Size-Dependent Buckling of Coated Functionally Graded Nanocylinders Surrounded by a Complex Elastic Foundation

This work introduces the first analytical framework for size-dependent buckling of bidirectional functionally graded (2D-FG) cylindrical nanoshells with Hardcore (HC) and Softcore (SC) configurations. The nanoshells are engineered with a distinctive bidirectional material gradation, in which ceramic and metallic constituents vary simultaneously along both axial and radial directions, enabling tailored stiffness profiles. To accurately capture scale-dependent effects, microstructure size influences, and transverse shear deformation, the formulation fully integrates first-order shear deformation theory (FSDT) with nonlocal strain gradient theory (NSGT). The governing equations are solved using the Galerkin method for a wide range of boundary conditions, while complex elastic foundations (Winkler, Pasternak, Kerr) are incorporated to reflect realistic nanoscale support environments. The governing equations are rigorously established through the principle of virtual work, and an efficient solution scheme — capable of handling a wide range of boundary conditions — is developed using the Galerkin method. A key finding is that HC nanocylinders consistently exhibit significantly higher buckling resistance than their SC counterparts, attributed to their ceramic-rich core. Furthermore, the analysis reveals the competing nature of size effects: the critical buckling load decreases with an increasing nonlocal parameter (softening effect) but increases with the strain gradient parameter (stiffening effect). The results also demonstrate that the elastic foundation substantially enhances stability, with the Pasternak model providing the most pronounced reinforcing effect, followed by the Kerr and Winkler models. This study provides critical insights for the design and optimization of advanced nanoscale cylindrical structures in aerospace and nano-engineering applications.