Resonant Willis metamaterials based on fractal geometry

A class of metamaterials that exhibit mass-momentum coupling are known as Willis materials. The mass-momentum coupling arises from hidden degrees of freedom that are asymmetrical with respect to the overall material. Resonant Willis metamaterials incorporate resonators, e.g., a Helmholtz resonator, along with material asymmetry to effectively obtain Willis coupling over a narrow frequency band. By incorporating fractal geometry into the resonator shapes Willis coupling can be obtained over a broader frequency range. This is accomplished by self-similar resonator shapes, which translates into resonance frequencies that exhibit a power-law dependence. Furthermore, manipulation of the internal asymmetry allows for tuning of the Willis coupling. This study reports on the measurement and modeling of resonant Willis metamaterials based on fractal geometry. Measurements are obtained with a transmission impedance tube, and predictions based on a lumped-element model. With resonator volume shapes based on the Sierpiński triangle, Willis coupling is obtained over a frequency range of 500 to 1500 Hz. Furthermore, for a subset of the resonators, Willis coupling is tuned by inverting the asymmetric location of the resonator necks.