Curvature dependent surface energy for free standing monolayer graphene: Geometrical and material linearization with closed form solutions

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. To introduce material linearity energy is assumed to have a quadratic dependence on the strain tensor, the curvature tensor, the shift vector, as well as to combinations of them. Hexagonal symmetry then reduces the overall number of independent material constants to nine. 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 for the geometrically and materially linear case. We start with the problem of in-plane motions only. By prescribing the displacement, the components of the shift vector are evaluated. This way the field equations are satisfied trivially. Outof-plane motions are treated as well; we assume in-plane tension/compression that leads to buckling/wrinkling and solve for the components of the shift vector as well as the function present in buckling's modeling. The assumptions of linearity adopted here simplifies the analysis and facilitates analytical results.

Graphene resting on substrate: Closed form solutions for the perfect bonding and the delamination case

“…Dependence on the shift vector result from well established theories of multilattices (Parry, 1978;Ericksen, 1970Ericksen, , 1979Fadda and Zanzotto, 2000;Fadda and Zanzotto, 2001;Pitteri, 1984Pitteri, , 1985Pitteri and Zanzotto, 2003). So, all in all, graphene's elastic energy has the form (Sfyris et al, 2014b) …”

“…For graphene we adopt the linear framework of Sfyris et al (2014b), so the whole theory is confined to geometrical and material linearities for both film and substrate. For this linear modeling the in-surface material constants are four: c 1 ; c 2 ; c 5 ; c 9 .…”

On configurational weak phase transitions in graphene

“…Freestanding monolayer graphene is a monoatomic hexagonal 2-net [19][20][21] made of carbon atoms. The lattice and shift vectors for graphene can be chosen as follows (see Fig.…”

“…Motivated by these approaches [5,6,11,12,15,16,18] in some recent papers [19][20][21], we view graphene as a monoatomic hexagonal 2-lattice (or 2-net). The "2" in 2-net refers to the presence of two carbon atoms in the unit translation cell of graphene, while "net" refers to the fact that graphene is a planar lattice, i.e., a two-dimensional lattice.…”

An improved molecular structure mechanics method and its application for graphene wrinkling

“…Compared with the studies based on MD ( Inui, Mochiji, & Moritani, 2008;Herrero & Ramírez, 2009; and CTM ( Wang, Mylvaganam & Zhang, 2009;Bathe, Chapelle & Lee, 2003;Gil, Adhikari, Scarpa & Bonet, 2010;Naderi & Saidi, 2014;Sfyris, Sfyris, & Galiotis, 2014 ), the study based on MSM can consider the features of atomistic scale structure but is not perplexed in time scales.…”