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

• Published in: Archive for rational mechanics and analysis 2018-02, Vol.227 (2), p.477-543
• Main Authors: Minotti, Luca, Savaré, Giuseppe
• Format: Article
• Language: English
• Subjects: Classical Mechanics; Complex Systems; Fluid- and Aerodynamics; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-017-1165-5

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 (for example 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 of covering a wide range of applications. The new refined Energy Balance condition compensates for 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.