On the basis of the couple stress piezoelectric theory including flexoelectric effects, a size-dependent model of a bilayer microbeam consisting of a piezoelectric material layer and an elastic layer has been established. The governing equations along with the boundary conditions of the microbeam model have been obtained from variation principle. The microbeam bending problems have been solved to analyse the electromechanical coupling responses to the applied concentrated force and voltage. Numerical results show that whether under the concentrated force or the voltage, the influence of flexoelectric effects on the equivalent piezoelectric response is larger than that of rotation gradient elastic effects when the size of the microbeam is much larger than the material length scale parameters. However, with the continuous decrease of the beam characteristic size scale, rotation gradient elastic effects have a stronger impact on the electromechanical responses than flexoelectric effects, which leads to a rapid decrease of the equivalent piezoelectric responses. The result also shows that the contribution of flexoelectric effects on the equivalent piezoelectric response increases when the beam size diminishes, which leads to a significant surge of the electromechanical responses compared to the model under the classical piezoelectric theory.

基于包括柔电效应的耦合应力压电理论，建立了由压电材料层和弹性层组成的双层微束的尺寸相关模型。从变分原理出发，得到了微束模型的边界条件和控制方程。解决了微束弯曲问题，以分析机电耦合对施加的集中力和电压的响应。数值结果表明，无论是在集中力还是在电压下，当微束的尺寸远大于材料长度尺度参数时，挠曲电效应对等效压电响应的影响要大于旋转梯度弹性效应。然而，随着梁特征尺寸尺度的不断减小，旋转梯度弹性效应对机电响应的影响要大于柔电效应，从而导致等效压电响应的迅速减小。结果还表明，当电子束尺寸减小时，挠性电效应对等效压电响应的贡献增加，与经典压电理论下的模型相比，这导致机电响应明显激增。