平面弹性力学充要的随机边界元

将平面弹性力学确定性的充分必要的边界积分方程推广到含材料常数随机的不确定问题中去，给出了位移的均值以及偏差的充分必要的边界积分方程。数值计算结果表明，和确定性的积分方程一样，习用的随机边界积分方程在退化尺度附近，无论是均值还是偏差都存在巨大的误差，而充要的随机边界积分方程则始终保持良好的精度     

一类新变量的Reissner板弯曲边界元法

文［４］导出了二维弹性力学平面问题的一类新型边界积分方程，本文将该理论和方法推广到三变量的Ｒｅｉｓｓｎｅｒ板弯曲中，给出边界场变量含广义位移和新型广义力的边界积分方程。从而边界弯矩应力张量可直接由离散边界积分方程求出。     

弹性力学问题中一个新的边界积分方程——自然边界积分方程

位移导数边界积分方程一直存在着超奇异积分计算的障碍,该文提出以符号算子δye和εye作用于位移导数边界积分方程,施用一系列变换将边界位移、面力和位移导数转成为新的边界张量,从而得到一个新的边界积分方程－－自然边界积分方程,自然边界积分方奇异性为强奇性,文中给出了相应的Cauchy主值积分算式,自然边界积分方程与位移边界积分方程联合可直接获取边界应力,几个算例表明了自然边界积分方程的正确性。     

Reissner型板表面裂纹应力强度因子的线弹簧-不连续位移边界积分方程解法

本文由Reissner型板的不连续位移基本解，根据Betti互换定理，导出了Reissuer型板的不连续位移边界积分方程；结合平面问题的不连续位移边界积分方程─—边界元方法和线弹簧模型，给出了Rrissner型板表面裂纹应力强度因子的线弹簧-不连续位移边界积分方程解法。     

Reissner裂板表面裂纹应力强度因子的线弹簧—不连续位移…

本文由Ｒｅｉｓｓｎｅｒ型板的不连续位移基本解，根据Ｂｅｔｔｉ互换定理，导出了Ｒｅｉｓｓｎｅｒ型板的不连续位移边界积分方程，结合平面问题的不连续位移边界积分方程－－边界元方法和线弹簧模型，给出了Ｒｅｉｓｓｎｅｒ型板表面裂纹应力强度因子的线弹簧－不连续位移边界积分方程解法。     

弹性理论几类导数边界积分方程之间的变换关系

导数场边界积分方程通常难以应用，因为存在着超奇异主值积分的计算障碍。弹性理论中有几类不同的位移导数边界积分方程，本文采用算子δij和∈ij(排列张量)作用于这些导数边界积分方程，做一系列变换，原有的超奇异积分被正则化为强奇异积分获解。从而建立了这些位移导数边界积分方程之间的转换关系，它们均可以归结为自然边界积分方程。自然边界积分方程仅存在容易计算的Cauchy主值积分。自然边界积分方程分析可直接获得边界应力和位移导数。     

弹性力学的一种边界无单元法

首先对移动最小二乘副近法进行了研究，针对其容易形成病态方程的缺点，提出了以带权的正交函数作为基函数的方法－改进的移动最小二乘副近法，改进的移动最小二乘逼近法比原方法计算量小，精度高，且不会形成病态方程组，然后，将弹性力学的边界积分方程方法与改进的移动最小二乘逼近法结合，提出了弹性力学的一种边界无单元法，这种边界无单元法法是边界积分方程的无网格方法，与原有的边界积分方程的无网格方法相比，该方法直接采用节点变量的真实解为基本未知量，是边界积分方程无网格方法的直接解法，更容易引入界条件，且具有更高的精度，最后给出了弹性力学的边界无单元法的数值算例，并与原有的边界积分方程的无网格方法进行了较为详细的比较和讨论。     

基于等几何边界元法的声学敏感度分析

基于非均匀有理B样条(NURBS)曲面建模技术,边界物理量同样用NURBS基函数插值,推导出三维声场等几何边界积分方程。进一步以控制点为设计变量,用直接微分法推导出等几何敏感度边界积分方程,给出声场声压对形状参量的敏感度。针对边界积分方程中的超奇异积分,使用奇异相消技术并结合Cauchy主值积分和Hadamard有限部分积分处理,给出了超奇异积分的NURBS插值半解析表达式。数值算例验证了本文算法求解声学结构形状敏感度的有效性,为声学结构的整体形状优化打下基础。     

第二类抛物型变分不等式中的MRM方法

本文讨论了第二类抛物型变分不等式中的MRM(多重互易方法)方法。首先采用时间项半离散和隐格式方法将抛物型变分不等式化解为一个椭圆变分不等式，然后利用MRM-边界积分方程，将其化解为MRM-边界混合变分不等式，并给出了MRM-边界混合变分不等式解的存在唯一性。说明了该MRM-边界混合变分不等式与常规边界积分方程得到的边界混合变分不等式是一致的，并且具有更容易编程实现。这为使用MRM边界元方法数值求解抛物型变分不等式提供了方法和理论依据。文末给出了数值算例。     

弹性力学平面问题的等价边界积分方程的边界轮廓法

基于边界积分方程中被积函数散度为零的特性,提出了弹性力学平面问题的等价边界积分方程的边界轮廓法,该方法无需进行数值积分,只需要计算单元两结点势函数值之差。实例计算说明,基于传统的边界积分方程的边界轮廓法所得到的面力结果是错误,而本文建立的边界轮廓法则可给出精确的结果。     

A penny shaped crack in a transversely isotropic material under non-axisymmetric impact loads

A solution is given for problems involving non-axisymmetric dynamic impact loading of a penny shaped crack in a transversely isotropic medium. Laplace and Hankel transforms are used to reduce the equations of elasticity to integral equations, and solutions are obtained for the three modes of fracture. The stress intensity factors are determined for a penny shaped crack loaded by concentrated normal impact forces and concentrated radial shear impact forces. The integral equations are solved by numerical methods, and the results are plotted showing how the dynamic stress intensity factors are influenced by the asymmetric loading.     

Equilibrium of a Prestressed Strip Reinforced with Elastic Plates

The contact problem for a prestressed elastic strip reinforced with equally spaced elastic plates is considered. The Fourier integral transform is used to construct an influence function of a unit concentrated force acting on the infinite elastic strip with one edge constrained. The transmission of forces from the thin elastic plates to the prestressed strip is analyzed. On the assumption that the beam bending model and the uniaxial stress model are valid for an elastic plate subjected to both vertical and horizontal forces, the problem is mathematically formulated as a system of integro-differential equations for unknown contact stresses. This system is reduced to an infinite system of algebraic equations solved by the reduction method. The effect of the initial stresses on the distribution of contact forces in the strip under tension and compression is studied     

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.     

A Prestressed Elastic Strip with Elastic Reinforcements

A contact problem is studied for a prestressed elastic strip with an elastic reinforcement. The integral Fourier transform is used to construct an influence function for an infinite strip with one face fixed. A unit concentrated force is applied to the strip at an arbitrary angle. The contact problem on force transfer from a thin infinite stringer to the prestressed strip is solved. The problem is mathematically formulated as a system of integro-differential equations for the unknown contact stresses on the assumption that the beam bending model and the uniaxial stress model are valid for the stringer, which is subjected to both vertical and horizontal forces. This system is solved in a closed form using the integral Fourier transform. The contact stresses are expressed in terms of Fourier integrals in a quite simple form. The influence of the initial stresses on the contact stress distribution is analyzed, and effects of concentrated load are revealed     

Minimizing stress concentrations around an elliptical hole by concentrated forces acting on the uniaxially loaded plate

When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy's integral formula. And then, based on the superposition principle, the analytical solutions for stress around an elliptical hole in an infinite plate subjected to a uniform far-field stress and concentrated forces, are obtained. Tangential stress concentration will occur on the hole boundary when only far-field uniform loads are applied. When concentrated forces are applied in the reversed directions of the uniform loads, tangential stress concentration on the hole boundary can be released significantly. In order to minimize the tangential stress concentration, we need to determine the optimum positions and values of the concentrated forces. Three different optimization methods are applied to achieve this aim. The results show that the tangential stress can be released significantly when the optimized concentrated forces are applied.     

Linear and geometrically nonlinear analysis of non-uniform shallow arches under a central concentrated force

In this paper an integral equation solution to the linear and geometrically nonlinear problem of non-uniform in-plane shallow arches under a central concentrated force is presented. Arches exhibit advantageous behavior over straight beams due to their curvature which increases the overall stiffness of the structure. They can span large areas by resolving forces into mainly compressive stresses and, in turn confining tensile stresses to acceptable limits. Most arches are designed to operate linearly under service loads. However, their slenderness nature makes them susceptible to large deformations especially when the external loads increase beyond the service point. Loss of stability may occur, known also as snap-through buckling, with catastrophic consequences for the structure. Linear analysis cannot predict this type of instability and a geometrically nonlinear analysis is needed to describe efficiently the response of the arch. The aim of this work is to cope with the linear and geometrically nonlinear problem of non-uniform shallow arches under a central concentrated force. The governing equations of the problem are comprised of two nonlinear coupled partial differential equations in terms of the axial (tangential) and transverse (normal) displacements. Moreover, as the cross-sectional properties of the arch vary along its axis, the resulting coupled differential equations have variable coefficients and are solved using a robust integral equation numerical method in conjunction with the arc-length method. The latter method allows following the nonlinear equilibrium path and overcoming bifurcation and limit (turning) points, which usually appear in the nonlinear response of curved structures like shallow arches and shells. Several arches are analyzed not only to validate our proposed model, but also to investigate the nonlinear response of in-plane thin shallow arches.     

Mode-III interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials

Summary  The problem of an interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials is considered under the conditions of anti-plane shear and in-plane electric loading. The crack surfaces are assumed to be impermeable to the electric field. An integral transform technique is employed to reduce the problem under consideration to dual integral equations. By solving the resulting dual integral equations, the intensity factors of the stress and the electric displacement and the energy release rate as well as the crack sliding displacement and the electric voltage across the crack surfaces are obtained in explicit form for the case of concentrated forces and free charges at the crack surfaces and at the boundary. The derived results can be taken as fundamental solutions which can be superposed to model more realistic problems. Received 10 November 2000; accepted for publication 28 March 2001     

Two-dimensional sliding frictional contact of functionally graded materials

A multi-layered model for sliding frictional contact analysis of functionally graded materials (FGMs) with arbitrarily varying shear modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be a linear function while the Poisson's ratio is assumed to be a constant. In the contact area, it is assumed that the friction is one of Coulomb type. With this model the fundamental solutions for concentrated forces acting perpendicular and parallel to the FGMs layer surface are obtained. Then the sliding frictional contact problem of a functionally graded coated half-space is investigated. The transfer matrix method and Fourier integral transform technique are employed to cast the problem to a Cauchy singular integral equation. The contact stresses and contact area are calculated for various moving stamps by solving the equations numerically. The results show that appropriate gradual variation of the shear modulus can significantly alter the stresses in the contact zone.     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

Solutions of the integral equations for shearing stresses in two-material curved beams

In the same way as shearing stresses for curved beams made of one material, the problem of evaluating the shearing stresses of composite curved beams is also reduced to one of solving the integral equations. Solving directly two integral equations can derive the formulae of shearing stresses, which satisfy not only the equilibrium equations but also the static boundary conditions on the boundary surfaces of the beams. The present analysis will be used to investigate the shearing stresses of a cantilevered curved beam made of two materials, which is loaded by a concentrated force at its free end. The comparison between the numerical results of shearing stresses obtained using the equations developed in this paper and a three-dimensional finite element analysis shows excellent agreement.