Full mode-converting transmission between longitudinal and bending waves in plates and beams

Abstract

Two major challenges are often encountered in wave manipulation involving coupled bending and longitudinal modes. First, the coupling phenomena are generally governed by cubic equations, which are critically complicated for theoretical exploration. Second, there is a significant impedance mismatch between these two modes, which hinders the transmodal energy transferring. In this work, we successfully overcome these challenges, and propose an interference theory that can realize a nearly full mode-converting transmission between bending and longitudinal modes. Specifically, we find an alternative way in which the coupling phenomena can be described using only a portion of the wave characteristics, and solution of the cubic equation is no more required. Besides, we consider locally resonant metamaterials that exhibit extreme anisotropy in effective mass density. The proposed theory consists of four conditions, namely, the interference condition, impedance condition, polarization condition, and supplementary condition. Requirements on the effective material properties are derived from these conditions, and locally resonant metamaterials are designed accordingly. The theory, applicable not only for thin plates, but also for Euler-Bernoulli beams, offers a new way to realize free manipulation of wave energy in elastic systems.