A new triple-layer phononic crystal (PC) composite beam model is proposed, which binds two piezomagnetic layers to conventional PCs coupling the microstructure and flexoelectric effects. A variational approach is employed to derive the relevant equations of motion and boundary conditions (BCs), followed by using the Bloch theory and transfer matrix method (TMM) to accurately calculate the bandgap of elastic waves. Numerical results show that the microstructure and flexoelectric effects elevate the bandgap frequency at the micro-and nano-scales, respectively, but both can be neglected at the macroscale. The thickness of the piezomagnetic layers and the volume fraction of the PC play an important role in the generation of the bandgap. With the help of the piezomagnetic layers, tunable bandgaps can be obtained by adjusting the external magnetic potential field at all scales. This paper provides an innovative idea for flexoelectric-based PC devices designing when magnetic fields are involved.

提出了一种新的三层声子晶体 (PC) 复合梁模型，该模型将两个压磁层结合到耦合微观结构和挠曲电效应的传统 PC 上。采用变分方法推导相关运动方程和边界条件（BCs），然后利用布洛赫理论和传递矩阵法（TMM）精确计算弹性波的带隙。数值结果表明，微结构和挠曲电效应分别提高了微米和纳米尺度的带隙频率，但在宏观尺度上两者都可以忽略不计。压磁层的厚度和 PC 的体积分数在带隙的产生中起重要作用。在压磁层的帮助下，通过在所有尺度上调整外部磁势场可以获得可调带隙。本文为涉及磁场的基于柔性电的 PC 设备设计提供了一种创新思路。