Dynamic direction-dependent mode coupling in elastic metamaterial plates

Elastic plates support a spectrum of guided-wave modes known as Rayleigh-Lamb waves. Euler-Bernoulli and Timoshenko beam theory are well-known approximate models that are often employed to describe wave motion of the lowest order symmetric and antisymmetric modes. However, these theories make simplifying assumptions that can yield non-physical interpretation of experimental observations in structured plates that are the subject of elastic metamaterials Lee and Kim, Smart Mater. Struct., 32 123001 (2023). This work investigates direction-dependent coupling between modes in elastic plates containing resonant asymmetric scatterers and discusses the limitations of approximate theories. We first discuss restrictions imposed by reciprocity for direction-dependent scattering in passive multi-mode elastic wave systems generalized to Lamb modes of arbitrary order. We then present a finite element model case study of an elastic beam containing a resonant asymmetric scatterer yielding direction-dependent coupling of symmetric and antisymmetric modes. Constitutive relationships of the Willis form are then proposed and implemented in Euler-Bernoulli and Timoshenko beam theories in order to consider coupling between strain and momentum fields using analytical models. Discrepancies between these two theories are discussed based on constraints imposed by reciprocity, and we show that Timoshenko theory is required to capture direction-dependent mode coupling.