On negative effective mass and negative group velocity in anharmonic seismic metamaterials

In this work, an anharmonic mass-in-mass system that can be employed as a nonlinear seismic metamaterial is represented as an equivalent anharmonic mass-spring system via an effective medium approach. The dispersion relation and the behavior of the effective mass as a function of the angular frequency obtained in the regime of weak anharmonicity deviate from those of the corresponding linear system because of the effect of the fourth-order potential anharmonicity. In the presence of anharmonic soft springs it is found a range of wave vectors close to the Brillouin border zone at which the group velocity of the acoustic and optical modes is negative, namely it is opposite to the phase velocity, and a wider band gap at the border of the first Brillouin zone with respect to that of the linear case. Both effects can be tuned by varying the anharmonicity strength. The huge band gap amplitude together with the strong reduction of the frequency of the acoustic mode could be exploited for the design of nonlinear seismic metamaterials at the basis of composite foundations operating in the stop band frequencies.