Curvature dependent surface energy for a free standing monolayer graphene: some closed form solutions of the nonlinear theory

TL;DR

This paper develops a continuum nonlinear model for free-standing graphene monolayers, deriving analytical solutions for simple loadings and analyzing complex deformations like wrinkling and buckling with mathematical rigor.

Contribution

It introduces a closed-form analytical approach to modeling curvature-dependent surface energy in graphene, including out-of-plane deformations and buckling phenomena.

Findings

Analytical solutions for shift vector under simple loadings

Conditions for existence and uniqueness of solutions with out-of-plane deformations

Classification of momentum equations for general problems

Abstract

Continuum modeling of a free-standing graphene monolayer, viewed as a two dimensional 2-lattice, requires specifications of the components of the shift vector that act as an auxiliary variable. The field equations are then the equations ruling the shift vector, together with momentum and moment of momentum equations. We present an analysis of simple loading histories such as axial, biaxial tension/compression and simple shear for a range of problems of increasing difficulty. We start by laying down the equations of a simplified model which can still capture bending effects. Initially, we ignore out of plane deformations. For this case, we solve analytically the equations ruling the auxiliary variables in terms of the shift vector; these equations are algebraic when the loading is specified. As a next step, still working on the simplified model, out-of-plane deformations are taken into account and the equations complicate dramatically. We describe how wrinkling/buckling can be introduced into the model and apply the Cauchy-Kowalevski theorem to get existence and uniqueness in terms of the shift vector for some characteristic cases. Finally, for the treatment of the most general problem, we classify the equations of momentum and give conditions for the Cauchy-Kowalevski theorem to apply.