Study on the Complex Band Structure and Auxetic Behavior of Fractal Re-Entrant Honeycomb Metamaterials

In order to break the limitation of metamaterials used in the vibration and sound reduction field, this work designed a two-dimensional metamaterial based on the re-entrant honeycomb lattice and using the fractal technique. The first, second, and third-order fractal re-entrant honeycomb metamaterials are analyzed, respectively, within the established numerical models responsible for predicting the effective Poisson's ratio, the real band structure, and the attenuation diagram. The effects of the fractal order, fractal ratio, and geometrical characteristics on these multiple functionalities are investigated simultaneously. Through adjusting the proposed fractal metamaterials, the results show that the transformation of auxetic performance, the number and location of multiple stop bands, the attenuation level inside the stop bands, and the wave decaying directionality can be flexibly tuned. This demonstrates that the compatibility of mechanical features and wave motion characteristics is successfully achieved in the present work. It provides a theoretical and technical basis for the development of multi-functional design methods of metamaterials in solving engineering problems.