Three-dimensional analytical solution for a transversely isotropic functionally graded piezoelectric circular plate subject to a uniform electric potential difference

This paper studies the problem of a functionally graded piezoelectric circular plate subjected to a uniform electric potential difference between the upper and lower surfaces. By assuming the generalized displacements in appropriate forms, five differential equations governing the generalized displacement functions are derived from the equilibrium equations. These displacement functions are then obtained in an explicit form, which still involve four undetermined integral constants, through a step-by-step integration which properly incorporates the boundary conditions at the upper and lower surfaces. The boundary conditions at the cylindrical surface are then used to determine the integral constants. Hence, three-dimensional analytical solutions for electrically loaded functionally graded piezoelectric circular plates with free or simply-supported edge are completely determined. These solutions can account for an arbitrary material variation along the thickness, and thus can be readily degenerated into those for a homogenous plate. A numerical example is finally given to show the validity of the analysis, and the effect of material inhomogeneity on the elastic and electric fields is discussed.