Balanced Viscosity Solutions to a Rate-Independent Coupled Elasto-plastic Damage System

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 system. This extends performed for same system in [V. Crismale G. Lazzaroni, Calc. Var. Partial Differential Equations, 55 (2016), 17], where regularization involving only variable was set forth. In present work, an additional approximation featuring vanishing plastic hardening introduced order deal viscosity variable. Different regimes are considered, leading different notions of Balanced Viscosity solutions perfectly system, its version hardening.