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.

本研究为读者带来了三种新的混合高阶剪切变形理论（HSDT）模型，并分析了功能梯度材料（FGM）板。这项工作的主要目的是通过多项式-双曲线-指数型和多项式-三角型形式开发三种独特的HSDT，提出三种新的HSDT模型，通过考虑准三维理论来研究厚度拉伸的影响，分析各向同性和FGM单层和夹层板（对称和非对称，具有硬核和软核）的自由振动，以证明模型的能力。因此，利用汉密尔顿原理来基于仅五个未知数的位移场来开发运动方程，其中三个通过板厚区分了横向位移膜（弯曲，剪切和拉伸位移）。此外，通过将Navier方法应用于简单支持的边界条件类型，可以找到解析解。该理论还认为，横向剪切变形效应满足了无板表面上的无应力边界条件，而无需任何剪力校正因子。所使用的机械性能遵循幂律，并且在整个板厚上均遵循Mori-Tanaka方案。所确定的结果说明了不同的无量纲参数的影响，并且拟议的HSDT预测了单层和三明治（对称和非对称，与文献参考文献中发现的其他不同板的理论解决方案相比，FGM板具有硬核和软核），因此可以确定本方法的可靠性和准确性。由此得出的结论是，由于与剪切变形效应有关的许多问题，多项式-双曲线-指数式和多项式-三角形式的当前公式可以进一步扩展到所有现有的HSDTs模型。