Elastic wave scattering and dynamic stress concentrations in exponential graded materials with two elliptic holes

Based on the elastodynamics, employing complex functions and conformal mapping methods, and local coordinates, the scattering of elastic waves and dynamic stress concentrations in infinite exponential graded materials with two holes are investigated. A general solution of the problem and expression satisfying the given boundary conditions are derived. The problem can be reduced to the solution of an infinite system of algebraic equations. As an example, numerical results of dynamic stress concentration factors for two elliptic holes in exponential graded materials are presented, and the influence of incident wave number and holes spacing on dynamic stress distributions is analyzed.     

圆形域多圆孔多裂纹反平面问题研究

本文运用复变函数及积分方程方法，求解了圆形域多圆孔多裂纹反平面问题，建立了两种类型的基本解。复叠加原理和所得的基本解并沿国圆孔和裂纹表面取待定的基本解密度函数，可得到一组以基本解密度函数为未知函数的Ｆｒｅｄｈｏｌｍ积分方程。通过该积分方程组的数值可以得到密度函数的离散值，进而得到了裂纹尖端的应力强度因子。     

3D electroelastic fields in a functionally graded piezoceramic hollow sphere under mechanical and electric loadings

Summary  The problem of a piezoceramic hollow sphere is investigated analytically based on the 3D equations of piezoelasticity. The functionally graded property of the material along the radial direction can be taken arbitrarily in the paper. Displacement and stress functions are introduced, and two independent state equations with variable coefficients are derived. By employing the laminate model, the two state equations are transformed into ones with constant variables from which the state variable solution is easily obtained. Two linear relationships between the state variables at the inner and outer spherical surfaces are established. Numerical calculations are performed for different boundary conditions imposed on the spherical surfaces. Received 28 February 2001; accepted for publication 26 June 2001     

Crack-tip field on mode Ⅱ interface crack of double dissimilar orthotropic composite materials

Two systems of non-homogeneous linear equations with 8 unknowns are obtained.This is done by introducing two stress functions containing 16 undetermined coefficients and two real stress singularity exponents with the help of boundary conditions.By solving the above systems of non-homogeneous linear equations,the two real stress singularity exponents can be determined when the double material parameters meet certain conditions.The expression of the stress function and all coefficients are obtained based on the uniqueness theorem of limit.By substituting these parameters into the corresponding mechanics equations,theoretical solutions to the stress intensity factor,the stress field and the displacement field near the crack tip of each material can be obtained when both discriminants of the characteristic equations are less than zero.Stress and displacement near the crack tip show mixed crack characteristics without stress oscillation and crack surface overlapping.As an example,when the two orthotropic materials are the same,the stress singularity exponent,the stress intensity factor,and expressions for the stress and the displacement fields of the orthotropic single materials can be derived.     

含功能梯度材料加强环的任意几何形状孔附近应力集中分析

基于复变函数理论,结合保角变换技术研究含功能梯度材料（FGM）加强环的任意几何形状孔附近应力集中。采用分层均匀化方法,给出了远场均布载荷作用下材料参数沿孔周法线方向任意变化的FGM加强环内的复势函数解。通过数值算例,详细讨论了加强环内杨氏模量不同变化规律对三角形、正方形、矩形等各种几何形状孔附近应力分布的影响。结果表明：通过在孔周衬入FGM加强环并合理选择加强环内材料参数的递变规律,可以有效缓解各种几何形状孔附近的应力集中。同时通过一些特例与已有文献比对验证了本文结果的正确性。     

反平面圆形夹杂和多圆孔多裂纹相互作用问题

动用复变函数及积分方法方法求解了反平面圆形夹杂和多圆孔多裂纹相互作用问题。为解决该问题,建立了两种类型的基本解。利用叠加原理和所得的基本解没圆孔和裂纹表面取待定的基本解密度函数,可得一组Ｆｒｅｄｈｏｌｍ积分方程,通过积分方程组的数值求解,可以得到密度函数的离散值,进而得到应力强度因子。     

含有非圆形双孔的无限平板中应力的解析解研究

无限平板中含有任意形状单个孔的问题可以使用复变函数方法获得其应力解析解.对于无限平板中含有两个圆孔或两个椭圆孔的双连通域问题,也可以利用多种方法进行求解,比如双极坐标法、应力函数法、复变函数法以及施瓦茨交替法等.其中复变函数中的保角变换方法是获得应力解析解的一个重要方法.但目前尚未见到用此方法求解无限板中含有一个正方形孔和一个椭圆孔的问题.当板在无穷远处受有均布载荷和孔边作用垂直均布压力时,利用保角变换方法可以求解板中含有两个特定形状孔的问题.该方法将所讨论的区域映射成象平面里的一个圆环,其中最关键的一步是找出相应的映射函数.基于黎曼映射定理,提出了该映射函数一般形式,并利用最优化方法,找到了该问题的具体映射函数,然后通过孔边应力边界条件建立了求解两个解析函数的基本方程,获得了该问题的应力解析解.运用ANSYS有限单元法与结果进行了对比.研究了孔距、椭圆形孔大小和两孔布置方位对边界切向应力的影响,以及不同载荷下两孔中心线上应力分布规律.      

厚壁圆柱壳的弹性静力分析

本文研究了考虑横向剪切影响的弹性厚壁圆柱壳的静力问题。利用变分原理得到平衡微分方程组和相应的边界条件。将平衡方程组归并成一个高阶微分方程,用数值法求出它的特征根,得到问题的解。     

The establishment of boundary integral equations by generalized functions

By the theory of generalized functions this paper introduces a specific generalized function _p by which, together with its various derivatives, the boundary integral equations and its arbitrary derivatives of any sufficiently smooth function can be established. These equations have no non-integral singularities. For a problem defined by linear partial differential operators, the partial differential equations of the problem can always be converted into boundary integral equations so long as the relevant fundamental solutions exist.This paper is partial fulfilment of the first author's doctorial dissertation and the second author is the endviser.     

Elasticity solution of clamped-simply supported beams with variable thickness

This paper studies the stress and displacement distributions of continuously varying thickness beams with one end clamped and the other end simply supported under static loads. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of the plane stress problem, the general expressions of displacements, which satisfy the governing differefitial equations and the boundary conditions attwo ends of the beam, can be deduced. The unknown coefficients in the general expressions are then determined by using Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence properties. Comparing the numerical results to those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.     

A Mode- II Griffith Crack in Decagonal Quasicrystals

By using the method of stress functions, the problem of mode-II Griffith crack in decagonal quasicrystals was solved. First, the crack problem of two-dimensional quasicrystals was decomposed into a plane strain state problem superposed on anti-plane state problem and secondly, by introducing stress functions, the18 basic elasticity equations on coupling phonon-phason field of decagonal quasicrystals were reduced to a single higher-order partial differential equations. The solution of this equation under mixed boundary conditions of mode-II Griffith crack was obtained in terms of Fourier transform and dual integral equations methods. All components of stresses and displacements can be expressed by elemental functions and the stress intensity factor and the strain energy release rate were determined. Biography: GUO Yu-cui (1962-), Associate professor, Doctor     

饱和土中的任意形状孔洞对弹性波的散射

根据Biot波动理论建立了求解饱和土中任意形状孔洞对弹性波散射问题的复变函数方法.首先通过引入位移势函数把稳态条件下的Biot波动方程解耦为势函数所满足的Helmholtz方程.利用分离变量方法即得到Helmholtz方程完备的通解.根据所得位移势函数的通解,可得骨架位移、流体相对骨架的位移、应力和孔压的表达式.通过保角变换方法,把物理平面上的孔洞映射到像平面上单位圆.利用土骨架和流体的边界条件,即可确定波函数展开式中的未知系数.给出了一些数值结果.     

两个相等圆孔板弯曲时的三维应力集中

根据Hellinger-Reissner原理建立了具有一个无外力圆柱表面的三维八节点杂交应力元,其假设应力场严格满足柱坐标下三维平衡方程及圆柱面上无外力边界条件;当元退化为二维时也满足协调方程。数值算例表明,这种特殊杂交应力元可高效地分析具有两个圆孔薄板和厚板的应力集中,特别是三维应力集中。     

弹性薄板分析的条形传递函数方法

提出一种用于矩形弹性薄板变形分析的条形传递函数方法．一个矩形区域首先沿某一个方向被剖分成若干个条形子域,分割这些子域的直线称为结线,在结线上定义位移函数,它是结线坐标的一维函数,结线的两个端点称为结点．为适应复杂边界条件,在边界结线上定义若干结点,该结线的位移函数用结点位移参数插值表示．每个条形子域的变形用结线位移函数和适当的插值函数（形函数）表示．结线位移函数和结点位移参数满足的平衡微分方程及代数方程由变分原理给出     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Interface crack problems for mode Ⅱ of double dissimilar orthotropic composite materials

The fracture problems near the interface crack tip for mode Ⅱ of double dissimilar orthotropic composite materials are studied. The mechanical models of interface crack for mode Ⅱ are given. By translating the governing equations into the generalized bi-harmonic equations,the stress functions containing two stress singularity exponents are derived with the help of a complex function method. Based on the boundary conditions,a system of non-homogeneous linear equations is found. Two real stress singularity exponents are determined be solving this system under appropriate conditions about himaterial engineering parameters. According to the uniqueness theorem of limit,both the formulae of stress intensity factors and theoretical solutions of stress field near the interface crack tip are derived. When the two orthotropic materials are the same,the stress singularity exponents,stress intensity factors and stresses for mode Ⅱ crack of the orthotropic single material are obtained.     

Development of the direct stress solution technique for three-dimensional hydrodynamic models using finite elements

Velocity varies rapidly near sheared boundaries. Therefore in many practical fluid problems it can be inefficient to solve discrete equations with velocity as the dependent variable. Conversely, shear stress varies slowly near sheared boundaries, suggesting that it may be well suited for use as the dependent variable in discrete equations. This paper describes a formulation of the internal mode equations for a three-dimensional hydrodynamic model using shear stress as the dependent variable. The resulting direct stress solution (DSS), coupled with a spatial discretization using linear finite elements, yields a system matrix that can be set up and solved with the efficiency of a banded matrix with bandwidth 8. If the eddy viscosity distribution is assumed to be piecewise linear over the depth (with an arbitrary number of time-varying segments), the recovery of velocity from stress can be easily accomplished in closed form, thereby avoiding any difficulty resulting from the logarithmic singularity in the velocity profile that occurs at a boundary. Results from tidal and wind-driven test cases with realistic boundary layers are used to demonstrate the accuracy and computational advantages of a DSS formulation versus a standard velocity-based formulation.     

板弯曲与平面弹性问题的多类变量变分原理

进一步完善板弯曲与平面弹性问题的多类变量变分原理,给出了相关边界积分项的具体表达式．多类交量变分原理涵盖了平衡、应力函数、应力、位移一应变、协调和物性共五大类基本方程和所有边界条件,是一个具有更加广泛意义的变分原理．     

用间接边界元方法求解不可压缩孤立翼型绕流的一种库塔条件处理方法

本文对不可压二维翼型势流绕流的边界元法求解作了分析,在对基本方程与边界条件进行数值离散化时,将库塔条件代入基本方程。按本方法编制的计算机程序对若干算例进行了验算。结果表明,本文提出的方法是可靠的,该法计算简捷、方便,占用计算机内存少,具有实用意义。     

变边界粘弹性轴对称问题的复变函数法

对边界几何形状、位置随时间变化的变边界结构,给出了用复变函数求解粘弹问题的解析方法。文中用拉普拉斯变换结合平面弹性复变方法,对内外边界变化时粘弹性轴对称问题进行求解。引入两个与时间、空间相关的解析函数,给出了变边界情况下应力、位移以及边界条件与解析函数的关系。当解析函数形式部分确定,则可用边界条件求解其中与时间相关的待定函数。求解待定函数的方程一般情况下为一系列积分方程,特殊情况可求得解析解。对轴对称问题中应力边值问题、位移边值问题以及混合边值问题,分别利用边界条件求得相关系数,从而得到了应力与位移的解析表达。当取Boltzmann粘弹模型时,进行不同边值问题的分析。分析显示,应力、位移的形态与大小均与边界变化过程相关,与固定边界粘弹性问题有较大不同。本文解答可用于粘弹性轴对称问题内外边界任意变化及各种边值问题的力学分析。此外,该法可进一步进行荷载非对称、复杂孔型变边界问题的求解。