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Deformation and stress theory of surrounding rock of shallow circular tunnel based on complex variable function method

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APPLIED MATHEMATICS AND NONLINEAR SCIENCES 2022年第1期7卷 629-640页

作者： Wang, Guizhen Zhou, Linglong Alotaibi, Reem Hdeib, Roaya Chongqing Vocat Inst Engn Sch Architecture Chongqing 402260 Peoples R China Edith Cowan Univ Sch Business & Law Joondalup WA 6027 Australia King Abdulaziz Univ Fac Comp & Informat Technol Dept Informat Technol Jeddah Saudi Arabia Appl Sci Univ Al Eker Bahrain

After reviewing many literature foundations, the thesis combines the basic methods of elastic mechanics with mathematical knowledge, sets the bipotential stress potential complex function and analyses the relationship between stress component, strain component and stress potential function, and applies the complex variable function. The expression of the relevant stress component is derived, and the displacement boundary conditions of the surrounding rock of shallow circular tunnel are obtained. Furthermore, the paper applies the basic theory of complex variable function to solve the boundary condition complex variable function for common tunnel sections, and obtains the analytical expression of the surrounding rock stress of shallow circular tunnel. The simulation is carried out by finite element method. The establishment of complex variable function has a good application value in solving the stress of surrounding rock of shallow tunnel.

关键词： complex variable function shallow-buried circular tunnel surrounding rock deformation analysis stress finite element analysis

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Application of the complex variable function method in solving the floor heave problem of a coal mine entry

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ARABIAN JOURNAL OF GEOSCIENCES 2018年第17期11卷 515-515页

作者： Yang, Jing-Hu Song, Gao-Feng Yang, Yi Yang, Zeng-Qiang China Univ Min & Technol Beijing Sch Resource & Safety Engn Beijing 100083 Peoples R China China Univ Min & Technol Beijing Beijing Key Lab Precise Min Intergrowth Energy & Beijing 100083 Peoples R China North China Univ Technol Sch Civil Engn Beijing 100144 Peoples R China

With the increase of mining depth in Chinese coal mines, the entry floor heave becomes one of the major engineering problems. This paper has studied the stress and displacement distribution around an entry in Buertai coal mine, China. The mechanism of floor heave is analyzed using the method of complex variable function. The influences of floor mechanical properties, longwall face advance, and mine water on the stability of the floor are then identified. The results show that (1) the floor moves towards the center under the concentration of the tangential stress, resulting in floor heave;(2) the floor stability is mainly influenced by the mechanical properties of the floor and the confining stress from the surrounding rock;and (3) the unnoticeable movement of the floor may be enlarged by the mining activity (face advance) and mine water. The proposed theoretical model is validated by the numerical modeling and the in situ. Good agreements are found between the theoretical analysis and the numerical modeling results and the in situ. It is thus believed that the complex variable function method can be used for analyzing the floor heave problem. Based on the mechanism of floor heave, we divide the control techniques into two categories: creation of a stress-relief entry and floor reinforcement. The techniques were practiced at two Chinese coal mines with good ground control effects.

关键词： Floor heave complex variable function Numerical modeling Control technique

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Research on Deformation and Failure Mechanism of Surrounding Rock in Inclined Strata with Soft-Hard Interbedded Roadways

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ROCK MECHANICS AND ROCK ENGINEERING 2024年第12期57卷 11319-11332页

作者： Ma, Xinmin Zhai, Zhonghua Xiang, Junjie Wei, Mengjie Tian, Zhihui Chen, Liying China Univ Min & Technol Sch Mech & Civil Engn Beijing 100083 Peoples R China

In order to address the challenge of stability control of surrounding rock in roadways under complex geological conditions such as deep high-stress inclined strata and soft-hard interbedding, a comprehensive approach combining similarity model tests and complex variable function analysis with the Adam optimization algorithm was employed. This approach was utilized to obtain the stress and failure characteristics of surrounding rock in roadways under inclined strata and soft-hard interbedding conditions, elucidating the stability failure mechanism of surrounding rock in roadways. The research results indicate that: the internal stress direction in inclined strata shows obvious orientation, tending to be perpendicular to the stratigraphic boundary;the stress distribution in roadway surrounding rock varies with the inclination direction of the strata. Specifically, when the strata inclination angle is clockwise, stress concentration tends to occur on the left sidewall and right floor of the roadway;conversely, when the strata inclination angle is counterclockwise, stress concentration tends to occur on the right sidewall and left floor of the roadway. The use of the Adam optimization algorithm for solving mapping functions demonstrates superiority, with a solution time of only 1.25 s and high accuracy. When the number of coefficient terms in the mapping function is 9, the average error is only 0.17%, which is 0.34 times lower compared to other optimization methods, with the required number of iterations reduced by 0.64, significantly reducing subsequent computational pressure. For cross-layer roadways with soft-hard interbedding surrounding rock, the deformation and failure of roadways are closely related to the interface position between the rock layers, with deformation more likely to occur in the soft rock area and with larger deformations. Under conditions of high stress, when facing both the soft-hard interbedding and inclined strata issues simultaneously, priorit

关键词： Inclined strata Soft-hard interbedding Model test complex variable function Optimization algorithm

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Analytical Solution of the Non-circular Tunnel with a Void Defect in the complex Stress Field

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KSCE JOURNAL OF CIVIL ENGINEERING 2024年第10期28卷 4717-4727页

作者： Wang, Dapeng Niu, Xing Wang, Jingchun Rao, Chenjie Xu, Qiang Shijiazhuang Tiedao Univ Sch Civil Engn Shijiazhuang 050043 Peoples R China Shijiazhuang Tiedao Univ Sch Safety Engn & Emergency Management Shijiazhuang 050043 Peoples R China

To address the issue of stress concentrations afflicting non-circular tunnels caused by void defects, a method of analytical solution is introduced, which is grounded in the theory of complex variables. Firstly, the mapping function of the non-circular tunnel containing a void was obtained;then the power series method was implemented to determine the coefficients of the stress function and formulate the solution equations. Finally, the analytical solution was verified by simulation through Flac3D software. The effects of pore depth, width, stress inclination and stress ratio on the stress and deformation of the tunnel are considered analytically using the method of this paper. The findings indicate a strong agreement between the analytical solution and the numerical solution, and there is an obvious stress concentration at the defect location. The void depth, width, main stress field inclination and stress ratio contribute remarkably to the stress concentration at the location of tunnel void defects, thus affecting the mechanical behavior and safety condition after tunnel excavation. The results of the study propose a fast and accurate elasticity calculation method for non-circular tunnel excavation problems, offering a valuable reference to similar tunnel projects in design, construction and stability analysis.

关键词： Tunnel engineering complex variable function Analytical solution Void defect Non-circular tunnel

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Surface motion of P/SV wave scattering by a semicircular canyon in saturated half-space by complex function and conformal mapping

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SOIL DYNAMICS AND EARTHQUAKE ENGINEERING 2023年 175卷

作者： Jin, Liguo Duan, Xueliang Zhou, Bengang Zhou, Zhenghua China Earthquake Adm Inst Geol Beijing 100029 Peoples R China Nanjing Tech Univ Coll Transportat Engn Nanjing 210009 Peoples R China

Based on the complex function theory of elastic dynamics, a complex function solution of P/SV-wave scattering by a semicircular canyon in an elastic porous saturated half-space is obtained. Firstly, through Biot theory and Helmholtz decomposition, the wave potential functions with unknown coefficients in the model are obtained based on the wave function expansion method. Secondly, the complex function expressions of stress and displacement in the model are obtained by using the complex function method and conformal mapping. According to the boundary conditions of the horizontal ground surface and semicircular canyon surface, this boundary value problem can be converted into a series of algebraic equations and solved numerically by truncation. Then, the correctness of the complex function solution in this paper is verified by comparing it with the published results through degenerating the scattering problem in the saturated porous elastic half-space in this paper into that in the elastic half-space. Finally, the influence of the boundary drainage conditions, the medium porosity, and the incident angle and frequency of the incident wave on the ground surface displacement is investigated. Although the P/SV-wave scattering problem is a traditional problem, there is currently no accepted analytical solution that strictly satisfies the zero-stress condition on the horizontal ground surface. This paper is an attempt at this problem, which may provide insight into the search for the strictly analytical solution for P/SVwave scattering.

关键词： Saturate porous elastic medium Canyon Wave scattering complex variable function Conformal mapping

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Stress Characteristics and Mechanical Behavior of Rock Masses with an Opening under complex Deep Underground Stress Conditions

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APPLIED SCIENCES-BASEL 2024年第16期14卷 7197页

作者： Cao, Mingyu Qiu, Xianyang Cao, Rihong Li, Zeyu Shi, Xiuzhi Tan, Lihai Cent South Univ Sch Resources & Safety Engn Changsha 410083 Peoples R China Guangdong Prov Dabaoshan Min Co Ltd Shaoguan 512127 Peoples R China Univ South China Sch Resource Environm & Safety Engn Hengyang 412001 Peoples R China

In this study, the impact of principal stress states on the stress characteristics and initial failure of the rock mass surrounding a three-center arch opening was investigated using complex variable function methods and Discrete Element Method (DEM) numerical modeling. First, the mapping function of the opening was determined using the trigonometric interpolation method, and the influence of the number of terms in the mapping function on its accuracy was revealed. Based on this, the far-field stress state of the underground rock mass was characterized by the ratio of the minimum to maximum principal stress (lambda) and the angle (beta) between the principal stress and the vertical direction. This stress state was then converted into normal and shear stresses. Using complex variable function theory, the stress characteristics at the boundary of the opening under different stress states were analyzed. Finally, DEM numerical modeling was employed to study the initial failure characteristics at the boundary of the opening and its relationship with the stress distribution. The results indicate that the lateral pressure coefficient significantly affects the stability of the opening by influencing stress concentration around the surrounding rock. Low lateral pressure coefficients lead to tensile stress concentration at the boundary perpendicular to the maximum principal stress. As the coefficient increases, tensile stress decreases, and compressive stress areas expand. While the principal stress direction has a minor effect on stress concentration, it notably impacts stress distribution at the boundary. When lambda < 1.0 and beta = 45 degrees, stress distribution asymmetry is most pronounced, with the highest compressive stress. The early failure distribution aligns with stress concentration areas, validating the use of stress analysis in predicting opening stability and failure characteristics.

关键词： geo-stress state stress solution underground opening complex variable function numerical modeling

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Research on online advertising attention evaluation decision based on the stability of delay differential equations and Hopf bifurcation analysis

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MATHEMATICAL BIOSCIENCES AND ENGINEERING 2021年第4期18卷 4761-4771页

作者： Xie, Xinzhou Wang, Zhongru Tian, Li Dekkiche, Hamoud Salama, Mohamad Peking Univ Sch New Media Beijing Peoples R China Chinese Acad Cyberspace Studies Beijing Peoples R China Canadian Univ Dubai Dubai U Arab Emirates Mohamad Salama Appl Sci Univ Al Eker Bahrain

This paper uses the stability of the delay differential equation to study its impact on online advertising, helps analyze Hopf branch characteristics in a big data environment, helps companies make online advertising decisions, and maximizes the benefits of product sales. The thesis fully considers various factors such as advertising volume, advertising schedule, and advertising investment level, discusses the singularity types of the advertising delay differential equation, and gives the best decision for advertising *** stability of the time-lag differential equation studied in this paper is to study its impact on online advertising, to help analyze the Hopf branch characteristics in the big data environment, and to help companies make online advertising decisions. structure of this article is also from the amount of advertising, the time of advertising, Advertising investment level gradually expands with a certain degree of continuity.

关键词： complex variable function shallow buried circular tunnel surrounding rock deformation analysis stress finite element analysis

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Analysis of stress intensity factor for a crack emanating from elliptical hole subjected to compressive stress and shear stress

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THEORETICAL AND APPLIED FRACTURE MECHANICS 2022年 120卷

作者： Huang, Jianwei Wang, Hanbing Zhou, Lei Zhu, Zheming Deng, Ze Deng, An Sichuan Univ Inst Disaster Management & Reconstruct MOE Key Lab Deep Earth Sci & Engn Chengdu 610065 Peoples R China Sichuan Univ Coll Architecture & Environm Failure Mech & Engn Disaster Prevent Key Lab Sichu Chengdu 610065 Peoples R China Sichuan Univ Coll Architecture & Environm State Key Lab Hydraul & Mt River Engn Chengdu 610065 Peoples R China Chengdu Surveying Geotech Res Inst Co Ltd MCC Chengdu 610023 Peoples R China

Underground caves often exist in karst areas, which will cause many difficulties in tunnel construction. The underground caves are often subjected to compressive stress, shear stress and internal pressure come from rock, faults and groundwater, respectively. The crack emanating from the cave will propagate under the effect of external forces, leading to the disaster of cave collapse and water gushing. To research the stability of crack emanating from the elliptical cave, a model of a plane containing an elliptical hole with a single crack under the compressive stress, shear stress and internal pressure was established. The theoretical solutions of stress intensity factors (SIFs) for crack tip were derived using complex variable function and conformal mapping function. The theoretical results were compared with the results of numerical simulation to verify the correctness of the theoretical solutions. The effects of crack length, ellipse shape, compressive stress, shear stress, internal pressure and crack inclination angle on SIFs were discussed. We can conclude that when the crack inclination angle is varied from 0 degrees to 30 degrees, the initiation pressure is small, and the cave is easily destroyed by groundwater. The flatter the elliptical hole, the easier the crack propagates. With an increasing of crack length, the value of SIFs at the crack tip increase quickly first and then increase linearly. When the crack inclination angle is 45 degrees, the initiation angle of the crack reaches the maximum value.

关键词： Stress intensity factor Crack Elliptical hole complex variable function Conformal function

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Stress intensity factor for bonded dissimilar materials weakened by multiple cracks

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APPLIED MATHEMATICAL MODELLING 2020年第Part1期77卷 585-601页

作者： Hamzah, K. B. Long, N. M. A. Nik Senu, N. Eshkuvatov, Z. K. Univ Putra Malaysia Inst Math Res Lab Computat Sci & Math Phys Serdang 43400 Selangor Malaysia Univ Putra Malaysia Math Dept Fac Sci Serdang 43400 Selangor Malaysia Univ Sains Islam Malaysia Fac Sci & Technol Negerisembilan 71800 Malaysia Univ Tekn Malaysia Melaka Fak Teknol Kejuruteraan Mekan & Pembuatan Hang Tuah Jaya Durian Tunggal 76100 Melaka Malaysia

The new hypersingular integral equations (HSIEs) for the multiple cracks problems in both upper and lower parts of the bonded dissimilar materials are formulated using the modified complex potential method, and with the help of the continuity conditions of the resultant force function and displacement. The crack opening displacement is the unknown and the traction along the crack as the right term of the equations. The appropriate quadrature formulas are used in solving the obtained HSIEs for the unknown coefficients. Numerical results for the multiple inclined or circular arc cracks subjected to the remote shear stress are presented. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Stress intensity factors Bonded dissimilar materials complex variable function Hypersingular integral equation

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FIELD INTENSITY FACTORS OF THREE CRACKS ORIGINATING FROM A CIRCULAR HOLE IN A THERMOELECTRIC MATERIAL

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JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES 2020年第5期15卷 605-617页

作者： Liu, Qing-Nan Ding, Sheng-Hu Ningxia Univ Sch Math & Stat Yinchuan 750021 Ningxia Peoples R China

The fracture behavior of three cracks originating from a circular hole in a thermoelectric material is studied. The basic theory of thermoelectric materials is given under the condition of the impermeable boundary of the heat flux and electric current. By introducing the conformal mapping function, the infinite plane on the physical plane is mapped into the inner unit circle on the mathematical plane. The formulas of the relevant temperature and stress fields are derived, and the analytical solutions of the complex stress functions are obtained by the complex variable function method. According to Cauchy integral formula, the analytical expressions of the electric current intensity factor and the stress intensity factor for three cracks originating from a circular hole are obtained. The effects of the hole radius and the crack lengths on the electric current intensity factor and stress intensity factors were investigated.

关键词： thermoelectric material circular hole three cracks complex variable function intensity factors

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