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The paper studies the asymptotic analysis of a model couplingelastoplasticity and damage depending on three parameters -- governingviscosity, plastic hardening, and convergence rate of plastic strain anddisplacement to equilibrium -- as they vanish in different orders. The notionof limit evolution obtained is proven to coincide in any case with a notionintroduced by Crismale and Rossi in 2019; moreover, such solutions are closelyrelated to those obtained in the vanishing-viscosity limit by Crismale andLazzaroni in 2016, for the analogous model where only the viscosity parameterwas present.

Singular limits of a coupled elasto-plastic damage system as viscosity and hardening vanish