Flexoelectricity refers to phenomena that polarization is coupled with strain gradient and stress is coupled with electric gradient. It occurs in all dielectrics, and becomes significant under a sharp conductive tip in piezoresponse force microscopy (PFM), under which large gradients in electromechanical field naturally arise. Here we analyze direct and converse flexoelectric effects under such a scanning probe, using mixed finite element method (FEM) developed under the phenomenological continuum framework. The FEM was first validated by analytic solutions for simple axially-symmetric tubes, and then applied to analyze two different modes of PFM experiments. It reveals that the flexoelectric effect accounts for less than 20% of the measured piezoresponse in a typical piezoelectric material, while mechanical switching via flexoelectricity is only possible for ferroelectric materials having upward polarization, with the switching zone confined to a small region near surface. These analyses explain a number of experimental observations well, and shed insight into complex electromechanical phenomenon under a sharp PFM tip. The theoretical and computational framework developed can also be applied to study flexoelectric effect in other structural configurations.

柔电是指极化与应变梯度耦合而应力与电梯度耦合的现象。它出现在所有电介质中，并在压电响应力显微镜（PFM）的尖锐导电尖端下变得非常重要，在该尖端下会自然产生机电磁场的大梯度。在这里，我们使用在现象学连续性框架下开发的混合有限元方法（FEM）在这种扫描探针下分析直接和相反的柔电效应。有限元法首先通过解析方法验证了简单的轴对称管，然后用于分析两种不同的PFM实验模式。结果表明，在典型的压电材料中，柔电效应占不到所测压电响应的20％，只能通过具有向上极化的铁电材料通过挠性电进行机械切换，而切换区域仅限于靠近表面的小区域。这些分析很好地解释了许多实验观察，并在尖锐的PFM尖端下深入了解了复杂的机电现象。所开发的理论和计算框架也可以用于研究其他结构配置中的柔电效应。