Exact solutions for flexoelectric response in elastic dielectric nanobeams considering generalized constitutive gradient theories

This paper deals with the derivation of the exact solutions for the static flexoelectric response of a simply supported dielectric nano-beam subjected to distributed mechanical and electrical loads. The governing differential equations and the boundary conditions are obtained based on the Gibbs free energy for linear dielectrics considering the strain and the electrical field gradients, and their conjugates in the form of the higher order stresses and higher order polarization fields. The trends and observations from the current study are compared with the literature. The electro-mechanical coupling observed from the current model is compared for different electrical boundary conditions. The polarization and the electric field profiles across the thickness, developed due to the direct effect are also presented. Due to the use of gradient field energies, and a subsequent evaluation of their conjugates, the size effects are better exhibited by the current model than the models in the literature derived without considering strain and electric field gradients. The present study suggests that upon considering strain gradient elasticity the sensitive nature of flexoelectric nanosensors, nano energy harvesters and nanoactuators is realized. The exact solutions developed in this paper may be used as benchmark solutions for further research on flexoelectric solids.