The Rothe Method for Variational-Hemivariational Inequalities with Applications to Contact Mechanics

Abstract

We consider a new class of first order evolutionary variational-hemivariational inequalities for which we prove an existence and uniqueness result. The proof is based on a time-discretization method, also known as the Rothe method. It consists of considering a discrete version of each inequality in the class, proving its unique solvability, and recovering the solution of the continuous problem as the time step converges to zero. Then we introduce a quasi-static frictionless problem for Kelvin--Voigt viscoelastic materials in which the contact is modeled with a nonmonotone normal compliance condition and a unilateral constraint. We prove the variational formulation of the problem cast in the abstract setting of variational-hemivariational inequalities, with a convenient choice of spaces and operators. Further, we apply our abstract result in order to prove the unique weak solvability of the problem.