Early investigations on piezoelectric plates were based on a priori mechanical and experimental considerations. They assume plane stress and consider only transverse components of electric displacement and field. Beside, these were supposed constant in the plate thickness. Through an asymptotic analysis, this paper shows that mechanical hypotheses follow Kirchhoff-Love theory of thin plates. However, electric assumptions are found to be strongly dependent on the electric boundary conditions. That is, two regular problems should be distinguished: (1) the short circuited plate, for which only transverse electric displacement and field have to be considered-the electric potential is then found to be the sum of a known part, which depends on prescribed potentials, and an unknown part, which represents an induced potential and cannot be a priori neglected; the mechanical and electrical problems may be uncoupled; (2) the insulated plate, for which only in-plane electric displacement and field components are to be considered; the mechanical and electrical problems may be uncoupled for orthorhombic plates but not in general. Based on the above asymptotic analysis, two variational and local two-dimensional static models are presented for heterogeneous anisotropic plates. They are then applied to homogeneous and orthorhombic piezoelectric plates. For homogeneous orthorhombic piezoelectric plates, the electromechanical problem can be uncoupled. Hence, a mechanical problem is first solved for the mechanical displacement, then electric potentials are explicitly deduced from this displacement. Classical finite element codes having multilayer plate facilities can be used for solving the plate problems obtained.

In order to develop a mathematical modeling of the smart structure, the study consists of using, in first time, the finite element method and Euler-Bernoulli theory applied to a flexible beam divided into a finite number of elements. For this purpose, we define the beam represented by the imperfect FG material, partially bonded on both surfaces, four thin layers of piezoelectric materials used as sensors/actuators. The effects of temperature and porosity on the dynamics and the active control are studied and discussed. The equations of motion are obtained via Hamilton principle. The structure is modeled analytically and numerically and the optimal LQG control with Kalman filtering is applied. The results of the simulation are presented to visualize the states of the dynamics of the several cases.