General Elasticity Theory for Graphene Membranes Based on Molecular Dynamics

Abstract

We have studied the mechanical properties of suspended graphene membranes using molecular dynamics (MD) and generalized continuum elasticity theory (GE) in order to develop and assess a continuum description for graphene. The MD simulations are based on a valence force field model which is used to determine the deformation and the elastic energy of the membrane (E MD ) as a function of external forces. For the continuum description, we use the expression E cont = E stretching + E bending for the elastic energy functional. The elastic parameters (tensile rigidity and Poisson ratio) entering E cont are determined by requiring that E cont = E MD for a set of deformations.Comparisons with the MD results show excellent agreement. We find that the elastic energy of a supported graphene sheets is typically dominated by the nonlinear stretching terms whereas a linear description is valid only for very small deflections. This implies that in some applications, i.e. NEMS, a linear description is of limited applicability.

Mechanical systems in the quantum regime

“…It is, however, known that for large, macroscopic objects this is unnecessary and that materials can be accurately described as a continuum with the mechanical behavior captured by a few parameters such as the elasticity tensor. Molecular dynamics simulations [129,130,131,132,133,134] and experiments [20,135,107] demonstrate that even for nanometer-sized objects continuum mechanics is, with some modifications, still applicable. This means that the dynamics of the individual particles is irrelevant when one talks about deflections and deformations; the microscopic details do, however, determine the material properties and therefore also the values of macroscopic quantities like the Young's modulus or the Poisson ratio.…”

Continuum Elastic Modeling of Graphene Resonators

“…Most research on graphene has hitherto focused on the electronic properties of graphene, and less attention has been directed to mechanical properties. For modeling NEMS, a reliable and efficient description of the mechanical response of nanocarbons to external forces is essential. , While continuum elasticity theory has been applied successfully to the study of mechanical properties of nanotubes for a long time, − it has only recently been applied to graphene membranes. − …”

“…Provided that the length scale of the deformation is large compared to the lattice spacing (long-wavelength limit), continuum theory can be used for graphene and we can write U sp 2 = ∫ .25em d x d y W 0 [ u̅ ( x , y ) ] by parametrizing the deformation of the surface as u̅ ( x , y ) = [ u ( x , y ), v ( x , y ), w ( x , y )]. The elastic energy density W 0 can be divided into stretching and bending contributions, W 0 = W 0 S + W 0 B .…”

Kinks in buckled graphene uncompressed and compressed in the longitudinal direction

“…Some billion-atom molecular dynamics simulations have been reported recently [5,6]. We note that the elasticity theory (more specifically, the classical non-linear theories for plates [7] with strain limitations) offers an alternative framework to the description of graphene elasticity [8,9]. An interesting future problem is to describe analytically the shape of graphene kinks, and find analytical expression for kink energy.…”