Willis metamaterial on a structured beam

Bianisotropy is common in electromagnetics 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.

3 Metamaterials, constructed with artificially designed microstructures, have been employed and developed in electromagnetism 1,2 , acoustics [3][4][5][6][7] , thermodynamics 8,9 and mechanics 10 -12 to give unique properties beyond those provided by natural and composite materials. Such a concept has recently been extended to elastic waves in solids 13 -17 . Compared to acoustic waves, the additional degrees of freedom in polarizations states require more sophisticated dispersion engineering [18][19][20] and also lead to non-trivial mode-matching at interfaces. Metamaterials are found particularly useful as a tool to explore the physics behind this complexity. For example, for in-plane waves, a trans-modal Fabry-Pérot condition is found necessary for maximum mode conversion between longitudinal and shear modes 21 . For flexural waves 22,23 , evanescent modes can be used to modify surface impedance for obtaining higher transmission than structures in acoustics 24 .
Peculiarly, classical elastic wave equations, i.e. the Hooke's law together with the Newton's second law, are not form-invariant under a coordinate transformation 25 , suggesting that we are currently exploring an unnecessarily limited palette of material properties. For example, an introduction of rotation modulus in Hooke's law is found necessary to describe biological composites such as wet bones 26 . This is called a Cosserat solid and it has been recently constructed using metamaterial approach 27 .
Another attempt is the proposal to realize Willis media 28 . These media introduce new constitutive terms not only in Hooke's law but also in Newton's second law [29][30][31][32][33] . These terms, which introduce coupling between stress and velocity and between momentum and strain, are typically small perturbations and difficult to realize. Currently, a similar 4 modification to both equations was experimentally introduced in airbone acoustics [34][35][36] , and a strategy to induce strong Willis coupling in suitably designed acoustic metamaterials was theoretically introduced 37 . The acoustic wave equations, when Willis coupling is considered, can be written in analogy to the electromagnetic scenario, and the additional constitutive terms correspond to bianisotropy in electromagnetism [38][39][40][41] .
These developments indicate that Willis media for elastic waves in solid may become practical through the notion of metamaterials, although the exact microstructural design has yet to be determined.
In this work, we consider the situation when a Willis medium is used to construct a plate or a beam for flexural wave propagation. Such a reduced version from three to two dimensions facilitates analysis and an intuitive understanding of Willis coupling.
Plates and beams are actually common in a wide range of length scales from building structures to micromechanical systems. Their theories can be traced back to the 1950s to 1980s in a series of works from Kirchhoff-Love plate theory to extensions with rotary and shear deformations, and from isotropic to anisotropic plates 42,43

Willis plate theory
We start from a Willis medium, with constitutive relation in its most general form 31 : where and are the stiffness and density tensors, is the radial frequency and , , are indices iterating the spatial coordinates. The additional term in Hooke's law, Willis coupling coefficient, couples stress to displacement and the same term (due to reciprocity) in Newton's second law couples momentum to strain .
Suppose that we now construct a plate ( = −ℎ/2 to ℎ/2 ) from this medium. By integrating Eq. (1) along (zeroth moment), we have We assume wave propagation along the x-direction ( → 0 for simplicity coupling. We also note that there can be possibly a higher-order Willis coupling term in the first moment (which will appear inside the two brackets in Eq. (3)), but its effect is negligible for subwavelength inclusions, see Supplementary Notes 1 and 2 for more details).

Metamaterial design with Willis coupling
As the Willis coupling term comes from of the bulk, a necessary condition to have non-zero is broken mirror symmetry in the x-direction. Figure 1a shows the unit cell of the metamaterial plate designed by perforating slots in a background acrylic plate. The inner disk is connected to the matrix by two thin ribs, either along the y-direction (as a reference case without Willis coupling in the next section), or both connected to the back interface (as in the case with Willis coupling).
The various dimensions of the structure are listed in the caption. In the latter case, there

Tunable asymmetry by varying the loss coefficient
Next, we show that the degree of asymmetry and also the frequencies (the FP dips) where the asymmetry occurs can be easily tuned by loading additional materials with loss to the Willis metamaterial. Here, soft porous silicon rubber synthesized by thermal polymerization 50 is pasted onto the upper and lower surfaces of the metamaterial layer (with 5 unit cells), as schematically shown in Fig. 3a. Figure 3b  Calculation of scattering coefficients. Once the frequency-domain wave field is tested, we first suppose that the wave pattern can be described as = • + • and = • + • on the incident side and transmitted side, respectively.
Here, the wave is supposed to propagate along the x direction, and the origin is set at the center of the metamaterial layer. Then the complex coefficients , , and can be calculated by data fitting of the wave pattern at each frequency, for both the forward and backward case. Thus the scattering matrix is determined by where subscripts " " and " " correspond to forward and backward incident waves, respectively.
Simulation. All full-wave simulations performed in the paper are obtained using