Three-dimensional free vibration of multi-layered piezoelectric plates through approximate and exact analyses

This article presents two models that have the aim of analysing three-dimensional freely vibrating plates made of an arbitrary combination of structural and/or piezoelectric layers. The first model is derived from a displacement-based variational statement, and it investigates the possibility of approaching exact three-dimensional results at any degree of accuracy. This model has been developed as if the plates were virtually made of a single layer, and it is herein referred to as the approximate analysis model. The second model is based on solving the set of three-dimensional linear equations coupling the relevant mechanical and electric quantities, and it therefore provides exact results. The latter model is derived from the transfer-matrix technique which, having shown numerical instability in the multiphysics problem being dealt with, was then successfully modified to provide exact and reliable results. Excellent agreement has been obtained between the models, and this shows how the exact approach here designed is stably able to overcome ill conditioning problems, while the first model, having been validated by the exact results, could be applied to effectively investigate multiphysics problems for general boundary conditions, and for cross- and/or angle-ply laminates, at any level of required accuracy.