Wave propagation in non-centrosymmetric beam-lattices with lumped masses: Discrete and micropolar modeling

The in-plane acoustic behavior of non-centrosymmetric lattices having nodes endowed with mass and rotational inertia and connected by massless ligaments with asymmetric elastic properties has been an- alyzed through a discrete model and a continuum micropolar model. In the \ufb01rst case the propagation of harmonic waves and the dispersion functions have been obtained by the discrete Floquet\u2013Bloch ap- proach. It is shown that the optical branch departs from a critical point with vanishing group velocity and is decreasing for increasing the norm of the wave vector. A micropolar continuum model has been derived through a continualization method based on a down-scaling law from a second-order Taylor ex- pansion of the generalized macro-displacement \ufb01eld. It is worth noting that the second order elasticity tensor coupling curvatures and micro-couples turns out to be negative-de\ufb01nite also in the general case of non-centrosymmetric lattice. The eigenvalue problem governing the harmonic propagation in the mi- cropolar 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 have been analyzed and their acoustic properties have been derived with the discrete and continuum models. The dependence of the Floquet\u2013 Bloch spectra on the lattice non-centrosymmetry is shown together with validity limits of the micropolar model. Finally, in consideration of the negative de\ufb01niteness of the second order elastic tensor of the micropolar model, the loss of strong hyperbolicity of the equation of motion has been investigated