Phonon spectra prediction in carbon nanotubes using a manifold-based continuum finite element approach

This work develops a tensor-based, reduced-order shell (two-manifold) finite element formulation for predicting phonon spectra in finite-length cylindrical and toroidal carbon nanotubes (CNTs). The formulation does not require an assumed tube thickness. Displacements referencing two covariant basis vectors lying in the tangent space, and one basis vector orthogonal to the tangent space, capture the systems' kinematics. These basis vectors compose a curvilinear coordinate system useful for capturing cylindrical, toroidal, and generically-curved nanotube configurations. 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 phonon spectra and energies to be predicted via a general eigenvalue problem. Results are generated for example cylindrical and toroidal CNTs documenting accurate prediction of phonon spectra, to include the expected longitudinal, torsional, bending, and breathing-like phonons.