We consider a variational model for damaged elastic materials. This model depends on three small parameters, the first related to the cost of the damage, the second to the width of the damaged regions, and the third preventing the material to be totally damaged. We first investigate the Gamma-limit of the corresponding functionals in the case of antiplane shear, as these parameters tend to zero. We find that the limit functional corresponds to a model for fracture mechanics or for plasticity, depending on the asymptotic ratios of the three parameters. The extension of some interesting results from the antiplane to the general case of linearized elasticity in dimension n is the object of the second part of the talk. As we shall see, this is not a straightforward generalization of the scalar case, requiring the involvement of the new functional space GSBD and the proof of suitable density properties.