Chapter 1 A Shell Theory for Carbon Nanotubes Based on the Interatomic Potential and Atomic Structure

Abstract

A finite-deformation shell theory for single-wall carbon nanotubes (CNTs) is established directly from the atomic structure of CNT and the interatomic potential by accounting for the important effect of moment and curvature (of CNTs) in the Cauchy–Born rule. The theory incorporates the effect of bending moment and curvature for a curved surface, and accurately accounts for the nonlinear, multibody atomistic interactions, as well as the carbon nanotube chirality. It avoids the ambiguous definition of nanotube thickness, and provides the constitutive relations among stress, moment, strain, and curvature via the interatomic potential. The constitutive behavior of a CNT is different from that of a classical shell, but its overall structural response at infinitesimal deformation may still be approximately represented by a linear elastic thin shell for some representative loadings such as tension, compression, bending, torsion, internal, and external pressure. The ratio of atomic spacing (Δ ≈ 0.14 nm) to CNT radius, Δ/R, is used to identify the order of error, where Δ/R ranges from zero (for graphene) to about 40% [for the (5,5) armchair CNT (R = 0.35 nm)]. For the order of error O[(Δ/R)3] (as compared to unity), which is about 6% for the (5,5) armchair CNT, the structural response of a CNTs cannot be represented by any classical shell. For the order of error O[(Δ/R)2], which is about 16% for the (5,5) armchair CNT, a CNT can be approximated by a linear elastic orthotropic thin shell. Only for the order of error O(Δ/R), which is about 40% for the (5,5) armchair CNT, a universal constant shell thickness and Young's modulus can be defined, and CNTs can be represented by an elastic isotropic thin shell. The instability of single-wall CNTs subject to tension, compression, internal and external pressure, and torsion is studied.