On a model of layered piezoelectric beams including transverse stress effect

Abstract

In this paper a Euler–Bernoulli-like model of layered piezoelectric beams is presented. It describes more accurately than the others already presented in the literature both transverse (Poisson and piezoelectrically induced) cross-sectional deformations and through-the-thickness variations of the electric field and electric displacement. A deductive approach based on a mixed variational formulation is adopted and distributions of deformation, stress, electric field and electric displacement are simultaneously prescribed. The attention is focused on the choice of the most fitting assumptions to recover complex 3D cross-sectional field distributions. In particular, transverse interactions between different layers are taken into account by enforcing specific conditions on transverse stress through the Lagrange multipliers method. The estimate of electromechanical beam constitutive coefficients is discussed and comparison with standard modelling approaches, which assume either vanishing transverse stresses or vanishing transverse strains, is emphasized. For a sandwich piezoelectric beam and for a two-layer beam, expressions of the beam constitutive coefficients are provided and the main features of the proposed model are highlighted by presenting the through-the-thickness distribution of the 3D state fields associated to beam-axis deformations and applied voltage. As a main peculiarity, the proposed beam model is able to coherently estimate the equivalent piezoelectric capacitance also when the thickness of elastic and piezoelectric layers is comparable.