Another aspect of Saint-Venant's principle in elasticity

An energy decay estimate of the Saint-Venant type is obtained for an elastic cylinder whose curved surface is displacement free, and a plane end of which is subjected to tractions which are not necessarily self-equilibrated.

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Résumé On obtient une appréciation du type St. Venant pour l'attenuation de l'énergie dans un cylindre élastique dont la surface courbée ne subit pas de déplacement et dont une extremité plane subit des tractions qui ne sont pas forcément en equilibre.

     

The mixed boundary value problem for the elasticity theory system in unbounded domains

The paper deals with the uniqueness and stability of generalized solutions to the mixed boundary-value problem for the elasticity theory system in an unbounded domain, coinciding with a cone in a neighborhood of infinity. It is assumed that the boundary of the domain consists of two parts: the Dirichlet condition is prescribed on one of them, denoted by Γ₁, and the Neumann condition is prescribed on the other. The paper contains sufficient conditions (in terms of metric properties of Γ₁) for the validity of the Korn and Hardy inequalities, which imply the uniqueness and stability of the solution to the considered problem in appropriate function spaces. Bibliography: 8 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 106–156, 1995.     

First boundary value problem for the equations of hydrodynamics in a domain with a piecewise-smooth boundary

One considers the first boundary value problem for the Stokes and Navier-Stokes equations in a bounded domain with a piecewise-smooth boundary without zero angles. In Theorems 1 and 2 one gives solvability conditions in certain weighted spaces of Sobolev or Hölder type. Pointwise estimates for the Green tensor are formulated. Theorem 3 represents a generalization of the Miranda-Agmon maximum principle, while in Theorem 4 one gives conditions under which one has an estimate in W² ₂ ().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematlcheskogo Instituta im. V. A. Steklova Akad Nauk SSSR, Vol. 96, pp. 179–186, 1980.     

A minimax principle for the critical value of a boundary value problem with positive and increasing nonlinearity

Summary For a non-linear boundary value problem with a positive and increasing non-linearity there exists a critical value* of the parameter, beyond which there are no solutions. We give a minimax characterization of*.

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Zusammenfassung In der Randwertaufgabe –u(x)=f(x, u(x)), u(a)=u(b)=0, seif positiv und wachsend im zweiten Argument. Dann gibt es einen Wert*, so dass keine Lösung existiert für>*. In dieser Arbeit wird* durch ein Minimaxprinzip charakterisiert. Der Beweis beruht auf der Anwendung von Ober- und Unterlösungen und monotonen Iterationen.

     

Bounds for solutions of multipoint boundary value problems

Solutions of an nth order differential inequality with n boundary conditions are compared to solutions of an nth order disconjugate differential equation. It is shown the ratios of certain of these solutions are monotone.     

Solution of the second boundary value problem for one variant of the nonclassical theory of elasticity

The existence and uniqueness of the solution of the second boundary value problem of the nonclassical theory of elasticity describing the behavior of media with memory are investigated.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 1, pp. 88–91, January–February, 1971.     

A fourth order nonlinear two point boundary value problem arising in linear elasticity

The shape of a length of wire confined to the surface of a sphere with its ends tangent to two given circles of lattitude and otherwise free, in its most relaxed state, is found by solving a variational problem to be the solution of a fourth order nonlinear two point boundary value problem. A scheme of numerical computation is outlined which avoids a nearby singularity.     

Viscoelastic boundary stabilization for a transmission problem in elasticity

We consider an anisotropic body which is constituted of twodifferent types of materials supporting a memory boundary conditionand we show that its energy decays uniformly as time goes toinfinity with the same rate as the relaxation function g, thatis, the energy decays exponentially when g decays exponentially,and polynomially when g decays polynomially.     

On the solvability of the three-dimensional first dynamic boundary value problem of hemitropic elasticity

We consider the first main dynamic boundary value problem for a three-dimensional piecewise homogeneous hemitropic micropolar medium. By using the Fourier method under sufficiently general assumptions, we prove the solvability of the problem in the classical sense.     

The boundary equation method in the third initial boundary value problem of the theory of elasticity. Part 2: Methods for approximate solutions

This work considers the methods for solving approximately the four types of boundary equations arising when the third initial boundary value problem of the theory of elasticity is solved with the help of retarded elastic potentials. The convergence of these methods is proved.     

Global uniqueness for an inverse boundary problem arising in elasticity

Summary We prove that we can determine the Lamé parameters of an elastic, isotropic, inhomogeneous medium in dimensionsn3, by making measurements of the displacements and corresponding stresses at the boundary of the medium.Oblatum 14-VI-1993 & 22-XII-1993Partially supported by NSF Grant DMS-9100178 and ONR grant N 0014-93-1-0295     

The mesa problem for Neumann boundary value problem

In this paper, we study the singular limit of the Porous Medium equation u_t=Δu^m+g(x,u), as m→∞, in a bounded domain with Neumann boundary condition.     

Optimal preconditioner for boundary element formulation of the Dirichlet problem in elasticity

An optimal preconditioning procedure for the numerical solution of two-dimensional Dirichlet problem for Lamé equations by boundary element method is constructed. An efficient algorithm for the above problem is also developed.     

Solvability analysis of a nonlocal boundary value problem by applying the contraction mapping principle

The existence and uniqueness of a classical solution to the nonlocal boundary value problem for Poisson’s operator on a two-dimensional rectangular domain is proved in detail by applying the contraction mapping principle.     

平面弹性力学问题边界元直接法中边界积分的解析处理

确立平面位势和弹性力学问题的边界元直接法中边界积分的解析计算框架系统,从而避免了传统的高斯近似求积分,数值算例表明它具有较高的精度和效率,特别是在边界量和边界附近区域内点物理量的计算可获得较高的精度。