An alternate numerical treatment for nonlinear PDE models of piezoelectric laminates

When piezoelectric laminates undergo large deformations, exhibiting a nonlinear stress-strain behavior, and the longitudinal vibrations are not neglected, linear models of piezoelectric laminates fail to represent and predict the governing dynamics. These large deformations are pronounced in certain applications such as energy harvesting. In this paper, first, a consistent variational approach is used by considering nonlinear elasticity theory to derive equations of motion for a three-layer piezoelectric laminate where the interactions of layers are modeled by the Rao-Nakra sandwich beam theory. The resulting equations of motion form into an unbounded infinite dimensional bilinear control system with nonlinear boundary conditions. The corresponding state-space formulation is shown to be well-posed in the natural energy space. With a particular choice of nonlinear feedback controllers, based on the nonlinearity of the model, the system dynamics can be stabilized to the equilibrium. Stabilization results are presented through the filtered semi-discrete Finite Difference approximations, and these results are compared to the ones of the linearized model

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