The current paper investigates the free vibration characteristics of the bidirectional functionally graded piezoelectric (BD-FGP) porous plate subjected to thermal and electrical loading. The properties of the plate vary as power-law distributions in two directions, that is, longitudinal and transverse directions. The BD-FGP plate is considered to be porous of two types even type and uneven type. The governing equation of the BD-FGP plate is attained with the Von Karman nonlinear strains associated with the FSDT displacement field and Hamilton’s principle. Also, the 9-noded Lagrange quadrilateral element with seven degrees of freedom (DOFS) discretized the BD-FGP plate and solved it by the higher-order finite element (HOFE) method. The present formulation's convergence and accuracy are confirmed by comparing the attained results with the literature. The influence of the various parameters includes thickness ratios, porous exponent, volume fraction exponent, electrical loading, thermal loading, and support conditions. The investigations of the current study show that the material variations in both directions, that is, longitudinal and transverse, have significant importance in the design and development of modern smart structures and their components of engineering applications.

In the present work, the vibration response of the porous functionally graded piezoelectric plates with electro-thermal loading using finite element formulations has been studied. A sigmoid law is used for the material property deviation along the thickness direction. The first-order shear deformation theory (FSDT) with von Karman nonlinear strains and Hamilton's principle is used to obtain the governing equations. The governing equations are solved by finite element methods with 9-noded iso-parametric rectangular elements with 7 degrees of freedom (DOFs) per node. The accuracy of the results is evaluated by comparing them with the results available in the literature. The analysis shows that the uneven type porosity distribution gives a larger nondimensional frequency than the even type porosity distribution FGP plate. The non-dimensional frequency decreases with increases in the $a/h$ ratio. Among all the conditions, CCCC boundary conditions have a larger frequency than SSSS, SFSF, SFCF, and CFFF. The obtained frequency under applied electric loading depends on the magnitude and the applied voltage sign. As the temperature change $(\Delta T)$ increases with the porosity exponent $(\alpha )$ of the FGP plate, the nondimensional frequency is also increasing. The obtained results can design a porous FGP material-based smart composite structure under a thermo-electric environment.