Size-dependent electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams

In this article, by applying the Euler–Bernoulli model and using the consistent size-dependent theory, the nonlinear formulation of functionally graded piezoelectric material nanobeam is developed. In this formulation, the nonlinear geometric effect resulting from mid-plane stretching for the nanobeam behavior is taken into account, and the nonlinear governing equations of the functionally graded piezoelectric material nanobeam are derived using Hamilton’s principle and the variational method. The power-law distribution rule is assumed for the mechanical properties in beam thickness. Afterwards, in the special case, analysis of nanobeam under mechanical and electrical loading for the clamped–clamped and cantilever functionally graded piezoelectric material nanobeams are investigated, and the effects of electrical force, mechanical force, and material properties of functionally graded piezoelectric material beam on the static responses, buckling, and free vibrations are discussed and some significant results are obtained.