Damage localization and rupture with gradient damage models

We propose a method of construction of non homogeneous solutions to the problem of traction ofa bar made of an elastic-damaging material whose softening behavior is regularized by a gradient damage model.We show that, for sufficiently long bars, localization arises on sets whose length is proportional to the materialinternal length and with a profile which is also characteristic of the material. The rupture of the bar occurs at thecenter of the localization zone when the damage reaches there the critical value corresponding to the loss ofrigidity of the material. The dissipated energy during all the damage process up to rupture is a quantity c G whichcan be expressed in terms of the material parameters. Accordingly, c G can be considered as the usual surfaceenergy density appearing in the Griffith theory of brittle fracture. All these theoretical considerations areillustrated by numerical examples