Keywords: finite element method, force field method, carbon nanotubes, nanoscale structures, stability problems.

It is obvious that experiments on the nanoscale level are very expensive and time consuming; hence there is a great need for the computational simulation of spatial structures at nanoscale. The interactions of atoms will be simulated here with so-called force field methods, which model the atoms as masses and the atomic bonds as springs, see Figure  135.1(a). In Figures  135.1(b), 135.1(c) and 135.1(d) the potentials for the interactions of carbon atoms in hexagonal patterns are given.

This approach can easily be implemented in a finite element code for the benefit of efficient solvers and comprehensive pre-/postprocessors. Therefore a new four-node element has been developed which allows for the computation of atomic structures such as carbon nanotubes. The material parameters for this element can be directly taken from the well-proven DREIDING force field, see [2], which is a great advantage compared to standard finite elements. The robustness and efficiency of the presented algorithm and stability problems on the nanoscale will be shown in bending and torsion tests of carbon nanotubes, see Figure  135.2(a), with and without Stone-Wales defects, see figure  135.2(b). The results correspond to those known as stability failure at macroscale analyses.

There is considerable interest in carbon nanotubes due to their extraordinary mechanical, electrical, thermal and chemical properties since their discovery in 1991. They consist of hexagonal rings of carbon atoms and thus can be imagined as a rolled-up sheet of graphene, what makes them a shell-like structure at nanoscale. Many applications for carbon nanotubes are possible, where their use as fiber reinforcement in polymers, ceramics etc. is only one facet, see [1].

In short, this paper provides an introduction into the novel nanotechnology, presenting some theory and numerical results in this new and highly rewarding field of research. The authors believe that engineering new nanomaterials might become the only way to meet the essentially stringent demands for light weight, high strength structural components in the next generation of shell and spatial structures.

1

P.M. Ajayan and O.Z. Zhou. "Applications of carbon nanotubes", In M.S. Dresselhaus, G. Dresselhaus, and P. Avouris, editors, Carbon Nanotubes, Topics in Applied Physics, volume 80, pages 391-425. Springer-Verlag, Berlin, 2001.

2

S.L. Mayo, B.D. Olafson, and W.A. Goddard (III). "(DREIDING): A generic force field for molecular simlations", J. Physical Chemistry, 94:8897-8909, 1990. doi:10.1021/j100389a010

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