size dependent nonlinear bending analysis of a flexoelectric functionally graded nanoplate under thermoelectromechanical loads

The effects of flexoelectricity on thermo-electro-mechanical behavior of a functionally graded electro-piezo-flexoelectric nanoplate 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 nanoplate and their associated boundary conditions are obtained.  the functionally graded nanoplate is modeled using a power law equation along the plate thickness direction. the nanoplate 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 nanoplate 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 nanoplate increases. also, the deflection and the generated electric potential along nanoplate thickness decreases. finally, induced polarization decreases as a linear temperature variation is applied on the nano-plate.