  A rate-independent model coupling small strain associative elasto-plasticity
and damage is studied via a 'vanishing-viscosity' analysis with respect to all
the variables describing the system. This extends the analysis performed for
the same system in [Crismale-Lazzaroni 2016], where a vanishing-viscosity
regularization involving only the damage variable was set forth. In the present
work, an additional approximation featuring vanishing plastic hardening is
introduced in order to deal with the vanishing viscosity in the plastic
variable. Different regimes are considered, leading to different notions of
Balanced Viscosity solutions for the perfectly plastic damage system, and for
its version with hardening.
