Metamaterial with anisotropic mass density for full mode-converting transmission of elastic waves in the ultralow frequency range

The bimodal quarter-wave impedance matching theory, with which an incident longitudinal (transverse) wave can be completely converted to a transmitted transverse (longitudinal) wave, requires that the matching element must exhibit specific anisotropy. Previously, the specific anisotropy was satisfied between components of the stiffness tensor, and the phenomenon was only realized in the ultrasonic frequency range. In this work, we find that such anisotropy can also be satisfied between components of the mass density tensor, which allows an ultralow frequency realization. Meanwhile, the stiffness should also exhibit special anisotropy. To meet such unique anisotropy, we propose to design ternary locally resonant metamaterials. The dipolar local resonance around the lowest bandgap allows us to deal with the effective stiffness and mass density separately. The requirement on stiffness is satisfied by designing the matrix, and the mass anisotropy is realized through design of the coating layer. With the designed metamaterials, the matching elements can convert wave modes, which have a wavelength much larger than the element's width. Considering that mode conversion is a fundamental phenomenon in the elastic field, our finds and design can be critically useful to extend its application in the ultralow frequency range.