Static analysis of functionally graded plate structures resting on variable elastic foundation under various boundary conditions

Functionally graded materials are widely utilized in several industrial applications, and their accurate modeling is challenging for researchers, principally for FGM nanostructures. This study develops and presents a quasi-3D analytical plate theory to explore the bending behavior of a new model of FG plate structures (FGPSs), resting on modified four parameters Winkler/Pasternak elastic foundations, under various boundary conditions. For this context, different types of functionally graded nanoplates (FGNPs), including (i) the classical FG nanoplate, (ii) the FG sandwich nanoplate, (iii) the trigonometric FG nanoplate of type A, and (4) the trigonometric FG nanoplate of type B as well as their macro-counterparts are also examined. Cosine functions describe the material gradation and material properties through the thickness of the FGNPs. The modified continuum nonlocal strain gradient theory is utilized to include the material and geometrical nanosize length scales. The kinematic relations of the plate are achieved according to hybrid hyperbolic-parabolic functions to satisfy parabolic variation of shear along the thickness of FGNP and zero shears at the inferior and superior surfaces. The equilibrium equations are obtained using the virtual work principle and solved using the Galerkin method to cover various boundary conditions. The results for the macro-counterparts of FGNPs are obtained by taking the small-scale parameters zero in the special cases. The precision and consistency of the generated analytical model are confirmed by comparing the findings to results from the scientific literature. Moreover, a comprehensive parametric study is also performed to determine the sensitivity of the bending response of FGPSs to boundary conditions, EF parameters, nonlocal length-scale, strain gradient microstructure-scale, and geometry.