Lamb waves in discrete homogeneous and heterogeneous systems: Dispersion properties, asymptotics and non-symmetric wave propagation

In this paper, we study Lamb waves propagating in a discrete strip, whose microstructure is represented by either a monatomic or a diatomic triangular lattice. In considering the in-plane vector problem, we derive an analytical solution for the dispersion relation of Lamb waves. Additionally, we investigate the main features of the eigenmodes of the system, which describe how the lattice strip vibrates at different frequencies. Further, we discuss how the dispersion properties depend on the number of the lattice's rows and on the chosen boundary conditions. For heterogeneous systems, we focus the attention on the internal stop-band and on the flat bands appearing in the dispersion diagram. Different asymptotic models are employed to approximate the low-frequency behaviour of the lattice strip, starting from the classical Euler–Bernoulli beam. The effective behaviour of a lattice strip with dense microstructure is also investigated, and we present a comparative numerical analysis with the analogous continuum for which the classical Lamb wave problem is posed. The theory developed is exploited here to design a structured medium capable of manipulating wavemodes, and, through conversion and selection, generating uni-directional wave phenomena. We envisage that the present work can fill a gap in the research field related to the analytical study of dispersive waves in microstructured media, whose dynamic performance is influenced by the presence of multiple external boundaries.