Inertial Balanced Viscosity solutions to rate-independent systems

Abstract: Rate-independent evolutions frequently occur in mechanics when the problem under consideration presents such small rate-dependent effects, as inertia or viscosity, that can be neglected. If the driving potential energy is nonconvex any (reasonable) solution necessarily exhibits time-discontinuities. To describe the behaviour of the jumps, in the last decades the notions of Energetic and Balanced Viscosity solutions have been developed, the latter as a refinement of the former. While for Energetic solutions the cost at jumps is described only in terms of the rate-independent dissipation potential, in the case of Balanced Viscosity solutions it arises from a suitable coupling between this dissipation potential and a smaller and smaller viscosity potential, which is reminiscent of an original presence of viscous damping. In this talk we present a novel notion of solution to rate-independent systems, named Inertial Balanced Viscosity (IBV) solution, which in addition takes into account small inertial effects. Differently from the previous cases, now the jump cost turns out to be rate-dependent. In finite dimension, we show how IBV solutions can be obtained as a vanishing inertia and viscosity limit of dynamic evolutions, and we introduce a multiscale Minimizing Movements algorithm which can be used to build such solutions. This is a joint work with G. Scilla and F. Solombrino.

MathematicsPhysics

Audience: researchers in the topic

Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor).