Dynamic History-Dependent Hemivariational Inequalities Controlled by Evolution Equations with Application to Contact Mechanics

Abstract

This paper is devoted the study of a generalized hybrid dynamical system, which consists of a history-dependent hemivariational inequality of parabolic type and a nonlinear evolution equation. The unique solvability for the system is established via applying surjectivity of multivalued pseudomonotone operators, fixed point theorem, and properties of the Clarke generalized gradient. As an illustrative application, a dynamic frictional contact problem for viscoelastic materials with history-dependent and adhesion is investigated, in which the friction condition is described by the Clarke generalized gradient of a nonconvex and nonsmooth function involving adhesion, and the normal damped response condition is expressed by a given nonnegative function depending on the normal velocity and adhesion.

Well-posedness of constrained evolutionary differential variational–hemivariational inequalities with applications

“…The literature on hemivariational inequalities has been significantly enlarged in the last forty years, see monographs [3,8,25,32,38], for analysis of various classes of such inequalities, see e.g. [2,9,11,13,14,15,16,18,21,20,30,43]. By our approach, we relax the assumptions on similar problems treated in the aforementioned papers and considerably improve some results by allowing the history-dependent operators to appear in the convex term and in the generalized directional derivative of a nonconvex potential.…”

“…If the operator A and a locally Lipschitz function j are assumed to be independent of x and R 1 , then the problem reduces to the parabolic hemivariational inequality studied, for example, in [14,17,22]. Very recently the system has been treated in [43] with K = V , ϕ = 0, and F , j independent of the history-dependent operators, and f independent of the variable x. A general dynamic variational-hemivariational inequality with history-dependent operators without constraints can be found in [11].…”