Non-reciprocal elastic wave propagation in spatiotemporal periodic structures

We study longitudinal and transverse wave propagation in beams with elastic properties that are periodically varying in space and time. Spatiotemporal modulation of the elastic properties breaks mechanical reciprocity and induces one-way propagation. We follow an analytic approach to characterize the non-reciprocal behavior of the structures by analyzing the symmetry breaking of the dispersion spectrum, which results in the formation of directional band gaps and produces shifts of the first Brillouin zone limits. This approach allows us to relate position and width of the directional band gaps to the modulation parameters. Moreover, we identify the critical values of the modulation speed to maximize the non-reciprocal effect. We numerically verify the theoretical predictions by using a finite element model of the modulated beams to compute the transient response of the structure. We compute the two-dimensional Fourier transform of the collected displacement fields to calculate numerical band diagrams, showing excellent agreement between theoretical and numerical dispersion diagrams.