Abstract
In this paper, the flexoelectric formulations of Timoshenko nanobeams are developed and the static deflection as well as the free vibration of nanobeams are extracted from the formulation under direct and inverse flexoelectric effects. Considering both the direct and the inverse flexoelectric effects, the governing equations and the associated boundary conditions of the nanobeams are developed. Afterwards, static and vibrational analyses are conducted on the Timoshenko and Euler-Bernoulli nanobeams subjected to cantilever and simply supported boundary conditions. Also, the exact solutions for the displacements and the electric potential are obtained in the static analysis. Finally, the effects of the flexoelectric coefficient and the size effect parameter on natural frequency, nanobeam deflection and electric field are investigated. The formulation developed in this study is validated by drawing a comparison between the present results and the results of previous studies. The results presented in this work might help to design advanced smart nanostructures in future studies.
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