On finite element formulation for a 3D piezoelectric beam structures
Résumé
Smart structures are systems that are capable of sensing and reacting to their environment, through the integration of sensors and actuators. They can vary their shape without using classical mechanical actuators, and even monitor their own structural health. Piezoelectric, piezomagnetic, electrostrictive, and magnetostrictive materials are usually of interest when designing smart structures. In this work the focus is primarily on piezoelectric materials and the improvement of the classical structural models that are able to deal with mechanical and electric field loading. More precisely, the presented development is related to piezoelectric Timoshenko beam finite element following the work of [1, 2]. In the presented ongoing research fundamental piezoelectricity equations are discussed starting from a more general thermodynamical consideration. Current formulation assumes linear constitutive relations, and small rotation of the beam cross section. The piezoelectric beam formulation in this work relays to classical kinematic hypothesis that describes the motion of the beam in terms of parameters which vary only along the axial coordinate. For the mechanical part a usual constraint of plane sections (see e.g. [3]) is placed on the displacement field. Likewise, the electric kinematic assumption predetermines the electric potential distribution. Different assumptions, namely linear and parabolic through thickness distribution, are implemented and compared. Plugging the given kinematic assumptions (both mechanical and electric) into the weak form of 3D piezoelectric governing equations, we obtain a reduced variational form with classical splitting of the volume integration into 'cross-sectional' and 'along beam' parts. Reduced variational form is further exploited to define the corresponding conjugate variables in terms of stress and electrical displacement (beam) resultants. Finite element approximation is based on a 2-node shear deformable Thimoshenko element. Thus, the geometry, displacements, rotations and electric potential are mapped to unit line element, and approximated using linear shape functions. Simplicity of the developed finite element ensures robustness and is a starting point for a number of future improvements. These improvements are related to standard beam theory upgrades to large deformation and rotations and incompatible modes, as well as to the development of the multi-coupled formulation.
Domaines
Mécanique [physics.med-ph]
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