Contact Problem For Viscoelastic Materials Research Articles

Fuzzy fractional delay differential inclusions driven by hemivariational inequalities in Banach spaces

This paper investigates a novel class of nonlinear dynamical fuzzy systems referred to as fuzzy fractional delay differential hemivariational inequalities (FFDDHVIs) in Banach spaces. These systems integrate fractional fuzzy differential inclusions with delay and hemivariational inequalities. The existence theorem for the HVIs is established based on the KKM theorem. Moreover, specific propositions are proved, covering the superpositional measurability and the upper semicontinuity for the HVIs. Next, by using the fixed points theorem, we establish the existence and compactness of mild solution sets for the FFDDHVIs under certain mild conditions. Finally, As an illustrative application, we investigate a frictional quasistatic contact problem for viscoelastic materials, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals.

Exponential stabilization of a dynamic frictional contact problem for viscoelastic materials with normal damped response and infinite memory

Exponential stabilization of a dynamic frictional contact problem for viscoelastic materials with normal damped response and infinite memory

A new class of differential quasivariational inequalities with an application to a quasistatic viscoelastic frictional contact problem

The overarching goal of this paper is to introduce and investigate a new nonlinear system driven by a nonlinear differential equation, a history-dependent quasivariational inequality and a parabolic variational inequality in Banach spaces. Such a system can be used to model quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear. By using the Banach fixed point theorem, we prove an existence and uniqueness theorem of solution for such a system under some mild conditions. As a novel application, we obtain a unique solvability of a quasistatic viscoelastic frictional contact problem with long memory, damage and wear.

A unilateral constraint contact problem for viscoelastic materials with wear

The aim of this work is to investigate a mathematical model that describes the frictional contact between viscoelastic materials with a long memory and a moving foundation. The contact condition is modeled with a version of normal compliance condition with unilateral constraint in which wear of foundation is considered. Friction contact condition follows a sliding version of Coulomb's law of dry friction. We derive a variational formulation of the problem, which couples a variational inequality and an integral equation. Then, we obtain the unique weak solvability of the contact problem by virtue of a fixed point theorem. By using the penalization method, we discuss the convergence of the contact problem. Further, we obtain the unique solution of the penalized problem and this solution converges to the solution of the original contact model as the penalization parameter converges to zero.

History-dependent fractional hemivariational inequality with time-delay system for a class of new frictionless quasistatic contact problems

We study a new frictionless quasistatic contact problem for viscoelastic materials, in which contact conditions are described by the fractional Clarke generalized gradient of nonconvex and nonsmooth functions and a time-delay system. In addition, our constitutive relation is modeled using the fractional Kelvin–Voigt law with long memory. The existence of mild solutions for new history-dependent fractional differential hemivariational inequalities with a time-delay system are obtained by the Rothe method, properties of the Clarke generalized gradient, and a fixed-point theorem.

Dynamic History-Dependent Hemivariational Inequalities Controlled by Evolution Equations with Application to Contact Mechanics

This paper is devoted the study of a generalized hybrid dynamical system, which consists of a history-dependent hemivariational inequality of parabolic type and a nonlinear evolution equation. The unique solvability for the system is established via applying surjectivity of multivalued pseudomonotone operators, fixed point theorem, and properties of the Clarke generalized gradient. As an illustrative application, a dynamic frictional contact problem for viscoelastic materials with history-dependent and adhesion is investigated, in which the friction condition is described by the Clarke generalized gradient of a nonconvex and nonsmooth function involving adhesion, and the normal damped response condition is expressed by a given nonnegative function depending on the normal velocity and adhesion.

A Class of Nonlinear Inclusions and Sweeping Processes in Solid Mechanics

We consider a new class of inclusions in Hilbert spaces for which we provide an existence and uniqueness result. The proof is based on arguments of monotonicity, convexity and fixed point. We use this result to establish the unique solvability of an associated class of Moreau’s sweeping processes. Next, we give two applications in Solid Mechanics. The first one concerns the study of a time-dependent constitutive law with unilateral constraints and memory term. The second one is related to a frictional contact problem for viscoelastic materials. For both problems we describe the physical setting, list the assumptions on the data and provide existence and uniqueness results.

Existence for a quasistatic variational-hemivariational inequality

<p style='text-indent:20px;'>This paper deals with an evolution inclusion which is an equivalent form of a variational-hemivariational inequality arising in quasistatic contact problems for viscoelastic materials. Existence of a weak solution is proved in a framework of evolution triple of spaces via the Rothe method and the theory of monotone operators. Comments on applications of the abstract result to frictional contact problems are made. The work extends the known existence result of a quasistatic hemivariational inequality by S. Migórski and A. Ochal [SIAM J. Math. Anal., 41 (2009) 1415-1435]. One of the linear and bounded operators in the inclusion is generalized to be a nonlinear and unbounded subdifferential operator of a convex functional, and a smallness condition of the coefficients is removed. Moreover, the existence of a hemivariational inequality is extended to a variational-hemivariational inequality which has wider applications.

Convergence and Optimization Results for a History-Dependent Variational Problem

We consider a mixed variational problem in real Hilbert spaces, defined on on the unbounded interval of time and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in our previous paper. Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis of a large class of nonlinear boundary value problems which, in a weak formulation, lead to history-dependent mixed variational problems. To provide an example, we illustrate our abstract results in the study of a frictional contact problem for viscoelastic materials with long memory.

Existence of a class of variational inequalities modelling quasi‐static viscoelastic contact problems

AbstractThis work concerns the existence for a class of variational inequalities arising in quasi‐static frictional contact problems for viscoelastic materials. We first consider dynamic contact problems described by evolutionary variational inequalities with a small parameter in the inertial term. Then we study the asymptotic behavior of their solutions when the parameter tends to zero. Based on a time‐discretization technique and monotone operators theory, the inequalities are solved in the form of evolutionary inclusions in the framework of evolution triples. We show that the limit functions of dynamic problems are solutions to the quasi‐static variational inequalities. Applications of the abstract result to frictional contact problems with multivalued constitutive law and normal damped response are given. Moreover, some comments on nonmonotone boundary problem are presented.

Variational–hemivariational inequality for a class of dynamic nonsmooth frictional contact problems

In this paper, a dynamic frictional contact problem for viscoelastic materials with long memory is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality for the velocity. A result on the unique weak solvability to this inequality is proved through a recent contribution on evolutionary subdifferential inclusions and a fixed point approach.

A quasistatic frictional contact problem for viscoelastic materials with long memory

Abstract The aim of this paper is to study a quasistatic frictional contact problem for viscoelastic materials with long-term memory. The contact boundary conditions are governed by Tresca’s law, involving a slip dependent coefficient of friction. We focus our attention on the weak solvability of the problem within the framework of variational inequalities. The existence of a solution is obtained under a smallness assumption on a normal stress prescribed on the contact surface and on the coefficient of friction. The proof is based on a time discretization method, compactness and lower semicontinuity arguments.

A class of fractional differential hemivariational inequalities with application to contact problem

In this paper, we study a class of generalized differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces. We use the Rothe method combined with surjectivity of multivalued pseudomonotone operators and properties of the Clarke generalized gradient to establish existence of solution to the abstract inequality. As an illustrative application, a frictional quasistatic contact problem for viscoelastic materials with adhesion is investigated, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals, and the constitutive relation is modeled by the fractional Kelvin–Voigt law.

Analysis and numerical approximation of a dynamic contact problem with friction and adhesion

We study a dynamic contact problem for viscoelastic materials with long-term memory. The contact is modelled with a coupled system of Tresca’s friction law and an ordinary differential equation which describes the adhesion effect. Under appropriate assump

A Dynamic Contact Problem with History-Dependent Operators

In this paper we present results on existence, uniqueness and convergence of solutions to the Cauchy problem for abstract first order evolutionary inclusion which contains two operators depending on the history of the solution. These results are applicable to a dynamic contact problem for viscoelastic materials with a normal compliance contact condition with memory and a friction law in which the friction bound depends on the magnitude of the tangential displacement. The proofs are based on recent results for hemivariational inequalities and a fixed point argument.

Analysis of an adhesive contact problem for viscoelastic materials with long memory

The goal of this paper is to study a mathematical model which describes the adhesive contact between a deformable body and a foundation. The body consists of a viscoelastic material with long memory and the process is assumed to be quasistatic. The adhesive contact condition on the normal plane is modeled by a version of normal compliance condition with unilateral constraint in which adhesion is taken into account, the adhesive contact condition on the tangential plane is described by an adhesive Clarke subdifferential condition, and the evolution of the bonding field is described by an ordinary differential equation. We derive the variational formulation of this problem which is a system of a variational–hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field. Then, we obtain existence and uniqueness results on abstract inclusions and abstract variational–hemivariational inequalities. Finally, we apply the abstract results to prove the existence of a unique weak solution to the contact problem.

Variational and numerical analysis of a quasistatic viscoelastic contact problem with normal compliance and friction

In this paper, a class of generalized evolution variational inequalities arising in quasistatic friction contact problem for viscoelastic materials is introduced and studied. Under some suitable assumptions, we obtain an existence and uniqueness theorem of the solution for the generalized evolution variational inequalities by using Banach’s fixed point theorem. Moreover, we study two numerical approximation schemes of the problem: semidiscrete scheme and fully discrete scheme. For both schemes, we prove the existence of the solution and derive the error estimations.

An evolutionary mixed variational problem arising from frictional contact mechanics

We consider an abstract mixed variational problem which consists of a system of an evolutionary variational equation in a Hilbert space X and an evolutionary inequality in a subset of a second Hilbert space Y, associated with an initial condition. The existence and the uniqueness of the solution is proved based on a fixed point technique. The continuous dependence on the data was also investigated. The abstract results we obtain can be applied to the mathematical treatment of a class of frictional contact problems for viscoelastic materials with short memory. In this paper we consider an antiplane model for which we deliver a mixed variational formulation with friction bound dependent set of Lagrange multipliers. After proving the existence and the uniqueness of the weak solution, we study the continuous dependence on the initial data, on the densities of the volume forces and surface tractions. Moreover, we prove the continuous dependence of the solution on the friction bound.

A Quasistatic Contact Problem for Viscoelastic Materials with Slip‐Dependent Friction and Time Delay

A mathematical model which describes an explicit time‐dependent quasistatic frictional contact problem between a deformable body and a foundation is introduced and studied, in which the contact is bilateral, the friction is modeled with Tresca’s friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a quasistatic integro‐differential variational inequality system. Based on arguments of the time‐dependent variational inequality and Banach′s fixed point theorem, an existence and uniqueness of the solution for the quasistatic integro‐differential variational inequality system is proved under some suitable conditions. Furthermore, the behavior of the solution with respect to perturbations of time‐delay term is considered and a convergence result is also given.

A class of generalized evolution variational inequalities in Banach spaces

In the present paper, a class of generalized evolution variational inequalities arising in a number of quasistatic frictional contact problems for viscoelastic materials is introduced and studied. Using Banach’s fixed point theorem, the existence and uniqueness theorem of the solution for the generalized evolution variational inequalities is proved under some suitable assumptions.