Modelling and well-posedness of evolutionary differential variational-hemivariational inequalities

In this paper, we study the well-posedness of a class of evolutionary variational-hemivariational inequalities coupled with a nonlinear ordinary differential equation in Banach spaces. The proof is based on an iterative approximation scheme showing that the problem has a unique mild solution. In addition, we established continuity of the flow map with respect to the initial data. Under the general framework, we consider two new applications for modelling of frictional contact with viscoelastic materials, where the friction coefficient $\mu$ depends on an external state variable $\alpha$ and the slip rate $|\dot{u}_\tau|$. In the first application, we consider Coulomb friction with normal compliance, and in the second, normal damped response. In both cases, we present a new first-order approximation of the Dieterich rate-and-state friction law