Recent developments in the analysis of carbon nanotubes and nonlinear shell theories

This paper has a two-fold objective: (a) to provide an overview of the modeling and analysis of carbon nanotubes using continuum mechanics models, including shell models; and (b) to present a computational model for the finite deformation analysis of shell structures. Great efforts have been made by researchers to develop continuum mechanics models as an alternative to atomistic simulations for the analysis of CNTs. Although CNTs are of few nanometers in diameter, continuum mechanics models have been found to describe their mechanical behavior surprisingly well under various circumstances. In the determination of mechanical properties such as Young's modulus, yield and fracture strengths by experiments, the experimental data fitting exercises have been mostly undertaken in association with continuum mechanics (beam or cylindrical shell) models. For the second part, we present a tensor-based finite element formulation which describes the mathematical shell model in a natural and simple way by using curvilinear coordinates. In addition, we utilize the spectral/hp method for the finite element discretization to avoid membrane and shear locking where no mixed interpolations are required. The first-order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the Lagrangian description. This approach takes into account thickness changes and, therefore, 3D constitutive equations are utilized. Numerical simulations and comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composites, as well as functionally graded shells are presented.