Vibration Of Single-layered Graphene Sheets Research Articles

Determination of vibration of single-layered graphene sheets using nonlocal modified couple stress theory

The free vibration of single-layered graphene sheet (SLGS) has been studied by nonlocal modified couple stress theory (NMCS), analytically. Governing equation of motion for SLGS is obtained via thin plate theory in conjunction with Hamilton’s principle for two cases: (1) using nonlocal parameter only for stress tensor, (2) using nonlocal parameter for both stress and couple stress tensors. Navier’s approach has been used to solve the governing equations for simply supported boundary conditions. It is found that the frequency ratios decrease with increasing nonlocal parameter and also with enhancing vibration modes, but increase with raising length scale parameter. The nonlocal and length scale parameters are more prominent in higher vibration modes. The obtained results have been compared with the previous studies obtained by using classical plate theory, the modified couple stress theory and nonlocal elasticity theory, separately.

Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field

Vibrations of single-layered graphene sheets subjected to a longitudinal magnetic field are considered. The Winkler-type and Pasternak-type foundation models are employed to reproduce the surrounding elastic medium. The governing equation is based on the modified couple stress theory and Kirchhoff–Love hypotheses. The effect of the magnetic field is taken into account due to the Lorentz force deriving from Maxwell’s equations. The developed approach is based on applying the Ritz method. The proposed method is tested by a comparison with results from the existing literature. The numerical calculations are performed for different boundary conditions, including the mixed ones. The influence of the material length scale parameter, the elastic foundation parameters, the magnetic parameter and the boundary conditions on vibration frequencies is studied. It is observed that an increase of the magnetic parameter, as well as the elastic foundation parameters, brings results closer to the classical plate theory results. Furthermore, the current study can be applied to the design of microplates and nanoplates and their optimal usage.

How does porosity affect the free vibration of single-layered graphene sheets?

This paper aims to investigate the influence of porosity and length size on the free vibration of single-layered graphene sheets (SLGSs). Frequency analysis is performed using a finite element based molecular structural mechanics (MSM) approach mimicking the SLGSs as frame-like structures constructed out of the beam elements. Defining a porous unit cell, 320 SLGSs with different arrangements and values of porosities and various length sizes ranging from 4 to 32 nm are considered. Results reveal that increasing porosity as well as length size both decrease the natural frequencies of SLGSs, significantly. To improve the applicability of the results, a nonlocal small scale parameter introduced by the analytical solutions for vibration of nanoplates in the literature is calibrated in such a way that the obtained frequencies by MSM match the analytical solutions based on the nonlocal theory of elasticity. Both neural network and genetic programming processes are successfully implemented for the calibration. The proposed calibrated parameter can be easily applied to evaluate the natural frequencies of SLGSs for certain values of porosities and length sizes.

Surface and Nonlocal Effects on Coupled In-Plane Shear Buckling and Vibration of Single-Layered Graphene Sheets Resting on Elastic Media and Thermal Environments using DQM

AbstractIn this article, surface and nonlocal effects are explored in the analysis of buckling and vibration in rectangular single-layered graphene sheets embedded in elastic media and subjected to coupled in-plane loadings and thermal conditions. The small-scale and surface effects are taken into account using the Eringen's nonlocal elasticity and Gurtin-Murdoch's theory, respectively. Using the principle of virtual work, the governing equations considering small-scale are derived for the nanoplate bulk and surface. The differential quadrature method (DQM) is utilized for the solution of the relevant problems and the results are validated against Navier's solutions. The impacts of the nonlocal parameter, Winkler and shear elastic moduli, temperature rise, boundary conditions, and the in-plane biaxial, uniaxial, and shear loadings on the surface effects of buckling and vibration are investigated. Numerical results show that increasing nonlocal parameter leads to enhanced surface effects on both buckling and vibration. This is in contrast to those reported elsewhere. Moreover, increasing in-plane loads are observed to enhance surface effects on vibration. On the other hand, the nonlocal parameter is observed to have more pronounced effects on shear buckling and vibration of plates subjected to coupled in-plane shear loads than those subjected to biaxial and uniaxial loads. This is while surface effects have greater impacts on biaxial buckling and vibration of nanoplates than on shear buckling and vibration.

Effects of physical boundary conditions on the transverse vibration of single-layer graphene sheets

The effects of various approaches for a comprehensive application of boundary conditions on the molecular dynamics of graphene nanosheets were studied in this paper. Fixing more than two rows of carbon atoms was tested for satisfaction of clamped boundary condition in dynamics problems, and it was demonstrated that a completely different view should be taken for clamped boundary conditions. To do this, through the frequency domain decomposition approach, operational modal analysis has been developed to carry out the Laboratory of Nanometric Operational Modal Analysis on a molecular dynamics platform. The theory of the mentioned approach was introduced, and some comparisons were made with experimental works. The modeling results have shown that for graphene sheets with simply supported edges, fixing two or more rows leads to the same response as fixing one row. For clamped edges, the use of a flexible base as a substrate satisfies the boundary condition with the best possible. At the end, as an example, it has been demonstrated that the second and third natural vibration frequencies increase with the increase in aspect ratio, while the first frequency remains unchanged.

Nonlocal mass-nanosensor model based on the damped vibration of single-layer graphene sheet influenced by in-plane magnetic field

Nano-materials such as graphene sheets have a great opportunity to be applied in development of a new generation of nanomechanical sensors and devices due to their unique physical properties. Based on the nonlocal continuum theory and vibration analysis, the single-layered graphene sheet with attached nanoparticles affected by in-plane magnetic field is proposed as a new type of the mass-nanosensor. The nonlocal Kirchhoff-Love plate theory is adopted to describe mechanical behavior of single-layered graphene sheet as an orthotropic nanoplate. The equation of motion of a simply supported orthotropic nanoplate is derived, where the influence of Lorentz magnetic force is introduced through classical Maxwell׳s equations. Complex natural frequencies, damped frequency shifts and relative shift of damping ratio for nanoplate with attached nanoparticles are obtained in the explicit form. The influences of the nonlocal and magnetic field parameter, different mass weights and positions of attached nanoparticles and damping coefficients on the relative damped frequency shift and relative shift of damping ratio are examined. The presented results can be useful in the analysis and design of nanosensors applied in the presence of strong magnetic field. Our results show that magnetic field could be successfully used to improve sensibility performances of nanomechanical sensors.

Nonlocal continuum model for vibration of single-layered graphene sheets based on the element-free kp-Ritz method

In this paper, an implementation of the kp-Ritz method with the nonlocal continuum model as an element-free computational framework has been performed to investigate the free vibration behavior of a single-layered graphene sheet (SLGS). The nonlocal continuum model, which combines the Eringen nonlocal constitutive equation with the classical plate theory, has the ability to take the small scale effect into account. The study has shown that the element-free kp-Ritz method is an efficient approach for solving nonlocal continuum model for the SLGS vibration. The accuracy of the kp-Ritz method has been validated through comparison with the MD computation. The values of nonlocal parameter used in the study have been derived by matching the frequency of nonlocal continuum model with that of the MD model for different sizes and boundary conditions. The simulated results have illustrated that the value of nonlocal parameter depends on the sizes and boundary conditions of SLGS.

Nanoscale mass sensing based on vibration of single-layered graphene sheet in thermal environments

Based on vibration analysis, single-layered graphene sheet (SLGS) with multiple attached nanoparticles is developed as nanoscale mass sensor in thermal environments. Graphene sensors are assumed to be in simply-supported configuration. Based on the nonlocal plate theory which incorporates size effects into the classical theory, closed-form expressions for the frequencies and relative frequency shifts of SLGS-based mass sensor are derived using the Galerkin method. The suggested model is justified by a good agreement between the results given by the present model and available data in literature. The effects of temperature difference, nonlocal parameter, the location of the nanoparticle and the number of nanoparticles on the relative frequency shift of the mass sensor are also elucidated. The obtained results show that the sensitivity of the SLGS-based mass sensor increases with increasing temperature difference. Open image in new window

Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium

In the present paper, the sinusoidal shear deformation plate theory (SDPT) is reformulated using the nonlocal differential constitutive relations of Eringen to analyze the bending and vibration of the nanoplates, such as single-layered graphene sheets, resting on two-parameter elastic foundations. The present SDPT is compared with other plate theories. The nanoplates are assumed to be subjected to mechanical and thermal loads. The equations of motion of the nonlocal model are derived including the plate foundation interaction and thermal effects. The governing equations are solved analytically for various boundary conditions. Nonlocal theory is employed to bring out the effect of the nonlocal parameter on the bending and natural frequencies of the nanoplates. The influences of nonlocal parameter, side-to-thickness ratio and elastic foundation moduli on the displacements and vibration frequencies are investigated.

Thermal Vibrations of Single-Layered Graphene Sheets by Molecular Dynamics

Dynamics problems in the thermal vibration of single-layered graphene sheets (SLGSs) are investigated using molecular dynamics (MD) method based on the Brenner's second-generation reactive empirical bond order (REBO) potential. The in plane stiffness and Poisson ratio of SLGSs are calculated by stretching SLGSs. The effective thickness of SLGSs is obtained by MD simulations for the thermal vibration of SLGSs through the natural frequency. The natural frequencies for SLGSs of different sizes with initial stress in different temperatures are calculated through MD. The thin plate theory can predict the MD results very well in a certain range of strain. For the nonlinear relation between stress and strain when the strain is very large, the deviation between the MD results and plate theory becomes larger when the strain increases. The difference between the plate theory and the MD results becomes more and more obvious, when the size of graphene sheet is very small.

Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory

Free vibration of single-layered graphene sheet (SLGS) resting on an elastic matrix as Pasternak foundation model is investigated by using the modified couple stress theory. Governing equation of motion for SLGS is obtained via thin plate theory in conjunction with Hamilton’s principle. All edges simply supported boundary condition is considered. Analytical solution of the resulting equation is obtained via Fourier series approach. Effects of the material length scale parameter and elastic matrix parameters on vibration frequencies of SLGS are investigated. The influence of the mode numbers on frequencies for two-different matrix parameters and aspect ratio of graphene sheet are also studied. Numerical results reveal that the frequency values increase significantly with the increase of the material length scale parameter. It has been shown that scale effects are quite significant on frequencies especially when length and width of the SLGS is smaller and in higher modes of vibration and need to be included in the mechanical modeling of SLGS.

Stochastically driven vibrations of single-layered graphene sheets

Thermal vibration of single-layered graphene sheets (SLGSs) is investigated using plate model together with the law of equi-partition of energy and the molecular dynamics (MD) method based on the condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies (COMPASS) force field. The in-plane stiffness and Poisson ratio of SLGSs are calculated by stretching SLGSs. The effective thickness of SLGSs is obtained by the MD simulations for the thermal vibration of SLGSs through the natural frequency. The root-mean-squared (RMS) amplitudes for SLGSs of differing temperatures and boundary conditions are calculated by the MD, and are compared with the results calculated by the thin plate model together with the law of equi-partition of energy. At the center of SLGSs, the thin plate theory can predict the MD results reasonably well. For the difference of bonding structure of the edge atoms, the deviation between the MD results and plate theory becomes more readily apparent near the edges of SLGSs.

Nonlinear membrane model for large amplitude vibration of single layer graphenesheets

The nonlinear vibrational properties of single layer graphene sheets (SLGSs) areinvestigated using a membrane model. The nonlinear equation of motion is considered forthe SLGSs by including the effects of stretching due to large amplitudes. The equation ofmotion is numerically solved utilizing the finite difference method for SLGSs with differentinitial and boundary conditions, sizes and pretensions. It is concluded that the nonlinearfundamental frequency of SLGSs increases by increasing the pretension and initial velocity.In addition, it is observed that an increase in the pretension weakens the effects ofthe initial velocity on the fundamental frequency, such that the fundamentalfrequency approximately becomes independent of the initial velocity. This is animportant feature of the vibrating systems consisting of SLGSs which are used inthe nano-electromechanical systems (NEMS), where resonators with a specificfundamental frequency and independent of the initial velocity are of interest.

Transverse vibration of single-layer graphene sheets

We investigate the vibrational properties of zigzag and armchair single-layer graphene sheets (SLGSs) using the molecular mechanics (MM) approach. The natural frequencies of vibration and their associated intrinsic vibration modes are obtained. Vibrational analysis is performed with different chirality and boundary conditions. The simulations are carried out for three types of zigzag and armchair SLGS. The universal force field potential is used for the MM approach. The first four natural frequencies are obtained for increasing lengths. The results indicate that the natural frequencies decrease as the length increases. The results follow similar trends with results of previous studies for SLGS using a continuum structural mechanics approach. These results have shown the applicability of SLGSs as electromechanical resonators.

Nonlocal plate model for free vibrations of single-layered graphene sheets

Vibration analysis of single-layered graphene sheets (SLGSs) is investigated using nonlocal continuum plate model. To this end, Eringens's nonlocal elasticity equations are incorporated into the classical Mindlin plate theory for vibrations of rectangular nanoplates. In contrast to the classical model, the nonlocal model developed in this study has the capability to evaluate the natural frequencies of the graphene sheets with considering the size-effects on the vibrational characteristics of them. Solutions for frequencies of the free vibration of simply-supported and clamped SLGSs are computed using generalized differential quadrature (GDQ) method. Then, molecular dynamics (MD) simulations for the free vibration of various SLGSs with different values of side length and chirality are employed, the results of which are matched with the nonlocal model ones to derive the appropriate values of the nonlocal parameter relevant to each boundary condition. It is found that the value of the nonlocal parameter is independent of the magnitude of the geometrical variables of the system.