Two-dimensional strain gradient damage modeling: a variational approach

In this paper we formulate a linear elastic second gradient isotropic two- dimensional continuum model accounting for irreversible damage. The failure is de ned as the condition in which the damage parameter reaches 1. The quasi-static approximation is done, i.e. the kinetic energy is neg- ligible. In order to deal with dissipation of energy, as damage increases, a dissipation term is considered in the deformation energy. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush Kuhn Tucker con- ditions. Finally, numerical simulations for exemplary problems are dis- cussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element Free Galerkin (EFG) method and Moving Least Square (MLS) shape functions have been employed.