*Accepted Paper*

**Inserted:** 13 jul 2018

**Last Updated:** 21 feb 2019

**Journal:** J. Nonlinear Sci.

**Year:** 2018

**Doi:** 10.1007/s00332-018-9511-9

**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.

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