A finite deformation hyper-elastic membrane theory based on inter-atomic potentials for crystalline films composed of a single atomic layer is developed. For this purpose, an extension of the standard Born rule that exploits the differential geometry concept of the exponential map is proposed to deal with the curvature of surfaces. The exponential map is approximated locally and strain measures based on the stretch and the curvature of the membrane arise. The methodology is first particularized to atomic chains in two dimensions, and then to graphene sheets. A reduced model for the transverse mechanics of carbon nanotubes is developed in detail. This model is a hyper-elastic constrained membrane which fully exploits the symmetry of the transverse deformation. Additionally, a continuum version of the non-bonded interactions is provided. The continuum model is discretized using finite elements and very good agreement with molecular mechanics simulations is obtained. Finally, several simulations illustrate the strong effect of the van der Waals interactions in the transverse deformation of carbon nanotubes.
Abstract
A finite deformation hyper-elastic membrane theory based on inter-atomic potentials for crystalline films composed of a single atomic layer is developed. For this purpose, an extension of the standard Born rule that exploits the differential geometry concept of the exponential [...]
We present a three-dimensional continuum model for layered crystalline materials made out of weakly interacting two-dimensional crystalline sheets. We specialize the model to multilayer graphene materials, including multi-walled carbon nanotubes (MWCNTs). We view the material as a foliation, partitioning of space into a continuous stack of leaves, thus loosing track of the location of the individual graphene layers. The constitutive model for the bulk is derived from the atomistic interactions by appropriate kinematic assumptions, adapted to the foliation structure and mechanics. In particular, the elastic energy along the leaves of the foliation results from the bonded interactions, while the interaction energy between the walls, resulting from van der Waals forces, is parametrized with a stretch transversal to the foliation. The resulting theory is distinct from conventional anisotropic models, and can be readily discretized with finite elements. The discretization is not tied to the individual walls and allows us to coarse-grain the system in all directions. Furthermore, the evaluation of the non-bonded interactions becomes local. We test the accuracy of the foliation model against a previously proposed atomistic-based continuum model that explicitly describes each and every wall. We find that the new model is very efficient and accurate. Furthermore, it allows us to rationalize the rippling deformation modes characteristic of thick MWCNTs, highlighting the role of the van der Waals forces and the sliding between the walls. By exercising the model with very large systems of hollow MWCNTs and suspended multilayer graphene, containing up to 109 atoms, we find new complex post-buckling deformation patterns.
Abstract
We present a three-dimensional continuum model for layered crystalline materials made out of weakly interacting two-dimensional crystalline sheets. [...]