Continuous modelling of single layer 2D carbon nanostructures based on a modified Cauchy–Born rule

Abstract

AbstractGraphene is the thinnest known structure. It consists of a single layer of carbon atoms and has unique mechanical properties. The carbon atoms form bonds with neighbouring atoms that can be described by interatomic potentials. A simulation of molecular mechanical processes can be performed by applying the formalism of the finite element method to the interatomic potentials. Neglecting local effects, a continuum formulation is an efficient alternative to a discrete model. The continuum formulation requires a connection between the discrete atomic lattice and the continuum. One way of doing this is to use the Cauchy–Born rule. This paper reviews the extensions of the Cauchy–Born rule, in particular, the exponential Cauchy–Born rule, and adds an alternative approach. It is shown that all contributions dealing with the exponential Cauchy–Born rule inevitably lead to shell models with Kirchhoff–Love kinematics. The approach presented here instead uses a shell model with Reissner–Mindlin kinematics, which has the major advantage of a much simpler mesh generation and a simpler application of the boundary condition.