An electromechanically coupled theory for piezoelastic beams taking into account the charge equation of electrostatics

SummaryThe present paper is devoted to the coupling between electrical and mechanical fields in piezoelastic structures. In the present contribution, an electromechanically coupled technical theory for flexural and extensional deformations of piezoelastic composite beams is developed. Such a technical theory should be of special interest for control applications, where a lower-order but sufficiently accurate modelling is required.In a first step, an equivalent single-layer theory of the Timoshenko-type for composite beams is utilized. The influence of shear, rotatory inertia as well as the influence of the electric field is taken into account in this technical beam theory. The electric field is unspecified so far in this formulation, but is coupled to the deformation by means of the charge equation of electrostatics. In order to incorporate this coupling, the electric potential is approximated by a power series in the thickness direction of the beam. Terms up to an order of two are considered in the approximation. The formulation then is adapted to the electric boundary conditions at the upper and lower sides of the electroded piezoelectric layers, namely that the electrodes have to be equipotential areas. Putting this distribution into an electrical variational principle, a weak one-dimensional formulation of the charge equation of electrostatics is obtained for the axial distribution of the electric potential. Prescribing the electric potential at the electrodes, and specifying the electrical boundary conditions at the vertical ends of the layer, this weak form completes the proposed electromechanically coupled technical theory for composite piezoelastic beams.In order to demonstrate the influence of the coupling between deformation and electric field, the quasi-static behavior and free flexural vibrations of a symmetrically laminated 3-layer beam are studied in detail. Results are compared to results of coupled finite element computations as well as to results obtained by a simplified theory, previously developed by the authors.

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