Achieving any desirable dispersion curves using non-local phononic crystals

Phononic crystals and vibro-elastic metamaterials are characterized by their dispersion relations—how frequency changes with wave number/vector. While there are many existing methods to solve the forward problem of obtaining the dispersion relation from any arbitrarily given design. The inverse problem of obtaining a design for any arbitrarily given dispersion bands has only had very limited success so far. Here, we report a new design scheme for arbitrary dispersion relations by incorporating non-local interactions between unit cells. Considering discrete models of one-dimensional mass-spring chains, we investigate the effects of both local (i.e., springs between the nearest neighbors) and non-local (i.e., springs between the next nearest neighbors and other longer-range springs) interactions. First, we derive the general governing equations of non-local phononic chains. Next, we examine all design constraints for a linear, periodic, passive, statically stable, non-gyroscopic, and free-standing system. Finally, we perform analytical calculations and numerical simulations to solve the inverse problem. The results illuminate a new path toward novel wave manipulation functionalities, such as ordinary and higher-order critical points with zero-group-velocity (ZGV) modes, as well as multi-wavelength and multi-speed propagations of the same mode at the same frequency.