Viscoelastic Contact Problem Research Articles

Existence and Uniqueness to a Dynamic Bilateral Frictional Contact Problem in Viscoelasticity

In this paper we examine an evolution problem which describes the dynamic bilateral contact of a viscoelastic body and a foundation. The contact is modeled by a friction multivalued subdifferential boundary condition which incorporates the Coulomb law of friction, the SJK model and the orthotropic friction law. The main result concerns the existence and uniqueness of weak solutions to the hyperbolic variational inequality when the friction coefficient is sufficiently small. The proof is based on a surjectivity result for multivalued operators and a fixed point argument.

An Incremental Adaptive Procedure for Viscoelastic Contact Problems

Contact pressure distribution throughout the contact interface has a vital role on the tribological aspects of the contact systems. Generally, contact of deformable bodies is a nonlinear problem. Viscoelastic materials have a time-dependent response, since both viscous and elastic characteristics depend on time. Such types of materials have the capability of storing and dissipating energy. When at least one of the contacting bodies is made of a viscoelastic material, contact problems become more difficult, and a nonlinear time-dependent contact problem is obtained. The objective of this paper is to develop an incremental adaptive computational model capable of handling quasistatic viscoelastic frictionless contact problems. The Wiechert model, as an effective model capable of describing both creep and relaxation phenomena, is adopted to simulate the linear behavior of viscoelastic materials. The resulting constitutive integral equations are linearized and, therefore, complications that arise during the direct integration of these equations, specially with contact problems, are avoided. In addition, the incremental convex programming method is adopted and modified to accommodate the contact problem of viscoelastic bodies. The Lagrange multiplier method is adopted to enforce the contact constraints. Two different contact problems are presented to demonstrate the efficient applicability of the proposed model.

Evolution Hemivariational Inequality for a Class of Dynamic Viscoelastic Nonmonotone Frictional Contact Problems

In this paper, we consider a class of hyperbolic hemivariational inequalities modeling the frictional contact between a viscoelastic body and a rigid foundation. The friction condition is described by a multivalued subdifferential relation which includes both the slip displacement and the slip rate dependent friction, and the Tresca models. The existence of weak solutions to the problem is proved by exploiting a surjectivity result for a class of pseudomonotone mappings.

Analysis of a viscoelastic contact problem with Normal damped response and damage

We consider a mathematical model for the process of contact between a viscoelastic body and a reactive foundation. The material is viscoelastic with internal state variable which may describe the damage of the system. We establish a variational formulation for the model and prove the existence and uniqueness result of the weak solution. Finally we prove a dependence result with respect to the data.

Numerical approximation of a viscoelastic frictional contact problem

We consider a fully discrete scheme for a quasistatic frictional contact problem between a viscoelastic body and an obstacle. The contact is bilateral, the friction is modeled with Tresca's law and the behavior of the material is described with a viscoelastic constitutive law with long memory. We state an existence and uniqueness result for the discrete solution, followed by error estimate results. Then, we present numerical simulations in the study of a two-dimensional test example. To cite this article: Á. Rodríguez-Arós et al., C. R. Mecanique 334 (2006).

Efficient stress evaluation of stationary viscoelastic rolling contact problems using the boundary element method: Application to viscoelastic coatings

The present work is a contribution towards the development of a new and novel technique to evaluate the contact stresses between linear-viscoelastic cylinders under stationary rolling contact loading. A two-dimensional numerical model is considered using a reference system fixed to the contact zone and including the effects of Coulomb friction between the rolling cylinders. The contact stresses are determined by solving a non-linear system of equations coming from the combination of the discretized boundary element viscoelastic equations and the fulfillment of the contact conditions, expressed using projection functions. The results obtained for viscoelastic rollers are compared with those from other authors for different rolling velocities, showing good agreement and consistency. Finally, some results for viscoelastic coatings are presented and commented.

A Unified Approach to Dynamic Contact Problems in Viscoelasticity

In this paper we consider mathematical models describing dynamic viscoelastic contact problems with the Kelvin–Voigt constitutive law and subdifferential boundary conditions. We treat evolution hemivariational inequalities which are weak formulations of contact problems. We establish the existence of solutions to hemivariational inequalities with different types of non-monotone multivalued boundary relations. These results are consequences of an existence theorem for second order evolution inclusions. In a particular case we deliver sufficient conditions under which the solution to a hemivariational inequality is unique. Finally, applications to several unilateral and bilateral problems in contact mechanics are provided.

Viscoelastic frictionless contact problems with adhesion

We consider two quasistatic frictionless contact problems for viscoelastic bodies with long memory. In the first problem the contact is modelled with Signorini's conditions and in the second one is modelled with normal compliance. In both problems the adhesion of the contact surfaces is taken into account and is modelled with a surface variable, the bonding field. We provide variational formulations for the mechanical problems and prove the existence of a unique weak solution to each model. The proofs are based on arguments of time-dependent variational inequalities, differential equations, and a fixed point theorem. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solutions of the contact problem with normal compliance as the stiffness coefficient of the foundation converges to infinity.

A dynamic viscoelastic contact problem with normal compliance and damage

We consider a model for dynamic frictionless contact between a viscoelastic body and a foundation. The material constitutive relation is assumed to be nonlinear and the contact is modelled with the normal compliance condition. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, the evolution of which is determined by a parabolic inclusion. We derive a variational formulation for the problem and prove the existence of its unique weak solution. We then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions. Finally, we report some numerical results in the simulation of two-dimensional test problems.

Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction

In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.

Boundary Hemivariational Inequalities of Hyperbolic Type and Applications

In this paper we examine two classes of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. We prove two existence results for multidimensional hemivariational inequalities: one for the inequalities with relation between reaction and velocity and the other for the expressions containing the reaction–displacement law. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. We present also an example of dynamic viscoelastic contact problem in mechanics which illustrate the applicability of our results.

Hemivariational inequality for viscoelastic contact problem with slip-dependent friction

In this paper we consider a class of hyperbolic hemivariational inequalities modeling the frictional contact between a viscoelastic body and a rigid foundation. The friction law is described by a multivalued subdifferential relation which includes the slip displacement dependent friction and the Tresca models. We establish the existence of weak solutions to the problem by using a surjectivity result for pseudomonotone mappings.

Numerical analysis of a frictionless viscoelastic contact problem with normal damped response

We consider a mathematical model which describes the frictionless contact between a viscoelastic body and a deformable foundation. We model the material's behavior with a nonlinear Kelvin-Voigt constitutive law. The process is assumed to be quasistatic and the contact is modeled with a general normal damped response condition. We present the variational formulation of the problem including the existence of a unique weak solution to the model. We then study the numerical approach to the problem using a fully discrete finite-elements scheme with an explicit discretization in time. We state the existence of the unique solution for the scheme and derive an error estimate on the approximate solutions. Finally, we present some numerical results involving examples in one and two dimensions.

Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion

A model for the adhesive, quasistatic and frictionless contact between a viscoelastic body and a deformable foundation is described. The adhesion process is modelled by a bonding field on the contact surface, and contact is described by a modified normal compliance condition. The problem is formulated as a coupled system of a variational equality for the displacements and a differential equation for the bonding field. The existence of a unique weak solution for the problem and its continuous dependence on the adhesion parameters are established. Then, the numerical analysis of the problem is conducted for the fully discrete approximation. The convergence of the scheme is established and error estimates derived. Finally, representative numerical simulations are presented, depicting the evolution of the state of the system and, in particular, the evolution of the bonding field.

A viscoelastic frictionless contact problem with adhesion

We consider a model for the quasistatic, bilateral, adhesive and frictionless contact between a viscoelastic body and a rigid foundation. The adhesion process on the contact surface is modeled by a surface internal variable, the bonding field, and the tangential shear due to the bonding field is included. The problem is formulated as a coupled system of a variational equality for the displacements and an integro-differential equation for the bonding field. The existence of a unique weak solution for the problem is established by construction of an appropriate mapping which is shown to be a contraction on a Hilbert space. We also consider the problem describing the bilateral contact between two viscoelastic bodies, and establish similar results.

Dual formulation of a quasistatic viscoelastic contact problem with tresca's friction law

We consider quasistatic evolution of a viscoelastic boby which is in bilateral frictional contact with a rigid foundation. We derive two variational formulations for the problem:the primal formulation in terms of the displacements and the dual formulation in terms of the stress field. We prove the existence of a unique solution to each one and establish the equivalence between the two variational formulations. We also prove the continuous dependence of the solution on the friction yield limit.

Time-dependent variational inequalities for viscoelastic contact problems

We consider a class of abstract evolutionary variational inequalities arising in the study of contact problems for viscoelastic materials. We prove an existence and uniqueness result, using standard arguments of time-dependent elliptic variational inequalities and Banach's fixed point theorem. We then consider numerical approximations of the problem. We use the finite element method to discretize the spatial domain and we introduce spatially semi-discrete and fully discrete schemes. For both schemes, we show the existence of a unique solution, and derive error estimates. Finally, we apply the abstract results to the analysis and numerical approximations of a viscoelastic contact problem with normal compliance and friction.

Analysis of a class of implicit evolution inequalities associated to viscoelastic dynamic contact problems with friction

For a class of implicit evolution inequalities, the existence of solutions is proved by using a new approach. The results are directly applied to dynamic viscoelastic contact problems with friction, using in particular the model of persistent contact with time depending Tresca (given) friction or the model of normal compliance for both contact and friction. Moreover, using a penalty method and a weaker formulation, an existence result is obtained for a unilateral contact problem with a non-local Coulomb friction law.

A quasistatic viscoelastic contact problem with friction

We consider a mathematical model which describes the bilateral frictional contact of a viscoelastic body with an obstacle. The viscoelastic constitutive law is assumed to be nonlinear and the friction is described by a nonlocal version of Coulomb's law. A weak formulation of the model is presented and an existence and uniqueness result is established when the coefficient of friction is small. The proof is based on classical results for elliptic variational inequalities and fixed point arguments. We also consider a model for the wear of the contacting surface due to friction. In the case of sliding contact we obtain a new nonstandard contact boundary condition. We prove the existence of the unique weak solution to the problem with the same restriction on the friction coefficient.

Evolutionary Variational Inequalities Arising in Viscoelastic Contact Problems

We consider a class of evolutionary variational inequalities arising in various frictional contact problems for viscoelastic materials. Under the smallness assumption of a certain coefficient, we prove an existence and uniqueness result using Banach's fixed point theorem. We then study two numerical approximation schemes of the problem: a semidiscrete scheme and a fully discrete scheme. For both schemes, we show the existence of a unique solution and derive error estimates. Finally, all these results are applied to the analysis and numerical approximations of a viscoelastic frictional contact problem, with the finite element method used to discretize the spatial domain.