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

Abstract

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.