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

Abstract

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.

Wave propagation and directionality in two-dimensional periodic lattices considering shear deformations

“…It should be noted that the X − Y plane is considered as the plane of periodicity, therefore, the periodic structure is formed by the tessellation of identical unit cells in this plane. 51…”

“…It should be noted that the X À Y plane is considered as the plane of periodicity, therefore, the periodic structure is formed by the tessellation of identical unit cells in this plane. 51 The matrix form of the governing equations of motion for the wave propagation in frame and grid elements can then be obtained by implementing their corresponding mass and stiffness matrices. These matrices are presented in Appendix 2.…”

Modulational instability and rogue waves in one-dimensional nonlinear acoustic metamaterials: case of diatomic model

“…Acoustic metamaterials (AMs) are actually promising candidates among many tracks of research to improve properties of materials and engineering [1][2][3][4][5][6][7][8][9][10]. They are artificial materials and present an unusual behavior of propagation of mechanical waves which one does not find in natural materials [11,12].…”

An analysis of axisymmetric Sezawa waves in elastic solids

“…with properties[27][28][29] Then by evaluating equation(27) with different wavenumbers, the phase velocity solutions can be obtained and plotted as in figure2. This is exactly the velocity, or dispersion relation, of Sezawa waves from a textbook[20].By choosing x kh 3.3622 = = in figure 2, within the given range there are four wave modes with corresponding velocities…”