The bending of single layer graphene sheets: the lattice versus continuum approach

Abstract

The out-of-plane bending behaviour of single layer graphene sheets (SLGSs) is investigated using a special equivalent atomistic-continuum model, where the C-C bonds are represented by deep shear bending and axial stretching beams and the graphene properties by a homogenization approach. SLGS models represented by circular and rectangular plates are subjected to linear and nonlinear geometric point loading, similar to what is induced by an atomic force microscope (AFM) tip. The graphene models are developed using both a lattice and a continuum finite element discretization of the partial differential equations describing the mechanics of the graphene. The minimization of the potential energy allows us to identify the thickness, elastic parameters and force/displacement histories of the plates, in good agreement with other molecular dynamic (MD) and experimental results. We note a substantial equivalence of the linear elastic mechanical properties exhibited by circular and rectangular sheets, while some differences in the nonlinear geometric elastic regime for the two geometrical configurations are observed. Enhanced flexibility of SLGSs is observed by comparing the nondimensional force versus displacement relations derived in this work and the analogous ones related to equivalent plates with conventional isotropic materials.

Transverse vibration of single-layer graphene sheets

“…Thus, in this work, we analyse two groups of SLG resonators, i.e. three zigzag SLGS (10,0), (14,0), (18,0) and three armchair SLGS (11,11), (15,15), (19,19), with increasing length. In this study, we computed our results using bridged (atoms at the two sides along width are restrained) and cantilevered (atoms at one side along width are restrained) boundary conditions.…”

“…A SLGS (shown in figure 1) may be approximated by a thin elastic plate [11]. The equation of motion of the transverse free vibration of a thin elastic plate [40,43] can be expressed as…”

“…The exfoliation technique has led to the successful production of graphene, opening a new era in the field of nanoelectronics [2][3][4]. The very high in-plane stiffness of graphene sheets [5] has suggested some possible use of graphite nanosheets for nanosensors and NEMS applications [6,7], due to their extremely high surface-to-volume ratio, as well as large deflection capability under point loading [8][9][10][11][12][13]. In this paper the out-of-plane or transverse vibration of singlelayer graphene sheets (SLGSs) is considered.…”

EELS and Raman: Nonbond Interactions

“…Therefore, the frequency shift corresponds to the square root of the stiffness of the particular non-bond interaction. The stiffness is the product of the elastic modulus Y (energy density) and the bond length d [13].…”

Nonlinear softening and hardening nonlocal bending stiffness of an initially curved monolayer graphene

“…They concluded that a large piece of graphene can be regarded as a thin plate by considering the nonlinear effects in the strain components. Scarpa et al [16] studied the out of plane bending behavior of single layer graphene sheets subjected to point loading by using an atomistic-continuum model.…”