Gradient damage models coupled with plasticity: a variational approach

par Jean-Jacques Marigo (École Polytechnique (LMS))

lundi 14 févr. 2022, 11:00 → 12:00 Europe/Paris

Université Paris Dauphine

It is now well established that gradient damage models are very efficient to account for the behavior of brittle and quasi-brittle materials. However, these "quasi-brittle" models are not able to account for residual strains and consequently cannot be used in ductile fracture. Moreover there is no discontinuity of the displacement in the damage strip before the loss of rigidity at its center, i.e. before the nucleation of a crack. In other words such a model cannot account for the nucleation of cohesive cracks, i.e. the existence of surface of discontinuity of the displacement with a non vanishing stress. The natural way to include such effects is to introduce plastic strains into the model and to couple their evolution with damage evolution. For that we will adopt a variational approach. The main ingredients are the following ones: (i) one defines the total energy of the body in terms of the state fields which include the displacement field and the internal variable fields, namely the damage, the plastic strain and the cumulated plastic strain fields; (ii) one postulates that the evolution of the internal variables is governed by the three principles of irreversibility, stability and energy balance. In particular, the stability condition is essential as well for constructing the model in a rational and systematic way as for obtaining and proving general properties. Besides, we have the chance that the variational approach works and has been already developed both in plasticity and in damage mechanics, even though only separately up to now. So, it "suffices" to introduce the coupling by choosing the form of the total energy to obtain, by virtue of our plug and play device, a model of gradient damage coupled with plasticity. Specifically, our model, presented here in a three-dimensional setting, contains three state functions, namely $E(\alpha)$, $d(\alpha)$ and $\bar\sigma_P(\alpha)$ which give the dependence of the stiffness, the local damage dissipated energy and the plastic yield stress on the damage variable. So, our choice of coupling is minimalist in the sense that it simply consists in introducing this dependence of the yield plastic stress $\sigma_P(\alpha)$ on the damage variable (with the natural assumption that $\sigma_P$ goes to $0$ when the damages goes to $1$). In turn, by virtue of the variational character of the model, the product $\sigma'_P(\alpha)\bar p$ the derivative of the state function $\sigma_P(\alpha)$ by the cumulated plastic strain $\bar p$ enters in the damage criterion and this coupling plays a fundamental role in the nucleation of a cohesive crack.