ElsevierMerli Gisbert, R.; Lazaro, C.; Monleón Cremades, S.; Domingo Cabo, A. (2013). A molecular structural mechanics model applied to the static behavior of single-walled carbon nanotubes: New general formulation. Computers and Structures. 127:68-87. doi:10.1016/j.compstruc.2012.11.023 AbstractA new general formulation for the mechanical behavior of Single-Walled Carbon Nanotubes is presented. Carbon atoms are located at the nodes of an hexagonal honeycomb lattice wrapped into a cylinder. They are linked by covalent C − C bonds represented by a truss or spring element, and the three-body interaction among two neighboring covalent bonds is reproduced by a rotational spring. The main advantage of our approach is to allow general load conditions (and any chirality) with no need of specific formulation for each load case, in contrast with previous works [26], [27], [31]. Four load configurations are adopted: tension, compression, bending and torsion of cantivelered SWCNTs. Calculations with our own codes for both AMBER and Morse potential functions have been carried out, aimed to compare their final results. Initial positions of the atoms (nodes) into nanotube cylindrical geometry has been reproduced in great detail by means of a conformal mapping from the planar graphene sheet. Therefore, the effect of initial SWCNTs curvature has been introduced explicitly through a system of initial stresses (prestressed state) which contribute to maintain their circular cross-section. Numerical results and deformed shapes for nanotubes with several diameters and chiralities under each load case are used to obtain their mechanical parameters with the only objective of checking the present formulation with previous works [28], [30], [20], [24]. Also, the significance of the atomistic discrete simulations at the nano-scale size against other continuum models is underlined.

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This article reviews the different aspects involved in computational form finding of bending-active structures based on the dynamic relaxation technique. Dynamic relaxation has been applied to form-finding problems of bending-active structures in a number of references. Due to the complex nature of large spatial deformations of flexible beams, the implementation of suitable mechanical beam models in the dynamic relaxation algorithm is a non-trivial task. Type of discretization and underlying beam theory have been identified as key aspects for numerical implementations. References can be classified into two groups depending on the selected discretization: finite-difference-like and finite-element-like. The first group includes 3-and 4-degree-of-freedom implementations based on increasingly complex beam models. The second gathers 6-degree-of-freedom discretizations based on co-rotational three-dimensional Kirchhoff-Love beam elements and geometrically exact Reissner-Simo beam elements. After reviewing and comparing implementation details, the advantages and drawbacks of each group have been discussed, and open aspects for future work have been pointed out.

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a b s t r a c tIn this paper, the post-critical behavior and buckling modes of single-walled carbon nanotubes are analyzed via a Molecular Mechanics model. The main target is to develop a general formulation for the model, which has been simplified under small strains assumption, and to implement a versatile tool for the structural analysis of carbon nanotubes in the framework of geometrical nonlinearity. For this purpose, a mechanical formulation able to reproduce any load configuration and supporting conditions has been derived by using an energy approach. Then, an incremental-iterative solution procedure has been implemented in order to trace several nonlinear equilibrium paths and to obtain the corresponding critical strains of clamped-clamped nanotubes under compressive, flexural and torsional loading distributions. The model shows a good numerical performance and results in agreement with previous atomistic works. Two interatomic potentials have been adopted in order to find out the influence of different constitutive relationships on the final nonlinear response. We have concluded that the choice of the potential function has no significant effect on the final buckling strains. Our results confirm that the final buckling response is strongly determined by geometrical imperfections in the nanotube, which can be well reproduced in the proposed model, but are much more difficult to handle in continuum models.

In this paper, the geometry of single-walled carbon nanotubes without any external loading is analyzed via an energy procedure. The nanotube is assumed to be inscribed into a perfect cylinder of unknown diameter, which is estimated by minimizing the total interatomic potential involved into a basic cell with several carbon atoms and their corresponding bonds. In this step, two interatomic potentials have been adopted in order to compare their influence on the obtained results. Our calculations show that the widely used conformal mapping is not the most suitable option to reproduce the geometry of single-walled nanotubes in absence of external loading. Likewise, a more accurate method to estimate the initial diameter of the nanotube is developed, yielding higher differences with smaller nanotubes in comparison with other published works. The present analysis can be useful in the framework of Molecular Mechanics or continuum models as an alternative way to introduce initial stresses (due to the curvature of the cylinder) in the mechanical analysis, against other involved methods.

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