  In this paper we analyze a system for brittle delamination between two
visco-elastic bodies, also subject to inertia, which can be interpreted as a
model for dynamic fracture. The rate-independent flow rule for the delamination
parameter is coupled with the momentum balance for the displacement, including
inertia. This model features a nonsmooth constraint ensuring the continuity of
the displacements outside the crack set, which is marked by the support of the
delamination parameter. A weak solvability concept, generalizing the notion of
energetic solution for rate-independent systems to the present mixed
rate-dependent/rate-independent frame, is proposed. Via refined variational
convergence techniques, existence of solutions is proved by passing to the
limit in approximating systems which regularize the nonsmooth constraint by
conditions for adhesive contact. The presence of the inertial term requires the
design of suitable recovery spaces small enough to provide compactness but
large enough to recover the information on the crack set in the limit.
