Balanced-Viscosity solutions to infinite-dimensional multi-rate systems

TL;DR

This paper develops a framework for analyzing the vanishing-viscosity limit in generalized gradient systems with mixed rate-independent and rate-dependent dissipation, capturing complex jump behaviors in continuum mechanics models.

Contribution

It introduces a comprehensive approach to derive Balanced-Viscosity solutions in infinite-dimensional systems, accounting for different relaxation time regimes and applying to continuum mechanics problems.

Findings

Different limits for relaxation times $eta$ in (0,1), 1, and >1.

Framework successfully models jump behavior in continuum mechanics.

Application to a delamination problem demonstrates practical relevance.

Abstract

We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable $u$ having a viscous damping with relaxation time $\varepsilon^\alpha$ and an internal variable $z$ with relaxation time $\varepsilon$ we obtain different limits for the three cases $\alpha \in (0,1)$, $\alpha=1$ and $\alpha>1$. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.