Numerical analysis of history-dependent quasivariational inequalities with applications in contact mechanics

Abstract

A new class of history-dependent quasivariational inequalities was recently studied in (M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22 (2011) 471-491). Existence, uniqueness and regularity results were proved and used in the study of several mathematical models which describe the contact between a deformable body and an obstacle. The aim of this paper is to provide numerical analysis of the quasivariational inequal- ities introduced in the aforementioned paper. To this end we introduce temporally semi-discrete and fully discrete schemes for the numerical approximation of the inequalities, show their unique solvabil- ity, and derive error estimates. We then apply these results to a quasistatic frictional contact problem in which the material's behavior is modeled with a viscoelastic constitutive law, the contact is bilat- eral, and friction is described with a slip-rate version of Coulomb's law. We discuss implementation of the corresponding fully-discrete scheme and present numerical simulation results on a two-dimensional example.