Design of non-linear seismic metamaterials - Invited talk by R. Zivieri

A model used to understand seismic metamaterials from a theoretical point of view is based on the concept of the periodic sub-wavelength resonant mass-in-mass system, see Fig. 1. We have already proposed a continuous implementation of those type of seismic metamaterials based on the use of isochronous mechanical oscillators. However, the bandgap of the this device is centered at the resonance frequency of the atomic mass-in-mass element. A key challenge is to achieve a broad extension of the bandgap and a bandgap starting at a frequency as low as possible To reach this result, it has been proposed to exploit the non-reciprocity feature of the chiral materials, introducing in the system hybrid modes. Here, we focus on the possible engineering of the non-linearity of the external spring of a mass-in-mass system, we evaluate three cases including an hysteretic ke value. Starting by the Lagrangian equation of the energy, we obtain the dynamical equations taking into account the relationship of ke relate to the displacement of the me. The dynamical equations are then solved numerically. Our results point out that the start frequency of the bandgap can be decreased by 25% by considering ke=A atan(Bue) with proper values of A and B.