A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint

We consider a system of two viscoelastic bodies attached on one side by an adhesive where a delamination process occurs. We study the dynamic of the system for small strains, subjected to external forces, suitable boundary conditions, and an unilateral constraint on the jump of the displacement at the interface between the bodies. The constraint arises in a graph inclusion, while the delamination coefficient evolves in a rate-independent way. We prove the existence of a weak solution to the corresponding system of PDEs.