SEMINARIO DEI DOTTORANDI Starting from the damage model for elastic material introduced by Francfort&Marigo, we will present results that combines efficiently the notion of quasi-static evolution with processes of homogenization. We will consider 3 different kind of energy derived from such model: a 1D oscillating energy describing the evolution of a two two-phase material, an elastic energy penalized by the perimeter of the damage region, and finally we study the problem in a dynamic framework considering also a kinetic term. Existence and convergence results will be shown. Moreover we wil present a different way to study such evolutions through a threshold criterion, showing some results in this direction.