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

Abstract

In this paper, we study the well-posedness of a class of evolutionary variationalhemivariational inequalities coupled with a nonlinear ordinary differential equation in a Banach space. The intended application is in modelling frictional contact between a deformable viscoelastic body and a rigid foundation. The system under consideration allows the friction coefficient µ to depend on an external state variable α, described by an ODE, and the slip rate | uτ |. In addition, the normal stress σν is a function of time and space. We base the proof on an iterative approximation scheme, showing that the problem has a unique weak solution. Moreover, we show that the flow map depends continuously on the initial data. Finally, we include two applications; the first is in the normal compliance setting, and in the second, we consider normal damped response. Both are relevant for rate-and-state friction laws.