Elasticity solutions for plane anisotropic functionally graded beams

This paper considers the plane stress problem of generally anisotropic beams with elastic compliance parameters being arbitrary functions of the thickness coordinate. Firstly, the partial differential equation, which is satisfied by the Airy stress function for the plane problem of anisotropic functionally graded materials and involves the effect of body force, is derived. Secondly, a unified method is developed to obtain the stress function. The analytical expressions of axial force, bending moment, shear force and displacements are then deduced through integration. Thirdly, the stress function is employed to solve problems of anisotropic functionally graded plane beams, with the integral constants completely determined from boundary conditions. A series of elasticity solutions are thus obtained, including the solution for beams under tension and pure bending, the solution for cantilever beams subjected to shear force applied at the free end, the solution for cantilever beams or simply supported beams subjected to uniform load, the solution for fixed–fixed beams subjected to uniform load, and the one for beams subjected to body force, etc. These solutions can be easily degenerated into the elasticity solutions for homogeneous beams. Some of them are absolutely new to literature, and some coincide with the available solutions. It is also found that there are certain errors in several available solutions. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a functionally graded anisotropic cantilever beam.     

Problems of non-uniqueness when interpreting the effect of fluid inertia on the complex viscosity function

A theoretical investigation is carried out into the interpretation of the effect of fluid inertia on the complex viscosity function as measured on a controlled stress rheometer. The problem of non-unique solutions to the governing equations is considered for the parallel plate geometry. The locations of these solutions are investigated by considering the critical points of the complex mapping associated with the linear viscoelastic equations of motion. It is shown that these critical points play an important role in determining where convergence problems are likely to occur when applying numerical methods of solution to the governing equations. Analytical approximations based on a series expansion about a critical point are developed as an alternative approach to a numerical solution in the neighbourhood of a critical point. In order to verify the theoretical predictions a numerical simulation of the behaviour of a single element Maxwell fluid on a controlled stress rheometer is carried out for a parallel plate geometry. Received: 27 July 1998 Accepted: 9 April 1999     

Stress concentration around a hole in a radially inhomogeneous plate

The stress concentration factor around a circular hole in an infinite plate subjected to uniform biaxial tension and pure shear is considered. The plate is made of a functionally graded material where both Young’s modulus and Poisson’s ratio vary in the radial direction. For plane stress conditions, the governing differential equation for the stress function is derived and solved. A general form for the stress concentration factor in case of biaxial tension is presented. Using a Frobenius series solution, the stress concentration factor is calculated for pure shear case. The stress concentration factor for uniaxial tension is then obtained by superposition of these two modes. The effect of nonhomogeneous stiffness and varying Poisson’s ratio upon the stress concentration factors are analyzed. A reasonable approximation in the practical range of Young’s modulus is obtained for the stress concentration factor in pure shear loading.     

黏弹性接触界面端附近的奇异应力场

研究蠕变加载条件下线黏弹性材料接触界面端附近的奇异应力场问题.考虑接触界面的摩擦,假设界面端的滑移方向不改变,相对滑移量微小,且其与位移同量级,由此线性化局部边界条件,根据对应原理得到Laplace变换域中的界面端应力场,导出时域中奇异应力场的卷积积分表达式.对卷积积分核函数进行数值反演,考虑接触材料的两类组合,一是持久模量具有量级上的差异,另一是持久模量接近相同.算例结果证实核函数可以用准弹性法求得的解析式较准确地近似.在此基础上,利用积分中值定理,并引入各应力分量的修正系数,得到黏弹性奇异应力场的简化式.结合核函数的数值反演结果分析修正系数表达式的取值范围,得到如下结论,若两相接触材料的持久模量相差很大,可以采用准弹性解的解析式较准确地描述界面端的奇异应力场;一般情况下,应力场不存在统一的奇异值和应力强度系数,当采用类似于准弹性解的表达式近似给出黏弹性应力场时,可以估计此近似描述的误差限.文中最后采用有限元分析黏弹性板端部嵌入部位的应力场,算例包括了黏弹性板与弹性金属支承、黏弹性板与黏弹性垫层所形成的滑移接触界面端,利用黏弹性有限元的数值结果验证理论分析所得结论的有效性.      

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.      

Heat and Mass Transfer in MHD Micropolar Flow Over a Vertical Moving Porous Plate in a Porous Medium

An analysis is presented for the problem of free convection with mass transfer flow for a micropolar fluid via a porous medium bounded by a semi-infinite vertical porous plate in the presence of a transverse magnetic field. The plate moves with constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface in which absorbs the micropolar fluid with a suction velocity varying with time. Numerical results of velocity distribution of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Also, the results of the skin-friction coefficient, the couple stress coefficient, the rate of the heat and mass transfers at the wall are prepared with various values of fluid properties and flow conditions.     

Forced and mixed convection boundary layer flow along a flat plate with variable viscosity and variable Prandtl number: new results

The steady laminar boundary layer flow along a flat plate is studied taking into account the variation of fluid viscosity and fluid Prandtl number with temperature. In the forced convection case the plate moves with constant velocity and its temperature varies in power law with x. In the mixed convection case the plate temperature is constant and the fluid moves upwards due to an external free stream and due to buoyancy forces. The results are obtained with the direct numerical solution of the boundary layer equations. The study concerns the wall heat transfer, the wall shear stress and velocity and temperature profiles across the boundary layer. The results of the present work are different from those existing in the literature, which have been obtained with the assumption of constant Pr number.     

On laminar bouudary layers with suction

In this paper, we obtain the asymptotic solution of the general equation for laminar boundary-layer flows with suction. Formulae for calculating the displacement thickness, momentum thickness, an’d skin friction are then derived. Furthermore, the problem of determining the separation point is dealt with. Finally, as a numerical example, we compute certain characteristic boundary-layer parameters for the case of uniform flow over flat plate with constant suction. Our numerical results obtained are in good agreement with those of Iglisch.     

AN ANALYSIS FOR DIFFRACTION OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN THICK PLATES WITH A CAVITY

By using the complex variable method and conformal mapping,the diffraction of flexu-ral waves and dynamic stress concentrations in thick plates with a cavity have been studied.A generalsolution of the stress problem of the thick plate satisfying the boundary conditions on the contour of anarbitrary cavity is obtained.By employing the orthogonal function expansion technique,the dynamicstress problem can be reduced to the solution of an infinite algebraic equation series.As an example,the numerical results for the dynamic stress concentration factor in thick plates with a circular,ellipticcavity are graphically presented.The numerical results are discussed.     

A solution for the vertical variation of stress,rather than velocity,in a three-dimensional circulation model

A simple technique is presented that allows a numerical solution to be sought for the vertical variation of shear stress as a substitute for the vertical variation of velocity in a three-dimensional hydrodynamic model. In its most general form the direct stress solution (DSS) method depends only upon the validity of an eddy viscosity relation between the shear stress and the vertical gradient of velocity. The rationale for preferring a numerical solution for shear stress to one for velocity is that shear stress tends to vary more slowly over the vertical than velocity, particularly near boundaries. Consequently, a numerical solution can be obtained much more efficiently for shear stress than for velocity. When needed, the velocity profile can be recovered from the stress profile by solving a one-dimensional integral equation over the vertical. For most practical problems this equation can be solved in closed form. Comparisons are presented between the DSS technique, the standard velocity solution technique and analytical solutions for wind-driven circulation in an unstratified, closed, rectangular channel governed by the linear equations of motion. In no case was the computational effort required by the velocity solution competitive with the DSS when a physically realistic boundary layer was included. The DSS technique should be particularly beneficial in numerical models of relatively shallow water bodies in which the bottom and surface boundary layers occupy a significant portion of the water column.     

Analysis of a laminar boundary layer flow over a flat plate with injection or suction

An analysis is performed to study a laminar boundary layer flow over a porous flat plate with injection or suction imposed at the wall. The basic equations of this problem are reduced to a system of nonlinear ordinary differential equations by means of appropriate transformations. These equations are solved analytically by the optimal homotopy asymptotic method (OHAM), and the solutions are compared with the numerical solution (NS). The effect of uniform suction/injection on the heat transfer and velocity profile is discussed. A constant surface temperature in thermal boundary conditions is used for the horizontal flat plate.     

Interlaminar hygrothermal stresses in laminated plates

Within the elasticity formulation the most general displacement field for hygrothermal problems of long laminated composite plates is presented. The equivalent single-layer theories are then employed to determine the global deformation parameters appearing in the displacement fields of general cross-ply, symmetric, and antisymmetric angle-ply laminates under thermal and hygroscopic loadings. Reddy’s layerwise theory is subsequently used to determine the local deformation parameters of various displacement fields. An elasticity solution is also developed in order to validate the efficiency and accuracy of the layerwise theory in predicting the interlaminar normal and shear stress distributions. Finally, various numerical results are presented for edge-effect problems of several cross-ply, symmetric, and antisymmetric angle-ply laminates subjected to uniform hygrothermal loads. All results indicate high stress gradients of interlaminar normal and shear stresses near the edges of laminates.     

Three dimensional unsteady convection and mass transfer flow through porous medium

The problem of three dimensional unsteady convection flow through a porous medium, with effect of mass transfer bounded by an infinite vertical porous plate is discussed, when the suction at the plate is transverse sinusoidal and the plate temperature oscillates in time about a constant mean. Assuming the free stream velocity to be uniform, approximate solutions are obtained for the flow field, the temperature field, the skin-friction and the rate of heat transfer. The dependence of solution on Pr (Prandtl number), Gr (Grashof number based on temperature), Gc (modified Grashof number based on concentration difference), Sc (Schimdt number), the frequency and the permeability parameter is also investigated.     

润滑力学中非牛顿流动的普遍Reynolds方程

本文导出了润滑力学中关于非牛顿流动的普遍 Reynolds 方程。这一方程适用于多种非牛顿流动模型,可以用于解算热流体动力润滑或热弹性流体动力润滑膜的压力分布。本文给出了一种同时求出剪应力、剪切率、速度和等效粘度的解法,并以两种润滑力学中常用的流变模型为例,应用这一方程得到了线接触热弹性流体动力润滑问题的数值解。     

Winkler地基上变厚度圆板的轴对称弯曲

本文提出了Winkler地基上变厚度圆板轴对称弯曲的传递矩阵算法。首先,根据贝塞尔函数理论获得了等厚度圆板和环板单元在任意荷载作用下轴对称弯曲的解析解,这些解均由通解和特解两部分组成。基于这些解析解,导出了等厚度圆板和环板单元的传递矩阵。然后沿径向将变厚度圆板划分成一个等厚度圆板单元和一系列等厚度环板单元,应用传递矩阵算法原理获得了变厚度圆板的整体传递矩阵。引入圆板的边界条件,给出了该板每条节线上的挠度、径向转角、径向弯矩和径向剪力。最后,讨论了受均布荷载作用的简支线性变厚度圆板的弯曲,将本文数值解与解析解进行比较,证实了本文方法的有效性,并简要地讨论了地基参数对板挠度和径向弯矩的影响。     

含孔软铁磁材料Mindlin板中弹性波散射问题

基于考虑磁弹相互作用的Mindlin板弯曲波动方程，采用波函数展开法，分析研究 了含孔软铁磁材料Mindlin板中弹性波散射与动应力集中问题，给出了问题的分析 解和数值算例. 通过分析发现：磁感应强度对动弯矩集中系数和动剪力集中系数有 增加的作用，特别是在低频的情况下.     

Solution of De Saint Venant flexure-torsion problem for orthotropic beam via LEM (Line Element-less Method)

In this paper the numerical technique, labelled Line Element-less Method (LEM), is employed to provide approximate solutions of the coupled flexure-torsion De Saint Venant problem for orthotropic beams having simply and multiply-connected cross-section. The analysis is accomplished with a suitable transformation of coordinates which allows to take full advantage of the theory of analytic complex functions as in the isotropic case.A boundary value problem is formulated with respect to a novel complex potential function whose real and imaginary parts are related to the shear stress components, the orthotropic ratio and the Poisson coefficients. This potential function is analytic in all the transformed domain and then expanded in the double-ended Laurent series involving harmonic polynomials.The solution is provided employing an element-free weak form procedure imposing that the squared net flux of the shear stress across the border is minimum with respect to the series coefficients.Numerical implementation of the LEM results in system of linear algebraic equations involving symmetric and positive-definite matrices. All the integrals are transferred into the boundary without requiring any discretization neither in the domain nor in the contour.The technique provides the evaluation of the shear stress field at any interior point as shown by some numerical applications worked out to illustrate the efficiency and the accuracy of the developed method to handle shear stress problems in presence of orthotropic material.     

Unsteady compressible boundary layer flow over a circular cone near a plane of symmetry

An analysis has been performed to study the unsteady laminar compressible boundary layer governing the hypersonic flow over a circular cone at an angle of attack near a plane of symmetry with either inflow or outflow in the presence of suction. The flow is assumed to be steady at time t=0 and at t>0 it becomes unsteady due to the time-dependent free stream velocity which varies arbitrarily with time. The nonlinear coupled parabolic partial differential equations under boundary layer approximations have been solved by using an implicit finite-difference method. It is found that suction plays an important role in stabilising the fluid motion and in obtaining unique solution of the problem. The effect of the cross flow parameter is found to be more pronounced on the cross flow surface shear stress than on the streamwise surface shear stress and surface heat transfer. Beyond a certain value of the cross flow parameter overshoot in the cross flow velocity occurs and the magnitude of this overshoot increases with the cross flow parameter. The time variation of the streamwise surface shear stress is more significant than that of the cross flow surface shear stress and surface heat transfer. The suction and the total enthalpy at the wall exert strong influence on the streamwise and cross flow surface shear stresses and the surface heat transfer except that the effect of suction on the cross flow surface shear stress is small.     

Three dimensional free convection flow and heat transfer along a porous vertical plate

The free convection flow along a vertical porous plate with transverse sinusoidal suction velocity distribution is investigated. Due to this type of suction velocity at the plate the flow becomes three dimensional one. For the asymptotic flow condition, the wall shear stress in the direction of main flow for different values of buoyancy parameter G is obtained. For G=0, the skin friction in the direction of free stream and the rate of heat transfer from the plate to the fluid are given. It is found that these results differ from those obtained by Gersten and Gross.     

板结构计算模型的新发展

现代复合材料层合板具有高强和轻型的突出优点，从而在军工和民用等诸多领域发挥着重要作用。这种板结构的特点是随着纤维走向的不同，层间材料的物理-力学特性发生剧烈变化。沿板厚方向变形的梯度比较陡峭，并在层间结合面处发生强不连续，呈现zig-zag （锯齿状）现象。这导致横向剪应变在板的静态和动态响应中发生重要作用，不计横向变形的经典组合板计算模型CLPT难以适应现代多层板计算分析的需要。考虑横向剪切变形影响的板的计算模型得到重视和发展。需要指出，现有各种考虑剪切变形影响的计算模型虽然有了很大的发展，但在全面和准确性上仍然存在一定的不足，难以适应现代多层组合板横向力和物理性能多变的情况。模型预测的沿板厚方向位移和应力的变化规律难以通过严格的检验。本文提出的以比例边界有限元为基础的正交各向异性板的数值计算模型，同时可适用于各种薄板与厚板的分析，对现代复合材料层合板的分析具有特殊的优越性。所得到的板的位移、正应力和剪应力沿板厚方向的变化，与三维弹性理论的标准解高度吻合。数值算例进一步表明，随着层间纤维走向的变化，板内位移场和应力场沿板厚方向剧烈变化所呈现的锯齿现象均可以精准地进行模拟。据此，本文建议方法对现代板分析的广泛适应性和高度准确性得到了充分论证。