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.

与对宏观晶格平面内运动的广泛研究不同，它们的平面外运动在机械工程界受到的关注少得多。然而，在小尺度结构中检测极高频面内振动的实际局限性极大地限制了他们的研究，从而导致研究人员进行面外分析。测量周期性晶格的平面外位移的必要性，尤其是在小尺度结构中，需要对其平面外行为有一个彻底的了解。因此，在本文中，使用有限元方法和 Bloch 定理研究了具有六边形、方形、三角形和 Kagome 拓扑结构的周期性微晶格的尺寸相关的面外波传播。为了包括大小依赖性，利用修正的偶应力理论，得到了小尺度微梁弯曲和扭转运动的控制方程。此外，为了分析面外波传播，将单位单元离散为网格单元，每个节点具有三个自由度（面外位移、扭转和面外弯曲） . 此外，还通过分析每个结构的等频轮廓来研究低频面外波的方向性。结果表明，尺寸依赖性在确定小尺度微晶格中波的传播、衰减和方向性方面具有重大贡献。然而，随着结构尺寸变大，