On the correction of the Bernoulli-Euler beam theory for smart piezoelectric beams

The present paper is devoted to the development of a simple Bernoulli-Euler type beam theory for smart piezoelectric composite beams. It is the scope of this paper to formulate a formally purely mechanical beam theory taking into account the coupling to the electric field by means of the direct piezoelectric effect. Different electric boundary conditions, electroded layers with either the electric potential or the total charge being prescribed, or non-electroded layers, are considered. Approximations for the electric field depending on the actually realized electric boundary conditions are then utilized in the present formulation. Finally a beam model is obtained taking into account the electromechanical coupling by means of effective stiffness parameters. Results obtained by using this simple model are compared to two-dimensional electromechanically coupled finite element calculations. A very good agreement for both the mechanical and the electrical response is found, showing that the accuracy of the present one-dimensional theory is comparable to that of the two-dimensional finite element calculations.