Possible Flexural Wave Propagation Regimes in Periodic Beams

Abstract

Flexural wave propagation in periodic beams has been the object of studies for decades and it remains an active field of research, particularly because of the possibility of generating bandgaps, i.e., frequency intervals at which wave propagation is not allowed. For three models of beams (Euler-Bernoulli, Rayleigh or Timoshenko), there exist two wave modes, and each one of them can be either propagating, evanescent (exponentially decaying) or decaying with oscillations. The present work establishes which combinations in mode character can emerge when the beam is assumed as infinitely periodic and the unit cell is piece-wise constant. Regardless of number of layers in the unit cell and material and cross-section properties, a priori guarantees on the wave propagation behavior are derived, showing that Euler–Bernoulli beams always feature an evanescent mode, while any combination of modes is possible for Rayleigh and Timoshenko, unless the design parameter space is constrained. Bloch’s theorem for infinite media and spectral analysis of transfer matrices are the mathematical levers we use to derive the results, once the relation between eigenvalues of the transfer matrix and wave propagation is established. Our approach relies on leveraging the symplectic structure of the transfer matrices and the notion of “dominant eigenvalue”. Practical consequences and the possible extension of the results towards continuous beam profiles are discussed. This work offers foundational results to guide bandgap optimization for flexural waves.