Non-reciprocal Rayleigh waves in elastic gyroscopic medium

The homogenized medium of the gyro lattice with the mechanical quantum Hall effect, here denoted as the gyroscopic elastic medium, is governed by a new set of elastodynamic equations with the chiral density. This paper presents a generic theory based on the Stroh formalism to disclose the non-reciprocal Rayleigh waves supported by the gyroscopic medium. The Stroh formulation is investigated in the general case of anisotropic elasticity, and after introducing the chiral density, it is extended to yield new forms of eigenvalue equations, which are distinct for surface waves traveling in opposite directions. Numerical examples are performed in the isotropic elasticity scheme, illustrating the non-reciprocal Rayleigh waves as well as the evolution of the particle motion trajectory and polarization at shallow depths as the gyro coupling magnitude increases. The developed theory confirms the prediction for the classical Rayleigh wave that is reciprocal when the gyro coupling term vanishes. Finally, a general effective-medium method that relies on the surface response to external excitation is proposed for the estimation of the overall properties of gyroscopic composite materials in the long-wavelength regime. Consistency in the wave response of gyroscopic composites with periodic inclusions and their effectively homogeneous materials is demonstrated for a large range of gyro coupling constants.