Higher-order piezoelectric plate theory derived from a three-dimensional variational principle

A three-dimensional mixed variational principle is used to derive a Kth-order two-dimensional linear theory for an anisotropic homogeneous piezoelectric (PZT) plate. The mechanical displacements, the electric potential, the inplane components of the stress tensor, and the in-plane components of the electric displacement are expressed as a finite series of order K in the thickness coordinate by taking Legendre polynomials as the basis functions. However, the transverse shear stress, the transverse normal stress, and the transverse electric displacement are expressed as a finite series of order (K + 2) in the thickness coordinate. The formulation accounts for the double forces without moments that may change the thickness of the plate. Results obtained by using the plate theory are given for the bending of a cantilever thick plate loaded on the top and the bottom surfaces by uniformly distributed 1) normal tractions and 2) tangential tractions. Results are also computed for the bending of a cantilever thick PZT beam loaded by 1) a uniformly distributed charge density on the top and the bottom surfaces and 2) equal and opposite normal tractions distributed uniformly only on a part of the beam. The seventh-order plate theory captures well the boundary-layer effects near the clamped and the free edges and adjacent to the top and the bottom surfaces of a thick orthotropic cantilever beam with the span to the thickness ratio of two. Also, through-the-thickness variation of the transverse shear and the transverse normal stresses agree well with those computed from the analytical solution of the three-dimensional elasticity equations. The governing partial differential equations are second order, so that Lagrange basis functions can be used to solve the problem by the finite element method.