This work presents the development of a two-way coupled flexoelectric plate theory starting from a 3D gradient electromechanical theory. The gradient electromechanical theory considers three mechanical length scale parameters and two electric length scale parameters to account for both mechanical and electrical size effects. Variational formulation is used to derive the plate governing equations and boundary conditions considering Kirchhoff’s assumptions. A computationally efficient \(C^2\) continuous non-conforming finite element is developed to solve the resulting plate equations. To assess the accuracy of the non-conforming finite element framework, the results are compared with Navier-type analytical solution for a simply supported flexoelectric plate. The finite element framework is also validated with experimental results in the existing literature for a passive micro-plate. The results show excellent agreement with both analytical and experimental results. Furthermore, computational efficiency of the non-conforming element is compared with the standard conforming element, which contains greater degrees of freedom and continuity across all elemental edges. It was observed that the non-conforming element is almost twice as fast as the conforming element without a significant loss of accuracy. The 2D finite element formulation is subsequently used to analyze the size-dependent response of flexoelectric composite plates operating in both sensor and actuator modes. Various parametric studies are performed to analyze the effect of boundary conditions, length scale parameters, size of the plate, flexoelectric layer thickness ratio, etc., on the response of flexoelectric plate-type sensors and actuators. It is found that the effective electromechanical coupling increases in a flexoelectric plate at microscale (due to the size effects), and it is higher than standard piezoelectric materials for plate thickness \(h \le 8\,{{\upmu }}\)m.

这项工作提出了从 3D 梯度机电理论出发的双向耦合挠曲电板理论的发展。梯度机电理论考虑三个机械长度尺度参数和两个电气长度尺度参数来解释机械和电气尺寸效应。考虑基尔霍夫的假设，变分公式用于推导板控制方程和边界条件。计算效率高的\(C^2\)开发连续非相容有限元来求解所得板方程。为了评估非一致性有限元框架的准确性，将结果与简支柔性板的纳维型解析解进行了比较。有限元框架还通过现有文献中被动微板的实验结果进行了验证。结果与分析和实验结果非常吻合。此外，将不合格单元的计算效率与标准合格单元进行比较，标准合格单元在所有单元边上包含更大的自由度和连续性。据观察，不合格元件的速度几乎是合格元件的两倍，而精度没有显着损失。随后使用二维有限元公式来分析在传感器和执行器模式下运行的柔性电复合板的尺寸相关响应。通过各种参数研究来分析边界条件、长度尺度参数、板尺寸、挠电层厚度比等对挠电板型传感器和执行器响应的影响。研究发现，柔性电板的有效机电耦合在微尺度上有所增加（由于尺寸效应），并且其板厚度高于标准压电材料 长度尺度参数、板尺寸、挠电层厚度比等对挠电板式传感器和执行器响应的影响。研究发现，柔性电板的有效机电耦合在微尺度上有所增加（由于尺寸效应），并且其板厚度高于标准压电材料 长度尺度参数、板尺寸、挠电层厚度比等对挠电板式传感器和执行器响应的影响。研究发现，柔性电板的有效机电耦合在微尺度上有所增加（由于尺寸效应），并且其板厚度高于标准压电材料\(h \le 8\,{{\upmu }}\) m。