Size Dependent Nonlinear Bending Analysis of a Flexoelectric Functionally Graded Nano-Plate Under Thermo-Electro-Mechanical Loads

The effects of flexoelectricity on thermo-electro-mechanical behavior of a functionally graded electro-piezo-flexoelectric nano-plate are investigated in this paper using flexoelectric modified and the Kirchhoff classic theories. Moreover, using the variation method and the principle of minimum potential energy for the first time, the coupled governing nonlinear differential equations of the nano-plate and their associated boundary conditions are obtained.  The functionally graded nano-plate is modeled using a power law equation along the plate thickness direction. The nano-plate behavior is analyzed under mechanical, electrical, and thermal loadings with different boundary conditions. It should be noted that the direct and reverse flexoelectric effects under different loading conditions were investigated.  Finally, the important quantities such as: the nano-plate deflection, the induced electrical voltage for different values of the length parameter, the power index related to the functionally graded behavior model and the geometric ratio parameter are determined. The results indicate that in the presence of flexoelectricity, the rigidity of the nano-plate increases. Also, the deflection and the generated electric potential along nano-plate thickness decreases. Finally, induced polarization decreases as a linear temperature variation is applied on the nano-plate.