Asymptotic modeling of thin piezoelectric plates

We study the asymptotic behavior of a three dimensional flat, heterogeneous and anisotropic piezoelectric body when its thickness seen as a parameter goes to zero. Depending on the type of the electrical loading two models are obtained which are related to the plate used as a sensor or as an actuator. These models are explicitly derived in any piezoelectric crystal symmetry class. For some of them, a striking structural switch-off of the piezoelectric effect occurs. The static case is solved through a unifying approach using techniques of singular perturbation while the transient situation is formulated in terms of evolution equations in Hilbert spaces of possible states with finite electromechanical energy, so that the study of these transient problems are easily deduced from the static case through the Trotter theory of convergence of semi-groups of operators acting on variable spaces.