Visco-Energetic solutions to one-dimensional rate-independent problems

TL;DR

This paper introduces Visco-Energetic solutions for one-dimensional rate-independent systems, providing a detailed variational characterization that interpolates between Energetic and Balanced Viscosity solutions, with a focus on jump behavior.

Contribution

It offers a full characterization of Visco-Energetic solutions in one dimension and demonstrates their intermediate behavior between existing solution concepts.

Findings

Visco-Energetic solutions can be finely tuned via viscous correction.

They exhibit an intermediate jump behavior between Energetic and BV solutions.

A broad class of energy functionals admits this characterization.

Abstract

Visco-Energetic solutions of rate-independent systems are obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation is reinforced by a viscous correction, 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.