Willis Metamaterial on a Structured Beam

Bianisotropy is common in electromagnetism whenever a cross-coupling between electric and magnetic responses exists. However, the analogous concept for elastic waves in solids, termed as Willis coupling, is more challenging to observe. It requires coupling between stress and velocity or momentum and strain fields, which is difficult to induce in non-negligible levels, even when using metamaterial structures. Here, we report the experimental realization of a Willis metamaterial for flexural waves. Based on a cantilever bending resonance, we demonstrate asymmetric reflection amplitudes and phases due to Willis coupling. We also show that, by introducing loss in the metamaterial, the asymmetric amplitudes can be controlled and can be used to approach an exceptional point of the non-Hermitian system, at which unidirectional zero reflection occurs. The present work extends conventional propagation theory in plates and beams to include Willis coupling and provides new avenues to tailor flexural waves using artificial structures.