  In this paper we analyze an isothermal and isotropic model for viscoelastic
media combining linearized perfect plasticity (allowing for concentration of
plastic strain and development of shear bands) and damage effects in a dynamic
setting. The interplay between the viscoelastic rheology with inertia,
elasto-plasticity, and unidirectional rate-dependent incomplete damage
affecting both the elastic and viscous response, as well as the plastic yield
stress, is rigorously characterized by showing existence of weak solutions to
the constitutive and balance equations of the model. The analysis relies on the
notions of plastic-strain measures and bounded-deformation displacements, on
sophisticated time-regularity estimates to establish a duality between
acceleration and velocity of the elastic displacement, on the theory of
rate-independent processes for the energy conservation in the dynamical-plastic
part, and on the proof of the strong convergence of the elastic strains.
Existence of a suitably defined weak solutions is proved rather constructively
by using a staggered two-step time discretization scheme.
