Mathematical justification of the obstacle problem in the case of piezoelectric plate

Abstract

This paper aims at properly justifying the modeling of a thin piezoelectric plate in unilateral contact with a rigid plane. In order to do that we start from the three-dimensional non-linear Signorini problem which couples the elastic and the electric effects. By an asymptotic analysis we study the convergence of the displacement field and of the electric potential as the thickness of the plate goes to zero. We establish that, at the limit, the in-plane elastic components and the electric potential are coupled and solve a bilateral linear piezoelectric problem. However the transverse mechanical displacement field, is independent of the electric effect and solves a two-dimensional elastic obstacle problem. We also investigate the very popular case of cubic crystals and show that, for thin plates, the piezoelectric coupling effect disappears.