Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits.: Models for inclusions that are small with respect to the thickness of the plate

This paper is devoted to the modelling of thin elastic plates with small, periodically distributed, piezoelectric inclusions, in view of active controlled structure design. The initial equations are those of linear elasticity coupled with the electrostatic equation. Different kinds of boundary conditions on the upper faces of inclusions are considered, corresponding to different ways of control: Dirichlet, Neumann, local or nonlocal mixed conditions. We compute effective models when the thickness a of the plate, the characteristic dimension epsilon of the inclusions, and epsilon/a tend together to zero. Other situations will be considered in two forthcoming papers.