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.

Popular Summary

Understanding how solids vibrate or how waves propagate through them is essential for applications such as earthquake protection, ultrasonic sensing, and evaluation of engineering structures. This behavior is mathematically governed by a set of elastic wave equations. Interestingly, the form of these equations depends on the choice of coordinates, which implies that researchers have yet to identify the full possible range of parameters that describe wave propagation. In the 1980s, physicist John Willis proposed an additional term—known as the Willis coupling term—for the wave equations. However, it has remained only a theoretical proposal. Here, we report the first experimental realization of a material that incorporates Willis coupling for elastic waves on a structured beam.

Willis’ coupling term introduces a coupling between stress and velocity as well as between momentum and strain. To realize a material with these properties, we first extend conventional theories that describe wave propagation in plates and beams to include Willis coupling. By incorporating a cantilever-type resonating structure, we then design and fabricate a Willis metamaterial for elastic waves in solid. We experimentally show that the metamaterial leads to asymmetric wave reflection, as seen in both the amplitudes and phases of the reflected waves. Moreover, the asymmetry can be controlled by adding different amounts of loss to the material. In the extreme, we can reduce wave reflection in one direction to zero.

Our work may stimulate new designs of beam and plate structures in mechanical engineering, from aerospace structures to cantilevers used in atomic force microscopy.

Figure 1

Symmetry breaking in reflection and Willis metamaterial. (a) Symmetric metamaterial is changed into the Willis one by breaking mirror symmetry. For this Willis material, $a_0=5.0\mathrm{mm}$, $h_0=2.0\mathrm{mm}$, $r_1=0.45a_0$, $r_2=0.25a_0$, and $d=0.5\mathrm{mm}$, with orientation angles of two thin ribs 150° and 210°, respectively. The displacement field $u_z$ is plotted by loading force $F_z$ (shown as black arrows). The field of $u_z$ on the two end surfaces is also presented by blue arrows. The Willis material is assembled as a metamaterial layer in a thin beam. Forward (backward) plane wave is incident into the left (right) side of the beam. (b) Amplitudes $|r|$ and phases $\arg (r)$ of reflected waves from both sides for lossless material (density ${\rho }_0=1190\mathrm{kg}/{\mathrm{m}}^3$, Young’s modulus $E_0=2.95\mathrm{GPa}$, and Poisson ratio ${\nu }_0=0.29$) obtained from finite element method (FEM). (c) Amplitudes $|r|$ and phases $\arg (r)$ of reflected waves for lossy material [Young’s modulus turns into $E_0=2.95(1-0.05i)\mathrm{GPa}$]. (d) Effective parameters at different frequencies. (e) Theoretical values of reflection amplitudes $|r|$ and phases $\arg (r)$ for lossy material obtained from the effective medium in (d). In (b), (c), and (e), the solid (dashed) lines correspond to the case of forward (backward) incident waves.

Figure 2

Experimental testing of asymmetric reflection and extraction of Willis coupling. (a) Photograph of Willis and symmetric metamaterials. (b),(c) The experimentally tested and simulated reflected amplitude of different cases, respectively. For the symmetric material, $r_1=0.4a_0$, $d=0.4\mathrm{mm}$, orientation angles of the two thin ribs being set along the $y$ direction. Other geometric and material parameters are same as the Willis case. (d)–(f) The theoretical (solid lines) and experimental (symbols) values of the even scattering coefficient $s_{ee}$, the odd scattering coefficient $s_{oo}$, and the cross-coupling scattering coefficient $s_c$ of the Willis metamaterial.

Figure 3

Tunable asymmetry by loading lossy porous rubber. (a) Schematic of adding loss by attaching soft porous rubber patches to the metamaterial layer. (b) Measured amplitudes of the reflected waves $|r|$ with and without porous rubber patches. (d) Eigenvalues of the $S$ matrix at different frequencies for a 5-unit Willis metamaterial layer, with soft porous rubber patches attached on surfaces. Green arrows represent the trajectories of both eigenvalues along with the increase of the frequency. (d) Eigenvalues of the $S$ matrix at different frequencies for a 1-unit Willis metamaterial layer, no soft porous rubber pasted. The case of ideal lossless layer is a perfect unit circle. (e) Amplitude difference of reflected waves $\mathrm{\Delta }|r|=|r_f|-|r_b|$. (f) Phase difference of reflected waves $\mathrm{\Delta }\varphi =Arg(r_f/r_b)$ for both cases.

Figure 4

Dimensionless effective parameters of the Willis metamaterial with respect to the frequency.

Figure 5

Theoretical values of reflection amplitude $|r|$ and phase $\arg (r)$ from both sides for the lossless case obtained from effective medium.

Figure 6

Physical meaning of the scattering coefficients for the cases of (a) symmetric incident waves and (b) antisymmetric incident waves.