Analysis of the dispersion spectrum of fluid-loaded anisotropic plates: flexural-type branches and real-valued loops

Abstract

The subsonic spectrum of complex velocity versus real frequency in immersed anisotropic plates is considered for the various options of presupposed choice of modes in the loading fluid half-spaces. Two principal implications of the latter prerequisite are investigated. The first is related to the family of flexural-type branches (with origin at v = 0 , k = 0 ), which evolve from the so-called A 0 and A 1 free-plate branches in a way that depends on the choice of fluid modes. By inspecting the low-frequency solutions of the dispersion equations corresponding to the different choices of fluid modes, a complete set of the flexural-type branches in an anisotropic fluid-loaded plate is identified. The result generalizes and also rectifies the interpretation commonly adopted for the isotropic case. The second issue is concerned with the real-valued loops on otherwise complex subsonic branches, involving the fluid mode increasing away from the plate. This phenomenon has been broadly discussed in numerical and experimental works on isotropic plates. The topological origin and shape of the real loops for an arbitrary plate can be readily viewed by way of the graphical layout of the sextic plate formalism. To this end, three possibilities are considered, namely, the sound velocity in the fluid, c f , being (i) less than the Rayleigh velocity, (ii) greater than that but less than the bulk-wave threshold in the plate, and (iii) greater than the bulk-wave threshold. These lead to three basic configurations. In case (i), a closed real loop exists provided than the fluid-to-solid density ratio is smaller than a certain critical value, estimated here in a simple explicit form. Case (ii) is typically characterized by the presence of two open real-valued arches with common high-frequency limits. Case (iii) produces an infinite sequence of progressively narrowing real-valued arches, whose upper arms rise up to c f and then transform into descending curves of solutions associated with the decreasing fluid modes (akin to the Sezawa continuum). A general overview of the locus of looping branches is supplied by analytical estimates and numerical examples.