弹性地基上的自由矩形板

在弹性地基上的自由矩形板的弯曲,在弹性薄板理论中也是个难题.本文以叠加法提供一个精确解.它满足微分方程,自由边界的条件以及自由角点条件.这样将导致一系列无穷联立方程.所解的问题为在板的中点作用一集中力这问题.我们并以地基反力应与这集中力相平衡.校核所作的计算是否正确.     

Modeling a swimming fish with an initial boundary value problem: Unsteady maneuvers of an elastic plate with internal force generation

In order to model unsteady maneuvers in swimming fish, we develop an initial-boundary value problem for a fourth-order hyperbolic partial differential equation in which the fish's body is treated as an inhomogeneous elastic plate. The model is derived from the three-dimensional equations of elastic dynamics, and is essentially a simple variant of the classical Kirchhoff model for a dynamic plate. The model incorporates body forces generating moment to simulate muscle force generation in fish. The initial-boundary value problem is reduced to a beam model in one spatial dimension and formulated computationally using finite differences. Interaction with the surrounding water is represented by nonlinear viscous damping. Two example applications using simple but physically reasonable physiological parameters are presented and interpreted. One models the acceleration from rest to steady swimming, the other a rapid turn from rest.     

Buckling of an annular plate under tensile point loading

In this paper, we address the stability of an elastic thin annular plate stretched by two point loads that are located on the outer boundary. A roller support is considered on the outer boundary while the inner edge of the plate is free. Muskhelishvili’s theory of complex potentials has been applied to obtain a solution of the plane problem in the form of a power series. The buckling problem has been solved using the Rayleigh–Ritz method, based on the energy criterion. The critical Euler force and the respective buckling mode have been computed. Dependence between the critical force and the relative orifice size has been illustrated. Analysis of the results has shown that a symmetric buckling mode takes place for a sufficiently large hole, with the greatest deflection observed around the hole along the force line. However, an antisymmetric buckling mode occurs for relatively small holes, with the greatest deflection being along a line that is orthogonal to the force line.     

集中载荷作用下层合厚圆板的轴对称弯曲

从三维弹性力学基本方程出发,建立了横观各向同性层合圆板轴对称弯曲问题的状态方程,并将板面的集中载荷展成付里叶贝塞尔级数,从而给出问题的解析解,此解满足弹性力学全部方程,计及了所有独立的弹性常数,并满足层间连续性条件。     

Determination of the Steady-State Response of Viscoelastically Point-Supported Rectangular Orthotropic Plates

In the present study, the steady-state response to a sinusoidally varying concentrated force acting at the center of a rectangular point-supported orthotropic elastic plate is analyzed. The equation of the plate is represented by finite-difference expressions. By using the ordinary finite-difference technique, the problem is reduced to the solution of a system of algebraic equations. The influence of mechanical properties of the plate material on the vibration modes and the steady-state response of the plate is investigated numerically. Also, the effect of location of the point supports is studied. The problems considered are solved within the framework of the Kirchhoff-Love hypothesis.     

Spannungsfeld der auf den Rand einer elastisch anisotropen Halbebene wirkenden Tangentialkraft

Summary The stress distribution obtained by solving the two-dimensional problem in an anisotropic medium, with boundary conditions of a concentrated tangential load acting on the boundary of a semi-infinite plate, is purely radial. The solution is given in closed form and is combined with the solution for a concentrated normal load to solve the problem of an inclined force acting on the boundary.     

Transition functions of the Il'yushin approximation method

Transition functions are constructed whose direct substitution for combinations of elastic constants in solutions of problems of elasticity theory permits writing the solution of the corresponding viscoelastic problems. Examples of using the transition functions for obtaining solutions of viscoelastic problems are given: circularly symmetric solid and anular plates, plate with a small circular hole, and viscoelastic half-plane loaded by a concentrated force.M. M. Gubkin Moscow Institute of the Petrochemical and Gas Industry. Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 405–416, May–June, 1973.     

Hertzian contact problem: Numerical reduction and volumetric modification

A technique for the analytical formulation and numerical implementation of an elastic contact model for rigid bodies in the framework of the Hertzian contact problem is described. The normal elastic force and the semiaxes of the contact area are computed so that the problem is sequentially reduced to a scalar transcendental equation depending on complete elliptic integrals of the first and second kinds. Based on the classical solution to the Hertzian contact problem, an invariant volumetric force function is proposed that depends on the geometric characteristics of interpenetration of two undeformed bodies. The normal forces computed using the force function agree with results obtained previously for non-Hertzian contact of elastic bodies. As an example, a ball bearing is used to compare the contact dynamics of elastic bodies simulated in the classical Hertzian model and its volumetric modification.     

Flexure of thin inhomogeneous isotropic plates

The problem of the flexure of an inhomogeneous plate with a hole may be reduced, by the small-parameter method, to the solution of a series of problems for a homogeneous plate with a hole. These problems are solved using classical complex-potential methods. For an infinite plate with a circular hole, the results of numerical calculations are given; from these results, it follows that inhomogeneity of the material has a considerable influence on the stress.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 43–49, 1987.     

弹性地基上矩形板弯曲的CC型级数解

本文利用双变量函数的Stockes变换,用CC型级数求弹性地基上矩形板弯曲问题的解析解.以弹性地基上四边自由矩形板中点作用一集中力为例给出数字计算结果.     

Effect of a Concentrated Normal Force on the Surface of an Orthotropic Half Space

An analytic solution is given for the stressed-deformed state of an orthotropic half space acted on by a normal concentrated force applied to its boundary. It is shown that the traditional choice of a general representation of the displacements in a gradient form, where the potential functions are in the form of double Fourier integrals with unknown densities, reduces the problem to solving two systems of linear algebraic equations – homogeneous and inhomogeneous. A solution obtained by taking a limiting transition is given for the case of an isotropic half space.     

Optimization of the stressed state by a distribution of elastic parameters in elastic bodies

For designing an isotropic inhomogeneous body having a variable elastic modulus and a prescribed shape under the influence of an external force load we give a formulation and solution of the problem of determining the design that is optimal with respect to stress. The problem of optimal design reduces to a certain problem in the theory of elasticity for a non-linearelastic material. As an example we consider the problem of optimal design of an inhomogeneous cylinder. Four figures. Bibliography: 9 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 78–82.     

Global wellposedness for quasi-linear symmetric hyperbolic systems with regularized coefficients

Following the abstract setting of [8] and using the global results of [2], global wellposedness and regularity results are proved for the solutions of quasi-linear symmetric hyperbolic systems with bounded coefficients which are regularized by a convolution in the space variables with a regularizing function. In the case of unbounded regularized coefficients, local existence of classical solutions is proved, as well as uniqueness and regularity of (not necessarily existing) global weak solutions with initial value in a Sobolev space. As the regularizing function tends to Dirac's δ, local-in-time convergence to the classical solution of the non-regularized problem is proved.     

Bending of a semi-infinite plate with elliptic inclusion under the effect of a concentrated moment

The article presents an approximate solution of the problem of the state of stress of a semi-infinite plate with an elliptic elastic inclusion deformed by a concentrated bending moment. It also presents the results of numerical investigations characterizing the dependence of the state of stress of the plate on its geometric characteristics and the point of application of the concentrated moment.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 66–69, 1988.     

两种材料组成的弹性楔的佯谬解

从Hellinger-Reissner变分原理出发,通过引入适当的变换可以将两种材料组成的弹性楔问题导入极坐标哈密顿体系,从而可以在由原变量和其对偶变量组成的辛几何空间,利用分离变量法和辛本征向量展开法求解该问题的解。在极坐标哈密顿体系下的所有辛本征值中,本征值-1是一个特殊的本征值。一般情况下本征值-1为单本征值,求解其对应的基本本征函数向量就直接给出了顶端受有集中力偶的经典弹性力学解。但当两种材料的顶角和弹性模量满足特殊关系时,本征值-1成为重本征值,同时经典弹性力学解的应力分量变成无穷大,即出现佯谬。此时重本征值-1存在约当型本征解,通过对该特殊约当型本征解的直接求解就给出了两种材料组成的弹性楔顶端受有集中力偶的佯谬问题的解。结果进一步表明经典弹性力学中弹性楔的佯谬解对应的就是极坐标哈密顿体系的约当型解。     

Duality theorem for a generalized Fermat-Weber problem

The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane tok given points in the plane. This problem was generalized by Witzgall ton-dimensional space and to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multi-facility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual.     

Effect of material properties on the bending of cantilevered plates under a concentrated load

In a continuation of a previous study [4] a solution is obtained for the problem of a transversely isotropic plate in the form of an infinite strip bent by a concentrated load applied at an arbitrary point. The solution obtained is investigated in relation to the geometric and physicomechanical properties of the plate. The results are compared with the corresponding classical Kirchhoff solution [7].Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov; Kamenets-Podol'skii General Engineering Department of Khmel'nitskii Technological Institute. Translated from Mekhanika Polimerov, No. 4, pp. 704–707, July–August, 1971.     

The dynamics of structures on an elastic base

The problem of the dynamics of structures interacting with an elastic homogeneous half-plane through a rigid unyielding plate is reduced to the construction of the matrix of transmission momentum functions (the Green's matrix), which establishes the dependence between the generalized coordinates of the oscillations of the plate and the force characteristics. The elements obtained for this matrix are represented by graphs as the result of numerical solution of the nonsteady-state dynamic contact problem. We conclude that approximating a Green's matrix of exponential type on an infinite time interval is unjustified.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 60–63.     

On the deformation of an elastic half-space with a thin slit for mixed conditions on its boundary

The problem of the state of stress and strain in an elastic half-space with a cutout in the shape of a circular slot is solved by Kelvin's method /1/. The conditions on the slot are satisfied in a suitable manner by selecting the scalar and vector mass force potentials as generalized functions concentrated at the slot. The problem reduces to a system of Fredholm integral equations of the second kind in a semi-infinite interval. The solution for an elastic space with a slot is obtained in final form in the limiting case, which enables an estimate to be made of the magnitude of the settling of the earth's surface as a result of oil or gas deposit development.     

Dirichlet problem for the Helmholtz equation in an exterior two-dimensional domain with cuts without matching conditions at the cut endpoints

The Dirichlet problem for the Helmholtz equation in a plane exterior domain with cuts is considered for the case in which functions defined on opposite sides of the cuts in the Dirichlet boundary condition do not necessarily satisfy the matching conditions at the cut endpoints and the solution of the problem is not necessarily continuous at the endpoints of the cuts. We give a well-posed statement of the problem, prove existence and uniqueness theorems for a classical solution, derive an integral representation of the solution, and use it to study its properties. We show that the Dirichlet problem in the considered setting does not necessarily have a weak solution, although there exists a classical solution. We derive asymptotic formulas describing the behavior of the gradient of the solution at the endpoints of the cuts.