Flexoelectricity – the effect where inhomogeneous polarization causes deformation – substitutes symmetry-forbidden piezoelectricity in nano-scale centrosymmetric materials. Its mathematical treatment in terms of continuum mechanics represents a challenge: correct models are subjects of debates during the past decades. In the present article we formulate a model with reasonable approximations for converse flexoelectric effect in a plate. Except for very specific boundary conditions the theory inevitably include nonclassical higher order terms. These are treated using variational calculus and are essentially reduced to a modification of the boundary conditions for the elastic variables. Analytical solutions are obtained for a circular plate with round electrodes, where deflections are proportional to the inverse square of plate thickness. For mechanically free boundary conditions at plate’s edges deflection grows monotonically with the electrode radius. In contrast, a plate with clamped edge does not deform at all when fully electroded, an optimal electrode size is found rendering maximal deflection.

挠曲电——不均匀极化导致变形的效应——替代了纳米级中心对称材料中的对称禁止压电。它在连续介质力学方面的数学处理代表了一个挑战：正确的模型是过去几十年争论的主题。在本文中，我们制定了一个模型，该模型具有合理近似的板中逆挠曲电效应。除了非常具体的边界条件外，该理论不可避免地包括非经典的高阶项。这些是使用变分微积分处理的，并且基本上简化为对弹性变量的边界条件的修改。对于带有圆形电极的圆形板，可以获得解析解，其中偏转与板厚的平方成反比。对于板边缘处的机械自由边界条件，挠度随电极半径单调增长。相比之下，具有夹紧边缘的板在完全电极时根本不会变形，因此可以找到最佳电极尺寸，从而实现最大偏转。