Dynamic stiffness analysis of graphene sheets and carbon nanotubes

Carbon based nanomaterials have great potential in advanced engineering applications on account of their exceptional mechanical, thermal and electrical properties. At the molecular level, these materials comprise two-dimensional hexagonal cells in which the carbon atoms are held together by massless covalent bonds. Arrays of cells can be formed into graphene sheets or rolled into carbon nanotubes. Li and Chou [1] presented a lattice model based on this topology, in which the carbon atoms are represented by point masses and are connected by massless space frame elements having extensional, flexural and torsional stiffness properties. Geometric and elastic properties of such elements have been obtained experimentally [2] and have previously been used in finite element analysis to find natural frequencies, which are typically of the order 1THz, for single layer graphene sheets [3] and single wall carbon nanotubes [4]. This paper applies lattice models to the same carbon nanostructures, but uses exact dynamic stiffness theory derived from analytical solutions of the governing differential equations. The analysis therefore avoids the use of arbitrary shape functions and discretisation into finite elements. The Wittrick-Williams algorithm ensures that any natural frequency can be found accurately and with certainty. Numerical results for the natural frequencies show good agreement with finite element results when Bernoulli-Euler beam theory is used, but they can be considerably reduced when Timoshenko theory is applied. The vibration modes show good agreement with published results. Solution times are acceptable, and can be reduced substantially by using a repetitive analysis in which only a single repeating portion is analysed. However, care must be taken to ensure satisfaction of the boundary conditions