Variational principles and vibrations of a functionally graded plate

In piezoelectromagnetism, the three-dimensional fundamental equations well known in differential form are alternatively expressed in variational form through a unified variational principle. The variational principle is deduced from Hamilton's principle by removing its constraints, the gradient equations and the constitutive relations, by use of Legendre's transformation. Then, a hierarchical system of the two-dimensional approximate equations is systematically derived for the vibrations of a functionally graded piezoelectromagnetic plate. In the derivation, the unified variational principle and the power series expansions in the thickness coordinate for the field variables are used for the plate. The system of the plate equations in invariant, differential and fully variational forms is capable of studying the direct problems of all the types of vibrations of the plate at both low and high frequency. The uniqueness is investigated and the conditions sufficient for the uniqueness are enumerated in solutions of the system of the plate equations. Further, the variational principle and the system of the plate equations are shown to recover some of the earlier ones, as special cases.