Strain gradient breaks inversion symmetry and induces flexoelectric polarization as well as electromechanical coupling in all material systems, though the effect is usually only significant at the nanoscale. Two-dimensional (2D) materials and thin membranes thus provide an ideal platform to explore flexoelectricity, which has been widely pursued, yet quantitative theoretical analysis is needed to guide the rapid experimental developments. In this work, we develop 2D flexoelectric model for suspended membrane based on von Kármán plate theory, and implement it into finite element computation using conforming BCIZ element. Numerical results and discussions on flexoelectric polarization in suspended membrane under uniform pressure or concentrated load are presented, which are validated by piezoresponse force microscopy (PFM) experiments under a range of membrane thicknesses and loading forces showing good agreement with computations. Since large strain gradient often exists in samples with small size in one or two dimensions, the method we develop provides a powerful tool to study a wide range of low-dimensional materials and structures with flexoelectric effect.

中文翻译：

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基于板材非线性弯曲理论分析悬浮膜的挠曲电极化


应变梯度打破了反转对称性，并在所有材料系统中诱导了挠曲电极化和机电耦合，尽管这种影响通常仅在纳米尺度上显着。因此，二维 （2D） 材料和薄膜为探索柔性电提供了一个理想的平台，这已被广泛追求，但需要定量理论分析来指导快速实验发展。在这项工作中，我们基于 von Kármán 板理论开发了悬浮膜的二维柔性电模型，并使用符合标准的 BCIZ 单元将其应用于有限元计算中。提出了在均匀压力或集中载荷下悬浮膜中挠曲电极化的数值结果和讨论，这些结果和讨论通过压电响应力显微镜 （PFM） 实验在一系列膜厚度和加载力下进行了验证，显示出与计算结果的良好一致性。由于大应变梯度通常存在于一维或二维小尺寸的样品中，因此我们开发的方法为研究具有挠曲电效应的各种低维材料和结构提供了强大的工具。