An active mechanical Willis meta-layer with asymmetric polarizabilities.

Willis materials exhibit macroscopic cross-coupling between particle velocity and stress as well as momentum and strain. However, Willis coupling coefficients designed so far are intrinsically coupled, which inhibits their full implementation in structural dynamic applications. This work presents a means to eliminate these limitations by introducing an active scatterer in a mechanical meta-layer that exploits piezoelectric sensor-actuator pairs controlled by digital circuits. We experimentally demonstrate abilities of the Willis meta-layer, in beams and plates, for independently engineering transmission and reflection coefficients of flexural waves in both amplitude and phase and nonreciprocal wave propagations. The meta-layer is described by a flexural wave polarizability tensor, which captures independent higher-order symmetric-to-symmetric and symmetric-to-antisymmetric couplings. The active meta-layer is adaptive in real time for reconfigurable broadband operation thanks to its programmability. This work sheds a new light on unsurpassed control of elastic waves, ranging from vibration protections to ultrasonic sensing and evaluation of engineering structures.