An atomistic-based continuum approach for calculation of elasticproperties of single-layered graphene sheet

Khalid I. Alzebdeh n

Department of Mechanical and Industrial Engineering, College of Engineering, Sultan Qaboos University, P.O. Box 33, Al-Khod 123, Oman

a r t i c l e i n f o

Article history:Received 4 July 2013Received in revised form10 September 2013Accepted 16 September 2013by Y.E. LozovikAvailable online 25 September 2013

Keywords:A. Graphene sheetC. NanostructureC. Atomistic-continuum modelD. Elastic properties

a b s t r a c t

The elastic deformation of a single-layer nanostructured graphene sheet is investigated using anatomistic-based continuum approach. This is achieved by equating the stored energy in a representativeunit cell for a graphene sheet at atomistic scale to the strain energy of an equivalent continuum mediumunder prescribed boundary conditions. Proper displacement-controlled (essential) boundary conditionswhich generate a uniform strain eld in the unit cell model are applied to calculate directly one elasticmodulus at a time. Three atomistic nite element models are adopted with an assumption that the forceinteraction among carbon atoms can be modeled by either spring-like or beam elements. Thus, elasticmoduli for graphene structure are determined based on the proposed modeling approach. Then, effectiveYoung's modulus and Poisson's ratio are extracted from the set of calculated elastic moduli.

Results of Young's modulus obtained by employing the different atomistic models show a goodagreement with the published theoretical and numerical predictions. However, Poisson's ratio exhibitssensitivity to the considered atomistic model. This observation is supported by a signicant variation inestimates as can be found in the literature. Furthermore, isotropic behavior of in-plane graphene sheetswas validated based on current modeling.

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1. Introduction

Recently, nanostructured graphene sheets have captured theattention of many researchers. This can be attributed to theirremarkable mechanical properties and cheap method of produc-tion as presented by Stankovich et al. [1]. In addition, character-ization of behavior of graphene facilities better understanding ofother fundamental nano-materials like Carbon nanotubes (CNTs)which are viewed as a deformed graphite sheets.

The complexity and high expense of investigating the mechanicalbehavior of graphene sheets via experiments stimulated the use ofnumerical simulation as proven tool capable of modeling nanostruc-tures with different dimensions. In this context, equivalentcontinuum-structural mechanics has been widely used to character-ize the mechanical behavior of nanostructured materials. In thisapproach, typical elements of structural mechanics such as rods,beams and shells are used to simulate the static and dynamicbehavior of monolayer graphene. The mechanical properties of suchstructural elements are derived from the equivalence between stericpotential of the carboncarbon (CC) bonds and mechanical strainenergies associated with tension, torsion and bending related to themechanical elements simulating the bonds themselves. A truss

model was proposed by Odegard et al. [2], wherein rods of differentdegrees of stiffness represent the stretching and in-plane bendingcapabilities of the CC bonds. Li and Chou [3] proposed an equivalentstructural beam capable of modeling interatomic forces of the carboncovalent bonds. They adopted a molecular structural mechanicsapproach to compute effective elastic constants of carbon nanotubes.Meo and Rossi [4] developed a nite element model based on the useof nonlinear central spring and linear torsional spring elements torepresent the modied Morse potential when simulating graphene.Cho et al. [5] carried out a molecular structural analysis to predict theelastic constants of graphite. The in-plane properties of graphite werederived by considering a single-layer graphene sheet subjected to anin-plane loading. Based on atomistic nite element approach,Shakhaee-Pour [6] investigated the elastic behavior of single-layergraphene sheets. By employing an equivalent structural beam, theelastic constants of graphene were calculated. Scrape et al. [7]proposed a truss-type model in conjunction with cellular materialmechanics theory to describe the in-plane elastic properties ofsingle-layer graphene sheets.

Analytically, some researchers [8] investigated Young's modulusof graphene and CNTs based on nanoscale continuum modeling.They employed frame elements to simulate CC bonds for whichthey obtained a closed form solution. Several others utilizedatomistic nite elements to simulate graphene sheet using linearinteratomic potential functions for bonds. Atomistic-based niteelement models have been used to analyze graphene sheets in many

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Solid State Communications

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Solid State Communications 177 (2014) 2528

recent studies in literature due to its simplicity, computational costeffectiveness and low CPU time. In contrast, atomistic models requirea signicant computational effort. Taking into account the advantagesof the atomistic models, FE incorporates actual atomistic character-istics and interatomic forces among carbon atoms while othercontinuum models fails to do so. Georgantzinos et al. [9] developeda formulation on the basis of spring element using linear interatomicpotentials to compute mechanical properties. Some other researchersincorporated non-linear springs for simulating both bond stretch andbond angle variation [10,11]. Based on an equivalent continuumapproach, Alzebdeh [12] used atomistic nite element simulation inconjunction with a beam model to evaluate elastic moduli andconstants for single-layered graphene sheets with different sizes.

In this paper, a continuum approach based on an atomisticmodeling is proposed to simulate mechanical behavior of gra-phene, in order to predict its mechanical properties. At nanoscale,we consider a unit cell representing a nite-size graphene sheetequivalent to a homogenized medium in a continuum sense. Threeatomistic nite element-based models for bond stretch and bondangle variation are adopted: (i) linear spring, (ii) nonlinear spring,and (iii) structural beam to model interatomic interactionsbetween carbon atoms, described by modied Morse potentialfunction.

2. Molecular interactions

From the viewpoint of molecular mechanics, the nanostructureof a graphene sheet is constituted by a monolayer of carbon atomsarranged periodically and uniformly in a hexagonal honeycombfashion (as shown in Fig. 1).

Their motion is governed by the molecular force eld, which isgenerated from electronnucleus and nucleusnucleus interac-tions. The total interatomic potential energy of a molecular systemis expressed as a sum of several energy terms after neglecting thenon-bonded interactions:

U UrUUU 1

where Ur stands for a bond stretch, U is for a bond angle bending,U is for dihedral angle torsion, and U is for an improper (out-of-plane) torsion. Several different potential functions for describ-ing the carboncarbon bond other than simple harmonic functionsare available [13,14]. The TersoffBrenner potential function isgenerally more accurate compared to other potential functions butit is complicated as presented by Jiang et al. [15]. Considering asingle-layered sheet, generally, the dominant parts of interatomicpotential are bond stretching and bond angle variations due totheir signicant contribution comparing to other interactions. Themodied Mores potential function in Belytschko et al. [16] issimple and therefore, will be adopted in the present study asgiven below:

U UstrechUangle 2

U Def1er21g 3

U 12k21ksextic4 4

where Ustrech is the bond energy due to bond stretch, and Uangle isthe bond energy due bond angle variation. The above parameters(constants) are calibrated by Belytsckho [16] with Brenner poten-tial as follows:

De 0:6031 nN nm; 26:25 nm1

k 8:7e10 nNUnm

rad2; ksextic 0:755 rad4

This set of parameters corresponds with the Brenner potentialfor strain below 10% and separation (dissociation) energy of124 kcal mol [17]. The modied Morse interatomic potential ener-gies and force elds will serve as a basis for the current atomisticmodeling and subsequent calculations.

3. Modeling technique

The proposed atomistic-based continuum approach for model-ing of graphene sheets is presented. The basic idea is to use arepresentative unit cell at atomistic scale for repeating honeycombgeometry of graphene sheet with nite dimensions (Fig. 2). Tocalculate effective elastic moduli, an equivalent homogenizedmedium of a triangular shape with equal mechanical propertiesis assumed. This equivalence is established by equating energies ofboth models under controlled boundary conditions which gener-ates a uniform (constant) strain eld over the domain of unit cell.Potential energy stored in the atomistic model is determined viaatomistic nite element simulations as will be outlined later.

3.1. Equivalent continuum model

For a homogeneous continuum medium, the constitutive rela-tionship under a plane stress assumption is expressed in terms ofthe stiffness matrix Cij by:

s11s22s12

8>:

9>=>;

C11 C12 0C21 C22 00 0 C66

264

375

1122

12

8>:

9>=>;

5

Each entry (C11, C22, C12) of the stiffness matrix is calculated byconducting a simulation run under prescribed loading conditionson boundaries. C66 is not an independent modulus; therefore itscalculation will not be considered. Equivalently, the simulationmodels are run by applying a uniform strain (oij) to the continuummodel generating prescribed displacements on the boundariesgiven by:

ui oijxj i; j 1;2 6

where ui is the prescribed displacement, and xj is the coordinate ofboundaries. The corresponding continuum stored strain energydensity is given by

U V2sijij

V2C11112C22222C66122C121122 7

Fig. 1. Geometry of a graphene sheet. Fig. 2. Representative unit cell concept.

K.I. Alzebdeh / Solid State Communications 177 (2014) 252826

Thus, three sets of loading conditions are required. These are:

Load case I: 11 is prescribed, 22120; Load case II: 22 is prescribed, 11120; Load case III: 11 and 22 are prescribed, 120.

In each simulation run, separately, the energy is found by theatomistic nite element simulation and is set equal to a corre-sponding special form of Eq. (7). For example, under loading case I,the strain energy density is expressed as

U V2C11112 8

From which we infer C11. Similarly, loading case II gives:

U V2C22222 9

From which we infer C22, while from loading case III withstored energy given by Eq. (7), we infer C12.

Once the values of the three elastic moduli are determined,Young's modulus and Poisson's ratio of the graphene sheet can bededuced as

E1 C11C22C212

C22

E2 C11C22C221

C11

12 C12C22

21 C21C11

10

In this study, entries of the stiffness matrix were calculatedwith respect to two cases of applied strains (oij 0.5% and 5%).

3.2. Atomistic nite element model

We consider the wireframe structure captured in the repre-sentative unit cell as depicted in Fig. 2. An atomistic nite elementmodel (AFEM) for such a wireframe is developed, wherein, nodesrepresent carbon atoms. Each node is characterized by six degreesof freedom (three translations: u1, u2, u3, and three rotations: 1,2, 3). The links which properly connect nodes are consideredeither spring-like or beam nite elements. Elements geometricand material properties are assigned according to interatomicforce eld assumed acting between carbon atoms. The two-nodeelements, which interconnect the atoms and their correspondingnodes, are illustrated in Fig. 3.

In particular, we adopt three atomistic models [4,9,12] tosimulate bond stretching associated with CC and bond anglevariation attributed to the adjacent CCC:

1. Linear spring elements for both stretching and angle variation:Constant spring stiffness is assigned.

2. Nonlinear spring elements for both stretching and angle varia-tion: The forcedisplacement relationships for spring elementsare derived from Eqs. (3) and (4).

3. Beam element for both stretching and angle variation. Thespecied element is a uni-axial element with tension, compres-sion, torsion and bending capabilities. It has six degrees offreedom at each node, translating in the x, y, and z directionsand rotations about the nodal x, y, and z-axes. The element isdened by the two nodes, cross-sectional area, two moments ofinertia, and material properties. Details concerning this ele-ment can be found in [12].

Finite element models were analyzed using the commercialnite element package ANSYS 12 [18]. The built-in linear springelement (Combin7), nonlinear spring element (Combin39) and 3Delastic BEAM4 element were used in the three atomistic models,respectively.

4. Results and discussion

By employing the described AFEMs, potential energy stored in theunit cell is calculated under the three loading cases I, II and III,respectively. The correspondent computed energy is equated withstrain energy of the equivalent continuum model as given by Eqs.(7)(9). Input parameters to current modeling are given in Table 1.

The in-plane elastic moduli (C11, C22, C12) are determined fromwhich the elastic constants (E1, E2, 12, 21) are extracted accordingto Eq. (10). Table 2 presents results under an applied strainoij 0:5%. The obtained equal values of E1 and E2 at suchinnitesimal strain demonstrate the in-plane isotropic behaviorfor the nanostructured graphene sheet. This isotropic property canbe attributed to the hexagonal array arrangement of carbon atoms.

Young's modulus values as determined from the three differentAFEMs indicate that a reasonable consistency among them exists.However, a signicant variation in values of Poisson's ratio can beobserved. This highlights the sensitivity of Poisson's ratio to thetype of model being used. On the other hand, the nonlinear springmodel gives the highest estimate for both elastic constants (E and )

Fig. 3. A spring-based nite element model.

Table 1AFEM input parameters.

Parameter Value

Bond length (l) 0.139 nmThickness (t) 0.34 nmo 1201Kr 6.5107 N nm1K 8.71010 N nm rad2

Table 2In-plane effective elastic properties of graphene sheets (in TPa) at 0.5%.

Model C11 C22 C12 E1 E2 12 21

Linear springs 1.1 1.1 0.09 1.1 1.1 0.09 0.09Nonlinear springs 1.6 1.6 0.26 1.2 1.2 0.15 0.195Beam 1.06 1.06 0.051 1.05 1.05 0.05 0.05

K.I. Alzebdeh / Solid State Communications 177 (2014) 2528 27

as given in Table 2. Also, Poisson's ratio obtained from the beammodel lies on the low side compared with other models. This mayindicate that beam element is less reliable compared to othermodels. Further investigation on beam model revealed that it failsto capture all angle variations associated with the CCC bonds in agiven honeycomb lattice of graphene sheet.

A comparison between elastic constants calculated herein andthe available numerical and analytical data found in literature wasperformed. The results of Young's modulus showed a good agree-ment with other estimates as given in Table 3. However, ourcalculated Poisson's ratios are lower than those published by otherresearchers except the one from nonlinear spring model. It isworthwhile to observe that a large discrepancy among publishedvalues of (Table 3) still exists.

Table 4 presents Young's Modulus and Poisson's ratio at oij 5%under which a geometrical nonlinearly is expected to occur. There-fore, large displacement option in the FEM analysis was used.Evaluation of Young's modulus at higher strain is essential whenconsidering a full stressstrain relationship for graphene structure.Based on results of Table 4 we can draw the following observations:

Young's modulus values calculated by the three atomisticmodels are comparable.

The in-plane isotropy of graphene is still preserved. Lower values of Young's modulus are obtained compared to the

quantities determined under oij 0:5% expect for the beammodel where values remain constant.

Values of Poisson ratio at both strains are comparable except forthe nonlinear spring model in which it decreases dramatically.

5. Conclusions

An atomistic-based continuum modeling approach is proposed topredict the in-plane mechanical properties of single-layered graphenesheets. The method utilizes the concept of equivalence of energystored in a representative unit cell for graphene structure at nanoscaleand in continuum sense. Correspondent elastic moduli are calculatedunder proper sets of displacement-controlled boundary conditionsgenerating a uniform strain eld in the unit cell. The advantage of thisapproach is its simplicity and efciency since analysis is independentof the size of graphene sheet. Besides, unlike other existing modelingtechniques, it calculates directly with accuracy the elastic moduli ofgraphene sheets. However, in the case of defected sheets, it is stillapplicable but on the full sheet size since periodic nanostructure doesnot exist any more.

In this work, three atomistic models were adopted in themodeling approach. Results of Young's modulus obtained from thethree models are generally within acceptable scatter (1.051.2 TPa)and concur with the published theoretical and numerical predic-tions. However, obtained Poisson's ratios exhibits high sensitivity tothe atomistic model being used. Thus, further investigation is stillneeded. Furthermore, in contrast with some previous researchworks, isotropic behavior of in-plane graphene sheets was validatedbased on current modeling.

The presented results demonstrate that the current atomistic-continuum modeling provides a valuable and efcient tool for perdi-tion of effective elastic moduli of graphene sheets. Therefore, it will beused for further analysis of graphene structures and extended tomodeling of mechanical behavior of CNTs and nanocomposites.

References

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Table 3Elastic constants of graphene sheet (t0.34 nm) from literature and present work.

Author Model E (TPa)

Meo and Rossi [4] 2b 0.94 Shokrieh and Raee [8] 6f 1.04 Georgantzinos et al. [9] 1a 1.367 0.603Georgantzinos et al. [10] 3c 0.737 0.219Sakhaee-Pour [6] 4d 1.025 1.285Scarpa et al. [7] 5e 1.668 0.574Chang and Gao [19] 6f 1.06 0.16Cho et al. [5] 6f 1.13 0.195Blakslee et al. [20] ng 1.00 0.16Lee et al. [21] ng 0.8531.147 Peng et al. [22] ng 1.057 Alzebdeh [12] 4d 1.02 0.12Present work 1a 1.10 0.09Present work 3c 1.20 0.195Present work 4d 1.05 0.05

a Linear central spring and linear torsional spring.b Nonlinear central spring and linear torsional spring.c Nonlinear central spring and nonlinear torsional spring.d Beam; Model.e Truss; Model.f Analytical.g Experiment.

Table 4In-plane effective elastic properties of graphene sheets (in TPa) at 5%.

Model C11 C22 C12 E1 E2 12 21

Linear springs 0.99 1.01 0.11 0.98 0.99 0.108 0.110Nonlinear springs 1.08 1.10 0.09 1.07 1.09 0.082 0.083Beam 1.05 1.05 0.055 1.05 1.05 0.053 0.053

K.I. Alzebdeh / Solid State Communications 177 (2014) 252828

An atomistic-based continuum approach for calculation of elastic properties of single-layered graphene sheetIntroductionMolecular interactionsModeling techniqueEquivalent continuum modelAtomistic finite element model

Results and discussionConclusionsReferences