  The paper studies the modes of vibrations of a lattice with rod-like
particles, in a continuum model where the sites of the lattice are the
connections among strings and rigid rods. In these structures then,
translational and rotational degrees of freedom are strongly coupled. We will
discuss in particular two-dimensional lattices with auxetic-like behaviour.
Auxetics are materials with a negative Poisson elastic parameter, meaning that
they have a lateral extension, instead to shrink, when they are stretched. We
assume as "auxetic-like" two-dimensional structures, structures which do not
collapse, when stretched in one of the in-plane directions. The presence of
rigid rod-like particles in the lattice prevents the shrinking of the membrane.
Complete bandgaps between acoustic and optical modes are observed in analogy
with the behaviour of crystalline materials.
