Modeling Of Deformations Of Single-Walled Carbon Nanotubes
This thesis presents a study of single-walled carbon nanotubes (SWNTs) using revised Cosserat rod models. A SWNT is a thin hollow cylinder, with diameter of the order of 1 nm and length possibly as large as 1[MICRO SIGN]m. Its wall is composed of a single layer of carbon atoms with an estimated thickness around 0.335 nm. A single walled CNT (SWNT) is classified as armchair, zig-zag or chiral, depending on the (regular) arrangement of atoms on its surface. In general, CNT models based on traditional continuum mechanics can be inconsistent or inaccurate [57, 44, 16], while atomistic simulations can be prohibitively expensive. Atomistic-continuum models e.g. the quasi-continuum approach, originally proposed for bulk crystals [47, 51], attempt to combine the accuracy of atomistic simulations with the efficency of continuum models. This approach has been applied to CNTs [61, 26, 8, 9, 7, 10, 2, 55, 11]. At relatively long length scales, it makes sense to propose a one-dimensional model for a SWNT. For such long nanotubes, one-dimensional models are attractive for both theoretical modeling as well as numerical simulation. Chandraseker et al.  proposed a Cosserat rod model  for a SWNT that can capture large deformations of SWNTs. This model includes deformation modes such as bending, twisting, extension and shear, as well as coupling between ex- tension and twist and between shear and bending. Kumar and Mukherjee  futher developed the Cosserat rod model to incorporate cross-sectional deformation and allow coupling between cross-sectional deformation with twist and extension. In this work, finite element simulations of both standard and revised Cosserat rod models are carried out. Using a standard Cosserat rod model combined with atomistic simulation, the governing equations of a rod model are tranformed into weak form and discritized. The weak form of the govening equations contain both geometric and material nonlinearity. A implicite iterative method, based on the Newtown-Raphson method is performed to find the converged solution. Several numerical verification cases are performed to validate the method and demostrate the capability of this numerical implementation. Finally, in the case of the revised Cosserat rod model, a similar process is carried out to find the finite element solution. In this case, material property is assumed to be linear and only geometric nonlinearity is incoporated. Finally, a mathematically consistent extension to the revised rod theory is developed that connects an isotropic and hemitropic rod by exploring material symmetry. This proposed approach is applied to model SWNTs with different chiralities. It effectively connects the modeling of different types of SWNTs using five material parameters. The connection between different types of SWNTs is controlled by the chirality angle in the this derivation. Future effort could involve solving the inverse problem to find these material parameters.
Van Loan, Charles Francis; Zehnder, Alan Taylor
Theoretical and Applied Mechanics
Ph. D., Theoretical and Applied Mechanics
Doctor of Philosophy
dissertation or thesis