Boundary Value Problems of Plane Elasticity Involving Orientations of Displacements and Tractions

This paper presents unconventional formulations of boundary problems of plane elasticity formulated in terms of orientations of tractions and displacements on a closed contour separating internal and external domains as the boundary conditions. These are combined with the conditions of continuity of tractions or displacements across the boundary. Therefore the magnitudes of neither tractions nor displacements are assumed on the contour. Four boundary value problems for both external and internal domains are investigated by analyzing the solvability of the corresponding singular integral equations. It is shown that all considered problems can have non-unique solutions expressed as linear combinations of particular solutions and, hence, contain free arbitrary parameters, the number of which is finite and determined by the contour orientations of tractions and/or displacements     

A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR THE SECOND-ORDER E2 CLASS ELLIPTIC SYSTEMS IN GENERAL FORM

IntroductionThispaperdiscussesaclassofnonlinearboundaryvalueproblems (BVP)forthesecond_orderE2 classellipticsystemsingeneralform ,whichareofgreatimportanceinmathematicalsciencesandengineering ,andattractmoreandmoreinterestsfrommanyresearchersdomesticandabroad .ThesenonlinearBVPwerefirstintroducedbyRussianresearcher,L .V .Wolfersdorf,whereaclassofnonlinearBVPforholomorphicfunctionswasstudiedandsomeefficientresultsderived[1,2 ].LaterSigridKenchtandAliSeifMashimba[3],R .P .Gilbertandthea…     

Boundary integral equations of unique solutions in elasticity

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弹性力学中一种新的边界轮廓法

利用基本解的特性,将面力积分方程化成仅含有Ｃａｕｃｈｙ主值积分的形式,基于这种边界积分方程,提出了一种新的边界轮廓法,对于三维问题,该方法只须计算沿边界单元界线的线积分,对二维问题,则只需计算边界单元两点的热函数之差,无须进行数值积分计算,实例计算说明该方法是有效的。     

A class of nonlinear boundary value problems for the second-order<Emphasis Type="Italic">E</Emphasis><Subscript>2</Subscript> class elliptic systems in general form

A class of nonlinear boundary value problems (BVP) for the second-order E₂ class elliptic systems in general form is discussed. By introducing a kind of transformation, this kind of BVP is reduced to a class of generalized nonlinear Riemann-Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods. Foundation items: the National Natural Science Foundation of China (19671056); Shanghai Municipal Natural Scientific Foundation (99ZA14030, 01ZA14023); Jiangxi Provincial Natural Scientific Foundation (981102, 0211014) Biographies: LI Ming-zhong (1935−); XU Ding-hua (1967−)     

应用边界积分法求圆形夹杂问题的解析解

边界元方法作为一种数值方法, 在各种科学工程问题中得到了广泛的应用.本文参考了边界元法的求解思路, 从Somigliana等式出发, 利用格林函数性质,得到了一种边界积分法, 使之可以用来寻求弹性问题的解析解.此边界积分法也可以从Betti互易定理得到. 应用此新方法, 求解了圆形夹杂问题.首先设定夹杂与基体之间完美连接, 将界面处的位移与应力按照傅里叶级数展开,根据问题的对称性与三角函数的正交性来简化假设, 减少待定系数的个数.其次选择合适的试函数(试函数满足位移单值条件以及无体力的线弹性力学问题的控制方程),应用边界积分法, 求得界面处的位移与应力的值. 然后再求解域内位移与应力.得到了问题的精确解析解, 当夹杂弹性模量为零或趋向于无穷大时,退化为圆孔或刚性夹杂问题的解析解. 求解过程表明,若问题的求解区域包含无穷远处时, 所取的试函数应满足无穷远处的边界条件.若求解区域包含坐标原点, 试函数在原点处位移与应力应是有限的.结果表明了此方法的有效性.      

Solution for dissimilar elastic inclusions in a finite plate using boundary integral equation method

This paper studies the boundary value problem for a finite plate containing two dissimilar inclusions. The matrix and the two inclusions have different elastic properties. The loadings applied along the outer boundary are in equilibrium. The mentioned problem is decomposed into three boundary value problems (BVPs). Two of them are interior BVP for the elastic inclusions, while the other is a BVP for the triply-connected region. Three problems are connected together through the common displacements and tractions along the interface boundaries. Explicit form for the complex variable boundary integral equation (CVBIE) is derived. After discretization of relevant BIEs, the solutions are evaluated numerically. Three numerical examples for different elastic constant combinations are provided.     

Degenerate scale problem in antiplane elasticity or Laplace equation for contour shapes of triangles or quadrilaterals

This paper provides several solutions to the degenerate scale for the shapes of triangles or quadrilaterals in an exterior boundary value problem (BVP) of the antiplane elasticity or the Laplace equation. The Schwarz-Christoffel mapping is used throughout. It is found that a complex potential with a simple form in the mapping plane satisfies the vanishing displacement condition (or w = 0) along the boundary of the unit circle when the dimension R reaches its critical value 1. This means that the degenerate size in the physical plane is also achieved. The degenerate scales can be evaluated from the particular integrals depending on certain parameters in the mapping function. The numerical results of degenerate sizes for the shapes of triangles or quadrilaterals are provided.     

Singular boundary method for 2D dynamic poroelastic problems

This paper extends a strong-form meshless boundary collocation method, named the singular boundary method (SBM), for the solution of dynamic poroelastic problems in the frequency domain, which is governed by Biot equations in the form of mixed displacement–pressure formulation. The solutions to problems are represented by using the fundamental solutions of the governing equations in the SBM formulations. To isolate the singularities of the fundamental solutions, the SBM uses the concept of the origin intensity factors to allow the source points to be placed on the physical boundary coinciding with collocation points, which avoids the auxiliary boundary issue of the method of fundamental solutions (MFS). Combining with the origin intensity factors of Laplace and plane strain elastostatic problems, this study derives the SBM formulations for poroelastic problems. Five examples for 2D poroelastic problems are examined to demonstrate the efficiency and accuracy of the present method. In particular, we test the SBM to the multiply connected domain problem, the multilayer problem and the poroelastic problem with corner stress singularities, which are all under varied ranges of frequencies.     

基于网格框架的非结构附面层网格生成技术

附面层网格质量是确保计算流体力学粘性计算精度的关键技术环节.本文针对复杂外形提出了全局一致的高质量附面层网格构造算法,该方法基于针对特征的表面网格区域分解技术,利用表面网格分片后的边界线及其法向量构造最终附面层网格的轮廓框架线,并通过径向基函数及线性插值算法生成完整的附面层网格.通过典型算例分析可以看出,该方法生成的非结构附面层网格精度和全局一致性较高,且能够有效避免复杂外形附面层网格局部及全局交叉现象.     

压电介质二维边界积分方程中的基本解

由于压电介质的变形－电场耦合效应及压电响应的各向异性，使解析求解压电介质问题的工作变量十分复杂，若采用边界元数值方法求解，必须具备积分方程中的基本解，本文根据电磁场方程及连续介质力学的耦合性质论层出了二维无限域中分别在单位力及单位电荷载作用下的位移场，电势场、应力场和电位移场的解，从而确立了边界积分方程中所必需的八个基本解。     

Modeling of Steady Flows in a Channel by Navier–Stokes Variational Inequalities

A mathematical model of a steady viscous incompressible fluid flow in a channel with exit conditions different from the Dirichlet conditions is considered. A variational inequality is derived for the formulated subdifferential boundary-value problem, and the structure of the set of its solutions is studied. For two-ption on the low Reynolds number is proved. In the three-dimensional case, a class of constraints on the tangential component of velocity at the exit, which guarantees solvability of the variational inequality, is found.     

弹性薄板弯曲问题的边界轮廓法

导出了弹性薄板弯曲问题边界积分方程的另一种形式,基于这种方程,提出了平板弯曲问题的边界轮廓法,讨论了三次边界单元边界轮廓法的计算列式,并给出了计算内力的边界轮廓法方程。该法无需进行数值积分计算,完全避免了角点问题和奇异积分计算。给出的算例,与解析解相比较,证实该方法的有效性。     

On the Plane Orthotropic Stress Problem of Quasi-Static Thermoelasticity

The plane stress boundary value problem of quasi-static linear orthotropic thermoelasticity is discussed. The thermoelastic system on a bounded simply-connected domain is decoupled. The decoupled temperature equation is investigated by using accurate estimation and the contraction mapping principle. Representations of solutions of the field equation are obtained, and some solvability results are proved. The results are of both theoretical and numerical interest.     

Singular solutions and Green's method in micropolar theory of elasticity

The matrices of fundamental solutions are constructed for a concentrated force as well as a concentrated couple varying harmonically in time and acting in an unbounded micropolar elastic continuum. These solutions are then used to obtain solutions for some other loading singularities. Integral representations, for the displacement and the rotation vectors are obtained by making use of the basic singular solutions. The formal solutions to two fundamental boundary value problems are expressed in terms of integrals which include given surface and body data and Green's functions.     

完备型二次边界轮廓法的研究及J积分和应力强度因子的计算

选择二次完全多项多作为位移形函数,对边界轮廓法作了进一步的发展,证明二维弹性断裂问题的Ｊ积分方程的被积分函数的散度等于零,将Ｊ积分化为边界点的势函数数值的计算,无需计算数值积分,算例表明,该方法较传统边界元法求得的结果精度更好。     

Integral equations of plane static boundary value problems of the elasticity theory for an inhomogeneous anisotropic medium

The static boundary value problems of plane elasticity for an inhomogeneous anisotropic medium in a simply connected domain are reduced to the Riemann–Hilbert problem for a quasianalytic vector. Singular integral equations over the domain are obtained, and their solvability is proved for a sufficiently wide anisotropy class. In the case of a homogeneous anisotropic body, the solutions of the first and second boundary value problems are obtained in closed form.     

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     

Out-of-plane loading of anisotropic bimaterials and semi-infinite materials

Summary Analytical closed-form solutions are proposed in a rather compact form for the stress and displacement fields induced by out-of-plane loading of a semi-infinite anisotropic material with inclined strata. The solutions are then extended to include the case of a bimaterial with a planar interface. Several boundary conditions are considered for the interface which may be between two anisotropic half-planes with different elastic properties, or two different orientations of the strata in the same material.     

Three-dimensional flow of Oldroyd-B fluid over surface with convective boundary conditions

The present study addresses the three-dimensional flow of an Oldroyd-B fluid over a stretching surface with convective boundary conditions. The problem formulation is presented using the conservation laws of mass, momentum, and energy. The solutions to the dimensionless problems are computed. The convergence of series solutions by the homotopy analysis method (HAM) is discussed graphically and numerically. The graphs are plotted for various parameters of the temperature profile. The series solutions are verified by providing a comparison in a limiting case. The numerical values of the local Nusselt number are analyzed.