Wave propagation in non-centrosymmetric beam-lattices with lumped   masses: discrete and micropolar modelling
  The in-plane acoustic behavior of non-centrosymmetric lattices having nodes endowed with mass and gyroscopic inertia and connected by massless ligaments with asymmetric elastic properties has been analysed through a discrete model and a continuum micropolar model. In the first case the propagation of harmonic waves and the dispersion functions have been obtained by the discrete Floquet-Bloch approach. It is shown that the optical branch departs from a critical point with vanishing group velocity and for the considered cases this branch is decreasing for increasing the norm of the wave vector from the long wave limit. A micropolar continuum model, useful to approximate the discrete model, has been derived through a continualization method based on a down-scaling law based on a second-order Taylor expansion of the generalized macro-displacement field. It is worth noting that the second order elasticity tensor coupling curvatures and micro-couples turns out to be negative defined also in the general case of non-centrosymmetric lattice. The eigenvalue problem governing the harmonic propagation in the micropolar non-centrosymmetric continuum results in general characterized by a hermitian full matrix that is exact up to the second order in the wave vector. Examples concerning square and equilateral triangular lattices and their acoustic properties have been analysed from both the exact Lagrangian model (within the assumed hypotheses) and the micropolar approximate model. Finally, in consideration of the negative definiteness of the second order elastic tensor of the micropolar model, the loss of strong hyperbolicity of the equation of motion has been investigated. 
Andrea Bacigalupo and Luigi Gambarotta
cond-mat.mtrl-sci