On a phase-field model of damage for hybrid laminates with cohesive interface

In this paper, we investigate a rate-independent model for hybrid laminates described by a damage phase-field approach on two layers coupled with a cohesive law governing the behaviour of their interface in a one-dimensional set-up. For the analysis, we adopt the notion of energetic evolution, based on global minimisation of the involved energy. Due to the presence of the cohesive zone, as already emerged in literature, compactness issues lead to the introduction of a fictitious variable replacing the physical one which represents the maximal opening of the interface displacement discontinuity reached during the evolution. A new strategy which allows to recover the equivalence between the fictitious and the real variable under general loading-unloading regimes is illustrated. The argument is based on time regularity of energetic evolutions. This regularity is achieved by means of a careful balance between the convexity of the elastic energy of the layers and the natural concavity of the cohesive energy of the interface.