Bulk dynamic response modeling of carbon nanotubes using an intrinsic finite element formulation incorporating interatomic potentials

This article presents an efficient explicit dynamic formulation for modeling curved and twisted carbon nanotubes (CNT’s) based on a recently-developed intrinsic continua description (i.e., the dynamic state given by curvatures, strains, and velocities only) [Hodges, D.H., 2003. Geometrically-exact, intrinsic theory for dynamics of curved and twisted, anisotropic beams. AIAA Journal 41(6), 1131–1137.] 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 center-lines, (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 bulk (equivalently, zero-temperature) 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 bulk 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.