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

Highlights

• MSM approach can accurately implemented for vibrational analysis of porous SLGSs.

• Increasing porosities and length sizes both decreases the natural frequencies.

• A nonlocal small scale parameter is calibrated to simply obtain natural frequencies.

Abstract

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.

Introduction

Graphene seems to be an appropriate candidate in the future of nanotechnology due to its exceptional properties in comparison to its carbon-based rivals such as fullerenes and carbon nanotubes [1]. Numerous reports on application of graphene in medicine, sensors and actuators, nano composites, nano separation as well as energy storage devices reveal the attraction and importance of this special nano material [[2], [3], [4], [5], [6]]. Having a thickness equals to one layer of carbon atoms, the single-layered graphene sheets (SLGSs) as an all surface nanostructure has shown significant mechanical, electrical, and thermal properties. For instance, Young's modulus and tensile strength of SLGSs are reported about 1 TPa and 100 GPa, respectively [7]. As the basic component of many newborn devices in nanotechnology, behavior of SLGSs under various load conditions such as vibration [8,9], buckling [10,11], bending [12,13] and fatigue [14] have been comprehensively studied [15,16].

Parallel to increasing the attraction to nanoscaled structures, atomistic computational methods have been developed to investigate and predict the responses of these structures, thanks to the cost advantages and simplicity in comparison to experimental efforts. Unlike continuum methods, atomistic approaches take into account the discrete nature of nanostructures to increase the reality and accuracy. The well-known computational physics methods i.e. ab initio, Monte Carlo and molecular dynamics simulations in spite of their precision impose huge computational expenses [[17], [18], [19]]. On the other side, to find out the mechanical behavior of nanostructure, performing the continuum mechanics based on the elasticity theory does not provide the required accuracy [18,20]. Accordingly, some researches focused on finding a class of midrange solutions called semi-atomistic methods to not only reduce the computational cost by implementing the continuum mechanics fundamentals but also provide the realistic point of view of atomistic approaches.

Molecular structural mechanics (MSM) approach was first introduced by Li and Chu as a semi-atomistic method [21]. The preliminary concept of this approach is to consider a nanostructure like SLGSs as a frame-like structure whereas the chemical bonds between atoms are replaced by common structural elements like beams and atoms act as concentrated masses in joints. This approach has been widely addressed in the literature, authenticating its capability for predicting the mechanical behavior of the SLGSs. Recently, some modifications have been proposed to improve the applicability of MSM approach for more complicated studies [22,23].

Implementing MSM, Sakhaee-Pour [24] investigated the mechanical properties of SLGSs and obtained Young's and shear modulus as well as Poisson's ratio. Results correlated with those obtained by atomistic approaches. Scarpa et al. [25] used truss elements to find the SLGSs mechanical properties. They provided the mechanism of mechanical deformation of SLGSs under small strain deformations and pure shear loading conditions. Also, the pure bending condition on the SLGSs was investigated in the similar study by Lu et al. [26]. In the mentioned studies SLGSs considered as ideal two dimensional hexagonal lattice with no defect, while some inevitable defects like Stone-Wales defects are reported that play a vital role in the mechanical properties specially in the failure mechanism [27]. Wang et al. [28] investigated the mechanical properties of the graphene nanofilms included Stone-Wales defects. In their study, the influence of number and types of Stone-Wales defects, the distance between two defects and the position of defects were studied by MSM approach.

Many researches approved the excellent performance of SLGSs as ultra-sensitive sensors. Generally, the graphene-based sensing can be categorized as chemical, electromechanical, photoelectrical, electrical, magnetic and finally mechanical sensors [29]. Among mechanical ones, resonant sensors have been received many attractions due to their high value of sensitivity. The fundamental rule is to measure the goal quantity by consideration of the frequency shift. The possibility of implementing SLGSs for strain, pressure, and dust resonant sensors have been investigated by Sakhaee-pour et al. [[30], [31], [32]] based on MSM approach. The results showed that SLGSs can successfully be used in resonant sensing applications. It was reported that the natural frequencies of SLGSs were independent of the chirality while it was related to the aspect ratio of the sheets. They also used different edge conditions to compare the accuracy of results in comparison with the atomistic methods. It was shown that MSM could predict within 3% different with respect to the atomistic simulation. A finite element study using MSM approach was performed by Mahmoudinezhad and Ansari [33] for vibration of circular and square SLGSs. They compared the results of MSM method with molecular dynamic simulation and reported that the results of the MSM approach for the vibrational characteristic of the SLGSs were correlated with the molecular dynamic ones. Their report demonstrated that the values of fundamental frequencies getting decreased by increasing the SLGSs size.

One of the other applications of SLGSs is in filtrations and desalinations because they are chemically and mechanically stable and flexible [34,35]. To achieve the good performance for filtration and desalination applications, the perfect SLGSs need to have some controlled defects and vacancies results in nanoporous SLGSs. In spite of wide investigations on the mechanical properties of defected SLGSs [[36], [37], [38]], few studies on the porous SLGSs have been carried out in the literature [39]. Therefore, in the present study, free vibrational behavior of porous SLGSs based on MSM approach is investigated. Besides, a comparison with analytical solution of the vibration of nanoplate based on the nonlocal theory of elasticity is performed for possibility of presenting a calibrated nonlocal small scale parameter.