Accurate modelling of piezoelectric plates: single-layered plate

The paper presents an efficient two-dimensional approach to piezoelectric plates in the framework of linear theory of piezoelectricity. The approximation of the through-the-thickness variations accounts for the shear effects and a refinement of the electric potential. Using a variational formalism, electromechanically coupled plate equations are obtained for the generalized stress resultants as well as for the generalized electric inductions. The latter are deduced from the conservative electric charge equation, which plays a crucial role in the present model. Emphasis is placed on the boundary conditions at the plate faces. The model is used to examine some problems for cylindrical bending of a single simply supported plate. Number of situations are examined for a piezoelectric plate subject to (i) an applied electric potential, (ii) a surface density of force, and (iii) a surface density of electric charge. The through-thickness distributions of electromechanical quantities (displacements, stresses, electric potential and displacement) are obtained, and compared with results provided by finite element simulations and by a simplified plate model without shear effects. A good agreement is observed between the results coming from the present plate model and finite element computations, which ascertains the effectiveness of the proposed approach to piezoelectric plates.