Electromechanical coupling correction for piezoelectric layered beams

This paper deals with the bending of layered piezoelectric beams (multimorphs) subjected to arbitrary electrical and mechanical loading. Weinberg [1] obtaind a closed-form solution to this problem using Euler-Bernoulli beam theory and integrated equilibrium equations. In his analysis, Weinberg assumes that the electric field is constant through the thickness of the piezoelectric layers. This approximation is valid for materials with small electromechanical coupling (EMC) coefficients. In this paper, we relax this constraint and obtain a solution which accounts for the effect of strain on the electric field in the layers. We find that Weinberg's solution can be extended to arbitrary EMC with a simple correction to the moment of inertia I of the piezoelectric layers. The EMC correction amounts to replacing I with (1 + xi) I, where xi is the square of the expedient coupling coefficient. The error in beam curvature introduced by neglecting the effect of EMC is shown to be proportional to xi. This effect can be quite significant for modern piezoelectric materials which tend to have large EMC coefficients. The formulation is applied to three example cases: a cantilever unimorph, an asymmetric bimorph and a three-layer multimorph with an elastic core. The theoretical predictions for the last two examples are compared to simulations using the finite-element method (FEM) and found to be in excellent agreement.