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Out-of-plane wave propagation in two-dimensional micro-lattices

Unlike the extensive studies on the in-plane motion of macroscale lattices, their out-of-plane motion has received drastically less attention in the mechanical engineering community. However, the practical limitations of the detection of very high-frequency in-plane vibrations in small-scale structures have restricted their investigations considerably leading the researchers toward their out-of-plane analysis. The necessity of measuring the out-of-plane displacements of the periodic lattices, especially in small-scale structures, demands a thorough perception of their out-of-plane behavior. As a result, in this paper, the size-dependent out-of-plane wave propagation of periodic micro-lattices with hexagonal, square, triangular, and kagome topologies is investigated using the finite element method and Bloch’s theorem. To include size-dependency, the modified couple stress theory is used and the governing equations of the flexural and torsional motion of the small-scale microbeams are obtained. Furthermore, for the analysis of the out-of-plane wave propagation, the unit cells are discretized into grid elements, with three degrees of freedom (out-of-plane displacement, torsion, and out-of-plane bending) at each node. Also, the directionality of the out-of-plane waves at low frequencies is investigated by analyzing the iso-frequency contours of each structure. The results reveal that the size-dependency has a major contribution in determining the propagation, attenuation, and directionality of waves in small-scale micro-lattices. However, As the dimension of the structure becomes larger, the impact of size-dependency fades away and the behavior of small-scale structures becomes closer to that of classical structures.