Discrete and continuous aspects of some metamaterial elastic structures with band gaps

We study three different 1D continuous models (extensional rods, Euler and Timoshenko beams) for addressing the dynamic properties of those microstructural materials containing a density of resonators. These models correspond to metamaterials which show interesting properties: In particular, the property that is the objective of this paper is the capacity of eliminating the vibration amplitude in a specific frequency range, which is called hereinafter band gap. The simplicity of these models emphasizes those microstructural properties having a relation with the band gap. We show that the rigidity of the hosting structure does not affect the values of the frequency band gap; it affects only the distance between the load-source of vibration and those points where the amplitude attenuation is visible. We also study, from a numerical point of view and using the Euler beam as the hosting structure, the case of a finite number of resonators. In particular, we study the minimum number of resonators which provides the same band gap as in the case of the presence of a density of resonators. We finally perform a numerical study on a periodic 2D elastic structure, which behaves like the Timoshenko beam model and for which an identification procedure is given.