Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium

Abstract

Based on nonlocal theory, this article discusses vibration of CDVGS¹ systems. The properties of each single layer graphene sheet (SLGS) are assumed to be orthotropic and viscoelastic. The two SLGSs are simply supported and coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak layer. This model is aimed at representing dynamic interactions in nanocomposite materials with dissipation effect. By considering the Kirchhoff plate theory and Kelvin–Voigt model, the governing equation is derived using Hamilton's principle. The equation is solved analytically to obtain the complex natural frequency. The parametric study is thoroughly performed, concentrating on the series effects of viscoelastic damping structure, aspect ratio, visco-Pasternak medium, and mode number. In this system, in-phase (IPV) and out-of-phase (OPV) vibrations are investigated. The numerical results of this article show a perfect correspondence with those of the previous researches.

Introduction

An SLGS is a two dimensional Nano-structure with excellent electronic and mechanical properties made up of a single layer of carbon atoms arranged in a hexagonal structure [1]. Graphene as a Nanoplate (NP) would be one of the prominent new materials for the next generation Nano-electronic devices. Similar to micro-plates, a technological extension of the single NP to the complex-NP-systems such as DLGSs, that may find applications in nano optomechanical systems (NOMS) and Nano-electromechanical systems (NEMS). High frequency resonators, mass and chemical sensors, semiconductor devices and vibration isolation systems are cases of this application [2], [3], [4], [5], [6], [7], [8], [9], [10]. Performance and control of experiments on the nano-scale level is a very difficult and expensive duty. This is one of the reasons why many researchers emphasis their investigation on the development of theoretical models.

There are three main methods that have been developed to simulate the dynamical behavior of nanostructures such as atomistic, atomistic–continuum and continuum mechanics methods. In continuum modeling, double-NP is viewed as a continuous system. Differential equations of motion were derived for different plate theories by using Eringen's nonlocal (NL) constitutive relation [11], [12], [13]. Choice of NL parameter in dimension of length is crucial to ensure the validity of NL model. Ansari et al. [12] analyzed vibration problem of SLGS utilizing the Mindlin plate theory. Solutions for natural frequencies were computed by General Differential quadrature (GDQ) method and validate by molecular dynamics (MD).

Two coupled NPs is one of the significant nano-structure systems which are very important for the design of nano-dynamical absorbers and nano resonators. The vibration of bonded double-plate system, which can be modeled as composite structures, is vital for both theoretical and practical reasons such as NOMS. Murmu and Adhikari [14] analyzed two elastic NPs coupled by an elastic layer represented by vertical springs. Expressions for natural frequencies were obtained by using nonlocal elasticity and Navier solutions. Wave propagation analysis in elastically connected double-NP system was performed by Wang et al. [15]. Also, recently Karličić et al. [16] analyzed Vibration and buckling of a multi-nanoplate system (MNPS) embedded in the Winkler elastic medium. Analytical solutions of dimensionless natural frequencies and critical buckling load were derived for a different number of simply supported nanoplates in MNPS by using Navier's and trigonometric method.

The fundamental characteristic of a damped Nano mechanical system is dissipation of total energy [17]. Two types of damping are more presented: (i) internal damping or structural damping which comes from constitutive relations and (ii) external damping from the influence of the surrounding media. In most of the articles, GSs are modeled as elastic structures while they representation viscoelastic structural damping. In the recent paper by Su et al. [18], viscoelastic properties of the graphene oxide NP have been proven by obtaining the hysteresis loops in tensile test. Hence, for more accurate modeling of dynamical behavior of GSs, it is necessary to take into account.

Any complex structures in contact with polymeric layers exhibit viscoelastic behavior during both static and dynamic loading regimes. Instances of composite structures with viscoelastic properties in nanoscale systems are recently discovered graphene–polymer composites [19], [20]. Croy et al. [21] offered NEMS graphene resonators with nonlinear damping effects, by using the continuum mechanical model. Eichler et al. [4] investigated the damping in mechanical resonators based on graphene sheets. They studied the system where damping depends on the amplitude of motion which was described by nonlinear damping force.

Arani and Roudbari [22] introduced visco-Pasternak foundation as a smart medium can be used in boron nitride nanotube system to control the stability and vibration. ViscoPasternak layer can denote specified polymer matrix or in case of optomechanical devices, springs (winkler and Pasternak modules) may represent van der Waals forces and dampers may indicate some external damping effects. Ghorbanpour Arani et al. [23] based on nonlocal orthotropic plate theory, studied vibration analysis of the coupled system of DLGS embedded in a visco-Pasternak foundation. The conclusions indicate that the frequency ratio of the system is higher than that of the SLGS. Vibration analysis of a viscoelastic orthotropic nanoplate resting on the viscoelastic foundation was implemented by Pouresmaeeli et al. [24], the authors found the complex eigenvalues in the closed analytical form. Besides, influences of the NL parameter, structural dumping and coefficient of foundation on the complex frequency were investigated. Such given continuum based model, that is a combination of space nonlocality and time dependent viscoelastic behavior of nanostructure, is more realistic due to the obvious presence of dissipation effects at all scales and it is appropriate to obtain closed form solutions.

In the present paper, we study the free transverse vibration of a nonlocal viscoelastic double graphene sheets (NVDGS) coupled with viscoPasternak layer. The discussions are limited to case of all boundary conditions are simply support. Governing Partial Differential Equations (PDEs) are obtained using Hamilton principle and nonlocal elasticity theory and the exact solution for vibration of system is determined. Dissipation effects due to the internal damping of nanoplates and external damping of a viscoPasternak layer are taken into account in our model of NVDGS. To justify the presented method, our results are validated with the corresponding researches. In the validation study, we obtained complex natural frequency which the real part is damped natural frequency and the imaginary part is damping ratio. Further, effects of NL parameter, aspect ratio, damping from viscoelastic NPs and layer on complex eigenvalues for the free transverse vibration of NVDGS are presented throughout numerical examples.