A three-dimensional (3D) asymptotic local elasticity theory is reformulated for the static analysis of a simply supported, single-walled carbon nanotube (SWCNT) embedded in an elastic medium under a transverse normal load at its inner and outer surfaces. Eringen's nonlocal constitutive relations are used to account for the small length scale effects in the formulation examining the static behavior of the SWCNT. The interaction between the SWCNT and its surrounding foundation is modeled as either a Winkler-type or a Pasternak-type model. The two-dimensional (2D) nonlocal classical shell theory (CST) is derived as a first-order approximation of the 3D nonlocal elasticity theory. The 2D nonlocal CST solutions can be modified order-by-order to asymptotically approach the exact 3D nonlocal elasticity solutions in a hierarchic and consistent manner. A parametric study related to some effects on the static behavior of the embedded SWCNT is undertaken, including the aspect ratio, the stiffness and shear modulus of the surrounding medium of the SWCNT, and the nonlocal parameter. The current 3D asymptotic solutions can provide a standard to assess the accuracy and convergence rate of various 2D nonlocal shell theories.
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