A string moving with geostationary angular velocity in its radial relative equilibrium configuration around the Earth, reaching from the surface of the Earth far beyond the geostationary height, could be used as track for an Earth to space elevator. This is an old dream of mankind, originating about 100 years ago in Russia. Besides the question of feasibility from a technological point of view also the question concerning the stability of such a configuration has not yet been completely solved. Under the assumption that a proper material (defective carbon nanotubes) is available, making the connection possible technologically, we address the question of stability of the radial relative equilibrium of a tapered string on a circular geosynchronous orbit around the Earth, reaching from the surface of the Earth far beyond the geostationary height.
A string moving with geostationary angular velocity in its radial relative equilibrium configuration around the Earth, reaching from the surface of the Earth far beyond the geostationary height, could be used as track for an Earth to space elevator. This is an old dream of mankind, originating about 100 years ago in Russia. Besides the question of feasibility from a technological point of view also the question concerning the stability of such a configuration has not yet been completely solved. Under the assumption that a proper material (defective carbon nanotubes) is available, making the connection possible technologically, we address the question of stability of the radial relative equilibrium of a tapered string on a circular geosynchronous orbit around the Earth, reaching from the surface of the Earth far beyond the geostationary height.
The effect of free edges of a monoatomic graphene sheet leads to excess edge energy due to the reconstruction of dangling bonds. Molecular static calculations show, that individual carbon atoms near the edge are displaced out of plane for relaxed nanoribbons [1]. In this work we are considering the effect of excess edge energy for almost circular graphene patches. To tackle this problem in the framework of continuum mechanics we are modelling the edge effect with a non-Euclidean plate model. A linear stability analysis of the flat configuration leads to the stability boundary in the parameter plane. Atomistic modelsAb initio calculations [2] give accurate values for the energy cost of creating free edges in a monoatomic graphene sheet. The same procedure was used for calculating the edge energies with two classical potentials AIREBO [3] and REBO [4]. The general form of the used AIREBO potential is Edge energy as a function of the applied strain is depicted in Fig. 1. The slope of the individual lines at = 0 represents the edge stresses. There is a significant difference between the results from ab initio calculations and the classical potentials. Ab initio calculations predict larger edge energies for zigzag edges than for armchair ones, whereas the two classical potentials (REBO & AIREBO) show the opposite order. The order of the edge stresses are interchanged as well. A comprehensive comparision of results from different atomistic models can be found in [6]. Empirical force fields are efficient for simulating larger systems, which makes them appropriate for studying the effect of the excess edge energy on the global configuration. Although the discrepance in the quantitative individual values, the results are qualitatively compareable. In particular the negative edge stress is remarkable, because this is the reason for destabilisation of the flat edge. Figure 2 shows the relaxed configuration of an almost circular graphene patch. The wavy disturbance of the edge decays towards the center.
We study the buckling behaviour of a single rectangular graphene layer by a molecular mechanics force field approach. The so called MM3-Potential [1] is used to model the atomistic interactions. The global minimum of the total potential energy is calculated for a prescribed linear displacement field at the edges of the plate. Various buckled configurations depending on the dimension of the plate are calculated and are compared with results from continuum mechanics.
Axially loaded cylindrical continuous shells collapse either globally like a rod (Euler buckling), or locally (local shell wall buckling), depending on the ratio of the length of the shell over the diameter [1]. There are many published investigations, which show that this behaviour is also true for Carbon Nanotubes CNTs [2]. In this work a systematic analysis of the problem is given in the framework of molecular statics. This approach has the advantage of taking care of the discrete structure of CNTs. The covalent bonds of the hexagonal carbon network are modelled as nonlinear springs, and the compressive load is applied quasistatically, excluding follower forces. The software package LAMMPS [3] offers the AIREBO potential [4] and is suitable for describing CNTs. To identify the stability boundary in the parameter plane, LAMMPS is extended to compute the definiteness of the Hessian.
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