Three new hybrid quasi-3D and 2D higher-order shear deformation theories for free vibration analysis of functionally graded material monolayer and sandwich plates with stretching effect

The present investigation brings to the readers three new hybrid higher-order shear deformation theory (HSDT) models and analyses the functionally graded material (FGM) plates. The major objective of this work is to develop three HSDTs in a unique formulation by polynomial–hyperbolic–exponential and polynomial–trigonometric forms, propose the three new HSDT models, investigate the effect of thickness stretching by considering a quasi-three-dimensional theory and analyse the free vibration of isotropic and FGM monolayer and sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) plates to demonstrate the models ability. Therefore, the Hamilton’s principle is exploited to develop equations of motion based on a displacement field of only five unknowns, of which three of them distinguished the transverse displacement membranes through the plate thickness (bending, shear and stretching displacements). In addition, the analytical solutions are found by applying the Navier approach for a simply supported boundary conditions type. The theory also considered that transverse shear deformation effect satisfied the stress-free boundary conditions on the plate-free surfaces without any requirement of shear correction factors. The used mechanical properties followed the power law and the Mori–Tanaka scheme distributions through the plate thickness. The determined results explained the effects of different non-dimensional parameters, and the proposed HSDTs predict the proper responses for monolayer and sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) FGM plates in comparison with other different plates’ theories solutions found in the literature references, thus the reliability and accuracy of the present approach are ascertained. It is obtained that the present formulations of polynomial–hyperbolic–exponential and polynomial–trigonometric forms can be further extended to all existing HSDTs models, for numerous problems related to the shear deformable effect.