Piezoelastic vibrations of composite Reissner–Mindlin-type plates

Abstract

Linear vibrations of Reissner–Mindlin-type composite plates in the presence of piezoelectric eigenstrains are studied. Piezoelectric eigenstrains are produced by applying electrical loads to piezoelectric layers embedded in or attached to substrate layers. The influence of the mechanical field upon the electric field is taken into account in the modelling, ending up with electro-mechanically coupled field equations and boundary conditions, which describe the mechanical and the electrical dynamic response of the plate. The mechanical displacements are approximated by means of the kinematic hypothesis of Hencky. The electric potential distribution is assumed to be composed of a superposition of a linear and a parabolic distribution in the thickness direction. The linear part accounts for the electric potential difference between the electrodes of the totally electroded piezoelectric layers. The parabolic part is considered in order to take into account the influence of the mechanical field upon the electric potential by means of the direct piezoelectric effect. A weak two-dimensional formulation of the three-dimensional field equations is obtained by utilizing mechanical and electrical variational principles. This formulation is characterized by resultants of stress and electric displacement. The electro-mechanically coupled behaviour comes into play by means of the constitutive relations. In case the electric potential difference is not prescribed, it can be calculated from a relation, which connects the total electric charge and the electric potential difference to each other. Because this relation is obtained from the Gauss law of electrostatics, requiring integration with respect to the area of the electrode, non-local constitutive relations for the plate are found. The non-local constitutive relations bring a new aspect into the theory of plates. An analysis for the practically interesting one-dimensional case of composite, piezoelectric plates in cylindrical motion completes the paper.