A strain gradient variational approach to damage. A comparison with damage gradient models and numerical results.

The global response of experimental uniaxial tests cannot be homogeneous, because of the unavoidable presence of localized deformations, which is always preferential from an energetic viewpoint. Accordingly, one must introduce some characteristic lengths in order to penalize deformations that are too localized. This is what leads to the concept of non-local damage models. The non-local approach employs non-local terms in the internal deformation energy in order to control the size of the localization region. In phase-field models, and, in general, in gradient models, dependence of the internal energy upon the first gradient of damage is assumed, while in our approach the non-locality is given by the dependence of the internal energy upon the second gradient of the displacement field. A discussion of the advantages and challenges of using the gradient of damage and of using the second gradient of the displacement field will be addressed in the present paper. A variational inequality is formulated and Partial Differential Equations (PDEs), Boundary Conditions (BCs) and Karush-Kuhn-Tucker (KKT) conditions will be derived within the framework of 2D strain gradient damage mechanics. A novel dependence of the stiffness coefficients with respect to the damage field will also be discussed. Besides, an explicit derivation of the damage field evolution in loading conditions will be provided. Finally, a numerical technique based on commercial softwares has been introduced and discussed for a couple of standard problems.