Atomistic continuum modeling of graphene membranes

The paper deals with the modeling of thin, monolayer graphene membranes, which have significant electrical and physical properties used for nano- or micro-devices, such as resonators and nanotransistors. The membrane is considered as a homogenized graphene monolayer on the macroscopic scale, and a continuum–atomistic multiscale approach is exploited, focusing the Tersoff–Brenner (TB) potential for the interaction between the carbonic bonds. The associated Representative atomistic Unit Lattice (RUL) is thereby considered as a micro-scale quasi-continuum placed in context of computational homogenization. In this development, the Cauchy–Born rule (CBN) is extended by the atomic fluctuation to allow for relaxation in the RUL. The paper discusses the handling of the TB-potential, both in the context of macro–micro homogenization, and in the context of numerical implementation perspectives. In particular, explicit expressions of the homogenized membrane forces and stiffness are expressed in terms of the first and second gradient of the potential, with due consideration to the involved “non-local” pairwise interaction in the model. In addition, the detailed resulting macroscopic non-linear and linearized finite element response is formulated in terms of the relaxed lattice level atomistic response. Numerical results are provided for the lattice response in terms of the apparent anisotropic behavior induced by the graphene atomic structure. An assessment of the convergence of RULs with respect to different deformation states of the lattice membrane is also carried out. Finally, a validation of an experiment of a circular graphene membrane, using atomic force microscopy (AFM) measurements, is provided based on standard TB-parameters available in the literature