Can a single-wall carbon nanotube be modeled as a thin shell?

Abstract

Single-wall carbon nanotubes (SWCNT) have been frequently modeled as thin shells, but the shell thickness and Young's modulus reported in literatures display large scattering. The order of error to approximate SWCNTs as thin shells is studied in this paper via an atomistic-based finite-deformation shell theory, which avoids the shell thickness and Young's modulus, but links the tension and bending rigidities directly to the interatomic potential. The ratio of atomic spacing ( Δ≈0.14 nm) to the radius of SWCNT, Δ/ R, which ranges from zero (for graphene) to 40% [for a small (5,5) armchair SWCNT ( R=0.35 nm)], is used to estimate the order of error. For the order of error O[( Δ/ R) 3], SWCNTs cannot be represented by a conventional thin shell because their constitutive relation involves the coupling between tension and curvature and between bending and strain. For the order of error O[( Δ/ R) 2], the tension and bending (shear and torsion) rigidities of SWCNTs can be represented by an elastic orthotropic thin shell, but the thickness and elastic modulus cannot. Only for the order of error O( Δ/ R), a universal constant shell thickness can be defined and SWCNTs can be modeled as an elastic isotropic thin shell.