On uniqueness and localization in elastic-damage materials

We consider isotropic elastic-damage behaviour such that the evolution law of the unique scalar internal variable characterizing material damage is associative (“ d -associativex201D; model). Two particular variants of modelling are considered. More precisely, and because of some analogies between these models and non-associative plasticity, we attempt to compare the loss of positiveness of second-order work and the localization criteria. These criteria are written in terms of critical damage. Thus, the damage value at loss of positiveness of second-order work is explicitly calculated in three-dimensional and plane strain cases. The procedure leading to the damage value at localization in the plane strain case is also presented. Both criteria are then compared for some loading paths. The results obtained indicate the localization occurring before loss of positiveness of the second-order work for some loading paths for one of two models. The same loading paths were tested with the other isotropic elastic-damage model; it is shown that localization always occurs after the loss of positiveness of second-order work (i.e. in softening phase). To endow this result with more generality, we then consider the problem of localization through a spectral analysis which finally shows that localization cannot take place in the hardening phase.