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.     

Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory

Higher order shear deformation theory (HSDT) is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. The developed equations of motion have been applied to study buckling characteristics of nanoplates such as graphene sheets. Navier's approach has been used to solve the governing equations for all edges simply supported boundary conditions. Analytical solutions for critical buckling loads of the graphene sheets are presented. Nonlocal elasticity theories are employed to bring out the small scale effect on the critical buckling load of graphene sheets. Effects of (i) nonlocal parameter, (ii) length, (iii) thickness of the graphene sheets and (iv) higher order shear deformation theory on the critical buckling load have been investigated. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories as applied to the stability analysis of nanoplates and nanoshells.     

Nonlocal vibration of coupled DLGS systems embedded on Visco-Pasternak foundation

In this paper, vibration analysis of the coupled system of double-layered graphene sheets (CS-DLGSs) embedded in a Visco-Pasternak foundation is carried out using the nonlocal elasticity theory of orthotropic plate. The two DLGSs are coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak foundation. Considering the Von Kármán nonlinear strain-displacement-relations, the motion equations are derived using the Hamilton's principle. Differential quadrature method (DQM) is applied to obtain the frequency ratio for various boundary conditions. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, aspect ratio, graphene sheet's size, boundary conditions and the elastic and viscoelastic medium coefficients on the frequency ratio of CS-DLGSs. In this coupled system, two case of DLGSs vibration are investigated and compared with each other: (1) In-phase vibration (2) Out-of-phase vibration. The results indicate that the frequency ratio of the CS-DLGSs is more than the single-layered graphene sheet (SLGS). The results are in good agreement with the previous researches.     

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.     

Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models

In the present work, vibration analysis of multilayered graphene sheets embedded in polymer matrix has been carried out employing nonlocal continuum mechanics. Governing equations have been derived using the principle of virtual work. It has been shown that nonlocal effect is quite significant and needs to be included in the continuum model of graphene sheet.     

Density functional theory study on the adsorption of alkali metal ions with pristine and defected graphene sheet

The graphene-based materials along with the adsorption of alkali metal ions are suitable for energy conversion and storage applications. Hence in the present work, we have investigated the structural and electronic properties of pristine and defected graphene sheet upon the adsorption of alkali metal ions (Li⁺, Na⁺, and K⁺) using density functional theory (DFT) calculations. The presence of vacancies or vacancy defects enhances the adsorption of alkali ions than the pristine sheet. From the obtained results, it is found that the adsorption energy of Li⁺ on the vacancies defected graphene sheet is higher (3.05?eV) than the pristine (2.41?eV) and Stone–Wales (2.50?eV) defected sheets. Moreover, the pore radius of the pristine and defected graphene sheets are less affected by metal ions adsorption. The increase in energy gap upon the adsorption of metal ions is found to be high in the vacancy defected graphene than that of other sheets. The metal ions adsorption in the defective vacancy sheets has high charge transfer from metal ions to the graphene sheet. The bonding characteristic between the metal ions and graphene sheet are analysed using QTAIM analysis. The influence of alkali ions on the electronic properties of the graphene sheet is examined from the Total Density of States (TDOS) and Partial Density of States (PDOS).     

Fracture analysis of monolayer graphene sheets with double vacancy defects via MD simulation

Carbon nanostructures such as carbon nanotubes (CNTs) and graphene sheets have attracted great attention due to their exceptionally high strength and elastic strain. These extraordinary mechanical properties, however, can be affected by the presence of defects in their structures. When a material contains multiple defects, it is expected that the stress concentration of them superposes if the separation distances of the defects are low, which causes a more reduction of the strength. On the other hand, it is believed that if the defects are far enough such that their affected areas are distinct, their behavior is similar to a material with single defect. In this article, molecular dynamics (MD) is used to explore the influence of separation distance of double vacancy defects on the mechanical properties of single-layered graphene sheets (SLGSs). To this end, critical stress and strain of SLGSs containing double vacancy with different distances are determined and the results are compared with those of perfect SLGSs and graphene sheets with single vacancy defect. The results reveal that the ultimate strength of the SLGS with double vacancy tends to the one with a single vacancy when the separation distance becomes further. In this regard, the threshold distance beyond which double defects behave like a single one is examined. Finally, Young’s modulus of perfect, single and double vacancy defected graphene sheets with different separation distances is determined. It is concluded that this property is slightly affected by the separation distance.     

Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model

In this article, the buckling behavior of nanoscale circular plates under uniform radial compression is studied. Small-scale effect is taken into consideration. Using nonlocal elasticity theory the governing equations are derived for the circular single-layered graphene sheets (SLGS). Explicit expressions for the buckling loads are obtained for clamped and simply supported boundary conditions. It is shown that nonlocal effects play an important role in the buckling of circular nanoplates. The effects of the small scale on the buckling loads considering various parameters such as the radius of the plate and mode numbers are investigated.     

Mechanical properties of defective single-layered graphene sheets via molecular dynamics simulation

In this paper, the effects of two main types of structural defects, i.e. Stone–Wales and single vacancy, on the mechanical properties of single-layered graphene sheets (SLGSs) are investigated. To this end, molecular dynamics simulations based on the Tersoff–Brenner potential function and Nose–Hoover thermostat technique are implemented. The results obtained have revealed that the presence of defects significantly reduces the failure strain and the intrinsic strength of SLGSs, while it has a slight effect on Young’s modulus. Furthermore, the examination of loading in both armchair and zigzag directions demonstrated that SLGSs are slightly stronger in the armchair direction and defects have lower effect in this direction. Considering the fracture mechanism, the failure process of defective and perfect graphene sheets is also presented.     

Effect of triangular vacancy defect on thermal conductivity and thermal rectification in graphene nanoribbons

We investigate the thermal transport properties of armchair graphene nanoribbons (AGNRs) possessing various sizes of triangular vacancy defect within a temperature range of 200–600 K by using classical molecular dynamics simulation. The results show that the thermal conductivities of the graphene nanoribbons decrease with increasing sizes of triangular vacancy defects in both directions across the whole temperature range tested, and the presence of the defect can decrease the thermal conductivity by more than 40% as the number of removed cluster atoms is increased to 25 (1.56% for vacancy concentration) owing to the effect of phonon–defect scattering. In the meantime, we find the thermal conductivity of defective graphene nanoribbons is insensitive to the temperature change at higher vacancy concentrations. Furthermore, the dependence of temperatures and various sizes of triangular vacancy defect for the thermal rectification ration are also detected. This work implies a possible route to achieve thermal rectifier for 2D materials by defect engineering.     

Structural evolution in CVD graphene chemically oxidized by sulphuric acid

The role of sulphuric acid (H₂SO₄) in fabrication graphene oxide besides as intercalant has not been well addressed. In this work, Raman spectroscopy is used to monitor structural evolution in chemical vapor deposition (CVD) graphene chemically oxidized by dilute H₂SO₄. From the analysis of Raman spectra of oxidized graphene, we propose that oxidation first initiates at preexisting defects, and vacancy‐like defects are formed. Following is the radial growth of the vacancy, and oxidation pits appear in graphene. This assumption is further confirmed by atomic force microscope measurement. It is also found that with increase of amounts of defects, G peak is blue shift, and this is explained by defect and hole doping effect. Hole doping in graphene is much stronger at hexagon regions near the oxidation pits. This work helps in understanding the role of H₂SO₄ in fabrication graphene oxide as oxidizer as well as helps in obtaining structure information of graphene oxide. Copyright © 2015 John Wiley & Sons, Ltd.     

Geometric stability and adsorption property of hydroxyl group on graphene sheets

The stable geometrics and adsorption behaviors of hydroxyl (OH) groups on graphene sheets are investigated using the first-principles calculations. The single hydroxyl adatom has small adsorption energy and diffusion barrier on pristine graphene. The binding strength of the hydroxyl group increases with the coverage, and the aggregations of the hydroxyl groups reduce the structural bucking of graphene sheet. On the graphene with single vacancy (SV-graphene), the large trapping zones mean the adsorbed OH would be easily trapped at the vacancy site. The hydroxyl groups prefer to aggregate on graphene surfaces and form the water molecule, leaving the epoxy group on pristine graphene or oxygen dopant in SV-graphene, which is used to constitute the structural model of oxidized graphene. These results would provide us a useful reference to understand the atomic structure and adsorption property of functional groups on graphene sheets.     

First-principles study of lithium intercalated bilayer graphene

Lithium intercalated bilayer graphene has been investigated using first-principles density functional theory calculations. Results show that there exist AB and AA stacking sequences for bilayer graphene in which the latter is more favorable for the Li storage and the former will evolve into the latter with the intercalation of Li ions. The relationship between the interlayer distance of two graphene sheets and the intercalated capacity of Li ions is discussed. It is found that structural defect is identified to store Li ions more favorably than pristine bilayer graphene and an isolated C atom vacancy in bilayer graphene can capture three Li ions between two graphene sheets.     

Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory

In this article, we use shear deformable ring theory (SDRT) for the analysis of free in-plane vibration of nanorings based on nonlocal elasticity theory. The equations of motion of the nanoring are derived for the aforementioned problem by considering the small scale effect. Analytical solutions for the natural frequencies of the nanorings are presented. It is shown that the nonlocal effects play an important role in the vibration of nanorings and cannot be neglected. The effects of the small scale on the natural frequencies considering various parameters such as the radius of the nanoring, the thickness of the nanoring and mode numbers are investigated.     

Computer simulation and study of radiation defects in graphene

Computer simulation has been used to study the properties of radiation defects in a graphene sheet. Several possible stable configurations of 3D radiation defects in graphene have been simulated. Several types of absorbed species were calculated involving a single carbon atom and a dumbbell absorbed at graphene sheet surface and a dumbbell configuration at a vacancy created by radiation knocking out a carbon atom. The calculations also included the effect of structural relaxation. The defect caused by the atom being displaced from the structure converted the sheet from metallic to semiconducting.     

分子空位缺陷对环三亚甲基三硝胺含能材料几何结构、电子结构及振动特性的影响

含能材料中的微观缺陷是导致“热点”形成并相继引发爆轰的重要因素. 然而, 由于目前人们对材料内部微观缺陷的认识不足, 限制了对含能材料中“热点”形成微观机理的理解, 进而阻碍了含能材料的发展和应用. 为了洞悉含能材料内部微观缺陷特性及探索缺陷引发“热点”的形成机理, 利用第一性原理方法研究了分子空位缺陷对环三亚甲基三硝胺(RDX) 含能材料的几何结构、电子结构及振动特性的影响, 探讨了微观缺陷对初始“热点”形成的基本机理. 采用周期性模型分析了分子空位缺陷对RDX几何结构、电子能带结构、电子态密度及前线分子轨道的影响. 采用团簇模型分析了分子空位缺陷对RDX振动特性的影响. 结果发现, 分子空位缺陷的存在使其附近的N–N键变长, 分子结构变得松弛; 使导带中很多简并的能级发生分离, 电子态密度减小, 并使由N-2p和O-2p轨道形成的导带底和价带顶均向费米面方向移动, 降低了能带隙值, 增加了体系活性. 前线分子轨道及红外振动光谱的计算分析表明, 分子缺陷使最高已占分子轨道电荷主要集中在缺陷附近的分子上, 且分子中C–H键和N–N键能减弱. 这些特性表明, 分子空位缺陷的存在使体系能带隙变小, 并使缺陷附近的分子结构松弛, 电荷分布增多, 反应活性增强; 在外界能量激发下, 缺陷附近分子将变得不稳定, 分子中的C–H键或N–N键较易先发生断裂, 发生化学反应释放能量, 进而成为形成“热点”的根源.     

Explicit analytical expressions for the critical buckling stresses in a monolayer graphene sheet based on nonlocal elasticity

Explicit expressions are given to study the biaxial buckling of monolayer graphene sheets. Based upon the continuum mechanics, a plate model is adopted in which the small length scale effect is incorporated into the governing equation through the nonlocal elasticity theory of Eringen. By employing the Galerkin method, analytical expressions are derived which allow quick and accurate calculation of the critical buckling loads of monolayer graphene sheets with various boundary conditions from the static deflection under a uniformly distributed load. The effectiveness of the present study is assessed by molecular dynamics simulations as a benchmark of good accuracy.     

Effects of structural defects on the compressive buckling of boron nitride nanotubes

In this paper, we examined the buckling of perfect and defective armchair boron nitride nanotubes with three types of vacancy defects, i.e. B- and N- single vacancy defects and B–N- double vacancy defect, using molecular dynamics simulations. To this end, all systems were modeled with a Tersoff-type potential, which is able to accurately describe covalent bonding of BN systems. We applied external uniaxial compressive forces to the nanotubes in vacuum and derived the critical buckling loads and strains, at room temperature in an NVT-ensemble. Our results showed significant differences between the critical buckling strengths of pristine and defective nanotubes. The resistance to axial buckling decreased with the introduction of one vacancy defect, and the B–N- double vacancy was the most seriously damaged structure, followed by B-vacancy and N-vacancy defects. Furthermore, the B-vacancy was shown to have the most significant effect on the decrease of the critical buckling strain. This can be attributed to the excessive asymmetries and perturbations induced in the structure of the nanotube and the local deformations around the defective site around the B-vacancy, even before loading. Moreover, results show that reduction in the buckling strength of the nanotube due to the presence of more than one B-vacancy defect depends on their distribution. If the two or three defects are close to each other, they act as a single point of weakness and the critical buckling load is only slightly reduced (similar to the existence of only one vacancy defect). However, if the defects are at more distant points, the critical buckling load may experience a higher decrease. Results show that vacancy defects play a critical role in the compressive buckling performance of boron nitride nanotubes and special attention must be paid to the presence of structural defects when designing members against buckling, especially for micro- and nano-electro-mechanical systems. On the other hand, defect engineering is a great means for tailoring the buckling strength of boron nitride nanotubes, in cases where the nanotube is expected to absorb energy through compressive buckling deformation and is not designed against, but for buckling.     

On correct nonlocal generalized theories of elasticity

The paper discusses nonlocal elasticity theories among which are models of media with defect fields, gradient elasticity theories, and hybrid nonlocal elasticity theories. Gradient theories are analyzed, and their correctness properties are examined. Applied theories that satisfy the correctness conditions are developed, and known applied gradient theories are verified for the correctness properties. A new nonlocal generalized theory has been developed for which the operator of balance equations is represented as the product of the equilibrium operator of classical elasticity theory and the Helmholtz operator. It is shown that this theory is one-parameter and is the only representative of hybrid models constructed by a complete system of equations for forces and moments. Unlike classical elasticity that is free from scale parameters characterizing the internal material structure, nonlocal elasticity theories naturally incorporate these parameters. That is why they are suitable for the modeling of scale effects and find application in the solution of numerous applied problems for heterogeneous structures with developed phase interfaces where the degree of influence of scale effects depends on the density of phase boundaries. Nonlocal continuum models are especially attractive for modeling the properties of various micro/nanostructures, elastic properties of composites and structured materials with submicron- and nanosized internal structures in which effective properties are to a great extent defined by the scale effects (short-range interaction effects of cohesion and adhesion). Generalized elasticity theories even for isotropic materials contain many additional physical constants that are difficult or impossible to determine experimentally. Applied models with a small number of additional physical parameters are therefore of great interest. However, the reduction of nonlocal theories aimed at reducing the number of additional parameters is a nontrivial task and may lead to incorrect theories. The goal of this paper is to study the symmetry properties in gradient theories, to analyze the correctness of gradient theories, and to develop applied one-parameter elasticity theories.     

Effect of Point and Line Defects on Mechanical and Thermal Properties of Graphene: A Review

New materials with distinctive properties are arising and attracting the scientific community at regular intervals. Stiffness and strength are the important factors in determining stability and lifetime of any technological devices, but defects which are inevitable at the time of production can alter the structural properties of any engineering materials. Developing graphene with specific structural properties depends upon controlling these defects, either by removing or deliberately engineering atomic structure to gain or tailoring specific properties. In this article, a comprehensive review of defective graphene sheets with respect to its mechanical and thermal properties are presented and examined.