocks under the former loading mode were all greater than the latter, among which the peak strength of the former was about 1.82 times that of the latter, while the peak strain of the former was about 2.15 times that of the latter

  • (2)

    According to the morphology of coal samples after failure and the digital speckle image processing results at the moment of failure, the macroscopic failure mode of coal samples under uniaxial compression perpendicular to the direction of bedding plane was dominated by shear failure, while the macroscopic failure mode of coal samples under uniaxial compression parallel to the direction of bedding plane was dominated by splitting failure

  • (3)

    The maximum shear deformation field of coal samples under uniaxial compression axially parallel to the bedding plane went through three stages, namely, uniform deformation stage, localization stage, and failure stage; due to the hard brittleness of raw coal, the characteristics of the maximum shear deformation field of coal samples under uniaxial compression perpendicular to the bedding plane were not obvious, without showing obvious localization characteristics

  • (4)

    This paper corrects the original statistical index describing the nonuniformity of deformation field and finds a statistical index calculation method that can better reflect the stress curve fluctuation of “hard brittle” raw coal. The curve of the corrected statistical index of coal samples under uniaxial compression perpendicular to and parallel to the direction of bedding plane showed a typical two-stage characteristic. The stress for initiating the deformation localization of coal and rocks under loading axially perpendicular to the bedding plane is closer to the peak strength, and coal and rocks under loading axially parallel to the bedding plane are easy to show localization characteristics

  • (5)

    This paper puts forward the concepts of equivalent cohesive force and equivalent tensile strength and establishes the three-dimensional yield criterion of transversely isotropic coal and rock mass considering the equivalent cohesive force and equivalent tensile strength

All data, models, and code generated or used during the study appear in the submitted article.

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China for Young Scholars (Grant numbers 52204105, 51804211, and 52004170), the Open Project of Engineering Research Center of Phosphorus Resources Development and Utilization of Ministry of Education (Grant numbers LKF2021006), and Liaoning Provincial Department of Education Project (LJKZ0356).

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