Rayleigh waves in micro-structured elastic systems

Non-reciprocity and energy symmetry breaking


Rayleigh waves have been extensively studied for potential engineering applications, such as materials characterization and design of electrical devices where transformation between mechanical and electrical energy is required. Rayleigh waves are common and appear in many scenarios. Typically, they are mechanical waves localized on the surfaces of a medium. Unlike other mechanical waves, they have relatively slow speed, higher amplitudes, and are more harmful.

The features of Rayleigh waves are extensively represented in literature. Numerous researchers before reported the propagation of these waves in different media, providing more insights into their effects and behaviors and, thereby, providing opportunities to discover more of their applications. Similarly, elastic gyroscopic systems have demonstrated several potential innovative applications and have been the focus of research in recent years. Broadening the applicability of elastic gyroscopic systems requires a thorough understanding of the gyroscopic coupling effects that are still underexplored.

The concept of gyricity, i.e. the summation of the gyroscope’s initial spin and precession rates, has been recently used to describe the coupling and dynamic behaviors of two-dimensional (2D) gyroscopic elastic systems. Motivated by the recent improvements, an international collaboration of Dr. Michael Nieves from Keele University, Dr. Giorgio Carta and Professor Michele Brun from the University of Cagliari, together with Professor Vincent Pagneux from Laboratoire d’Acoustique de l’Université du Maine (LAUM), investigated the propagation of Rayleigh waves on the free boundary of a 2D gyroscopic elastic triangular lattice. Their aim was to demonstrate the energy symmetry breaking property of the system by analyzing the gyricity effects of the spinners on the eigenmodes and dispersion curves. The work is currently published in the International Journal of Engineering Science.

In their approach, a dispersion analysis was developed for an elastic lattice coupled with inertial devices represented by gyroscopic spinners. A closed-form explicit expression was derived to describe the dispersion of Rayleigh waves in gyroscope lattices. The influence of gyricity on the dispersion curves, displacement of the lattice particles, and eigenmodes of the discrete system were discussed. Also, an analysis of the continuum approximation of a gyroscopic lattice was presented, and results were compared to those of discrete systems. In addition, the non-reciprocity for the particular continuum was demonstrated. Finally, the response of the gyroscopic system to an external force was determined.

The authors found out that in the gyroscopic elastic lattice, the shapes of the elliptical trajectories were different for surface waves propagating in opposite directions. Even though the dispersion curves were symmetric with respect to the zero wavenumber on the frequency-wavenumber diagram, numerical computations revealed that in-plane inertial coupling due to the gyroscopic effect broke down the symmetry of the eigenmodes making the system non-reciprocal. This uncommon effect was attributed to the presence of the gyroscopic system, affecting the behavior of surface and bulk waves generated by the external force on the boundary. An example of the response of a gyroscopic elastic lattice is shown in the featured figure.

In summary, the authors reported the propagation of Rayleigh waves in a system involving an elastic lattice coupled with gyroscopic spinners. The explicit solutions for both continuous and discrete systems were obtained through the analytical derivation of the Rayleigh dispersion relation. The effects of gyricity were discussed, and the results were verified by comparing them to asymptotic estimates and numerical simulations. The non-reciprocity property, dependent on the asymmetry of the internal eigenmodes rather than the dispersion curves, was demonstrated. The proposed gyroscopic system provided more insights into applications of Rayleigh waves. Notably, Dr. Michael Nieves, in a statement to Advances in Engineering, explained that the energy symmetry breaking could be researched further for potential design of special smart devices.

Rayleigh waves in micro-structured elastic systems - Advances in Engineering
Figure: The response of a gyroscopic lattice system subjected to a sinusoidal force on the boundary. The load, indicated by the black arrow, is located at the origin and acts horizontally parallel the system’s surface at at y = 0 . The color plot shown represents the total displacements of the nodes in the lattice associated with their motion in the xy-plane. Here the blue color corresponds to a large displacement whereas the dark red color can be identified with small displacements. In this particular example, the gyricity of the lattice is chosen to prevent the propagation of surface waves along the boundary to the left of the sinusoidal load. Instead, one can observe a surface wave propagating to the right of the load and only waves of the shear-type propagating within the bulk of the medium.


Nieves, M., Carta, G., Pagneux, V., & Brun, M. (2020). Rayleigh waves in micro-structured elastic systems: Non-reciprocity and energy symmetry breakingInternational Journal of Engineering Science, 156, 103365.

Go To International Journal of Engineering Science

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