An equivalent single-layer approach for free vibrations analysis of smart laminated thick composite plates

An equivalent single-layer model for the free vibration analysis of smart laminated plates is presented. The electric and magnetic fields are assumed to be quasi-static and third-order in-plane kinematics is employed to adequately take the shear influence into account when the plate thickness increases. The model governing equations are the plate equations of motion written in terms of mechanical primary variables and effective stiffness coefficients, which take the multifield coupling effects into account. The model shows that the surfaces magneto-electric boundary conditions enter the definitions of the laminate forces and moments resultants. Moreover, it reveals that new stiffness terms, which are related to the derivatives of the transverse displacement component and are exclusively associated to the piezoelectric and piezomagnetic couplings, are involved. Free vibrations solutions for simply-supported plates are presented to validate the model by comparing the present results with benchmark 3-\emph{D} solutions. Comparison of the results obtained by lower order models, namely zero and first order shear deformation theories, is presented and discussed focusing on the adequateness of the obtained models with respect to the plate thickness. Some characteristics features of smart laminates behavior have also been addressed.