Variational and numerical analysis of a dynamic viscoelastic contact problem with friction and wear

Abstract

In this paper, we consider a dynamic viscoelastic contact problem with friction and wear, and describe it as a system of nonlinear partial differential equations. We formulate the previous problem as a hyperbolic quasi-variational inequality by employing the variational method. We adopt the Rothe method to show the existence and uniqueness of weak solution for the hyperbolic quasi-variational inequality under mild conditions. We also give a fully discrete scheme for solving the hyperbolic quasi-variational inequality and obtain error estimates for the fully discrete scheme.As a generalization of the contact problem considered in [5], Chau et al. [4] introduced and studied a dynamic frictionless contact problem and gave a fully discrete scheme for solving such problem. Bartosz[3] considered a dynamical viscoelastic contact problem to modify the model treated by Ciulcu et al. [7].Recently, Cocou [8] extended the static contact problem considered by Rabier et al.[18] to a dynamic viscoelastic contact problem with friction and obtained an existence and uniqueness of the weak solution for such problem. However, to our best knowledge, there is no study for the dynamic viscoelastic contact problem with friction and wear in the existing literature. The motivation of this paper is to make a new attempt in this direction.It is well known that Duvaut and Lions [9] were the first to study quasi-static frictional contact viscoelastic problems within the framework of variational inequalities. From then on, various variational inequalities, hemivariational inequalities and other related problems have been derived from different physical phenomena in contact mechanics and abundant research results have been obtained for their studies ([15, 20, 21, 27, 28, 30, 31, 33, 36]). Recently, Migórski and Zeng [23] studied a class of hyperbolic variational inequalities and applied their results to study the existence of weak solutions for the dynamic frictional contact problem without wear.In this paper, we consider a mathematical model which describes a dynamic viscoelastic contact problem with friction and wear, in which the material behavior is followed by the Kelvin-Voigt viscoelastic constitutive law and the frictional contact is modelled with a wear governed by a simplified version of Archard's law [33,34] for the velocity field associated to a version of Coulombs law of dry friction.The rest of this paper is organized as follows. Section 2 presents some necessary preliminaries and the weak formulation of the dynamic viscoelastic contact problem with friction and wear. Inspired by Migórski and Zeng [23], we prove the existence and uniqueness of the solution for the hyperbolic quasi-variational inequality under mild conditions by applying the Rothe method [19] in Section 3. We obtain the existence and uniqueness of weak solution for the dynamic viscoelastic contact problem with friction and wear in Section 4. Finally, we present the fully discrete scheme for solving the hyperbolic quasi-variational inequality and derive the error ...

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

“…Recently, Chen et al [6] introduced a hyperbolic quasi-variational inequality to characterize a dynamic viscoelastic contact problem with friction and wear. Very recently, in order to model an elastic frictional contact problem with long memory, damage and wear, Chen et al [5] considered a new class of differential nonlinear system driven by a differential equation, a history-dependent hemivariational inequality and a parabolic variational inequality in Banach spaces.…”