Existence Results for Quasistatic Contact Problems with Coulomb Friction

We prove the existence of a solution for an elastic frictional, quasistatic, contact problem with a Signorini non-penetration condition and a local Coulomb friction law. The problem is formulated as a time-dependent variational problem and is solved by the aid of an established shifting technique used to obtain increased regularity at the contact surface. The analysis is carried out by the aid of auxiliary problems involving regularized friction terms and a so-called normal compliance penalization technique. \par Accepted 15 May 2000. Online publication 6 October 2000.     

Unilateral Contact with Coulomb Friction and Uncertain Input Data

Abstract

A quasivariational inequality (QVI) in R ^d , d = 2, 3, with perturbed input data is solved by means of a worst scenario (anti-optimization) approach, using a stability result for the solution set of perturbed QVI-problems. The theory is applied to the dual finite element formulation of the Signorini problem with Coulomb friction and uncertain coefficients of stress-strain law, friction, and loading.     

具有非单调摩擦热弹性接触问题的有限元法

给出了一个变形体和刚性基础之间用双边摩擦表达其接触性质的、静态热弹性问题的方程式及其近似解法.以非单调、多值性表示该摩擦定律.忽略了问题的耦合效应,则问题的传热部分与弹性部分各自独立处理.位移矢量公式化为非凸的次静态问题,用局部Lipschitz连续函数来表示变形体的总势能.用有限单元法近似求解全部问题.     

A Boundary Element Approximation of a Signorini Problem with Friction Obeying Coulomb Law

In this work, a Signorini problem with Coulomb friction in two dimensional elasticity is considered. Based on a new representation of the derivative of the double-layer potential, the original problem is reduced to a system of variational inequalities on the boundary of the given domain. The existence and uniqueess of this system are established for a small frictional coefficient. The boundary element approximation of this system is presented and an error estimate is given.     

A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz's representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.     

带摩擦的弹性接触问题广义变分不等原理的简化证明

在弹性摩擦接触问题中 ,从变分原理出发来研究接触问题 ,可以将摩擦力纳入问题的能量泛函 .为了得到摩擦约束弹性接触问题的能量泛函 ,日前大多是用拉格朗日乘子法 ,但拉格朗日方法用在变分不等问题中 ,要利用非线性泛函分析和凸分析来证明 ,证明复杂 .本文利用向量分析的工具及巧妙的变换 ,对带摩擦约束的弹性接触问题的广义变分不等原理进行了严格的证明 ,由于只用到向量分析 ,简化了证明 .     

准静力学双面接触支承问题及其粘弹性材料的非局部摩擦

建立了描述变形体和基础间接触问题的数学模型.接触是双面的,并采用非局部摩擦定理建模,支承列入计算.粘结场(bonding field)的变化用一个一阶的常微分方程来表示,材料特性用一个非线性粘弹性本构关系建模.导出了该力学问题的变分公式,当摩擦因数充分小时,证明了其弱解的存在性和唯一性.依赖于时间的变分不等式、微分方程和Banach不动点理论,是该证明依据的基础.     

Generalized Contact Manifolds

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Vibrations of a Linear Oscillator with Generalized Friction

A mathematical model of a special class of vibration isolation systems is investigated. The model contains formal (generalized) functions and reduces to the successive solution of boundary-value problems for differential equations with coupling conditions, ultimately yielding transcendental equations that can be solved numerically. Analytical and numerical solutions for autonomous systems are obtained, providing a means for the solution of problems in choosing the right parameters for damping systems to satisfy specified conditions for the normal operation of such systems in the presence of transient or abrupt inertial forces.     

广义F-互补问题及其与变分不等式问题的等价性

在Banach空间中引入了一类广义F-互补问题,研究了它与一类变分不等式问题的等价性关系,给出了此类变分不等式问题的解的存在唯一性定理.     

Generalized Weber Problem with Rectangular Regions

The problem of locating one or more new facilities relative to a number of existing rectangular regions is treated for the case where the rectilinear norm is used. The new facilities are to be located such that the total weighted distance is minimized. If there is interfacility interaction among the new facilities, a gradient-free nonlinear search algorithm is utilized. Computational experience suggests that this algorithm is expedient even in the solution of large problems.     

Generalized Pinwheel Problem

This paper studies a non-preemptive infinite-horizon scheduling problem with a single server and a fixed set of recurring jobs. Each job is characterized by two given positive numbers: job duration and maximum allowable time between the job completion and its next start. We show that for a feasible problem there exists a periodic schedule. We also provide necessary conditions for the feasibility, formulate an algorithm based on dynamic programming, and, since this problem is NP-hard, formulate and study heuristic algorithms. In particular, by applying the theory of Markov Decision Process, we establish natural necessary conditions for feasibility and develop heuristics, called frequency based algorithms, that outperform standard scheduling heuristics.     

Coulomb Problem in a One-Dimensional Space with Constant Positive Curvature

We construct a complex transformation generalizing the known Hurwitz transformation in the Euclidean space for one-dimensional quantum mechanics. As in the case of the flat space, this transformation allows establishing the connection between the Coulomb problem and the oscillator problem with the Calogero–Sutherland potential added. We fully describe the motion in a Coulomb field in S ₁ and determine the energy spectrum and the wave functions with the correct normalizing constant.     

关于带两个位移的广义Hilbert问题

在文章中提出了带位移的广义Hilbert问题,并对这一问题建立了Noether性条件。在文章中又利用带位移的奇异积分方程理论得到了这一问题可解的充分必要条件以及指数计算公式。显然,这些结果推广了一般Hilbert问题的相应结论。本文目的是利用作者在中建立的理论解决带两个位移的广义Hilbert问题,并得到了相应的可解条件和指数计算公式。     

The Generalized Moment Problem with Complexity Constraint

In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.     

3‐D Contact problems for elastic wedges with Coulomb friction

3‐D quasi‐static contact problems for elastic wedges with Coulomb friction are reduced to integral equations and integral inequalities with unknown contact normal pressures. To obtain these equations and inequalities, Green's functions for the wedges, where one face of the wedges is either stress‐free or fixed, are needed. Using Fourier and Kontorovich–Lebedev integral transformations, all the stresses and displacements in the wedges can be constructed in terms of solutions of Fredholm integral equations of the second kind on the semiaxis. The Green's functions can be calculated as uniformly convergent power series in (1‐2ν), where νis Poisson's ratio. An exponential decay of the kernels and right‐hand sides of the Fredholm integral equations provides the applicability of the collocation method for simple and fast calculation of the Green's functions. For a half‐space, which is a special case of an elastic wedge, the kernels degenerate and the functions reduce to the well‐known Boussinesq and Cerruti solutions. Analysing the contact problems reveals that the Green's functions govern the kernels of the above mentioned integral equations and inequalities. Under the assumption that the punch has a smooth shape, the contact pressure is zero on the boundary of the unknown contact zone. Solving the contact problems with the help of the Galanov–Newton method, the normal contact pressure, the contact zone and the normal displacement around the contact zone can be determined simultaneously. In view of the numerical results, the influence of the friction forces on the punch force and the punch settlement is discussed. Copyright © 2004 John Wiley & Sons, Ltd.