The multi-mode and dispersion natures of guided waves in plate bring big challenges of functional phononic crystal (PC) plate design for wave control. For designing a periodic multilayer piezoelectric plate to control specific guided wave propagation, it is vital to study the dispersion relation of each guided wave mode. In this paper, a plane wave expansion method is developed to calculate the dispersion relation of each guided wave mode in the periodic multilayer piezoelectric plate with a mirror plane. To quantitatively demonstrate the effectivity of the developed method, the band structures of PC plates with different kinds of piezoelectric scatterers are discussed to reveal the influences of geometry and polarization effect of piezoelectric scatterer on the guided wave propagation. Results show that the mode bandgaps of symmetric mode Lamb waves, anti-symmetric mode Lamb waves, and shear horizontal waves can be obtained separately by the proposed method, and they exhibit different responses with the changes of piezoelectric scatterer. Therefore, the proposed method can provide an effective analysis way of band structures for designing smart guided wave control devices based on multilayered piezoelectric PC plate.

板中导波的多模和色散性质给功能性声子晶体 (PC) 板设计带来了巨大的挑战，用于波控制。为了设计周期性多层压电板来控制特定导波的传播，研究每种导波模式的色散关系至关重要。本文提出了一种平面波展开法来计算带镜面的周期性多层压电板中各导波模式的色散关系。为了定量证明所开发方法的有效性，讨论了具有不同类型压电散射体的 PC 板的能带结构，以揭示压电散射体的几何形状和极化效应对导波传播的影响。结果表明，该方法可以分别获得对称模式兰姆波、反对称模式兰姆波和水平剪切波的模式带隙，并且它们随着压电散射体的变化表现出不同的响应。因此，所提出的方法可以为设计基于多层压电PC板的智能导波控制装置提供一种有效的能带结构分析方法。