On one contact problem of plane elasticity theory with partially unknown boundary

Applications of elastic plates weakened with full-strength holes are of great interest in several mechanical constructions (building practice, in mechanical engineering, shipbuilding, aircraft construction, etc). It's proven that in case of infinite domains the minimum of tangential normal stresses (tangential normal moments) maximal values will be obtained on such contours, where these values maintain constant(the full strength holes). The solvability of these problems allow to control stress optimal distribution at the hole boundary via appropriate hole shape selection. The paper addresses a problem of plane elasticity theory for a doubly connected domain S on the plane z = x + iy, which external boundary is an isosceles trapezoid boundary; the internal boundary is required full-strength hole including the origin of coordinates. In the provided work the unknown full-strength contour and stressed state of the body were determined. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On one contact problem of plane elasticity for a doubly connected domain: application to a hexagon

The paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a regular hexagon boundary, and the internal boundary is the required full-strength hole including the origin of coordinates. Hexagon’s two vertices are laid at the axis Oy, and the middle points of its two opposite sides are laid at the axis Ox. This full-strength hole is cycle symmetric. It is assumed that to every link of the broken line of the outer boundary of the given body are applied absolutely smooth rigid stamps with rectilinear bases, which are under action of the force P that applies to their middle points. There is no friction between the surface of given elastic body and stamps. The unknown full-strength contour is free from outer actions. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body) and unknown parts of its boundary are determined under the condition that the tangential normal moment arising at it takes a constant value. Such holes are called full-strength holes. Numerical analysis are also performed and the corresponding graphs are constructed.     

双周期裂纹场平面弹性焊接的数学问题

本文讨论双周期胞腔中含任意形状裂纹的不同材料的弹性平面焊接(焊线为任意形状的封闭光滑曲线)的第二基本问题.运用Мусхелишвили复变函数方法,对这类弹性平面问题建立起了数学模型,将求解弹性平衡问题化归为寻求复应力函数满足一定边界条件的边值问题,然后构造其解的形式,再将其转化为正则型的奇异积分方程,数学上严格证明其解的存在与唯一.     

平面弹性第二基本问题的新提法*

本文给出了平面弹性第二基本问题的一般提法,这里在多连通弹性区域的各个闭边界围线上给出的位移只是相对的,各允许有一不同的刚性移动.在这种情况下,为了解的唯一性,证明了必须另外给出每一边界围线上外力的主矢量与主力矩.还给出了求解的方法以及一些例题.     

SINGULAR INTEGRAL EQUATIONS WITH COSECANT KERNEL IN SOLUTIONS WITH SINGULARITIES OF ORDER ONE

In this article, periodic Riemann boundary value problem with period 2aπalong closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted in the strip 0< Rez相似文献   

Torsion of multiconnected bodies with cracks

A general expression is obtained for a multiconnected anisotropic (isotropic) body with rectilinear cracks, for the complex torsion potential that exactly satisfies conditions on the cracks and contains unknown functions determined from the boundary conditions on closed contours. A solution is given for the torsion problem of an elliptical rod with a crack. Results are presented of investigation on the clarification of the influence of geometric and elastic characteristics of the rod on the magnitude of the stress intensity coefficient near an outer edge.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 34–38, 1987.     

Diffraction of water waves due to a presence of two headlands

This paper is concerned with a boundary value problem for the Helmholtz equation on a horizontal infinite strip with obstacles. The derivation of Helmholtz equation from shallow water equations is given and the boundary value problem with an arbitrary shap of headland is stated. The boundary conditions are of the general Neumann type, and thus we use the finite difference method in numerical solution. Helholtz equation is replaced by the five-points formula and for the points close to the boundary, Taylors expansions are made useful with non-uniform spacing. For solving the resulting system of linear equations, the “Mathematica” package is used. The graphs show the velocity potential contours in the cases, of semielliptic, semicircular and narrow headland. Also, we discuss the problem in the presence of two headlands.     

The Schwarz problem for infinite sets of intervals

We solve the Schwarz problem for boundary contours consisting of countable sets of segments with limit point at infinity, including the periodic case. The solution is a result of a reduction to corresponding Riemann boundary-value problems.     

On some generalizations of the classical Riemann problem

The special class of the homogeneous vector boundary Riemann problems on the finite sequence of algebraic surfaces is investigated completely. Its coefficients are the noncommutative permutative matrices of the arbitrary but not prime order, and boundary conditions are given on the system of open contours. The constructive solution procedure and definite structure of the canonical solution matrix are obtained and present some generalizations of the classical Riemann problem. Simultaneously the corresponding class of algebraic equations for the appropriate covering surfaces is formed explicitly too. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress analysis near holes in shells of revolution: Geometrically nonlinear problem

An approach is proposed to numerical analysis of the stress state of shells of revolution in the geometrically nonlinear setting. It is based on a combination of linearization and discrete orthogonalization methods. Stress concentration is analyzed for a flexible spherical shell with a central hole, depending on load intensity, hole size, and the boundary conditions on the contours in the precritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 97–101, 1985.     

Computational homogenization of materials with small deformation to determine configurational forces

In this work the mechanical boundary condition for the micro problem in a two-scaled homogenization using a FE² approach is discussed. The strain tensor is often used in the literature for small deformation problem to determine the boundary conditions for the boundary value problem on the micro level. This strain tensor based boundary condition gives consistent homogenized mechanical quantities, e.g. stress tensor and elasticity tensor, but the present work points out that it leads to unphysical homogenized configurational forces. Instead, we propose a displacement gradient based boundary condition for the micro problem. Results show that the displacement gradient based boundary condition can give not only the consistent homogenized mechanical quantities but also the appropriate homogenized configurational forces. The interpretation of the displacement gradient based boundary condition is discussed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A problem of the bending of a plate for a doubly connected domain with a partially unknown boundary

The problem of the bending of an isotropic elastic plate, bounded by two rectangles with vertices lying on the same half-line, drawn from the common centre, is considered. The vertices of the inner rectangle are cut by convex smooth arcs (we will call the set of these arcs the unknown part of the boundary). It is assumed that normal bending moments act on each rectilinear section of the boundary contours in such a way that the angle of rotation of the midsurface of the plate is a piecewise-constant function. The unknown part of the boundary is free from external forces. The problem consists of determining the bending of the midsurface of the plate and the analytic form of the unknown part of the boundary when the tangential normal moment acting on it takes a constant value, while the shearing force and the normal bending moments and torques are equal to zero. The problem is solved by the methods of the theory of boundary-value problems of analytical functions.     

Local Artificial Boundary Conditions for the Incompressible Viscous Flow in a Slip Channel

1.IntroductionManyboundaxyvaJueproblemsofpartialdiffereotialequationsinvo1vingunboundeddomainoccurinmanyareasofapplications,e-g.lfluidflowaroundobstacles,couplingofstructureswithfoundationandsoon.Forgettingthenumericalsolutionsoftheproblemsonunboundeddomian,anaturalapproachistocutoffanunboundedpartofthedomainbyintroducinganartificialboundaryandsetupanaPpropriatear-tificialboundaryconditiononthearti%ialboundaryThentheoriginalproblemisapproximatedbyaproblemonbou.d.dfdomain.Inthelastteny6aJrs,b…     

Axisymmetric nose shapes of specified aspect ratio,optimum or close to optimum with respect to wave drag

The problem of constructing optimum or close to optimum nose-shapes of bodies of revolution of fixed aspect ratio in a supersonic flow is solved within the framework of a perfect (inviscid and non-heat-conducting) gas. Their contour includes the front face, that is, the boundary extremum section with respect to the length and, adjacent to it, the smooth, slightly sloping section that makes a corner. In case of low aspect ratios, the slightly sloping section is the result of the exact solution of a variational problem. In the case of aspect ratios which exceed a certain value, depending on the free-stream Mach number M_∞, the exact solution requires the introduction of small internal breaks with corner points where even the dominant one of these only has a weak effect on the drag value. Contours which are referred to as “close to optimum” do not satisfy the optimality condition, which defines the dominant corner. In the examples (1.2 ≤ M_∞ ≤ 10) for which calculations were carried out, conical nose shapes were found to be far worse that the optimum ones. For contours which are optimum in the approximation of Newton's formula and also, optimum blunt and pointed, power-law nose shapes, the situation occurs for low-aspect ratios and low supersonic Mach numbers (pointed, power-law contours can only be successfully constructed for fairly high aspect ratios). The fact that the front face is a section of a boundary extremum is shown by comparing the drags of bodies obtained with different permissible variations of the front face. An alternative proof, which is not limited by the actual form of the variation in the front face, can be obtained from the solution of the conjugate problem, formulated within the framework of the general method of Lagrange multipliers. This problem is also of interest in its own right, in particular, on account of the singularities, revealed during its formulation, in the reflection of the discontinuities of the Lagrange multipliers from the sonic line with parts of them becoming infinite at the point of reflect.     

Analyticity for the Navier-Stokes equations governed by surface tension on the free boundary

A free boundary problem for an incompressible viscous fluid is considerd; the boundary is to be determined by equilibrium conditions involving the fluid's stress tensor and its surface tension. It is proved that if the data of the problem are analytic then the free boundary, the velocity vector, and the pressure are all analytic.     

Regularity for minimizers of some variational problems in plasticity theory

A variational problem for a functional depending on the symmetric part of the gradient of the unknown vectorvalued function is considered. We assume that the integrand of the problem has power growth with exponent less than two. We prove the existence of summable second derivatives near a flat piece of the boundary. In the two-dimensional case, Hölder continuity up to the boundary of the strain and stress tensors is established. Bibliography: 6 titles.     

圆外平面弹性问题的边界积分公式

将边界上的应力函数及其法向导数展开为罗朗级数,与复应力函数的罗朗级数的表达式对比,可以确定罗朗级数的各系数,再利用傅利叶级数和卷积的几个公式进行计算,得到应力函数边界积分公式．通过边界的应力函数及其法向导数的积分,直接得到圆外应力函数值,并给出几个算例,表明结果用于求解单位圆外平面弹性问题十分方便．     

工程中一类拼接问题的复变方法

讨论工程中一类含边裂缝弹性材料补强的拼接问题 .根据平面弹性复变方法 ,问题归结为一类解析函数的边值问题 ,通过有效的分析方法和积分变换 ,进一步将问题简化为一类奇异积分方程 ,证明了方程解的存在唯一 ,并对方程解的简化进行了研究 ,得到了弹性材料体内应力分布的封闭形式解 ,并导出一直裂缝情况裂缝尖端应力强度因子的表达式     

The diffusion of a discontinuity of the shear stress at the boundary of a viscoplastic half-plane

For the problem of the diffusion of a discontinuity of the shear stress at the boundary of a half-plane, which is a special case of the general problem of the diffusion of a vortex layer, the classes of media and types of assignment of boundary conditions for which self-similar solutions exist are discussed. For a viscoplastic medium in a half-plane the problem reduces to the problem in a layer of time-variable thickness, the solution of which does not possess the property of analyticity. The long-term asymptotic of this problem are investigated. In the case where, at an accessible boundary, it is possible simultaneously to measure both the shear stress and the horizontal velocity, an algorithm is proposed for finding a quantity that is difficult to measure, A namely, the thickness of the zone of viscoplastic flow.     

对称迭层矩形板的平面应力分析

常用的对称迭层板为各向异性板．根据平面应力问题的基本方程精确地用应力函数解法求得了各向异性板的一般解析解．推导出平面内应力和位移的一般公式,其中积分常数由边界条件来决定．一般解包括三角函数和双曲函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式解,它能满足4个角的边界条件．因此一般解可用以求解任意边界条件下的平面应力问题．以4边承受均匀法向和切向载荷以及非均匀法向载荷的对称迭层方板为例,进行了计算和分析．