Biharmonic Split Ring Resonator Metamaterial: Artificially dispersive effective density in thin periodically perforated plates

摘要

We present in this paper a theoretical and numerical analysis of bendingwaves localized on the boundary of a platonic crystal whose building blocks aresplit ring resonators (SRR). We first derive the homogenized parameters of thestructured plate using a three-scale asymptotic expansion in the linearizedbiharmonic equation. In the limit when the wavelength of the bending wave ismuch larger than the typical heterogeneity size of the platonic crystal, weshow that it behaves as an artificial plate with an anisotropic effective Youngmodulus and a dispersive effective mass density. We then analyze dispersiondiagrams associated with bending waves propagating within an infinite array ofSRR, for which eigen-solutions are sought in the form of Floquet-Bloch waves.We finally demonstrate that this structure displays the hallmarks ofAll-Angle-Negative-Refraction(AANR) and it leads to superlensing andultrarefraction effects, interpreted thanks to our homogenization model as aconsequence of negative and vanishing effective density, respectively.