Complex function solution for deformation and failure mechanism of inclined coal seam roadway

The stressed environment of the inclined coal seam roadway is complex and changeable, and the damage degree of surrounding rock increases, threatening the safe mining of coal mines. In order to take targeted support measures to control the stability of roadway surrounding rock, it is very important to study the stress and deformation characteristics of roadway surrounding rock in inclined coal seam. Therefore, this paper analyzes the deformation and failure law of inclined coal seam roadway according to the theory of complex variable function. It optimizes the solution process and accuracy of the mapping function coefficient and deduces the analytical solution of surrounding rock stress and deformation inclined coal seam roadway. The deformation and failure mechanism of surrounding rock in inclined coal seam roadway is revealed theoretically, and further use numerical simulation and physical simulation tests for supplementary analysis and verification. The results show that the stress and deformation of roadway surrounding rock in inclined coal seam show obvious asymmetric distribution characteristics. The stress and deformation of roadway surrounding rock on the right side are greater than on the left side. The two sides of the roadway, the right side of the roof and the roof angle of the right side, are the key positions of roadway stress concentration and deformation. According to the variation law of stress and deformation distribution of roadway surrounding rock, roadway cyclic deformation and failure theory is put forward. The numerical simulation and physical simulation tests show that the deformation and failure law of roadway is consistent with the theoretical analysis results, and there are differences in numerical values. The cyclic deformation and failure mechanism of roadway in inclined coal seam is verified, which can provide theoretical guidance for roadway support design.

The reserves of inclined coal seams account for more than 35% of the total coal resources in China, and the coal quality is excellent, and the mining value is large. Under the joint influence of dip angle and various geological conditions, the surrounding rock of the inclined coal seam roadway has asymmetric deformation and failure, making the roadway's stability difficult to control and restricting the safe and efficient mining of inclined coal resources [1][2][3][4][5][6] . Therefore, the study on the deformation and failure mechanism of inclined coal seam roadway is of great significance to the design of the roadway support scheme.
The stress and deformation characteristics of roadway surrounding rock in inclined coal seam are an important basis for roadway support design. Most previous studies have simplified the roadway surrounding rock into beam, plate or arch structure for separate analysis. Rarely analyze the roof, floor and two sides of roadway surrounding rock as a whole. It is difficult to obtain the analytical solution of stress and deformation of roadway surrounding rock in inclined coal seam, so it is impossible to establish the essential relationship between roadway roof, floor and two sides of deformation. Due to the complex section of roadway, the complex function theory provides a new idea for this. The stress-deformation analysis of surrounding rock of roadway with arbitrary section can be reduced to the stress-deformation distribution problem of infinite plane with complex orifice in elasticity, which can be solved according to the complex function method.
Based on the complex function and conformal transformation theory, the complex orifice boundary can be mapped to the unit circle boundary to solve the plane problem in elasticity. The unit circle boundary corresponds www.nature.com/scientificreports/ where σ ρ、 σ θ and τ ρθ are the radial, circumferential and tangential stresses at the points (ρ, θ) of the polar coordinate system, u ρ and u θ are the projections of the displacement vector on ρ and θ, E is the elastic modulus, v is poisson's ratio, and the sum of the x and y axis directions on the boundary s is x and y . ω(ζ) is the mapping function, and Φ(ζ), Ψ(ζ), φ(ζ), ψ(ζ) are complex potential analytical functions.
The equations of φ(ζ) and ψ(ζ) on the ζ plane are: where η is the main stress direction, a n and b n are proportional constants; φ 0 (ζ) and ψ 0 (ζ) are the analytic functions of the complex; function ζ = ρe iθ in the central unit circle; ρ is the distance from any point to the center of the circle, and θ is the angle between any point and the center of the circle in the positive x direction; α, α′, and β′ are constants related to the far field stress σ 1 and σ 2 .
According to the stress boundary condition Eq. (3), the basic equations of φ 0 (ζ) and ψ 0 (ζ) can be obtained: ϕ 0 (ζ ) = ∞ � n=1 a n ζ −n , ψ 0 (ζ ) = ∞ � n=1 b n ζ −n , # 4 C o a l S e a m M u d s to n e M e d iu m S a n d s to n e P a c k s a n d S il ts to n e ¦Á a b P 23¡ã S il ts to n e S il ts to n e P a c k s a n d M u d s to n e # 5 C o a l S e a m www.nature.com/scientificreports/

Solution of mapping function coefficients. Theory of mapping function.
Assuming that Z is a bounded connected region, its complement Z C is simply connected on the extended plane. The conformal mapping function Φ can be used to map Z C to the outer ζ plane of the unit circle, where the Laurent expansion of z is a faber polynomial, and the faber polynomial can be calculated through the recursive relationship f I function to determine the coefficients of the mapping function.
(1) Schwarz-Christoffel mapping function The arbitrary conformal mapping functions from the unit circle to the inside and outside of the bounded polygon P are f I (z) and f ' I (z): where A and C 1 are constants.
(2) Coefficients of faber polynomials Assuming that the complement Z C of Z is simply connected on the extended plane (on the Riemannian sphere). According to Riemannian mapping theorem, there is a conformal mapping from Z C to the outside of the unit circle Φ(z), so Φ(∞) = ∞, then Φ Laurent expansion: where |C1|> 0, the horizontal curve of Z is ϕ − 1. For a circle with radius ρ > 1, the mapping relationsip is shown in Fig. 4.
Assuming ζ = Φ(z), u = 1/ζ, f is the Schwarz-Christoffel mapping function: Then the inverse mapping of the Laurent expansion of Φ is: www.nature.com/scientificreports/ By deriving the inverse mapping of Laurent expansion and combining binomial theorem, the recursive formula of faber polynomial is obtained: The recurrence formula of the Faber polynomial is programmed through Matlab software, and the coefficient of the mapping function is calculated up to the nth order. The mapping effect of the mapping function of different orders is shown in Fig. 5. As n increases, the average absolute error of the mapping function decreases. When n = 9, the mapping figure represented by the mapping function is very similar to the roadway section.
Calculation of mapping function coefficients. The inclined coal seam roadway boundary is mapped to the unit circular boundary in the ζ plane through conformal transformation to solve the stress and deformation of the surrounding rock, that is, the unit circle radius ρ = 1, as shown in Fig. 6.
According to Laurent expansion and complex function theory, combined with the characteristics of inclined coal seam roadway (as shown in Fig. 6), the basic type of mapping function is further determined: where C j (j = 0,1,2,3,…,n) is a complex constant, which is determined by the size and boundary shape of the roadway, and n is the number of terms of the series, so that C j = a j + id j , z = x + iy, ζ = ρ(cosθ + isinθ) = ρe iθ , ρ = 1.
According to the Schwarz-Christoffel mapping function, combined with the recursive formula of the Faber polynomial, the calculation process of the coefficients in the mapping function is programmed through Matlab software to solve the coefficients of the roadway mapping function. In the calculation process, 1000 points are taken on the roadway section for mapping. When n is 9, the approximate roadway section obtained by the   www.nature.com/scientificreports/ mapping is the same as the actual roadway section. The coefficients of the mapping function of the inclined coal seam roadway are shown in Table 1.
Example analysis of roadway. Roadway calculation azimuth layout. The inclination angle of the inclined coal seam in Shitanjing No. 2 mining area is 18°-27°, with an average of 23°. Take the overlying rock layer pressure (in-situ stress) P as 10 MPa, assuming that the rock layer is homogeneous, its elastic modulus is 3.5 GPa, and Poisson's ratio is 0.24. Substituting the coefficients of the mapping function in Table 1 into Eqs. (26) and (27) respectively, the surface hoop stress and radial displacement of the roadway can be obtained. The layout of the calculation points for the stress and displacement of the roadway surrounding rock is shown in Fig. 7.
Stress distribution characteristics of roadway surrounding rock. Figure 8 shows the stress distribution of roadway surrounding rock. It can be seen from the figure that the peak stress on the right side of the roadway (sharp corner, two sides, roof, and floor) is greater than that on the left, and the peak stress at the two sharp corners of the roadway roof is greater than that at the two sharp corners of the floor. Overall, the stress of roadway surrounding rock shows the changing trend of sharp angle > two sides > roof > floor. The maximum stresses of roadway right side roof angle, right side, left side, roof, and floor are 22 Deformation distribution characteristics of roadway surrounding rock. Figure 9 shows the displacement distribution of roadway surrounding rock. It can be seen from the figure that the deformation of roadway surrounding (30)  www.nature.com/scientificreports/ rock presents asymmetric distribution characteristics, that is, the deformation on the right side of roadway sharp corner, two sides and roof, and the floor is greater than that on the left. On the whole, the deformation at the top corner of the right side of the roadway is the largest, followed by the two sides of the roadway, and the smallest is the roadway roof and floor. The displacements of the top corner of the right side of the roadway, the left side, the right side, the roof, and the floor are 161.0 mm, 112.0 mm, 121.8 mm, 114.4 mm, and 52.5 mm, respectively.
Theory of cyclic deformation and failure of roadway surrounding rock. Shape of roadway stress and deformation zone. The contour of stress and deformation of surrounding rock of roadway in inclined coal seam presents a butterfly shape, and the protruding part is called butterfly leaf. As shown in Figs. 10, 11 and 12, in the coordinate system, it presents the shape of mutual four quadrant protruding and coordinate axi + s depression, in which r D is the length of butterfly leaf, (r x ,r y ) is the coordinate of any point on the stress area and deformation boundary of roadway surrounding rock, with |r D | >|r y | >|r x |.
Cyclic deformation and failure characteristics of roadway surrounding rock. As shown in Fig. 13, the stress concentration occurs at the butterfly leaf position in the stress area of the roadway surrounding rock, and the two sharp corners of the roadway roof are damaged. With the increase of the butterfly leaf in the stress area, the damage of the two sharp corners of the roof is intensified, increasing the roof span. It increases the stress of the two sides of the roadway, resulting in the bulging of the side of the roadway. With the continuous increase of stress, butterfly leaves in the stress area are generated at the two sharp corners of the roadway floor, and a slight floor heave occurs. Then the roof, floor, two sides, and sharp corners of the roadway surrounding rock interact,     www.nature.com/scientificreports/ Put z = x + iy, ζ = ρ(cosθ + isinθ) = ρe iθ , C j = a j + id j , ρ = into Eq. (15), we can get:: where L AnBn , L BnCn , L CnDn and L DnAn are the lengths of the roof, sides and bottom of the roadway, respectively. The angle θ is 0°-360°, which are the measuring points for the stress and deformation of the roadway. The abscissas of A, B, C, D around the roadway are x An , x Bn , x Cn , x Dn , and the ordinates are y An , y Bn , y Cn , y Dn , n = 1,2,3,4,5…. The roadway section is mapped to the unit circle through conformal transformation, and the mapping function coefficients are unchanged at this time. With the increase of θ, the values of cosθ and sinθ in the equation (2.58) change. When θ increases near kπ/2 + π/4 (k = 0, 1, 2, 3), the boundary of the stress zone and the deformation zone presents a butterfly-shaped contour. The larger position of the butterfly leaf in the butterfly stress zone indicates that the sharp corner of the roadway roof is the source of damage, which is prone to stress concentration. As the load increases, the two sides of the roadway are damaged, which leads to the increase of the roof span L AnBn and the stress of the roof. At the same time, the damage of the roof increases the length of the two sides L BnCn and L CnDn , which intensifies the damage of the two sides, resulting in the continued deterioration of the stress state of the roadway sharp corners and aggravating the roof sinking. The roof span L AnBn is further increased, and the damage of the two sides of the roadway extends to the deep part, which deteriorates the stress condition of the bottom plate, and causes the span of the bottom plate L DnAn to increase, which enters a vicious circle of destruction.  Table 2.

Result analysis. Stress distribution characteristics of roadway.
(1) Vertical stress Figure 17 shows the vertical stress distribution law of roadway in inclined coal seam. It can be seen from the figure that the vertical stress concentration peak of roadway appears in the side of roadway and www.nature.com/scientificreports/ presents asymmetric distribution characteristics, that is, the range of stress peak and stress concentration area of roadway on the right side is larger than that on the left side, and the stress distribution of roadway roof and floor deviates to the right. With the increase of load, the stress of two sides of roadway increases. When the load is 10 MPa, the peak stress of the left and right sides of the inclined coal seam roadway is 15.20 MPa and 15.73 MPa respectively.
(2) Horizontal stress Figure 18 shows the horizontal stress distribution law of roadway surrounding rock in inclined coal seam. From the figure, it can be seen that the horizontal stress concentration of roadway surrounding rock occurs at four sharp corners and presents asymmetric "Butterfly" distribution characteristics, that is, the range of stress peak and stress peak area at the sharp corner on the right side of the roadway is greater than that on the left side. With the increase of load, the stress at the sharp corner of the roadway increases. When the load is 10 MPa, the peak stress of the right side roof angle (RSRA), the left side roof angle (LSRA), the right side floor angle (RSFA) and the left side floor angle (LSFA) of the roadway are 9.57 MPa, 9.51 MPa, 9.03 MPa and 8.94 MPa respectively.

Distribution characteristics of roadway displacement.
(1) Vertical displacement Figure 19 shows the vertical displacement distribution law of roadway. It can be seen from the figure that the deformation of roadway roof and floor is skewed on the right side, showing asymmetric distribution characteristics. The right side of roadway roof and floor is the key part of roadway deformation. With  (2) Horizontal displacement Figure 20 shows the horizontal displacement nephogram of the roadway. It can be seen from the figure that the deformation of two sides of the roadway presents asymmetric characteristics, that is, the deformation of the right side of the roadway is greater than that of the left side. With the increase of load, the deformation of two sides of roadway is greater. When the load is 10 MPa, the maximum deformation of the left and right sides of the inclined coal seam roadway is 16.8 and 17.3 mm, respectively.
Distribution characteristics of roadway plastic zone. Figure 21 shows the cloud diagram of the plastic area of the roadway under different loads. It can be seen from the figure that the plastic area of the roadway is distributed in an asymmetric "Butterfly" along the inclined direction of the coal seam. Under the action of low load, the roadway plastic zone appears in the sharp corner and side of the roadway. With the increase of load, the plastic zone at the sharp corner of the roadway first expands to the roof, floor and the right side roof angle of the roadway. The interaction of various parts of the roadway leads to increased roadway stress, decreased strength, and finally asymmetric cyclic deformation and failure. www.nature.com/scientificreports/

Comparative analysis of theoretical calculation and numerical simulation results.
To verify the rationality and accuracy of theoretical calculation, the stress and displacement calculated by numerical simulation are converted into circumferential stress and radial displacement, compared with theoretical analysis and calculation results. As shown in Fig. 22, the theoretical analysis shows that the distribution law of roadway stress and displacement is consistent with the numerical simulation. There are some numerical differences, and the maximum numerical difference is less than 10%. This may be related to the calculation process of the two algorithms, in which the theoretical analysis is based on elasticity. In contrast, the numerical simulation analysis adopts the elastic-plastic constitutive model. It is also related to the size of computational boundary conditions and the difference of mesh generation of a numerical model. Therefore, there are some differences between theoretical calculation and numerical simulation results.

Experimental study on deformation and failure mechanism of roadway
Physical simulation test scheme. Taking the engineering geological conditions of Shitanjing No. 2 mining area as the background (the physical and mechanical parameters of coal seams are shown in Table 1), a variable-angle physical model test frame is used to establish a physical similarity model for inclined coal seam roadways, as shown in Fig. 23. The establishment of the similarity model meets the conditions of geometric, bulk density, material and stress similarity, and the size of the roadway model shall meet the requirement that the ratio of the distance from the roadway to the model boundary to the radius of the roadway is ≥ 3. The similarity parameters of the physical model are shown in Table 3. Similar materials are mainly composed of sand, gypsum, CaCO 3 and water in a certain proportion. The model consists of 10 layers, which are stacked in layers. The ratio of similar materials is shown in Table 4. In this study, two hydraulic jacks are used to apply uniform load on the top surface of the physical model. Starting from 0.021 Mpa, execute the graded loading method, as shown in Table 5.
Physical simulation test process. Miniature earth pressure sensor (L-YB-150) is adopted (φ 28 mm × 9 mm, 0.5 (% F.S.) and DH3818-1 static strain tester were used to record the stress changes, and DIC was used to record the displacement and failure law of roadway. Figure 24 shows the DIC test system. Its image consists of two sets with a resolution of 2648 × 2448. The sub-pixel accuracy of pixel CCD digital camera acquisition can be realized by using 3D-DIC software (GOM Aramis, version 8, related solutions). The shooting format of this experiment is 1.2 × 1.1 m, the magnification is 3.383 pixels/mm, and the DIC displacement measurement accuracy is expected to be 0.0296 mm [26][27][28][29][30][31][32][33] . Compared with traditional monitoring methods, DIC test system has short monitoring time intervals and high precision. It can realize high-precision testing of large-scale and detailed parts of rock stratum at the same time, and master the dynamic process of rock stratum deformation and failure.
Based on the deformation and failure of inclined coal seam roadway, the stress and displacement monitoring points are arranged, as shown in Fig. 25. Three stress monitoring points are arranged on the roadway roof, two side walls, floor and four sharp corners respectively. A total of 12 stress sensors are embedded in the roadway surrounding rock, and the measurement error is 0.5%. Displacement monitoring points are arranged in the influence range of roadway surrounding rock surface deformation for monitoring by DIC. At the same time, in order to eliminate the rigid body displacement, eight fixed control marks are placed on the physical model frame to correct the DIC measurement error. After the model is cured for one month to reach the expected mechanical strength and its water content meets 1.6-2.7%, the loading test can be carried out [34][35][36][37][38] . During the loading process, the stress, deformation and failure characteristics of two sides, roof, floor and sharp corners are recorded.     www.nature.com/scientificreports/

Results and interpretation. Stress analysis.
(1) Stress of roadway sidewall Figure 26 shows the stress distribution law of the roadway's two sides. It can be seen from the figure that in the loading stage of 0-0.063 MPa, the stress of two sides of the roadway increases rapidly with the increase of load, resulting in stress concentration. The stress concentration on the two sides reaches the maximum when the load is 0.063 MPa, and the stress concentration factors on the left and right sides are 2.0 and 4.1 respectively. At this time, cracks appear on two sides of the roadway, with slight wall spalling. In the loading stage of 0.063-0.112 MPa, the stress of the two sides of the roadway decreases rapidly, indicating that the roadway's two sides have been greatly damaged. There are large cracks on the inner side of the two sides of the roadway, and the two sides bulge seriously. The damage to the right side of the roadway is greater than that of the left side. The peak stress of the left and right sides of the roadway are 0.12 and 0.24 MPa, respectively.
(2) Stress of roadway roof   (1) Surface displacement of two sidewalls of roadway Figure 30 shows the displacement nephogram of each measuring point on the two sides of the roadway. It can be seen from the figure that the displacement of the right side of the roadway is greater than that of the left side, showing asymmetric characteristics, in which the maximum displacement of the right side of the roadway is 2.10 mm and the maximum displacement of the left side is 1.93 mm. The deep surrounding rock deformation trend of the two sides of the roadway is consistent with the shallow part. With the increase of the depth of the surrounding rock of the two sides of the roadway, the deformation of the two sides is smaller.
(2) Roof displacement of roadway Figure 31 shows the displacement nephogram of each measuring point of roadway roof deformation. It can be seen from the figure that the displacement on the right side of the roof is greater than that on the left, showing asymmetric distribution characteristics, in which the maximum displacement of roadway roof is 2.77 mm. The deformation trend of deep surrounding rock of roadway roof is basically consistent with that of the shallow part. With the increase of surrounding rock depth of roadway roof, the deformation of roadway roof is smaller.  Cyclic deformation and failure of roadway surrounding rock. Under the load action, the inclined coal seam roadway has cyclic deformation and failure, showing asymmetric characteristics. Stress concentration is easy to appear at the sharp corner of the roadway roof. With the increase of load, the stress concentration of roadway roof, two sides, and floor also reach the maximum. At this time, cracks appear in the roof, two sides, and floor. With the increasing load, the roof slightly separates from the layer. The stress state at the sharp corner continues to deteriorate as a connecting part, intensifying the roadway's deformation and failure. The interaction between the two sides of the roadway, the roof, and the two sharp corners of the roof leads to the increase of roadway stress, the decrease of strength, and the entry into a vicious circle of damage. Finally, the two sides of the roadway are seriously divided, the roof presents asymmetric "Beret" type caving arch damage, and the floor heaves slightly, as shown in Fig. 33.

Comparative analysis of theoretical calculation and experimental results.
To verify the rationality and accuracy of the theoretical calculation, the physical test results are converted by geometric similarity ratio and stress similarity ratio and compared with the theoretical calculation results. As shown in Fig. 34, the theoretical analysis shows that the distribution law of roadway stress and displacement is basically consistent with the physical test results. There are some differences in values, and the maximum difference is less than 32%. Because the results of theoretical analysis are elastic solutions based on complex function, while the results of numerical simulation are elastic-plastic solutions, the results of model tests are related to similar materials, the accuracy of test methods and test conditions, and the roadway is deformed and damaged. Therefore, some data are different. www.nature.com/scientificreports/

Conclusion
Based on the theory of complex function and elasticity, this paper establishes the mechanical model of roadway surrounding rock in inclined coal seam under the influence of dip angle, optimizes the solution process and accuracy of mapping function coefficient, and deduces the calculation equation of roadway surrounding rock stress and deformation. It reveals the deformation and failure mechanism of roadway in inclined coal seam, and is verified by numerical simulation and physical similarity simulation. The main conclusions are as follows: (1) The stress and deformation of roadway surrounding rock in inclined coal seam show obvious asymmetric distribution characteristics, that is, the stress and deformation on the right side of roadway surrounding rock are greater than that on the left side. On the whole, the stress and deformation of roadway surrounding rock show the change trend of side > roof > floor, and the two sides of roadway, the right side of roof and the top angle of right side are the key positions of stress concentration and deformation. The evolution mechanism of cyclic deformation and failure of roadway surrounding rock is further revealed, and the essential relationship between roadway roof and floor and deformation on both sides is established. (2) Using the numerical simulation analysis method, the stress, displacement and plastic zone distribution law of roadway surrounding rock in inclined coal seam are further analyzed. The roadway stress distribution and deformation law are basically consistent with the theoretical analysis results, and the numerical difference is less than 10%. The cyclic failure mechanism of inclined coal seam is verified. The stress concentration and deformation of surrounding rock in inclined coal seam roadway show asymmetric characteristics, that is, the stress and deformation on the right side are greater than that on the left side. The plastic zone of the roadway is asymmetrically "Butterfly" distributed along the inclined direction of the coal seam, that is, the range of the plastic zone on the right is greater than that on the left. (3) Through the physical model test, the stress and deformation law of the surrounding rock of the roadway in inclined coal seam are further analyzed. The test results are basically consistent with the theoretical analysis and numerical simulation. The maximum numerical difference is less than 32%, and the cyclic deformation failure theory of the roadway is verified. That is, the roof of the roadway and the sharp corner of the roof are damaged first. With the increase of load, the two sides of the roadway are divided, and the roof collapses slightly, resulting in the increase of roadway span, the pressure of the two sides is increased, and the damage of the two sides is increased. The stress state at the sharp corner continues to deteriorate. The two sides of the roadway, the two sharp corners of the roof and the sinking of the roof increase the stress, reduce the strength and destroy the vicious circle. Finally, the roadway roof is damaged by asymmetric "Beret" type caving arch, two sides are divided, and the floor heaves slightly.