Analysis of Quasistatic Viscoplastic Contact Problems with Normal Compliance

We consider two quasistatic problems that describe the contact between a viscoplastic body and an obstacle, the so-called foundation. The contact is frictionless and is modelled with normal compliance of such a type that the penetration is not restricted in the first problem, but is restricted with unilateral constraint, in the second one. For each problem we derive a variational formulation, then we prove its unique solvability. The proofs are based on a recent result on history-dependent quasivariational inequalities obtained in (Sofonea and Matei, Eur. J. Appl. Math. 22 (2011)). Next, we prove the convergence of the weak solution of the first problem to the weak solution of the second problem, as the stiffness coefficient of the foundation converges to infinity. Finally, we provide a numerical validation of this convergence result. To this end we introduce fully discrete schemes for the numerical approximation of the contact problems, implement them on a computer code and present numerical simulation results in the study of a two-dimensional example.