Localized versus diffuse damage: A variational approach

Abstract

A variational model based on incremental energy minimization is proposed to describe the evolution of damage in elastic materials. The model accounts for an elastic energy, depending on the damage variable, and for a damage energy, which has a local and a non-local term. The evolution of the displacement and damage fields, representing the problem unknowns, is determined by an incremental energy minimization problem, which is analytically solved in a special one-dimensional case, and numerically in a two-dimensional setting. By properly assigning the shapes of the elastic and damage energies, two different damage modes are reproduced: localized damage, consisting in a stress-softening process, in which damage forms in thin body portions and coalesces in fracture surfaces, and diffuse damage, which is characterized by a stress-hardening response, with damage spreading in large zones of the body. The former mechanism is typical of quasi-brittle materials, while the latter one is proper for ductile materials. Results of numerical simulations are presented, where the behaviors of both quasi-brittle and ductile materials are reproduced.