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We consider generalized gradient systems with rate-independent andrate-dependent dissipation potentials. We provide a general framework forperforming a vanishing-viscosity limit leading to the notion of parametrizedand true Balanced-Viscosity solutions that include a precise description of thejump behavior developing in this limit. Distinguishing an elastic variable $u$having a viscous damping with relaxation time $\varepsilon^\alpha$ and aninternal variable $z$ with relaxation time $\varepsilon$ we obtain differentlimits for the three cases $\alpha \in (0,1)$, $\alpha=1$ and $\alpha>1$. Anapplication to a delamination problem shows that the theory is general enoughto treat nontrivial models in continuum mechanics.

Balanced-Viscosity solutions to infinite-dimensional multi-rate systems