Nanoscale vibration characterization of multi-layered graphene sheets embedded in an elastic medium

Nanoscale vibration characteristics of multi-layered graphene sheets

Nonlocal vibration and buckling analysis of single and multi-layered graphene sheets using finite strip method including van der Waals effects

A stacked plate model for the vibration of multi-layered graphene sheets (MLGSs),in which the van der Waals (vdW) interaction between layers is described byan explicit formula, is presented. Explicit formulae are derived for predictingthe natural frequencies of double- and triple-layered graphene sheets, and theyclearly indicate the effect of vdW interaction on the natural frequencies. Thenatural frequencies are calculated for various numbers of layered graphene sheets,and the results show that the vdW interaction has no influence on the lowestnatural frequency (classical frequency) of an MLGS but plays a significant role inall higher natural frequencies (resonant frequencies) for a given combination ofm and n. The vibration modes that are associated with the classical frequencies for each sheet of anMLGS are identical. In contrast, the vibration modes that are associated withthe resonant frequencies are non-identical and give various vibration patterns,which indicates that MLGSs are highly suited to use as high frequency resonators.

Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium

In this paper, nanoscale vibrational analysis of a multilayered graphene sheet embedded in an elastic medium is investigated and the corresponding resonant modes and frequencies are determined. It is known that the elastic moduli of a graphene sheet in two orientations x, y are different, so the graphene sheet is assumed to be a general form of an orthotropic plate. The orthotropic sheets stacking at the top of each other bond with carbon-carbon van der Waals forces, also the whole multi-layered graphene sheet is influenced by polymer-carbon van der Waals forces from the surrounding elastic medium.

In this paper, it is tried to find an approximate single layer equivalent for multi-layer graphene sheets based on third order non-local elasticity theory. The plates are embedded in two parameter Winkler-Pasternak elastic foundation, and also the thermal effects are considered. A uniform transverse load is imposed on the plates. Applying the non-local theory of Eringen based on third order shear deformation theory and considering the van der Waals interaction between the layers, the governing equations are derived for a multi-layer graphene sheet. The governing equations for single layer graphene sheet are obtained by eliminating the van der Waals interaction. In this study, two different methods are applied to solve the governing equations. First, the results are obtained applying the differential quadrature method (DQM), which is a numerical method, and then a new semi-analytical polynomial method (SAPM) is presented. The results from DQM and SAPM are compared and it is concluded that the SAPM results are satisfactorily accurate in comparison with DQM. Since analyzing a multi-layer graphene sheet needs a time-consuming computational process, it is investigated to find an appropriate thickness for a single layer sheet to equalize the maximum deflections of multi-layer and single layer sheets. It is concluded that by considering a constant value of the van der Waal interaction between the layers, the maximum deflections of multi and single layer sheets are equal in a specific thickness of the single layer sheet.

In this study, a multiplate shear model is developed for dynamic analysis of multilayer graphene sheets with arbitrary shapes considering the interlayer shear effect. By utilizing the model, then some free-vibration analysis is presented. According to the experimental results, the weak interlayer van der Waals interaction cannot maintain the integrity of carbon atoms in adjacent layers. Therefore, it is required that the interlayer shear effect is accounted to study multilayer graphene mechanical behavior. The governing differential equation of motion is derived for the multilayer graphene sheets utilizing a variational approach based on the Kirchhoff plate model. The essential and natural boundary conditions are also obtained at both the smooth periphery parts of the multilayer graphene sheets and the possible sharp corners. By considering cantilever and simply supported multilayer rectangular graphene sheets as two case studies, the results for the free-vibration analysis are presented based on the developed model, and these results are compared with those of molecular dynamics simulations as some sort of verification. These results show that when the layers number increases, the natural frequency also increases up to a specific number, and afterward the influence of layers number on the natural frequency significantly decreases. Moreover, the natural frequency decreases with increase in the sheet aspect ratio up to a specific value, then the changes in the aspect ratio have no considerable effect in the natural frequency.

Nanoscale vibration analysis of embedded multi-layered graphene sheets under various boundary conditions

The vibration analysis of multilayered graphene sheets (MLGSs) using a continuum model is reported in this paper. An explicit formula is derived to predict the van der Waals (vdW) interaction between any two sheets of a MLGS. Based on the derived formula, a continuum-plate model is developed for the vibration of MLGSs. Our investigation indicates that the lowest natural frequency (classical natural frequency) of a MLGS for a given combination of $m$ and $n$ is independent of the vdW interaction, but that all of the other higher natural frequencies (resonant frequencies) are significantly dependent on this interaction. The mode shapes that are associated with the natural frequencies are investigated for double-layered and ten-layered graphene sheets. We find that the vibration modes that are associated with the classical natural frequency of all the sheets are in the same direction and have the same amplitude, whereas the vibration modes of the sheets that are associated with the resonant frequencies are different due to the influence of the vdW interaction. Thus various resonance modes can be obtained by varying the number of layers of a MLGS.

Nanoparticle mass detection by single and multilayer graphene sheets: Theory and simulations

Elastic buckling behaviour of multi-layered graphene sheets is rigorously investigated. Van der Waals forces are modelled, to a first order approximation, as linear physical springs which connect the nodes between the layers. Critical buckling loads and their associated modes are established and analyzed under different boundary conditions, aspect ratios and compressive loading ratios in the case of graphene sheets compressed in two perpendicular directions. Various practically possible loading configurations are examined and their effect on buckling characteristics is assessed. To model more accurately the buckling behaviour of multi-layered graphene sheets, a physically more representative and realistic mixed boundary support concept is proposed and applied. For the fundamental buckling mode under mixed boundary support, the layers with different boundary supports deform similarly but non-identically, leading to resultant van der Waals bonding forces between the layers which in turn affect critical buckling load. Results are compared with existing known solutions to illustrate the excellent numerical accuracy of the proposed modelling approach. The buckling characteristics of graphene sheets presented in this paper form a comprehensive and wholesome study which can be used as potential structural design guideline when graphene sheets are employed for nano-scale sensing and actuation applications such as nano-electro-mechanical systems.

In this paper, the small scale effect on the buckling analysis of bi-axially compressed orthotropic Single-Layered Graphene Sheets (SLGS) supported on elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and supporting elastic medium. Using the principle of virtual work the governing differential equations are derived for rectangular orthotropic graphene sheets supported on elastic medium. Solutions for buckling loads for various boundary conditions are computed using Differential Quadrature Method (DQM). Parametric study has been performed to investigate the dependence of small scale effect on various graphene sheet parameters. It is observed that the nonlocal effect is significant in graphene sheets supported on elastic medium and has a decreasing effect on the buckling loads.

ABSTRACTIn this article, the small-scale effect on the vibration behavior of orthotropic single-layered graphene sheets is studied based on the nonlocal Reddy's plate theory embedded in elastic medium considering initial shear stress. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. To simulate the interaction between the graphene sheet and surrounding elastic medium we used both Winkler-type and Pasternak-type foundation models. The effects of initial shear stress and surrounding elastic medium and boundary conditions on the vibration analysis of orthotropic single-layered graphene sheets are studied considering five different boundary conditions. Numerical approach of the obtained equation is derived by differential quadrature method. Effects of shear stress, nonlocal parameter, size of the graphene sheets, stiffness of surrounding elastic medium, and boundary conditions on vibration frequency rate are investigated. The results reveal that as the stiffness of the surrounding elastic medium increases, the nonlocal effect decreases. Further, the nonlocal effect increases as the size of the graphene sheet is decreased. It is also found that the frequency ratios decrease with an increase in vibration modes.

Application of modified couple-stress theory to stability and free vibration analysis of single and multi-layered graphene sheets

This study focuses on the van der Waals (vdW) interactions and oscillatory behaviour of nested spherical fullerenes (carbon onions) in the vicinity of a single-layer graphene (SLG) sheet. The carbon onions are of Ih symmetries and the graphene sheet is modelled as a fully constrained flat surface. Employing the continuum approximation along with the 6–12 Lennard-Jones (LJ) potential function, explicit analytical expressions are determined to calculate the vdW potential energy and interaction force. The equation of motion is solved numerically based on the actual force distribution to attain the displacement and velocity of the carbon onion. Using the conservation of mechanical energy principle, a semi-analytical expression is also derived to accurately evaluate the oscillation frequency. Numerical results are presented to examine the influences of size of carbon onion and initial conditions (initial separation distance and initial velocity) on the operating frequency of carbon onion–SLG sheet oscillators. It is shown that carbon onion executes oscillatory motion above the graphene sheet with frequencies in the gigahertz (GHz) range. It is further observed that smaller structures of carbon onions produce greater frequencies. We comment that the presented results in this study would contribute to the development of new generation of nano-oscillators.