A consistently efficient and accurate higher order shear deformation theory based finite element to model extension mode piezoelectric smart beams

Abstract

The accuracy and efficiency of the conventional higher order shear deformation theory based piezoelectric beam finite elements are shown to be affected by the geometric and material parameters of piezoelectric smart beams. The accuracy is affected by the induced potential effects and the efficiency is affected by asymmetric distribution of material over the beam cross-section. In this work, a novel higher order shear deformation theory based extension mode piezoelectric beam finite element is proposed, which consistently maintains the same level of accuracy and efficiency over a wide range of applicable material properties and geometric configurations of the beam. A consistent higher order through-thickness electric potential distribution, obtained from the electrostatic equilibrium equation is used in the variational formulation to derive the governing equations. These equations are solved to establish relationships between the field variables. Starting with assumed polynomials for the transverse displacement and a layerwise electric potential, coupled polynomial expressions for axial displacement and section rotation are derived using these relations. The set of coupled shape functions obtained using these polynomials accommodates the bending–extension, bending–shear and induced potential couplings in a variationally consistent manner, at the field interpolation level itself, and is shown to improve the performance of the proposed element. Moreover, the number of mechanical degrees of freedom per node is reduced from four (for the conventional higher order shear deformation theory beam elements) to three for the present higher order shear deformation theory beam element, without affecting the applicability.