Fatigue effects in elastic materials with variational damage models: A vanishing viscosity approach

TL;DR

This paper investigates the existence of quasistatic evolutions in gradient damage models that incorporate fatigue effects, using a vanishing viscosity approach to handle the mathematical challenges of damage accumulation in elastic materials.

Contribution

It introduces a novel vanishing viscosity method to prove the existence of quasistatic evolutions in damage models accounting for fatigue effects.

Findings

Established existence of quasistatic evolutions with fatigue effects

Developed a two-step vanishing viscosity approach

Provided a rigorous mathematical framework for damage accumulation

Abstract

We study the existence of quasistatic evolutions for a family of gradient damage models which take into account fatigue, that is the process of weakening in a material due to repeated applied loads. The main feature of these models is the fact that damage is favoured in regions where the cumulation of the elastic strain (or other relevant variables, depend on the model) is higher. To prove the existence of a quasistatic evolution, we follow a vanishing viscosity approach based on two steps: we first let the time-step $\tau$ of the time-discretisation and later the viscosity parameter $\epsilon$ go to zero. As $\tau \to 0$, we find $\epsilon$-approximate viscous evolutions; then, as $\epsilon \to 0$, we find a rescaled approximate evolution satisfying an energy-dissipation balance.