A geometrically non-linear beam theory for active beams with arbitrary cross-section

Abstract

Non-linear active beam theories can be used to model many types of prismatic structures that contain piezoelectric material. In this work, we show that the 3D governing continuum equations of active prismatic structures loaded only at their ends can, without any simplifying assumptions on the stress state or geometry of the cross-section, be decomposed into a large deflection active beam theory and deformation modes that exponentially decay in magnitude from both ends of the prismatic structure. This is a manifestation of Saint-Venant’s principle for active beams, where the contribution of the decaying solutions is only relevant near the ends of the beam and can thus be safely neglected if the structure is significantly longer than the dominant decay length. A finite element discretisation of the cross-section is used to handle arbitrary prismatic geometry. By directly discretising the cross-sectional Hamiltonian, the beam constitutive coefficients of the large deflection beam theory can be found efficiently by solving a sparse system of equations. Furthermore, the dominant decay length of the exponentially decaying solutions can be estimated with limited computational effort. The approach is validated against analytical solutions, other numerical cross-section analysis approaches and the 3D finite element software COMSOL for various validation cases.