Gradient damage models coupled with plasticity: Variational formulation and main properties

A variational formulation is proposed for a family of elastic–plastic–damage models within the framework of rate-independent materials. That consists first in defining the total energy which contains, in particular, a gradient damage term and a term which represents the plastic dissipation but depends also on damage. Then, the evolution law is deduced from the principles of irreversibility, stability and energy balance. Accordingly, the plastic dissipation term which appears both in the damage criterion and the plastic yield criterion plays an essential role in the damage–plasticity coupling. Suitable constitutive choices on how the plastic yield stress decreases with damage, allows us to obtain a rich variety of coupled responses. A particular attention is paid on the equations which govern the formation of cohesive cracks where the displacement is discontinuous and the plasticity localizes. In the one-dimensional traction test where the solution is obtained in a closed form, we show that, because of damage localization, a cohesive crack really appears at the center of the damage zone before the rupture and the associated cohesive law is obtained in closed form in terms of the constitutive parameters. A Finite Element discrete version of the energy functional is used to simulate a two-dimensional traction test over a rectangular domain with mixed boundary conditions; again a localized band of plastic strain is generated seemingly independent of the mesh size.