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In this paper, the free vibration of piezoelectric nanobeams considering static flexoelectric, dynamic flexoelectric, and surface effects is studied. Based on the theories of the Timoshenko beam and Euler-Bernoulli beam, a theoretical model of flexoelectric nanobeams is established and the governing equations and boundary conditions of this model are derived using the variational principle. Then, the analytical solution of the frequency equation is obtained by using the Navier method. Numerical results indicate that the size dependence of the dynamic flexoelectric effect is stronger than the surface effect. The surface and dynamic flexoelectric effects exhibit a competitive mechanism on the natural frequency.
Wang et al. (Wed,) studied this question.