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With the development of micro-nanotechnology, smart electronic devices are being updated and developed, and more and more flexoelectric sensors, actuators, and energy harvesters attached to elastic substrates have attracted a surge of interest due to unique features at the nano-scale. In this paper, the static bending behavior and vibration characteristics of a flexoelectric beam structure based on a linear elastic substrate under a magnetic field environment are investigated. Based on the electrical Gibbs free energy density, the governing equations and boundary conditions of structures are derived by using the Euler-Bernoulli beam theory and the Hamilton's variational principle. The expressions of the deflection and the induced electric potential of the beam structure are expressed analytically. The natural frequency of the beam under the open-circuit electrical conditions with surface electrodes (OCI) are obtained after further extending the solution. The results show that the flexoelectric effect, the linear elastic substrate, and the magnetic field have significant effects on the static bending and vibration behaviors of the flexoelectric beam which are beneficial for designing and developing flexoelectric devices with elastic substrates.

Bending and Vibration Analysis of Flexoelectric Beam Structure on Linear Elastic Substrates