基于Schwarz-Christoffel变换的圆形隧道围岩应力分布特征研究

隧道围岩应力还原到实际区域的变换方法被越来越多地应用到工程实践中,对复杂形状围岩体内的隧道围岩应力计算方法和隧道围岩稳定性的研究具有重要的实践指导意义.该文建立了不规则岩体中开挖圆形隧道的力学模型,运用复变函数理论,通过映射函数Schwarz-Christoffel变换,得到复平面单位圆到隧道所处多边形岩体的映射函数,在复变函数数域内分析了多边形岩体应力的求解步骤,运用弹性力学理论等推导了不规则岩体中圆形隧道的复变应力函数Φ(ξ)和φ(ξ)的表达式,并得出围岩体任一点应力分量σρ和σθ的解析通式.通过具体算例分析可知,岩体形状对圆形隧道的稳定性有较大的影响,隧道所处4种形状下的围岩体最大应力值分布规律为:顶底板的最大应力值从六边形、五边形、四边形、圆形围岩体依次减小,帮部的最大应力值从圆形、四边形、五边形、六边形围岩体依次减小.     

Investigation on a rotating FGPM circular disk under a coupled hygrothermal field

In this paper, the distributions of the temperature, moisture, displacement and stress of a functionally graded piezoelectric material (FGPM) circular disk rotating around its axis at a constant angular velocity under a coupled hygrothermal field are presented by a numerical method. The material properties of the FGPM circular disk are assumed to vary along the radial coordinate exponentially. First, the coupled hygrothermal field along the radius of a rotating circular disk is achieved by solving the coupled hygrothermal equations, and then the dynamic equilibrium is solved by utilizing the finite difference method. Finally, numerical results show the effects of functionally graded index, inner radius, angular speed and hygrothermal index on the hygrothermal behaviors of the FGPM circular disk. The results can be useful for the optimal design of rotating FGPM circular disks under a coupled hygrothermal field.     

三维非局部弹性场中裂纹问题的分析方法

通过求解得到了三维非局部弹性力学对称情形的单位集中不连续位移基本解·基于该基本解和三维局部（经典）弹性力学的不连续位移边界积分方程———边界元方法·提出了三维非局部弹性力学中的平片裂纹Ⅰ型问题的通用解法,并给出了算例     

Application of a biparametric perturbation method to large-deflection circular plate problems with a bimodular effect under combined loads

The large deflection condition of a bimodular plate may yield a dual nonlinear problem where the superposition theorem is inapplicable. In this study, the bimodular Föppl–von Kármán equations of a plate subjected to the combined action of a uniformly distributed load and a centrally concentrated force are solved using a biparametric perturbation method. First, the deflection and radial membrane stress were expanded in double power series with respect to the two types of loads. However, the biparametric perturbation solution obtained exhibited a relatively slow rate of convergence. Next, by introducing a generalized load and its corresponding generalized displacement, the solution is expanded in a single power series with respect to the generalized displacement parameter, thereby leading to the better convergence on the solution. A numerical simulation is also used to verify the correctness of the biparametric perturbation solution. The introduction of a bimodular effect will modify the stiffness of the plate to some extent. In particular, the bearing capacity of the plate will be strengthened further when the compressive modulus is greater than the tensile modulus.     

On rotating circular disks with varying material properties

Taking Young’s modulus, thermal expansion coefficient and density to be the functions of the radial coordinate, a closed form solution of rotating circular disks made of functionally graded materials subjected to a constant angular velocity and a uniform temperature change is proposed in this paper. Excellent agreement with the solution from Mathematica 5.0 indicates the correctness of the proposed closed form solution. Distributions of the radial displacement and stresses in the disks are determined with the proposed approach and how material properties, temperature change, geometric size and different material coefficients affect deformations and stresses is investigated.     

On rotating circular disks with varying material properties

Taking Young’s modulus, thermal expansion coefficient and density to be the functions of the radial coordinate, a closed form solution of rotating circular disks made of functionally graded materials subjected to a constant angular velocity and a uniform temperature change is proposed in this paper. Excellent agreement with the solution from Mathematica 5.0 indicates the correctness of the proposed closed form solution. Distributions of the radial displacement and stresses in the disks are determined with the proposed approach and how material properties, temperature change, geometric size and different material coefficients affect deformations and stresses is investigated.     

微分求积法分析平面接头应力奇异性北大核心CSCD

对于双材料平面接头问题提出了一个分析应力奇性指数的新方法:微分求积法(DQM).首先,将平面接头连接点处位移场的径向渐近展开格式代入平面弹性力学控制方程,获得了关于应力奇性指数的常微分方程组(ODEs)特征值问题.然后,基于DQM理论,将ODEs的特征值问题转化为标准型广义代数方程组特征值问题,求解之可一次性地计算出双材料平面接头连接点处应力奇性指数,同时,一并求出了接头连接点处相应的位移和应力特征函数.数值计算结果说明该文DQM计算平面接头连接点处应力奇性指数的结果是正确的.     

Mixed dynamic problem for a half plane and its application to rock mechanics

The mixed dynamic problem of the theory of elasticity is solved for an isotropic half plane. The dynamic equations are reduced to integration of fourth-degree equations in partial derivatives with constant coefficients, after whose solution, the components of the stress tensor and displacement vector are written in a form similar to that introduced by Lekhnitskii for an anisotropic body. The stress state of a rock mass subjected to rapid face advance in a seam is investigated using the solution obtained. The stress distribution is analyzed numerically. The existence of a critical rate at which the stress increases without restriction is demonstrated.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 56–61, 1990.     

Analysis of post buckling behavior of circular plates with non-concentric hole using the Rayleigh–Ritz method

This study is conducted to determine the post buckling behavior of circular homogenous plates with non-concentric hole subjected to uniform radial loading using Rayleigh–Ritz method. In order to implement the method, a computer program has been developed and several numerical examples for different boundary conditions are presented to illustrate the scope and efficacy of the procedure. The integration is carried out in natural coordinates through a proper transformation. Consequently, the displacement fields respect to natural coordinates are expressed using the Hierarchical, Hermitian and Fourier series shape functions for interpolating the out-of-plane displacement field and Fourier series and Hierarchical, Lagrange shape functions for interpolating the in-plane displacement field of plate. The Kirchhoff theory is used to formulate the problem in buckling condition. Due to the asymmetry in geometry, the in-plane solution is required to find the stress distribution. Finally, the problem is formulated in post buckling condition using Von-Karman non-linear theory, and a proper Hookean displacement field is presented to analyze the post buckling behavior.     

Remarks on flows of elastic fluids along cylindrical boundaries

A study is undertaken to ascertain non-Newtonian effects in steady flows of elastic fluids along circular cylinders when there is extraction across the boundary surfaces. Exact representations for the velocity and stress fields are derived when the fluid prototypes are generalized Maxwell prototypes (exhibiting stress relaxation). Whilst the scheme of study has physical importance, it is the nature of the solution which is of foremost interest here, it being shown that the presence of elasticity (of the type considered) has a marked effect on the transport velocity.     

Stress field around an arbitrary thin inclusion in a transversely isotropic elastic half-space

We consider a thin flat inclusion of arbitrary shape located inside a transversely isotropic elastic half-space in the plane parallel to its boundary z = 0. An arbitrary tangential displacement is prescribed on the inclusion. The boundary of the half-space is stress-free. We need to find the complete field of stresses and displacements in this half-space. A governing integral equation is derived by the generalized method of images, introduced by the author. The case of circular inclusion is considered as an example. Two methods of solution of the governing integral equation are derived. A detailed solution is presented for the particular cases of radial expansion, torsion and lateral displacement of the inclusion. The solution is also valid for the case of isotropy. The governing integral equation for the case of isotropy is derived.     

On a partially debonded rigid line inclusion penetrating a circular inhomogeneity

We derive closed-form solutions to the mixed boundary value problem of a partially debonded rigid line inclusion penetrating a circular elastic inhomogeneity under antiplane shear deformation. The two tips of the rigid line inclusion are just mutual mirror images with respect to the inhomogeneity/matrix interface, and the upper part of the rigid line inclusion is debonded from the surrounding materials. By using conformal mapping and the method of image, closed-form solutions are derived for three loading cases: (i) the matrix is subjected to remote uniform stresses; (ii) the matrix is subjected to a line force and a screw dislocation; and (iii) the inhomogeneity is subjected to a line force and a screw dislocation. In the mapped ξ-plane, the solutions for all the three loading cases are interpreted in terms of image singularities. For the remote loading case, explicit full-field expressions of all the field variables such as displacement, stress function and stresses are obtained. Also derived is the near tip asymptotic elastic field governed by two generalized stress intensity factors. The generalized stress intensity factors for all the three loading cases are derived.     

Effect of material in-homogeneity on electro-thermo-mechanical behaviors of functionally graded piezoelectric rotating shaft

In this article, a hollow circular shaft made from functionally graded piezoelectric material (FGPM) such as PZT_5 has been studied which is rotating about its axis at a constant angular velocity ω. This shaft subjected to internal and external pressure, a distributed temperature field due to steady state heat conduction with convective boundary condition, and a constant potential difference between its inner and outer surfaces or combination of these loadings. All mechanical, thermal and piezoelectric properties except for the Poisson’s ratio are assumed to be power functions of the radial position. The governing equation in polarized form is shown to reduce to a system of second-order ordinary differential equation for the radial displacement. Considering six different sets of boundary conditions, this differential equation is analytically solved. The electro-thermo-mechanical stress and the electric potential distributions in the FGPM hollow shaft are discussed in detail for the piezoceramic PZT_5. The presented results indicate that the material in-homogeneity has a significant influence on the electro-thermo-mechanical behaviors of the FGPM rotating shaft and should therefore be considered in its optimum design.     

非圆形水工衬砌隧洞与横观各向同性岩体在光滑接触下的解析分析

提出了横观各向同性岩体中的非圆形水工衬砌隧洞在各向同性衬砌与岩体处于光滑接触条件下的解析方法.基于复变函数理论,通过建立两种介质在光滑接触边界上的力和位移连续关系以及衬砌自由边界的水压力边界条件,考虑衬砌支护滞后效应并使用幂级数解法获得解析解.针对倾斜结构面岩体中的马蹄形水工衬砌隧洞,使用解析和数值方法验证了解析解的正...     

Dynamic proper colorings of a graph

We solve an axisymmetric problem of the interaction of harmonic waves with a thin elastic circular inclusion located in an elastic isotropic body (matrix). On both sides of the inclusion, between it and the body (matrix), conditions of smooth contact are realized. The method of solution is based on the representation of displacements in the matrix in terms of discontinuous solutions of Lamé equations for harmonic vibrations. This enables us to reduce the problem to Fredholm integral equations of the second kind for functions related to jumps of normal stress and radial displacement on the inclusion.     

Temperature dependence of an elastic modulus in generalized linear micropolar thermoelasticity

The model of the equations of generalized linear micropolar thermoelasticity with two relaxation times in an isotropic medium with temperature-dependent mechanical properties is established. The modulus of elasticity is taken as a linear function of reference temperature. Laplace and exponential Fourier transform techniques are used to obtain the solution by a direct approach. The integral transforms have been inverted by using a numerical technique to obtain the temperature, displacement, force and couple stress in the physical domain. The results of these quantities are given and illustrated graphically. A comparison is made with results obtained in case of temperature-independent modulus of elasticity. The problem of generalized thermoelasticity has been reduced as a special case of our problem.     

隧洞围岩应力复变函数分析法中的解析函数求解

利用复变函数理论进行地下任意开挖断面隧洞围岩应力分析的前提,是根据围岩应力边界条件方程推导出两个解析函数．从Harnack定理出发,将隧洞围岩应力边界条件方程转化成积分方程;把Laurent级数有限项表示的映射函数引入积分方程中,将以任意开挖断面为边界条件的解析函数求解转化成以单位圆周线为边界条件的求解问题．对积分方程中各被积函数在讨论域内的解析性进行了分析,在此基础上利用留数理论求解了方程中各项积分值,并获得了用来表示任意开挖断面隧道围岩应力的两个解析函数通式．给出了圆形和椭圆形隧道的两个解析函数求解算例,所获得的结果与文献中的结果一致．利用留数理论推导出的两个解析函数通式,适用于任意开挖断面隧洞的围岩应力解析解的计算,且计算过程更为简单,计算结果更为精确．     

Determination of the stress state of a circular disk with a crack

With the use of complex potentials from the solution of a contact problem for slits in a multiply-connected region, a solution is found for a problem of the theory of elasticity for an isotropic circular disk with an arbitrary radial crack. The case of an edge crack is among the cases for which a solution is found. The types of loading examined are uniform tension on an outside edge, internal pressure on the edges of cracks, and concentrated forces at arbitrary points of a disk. The unknown coefficients in the complex potentials are found from the boundary conditions on the outside edge of the disk by the series method, the colocation method, or the least squares method. Detailed numerical studies are conducted to determine the effect of the geometric characteristics and the points of application of concentrated forces on the character of the stress distribution and the stress intensity factor. A comparative characteristic of the methods used to find the coefficients is presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19 pp. 50–61, 1988.     

带球形空腔的广义热弹性无限大材料的弹性模量和传热系数与材料参考温度的相关性

利用具某一松弛时间的广义热弹性方程求解了带球形空腔的无限大材料问题．该材料的弹性模量和传热系数是可变的．空腔的内表面没有力作用,但有热冲击作用．利用Laplace变换求得直接逼近解．数值求解了Laplace逆变换．给出了温度、位移和应力的分布图．     

逆冲断层地震主震的突变理论分析

逆冲断层系统作准静态形变时的功、能量增量关系可以分解为关于体积应变能的功能增量关系和关于偏应力能的功能增量关系．采用突变理论方法对逆冲断层系统的偏应力功能量增量关系进行的分析表明:折迭突变模型展示的性状可对逆冲断层地震主震的发震条件、演化过程和若干震后特性作恰当描述．围岩的围压大,最大主应力大,逆冲潜断层面倾角小,围岩切向刚度与断层抗剪强度曲线软化段拐点处斜率的比值小, 则震时围岩弹性能释放量大、 震级高, 断层破裂半错距大, 围岩端面位移振幅也大．断层岩体破裂扩容和震时围岩体积应变能释放, 增强了前述效应．