Construction and analysis of localized responses for gradient damage models in a 1D setting

We propose a method of construction of non homogeneous solutions to the problem of traction of a 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 material internal length and with a profile which is also characteristic of the material. We point out the very sensitivity of the responses to the parameters of the damage law. All these theoretical considerations are illustrated by numerical examples