Three-dimensional asymptotic approach to inhomogeneous and laminated piezoelectric plates

Abstract

An asymptotic theory is developed for anisotropic inhomogeneous and laminated piezoelectric plates on the basis of three-dimensional linear piezoelectricity. The inhomogeneity is assumed in the thickness direction and includes the important piezoelectric laminates as a special case. Through asymptotic expansions, the resulting two-dimensional differential equations are of the same form for each order, with different nonhomogeneous terms being determined systematically by preceding-order solutions. The governing equations of the leading-order, when degenerated to pure elasticity, are shown to be the same as those for equivalent classical thin elastic plates. The proposed methodology is illustrated by considering a rectangular piezoelectric plate subject to both mechanical and electric loadings with its edges simply supported and grounded. A three-dimensional solution for the fully electromechanically coupled problem is obtained by successively solving the two-dimensional field equations from the leading order to higher orders. Excellent agreement is observed with established results and new results are presented, from which significant physical insights are discussed.