A Comprehensive Study on Mechanical Responses of Non-uniform Thickness Piezoelectric Nanoplates Taking into Account the Flexoelectric Effect

For the first time, this paper carried out a comprehensive study on static bending, free vibration, and dynamic responses of non-uniform-thickness piezoelectric nanoplates resting on variable elastic foundations with the flexoelectric effect, where the thickness of the structure varies in both length and width directions. The whole structure is supported by a two-parameter elastic foundation, the stiffness of which varies in the direction of the plate's length. The overall equation governing the static bending, free vibration, and forced vibration of the plate is established using four-unknown high-order shear deformation theory and the Lagrange principle. A collection of Matlab-coded computer algorithms is developed for determining the nanoplate's response to any boundary condition. The approach and program's correctness are proved numerically against the dependable assertions in special instances acquired from the article's model. The effect of geometrical and material characteristics on the mechanical responses of nanoplates is investigated in detail. The results reveal that the static bending, specific vibration, and forced vibration responses of the nanoplate change significantly when compared to a conventional plate due to the influence of variable thickness, altering elastic matrix stiffness, and especially the flexoelectric effect.