In this paper, the complete explicit solution for physical fields around a micro-hole is solved with the simultaneous consideration of the strain gradient elasticity, the direct flexoelectricity, and the converse flexoelectricity. First, the higher-order Navier-like governing equations are proposed for an isotropic flexoelectric solid by using an extended linear theory of flexoelectric materials considering the coupling between the strain gradient and the polarization, and conversely between the polarization gradient and the strain. Second, the displacement, the electric potential, and the polarization are successfully obtained by solving the corresponding boundary value problems for a micro-hole model. Finally, the influence of the strain gradients and the flexoelectric effect on the mechanical fields and the electric responses are studied around the micro-hole in flexoelectric solids, and the size effects of the direct flexoelectricity and the converse flexoelectricity are also investigated. The results indicate that an electric response can be obviously induced by a mechanical loading due to the direct flexoelectricity. Conversely, a mechanical strain can be produced by an electric field through the converse flexoelectricity. The size dependence of both the direct flexoelectricity and the converse flexoelectricity is also successfully predicted in this paper, in the form that the flexoelectricity significantly increases with the decrease in the sample size.

中文翻译：

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微孔周围与尺寸相关的直接和反向挠电性

在本文中，在同时考虑应变梯度弹性、直接挠曲电和逆挠曲电的情况下，求解了微孔周围物理场的完全显式解。首先，考虑到应变梯度和极化之间的耦合，以及极化梯度和应变之间的耦合，通过使用柔性电材料的扩展线性理论，提出了各向同性柔性电固体的高阶类纳维控制方程。其次，通过求解微孔模型的相应边界值问题，成功获得位移、电位和极化。最后，研究了应变梯度和挠曲电效应对挠曲电固体微孔周围机械场和电响应的影响，并研究了直接挠曲电和逆挠电的尺寸效应。结果表明，由于直接挠曲电，机械负载可以明显诱导电响应。相反，机械应变可以由电场通过逆挠曲电产生。本文还成功地预测了直接挠曲电和逆挠曲电的尺寸依赖性，即挠曲电随着样品尺寸的减小而显着增加。并且还研究了直接挠曲电和逆挠曲电的尺寸效应。结果表明，由于直接挠曲电，机械负载可以明显诱导电响应。相反，机械应变可以由电场通过逆挠曲电产生。本文还成功地预测了直接挠曲电和逆挠曲电的尺寸依赖性，即挠曲电随着样品尺寸的减小而显着增加。并且还研究了直接挠曲电和逆挠曲电的尺寸效应。结果表明，由于直接挠曲电，机械负载可以明显诱导电响应。相反，机械应变可以由电场通过逆挠曲电产生。本文还成功地预测了直接挠曲电和逆挠曲电的尺寸依赖性，即挠曲电随着样品尺寸的减小而显着增加。机械应变可以由电场通过反向弯曲电产生。本文还成功地预测了直接挠曲电和逆挠曲电的尺寸依赖性，即挠曲电随着样品尺寸的减小而显着增加。机械应变可以由电场通过反向弯曲电产生。本文还成功地预测了直接挠曲电和逆挠曲电的尺寸依赖性，即挠曲电随着样品尺寸的减小而显着增加。