Contradirectional coupling and complete band gap formation in periodically folded beams and plates

Periodic structures exhibit frequency ranges, known as band gaps, in which wave propagation is strongly attenuated due to Bragg scattering. In addition to this well-known mechanism, other band gaps can arise from wavemode coupling. This study investigates the contradirectional coupling between waves with opposite group velocities as a key mechanism for creating complete band gaps in periodically folded structures. Using the Dynamic Stiffness Method (DSM) and the Wave Finite Element (WFE) formulations, the analyses demonstrate how periodic folding couples flexural, longitudinal, and torsional wavemodes. Starting from the planar configuration, reference dispersion curves are used to identify critical points where the formation of the contradirectional coupling begins. A parametric study, supported by dispersion maps, identifies and quantifies the influence of geometric parameters on the position and width of the band gaps. WFE simulations show wavemode hybridisation by analysing energy distribution and visualising wavemode displacements near coupling and uncoupling points. The effects of modelling assumptions, such as finite thickness and fold curvature, are further studied, indicating that modelling simplifications can affect the hierarchy of wavemodes coupling and the edges of the band gaps. Experimental tests on a finite steel folded plate validate the numerical predictions, confirming the presence, location, and width of the contradirectional band gap.