Balanced-Viscosity solutions for multi-rate systems

TL;DR

This paper studies the vanishing-viscosity limit of multi-rate systems with static and rate-independent components, showing convergence to Balanced Viscosity solutions and analyzing jump dynamics for different relaxation rates.

Contribution

It introduces a finite-dimensional framework for multi-rate systems with viscous regularization, proving convergence to Balanced Viscosity solutions and characterizing jump behavior based on relaxation rate parameters.

Findings

Convergence of viscous solutions to Balanced Viscosity solutions as viscosity vanishes.

Reformulation of Balanced Viscosity solutions via subdifferential inclusions.

Different jump dynamics depending on the relation between b5 and 1.

Abstract

Several mechanical systems are modeled by the static momentum balance for the displacement $u$ coupled with a rate-independent flow rule for some internal variable $z$. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients $\varepsilon^\alpha$ and $\varepsilon$, where $0<\varepsilon \ll 1$ and $\alpha>0$ is a fixed parameter. Therefore for $\alpha \neq 1$ $u$ and $z$ have different relaxation rates. We address the vanishing-viscosity analysis as $\varepsilon \downarrow 0$ of the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system, and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in $u$ and the one in $z$ are involved in the jump dynamics in different ways, according to whether $\alpha>1$, $\alpha=1$, and $\alpha \in (0,1)$.