A finite element model for extension and shear modes of piezo-laminated beams based on von Karman's nonlinear displacement-strain relation

Piezoelectric actuators and sensors have been broadly used for design of smart structures over the last two decades.
Different theoretical assumptions have been considered in order to model these structures by the researchers. In this paper, an enhanced piezolaminated sandwich beam finite element model is presented. The facing layers follow the Euler-Bernoulli assumption while the core layers are modeled with the third-order shear deformation theory (TSDT). To refine the model, the displacement-strain relationships are developed by using von Karman's nonlinear displacement-strain relation. It will be shown that this assumption generates some additional terms on the electric fields and also introduces some electromechanical potential and non-conservative work terms for the extension piezoelectric sub-layers. A variational formulation of the problem is presented. In order to develop an electromechanically coupled finite element model of the extension/shear piezolaminated beam, the electric DoFs as well as the mechanical DoFs are considered. For computing the natural frequencies, the governing equation is linearized around a static equilibrium position. Comparing natural frequencies, the effect of nonlinear terms is studied for some examples.