We propose the new notion of Visco-Energetic solutions to rate-independent systems (X, E, d) driven by a time dependent energy E and a dissipation quasi-distance 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 d is incremented by a viscous correction δ (e.g. proportional to the square of the distance 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.

Visco-Energetic solutions of rate-independent systems (recently introduced in [14]) are obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation is reinforced by a viscous correction δ, typically a quadratic perturbation of the dissipation distance. Like Energetic and Balanced Viscosity solutions, they provide a variational characterization of rate-independent evolutions, with an accurate description of their jump behaviour.In the present paper we study Visco-Energetic solutions in the one-dimensional case and we obtain a full characterization for a broad class of energy functionals. In particular, we prove that they exhibit a sort of intermediate behaviour between Energetic and Balanced Viscosity solutions, which can be finely tuned according to the choice of the viscous correction δ. * Università di Pavia. email: luca.minotti01@universitadipavia.it.When the loading ℓ ∈ C 1 ([a, b]) is strictly increasing, Ψ(v) := α|v| with α > 0, and u(a) is choosen carefully, it is possible to prove, [17], that an Energetic solution u is an increasing selection of the equation α + W * * (u(t)) ∋ ℓ(t)for every t ∈ [a, b], (1.2)

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