Curve beams are widely used in structural engineering like civil engineering and aircraft structures. Due to their coupling effects of bending, shear, and torsion, curve beams have a significant role in wave propagation fields as complete bandgaps can be generated. With the miniaturization of devices, it becomes increasingly imperative to investigate wave characteristics of curve beams at the nanoscale, taking into account the flexoelectric effect. In this study, a theoretical model for the periodic flexoelectric curve beams is established under the strain-gradient electro-elasticity theory. Subsequently, a customized state-space-based transfer matrix method for flexoelectric curve beam is specifically proposed, referred to as FCB-TMM. According to the Floquet-Bloch theorem, the dispersion relations for the periodic flexoelectric curve beams with intriguing coupling characteristics can be obtained and verified. Additionally, the influence of the flexoelectric effect and strain gradient effect on its complete bandgaps is analyzed. The results indicate that the flexoelectric effect plays a vital role in wave propagation at small scales, causing a shift in the bandgaps towards higher frequencies. Ultimately, the synergistic effects of significant geometric and material parameters combined with the flexoelectric effect on the bandgaps are systematically discussed. It is discovered that changing flexoelectric coefficients may alter the pattern of bandgap variation with these geometric and material parameters, while the influence of flexoelectric coefficients on bandgaps varies depending on the values of these parameters. Our investigation offers an insight into understanding the wave propagation of the curve nanobeams, and thus further guides the application and development of flexoelectric wave components in MEMS/NEMS.

中文翻译：

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周期性挠电曲线纳米梁中的耦合带隙和波衰减

曲线梁广泛应用于土木工程、飞机结构等结构工程中。由于弯曲、剪切和扭转的耦合效应，曲线梁在波传播场中具有重要作用，因为可以产生完整的带隙。随着设备的小型化，考虑挠曲电效应，研究纳米尺度的曲梁的波动特性变得越来越重要。本研究基于应变梯度电弹性理论建立了周期性挠电曲线梁的理论模型。随后，具体提出了一种基于定制状态空间的挠电曲梁传递矩阵方法，简称FCB-TMM。根据Floquet-Bloch定理，可以得到并验证具有有趣耦合特性的周期性挠电曲线梁的色散关系。此外，还分析了挠曲电效应和应变梯度效应对其完整带隙的影响。结果表明，挠曲电效应在小尺度的波传播中起着至关重要的作用，导致带隙向更高频率移动。最终，系统地讨论了重要几何和材料参数与挠曲电效应相结合对带隙的协同效应。研究发现，改变挠曲电系数可以改变带隙随这些几何和材料参数变化的模式，而挠曲电系数对带隙的影响根据这些参数的值而变化。我们的研究为理解曲线纳米梁的波传播提供了深入的了解，从而进一步指导挠曲电波组件在 MEMS/NEMS 中的应用和开发。