A computational framework for the analysis of linear piezoelectric beams using hp-FEM. (May 2015) - Poya, Roman; Gil, Antonio J.; Ledger, Paul D.
Highlights: A new computational framework for 3D linear piezoelectric beams using hp-FEM. Framework suitable for statics, dynamics, actuation and energy harvesting problems. Framework suitable for any anisotropy or electric polarisation orientation. New set of beam equations with new mechanical and electrical cross-sectional area resultants. In order to aid prospective researchers, a new closed-form solution is presented. Abstract: In this paper, a new computational framework is introduced for the analysis of three dimensional linear piezoelectric beams using hp -finite elements. Unlike existing publications, the framework is very general and suitable for static, modal and dynamic scenarios; it is not restricted to either actuation or energy harvesting applications and, moreover, it can cope with any anisotropy or electric polarisation orientation. Derived from first principles, namely the fundamental equations of continuum piezoelectricity, a new set of beam balance equations is presented based on a Taylor series expansion for the displacement and electric potential across the cross section of the beam. The coupled nature of the piezoelectric phenomenon at a beam level arises via a series of mechanical (and electrical counterparts) stress and strain cross sectional area resultants. To benchmark the numerical algorithm, and in order to aid prospective researchers, a new closed-form solution is presented for the case of cantilever type systems subjected to end tip mechanical/electrical loads. To the best of the authors' knowledge, the analytical solution for this prototypical example has not been previously presented. Finally, some numerical aspects of the hp -discretisation are investigated including the exponential convergence of the hp -refinements and the consideration of linear or quadratic electric potential expansions across the cross section of the beam.