L.A. Kalutsky1
1 Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences (Novosibirsk, Russia)
leon199703@gmail.com

The growing application of micro- and nanoscale structures in MEMS, NEMS, and aerospace engineering has intensified the need for reliable models of functionally graded (FG) porous plates. These elements often operate on elastic foundations. However, despite their high practical relevance, the combined influence of geometrical nonlinearity, size-dependent effects, and a spatially non-homogeneous elastic foundation on the stress-strain state of porous functionally graded micro-/nanoplates remains insufficiently studied. Most existing works consider foundation parameters as constant, neglecting their coordinate-dependent nature and the synergistic effects of porosity and scale.

Objective – the primary objective of this work is to develop a new mathematical model for flexible porous functionally graded Kirchhoff micro-/nanoplates resting on a non-homogeneous Winkler-Pasternak elastic foundation. The model consistently accounts for geometrical nonlinearity using the von Kármán theory, size-dependent effects via the modified couple stress theory (non-local elasticity), and material gradation and porosity (U-PFGM). A secondary goal is to conduct a comprehensive parametric analysis to investigate the influence of porosity, foundation parameters, and the scale effect on the static bending response and load-carrying capacity of the plates.

The governing system of nonlinear partial differential equations was derived from Hamilton's principle in a mixed formulation. Three distinct numerical methods were applied and critically compared to solve this system: the Variational Iteration Method (VIM), interpreted as an extended Kantorovich method; the Bubnov-Galerkin Method (BGM) in higher approximations; and a second-order accuracy Finite Difference Method (FDM). The VIM algorithm involved sequential separation of variables and reduction to systems of nonlinear ODEs, discretized by FDM and solved via the Newton-Raphson method. The reliability of all solutions was rigorously verified by comparison with established benchmark results from the literature.

The comparative analysis revealed the superior performance of the Variational Iteration Method (VIM). It demonstrated exceptional computational efficiency, requiring significantly less time (seconds versus hundreds of seconds for FDM) to achieve comparable accuracy. The maximum deviation of VIM solutions from the reference data did not exceed 1%. Using the validated VIM, a detailed parametric study was performed. It was found that the size-dependent parameter significantly increases the plate's stiffness, reducing the maximum deflection by up to 40%, with the magnitude of this effect being highly dependent on boundary conditions. Furthermore, the study investigated the effect of localizing the elastic foundation in the corner and center of the plate. It was shown that the size and position of the foundation patch not only quantitatively change the deflection value but also lead to qualitative changes in the deformed surface shape, breaking its symmetry.

The developed mathematical model and the highly efficient computational algorithm based on the Variational Iteration Method provide a powerful tool for the fast and accurate analysis of next-generation micro-/nanoelectromechanical systems (MEMS/NEMS) and advanced functionally graded structures. This capability is crucial for their optimal design, performance prediction, and reliability assessment in modern engineering applications.

Kalutsky L.A. Mathematical modeling of flexible porous functionally gradient plates on a local elasti c Winkler–Pasternak substrate. Nonlinear World. 2026. V. 24. № 1. P. 46–61. DOI: https:// doi.org/10.18127/ j20700970-202601-04 (In Russian)

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