Viscous corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution Problems

We propose the new notion of Visco-Energetic solutions to rate-independent systems $(X,\mathcal E,\mathsf d)$ driven by a time dependent energy $\mathcal E$ and a dissipation quasi-distance $\mathsf d$ in a general metric-topological space $X$. As for the classic Energetic approach, solutions can be obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation (quasi-)distance $\mathsf d$ is incremented by a viscous correction $\delta$ (e.g.~proportional to the square of the distance $\mathsf d$), which penalizes far distance jumps by inducing a localized version of the stability condition. We prove a general convergence result and a typical characterization by Stability and Energy Balance in a setting comparable to the standard energetic one, thus capable to cover a wide range of applications. The new refined Energy Balance condition compensates the localized stability and provides a careful description of the jump behavior: at every jump the solution follows an optimal transition, which resembles in a suitable variational sense the discrete scheme that has been implemented for the whole construction.