@isola3; @Piezo]).

Motivation and originality of the present work
----------------------------------------------

Different modelling efforts were recently made trying to account for the observed band-gaps in a reliable manner. The most common models are based on homogenization procedures on simple periodic microstructures with strong contrast giving rise to Cauchy-type equivalent continua. Often, it is remarked that the equivalent continua obtained by means of these homogenization techniques are able to predict band-gaps when the equivalent mass of the macroscopic system becomes negative (see e.g.[@BoutinNegM; @Huang0; @Huang]). Homogenized systems with negative equivalent mass can be successfully replaced by suitable generalized continua. Indeed, it is known that homogenization of strongly heterogeneous periodic systems may lead to generalized continua as equivalent macroscopic medium (see e.g.[@isolaHomog]). Some rare papers (see e.g. [@HuangMicrostr; @Huang; @Vasiliev]) can be found in the literature in which lattice models with enriched kinematics are used to model band-gaps without the need of introducing negative equivalent masses. Nevertheless, to the authors’ knowledge, a systematic treatment of band-gap modelling based on the spirit of micromorphic continuuum mechanics is still lacking and deserves attention. Indeed, the idea of using generalized continuum theories to describe microstructured materials needs to be fully developed in order to achieve a 