Theory of multiresonant metamaterials forA0Lamb waves

Abstract

We develop an analytical wave approach to describe the physics properties of multiresonant metamaterials for Lamb waves propagating in plates. The metamaterial that we characterize consists of a 10 by 10 uniform, periodic array of long rods attached to the surface of the plate that forms the substrate in which antisymmetric ${A}_{0}$ Lamb waves are excited. We show that the ${A}_{0}$ Lamb wave propagation through the metamaterial can be accurately modeled using a simplified theory that replaces the two-dimensional array with a one-dimensional beam with a linear array of 10 rods. The wave propagation problem is solved rigorously for this one-dimensional system using the scattering matrix for a single rod. The exact eigenvalues of the system are approximated in a long wavelength expansion to determine a simple expression for the effective wave number and dispersion of the metamaterial. The modeled dispersion is compared with an experimental measurement of the dispersion inside the metamaterial with excellent agreement. The multiresonant rods, restricted to longitudinal vibration consistent with ${A}_{0}$ Lamb waves excited in the plate, produce two wide stop bands in the frequency domain from 0 to 10 kHz where the stop or passband boundaries align with the minima and maxima of the rod's impedance. We show that a negative effective density is obtained in the stop band. With the simple yet highly accurate relations given in this paper we have a tool to develop more complex metamaterials with rods and plates of different properties.