The notions of Inertial Balanced Viscosity and Inertial Virtual Viscosity solution for rate-independent systems

TL;DR

This paper introduces Inertial Balanced Viscosity (IBV) and Inertial Virtual Viscosity (IVV) solutions for rate-independent systems, capturing inertial effects and asymptotic behaviors in mechanical evolutions with dissipation.

Contribution

It defines IBV and IVV solutions, proves their convergence in finite dimensions, and extends the Minimizing Movements algorithm to incorporate inertial effects.

Findings

IBV solutions characterize asymptotic inertial effects in rate-independent systems.

Finite-dimensional evolutions converge to IBV and IVV solutions.

Extension of the Minimizing Movements algorithm incorporates inertial terms.

Abstract

The notion of Inertial Balanced Viscosity (IBV) solution to rate-independent evolutionary processes is introduced. Such solutions are characterized by an energy balance where a suitable, rate-dependent, dissipation cost is optimized at jump times. The cost is reminiscent of the limit effect of small inertial terms. Therefore, this notion proves to be a suitable one to describe the asymptotic behavior of evolutions of mechanical systems with rate-independent dissipation in the limit of vanishing inertia and viscosity. It is indeed proved, in finite dimension, that these evolutions converge to IBV solutions. If the viscosity operator is neglected, or has a nontrivial kernel, the weaker notion of Inertial Virtual Viscosity (IVV) solutions is introduced, and the analogous convergence result holds. Again in a finite-dimensional context, it is also shown that IBV and IVV solutions can be obtained via a natural extension of the Minimizing Movements algorithm, where the limit effect of inertial terms is taken into account.