A numerically accurate and efficient coupled polynomial field interpolation for Euler–Bernoulli piezoelectric beam finite element with induced potential effect

Abstract

An accurate and efficient coupled polynomial-based interpolation scheme is proposed for the Euler–Bernoulli piezoelectric beam finite element which accommodates induced potential effects and is free from material-locking due to asymmetric distribution of material in the beam cross-section. The consistent through-thickness potential derived from electrostatic equilibrium equation is used, unlike conventional formulations which use assumed linear through-thickness potential. The relationship between mechanical and electrical field variables involved in the formulation is established using governing equations derived from the variational formulation. This relationship is used to derive a coupled polynomial for the axial displacement field with contributions from an assumed cubic polynomial for transverse displacement and linear polynomials for layerwise electric potential. A set of coupled shape functions obtained using these polynomials handles the effects of extension–bending coupling and induced potential in an efficient manner at the field interpolation level itself. The accuracy of the present formulation is proved by comparison of results obtained for test problems with those from ANSYS 2D simulation and conventional formulations. Convergence studies prove the merit of the present coupled polynomial interpolation over the conventional independent polynomial interpolation. This improved performance is achieved with the same number of nodal degrees of freedom as used by conventional formulations.