Piezoelectric materials embedded into plates and shells make the structures being capable of sensing and actuation, usually called smart structures, which are frequently used for shape and vibration control, noise control, health monitoring, and energy harvesting. To give a precise prediction of static and dynamic behavior of smart structures, the linear/nonlinear multi-physics coupled modeling technique is of great importance. The article attempts to present the available research on modeling of piezoelectric integrated plates and shells, including (1) through thickness hypotheses for beams, plates, and shells; (2) geometrically nonlinear theories for plates and shells; (3) electroelastic material linear/nonlinear modeling; (4) multi-physics coupled modeling; and (5) modeling of advanced piezo-fiber composite bonded structures.

In this article, we focus on static finite element (FE) simulation of piezoelectric laminated composite plates and shells, considering the nonlinear constitutive behavior of piezoelectric materials under large applied electric fields. Under the assumptions of small strains and large electric fields, the second-order nonlinear constitutive equations are used in the variational principle approach, to develop a nonlinear FE model. Numerical simulations are performed to study the effect of material nonlinearity for piezoelectric bimorph and laminated composite plates as well as cylindrical shells. In comparison to the experimental investigations existing in the literature, the results predicted by the present model agree very well. The importance of the present nonlinear model is highlighted especially in large applied electric fields, and it is shown that the difference between the results simulated by linear and nonlinear constitutive FE models cannot be omitted.