ALEX 2018 Workshop: Abstracts

Visco-Energetic solutions to rate-independent systems, with applications to finite-strain plasticity and brittle fracture

Riccarda Rossi

Università degli studi di Brescia (Italy)

Visco-Energetic solutions to rate-independent systems have been recently obtained [3] by passing to the time-continuous limit in a time-incremental scheme, akin to that for Energetic solutions [1], but perturbed by a ‘viscous’ correction term, as in the case of Balanced Viscosity solutions [2]. However, for Visco-Energetic solutions this viscous correction is tuned by a fixed parameter. The resulting solution notion is characterized by a stability condition and an energy balance analogous to those for Energetic solutions, but, in addition, it provides a fine description of the system behavior at jumps as Balanced Viscosity solutions do. Visco-Energetic evolution can be thus thought as ‘in-between’ Energetic and Balanced Viscosity evolution, cf. [4].

We will explore these aspects in a general metric framework. We will then illustrate the application of the Visco-Energetic concept to models for finite-strain plasticity (cf. [5]) and for the evolution of brittle fractures. The talk reflects joint collaborations with Gianni Dal Maso, Giuseppe Savaré, and Rodica Toader.

References

  • 1 A. Mielke, Evolution of rate-independent systems. In: Evolutionary equations. Vol. II, 461–559, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005.
  • 2 A. Mielke, R. Rossi, and G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. (JEMS) 18 (2016), 2107–2165.
  • 3 L. Minotti and G. Savaré, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems. Arch. Ration. Mech. Anal. 227 (2018), 477–543.
  • 4 R. Rossi and G. Savaré, From Visco-Energetic to Energetic and Balanced Viscosity solutions of rate-independent systems. In: Solvability, Regularity, Optimal Control of Boundary Value Problems for PDEs, Springer INdAM Ser. vol. 22, 489–531, Springer International Publishing, 2017.
  • 5 R. Rossi, Visco-Energetic solutions to some rate-independent systems in damage, delamination, and plasticity, Submitted. Preprint arXiv: 1803.03768 (2018).