Anisotropic guided waves in isotropic metaplate with orthogonal surface perturbation

Abstract

Analytical solution of elastic wave propagation in anisotropic media is of great interest in various branches of engineering and applied sciences. In this study, a derivation of a generalized elastic wave dispersion equation and its solution is presented for a doubly corrugated metaplate made of isotropic material. The newly formulated dispersion equation resulted the well-known Rayleigh-Lamb (RL) dispersion equations when the surface perturbations were made to zero, resembling a flat plate. However, when the surface perturbation of the isotropic metaplate was non-zero, the solution resulted an anisotropic wave propagation behavior. Helmholtz decomposition of the displacement function with a scalar and three vector potential is the conventional method of finding dispersion equations for Rayleigh-Lamb (RL) wave and Shear Horizontal (SH) waves in isotropic material. However, the presence of orthogonal surface perturbations in isotropic material hinders the use of Helmholtz decomposition. Although the material is isotropic, in this study, the generalized analytical dispersion equations in the metaplate were developed assuming three potential functions first introduced by Buchwald (Buchwald, 1961) for anisotropic materials. Next a logical root-finding algorithm was employed to solve the generalized dispersion equation. A numerical analysis of wave propagation in a flat plate and a corrugated plate was conducted to prove the physics obtained from the analytical solutions. Spatiotemporal displacement data and transformed frequency-wavenumber data are analyzed and verified to show anisotropic wave propagation behavior in orthogonal perturbated waveguide with isotropic material. Finally, comparison of analytical and numerical wave field in passband and bandgap frequencies in a corrugated waveguide are verified.