Dynamic Finite Element Modeling of Carbon Nanotubes Using an Intrinsic Formulation

This article presents an efficient explicit dynamic formulation for modeling curved and twisted Carbon Nanotubes (CNT's) based on a recently developed intrinsic beam description (i.e. the dynamic state given by curvatures, strains, and velocities only) [Hodges, 2003] together with a finite element discretization incorporating atomistic potentials. This approach offers several advantages primarily related to the model's computational efficiency: 1) the resulting partial differential equations governing motion are in first-order form (i.e. have first-order time derivatives only), 2) the system nonlinearities appear at low order, 3) the intrinsic description incorporating curvature allows low-order interpolation functions to describe generally curved and twisted nanotube centerlines, 4) inter-element displacements, slopes, and curvatures are matched at the element boundaries, and 5) finite rotational variables are absent, along with their inherit complexities. In addition, the developed model and finite element discretization are able to capture the nanotube's dynamic response, without the expense of calculating the dynamic response of individual atoms as per Molecular Dynamics models. Simulation results are presented which illustrate the dynamic response of a typical CNT to axial, bending, and torsional loading. Results from the simulations are compared to similar results available in the literature, and close agreement is documented.