Based on nonlocal continuum mechanics, the authors develop an asymptotic theory by which to examine the buckling behavior of simply supported, single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium under combined hydrostatic pressure and axial compression using the perturbation method. In the formulation, the Eringen nonlocal constitutive relations are used to account for the small length scale effect; the interactions between the SWCNT and its surrounding medium are modelled as a Winkler foundation model, and the half-thickness- to- the mid-surface radius ratio is selected as the perturbation parameter. After performing some mathematical processes, including non-dimensionalization, asymptotic expansion, and successive integration, among others, the authors separate the three-dimensional (3D) nonlocal elasticity equations into recursive sets of governing equations (GEs) based on the Donnell-type nonlocal classical shell theory (CST) for various order problems. The Donnell-type nonlocal CST is derived as a first-order approximation of the 3D nonlocal elasticity problem, where the GEs for higher-order problems retain the same differential operators as those of the Donnell-type nonlocal CST although with different nonhomogeneous terms. The current asymptotic solutions for the critical load parameters of the embedded SWCNT are obtained in order to assess the accuracy of various nonlocal CSTs and nonlocal beam theories available in the literature.
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