Emergence of negative-dispersion passbands below the ring frequency of a piezoelectric meta-shell

Abstract

The local-resonance (LR) mechanism is usually employed to induce bandgaps in the scope of metamaterials. On the contrary, we investigate LR passbands in this paper, which are located below the ring frequency of a piezoelectric meta-shell shunted with resonant circuits. The flexural wave properties of the meta-shell are investigated analytically through dispersion relations, and also verified by numerical results. It is observed that the piezoelectric shunting could induce longitudinal bandgaps above the ring frequency, while passbands below the ring frequency. Interestingly, negative dispersions may appear in this longitudinal passband when negative bending rigidity is induced through the piezoelectric bending effect, implying that the group velocity of flexural wave is opposite to its phase velocity. We demonstrate such rigidity-correlated dispersion behaviors through energy flow analysis based on the Poynting vector. It is also revealed that the curvature of shells impacts the width of negative-dispersion passbands, with the smaller radius leading to narrower passbands. Since the curvature couples in-plane vibrations and out-of-plane vibrations, it facilitates flexural-wave passbands below the ring frequency as well by introducing the piezoelectric tensional effect, but with dispersions maintaining positive. Moreover, we find that if the piezoelectric shunting induces the bending effect and the tensional effect simultaneously, their interplay at the overlap frequency range may create negative-dispersion passbands above the ring frequency, whereas cancels the impacts with each other below the ring frequency. The results in this research break the limitation that flexural wave propagation below ring frequencies of shells is conventionally not allowed in the longitudinal direction. It also offers the opportunity to implement unprecedent wave properties related to negative dispersions, such as negative refraction and superlensing, with single-negative curved metamaterials.