Characterisation of Anisotropic, Non-Homogeneous Beam Sections with Embedded Piezo-Electric Materials

Abstract

The paper discusses the problem of the characterisation of untwisted, untapered, anisotropic and non-homogeneous straight beams with embedded piezo-electric devices. The determination of the section properties, i.e., the generalised electro-elastic stiffness and compliance matrices, is performed through an in-plane discretisation in a finite element way, in which the axial dependence of the unknown is handled in analytical form. This leads to a semi-analytical formulation that allows the solution of both the "central" and the "extremity" problems, namely, the indefinite and the boundary solutions. The piezo-electric generalisation implies the modelling of electro-static potential field. The peculiar bound for piezo-electric devices, the conductive laminae that allow sensing and actuation, implies a distributed non-holonomic constraint to the related electric variables. The section finite element discretisation allows detailed stress and electric displacement recovery, thus, giving the designer a powerful analysis tool. A Finite Volumes formulation is used in order to simplify the determination of the three-dimensional beam properties. This technique leads to punctual stiffness matrix evaluation, skipping numerical integration, and is intrinsically locking free. A complete set of numerical simulations and comparisons with three-dimensional models demonstrate the effectiveness and soundness of the proposed formulation.