An asymptotic theory for vibrations of inhomogeneous/laminated piezoelectric plates

An asymptotic theory for the vibration analysis of inhomogeneous monoclinic piezoelectric plates is developed by using small parameter expansion. The theory includes the important special case of a laminated plate in which each layer is homogeneous and orthotropic, but distinct from the other layers by having a different material or a different orientation. A hierarchy of two-dimensional equations of the same homogeneous operator for each order is reduced from the three-dimensional framework of linear piezoelectricity. The elasticity version of the leading-order equation is the same as that of the classical Kirchhoff inhomogeneous plate theory and, therefore, is easily solvable. By contrast, it is not straightforward to find solutions of the higher-order equations. The solvability condition is thus established for this purpose, by which higher-order frequency parameters are derived. The present theoretical formulation is examined by comparing the present asymptotic results with an exact three-dimensional solution for a piezoelectric bimorph strip, and excellent agreement is reached. Some new results are presented.