Phonon Prediction in Toroidal Carbon Nanotubes Using a Continuum Finite Element Approach

This work develops a tensor-based, reduced-order shell finite element formulation used to predict the phonon behavior of toroidal carbon nanotubes (CNTs). Displacements referencing two covariant basis vectors lying in the toroid’s tangent space, and one basis vector orthogonal to the tangent space, capture the kinematics of the toroidal CNT. These basis vectors compose a curvilinear coordinate system. Although specific attention is on toroidal CNTs, the formulation can be quickly adapted to cylindrical or other curvilinear CNTs by appropriate replacement of the metric tensor components and Christoffel symbols. The finite element procedure originates from a variational statement (Hamilton’s Principle) governing virtual work from internal, external (not considered), and inertial forces. Internal virtual work is related to changes in atomistic potential energy accounted for by an interatomic potential computed at reference area elements. Small virtual changes in the displacements allow a global mass and stiffness matrix to be computed, and these matrices then allow phonons to be predicted via the general eigenvalue problem. Results are generated for example toroidal CNTs documenting zero-energy behavior (rigid body motion) and the lowest phonons, which include the expected breathing-like and bending-like phonons.