A frictionless contact problem for elastic-viscoplastic materials with normal compliance: Numerical analysis and computational experiments

Abstract

In this paper we consider a frictionless contact problem between an elastic-viscoplastic body and an obstacle. The process is assumed to be quasistatic and the contact is modeled with normal compliance. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using strongly monotone operators arguments and Banach's fixed point theorem. We also study the numerical approach to the problem using spatially semi-discrete and fully discrete finite elements schemes with implicit and explicit discretization in time. We show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions. Finally, we present some numerical results involving examples in one, two and three dimensions. (2000): 65N30, 74C10, 74M15, 74S05 Mathematics Subject Classification

Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage

“…Rate-type viscoplastic constitutive laws of the form (1.1) in which the function G does not depend on β were considered by many authors, see for instance, [8,23] and the references therein. Frictionless contact problems for such kind of materials were studied in [10,12,22,28]. A frictionless contact problem for materials of the form (1.1) in which β is an internal state variable whose evolution is described by an ordinary differential equation has been recently considered in [11].…”

A Dynamic Problem of Frictionless Contact for Elastic-Thermo-Viscoplastic Materials With Damage

“…Dynamic and quasistatic contact problems are the topic of numerous papers, e.g. [1,2,4,13,32], and the comprehensive references [19,36]. However, the mathematical problem modelled the quasi-static evolution of damage in thermo-viscoplastic materials has been studied in [26].…”

A quasistatic viscoplastic contact problem with normal compliance and friction

“…We investigated recently a number of problems related to quasistatic contact for viscoplastic materials. In particular, models for frictionless contact were considered in [18], in the case of Signorini contact conditions, and in [8] in the case of normal compliance contact conditions. The analysis of the bilateral contact with Tresca friction law was provided in [2] and its numerical approximation was provided in [5].…”

“…Frictional problems with regularized Coulomb's law were study recently in [1] both in the case of bilateral and Signorini unilateral contact. The results obtained in [1,2,8,18,19] deal with the variational analysis of the mechanical problems. They involve existence and uniqueness of weak solutions, i.e.…”

“…They were obtained using arguments of evolutionary variational inequalities and fixed point. The results in [5,8] concern the numerical analysis of the models, including error estimates for the approximate solutions.…”