The modes of vibrations in honeycomb and auxetic structures are studied, with models in which the lattice is represented by a planar network where sites are connected by strings and rigid rods. The auxetic network is obtained modifying a model proposed by Evans et al. in 1991, and used to explain the negative Poissons ratio of auxetic materials. This relevant property means that the materials have a lateral extension, instead to shrink, when they are stretched. For what concerns the acoustic properties of these structures, they absorb noise and vibrations more efficiently than non-auxetic equivalents. The acoustic and optical dispersions obtained in the case of the auxetic model are compared with the dispersions displayed by a conventional honeycomb network. It is possible to see that the phonon dispersions of the auxetic model possess a complete bandgap and that the Goldstone mode group velocity is strongly dependent on the direction of propagation. The presence of a complete bandgap can explain some experimental observations on the sound propagation properties of the auxetic materials.
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