Three-dimensional exact solution for inhomogeneous and laminated piezoelectric plates

Abstract

This paper presents an exact three-dimensional method of solution for the distribution of mechanical and electric quantities in interior points of a symmetrically inhomogeneous and laminated piezoelectric plate in the framework of linear theory of piezoelectricity. A transfer matrix method and asymptotic expansions are used as the elements of the formulation. The full three-dimensional electroelasticity solution is generated from a set of two-dimensional differential equations on the midplane which, when degenerated to pure elasticity, are found to be the same as those for an equivalent classical elastic plate model. As an illustrative example, the analysis considers a rigidly clamped, elliptic monoclinic piezoelectric plate which is inhomogeneous through the thickness but symmetric with respect to the midplane and under uniform normal loading both mechanically and electrically. Excluding boundary layer effects, an exact closed-form analytical solution for the fully electromechanically coupled problem is developed as a consequence of the asymptotic expansion's termination after a few terms. Apart from some important properties already noted for a purely elastic plate in the same configuration, this exact solution reveals that certain significant electric quantities have been unjustifiably oversimplified by some of the existing piezoelectric plate models.