Strain gradient damage and fracture mechanics for granular heterogeneous materials

Localized deformations are often encountered in engineering applications and lead to stress concentrations, damage and fracture mechanisms. Thus, a scalar damage field, ranging from zero to one, is generally introduced to describe the internal state of structural degradation of the material. Besides, in order to control the size of localization regions, some characteristic lengths, in the constitutive model, must be introduced and, to penalize deformations that are too localized, non-local terms are used in the internal deformation energy. Damage and fracture phenomena that are here intended to be modeled are clearly of irreversible nature. In order to deal with dissipation of energy, as damage increases, a dissipation term is considered in the deformation energy. Therefore, in order to use the principle of least action, we show a variational inequality formulation, that leads to Partial Differential Equations (PDEs), Boundary Conditions (BCs) and Karush-Kuhn-Tucker (KKT) conditions. We remark that KKT conditions are associated to an explicit damage field evolution. The framework is that of 2D strain gradient damage mechanics, where the elastic strain energy density of the body has been assumed to be geometrically non-linear and to depend upon the strain gradient. The dependence of Lamé and second gradient elastic coefficients with respect to damage is not prescribed by the variational principle and need to be experimentally identified. A novel discussion on this point will be done. Generally, these constitutive coefficients are all assumed to decrease as damage increases and to be locally zero if the value attained by damage is one. This last situation is associated with crack formation and/or propagation. A numerical technique based on commercial software will be presented and discussed for a couple of exemplary problems, where a discussion will be performed as some constitutive parameters are varying, with the inclusion of mesh-independence evidence. Finally, the case of an obliquely notched rectangular specimen subject to monotonous tensile and shear loading tests and brittle fracture propagation is discussed.