Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials

Dumitru Motreanu,Mircea Sofonea

doi:10.1155/s1085337599000172

Abstract

We consider a class of evolutionary variational inequalities arising in quasistatic frictional contact problems for linear elastic materials. We indicate sufficient conditions in order to have the existence, the uniqueness and the Lipschitz continuous dependence of the solution with respect to the data, respectively. The existence of the solution is obtained using a time-discretization method, compactness and lower semicontinuity arguments. In the study of the discrete problems we use a recent result obtained by the authors (2000). Further, we apply the abstract results in the study of a number of mechanical problems modeling the frictional contact between a deformable body and a foundation. The material is assumed to have linear elastic behavior and the processes are quasistatic. The first problem concerns a model with normal compliance and a version of Coulomb's law of dry friction, for which we prove the existence of a weak solution. We then consider a problem of bilateral contact with Tresca's friction law and a problem involving a simplified version of Coulomb's friction law. For these two problems we prove the existence, the uniqueness and the Lipschitz continuous dependence of the weak solution with respect to the data.

Highlights

This work concerns the study of a class of abstract evolutionary variational inequalities modeling the frictional contact between an elastic body and a foundation
We model the contact with a general normal compliance condition, similar to the one in [7, 15]
We prove that the mechanical problem leads to a variational formulation of the form (1.1) and (1.2) in which u represents the displacement field

Summary

Introduction

This work concerns the study of a class of abstract evolutionary variational inequalities modeling the frictional contact between an elastic body and a foundation. Using the abstract results obtained in the study of the Cauchy problem (1.1) and (1.2), we establish the existence of a weak solution of the model, under a smallness assumption concerning the contact and frictional boundary conditions This result completes the results obtained in [2, 3, 9] where quasistatic contact problems with normal compliance and friction involving linear elastic materials were considered. We present a quasistatic elastic problem modeling the bilateral contact with Tresca’s friction law as well as a quasistatic contact problem with a simplified version of Coulomb’s law For both these problems we prove the existence, the uniqueness and the Lipschitz continuous dependence of the solution with respect to the data and we extend some results presented in [6, 13], where the corresponding static problems are considered.

The abstract problem

A frictional contact problem with normal compliance

Other quasistatic frictional contact problems