An Asymptotic Theory of Thin Piezoelectric Plates

Asymptotic integration theory in its variational form that exploits the ‘zoom’ technique, is used to establish the first two orders of an asymptotic theory of thin piezoelectric plane plates in the framework of electroelastostatics. The crystals considered belong to class 6 mm with sixth-order symmetry axis directed along the normal to the medium plane. At the first order of approximation a purely mechanical Love-Kirchhoff theory emerges while the electric potential satisfies a two-dimensional Poisson-Neumann problem with an effective dielectric constant accounting for electromechanical couplings. At the next order (which cannot be entirely solved because of a lack, at that order, of mechanical boundary conditions along the contour of the plate) electric and mechanical fields couple in a more symmetric manner with coupling coefficients characteristic of Bleustein-Gulayev surface waves. In particular, a Love-Kirchhoff scheme is also obtained at that order but with a deflection which includes and electric contribution. An appropriate functional framework is given for the validity of the asymptotic expansion.