Elastic Buckling Behaviour of General Multi-Layered Graphene Sheets

Nanoscale vibration characteristics of multi-layered graphene sheets

Article history: In this paper, a description is given of finite element method (FEM) simulations of the elastic bending modulus of multi-layered graphene sheets that were carried out to investigate the mechanical behavior of graphene sheets with different gap thicknesses through molecular mechanics theory. The interaction forces between layers with various gap thicknesses were considered based on the van der Waals interaction. A finite element (FE) model of a multi-layered rectangular graphene sheet was proposed with beam elements representing bonded interactions and spring elements representing non-bonded interactions between layers and between diagonally adjacent atoms. As a result, the average elastic bending modulus was predicted to be 1.13 TPa in the armchair direction and 1.18 TPa in the zigzag direction. The simulation results from this work are comparable to both experimental tests and numerical studies from the literature.

The morphology of graphene sheets overlying a substrate plays an important role in controlling the electronic behaviour of graphene-based electronic devices. Atomistic simulations and energetic analysis are performed to study the morphology of graphene edges on a model substrate. We find that the van der Waals interactions are able to significantly reduce the amplitude, penetration depth and wavelength of edge ripples for a single-layer graphene sheet, but suppression of the ripples is not observed in multilayer graphene sheets with unsaturated edges. We explain this difference using an analytical model that accounts for the energetics of rippling and van der Waals interactions.

In this paper,we study the quantum size effects in multilayer graphene sheets using first principles methods within the framework of density functional theory. Four different types of functionals are adopted respectively to describe the van der Waals interactions between graphene layer sheets: the DFT-GGA(PBE), the DFT-D2, the vdW-DF and the optPBE-vdW. By inspecting the binding energy as a function of increasing graphene layers, we find that the PBE functional can not well describe the van der Waals interactions between different layers of graphene sheets. In contrast, the other three methods exhibit similar results with monotonic increasing binding energy as a function of graphene layers towards the bulk limit, concluding that the layered graphene structure is stabilized by van der Waals interactions. The density of states at zero temperature indicate that the multilayer graphene sheets is a semi-metal, which is independent of sheet layers number. The finite temperature (about 200 K) density of states at Fermi surface are studied as a function of the number of stacking graphene layers. The systematic oscillating behavior of finite temperature density of states between odd and even number of layers is a demonstration of quantum size effects. The Fermi wavelength will converge to two times the inter-layer distance of graphite, which is consistent with the theory describing the motion of particles in a quantum well. Finally, we study the adsorption of single H atom on multilayer graphene sheets to test the role of quantum size effects. The adsorption energies and the vibration frequencies are calculated for comparison with experiments. Our results shed light on understanding the stacking process of multilayer graphene in vacuum both theoretically and experimentally.

The mechanical behaviour of a single-layer nanostructured graphene sheet is investigated using an atomistic-based continuum model. This is achieved by equating the stored energy in a representative unit cell for a graphene sheet at atomistic scale to the strain energy of an equivalent continuum medium under prescribed boundary conditions. Proper displacement-controlled (essential) boundary conditions which generate a uniform strain field in the unit cell model are applied to calculate one elastic modulus at a time. Three atomistic finite element models are adopted with an assumption that force interactions among carbon atoms can be modeled by either spring-like or beam elements. Thus, elastic moduli for graphene structure are determined based on the proposed modeling approach. Then, effective Young’s modulus and Poisson’s ratio are extracted from the set of calculated elastic moduli. Results of Young’s modulus obtained by employing the different atomistic models show a good agreement with the published theoretical and numerical predictions. However, Poisson’s ratio exhibits sensitivity to the considered atomistic model. This observation is supported by a significant variation in estimates as can be found in the literature. Furthermore, isotropic behaviour of in-plane graphene sheets was validated based on current modeling.

An atomistic-based continuum approach for calculation of elastic properties of single-layered graphene sheet

In the present study, the free vibration, mechanical buckling and thermal buckling analyses of multi-layered graphene sheets (MLGSs) are investigated. Eringen’s nonlocal elasticity equations are incorporated in new two-variable plate theories accounting for small-scale effects. The MLGSs are taken to be perfectly bonded to the surrounding medium. The governing differential equations of this model are solved analytically under various boundary conditions and taking into account the effect of van der Waals forces between adjacent layers. New functions for the displacements are proposed here to satisfy the different boundary conditions. Comparison of the results with those being in the open literature is made. This comparison illustrates that the present scheme yields very accurate results. Furthermore, the influences of nonlocal coefficient, moduli of the surrounding elastic medium and aspect ratio on the frequencies and buckling of the embedded MLGSs are examined.

In this paper, buckling analysis of biaxially compressed multi-layered graphene sheets with a continuum plate model is reported. The equations of motion are analytically solved to obtain closed-form solution for buckling loads of all edges simply supported multi-layered graphene sheets. The interaction of van der Waals (vdWs) pressure between the layers is incorporated in the formulation to determine the buckling behavior of simply supported MLGSs. Explicit formulae are derived for predicting the critical buckling loads of double- and triple-layered graphene sheets, and they clearly indicate the effect of vdW interaction on the critical buckling loads. The critical buckling loads are calculated for various numbers of layered graphene sheets, and the obtained results show that the vdW force has no effect on the first critical buckling load of an MLGS, but plays a significant role in all higher first critical buckling loads for all combinations of m and n.

A study on large amplitude vibration of multilayered graphene sheets

Based on an explicit formula used for describing the van der Waals (vdW) interaction between any two layers of multi-layered graphene sheets (MLGSs), the vibration properties of five- and eight-layered graphene sheets are analyzed with considering the influence of vdW interaction on the natural frequences. The present study shows that for a given combination n and m the lowest natural frequency (classical natural frequency) of a MLGSs is independent of vdW interaction, but that it is very significant for the influence of vdW interaction on the other higher frequencies (resonant frequencies). The different vibration modes, including in-phase and anti-phase ones, are given for exhibiting clearly the influence of vdW interaction.

Analysis of nonlinear forced vibration of multi-layered graphene sheets

Analytical investigation on free vibration of circular double-layer graphene sheets including geometrical defect and surface effects

In this paper, a multiplate nonlocal shear model and molecular dynamic simulations are presented to investigate the effects of interlayer shear and nonlocality on the natural frequencies of multilayer graphene sheets (MLGSs). From one aspect in the optimal design of such structures, the interaction between graphene layers, which can significantly vary the static and dynamic behavior due to lack of solidity of layers stack, should be considered. On the other hand, it is requied that the nonlocality phenomenon which has an effective role in the mechanical analysis of nanostructures is taken into account. To this aim, the equation of motion along with corresponding boundary conditions is derived by using the Kirchhoff plate model and nonlocal continuum theory to capture these effects. For a case study, the free vibration problem of simply supported MLGSs is studied by presenting the closed-form Navier’s solution of natural frequencies and carrying out Molecular dynamics (MD) simulations to estimate the value of nonlocal parameter. Finally, the influences of interlayer shear modulus, number of layers, foundation stiffness, and nonlocal parameter on the natural frequencies of such MLGSs are examined in detail. Findings show that natural frequencies including nonlocality effect become smaller than ones of the classical continuum theory and also have good agreements with MD simulations. Moreover, it is seen that the sensitivity of natural frequency to the nonlocal parameter becomes more significant when the number of layers or interlayer shear modulus increases as well as the aspect ratio or width of sheets decreases.

Underwater adhesion of abalone: The role of van der Waals and capillary forces

The ability of scanning force microscopy to detect minute forces at high spatial resolution offers the possibility to systematically map the ever-present van der Waals (VDW) forces between probe and sample. Rigorously based on macroscopic quantum field theory the presented analysis involves an ab initio calculation of VDW forces in close relation to the experimental situation. It is shown that VDW forces directly reflect surface dielectric properties of the sample under investigation. Highly polar immersion liquids present between probe and sample are shown to significantly reduce VDW forces in magnitude. Additionally, some immersion media are found to cause a transition from attractive to repulsive interactions. The lateral resolution of VDW microscopy is estimated to be in the sub-100 nm range. First experimental aspects concerning VDW force detection are presented.