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.     

An Elastic Frictional Contact Problem with Unilateral Constraint

We consider a mathematical model which describes the equilibrium of an elastic body in contact with two obstacles. We derive its weak formulation which is in a form of an elliptic quasi-variational inequality for the displacement field. Then, under a smallness assumption, we establish the existence of a unique weak solution to the problem. We also study the dependence of the solution with respect to the data and prove a convergence result. Finally, we consider an optimization problem associated with the contact model for which we prove the existence of a minimizer and a convergence result, as well.     

A Mathematical Programming Incremental Formulation for Unilateral Frictional Contact Problems of Linear Elasticity

In this article a detailed analytical formulation of the unilateral contact boundary conditions with Coulomb's law of dry friction is first attempted and the quasi-static contact problem between 3-D elastic bodies is studied thereafter. Discretizing the bodies by the Finite Element Method, introducing fictitious contact bonds and using the concept of the equivalent structural system, an incremental Nonlinear Complementarity Problem is finally formulated. Then, using additional simplifying assumptions, this problem can be transformed into an incremental Linear Complementarity Problem.     

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 Variational Approach Via Bipotentials for a Class of Frictional Contact Problems

In the reference (Cui and Yin, Pacific J. Math. 233:257–289, 2007), under the assumptions that the supersonic incoming flow is isothermal and symmetrically perturbed with respect to a uniform supersonic constant state, the authors have shown the global existence and stability of a symmetric supersonic conic shock for such a supersonic flow past a circular cone. In this paper, we will remove all the symmetric assumptions in the previous paper and study the global existence problem on a really multidimensional shock wave. More concretely, we establish the global existence and stability of a three-dimensional supersonic conic shock wave for a perturbed steady supersonic isothermal flow past an infinitely long conic body.     

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.     

Analysis of a Class of Frictional Contact Problems for the Bingham Fluid

We consider a mathematical model which describes the stationary flow of a Bingham fluid with friction. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive a weak formulation of the model which consists in a variational inequality for the velocity field. We establish the existence and uniqueness of the weak solution as well as its continuous dependence with respect to the contact condition. Finally, we describe a number of concrete friction conditions which may be set in this general framework and for which our results apply.     

A Peano Type Theorem for a Class of Nonconvex-Valued Differential Inclusions

We study a particular class of nonconvex-valued differential inclusions. We prove the existence of a continuously differentiable solution under a continuity condition on the associated multiple-valued function in the sense of Almgren.     

Properties of Attainable Sets of Evolution Inclusions and Control Systems of Subdifferential Type

In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and the evolution operators that are the compositions of a linear operator and the subdifferentials of a time-dependent proper convex lower semicontinuous function. Alongside the initial inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are the compositions of the same linear operator and the subdifferentials of the Moreau–Yosida regularizations of the initial function. We demonstrate that the attainable set of the initial inclusion as a multivalued function of time is the time uniform limit of a sequence of the attainable sets of the approximating inclusions in the Hausdorff metric. We obtain similar results for evolution control systems of subdifferential type with mixed constraints on control. As application we consider an example of a control system with discontinuous nonlinearities containing some linear functions of the state variables of the system.     

Solvability Theory and Projection Methods for a Class of Singular Variational Inequalities: Elastostatic Unilateral Contact Applications

The mathematical modeling of engineering structures containing members capable of transmitting only certain type of stresses or subjected to noninterpenetration conditions along their boundaries leads generally to variational inequalities of the form , where C is a closed convex set of (kinematically admissible set), (loading strain vector), and (stiffness matrix). If rigid body displacements and rotations cannot be excluded from these applications, then the resulting matrix M is singular and serious mathematical difficulties occur. The aim of this paper is to discuss the existence and the numerical computation of the solutions of problem (P) for the class of cocoercive matrices. Our theoretical results are applied to two concrete engineering problems: the unilateral cantilever problem and the elastic stamp problem.     

A Perturbation Result for Dynamical Contact Problems

This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.     

On the Stability of Sizing Optimization Problems for a Class of Nonlinearly Elastic Materials

In this work we deal with a stability aspect of sizing optimization problems for a class of nonlinearly elastic materials, where the underlying state problem is nonlinear in both the displacements and the stresses. In [14] it is shown under which conditions there exists a unique solution of discrete design problems for a body made of the considered nonlinear material, if the nonlinear state problem is solved exactly. In numerical examples the nonlinear state problem has to be solved iteratively, and therefore it can be solved only up to some small error \eps . The question of interest is how this affects the optimal solution, respectively the set of solutions, of the design problem. We show with the theory of point-to-set mappings that if the material is not too nonlinear, then the optimal design depends continuously on the error \eps . Accepted 15 March 2001. Online publication 14 August 2001.     

一类φ-强增生型变分包含解的存在和逼近问题

研究Banach空间中一类带有φ-强增生型映象的集值变分包含解的存在性及其具随机误差的Ishikawa迭代逼近问题,得到了迭代序列强收敛于变分包含问题的唯一解的若干等价条件,所得结果统一和推广了一些文献中的最新成果.     

A Kind of Nonconvex—valued Functional Differential Inclusions Problems

51.IntroductionLetXbeaBanachspaceandI=[O,7'](7'>O),L=[-a,Oj,J=[-a,7']bethreeinter-valsofR.WesetF:IXC[L;Xj-2"\diisaset-valuesmap.Weintroducethetimelagopera-tor7'.):CLJ;Xj-CLLiXj,definedby7'(t)x(s)=X(t s)foreveryseL-LetusassumethatA:Dom(A)-2"\diisaset-values))l-accretiveoperator.Inthispaper,w3discusstheini-tialvalueprob1emsasfoIlows:Sincethedevelopmentoftl1eControlTheoryandEconomics,theinitialva1ueproblem(1.1)havebeendiscussinginrecentyears,someimportantresultshavebeenobtainecl,for…     

Quadratic Growth and Strong Metric Subregularity of the Subdifferential for a Class of Non-prox-regular Functions

Journal of Optimization Theory and Applications - This paper mainly studies the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as...