Longitudinal waves in two-dimensional quasi-periodic lattices

We study longitudinal wave propagation in a two-dimensional elastic lattice formed by parallel bars coupled by slender beams whose out-of-plane thickness is modulated according to an Aubry–André–Harper profile. This modulation can yield periodic or quasi-periodic architectures depending on the choice of the parameters. Starting from the continuous bar–beam model, we derive a discrete formulation and analyze the existence of bounded solutions and band-gaps as functions of the modulation amplitude and geometry. We provide analytical criteria for band-gap nucleation and explicit estimates of gap widths, and we show how quasi-periodicity can both create new gaps and shrink existing ones relative to the periodic case. The results offer new insights on the influence of quasi-periodicity in 2D elastic lattices. We show numerically how this class of structures can be exploited to achieve topological pumping of elastic waves. • Longitudinal waves in bar-and-beam lattices with periodic and quasi-periodic modulation are analyzed. • Exact solution of the dispersion relation for the periodic lattice is given. • Analytical criteria for band-gap nucleation and width are derived for the quasi-periodic case. • Quasi-periodicity can both create new band-gaps and shrink existing ones. • The quasi-periodic arrangement allows to obtain topological pumping.