Numerical simulation of pressure relief stress distribution of diamond beaded rope saw cutting in low permeability coal seam

Abstract

The diamond bead slit is a practical method for changing the stress distribution of low permeability coal seam and achieving pressure relief and reflection improvement. The stress distribution of coal seam in large scale is not clear due to the influence of diamond bead slit parameters and geological parameters, making it difficult to identify. In this paper, the finite element model with built-in Coulomb friction contact surface is used to simulate the stress distribution in coal with different angles, seam length, working face length, seam friction coefficient and different in-situ stress difference. This investigation is conducted to examine the stress distribution of parallel working face, vertical coal seam, and advance working face. The simulation results show that the mechanism of stress transfer in large scale diamond beaded rope saw cutting coal seam is mainly due to the tangential slip of coal body on both sides of seam surface, forming concentrative zone and pressure relief zone with axial distribution, center symmetry and phase. The pressure relief range and maximum pressure relief range of all three direction present a "single peak" distribution with the change of angel α between slit and maximum in-situ stress, i.e. when α = 45°, both of them are maximum. The slit length mainly affects the stress distribution in the advancing direction of the working face, and the length of working face mainly affects the stress distribution in the direction of parallel working face and vertical coal seam, both of which are positively correlated with the pressure relief range and the maximum pressure relief amplitude. The friction coefficient of seam surface and the difference of in-situ stress affect the relative dislocation of coal body on both sides of seam surface, and they inhibit and promote the pressure relief range and the maximum pressure relief amplitude respectively, and are greatly affected by α. The simulation results above suggest that it is reasonable to select fracture and geological parameters in practical engineering.

Highlights

In the calculation of solids with cracks, most of the existing literatures regard cracks as rectangular or elliptical hole structures, which may have little impact on small-scale specimens. However, for large-scale specimens, We believe that cracks are more consistent with a group of contact surfaces with negligible spacing. This paper is also based on this assumption. The idea of limit is used to approach the problem of hole to the problem of crack. The innovation of this paper is that the influence of the normal displacement caused by the seam width on the distribution of stress field is ignored in the calculation of this kind of problem, and the influence of the staggered displacements on the stress distribution on both sides of the coal body along the parallel direction of the seam plane is considered, which is more consistent with the actual displacement of rock mass on both sides of the narrow seam in large scale. Based on this, We think this paper has a unique significance for the study of this field.

1 Introduction

Coal seam gas outburst poses a significant threat to underground safety production in coal mines, threatening the safety of coal mines and causing significant harm to the ecosystem (Liu et al. 2020a, b). Taking effective measures to prevent and control gas explosions is the top priority in the safe production of coalmines (Xie et al. 2019). Gas extraction plays a significant role in preventing and controlling gas disasters. For low permeability coal seam, it is difficult to extract gas by traditional methods, so it is necessary to relieve pressure of coal seam by auxiliary means to improve the permeability of coal seam (Li 2021). Common methods for depressurizing and improving coal seam reflection include water injection, hydraulic fracturing, and hydraulic slit (Feng et al. 2004; Cai and Xiong 2004; Li 2020a, b; Zhang et al. 2021). Wu et al. (2009a, b) discussed the mechanism of pressure relief and reflection improvement of seam grooves after seam cutting, and believed that the pressure relief effect of seam grooves was related to the shape and size of seam grooves, and the length of seam grooves was the dominant factor. Cheng et al. (2021) studied the influence of hole diameter slit width and slit length on the stress distribution of coal seam by using FLAC3D software. They believed that slit width had a great influence on the stress release of the coal seam, while hole diameter and slit length did not. Gao et al. (2011, 2019), Zou et al. (2020) used discontinuous deformation analysis (DDA) method to study the influence of seam length, seam width, ground stress and other factors, and believed that when the slit is not closed, the seam width has little influence on the pressure relief effect, but the seam length has a certain influence on the pressure relief range. Liu et al. (2015) studied the influence of slit angle on mechanical properties of coal based on PFC2D, and believed that with the increase of crack Angle, both peak strength and crack initiation stress increased. Song et al. (2014) and Liu et al. (2020a, b) studied the influence of the number and spatial arrangement of grooves on the stress distribution in coal, believed that reasonable design and optimization of the joints and their combinations could significantly reduce the range of stress concentration and energy aggregation, and force the stress to transfer to the deep. Zinovyev et al. (2012) established a mathematical model of rock mass stress release process through FLAC3D, and gave the geometry shape and position of the slit corresponding to the maximum pressure relief range. Si et al. (2019) and Li et al. (2020b) considered the relationship between the geometrical shape of in-situ stress fractures and the spacing of multiple fractures with the scope of coal failure area, and believed that the internal friction angle, the slot radius and the ratio of maximum principal stress to minimum principal stress were the top three factors affecting the effect of seam cutting. Xue et al. (2017) studied the influence of the slit radius of high-pressure water jet on gas flow rate. Although the above results have guiding significance for practical production, most of the slit height studied by the above research results is more than 5 mm, and the high efficiency of the slit should be particularly obvious under large scale hard coal rock.

In recent years, some scholars have tried to apply diamond beaded rope saw cutting technology to coal seam pressure relief and reflection improvement, and carried out relevant experimental research and numerical simulation. Diamond beaded rope saw cutting technology was first proposed by The British in 1968. At present, rope saw seam cutting has been successfully applied in mining engineering practice such as large-area cutting of salt-dissolved ore cracks and mining of very thin ore bodies, and has achieved good technical and economic indexes. Li et al. (2021) studied the stress distribution and permeability of coal seam after rope saw cutting, and compared it with hydraulic seam cutting. They believed that rope saw cutting formed a pressure relief zone in upper and lower coal seams, which could realize effective pressure relief of a coal seam with higher efficiency than hydraulic seam cutting. Tang et al. (2021) carried out numerical simulation on pressure relief of rope saw slit, and the results showed that the cutting efficiency and slit depth were higher than those of hydraulic slit due to the large cutting coal seam area. The stress concentration area is formed in the front of the working face of the rope saw joint, the stress reduction area and the stress recovery area are formed in the rear, and the pressure relief range shows a trend of first increasing and then decreasing. The diamond bead slot has the characteristics of large surface continuity and closing of the seam surface, so that its pressure relief and reflection enhancement mechanism is different from that of the hydraulic slot.

To better explain the difference between diamond beaded rope saw cutting and hydraulic cutting, we show the stitching characteristics of the two cutting techniques through Fig. 1. The principle of hydraulic slit is based on high-pressure water jet breaking rock to form a cut. As the distance of water jet from the nozzle increases, the energy of the jet decreases continuously, and the jet produces a gyrohollowing effect, which increases the width of the slit and can reach the centimeter level (Fig. 1a). In the process of numerical calculation, a disc with a certain thickness is often used to represent the hydraulic slit (Fig. 1b). Due to the large deformation space along the wide direction of the slit, the coal displacement (especially normal to the slot) is easy to occur under the action of earth stress, which is the principle of hydraulic slit pressure relief. By contrast, the principle of diamond beaded rope saw cutting seam is to tell the rotating diamond rope saw to form cutting effect on coal and rock, like thin line cutting bean curd (but coal and rock facies are much harder). Since the rope saw is extremely thin (Fig. 1c), and the cutting process is slower than that of water jet breaking rock, the coal and rock surfaces on both sides of the joint have enough time to close and form contact. The normal displacement in this process is very small (millimeter). Since then, although the change of coal rock is not obvious from the apparent point of view, the coal rock itself has undergone essential changes, from a complete rock mass to two fitted rock masses, as shown in Fig. 1d. Since then, the stress redistribution of coal seam is mainly caused by the dislocation between coal and rock on both sides driven by ground stress. In spite of the feasibility of this technique has been proved in low permeability coal seam, there is little research on the effect of the parameters of rope saw on the stress distribution of the coal seam and the surrounding rock. In view of this, numerical simulation method is employed to investigate the stress distribution law of diamond beaded rope saw seam cutting, utilizing contact surface theory.

Fig. 1

Comparison between traditional hydraulic cutting and diamond rope saw cutting (Lei et al. 2017)

2 Numerical modeling method for stress distribution of diamond beaded rope saw

2.1 Difference of dominant displacement field of coal body under wide seam and narrow seam

For the coal seam numerical simulation method, the traditional method of calculating the crack width for the coal seam numerical simulation method employs the actual width crack. This method is practical for the study of the hydraulic coal seam with the same width in small scale, because the displacement field is dominated by normal deformation (Fig. 2), and its stress equivalent envelope is shown in Fig. 3 (Cheng et al. 2012; Tian et al. 2020). But there are some limitations of this kind of situation in the numerical simulation process of diamond beads cutting under large-scale coal, diamond beads cutting seam width can be neglected, and small scale hydraulic slotted in essence difference, for the contact and relative sliding of coal surface on both sides of the slot after the seam face closed. At this time, the coal and rock mass displacement field is dominated by the sliding of the parallel fracture plane, as shown in Fig. 2. It can be found from Fig. 2 that there is a significant difference in the displacement fields of coal and rock mass between the slit with small and large crack width and the stress field distribution of the two is naturally different.

Fig. 2

The difference of displacement field between wide slit and contact surface

Fig. 3

Contour of stress field around wide slot

2.2 Control equation of numerical simulation of diamond beaded rope saw cutting in low permeability coal seam

2.2.1 Seam surface contact

The discussion in Sect. 2.1 indicates that the displacements of coal and rock on both sides of diamond bead grooves in large scale are mainly caused by tangential slip. In order to accurately describe the contact properties in the process of tangential displacement, the coulomb friction model is adopted in this paper to characterize the contact properties of coal and rock grooves (Tahmasebinia et al. 2020; Sultanov et al. 2020; Wang et al. 2018; Wanrui et al. 2018). As for the physical mechanism of this model describing the contact behavior between rock and soil masses, a large number of scholars (Khusanov and Rikhsieva 2019; Mirsalimov 2020; Nguyen et al. 2018) have studied it from three aspects: macroscopic shear test, microstructure morphology of the contact surface and shear numerical analysis of the contact surface, which will not be described in this paper.

The Coulomb friction model in ABAQUS infers the friction on the contact surface to the pressure on the contact surface (Liu et al. 2018). The constitutive model relationship is shown in Fig. 4. When two parts contact each other, their normal behavior is also a pressure interference model. When there is a gap between the contact interfaces, the normal contact pressure is zero and no contact pressure will be transmitted (Fig. 4a). It is considered that before the two contact surfaces start to slide relative to each other, a certain amount of shear stress is applied on the contact surface, and the friction force follows Formula (1). Slitting occurs when the actual shear force on the contact surface exceeds the critical shear stress. During sliding, the critical shear stress is the interfacial friction, which remains the critical shear stress throughout sliding (Fig. 4b).

$$\tau_{{\text{crit }}} = \mu F^{n}$$

(1)

where $\tau_{{\text{crit }}}$ is the critical shear stress of the contact surface; $\mu$ Is the friction coefficient; $F^{n}$ is the normal pressure. The value of friction coefficient is related to the surface roughness of rock mass on both sides of the fracture surface, the hardness of coal rock, the moisture content of coal rock and other factors (Chen and Wu 2018).

Fig. 4

Contact model definition

2.2.2 Load analysis

In an infinite semi-plane elastic body, the governing equation of its stress component satisfies:

$$\frac{{\partial \sigma_{x} }}{\partial x} + \frac{{\partial \tau_{xy} }}{\partial y} = 0$$

(2)

$$\frac{{\partial \tau_{xy} }}{\partial x} + \frac{{\partial \sigma_{y} }}{\partial y} = 0$$

(3)

As shown in Fig. 5a, concentrated normal stress $F^{n}$, concentrated tangential stress $F^{s}$, normal surface force $q^{n}$ and tangential surface force $q^{s}$ are applied at the top of the half plane. Under the concentrated force $F^{n}$ and $F^{s}$ at the origin $o$, the stress at any point in the plane can be expressed as (Mindlin 1936):

$$\sigma_{\eta v} (x,y) = H_{\eta v}^{n} (x,y)F^{n} + H_{\eta v}^{s} (x,y)F^{s}$$

(4)

where, $H_{\eta v}^{n}$ and $H_{\eta v}^{s}$ are stress influence coefficients,which are defined by the following formula:

$$H_{xx}^{n} (x,y) = - \frac{{2x^{2} y}}{{\pi \left( {x^{2} + y^{2} } \right)^{2} }};H_{xx}^{s} (x,y) = - \frac{{2x^{3} }}{{\pi \left( {x^{2} + y^{2} } \right)^{2} }}$$

(5)

$$H_{yy}^{n} (x,y) = - \frac{{2y^{3} }}{{\pi \left( {x^{2} + y^{2} } \right)^{2} }};H_{yy}^{s} (x,y) = - \frac{{2xy^{2} }}{{\pi \left( {x^{2} + y^{2} } \right)^{2} }}$$

(6)

$$H_{xy}^{n} (x,y) = - \frac{{2xy^{2} }}{{\pi \left( {x^{2} + y^{2} } \right)^{2} }};H_{xy}^{s} (x,y) = - \frac{{x^{2} y}}{{\pi \left( {x^{2} + y^{2} } \right)^{2} }}$$

(7)

Fig. 5

Force analysis of infinite half plane

With the distribute surface force $q^{n}$ and $q^{s}$ acts on the boundary section AB of elastic half-plane body (Fig. 5b), the stress at any point in the body can be obtained by integrating the analytical solution of the elastic stress of the half-plane body under the action of surface concentrated force:

$$\sigma_{\eta \nu } (x,y) = \int_{oA}^{oB} {\left[ {H_{\eta \nu }^{n} (x - \xi ,y)q^{n} (\xi ) + H_{\eta \nu }^{s} (x - \xi ,y)q^{s} (\xi )} \right]} d\xi$$

(8)

where, $\xi$ is the $x$ coordinate on the boundary of the half-plane body; $q^{n} (\xi )$ and $q^{s} (\xi )$ are the normal surface force and tangential surface force respectively.

2.2.3 Plane partitioning

There is a rectangular hole in an infinitely two-dimensional plane, which can be divided into four infinitely half-planes, namely the half-planes A, B, C and D. There are common areas between the adjacent semi-planes, and the upper surface of the semiplane is divided into three sections, as Fig. 6a.When subjected to far-site stress, according to the principle of stress superposition, it can be regarded as the result of applying load and far-site stress on the inner surface of the hole respectively. In the case of half plane A, the load is simplified as uniform distributed load acting on segment A₁, A₂ and the zero load acting on segment A, as shown in Fig. 6b. If the lengths of sections B and D approach zero infinitely, the problem of holes evolves into a problem of cracks (Mirsalimov 2020).

Fig. 6

Transformation between plane problems with holes and problems with cracks

2.2.4 Solution steps

Define the local coordinate system $x_{i} o_{i} y_{i}$ (i = A, B, C, D) with the midpoint of the inner surface of the hole as the origin, as shown in Fig. 5b. In the local coordinate system $x_{i} o_{i} y_{i}$, it is assumed that there is an initial distribution of forces on the extended top half plane $i_{\delta }$ ($\delta$ = 1, 2) on half-plane, which treats $i_{1} ii_{2}$ as its top surface. According to the theoretical Eqs. (4)–(8) of the elastic half-plane, in the local coordinate system $x_{i} o_{i} y_{i}$, the stress at any point ($x,y$) in the elastic half-plane,which treats $i_{1} ii_{2}$ as its top surface, is as follows:

$$\begin{aligned} \sigma_{i,\eta \nu } (x,y) & = \int_{i} {\left[ {H_{\eta v}^{n} \left( {x - \xi_{i1} ,y} \right)T_{i1}^{n} + H_{\eta \nu }^{s} \left( {x - \xi_{i1} ,y} \right)T_{i1}^{s} } \right]} d\xi_{i1} \\ & \quad + \int_{{i_{2} }} {\left[ {H_{\eta v}^{n} \left( {x - \xi_{i2} ,y} \right)T_{i2}^{n} + H_{\eta v}^{s} \left( {x - \xi_{i2} ,y} \right)T_{i2}^{s} } \right]} d\xi_{i2} \\ & \quad + \int_{i} {\left[ {H_{\eta v}^{n} \left( {x - \xi_{i} ,y} \right)F_{i}^{0n} + H_{\eta v}^{s} \left( {x - \xi_{i} ,y} \right)F_{i}^{0s} } \right]} d\xi_{i} \\ \end{aligned}$$

(9)

where, $\xi_{i}$,$\xi_{i\delta }$ ($\delta$ = 1, 2) are the $x_{i}$ coordinates of the top surface of the semi-plane body located at the inner edge $i$ of the hole and the top surface $i_{\delta }$ extending on both sides respectively; $T_{i\delta }^{n}$,$T_{i\delta }^{s}$ are the normal and tangential distribution forces on the extended top surface $i_{\delta }$; $F_{i}^{0n}$,$F_{i}^{0s}$ are the normal and tangential components of arbitrary loads $F_{i}^{0}$ on the inner boundary $i$ of the hole.

For segment A and C, both normal and tangential components of $F_{i}^{0}$ are existing, and fulfil the following criteria:

$$F_{i}^{0s} =\left\{ {\begin{array}{ll}0 ,&\quad u^{s} = 0 \\ \mu F_{i}^{0n} , &\quad u^{s} \ne 0 \\ \end{array} } \right.$$

(10)

For sections B and D, only normal loads of $F_{i}^{0}$ is existing. The extended top surface $j_{\delta }$ of one side of the adjacent half plane body is located in the half plane taking $i_{1} ii_{2}$ as top surface, and the elastic stress distribution at the position $j_{\delta }$ calculated by Formula (9) can be further converted into surface force:

$$T_{j\delta }^{n} = - \sigma_{i,yy} \cos^{2} \theta_{j\delta } - \sigma_{i,xx} \sin^{2} \theta_{j\delta } + 2\tau_{i,xy} \cos \theta_{j\delta } \sin \theta_{j\delta }$$

(11)

$$T_{j\delta }^{s} = \left( {\sigma_{i,xx} - \sigma_{i,yy} } \right)\cos \theta_{j\delta } \sin \theta_{j\delta } + \tau_{i,xy} \left( {\sin^{2} \theta_{j\delta } - \cos^{2} \theta_{j\delta } } \right)$$

(12)

where $\theta_{j\delta }$ is the included angle between the extended top surface $j_{\delta }$ and the axis $x_{i}$ the; $T_{j\delta }^{n}$ and $T_{j\delta }^{s}$ are the normal and tangential components of the force $T_{j\delta }$ above the extended top surface $j_{\delta }$ under the local coordinate system $x_{j} o_{j} y_{j}$.

It is assumed that the surface force extending to the top surface $i_{\delta }$ of each semi-plane body has an initial distribution (Fig. 7), and the surface force is generally taken as zero, i.e. $T_{i\delta }^{(0)} = 0$. Firstly, take the semi-plane body with top surface A₁AA₂ as a subject of study, according to the known surface force $F_{A}^{0}$ on the top surface A of the hole and the assumptive surface force $T_{A\delta }^{(0)}$ on the extended top surface $A_{\delta }$ ($\delta$ = 1, 2), the iterative surface forces $T_{B1}^{(1)}$, $T_{D2}^{(1)}$ of surface B₁ and D₁ can be calculated by Eqs. (9)–(12). Secondly, select the semi-flat body with top surface B₁BB₂ as a subject. According to the known concentrated force $F_{B}^{0}$ on B, the updated surface force $T_{B1}^{(1)}$ on extended top surface B₁ and the assumptive surface force $T_{B2}^{(0)}$ on B₂, the iterative surface force $T_{C1}^{(1)}$, $T_{A1}^{(1)}$ on C₁ and A₂ is calculated from Eqs. (9)–(12). Next choose the half-plane body which views C₁CC₂ as the top surface, the surface forces $T_{D1}^{(1)}$, $T_{B2}^{(1)}$ on surface D₁ and B₂ can be obtained during iteration of Eqs. (9)–(12) by taking known surface force $F_{C}^{0}$ from C, improved surface force $T_{C1}^{(1)}$ on C₁ is extended and assuming surface force $T_{C2}^{(0)}$ above C₂. Then regards semi-plane body with top surface D₁DD₂ as research object, and the known concentrated force $F_{D}^{0}$ on D and the improved surface force $T_{D1}^{(1)}$ on the extended surface $D_{\delta }$ as iterative condition, $T_{A1}^{(1)}$ and $T_{C2}^{(1)}$ on A₁, C₂ can be calculated by Eq. (9)–(12). The iterative calculation above is repeated until the difference between the unknowns computed in the two subsequent iterations is within the defined small range of values.

Fig. 7

Iterative process of surface force on partitioned half plane

According to the principle of stress superposition, when the far-field stress is $\sigma_{xx}^{\infty }$, $\sigma_{yy}^{\infty }$ and $\tau_{xy}^{\infty }$, the excess surface force $t_{i}^{\infty }$ generated by the far-field stress on the inner boundary $i$ is equal to the surface force $F_{i}^{0}$ on the inner boundary, namely

$$F_{i}^{0} = - t_{i}^{\infty }$$

(13)

And satisfy the following relations:

$$t_{i}^{\infty n} = - \sigma_{xx}^{\infty } \sin^{2} \chi_{i} - \sigma_{yy}^{\infty } \cos^{2} \chi_{i} + 2\tau_{xy}^{\infty } \cos \chi_{i} \sin \chi_{i}$$

(14)

$$t_{i}^{0s} = \left( {\sigma_{xx}^{\infty } - \sigma_{yy}^{\infty } } \right)\cos \chi_{i} \sin \chi_{i} + \tau_{xy}^{\infty } \left( {\sin^{2} \chi_{i} - \cos^{2} \chi_{i} } \right)$$

(15)

where $t_{i}^{\infty n}$ and $t_{i}^{\infty s}$ are the normal and tangential components of $t_{i}^{\infty }$.

The superimposed stress field of the two can be expressed as:

$$\sigma_{ij} = \overline{\sigma }_{ij} + \sigma_{ij}^{\infty } (i,j = x,y)$$

(16)

where $\sigma_{ij}$ is the superimposed stress field, $\overline{\sigma }_{ij}$ is the stress field of load on inner surface $i$, and $\sigma_{ij}^{\infty }$ is the stress field of the far field stress.

When there is an included angle between the local coordinate of the joint surface and the global coordinate, Eq. (9) can be further converted into the coordinate and stress in the global coordinate system according to Eqs. (17)–(21):

$$x = y_{i} \sin \varphi_{i} + x_{i} \cos \varphi_{i} + c_{i}$$

(17)

$$y = y_{i} \cos \varphi_{i} - x_{i} \sin \varphi_{i} + d_{i}$$

(18)

$$\overline{\sigma }_{xx} = \sigma_{i,yy} \sin^{2} \varphi_{i} + \sigma_{i,xx} \cos^{2} \varphi_{i} + 2\tau_{i,xy} \sin \varphi_{i} \cos \varphi_{i}$$

(19)

$$\overline{\sigma }_{yy} = \sigma_{i,yy} \cos^{2} \varphi_{i} + \sigma_{i,xx} \sin^{2} \varphi_{i} - 2\tau_{i,xy} \sin \varphi_{i} \cos \varphi_{i}$$

(20)

$$\overline{\tau }_{xy} = \tau_{i,xy} \left( {\cos^{2} \varphi_{i} - \sin^{2} \varphi_{i} } \right) + \left( {\sigma_{i,yy} - \sigma_{i,xx} } \right)\sin \varphi_{i} \cos \varphi_{i}$$

(21)

where ($c_{i}$,$d_{i}$) is the coordinates of the origin $o_{i}$ of the local coordinate system $x_{i} o_{i} y_{i}$ in the global coordinate system $xoy$, and $\varphi_{i}$ is the angle from the axis $y_{i}$ to the axis y counterclockwise. $\left\{ {x_{i} ,y_{i} ,\sigma_{i,xx} ,\sigma_{i,yy} ,\tau_{i,xy} } \right\}^{T}$ is the coordinates and stress of any point in the semi-plane with the top surface $i_{1} ii_{2}$ under the local system $x_{i} o_{i} y_{i}$, and $\left\{ {x,y,\overline{\sigma }_{xx} ,\overline{\sigma }_{yy} ,\overline{\tau }_{xy} } \right\}$ is the coordinates and stress of any point around the hole under the global coordinate system $xoy$.

2.3 Numerical model and simulation scheme

Based on ABAQUS, a cube model with size of 500 m × 500 m × 500 m was established in this paper to characterize large-scale coal rock and surrounding rock. The material properties were all Mohr Coulomb model, and the parameters of the model were shown in Table 1. Slot with different sizes parallel to the strike of the coal seam is prefabricated in the center of coal rock. The model and the characterization meaning of each parameter are shown in Fig. 8. Where, $d_{l}$ is the length of the slot, $d_{s}$ is the length of the working face, $\alpha$ is the angle between the slot and the maximum in-situ stress, $\mu$ is the friction coefficient of the seam surface, and $\Delta \sigma = \sigma_{H} - \sigma_{h}$ is the in-situ stress difference. Table 2 provides a list of values for each parameter. The three-dimensional ground stress ($\sigma_{h} = 10~{\text{MPa}}$, $\sigma_{H} = \sigma_{h} + \Delta \sigma$) is applied, and the stress distribution is described by the pressure relief range and the maximum pressure relief amplitude.

Table 1 Material parameters of coal and surrounding rock models

Fig. 8

Model and parameter schematic diagram

Table 2 Calculation scheme of each parameter

This paper will contain different slot length, length of working face, different angle, different seam surface friction coefficient, and different in-situ stress difference in large scale model of coal and rock into the three-dimensional in-situ stress numerical simulation test, based on the theory in joint 2.2, to determine the rule of how slot and geological parameters effect on the stress distribution of diamond wire saw cutting beading unloading in low permeability coal seam.

In this paper, the pressure relief amplitude is defined as the ratio of the effective stress difference before and after the slit to the effective stress before the slit (Eq. 22). In order to facilitate the description, the effective stress values of the model were monitored in three directions which are parallel to the working face, perpendicular to the coal seam and the advance of the working face, with the monitoring range of 0–250 m and the interval of 12.5 m. This article specifies that the zone with a pressure discharge amplitude of at least 10% (effective stress value less than 0.9 $\Delta \sigma$) is selected when the pressure discharge range is described. The maximum pressure relief amplitude of all monitoring points in three directions is selected, which is called the maximum pressure relief amplitude under the corresponding condition.

$$pressure\;relief\;amplitude = \frac{effective\;stress\;before\;slit - effective\;stress\;after\;slit}{{effective\;stress\;before\;slit}}$$

(22)

3 Influence of diamond bead slot parameters and in-situ stress on stress distribution of coal

3.1 The basic law of stress distribution in slotted coal body

Before analyzing the basic law of coal seam stress distribution, the pressure relief mechanism of diamond bead seam is analyzed. Contrast not slot and slotting seam stress distribution of coal seam (Fig. 9), the results show that the displacement on both sides of the rock mass are increased after the slot in the parallel direction of working face, vertical direction of coal seam and seam working face advancing direction, related to that of not slot situation in corresponding area, the maximum displacement to place in the center of the slot is 9.4 mm, and displacement occurs mainly in the working face advancing direction, The influence range of the displacement zone formed by the slit exceeds 100 m in this direction.

Fig. 9

Displacement comparison between slot and non-slot seams

The relationship between displacement and stress distribution is further analyzed. Taking vertical coal seam direction as an example (Fig. 10), the displacement field of un-slit coal remains close to 0 mm, and the corresponding effective stress remains at 5 MPa. In slotted coal, within the range of 0–20 m from the center of slotted seam, the decrease trend is fast, and the corresponding effective stress increases rapidly; within 20–100 m, the displacement decreases slowly, and the corresponding effective stress increases slowly; after 100 m, the displacement changes gradually become gentle, and the corresponding effective stress changes slowly. Therefore, it can be inferred that when the variation trend of the displacement field is opposite to that of the effective stress field, the effective stress can be inversely represented by the displacement field. The effective stress value is smaller when the displacement field is larger, which is the case when the pressure relief amplitude is larger, and vice versa.

Fig. 10

The relationship between displacement field and stress distribution

Figure 11 is a schematic diagram of the distribution law of the coal stress field. Depending on the increase and decrease of stresses in slotted coal relative to that in un-slotted coal, the stress concentration zone and stress release zone are defined. It can be seen from Fig. 11 that the coal seam is squeezed by in-situ stress, rock mass on both sides of the seam surface is relatively uncoordinated, and the stress inside the rock mass is transferred, forming the stress concentration zone and stress release zone with phase distribution. The stress concentration area is mainly distributed at the tip of the fracture and the distal end of the middle part of the fracture surface, which is distributed along the axis I and II of the concentration area. The stress release zone is distributed along the fracture surface in a spindle shape, and the overall distribution is along the I and II axes of the pressure relief zone. The angle between lines I and II in the zone of concentration is essentially equal to the angle between axes I and II in the relief zone.

Fig. 11

Distribution of stress concentration zone and stress relief zone in slotted coal

The spatial distribution of stress is shown in Fig. 12. It can be seen from the Figure that the stress concentration zone and the stress release zone are cross-distributed in the shape of "X", forming the stress envelope surface in the shape of "star peach" in space, which is obviously different from the stress envelope distribution of the wide crack mentioned in Sect. 2.1.

Fig. 12

Spatial distribution of stress envelope surface

Figure 13 shows the mechanism of diamond bead slit pressure relief. The blue area and red area in the left are the stress release area and the stress concentration area respectively. The blue arrows indicat the downward displacement, the red arrows indicate the upward displacement, and the green arrows indicate the direction of the principal stress. The formation of slit surface destroys the integrity of the coal body, resulting in relative dislocation of the coal body along the slit surface under the action of crustal stress. It can be seen that the displacement around the stress release zone (blue) remains in the same direction, while the displacement direction in the stress concentration zone (red) is opposite. The figure on the right shows the formation mechanism of stress concentration zone and stress release zone at the slit tip. The orange and green spheres represent the particles in the stress concentration zone and stress release zone respectively, the blue spheres represent the surrounding particles, and the arrows represent the displacement of each particle. Due to the downward extrusion of the coal body on the top of the seam, a shunt is formed at the tip of the seam. One part moves to the left and down along the direction of the seam groove, and the other part moves to the right and down due to the extrusion of the coal body under the seam groove. The orange particles above the slot tip are squeezed to form a local stress concentration, while the green particles below the slot tip are dragged by the blue particles in the lower right direction to form a pressure relief zone.

Fig. 13

Mechanism of coal stress redistribution caused by cutting seam with diamond beaded rope saw

3.2 Influence of Angle between slit and maximum in-situ stress direction on coal stress distribution

The stress distribution of the coal seam is affected by the angle between the slit and the maximum in-situ stress direction, as shown in Fig. 14. As can be seen from the Figure, when the included angle is between 0° and 45°, the effective pressure relief range increases gradually in the direction of parallel working face, vertical coal seam direction and working face advance direction, and the maximum pressure relief range also increases gradually. When the included angle is between 45° and 90°, the effective pressure relief range of the three directions decreases gradually, and the maximum discharge range also decreases progressivelly. Taking the data of 50 m seam length, 200 m working face length, zero friction coefficient of seam surface and 5 MPa ground stress difference as an example, when α = 45°, pressure relief range of 10% relief rate in the excavation direction of working face can reach 100 m, and the maximum pressure relief range can reach 50%. On the whole, the effective pressure relief range and maximum pressure relief range in three directions show a "single peak" state, which increases first and then decreases, and is basically symmetric with respect to α = 45°.

Fig. 14

Influence of Angle between slot and maximum in-situ stress direction on stress distribution of coal seam

As can be seen from Fig. 15, the stress concentration zone and stress release zone rotate in a "windmill" shape with angle α changes, and the stress distribution is almost uniform when α = 0° or 90°, which is consistent with the theoretical result in Sect. 2.2.

Fig. 15

Stress distribution of coal seam at Angle between different slot and maximum in-situ stress direction

3.3 Impact of slit length on coal stress distribution

The working face length is 200 m, the coal seam inclination angle is 20°, the friction coefficient of the seam surface is zero and the ground stress difference is 5 MPa are selected for analysis. Other working conditions can be analogous to this working condition. The influence rule of slit length on stress distribution of coal seam is shown in Fig. 16.With the increase of slit length, the pressure relief range corresponding to 10% pressure relief amplitude in the direction of parallel working face and vertical coal seam has little change and is stable at about 16 m. The pressure relief range of 10% in the advancing direction of the working face gradually increases, and its value approximately remains half of the length of the slit. In Fig. 16, it is shown that the maximum pressure relief amplitude remains within the limits of 17% despite changes in slit length.

Fig. 16

Effect of slit length on stress distribution of coal seam

Figure 17 shows the effect of increasing slit length on stress distribution in the advancing direction of the working face. With $d_{l}$ increase, the affected range of stress distribution along the advancing direction of the working face increases. While $d_{l} = 150~{\text{m}}$, The stress distribution at 150 m advance direction of working face is almost not affected. While $d_{l} = 225~{\text{ m}}$, the stress is obviously redistributed in the place at 150 m.

Fig. 17

Stress distribution of coal seam with different slot lengths

3.4 Influence of working face length on stress distribution of slotted coal body

Figure 18 illustrates the influence of working face length on the stress allocation in the coal seam. It can be seen from the figure that the main influence directions of working face length on the stress distribution of coal body are parallel working face direction and vertical coal seam direction, and the influence on the advancing direction of working face and the maximum pressure relief amplitude is very small. Taking the group data of 50 m seam length, 20° coal seam dip angle, zero seam friction coefficient and 5 MPa ground stress difference as an example, with the increase of working face length, the pressure relief range corresponding to 10% pressure relief amplitude in the direction of parallel working face increases from 15 to 85 m, and the pressure relief range corresponding to the traveling pressure relief in the direction of vertical coal seam increases from 11 to 50 m. The effective pressure discharge range in the forward direction of the working face is virtually unchanged, stable at approximately 95 m. With the increase of $d_{s}$, the maximum pressure relief range is slightly reduced. When $d_{s} = 50{\text{ m}}$, the maximum pressure relief amplitude is 48%. This value is 45% while $d_{s} = 200{\text{ m}}$. Therefore, it can be considered that the influence of working front length on the maximum pressure relief magnitude can be ignored.

Fig. 18

Effect of working face width on stress distribution of slotted coal seam

The impact of changing working face length is directly reflected by the stress distribution, as demonstrated in Fig. 19. The difference of stress redistribution is mainly reflected in the plane composed of parallel working face and vertical coal seam direction.

Fig. 19

Stress distribution of coal seam with different working face widths

3.5 Influence of seam friction coefficient on coal stress distribution

The friction coefficient generally reduces when the seam surface is smoother and the f value of the coal seam is higher, as the seam surface becomes smoother. The size of friction coefficient has a direct impact on the difficulty of sliding on both sides of the coal seam. According to the characterization relationship between displacement field and effective stress in 3.1, the larger the friction coefficient of fracture surface is, the more unfavorable the pressure relief of coal and rock mass is. The stress redistribution range expands as the friction coefficient decreases, as demonstrated by Fig. 20, which suggests that the greater the friction coefficient, the more stress redistribution there is. It can be seen from Fig. 21 that the influence of friction coefficient on the stress distribution of coal body is mainly reflected in the vertical coal seam and the advancing direction of the working face. The 10% pressure relief range of the two directions decreases slightly with the increase of friction coefficient μ, and the maximum pressure relief range also decreases with the increase of friction coefficient. Taking the group data of 50 m slit length, 200 m working face length, 20° coal seam inclination angle and 5 MPa ground stress difference as an example, when μ increases between 0 and 0.1, the pressure relief range of 10% vertical working face direction decreases from 16 to 13 m, and the advancing direction of working face decreases from 97 to 95 m. When μ is greater than 0.1, the pressure relief amplitude in both directions is less than 10%. When μ increases from 0 to 0.13, the maximum pressure relief amplitude decreases from 82 to 8%. Therefore, it can be inferred that the smoother the seam surface and the harder the coal, the better the pressure relief effect of the seam is.

Fig. 20

Stress distribution of coal seam with different friction coefficient of seam surface

Fig. 21

Influence of seam friction coefficient on stress distribution of coal seam

3.6 Influence of in-situ stress difference on coal stress distribution

The in situ stress difference is the major dynamic source of relative dislocation of the coal seam. The greater the in situ stress difference, the greater the vertical distortion of the coal layer. However, the dislocation displacement also depends on the angle between the slit and the maximum in-situ stress. When the coal seam dip angle is 20° ($\alpha = 70^\circ$), $\Delta \sigma$ change between 0 and 10 MPa has little effect on the increase of pressure relief range; When $\Delta \sigma$ exceeds 10 MPa, the pressure relief range of 10% in the direction of parallel working face increases slightly. The 10% pressure relief range and the maximum pressure relief amplitude in the other two directions have little change, and the maximum pressure relief amplitude is stable at about 50% (Fig. 22). Therefore, for the seam where the angle between the slit and the maximum in-situ stress is greater than 70°, the in-situ stress difference has no obvious influence on the maximum pressure relief amplitude. However, when the angle between the slot and the maximum in-situ stress is small, as shown in Fig. 23 (α = 45°), with the increase of $\Delta \sigma$, the relative displacement on both sides of the slot surface increases, and the pressure relief range and maximum pressure relief amplitude in each direction increase significantly.

Fig. 22

Influence of ground stress difference on stress distribution of coal seam

Fig. 23

Stress distribution of coal seam with different geo-stress difference

4 Discussion

As a new technology for relieving pressure and enhancing reflection of low permeability coal seam, the engineering practicability of diamond beaded rope saw is still in the demonstration stage. The results of previous studies are compared with ours. Li et al. (2021), using the test device shown in Fig. 24c, studied the displacement and pressure relief and reflection improvement effects of coal seam under diamond beaded rope saw. The monitoring results showed that the displacement value decreased with the increase of the monitoring distance (Fig. 24a), which was consistent with the analysis rule of the calculated results in this paper. In addition, we also refer to the literature on the use of numerical simulation methods to study traditional hydraulic slit pressure relief, Wu et al. (2009a, b) studied the stress distribution of hydraulically slotted coal seams at different angles, as shown in Fig. 24b. The figure shows that the stress distribution extends outward in an obvious ellipsoidal shape, and local stress concentration is formed at the slit tip, and the influence of slit angle on the stress distribution is similar to that analyzed in this paper (Fig. 11). Similar conclusions are also reflected in Lin et al.'s paper (2010), indicating that the conclusions obtained by using such models are widely recognized in the academic community.

Fig. 24

Comparison with previous research results

However, there are still some problems to be solved in our research. For example, the existing model cannot be used to calculate the crushing effect of diamond beaded rope saw on coal and rock in the process of cutting seams. In reality, the coal body fracture network caused by rope saw cutting plays a non-negligible role in coal seam gas unloading (Fig. 24d). This point is particularly important for the interpretation of the pressure relief mechanism of 0° inclination and 90° inclination slot. In the future, it can be combined with discrete element simulation to simulate the fracture characteristics under the cutting load of coal and rock, which is helpful to further deepen the understanding of the cutting slit relief technology of diamond beaded rope saw.

5 Conclusion

• 1. Tangential displacement on both sides of coal body is the basic mechanism of stress transfer of rock mass after diamond bead seam cutting in low permeability coal seam. The displacement of the slotted coal body is larger than that of un-slotted coal body in the parallel working face, vertical coal seam and advance direction of the working face. The displacement field of coal and rock mass is correlated with the stress field: the variation trend of displacement field is basically opposite to that of the effective stress field. The effective stress value decreases as the displacement field increases, and vice versa.

• 2. The action law of the angle between the slit and the direction of maximum in-situ stress on the stress distribution of coal body is as follows: when α varies from 0° to 90°, the pressure relief range and the maximum pressure relief rate in the direction of parallel working face, vertical coal seam and advance direction of working face show a "single peak" state, which increases first and then decreases, and reaches the peak value when α = 45°, and is basically symmetric with respect to α = 45°.

• 3. The influence of slit length on the stress distribution of coal body mainly affects the advancing direction of the working face, and the influence of working face length is mainly reflected in the direction of parallel working face and vertical coal seam. The effective pressure relief range in the corresponding direction increases with the increase of slit length and working face length. Both have little effect on the maximum pressure relief amplitude.

• 4. The friction coefficient of fracture surface and the difference of in-situ stress determine the ease of coal and rock sliding on both sides of the fracture surface and the magnitude of sliding load, and then affect the tangential displacement and stress field distribution. When the friction coefficient of seam surface increases, the pressure relief range and maximum pressure relief range of vertical coal seam direction and working face advance direction decrease, while the pressure relief range of parallel working face direction does not change obviously. The effect of stress difference on pressure relief range and maximum pressure relief amplitude is opposite, and the influence of the Angle between the slit and the direction of maximum stress is greater, the smaller α is, the more obvious influence is.

• 5. It can be known from the above conclusion, the closer the slot and in-situ stress angle is to 45°, the longer the slot length and the length of working face are, the smoother the cutting surface is, the greater the in-situ stress difference is, the better the effect of pressure relief is, but in the practical engineering of coal seam dip angle, the length of working face, the ground stress size factors such as human intervention is difficult, so that in a certain coal quality, the working face length, coal seam dip angle, and under stress, the measures such as increasing cutting length, decreasing bead diameter, increasing bead density and speeding up the speed of rope saw can effectively improve the relief effect of coal seam.

Appendix: Analytical expression and numerical calculation method of plane stress distribution with infinite crack

The method of deriving the internal force of plane crack problem is given, the difficulty of analytical solution is explained, and the numerical solution of the method is given.

Taking dip angle $\theta$ as an example, the stress expression of point $(x_{0} ,y_{0} )$ with local coordinate of edge $B_{1}$ is derived. As shown in Fig. 25, the elastic infinite plane contains quadrilateral holes with free boundaries inside the holes and is subjected to uniform stress $\left\{ {\sigma_{x}^{\infty } ,\sigma_{y}^{\infty } } \right\}^{T}$ at infinite distances. According to the principle of stress superposition, the problem shown in Fig. 25a can be formed by superposition of Fig. 25b, c, that is, an infinite plane without holes is subjected to far-field stress, and the stress distribution in the plane is uniform. The other is that the infinite plane contains the hole and the far field stress is zero due to the reaction force generated by the rebound around the hole. For (c) the force $F_{i}^{0}$ on the inner boundary at this time satisfies:

$$F_{i}^{0} = - t_{i}^{\infty }$$

(23)

where $t_{i}^{\infty }$ is the excess surface force caused by the inner boundary of the far field stress, the calculation formula is as follows:

$$t_{i}^{\infty n} = - \sigma_{xx}^{\infty } \sin^{2} \theta_{i} - \sigma_{yy}^{\infty } \cos^{2} \theta_{i} + 2\tau_{xy}^{\infty } \cos \theta_{i} \sin \theta_{i}$$

(24)

$$t_{i}^{0s} = \left( {\sigma_{xx}^{\infty } - \sigma_{yy}^{\infty } } \right)\cos \theta_{i} \sin \theta_{i} + \tau_{xy}^{\infty } \left( {\sin^{2} \theta_{i} - \cos^{2} \theta_{i} } \right)$$

(25)

where $t_{i}^{\infty n}$ and $t_{i}^{\infty s}$ are the normal and tangential components of $t_{i}^{\infty }$ respectively, as the initial values of the load on the boundary A. At this time, the load situation on the half plane A₁AA₂ is shown in Fig. 26. At this time, $T_{A1}^{n} ,T_{A1}^{s} ,T_{A2}^{n}$ and $T_{A2}^{s}$ are unknown quantities. Assuming initial $T_{A1}^{n} = T_{A1}^{s} = T_{A2}^{n} = T_{A2}^{s} = 0$, it can be obtained according to Eq. (26):

$$\begin{aligned} \sigma_{A,\eta \nu } (x_{0} ,y_{0} ) &= \int_{{A_{1} }} {\left[ {H_{\eta \nu }^{n} \left( {x_{0} - \xi_{A1} ,y_{0} } \right)T_{A1}^{n} + H_{\eta \nu }^{s} \left( {x_{0} - \xi_{A1} ,y_{0} } \right)T_{A1}^{s} } \right]} d\xi_{A1} \hfill \\ &\quad + \int_{{A_{2} }} {\left[ {H_{\eta \nu }^{n} \left( {x_{0} - \xi_{A2} ,y_{0} } \right)T_{A2}^{n} + H_{\eta \nu }^{s} \left( {x_{0} - \xi_{A2} ,y_{0} } \right)T_{A2}^{s} } \right]} d\xi_{A2} \hfill \\ &\quad + \int_{A} {\left[ {H_{\eta \nu }^{n} \left( {x_{0} - \xi_{A} ,y_{0} } \right)F_{A}^{0n} + H_{\eta \nu }^{s} \left( {x_{0} - \xi_{A} ,y_{0} } \right)F_{A}^{0s} } \right]} d\xi_{A} \hfill \\ &= \int_{a}^{b} {\left[ {H_{\eta \nu }^{n} \left( {x_{0} - \xi_{A} ,y_{0} } \right)t_{A}^{\infty n} + H_{\eta \nu }^{s} \left( {x_{0} - \xi_{A} ,y_{0} } \right)t_{A}^{\infty s} } \right]} d\xi_{A} \hfill \\ \end{aligned}$$

(26)

enter the expressions of $H_{\eta \nu }^{n}$ and $H_{\eta \nu }^{s} (\eta ,\nu = x,y)$ respectively to find $\sigma_{xx} (x_{0} ,y_{0} )$, $\sigma_{yy} (x_{0} ,y_{0} )$ and $\tau_{xy} (x_{0} ,y_{0} )$, and convert them into the surface force on the extended top surface B₁ on one side of the adjacent half plane according to the coordinate transformation formula:

$$T_{B1}^{n} = - \sigma_{yy} \cos^{2} \alpha_{B1} - \sigma_{xx} \sin^{2} \alpha_{B1} + 2\tau_{xy} \cos \alpha_{B1} \sin \alpha_{B1}$$

(27)

$$T_{B1}^{s} = \left( {\sigma_{xx} - \sigma_{yy} } \right)\cos \alpha_{B1} \sin \alpha_{B1} + \tau_{xy} \left( {\sin^{2} \alpha_{B1} - \cos^{2} \alpha_{B1} } \right)$$

(28)

where $\alpha_{B1}$ is the angle between side B₁ and the local coordinate system $x_{A}$. Since side B₁ is perpendicular to A, $\alpha_{B1} = 90^\circ - \theta_{A}$ is satisfied.

Fig. 25

Decomposition of stress superposition principle for plane hole problem

Fig. 26

Half plane partition and load distribution

The problem is that there are many points on B₁, and it is difficult to get an analytical solution to Eq. (26) due to the complexity of surface force distribution. In the actual solution process, only numerical integration method can be used to solve the problem. For the extended plane (A₁, A₂), m + 1 configuration points (divided into m segments) are inserted according to the geometric series. Usually, the corner point is the first configuration point ($j = 1$), and the distance between the following points and the first point is inserted according to Eq. (29):

$$d_{j} = d_{j - 1} + eq^{j - 1} ,d_{1} = 0 \, j = 2,3, \ldots ,m + 1$$

(29)

for the inner surface (A), due to its dense internal force distribution, insert it into a configuration point m₀ + 1 according to the cosine function, and the coordinates of the configuration point in the $x_{i}$ direction are:

$$x_{A,j} = \frac{{a_{A} }}{2}\cos \frac{{\left( {j - m_{0} - 1} \right)\pi }}{{m_{0} }} \, j = 1,2, \ldots ,m_{0} + 1$$

(30)

where $a_{A}$ is the length of the inner boundary A. Gaussian integral of order n is adopted. For the extended plane and the inner boundary with $N = mn$ and $N_{0} = m_{0} n$ Gaussian points, respectively, the Eq. (31) discretes the integral into summation form:

$$\begin{aligned} \sigma_{A,\eta \nu } (x,y) &= \sum\limits_{\beta = 1}^{N} {\left[ {\Lambda_{\eta \nu }^{n} \left( {x - \xi_{A1,\beta } ,y} \right)T_{A1}^{n} \left( {\xi_{A1,\beta } } \right) + \Lambda_{\eta \nu }^{s} \left( {x - \xi_{A1,\beta } ,y} \right)T_{A1}^{s} \left( {\xi_{A1,\beta } } \right)} \right]} \hfill \\ &\quad + \sum\limits_{\beta = 1}^{N} {\left[ {\Lambda_{\eta \nu }^{n} \left( {x - \xi_{A2,\beta } ,y} \right)T_{A2}^{n} \left( {\xi_{A2,\beta } } \right) + \Lambda_{\eta \nu }^{s} \left( {x - \xi_{A2,\beta } ,y} \right)T_{A2}^{s} \left( {\xi_{A2,\beta } } \right)} \right]} \hfill \\ &\quad + \sum\limits_{\alpha = 1}^{{N_{0} }} {\left[ {\Lambda_{\eta \nu }^{n} \left( {x - \xi_{A,\alpha } ,y} \right)F_{A}^{0n} \left( {\xi_{A,\alpha } } \right) + \Lambda_{\eta \nu }^{s} \left( {x - \xi_{A,\alpha } ,y} \right)F_{A}^{0s} \left( {\xi_{A,\alpha } } \right)} \right]} \hfill \\ \end{aligned}$$

(31)

where $\xi_{A,\alpha }$ and $\xi_{A\delta ,\beta }$ are the coordinates of Gauss points on the inner boundary and the extended plane in the local coordinate system respectively, that is:

$$\xi_{A\delta ,\beta } = \frac{{\left( {\xi_{A\delta ,j} - \xi_{A\delta ,j - 1} } \right)}}{2}\tau_{k} + \frac{{\left( {\xi_{A\delta ,j} + \xi_{i\delta ,j - 1} } \right)}}{2}$$

(32)

$$\xi_{A,\alpha } = \frac{{\left( {\xi_{A,j} - \xi_{A,j - 1} } \right)}}{2}\tau_{k} + \frac{{\left( {\xi_{A,j} + \xi_{A,j - 1} } \right)}}{2}$$

(33)

$$\delta = 1,2$$

$\Lambda_{\eta \nu }^{n}$ and $\Lambda_{\eta \nu }^{s}$ are stress influence coefficients:

$$\Lambda_{\eta \nu }^{n(s)} \left( {x - \xi_{A\delta ,\beta } ,y} \right) = \frac{{\left( {\xi_{A\delta ,j} - \xi_{A\delta ,j - 1} } \right)}}{2}A_{k} H_{\eta \nu }^{n(s)} \left( {x - \xi_{A\delta ,\beta } ,y} \right)$$

(34)

$$\Lambda_{\eta \nu }^{n(s)} \left( {x - \xi_{A,\alpha } ,y} \right) = \frac{{\left( {\xi_{A,j} - \xi_{A,j - 1} } \right)}}{2}A_{k} H_{\eta \nu }^{n(s)} \left( {x - \xi_{A,\alpha } ,y} \right)$$

(35)

$$k = 1,2, \ldots ,n$$

where $\xi_{A,j}$ and $\xi_{A\delta ,j}$ are the coordinates of the configured points on the inner boundary and extension plane in the local coordinate system, respectively. $\tau_{k}$ and $A_{k}$ are the nodes and weight coefficients of Gaussian points respectively. The solution of the final problem is based on the principle of stress superposition, which can be obtained:

$$\sigma_{ij} = \overline{\sigma }_{ij} + \sigma_{ij}^{\infty } \quad (i,j = x,y)$$

(36)

where $\sigma_{ij}$ is the solution of Fig. 26a, $\sigma_{ij}^{\infty }$ and $\overline{\sigma }_{ij}$ are the solutions of Fig. 26b, c, respectively.