On Some Contact Problems for Bodies with Elastic Inclusions

The paper deals with the contact problems of the theory of elasticity. The problems are reduced to Prandtl-type integral differential equations with a coefficient at the singular operator which has higher-order zeros at the ends of the integration interval. In some concrete cases the solution is constructed efficiently. Asymptotic representations are obtained.     

Instable Waves between Rough Elastic Bodies in Sliding Contact

Most sliding bodies are not perfectly flat, but show a rough surface topography or tribological layer. This paper analyses a sliding system consisting of two elastic bodies with continuous contact and steady sliding. Surface topographies are taken into account by an inertia film on one of the sliding surfaces. A linear model is developed that allows an analytical solution. As a typical tribological system, the contact between pad and disc in a brake system is discussed. It can be shown that unstable elastic waves travel through the contact between both bodies. This instability is opened only when a tribological layer on the brake pad is taken into account. This microscopic excitation can cause a loss of stability of the entire (brake) system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of Motion Stability for Elastic Cylindrical Bodies in Two Measures

We obtain sufficient conditions for stability, in two specific measures, of the zero solution of the linearized equation of motion of isotropic elastic bodies subjected to the action of a force load under kinematic, dynamic, and mixed boundary conditions. For cylinder bodies, the establishment of the conditions for stability is reduced to solving the corresponding problem of variational calculus with fixed edges. We find the critical values of parameters in the particular case of loading of a cylindrical body under various conditions of fixing.     

Preconditioned Uzawa Algorithm for Elastic Contact Problems

After finite element discretization of the elastic contact problems with friction, we obtain a big sparse non–symmetric and nonlinear systems of equations, and in many cases ill–conditioned. Solving these systems by direct methods or classical iterative methods are non efficient and with bad convergence properties. One way to overcome these difficulties is to use the preconditioned Uzawa–type algorithms. On this paper we focus on the transformation of the generalized Signorini elastic contact problems into a saddle point problem of some augmented Lagrangian functional and give a preconditioning technique for Uzawa algorithm. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact Modeling in Multibody Systems with Elastic Bodies in High-Frequency Applications

This paper discusses the refinement of multibody models by integration of flexible bodies and by considering nonlinearities from contacts. It presents common approaches for contact modeling in multibody simulations and strategies to include flexible bodies. A contact model is implemented in the elastic multibody model. Experimental results show that significant effects of system dynamics can be modeled by use of a multibody model including elastic bodies and contacts. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Two Dimensional Model for the Dynamical Contact of Elastic Rough Surfaces

To model the contact behavior including dynamical effects, a two dimensional mechanical model of elastic rough contact is developed. This model can simulate the contact behaviour between two rough surfaces depending on normal pressure, sliding speed and roughness profiles. The contact between two rough surfaces is reduced to a rough rigid and a rough elastic layer. The elastic layer is modeled by point masses connected by spring-damper elements. The total system is described by coupled ODEs. The number of ODEs and thus the degree of freedom of the model depends on the varying contact conditions. The contact conditions are monitored during the simulation and the simulation interrupts, in case the contact conditions change. The equations of motion are then adapted with respect to the contact constraints. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Unilateral Boundary-Value Problems for Elastic Bodies with Rugged Boundaries

The system of linear elasticity is considered in a domain whose boundary depends on a small parameter > 0 and has a part with a rugged structure. The rugged part of the boundary may bend sharply and embrace cavities or channels, and as 0, it approaches a limit surface on the boundary of the limit domain. On the rugged part of the boundary, conditions of two types are considered: (I) contact with rigid obstacles (conditions of Signorini type); (II) reaction forces involving the parameter and nonlinearly depending on displacements. We investigate the asymptotic behavior of weak solutions to such boundary-value problems as 0 and construct the limit problem, according to the geometric structure of the rugged part of the boundary and the external surface forces and their dependence on the parameter . In general, the limit problem has the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone characterizes the boundary conditions of the limit problem and is described in terms of the functions involved in the nonlinear boundary conditions on the rugged boundary. As shown by examples, in the limit, the type of boundary condition may change. To justify these asymptotic results, we give a detailed exposition of some facts about extensions, Korn's inequalities, traces, and nonlinear boundary conditions in partially perforated domains with Lipschitz continuous boundaries. Bibliography: 16 titles.     

A Class of Subdifferential Inclusions for Elastic Unilateral Contact Problems

We consider a class of stationary subdifferential inclusions in a reflexive Banach space. We reformulate the problem in terms of a variational inequality with multivalued term and prove an existence result using the Kakutani-Fan-Glicksberg fixed point theorem. This approach allows to consider, in a natural way, a dual variational formulation of the problem. Next, we study the link between the primal and dual formulations and provide an equivalence result. Then, we consider a new mathematical model which describes the contact of an elastic body with a foundation. We apply the abstract formalism to derive the primal and the dual variational formulations of the problem, in terms of displacement and stress, respectively. Finally, we present existence and equivalence results in the study of this contact model.     

Augmented Macro-Hybrid Mixed Finite Element Schemes for Elastic Contact Problems

On a setting of subdifferential models, variational augmented macro-hybrid mixed finite element schemes are formulated and analyzed for elastic unilateral contact problems with prescribed friction. Composition duality principles determine primal and dual mixed solvability, adopting coupling surjectivity for dualization. Macro-hybridization corresponds to nonoverlapping decompositions of elastic solid body systems, with displacement continuity and traction equilibrium transmission conditions dualized. In general, traction and displacement multipliers synchronize sub-bodies through nonmatching finite element interfaces. Three-field formulations give the basis for variational augmentation, in a sense of exact penalization, allowing speed-up of rates of convergence as well as proximation procedures of parallel numerical resolution algorithms.     

General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Basic Interface Problems)

Here we discuss three-dimensional so-called basic and mixed boundary value problems (BVP) for steady state oscillations of piecewise homogeneous anisotropic bodies imbedded into an infinite elastic continuum. Uniqueness is shown with the help of generalized Sommerfeld–Kupradze radiation conditions, while existence follows for arbitrary values of the oscillation parameter by the reduction of the original interface transmission BVPs to equivalent uniquely solvable boundary integral or pseudodifferential equations on the interfaces. For the basic BVPs, we show classical regularity and, in addition for the mixed BVPs that the solutions are Hölder continuous with exponent α ∈ (0, 1/2) in the neighbourhood of the curves of discontinuity of the boundary and transmission conditions.     

弹性摩擦接触问题的边界元—线性互补解法

本文对Coulomb摩擦的弹性接触问题给出了边界元一线性互补解法.这是对固体力学中自由边界问题进行边界元数学规划求解的新尝试,文中附有算例     

Comparison of Different Error Indicators for Contact Problems Involving Large Elastic Strains

Finite element methods (FEMs) are flexible tools which can be applied to solve contact problems with arbitrary geometries. As in any numerical method, the solutions obtained with FEM are only approximate. Errors occur e.g. due to the choice of the ansatz space and the approximation of the geometry. In general it is necessary to automatically control the error inherited in the method to obtain reliable solutions. This is especially true in case of non-linear contact problems since geometry and contact surface can change substantially during the deformation process. Thus the refinement process needed for an accurate analysis cannot be controlled by the user beforehand. Here an adaptive FEM is developed for large strain problems of two or more deformable bodies being in contact. The main focus is the comparison of different error indicators and error estimators related to the contact problem. In detail residual based, error estimators, error indicators relying on superconvergence properties and error estimators based on duality principles are investigated. Finally, examples show the convergence behaviour of the error measures.     

Triple Hilbert Transforms Along Polynomial Surfaces

Given W ì \mathbbZ₊³\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in L^p(\mathbbR⁴)L^p({\mathbb{R}}^4).     

Slightly Irregular Elastic Bodies Under Pressure

Exact solutions exist in linear elasticity theory generallyfor bodies whose boundary equations have a simple analytic form.This paper is concerned with the perturbation of these solutionswhen the boundaries are perturbed. Only uniform normal pressureon each perturbed boundary is considered. Two dimensional systemsand hollow spheres are treated in detail and right-circularcylinders in outline, with application to nearly circular holesin plates and to the initial yielding of imperfect sphericalpressure vessels.     

弹性接触问题参数变分原理的有限元并行算法*

本文基于弹性接触问题的参数变分原理的有限元解法,利用并行计算机的特性和并行处理结构,建立了相应的并行算法.该算法从刚度阵的生成和组集,静凝聚过程,求应力过程等多方面实现了并行化.该算法在西安交通大学ELXSI-6400并行计算机上程序实现,计算结果表明能有效地节省计算时间,是一种分析接触问题的有效的并行算法.     

Minimal Surfaces with an Elastic Boundary

LetD:= { C ³ (