Vibration of thermo-electrically post-buckled rectangular functionally graded piezoelectric beams

The present paper deals with the small free vibration of functionally graded piezoelectric material (FGPM) beams with rectangular cross sections in pre/post-buckling regimes. The in-plane mechanical boundary conditions are considered to be immovable and various out-of-plane boundary conditions are considered. The Beam is assumed to be under in-plane thermal and electrical excitations. Each thermo-electro-mechanical property of the beam is graded across the thickness based on a power law model. The von-Karman type geometrical non-linearity is implemented to account the large deflection behavior of the beam under in-plane loadings. A Ritz-based finite element formulation is developed to discrete the motion equations. The resulted system of non-linear equations is solved via the iterative Newton–Raphson scheme. Plots of frequency in terms of loading parameter reveals the existence of bifurcation or critical states in some cases. Vibration of the beam in both pre-buckling and post-buckling states are validated with the available data in the open literature. The effects of boundary conditions, beam geometry, composition rule of constituents, actuator voltage, and thermal environment are examined through the various parametric studies.