Flexoelectricity is an interesting electromechanical coupling that exists in all dielectrics when the deformation is inhomogeneous. The intensity of flexoelectricity is highly related to the deformation gradient and the dimensions of dielectric structures. Although flexoelectricity plays a significant role in micro-/nano- devices' behaviors, the complexity of electromechanical coupling impedes the utilization of flexoelectricity in engineering. In this paper, we revisit the strain gradient theory in three-dimensional spaces and decompose the strain gradient into four parts. Based on these decomposed strain gradients, we reformulate the general isotropic flexoelectric theory. To better illustrate the electromechanical coupling behaviors, we specialize the flexoelectric effects from three dimensional (3D) deformations to a few 1D deformations, including torsion, tension/compression and bending as buckling of columns in mechanics of materials. These deformations are commonly encountered in engineering, and we hope that this paper can help the design of micro-/nano- devices by harnessing flexoelectricity.

柔性电是一种有趣的机电耦合，当变形不均匀时，它存在于所有电介质中。挠曲电的强度与变形梯度和介电结构的尺寸高度相关。尽管柔性电在微/纳米器件的行为中起着重要作用，但机电耦合的复杂性阻碍了柔性电在工程中的利用。在本文中，我们重新审视了三维空间中的应变梯度理论，并将应变梯度分解为四个部分。基于这些分解的应变梯度，我们重新制定了一般的各向同性挠曲电理论。为了更好地说明机电耦合行为，我们专门研究了从三维（3D）变形到几个一维变形的挠电效应，包括扭转、拉伸/压缩和弯曲作为材料力学中柱的屈曲。这些变形在工程中很常见，我们希望本文可以通过利用柔性电来帮助微/纳米器件的设计。