Non-local plate and shell models have attracted much interest in the area of graphene and carbon nanotube simulations. This work explores further into this area and aims to provide more accurate and reliable non-local modeling methods to graphene and carbon nanotubes.

At first, a semi-analytical model for determining the equilibrium configuration of single wall carbon nanotubes is presented. By taking advantage of the symmetry characteristics, a carbon nanotube structure is represented by five independent variables. A line search optimization procedure is employed to determine the equilibrium values of these variables by minimizing the potential energy. With the equilibrium configuration obtained, the semi-analytical model enables a straightforward calculation of the radial breathing mode frequency of carbon nanotubes. The radius and radial breathing mode frequency results obtained from the semi-analytical approach are compared with those from molecular dynamics and ab initio calculations. The results demonstrate that the semi-analytical approach offers an efficient and accurate way to determine those properties.

Next, we investigate several issues in the local and non-local plate models of single layer graphene sheets. The issues include the ambiguity of the plate thickness in the moment-curvature relation, the definition of clamped boundary condition at graphene edges, and the value of the non-local parameter. For error analysis, the results obtained from a REBO potential based atomic lattice mechanics model are used as reference results. Errors of the plate models are analyzed and remedies are proposed within the framework of the non-local plate model. Numerical results of static and modal analysis of graphene are presented to demonstrate the effectiveness of the remedies.

In the last part of this work, a non-local finite element shell model is established for single-walled carbon nanotubes. Based on the accurately relaxed radius, bond lengths and angles obtained from the semi-analytical model, it is possible to calculate more accurate elastic constants directly from the interatomic potentials. Then through the combination of the classical first order shell theory, the non-local elasticity, and the potential-based elastic properties, a more accurate shell representation of single wall carbon nanotubes is established. The improvement in accuracy is demonstrated by comparing the spectral frequency analysis and dispersion relation results with those obtained from lattice mechanics and molecular dynamics simulations, respectively.