The significance of non-local constitutive relations for composite thin plates including piezoelastic layers with prescribed electric charge

The present paper presents a novel approach for the modeling of piezoelastic composite thin plates. The formulation is characterized by the consideration of mechanical and electrical field equations. These equations are coupled to each other by means of the direct piezoelectric effect and the converse piezoelectric effect. Owing to the thinness of the plate, the mechanical assumptions are based on classical lamination theory and the electrical field equations reduce to the one-dimensional charge equation of electrostatics. The direct piezoelectric effect that characterizes the conversion of mechanical energy into electrical energy naturally leads to the necessity of different modeling strategies for different electric conditions. Electroded layers with either the electric potential or the total charge being prescribed, or non-electroded layers, are considered in this paper. A plate model is obtained that takes into account the electro-mechanical coupling by means of effective stiffness parameters. In the case of electroded layers with the total charge prescribed, constitutive relations are found to be non-local; this brings in a new aspect into thin-plate theory. This model, although simple in its mathematical form, can be advantageously utilized in the study of the electro-mechanically coupled behavior of thin piezoelastic plates. A detailed analysis of simply supported, symmetrically laminated, transversely isotropic plates with arbitrary polygonal edges is presented. Classical methods for thin plates under the influence of eigenstrains are extended in order to account for the effect of electro-mechanical coupling. These extensions are mainly based on an analogy to thermal stress problems. Results are presented for a square plate in the case of quasi-static loading, free vibrations and harmonically excited vibrations.