Approximate analytical solutions for piezoelectric rectangular beams by using Boley-Tolins method

Abstract

The Airy stress function satisfies the bipotential equation which is a fourth order partial differential equation in two-dimensional elasticity for isotropic materials. In 1956 Bruno Boley published an iterative procedure for the solution of the Airy stress function. Nowadays this approximation method would be classified as homotopy analysis method (HAM). Boley presented analytical formulas for the deflections and the stresses of two-dimensional, isotropic, rectangular strips and beams. In this contribution Boley's method is extended by considering a piezoelectric material (PZT-5A). After defining the stress function and the electric potential and substituting the constitutive relations into the compatibility equation and Gauss’ law of electrostatics, one obtains two coupled partial differential equations. Adapting Boley's approximation method one finds an iteratively improving solution series where the higher iterations are considered as correction terms. In particular, it is interesting to note that the first iteration is identical to the one-dimensional piezoelectric beam theory within the framework of Bernoulli-Euler if the x-component of the electric displacement is neglected in the charge equation. Analytical formulas are presented for a piezoelectric rectangular beam (unimorph) and a piezoelectric bimorph in case of electrical actuation. Numerical results for the displacement and the electric potential are presented for a cantilever with various thickness-to-length ratios. The analytical outcome of this novel approach is compared to two-dimensional finite element results, showing a very good agreement even for very thick beams if the second iteration is considered.