Modeling of the flexoelectric annular microplate based on strain gradient elasticity theory

A size-dependent annular microplate model is established based on the isotropic flexoelectric theory, which considering the effects of strain gradient, polarization gradient, and microscopic electrical field gradient. The displacement mode of the Kirchhoff plate theory is used to obtain the detailed expressions of the electric enthalpy, dynamic energy, and the work done by external forces. The governing equations, boundary conditions, and initial conditions are derived using the Hamilton's principle. By solving the governing equations with only mechanical and only electrical boundary conditions, the analytical solutions of deflection and polarization in the static direct and converse flexoelectric responses are obtained, respectively. Meanwhile, the free vibration problem is also solved. Finally, based on the analytical solutions, some numerical results are given to illustrate the unique flexoelectric properties of the annular microplate compared with the solid circular microplate and influence of the strain gradient elastic effect on the static direct and converse flexoelectric responses as well as the natural frequency.