Quasi-Static Hemivariational Inequality via Vanishing Acceleration Approach

We study an abstract second order nonlinear evolution inclusion in a framework of evolution triple of spaces. We consider time-dependent possibly nonconvex nonsmooth functions and their Clarke subdifferentials operating on the unknown function. First we prove the existence of a weak solution. Then we study the asymptotic behavior of a sequence of solutions when a small parameter in the inertial term tends to zero. We prove that the limit function is a solution of a parabolic hemivariational inequality. Finally, we give an application of the abstract theorem to a quasi-static viscoelastic contact problem.