2DMater. 3 (2016) 025033 doi:10.1088/2053-1583/3/2/025033

PAPER

Uniaxial compression of suspended single andmultilayer graphenes

APSgouros1,2,3, GKalosakas1,3,4, CGaliotis1,5 andKPapagelis1,3

1 Institute of Chemical Engineering Sciences—Foundation of Research andTechnologyHellas (FORTH/ICE-HT), GR-26504 Patras,Greece

2 School of Chemical Engineering, National Technical University of Athens (NTUA), GR-15780 Athens, Greece3 Department ofMaterials Science, University of Patras, GR-26504 Patras, Greece4 Crete Center forQuantumComplexity andNanotechnology (CCQCN), PhysicsDepartment, University of CreteGR-71003Heraklion,

Greece5 School of Chemical Engineering, University of Patras, GR-26504 Patras, Greece

E-mail: georgek@upatras.gr

Keywords: graphene, uniaxial compression, critical buckling,molecular dynamics

Supplementarymaterial for this article is available online

AbstractThemechanical response of single andmultiple graphene sheets under uniaxial compressive loads wasstudiedwithmolecular dynamics (MD) simulations, using different semi-empirical forcefields atdifferent boundary conditions or constrains. Compressive stress–strain curves were obtained and thecritical stress/strain values were derived. TheMD results are compared to the linear elasticitycontinuum theory for loaded slabs. Concerning the length dependence of critical values, qualitativelysimilar behavior is observed between the theory and numerical simulations for single layer graphenes,as the critical stress/strain for bucklingwas found to scale to the inverse squared length.Howeverdiscrepancies were noted formultilayer graphenes, where the critical buckling stress also decreasedwith increasing length, though at a slower rate than expected from elastic buckling analysis.

1. Introduction

Graphene is an ultrathin membrane of atomic thick-ness that possesses exceptional electronic [1, 2], ther-mal [3–5] and mechanical properties [6] which farexceed those of conventional 3D materials. Graphenebeing the first two-dimensional material discovered[7], due to its unique properties could form a basis fornumerous applications in several branches of technol-ogy, such as in polymer matrix composite materials[8–11], optoelectronics [12, 13], drug delivery [14–16], super capacitors [17] andmanymore. By applyingstress or strain on graphene (the so called ‘strainengineering’) one can modify its electronic properties[18], and this can make an impact to numerousapplications, ranging from strain sensors [19, 20] toelectric circuits made from graphene [21]. Grapheneribbons under compressive loads fail by the formationof wrinkles, the wavelength, amplitude and directionof which can be controlled bymodifying the boundaryconditions along their edges [22]. Those wrinklescan affect drastically the local fields and the electronic

properties by inducing effective magneticfields [23, 24].

The experimental Young’s modulus and tensileintrinsic strength of graphene have been estimated to1 TPa and 130 GPa, respectively, by nanoindentingsuspended graphene sheets with an AFM tip [6].Another method for inducing tensile (compressive)strain to graphene is by placing graphene flakes onplastic beams [11, 25–27], and then, by bending thebeams [28], strains up to εt ∼ 1.5% (εc = 0.7%) havebeen achieved [29]. Graphene of monoatomic thick-ness freely suspended in air is expected to have extre-mely low resistance to compressive loading. However,the usage of substrate can drastically change themechanical properties of graphene, since the criticalbuckling strain of supported graphenes can be ordersof magnitude higher than suspended graphenes [11].This behavior is indeed very important since it con-firms the role of graphene as a reinforcing agent incomposite applications.

Themechanical behavior under uniaxial compres-sion of single layer graphene (SLG) has been

RECEIVED

2 February 2016

REVISED

19May 2016

ACCEPTED FOR PUBLICATION

31May 2016

PUBLISHED

10 June 2016

© 2016 IOPPublishing Ltd

investigated in a few theoretical works, using mole-cular mechanics [30], molecular dynamics (MD)simulations [31–33] and continuum mechanics [34].The critical buckling stress/strain of SLG has beenfound to scale inversely proportional to the square ofits length [30, 32, 33], in agreement with the linearelasticity theory of loaded slabs [35]. Additionally ithas been found [30, 33] that the critical buckling strainis the same for uniaxial compression along either thezigzag or the armchair directions. Similar theoreticalstudies of uniaxial compression on multilayer gra-phene (MLG) are limited; a finite element approachhas been used for the buckling instability of bilayergraphene [36].

In this work we performed MD simulations inorder to study the mechanical response of suspendedSLG and MLG under constant compressive loads(stress controlled simulations). Different semi-empirical force fields are used in order to verify therobustness of the observed behavior. The main resultsobtained from our simulations were the compressivestress–strain curves. The critical buckling stress (σcrit)and strain (εcrit), the Young’s modulus (E) and thePoisson’s ratio ( )v were calculated. The critical buck-ling values presented here concern intrinsic propertiesof suspended graphenes. As mentioned above, thecorresponding values for graphenes embedded inmatrices, or supported on beams are expected to besignificantly larger. The results of the simulationsregarding the critical buckling values were comparedwith the linear elasticity theory for wide plates withunsupported sides under uniaxial compressive forces.We find that the continuum elasticity theory fails todescribe the behavior observed inMLGs.

2.Methods

The MD simulations were performed with theLAMMPS package [37, 38] in the isobaric–isothermalensemble using periodic boundary conditions and atime step equal to 1 fs. In this study we presentextensive results for SLGs and MLGs (up to six layers)for various lengths at low temperatures. The temper-ature (pressure) of the systemwasmaintained constantto 1 K (0 bar) using the Nosé–Hoover thermostat(barostat) [39, 40] with an effective relaxation timeequal to 0.1 ps (1 ps). Simulations have also beenperformed at room temperature for single layer andbilayer graphenes and qualitatively similar results werefound as those presented here; however, the datashowed a considerable scatter and longer relaxationtimes are normally needed. We note that the dimen-sions of the studied graphenes (see below) at 1 K arelarger than the characteristic thermal length at thistemperature (around 2.5 nm), representing the lengthscale above which thermal fluctuations becomeimportant [41].

To obtain force field independent results, calcula-tions were performed for SLGs with three semi-empirical potentials: the Tersoff [42, 43], the REBO[44] and the LCBOP I [45]. Since we found qualita-tively similar behavior for these potentials concerningthe length dependence of the critical buckling values,for the computationally more intensive case of MLGswe used the LCBOP potential, though some examplecases were checked with the AIREBO [46] potential aswell. It has been shown that the LCBOP potential pro-vides a relatively accurate overall description of thephonon dispersion in graphene [47]. The atomisticrepresentations and the data visualization were aidedwith relevant software [48, 49].

In order to design the graphene sheets, the 4-atomorthogonal cell in figure 1(a) was duplicated severaltimes along the armchair and zigzag directions.Figure 1(b) displays a SLGwith dimensions lx (length),ly (width) and lz (thickness) inside a simulation boxwith dimensions Lx, Ly and Lz, where lx < Lx, ly = Lyand lz < Lz. The typical dimensions of the graphenesheets studied in this work had lengths (lx) 2.3–85 nmand widths (ly) 6 nm, while the vacuum gap along thex- and z-axis was set to about 10 nm. The thickness ofSLGs was assumed to be equal to the interlayer dis-tance ofMLG, lz= 0.335 nm [50, 51]. In order to avoidspurious interactions along the directions of x- and z-axis the differences Lx – lx and Lz – lz were larger thanthe cut off distance of the used potentials, hence oursystem was periodic only along the y-axis direction asit is illustrated in figure 1(b). In practice, this meansthat our results concern graphenes of relatively largewidths. By replicating the graphene sheets along the z-axis direction, MLGs with up to 6 layers were con-structed (as shown in figures 1(c)–(e) for 2, 3 and 4 lay-ers, respectively). It should be noted that the ABstacking was found to be stable in contrast to the caseof AA stacking, in accordancewith [52].

The applied constrains at the loading edges of gra-phene play an important role to its mechanicalresponse, as it has been demonstrated by continuummodels [35, 53]. In the case where no geometrical con-straints are applied (free edges) the graphene sheetscan simply rotate under compressive uniaxial loads, asit is illustrated in the schematic of figure 2(a) and theatomistic representation of a MD simulation infigure 2(d). One way to prevent rotations of the gra-phenes in MD is to constrain the edges of graphene inthe xy-plane (fixed edges, see figure 2(e)). The beha-vior of those systems corresponds to the case of slabswith clamped (fixed) edges in the continuum model(see schematic of figure 2(b)), since in both cases thedisplacement w of the edges along the z-axis and itsderivative (dw/dx) equal to zero [35]. Anothermethodimplemented in LAMMPS [38] to prevent rotations ofa nanostructure is to maintain its angular momentumto zero, by subtracting at each time step of the simula-tion from each atom the average angular velocity withrespect to the center of mass. In this case, the

2

2DMater. 3 (2016) 025033 AP Sgouros et al

graphenes under the compressive forces were bendedas shown in figure 2(f). This condition will be referredas ‘angular momentum conserving’ (AMC) constrain.It should be noted that this approach cannot beapplied effectively to graphenes where ly is much largerthan lx since they are able to respond by twistinginstead of bending (see for example the figure S1 in thesupplementary material information). AMC con-strained graphenes feature a shape similar with that of

pinned plates at the edges, in the sense that the dis-placement w exhibits an almost linear behavior nearthe edges indicating that d2w/dx2 ≈ 0. In plates withpinned edges the displacement w and its second deri-vative (d2w/dx2) at their edges are zero (see schematicof figure 2(c)). One strict difference is that in AMC thedisplacements w of the edges are not zero. However,due to AMC and the symmetry of the structure, therelative displacement between the two edges of

Figure 1. (a)A4-atomorthogonal cell replicated 2 times along the zigzag and armchair directions. The displayed lengths are in units ofC–Cbond lengths. (b)A single graphene layer that is periodic only along the y-axis with constant compressive loads applied to theedges of the graphene at the x direction (black atoms). The graphene layer from (b) replicatedmultiple times along the z-axis, resultingin (c) bilayer, (d) trilayer and (e) quadlayer graphene.

Figure 2. Schematic representations of loaded slabswith (a) free, (b)fixed (clamped) and (c) pinned (simply supported) edges.Configurations of (d) free, (e)fixed and (f)AMC-constrained graphenes under uniaxial compressive forces, as obtained fromMDsimulations. Thewhite arrows display the direction of the constant applied forces. In (d) snapshots of the evolution (indicated by theblack arrows) of the process that leads to the rotation of the SLG are shown. Equilibrium configurations for different loads arepresented in (e) and (f): the applied forces are slightly smaller (top), slightly larger (middle) andmuch larger (bottom) than the criticalbuckling stressσcrit.

3

2DMater. 3 (2016) 025033 AP Sgouros et al

graphene is zero, as happens with the case of pinnedplates. Graphenes with either fixed edges or underAMC constrains are considered in the MD simula-tions here.

Initially, the graphenes free of any load were left torelax for 100 ps to achieve equilibrium at the desiredtemperature and pressure, and acquire the initialdimensions. Next, compressive forces of constantmagnitude were applied to the atoms at the edges ofgraphene (black atoms in figure 1(b)) for up to 10 ns.We have tested that this time is sufficient for achievingtime independent results for the considered cases. Theobtained strain was calculated by time averaging thevalue of the strain during the last 0.5 ns of the simula-tion. By performing MD simulations for a wide rangeof the applied forces, the stress–strain curves wereextracted. We have checked that the depth of theregion of atoms that are constrained in-plane in thecase of fixed edges, does not affect the response of thegraphenes as long as the distance between thoseregions (the effective length of graphene) remainsconstant.

3. Results and discussion

3.1. Uniaxial compression ofmonolayer grapheneFigure 3 displays the compressive stress–strain curvesof SLGs with different lengths lx using the boundarycondition of fixed edges for the three different forcefields. In the limit of small strains theYoung’smodulus(Poisson’s ratio) was calculated to ∼0.93 TPa (0.2),1.26 TPa (−0.1) and 0.86 TPa (0.3) for the LCBOP,Tersoff andREBO force fields, respectively.

The value of the obtained elastic modulus is moreor less close to the experimental value of 1.0± 0.1 TPa[6] for the LCBOP and REBO potentials, while theTersoff overestimates it. The Poisson’s ratios in theLCBOP and REBO cases are also in agreement withother calculations [54–57]. The negative Poisson ratioobtained by the Tersoff potential is an artifact that hasbeen reported in literature [58] and can be attributedto inaccuracies of the potential. As it can be seen fromfigure 3, the behavior of the stress–strain curves is qua-litatively similar for all the tested potentials. Above acritical stress value the strain abruptly increases

Figure 3.Compressive stress (σx)—strain (|εx|) diagrams formonolayer graphenes withfixed edges, using the (a) LCBOP, (b)Tersoffand (c)REBO forcefields. Different lengths of graphene are shown: lx= 2.3 nm (circles), 4.0 nm (squares), 5.7 nm (triangles) and7.4 nm (diamonds). The right columndepicts an enlargement of the corresponding plots at low applied strains, showing the linearresponse. The error bars represent standard deviations of the obtained time-averaged strains.

4

2DMater. 3 (2016) 025033 AP Sgouros et al

indicating the buckling of graphene. The critical buck-ling value depends on the length lx of graphene.

Figures 4(a) and (b) present the critical bucklingstress and strain with respect to the length lx of gra-phene for the three force fields used. The critical buck-ling stress was calculated with a resolution lower than0.01 GPa (the corresponding error bars in figure 4(a)are negligibly small). The critical buckling strainequals to the value of strain just before the bucklinginstability (see the figure S2(a) in the supplementarymaterial). For all the used force fields, the lengthdependence of the critical values σcrit and εcrit areaccurately described by an inverse square law:

( )sa

ea

= =¢

l l, , 1

x xcrit 2 crit 2

where α and α′ are constants. This is in agreementwith [30, 32, 33] where the critical buckling valueswere found to be inversely proportional to the lengthsquare. As we examined periodic systems in thedirection of y-axis, simulations at different values of ly,up to 25 nm, show no dependence of the criticalbuckling values on ly, at least for the low temperaturesconsidered here.

Qualitatively, the behavior of SLGs under theAMC constrain was quite similar with that of SLGswith fixed ends. This can be seen from figures S3 andS4 in the supplementary material information, whichdisplay the stress–strain diagram and the plots of σcritand εcrit versus lx, respectively, for this case using theLCBOPpotential. However, the critical buckling stresswas found to be about 3–4 times smaller than thecorresponding value calculated for SLG with fixedends, a result that is related with continuum models(see below).

Figure 5 displays the variation of the C–C bondlengths along the loading direction (x-axis) for a fixed-edged SLG with lx = 2.7 nm, for various values of theapplied force above the critical buckling σcrit, using theLCBOP force field. The C–C bonds in graphenes that

are not subjected to compressive loads (figure 5, blackcircles) are equal to 1.42 Å in the central region of thesheet and about 1.6% larger at the boundary due toedge effects. However graphene sheets under loadsthat exceed the σcrit value (onset of buckling) show anincrease of the bond length near the central region.This is in accordance with the observed relaxation ofthe frequency shift of certain Raman bands after theonset of buckling ofmonolayer graphene [29, 59].

The critical buckling strains (εcrit) were found tobe the same along the armchair and zigzag directionsof graphene in [30, 33]. In the current study graphenesheets with loading direction at various angles wereexamined, in order to investigate the dependence ofσcrit on the chiral angle (θchiral) of the applied stress. Itshould be noted that the chiral angle can only take spe-cific values in order to maintain proper periodicity ofthe lattice along the periodic boundaries of the simula-tion box. Figure 6 displays results of σcrit for graphenesheets with lx ≈ 4 nm (circles) and lx ≈ 7 nm (trian-gles), uniaxially compressed at different chiral anglesθchiral using the LCBOP potential. In both casesly = 6 nm. These results are in agreement with [30]where the REBO potential was used in molecularmechanics simulations and the critical buckling wasfound insensitive to the chirality of the loadingdirection.

3.2. Uniaxial compression ofmultilayer grapheneHere, the compressive response of MLGs is discussed,throughMD simulations using the LCBOP force field.Figure 7 shows that the stress–strain curves of MLGswith fixed edges exhibit a similar behavior with that ofSLGs (see figure 3). The buckling instability in this caseis similar to that occurring in SLGs, as can be seen fromfigure S2(b) in the supplementary material section. Inall presented cases, the Young’s modulus was calcu-lated in the regime 0.90–0.95 TPa.

Figure 4.The critical buckling (a) stress and (b) strain as a function of the length lx of single layer graphene using the LCBOP (circles),Tersoff (triangles) andREBO (squares) force fields. The error bars in (b) represent standard deviations of the time-averaged value.Lines are fittings of the numerical results with equation (1).

5

2DMater. 3 (2016) 025033 AP Sgouros et al

As excpected, for σ > σcrit, there seems to be anincreased resistance of the sheets for further bendingas the number N of layers increases. For example, forsheets with length lx= 2.3 nm, when σx= 4 GPa the εxof SLG is about 65% (derived from the data shown infigure 3(a) with circles), while the corresponding oneof MLGs withN= 2, 3 and 4 layers is about εx= 25%,17% and 10% (derived from the data shown with cir-cles in figures 7(a)–(c), respectively). Similar behavioroccurs for other values of lx.

The critical buckling stress σcrit as a function of thelength lx of MLGs with fixed edges and unsupportedsides is presented in figure 8, forN= 2 up toN= 6 lay-ers. The SLG case with N = 1 (circles in figure 4(a)) isalso shown for comparison. The solid line shows thefitting of SLG results with the inverse square law,equation (1). Figure 8(a) clearly demonstrates that thecritical buckling stresses of MLGs do not follow aninverse square law, but they exhibit a slower decrease

with the length that can not be described through asimple power law. Similar results have been obtainedfor bilayer graphenes (N = 2) using the AIREBOpotential as well (see figure S5 in the supplementarymaterial).

Multilayers under the AMC constrain displayed asimilar behavior, regarding the σcrit dependence on lx,as illustrated in figure S6 of the supplementary infor-mation. Additionally, for AMC-constrained MLGs,the σcrit is about 2–4 times smaller than the corresp-ondingσcrit value ofMLGswithfixed ends.

3.3. Comparisonwith the linear elasticity theory forloaded slabsIn the linear elasticity theory of continuummechanics,the critical buckling stress of a wide plate withunsupported sides, length lx, thickness lz, Poisson’sratio v and Young’s modulus E is provided from the

Figure 5.Variation of theC–Cbond lengths along the loading direction for graphene sheets with lx= 2.7 nmunder compressiveuniaxial loads equal to 0.0 (circles), 2.1 (triangles), 2.3 (squares) and 2.5 (diamonds)GPa. In this caseσcrit= 1.8 GPa. The inset displaystwo graphene equilibrium configurationswithσx= 2.1 GPa (top) andσx= 2.5 GPa (bottom). The error bars represent standarddeviations of the time- andwidth-averaged values shown in thefigure.

Figure 6.The critical buckling stressσcrit versus the chiral angle (θchiral) of the loading direction for graphene sheets with lx≈ 4 nm(circles) and lx≈ 7 nm (triangles).

6

2DMater. 3 (2016) 025033 AP Sgouros et al

following relation [35]:

( ) ( )( )s

p E=

-=

v

l

l p

d

l p12 1, 2z

x xcrit

2

2

2

2 2

where =p 2 for slabs with fixed edges, while =p 1for slabs with pinned edges. This difference in thevalue of p results in s = 4crit

fix · s ,critpin were scrit

fix ( )scritpin is

the critical buckling stress for plates with fixed(pinned) edges. Note that the equation (1) of [11]giving the critical buckling value for plates with simplysupported sides, is reduced for wide plates to the l1 x

2

dependence of equation (2), taking into account thatwhen the widthw is much larger than the length l, thenthe geometric term K tends to w2/l2. Moreover, thesame equation contains the thickness dependence(∼lz2) appeared in equation (2), considering that theratio D/C of the flexural and tension rigidities equalsto h2/12 (see equation (2) of [29]), where h is thethickness of the plate. For wide plates such a coin-cidence of formulae corresponding to differentboundary conditions at the sides of the plate isexpected, since in this case (where the width is verylarge) the boundaries at the sides play a negligible role.

The results from the MD simulations in SLGs arein qualitative agreement with the continuum modelsince they give s µ -lxcrit

2 (see figure 4(a) andequation (1) or figure 8(a)). However, there is a dis-crepancy at a quantitative level: for a SLG withE= 1 TPa [6], lz= 0.335 nm [50] and ν= 0.22 [54], theconstant d in equation (2) equals to 97 GPa nm2,which ismuch larger than the value extracted from theMD simulations. The continuum model could be fit-ted to the results derived from the MD simulations byadjusting the conventional thickness of the SLG(lz= 0.335 nm) to an effective thickness that equals toabout 0.04 nm. Such an unrealistic requirement forthe graphene thickness, in order to fit atomistic resultswith the predictions of the continuum theory forplates, also appears in the case of graphene bending[60]. We also note that the ratio s scrit

fcritAMC of the cri-

tical values for fixed-edged and AMC-constrained(which show some analogy with the behavior of pin-ned edges as discussed in section 2) SLGs is close to 4 asit is displayed infigure 9.

In the case of MLGs qualitatively different beha-viors are obtained between the results of MD

Figure 7.Compressive stress (σx)—strain (|εx|) diagrams forMLGswith fixed edges, consisting of (a) two, (b) three and (c) four layers,using the LCBOP force field. Results for different lengths ofMLGs are shown: lx= 2.3 nm (circles), 4.0 nm (squares), 5.7 nm(triangles) and 7.4 nm (diamonds). The right column depicts an enlargement of the corresponding plots at low applied strains,showing the linear response. The error bars represent standard deviations of the obtained time-averaged strains.

7

2DMater. 3 (2016) 025033 AP Sgouros et al

simulations and the continuum theory for loadedslabs with unsupported sides. In particular, asfigure 8(a) shows, the MD calculations reveal sig-nificant deviations from the l1 x

2 law of equation (2),concerning the length dependence of critical bucklingvalues. Moreover, in the linear elasticity continuumtheory the σcrit is proportional to the square of theplate thickness lz (see equation (2)), while the MDsimulations show that σcrit only slightly increases withlz, as can be seen from figure 8(b). Both fixed edgedand AMC constrained graphenes reveal these featuresin the numerical simulations. The ratio s scrit

fcritAMC for

MLGs is also shown infigure 9.In MLGs the obtained MD results, concerning the

σcrit variation with the graphene length, are due totheir discrete nature and the interlayer interactions.Modifying the interlayer interactions, affects thelength dependence of critical buckling. This is demon-strated in figure S7 of the supplementary informationfor graphene bilayers. In this case the interlayer inter-action is described through interatomic Lennard–Jones potentials and we have varied the depth ε of theLennard–Jones potential from tenths up to hundredsof the corresponding normal value [61]. As expected,

when the strength of the interlayer interaction decrea-ses, the results tend to those of the SLG case, depictedalso in figure S7 for comparison. By increasing thestrength of this interaction, the behavior of σcrit chan-ges and the exponent of the power law that was fittedto the data decreases towards the value−2 (at least forthe lengths shown infigure S7).

4. Conclusions

The behavior of single layer and multilayer graphenesof large width under uniaxial compressive loads wasinvestigated, through MD simulations. The compres-sive tests were stress-controlled and the simulationstook place in the isothermal–isobaric ensemble. Thestudied graphene sheets in this work had typicaldimensions of length 2 nm < lx < 100 nm andthickness lz ≈ 0.3–2 nm (from one up to six graphenelayers), while periodic boundary conditions were usedalong the y direction. Different force fields andboundary conditions/constrains were used in ourMDsimulations. For SLGs the calculated elastic moduliwere relatively close to the experimental value

Figure 8. (a) Logarithmic plot of the critical buckling stress with respect to the length lx ofN-layered graphenes with fixed edges, forN= 1 (filled circles),N= 2 (triangles),N= 3 (squares),N= 4 (diamonds),N= 5 (crosses) andN= 6 (open circles) layers, using theLCBOP force field. (b)The corresponding linear plot for graphene lengths up to 10 nm. Lines are fittings of the numerical results forthe case of SLGswith an inverse square law, equation (1).

8

2DMater. 3 (2016) 025033 AP Sgouros et al

E= 1 TPa, while the Poisson ratios of the LCBOP andREBO potentials were in the range 0.2–0.3, in agree-mentwith values reported in the literature.

For SLGs, the critical buckling stress/strain isinversely proportional to the square of the length ofgraphene along the loading direction. Additionally,the σcrit was found to be independent on the loadingdirection (armchair, zigzag or any other chiral direc-tion). TheMD results for SLGs show qualitative agree-ment with the continuum theory for uniaxialcompressive deformation of wide plates of unsup-ported sides.

However, in the case of MLGs discrepancies werefound between the MD calculations and the con-tinuum linear elasticity theory. The latter one predictsthat σcrit increases with the square of the thickness,while the MD simulations show a significantly smallerincrease. Furthermore, the inverse squared lengthdependence of the critical buckling values, as antici-pated in the continuum theory for plates, is not ver-ified by the numerical simulations, where deviationsfrom this behavior are obtained for MLGs with N= 2up toN= 6 layers.

Acknowledgments

We would like to thank D N Theodorou and CTzoumanekas for valuable discussions. This work hasbeen partially supported from the Thales projectGRAPHENECOMP co-financed by the EuropeanUnion (ESF) and the Greek Ministry of Education(through ΕΣΠΑ program), and by European Union’sSeventh Framework Programme (FP7-REGPOT-2012-2013-1) under grant agreement n° 316165.Financial support of Graphene FET Flagship (Gra-phene-Based Revolutions in ICT and Beyond, GrantNo. 604391) is also acknowledged.

References

[1] NovoselovK S, GeimAK,Morozov SV, JiangD,KatsnelsonM I,Grigorieva I V,Dubonos SV and Firsov AA2005Two-dimensional gas ofmassless Dirac fermions ingrapheneNature 438 197–200

[2] ZhangY, TanY-W, StormerHL andKimP2005 Experimentalobservation of the quantumHall effect and Berry’s phase ingrapheneNature 438 201–4

[3] CaiW,MooreA L, ZhuY, Li X, Chen S, Shi L andRuoff R S2010Thermal transport in suspended and supportedmonolayer graphene grown by chemical vapor depositionNano Lett. 10 1645–51

[4] Faugeras C, Faugeras B,OrlitaM, PotemskiM,Nair RR andGeimAK2010Thermal conductivity of graphene in corbinomembrane geometryACSNano 4 1889–92

[5] BalandinAA,Ghosh S, BaoW,Calizo I, TeweldebrhanD,Miao F and LauCN2008 Superior thermal conductivity ofsingle-layer grapheneNano Lett. 8 902–7

[6] LeeC,Wei X, Kysar JWandHone J 2008Measurement of theelastic properties and intrinsic strength ofmonolayer grapheneScience 321 385–8

[7] GeimAK2011Randomwalk to graphene (Nobel lecture)Angew. Chem., Int. Ed. Engl. 50 6966–85

[8] Stankovich S, DikinDA,Dommett GHB,Kohlhaas KM,Zimney E J, Stach EA, Piner RD,Nguyen ST andRuoff R S2006Graphene-based compositematerialsNature 442 282–6

[9] RamanathanT et al 2008 Functionalized graphene sheets forpolymer nanocompositesNat. Nanotechnol. 3 327–31

[10] RafieeMA,Rafiee J, Yu ZZ andKoratkarN 2009 Bucklingresistant graphene nanocompositesAppl. Phys. Lett. 95 223103

[11] FrankO, Tsoukleri G, Parthenios J, Papagelis K, Riaz I, Jalil R,NovoselovK S andGaliotis C 2010Compression behavior ofsingle-layer graphenesACSNano 4 3131–8

[12] Avouris P and FreitagM2014Graphene photonics,plasmonics, and optoelectronics IEEE J. Sel. Top. QuantumElectron. 20 3677–94

[13] CooperDR et al 2012 Experimental review of graphene ISRNCondens.Matter Phys. 2012 1–56

[14] Zhang L, Xia J, ZhaoQ, Liu L andZhangZ 2010 Functionalgraphene oxide as a nanocarrier for controlled loading andtargeted delivery ofmixed anticancer drugs Small 6 537–44

[15] Liu Z, Robinson J T, SunX andDaiH2008 PEGylatednanographene oxide for delivery of water-insoluble cancerdrugs J. Am.Chem. Soc. 130 10876–7

[16] YangX, ZhangX, Liu Z,MaY,HuangY andChenY 2008High-efficiency loading and controlled release of doxorubicinhydrochloride on graphene oxide J. Phys. Chem.C 11217554–8

Figure 9.The ratio s scritf

critAMC (see text) ofN-layered graphenes of various lengths (lx), forN= 1 (filled circles),N= 2 (triangles),

N= 3 (squares),N= 4 (diamonds),N= 5 (crosses) andN= 6 (open circles) layers, as obtained by theMD simulations. The dashedline displays the ratio of the critical buckling values for slabswithfixed and pinned edges in the continuummodel.

9

2DMater. 3 (2016) 025033 AP Sgouros et al

[17] Zhiwei P, Lin J, Ye R, Samual E LG andTour JM2015 Flexibleand stackable laser-induced graphene supercapacitorsACSAppl.Mater. Interfaces 7 3414–9

[18] CastroNetoAH, PeresNMR,NovoselovK S andGeimAK2009The electronic properties of grapheneRev.Mod. Phys. 81109–62

[19] Sakhaee-PourA, AhmadianMT andVafai A 2008 Potentialapplication of single-layered graphene sheet as strain sensorSolid State Commun. 147 336–40

[20] WangY, YangR, Shi Z, Zhang L, Shi D,Wang E andZhangG2011 Super-elastic graphene ripples forflexible strain sensorsACSNano 5 3645–50

[21] Pereira VMandCastroNeto AH2009 Strain engineering ofgraphene’s electronic structure Phys. Rev. Lett. 103 046801

[22] BaoW,MiaoF,ChenZ,ZhangH, JangW,DamesCandLauCN2009Controlled ripple texturing of suspended graphene andultrathin graphitemembranesNat.Nanotechnol.4 562–6

[23] Guinea F,Horovitz B and LeDoussal P 2008Gauge fieldinduced by ripples in graphene Phys. Rev.B 77 205421

[24] IsacssonA, Jonsson LM,Kinaret JM and JonsonM2008Electronic superlattices in corrugated graphene Phys. Rev.B 77035423

[25] NiZH, YuT, LuYH,WangYY, Feng YP and ShenZX2008Uniaxial strain on graphene: Raman spectroscopy study andband-gap openingACSNano 2 2301–5

[26] Mohiuddin TMG et al 2009Uniaxial strain in graphene byRaman spectroscopy: G peak splitting, Grüneisen parameters,and sample orientation Phys. Rev.B 79 205433

[27] YuT,Ni Z,DuC, YouY,WangY and ShenZ 2008Ramanmapping investigation of graphene on transparent flexiblesubstrate: the strain effect J. Phys. Chem.C 112 12602–5

[28] Vlattas C andGaliotis C 1991Monitoring the behaviour ofpolymerfibres under axial compression Polymer 32 1788–93

[29] TsoukleriG,Parthenios J, PapagelisK, Jalil R, Ferrari AC,GeimAK,NovoselovKSandGaliotisC2009Subjecting agraphenemonolayer to tension and compression Small52397–402

[30] LuQ andHuangR 2009Nonlinearmechanics of single-atomic-layer graphene sheets Int. J. Appl.Mech. 01 443–67

[31] Neek-AmalMandPeeters FM2010Defected graphenenanoribbonsunder axial compressionAppl. Phys. Lett.97153118

[32] Neek-AmalM and Peeters FM2010Graphene nanoribbonssubjected to axial stressPhys. Rev.B 82 085432

[33] Zhang Y andLiu F 2011Maximumasymmetry in straininducedmechanical instability of graphene: compressionversus tensionAppl. Phys. Lett. 99 241908

[34] Pradhan SC andMurmuT2009 Small scale effect on thebuckling of single-layered graphene sheets under biaxialcompression via nonlocal continuummechanicsComput.Mater. Sci. 47 268–74

[35] ReesDWA2009Mechanics of Optimal Structural Design:MinimumWeight Structures (NewYork:Wiley)

[36] Chandra Y, ChowdhuryR, Adhikari S and Scarpa F 2011Elastic instability of bilayer graphene using atomistic finiteelementPhysica E 44 12–6

[37] Plimpton S 1995 Fast parallel algorithms for short-rangemolecular dynamics J. Comp. Phys. 117 1–19

[38] LAMMPSMolecularDynamics Simulator, Available from:(http://lammps.sandia.gov/)

[39] Nosé S 1984Aunified formulation of the constant temperaturemolecular dynamicsmethods J. Chem. Phys. 81 511–9

[40] HooverWG1985Canonical dynamics: equilibriumphase-space distributions Phys. Rev.A 31 1695–7

[41] Košmrlj A andNelsonDR2016Response of thermalizedribbons to pulling and bending Phys. Rev.B 93 125431

[42] Tersoff J 1988New empirical approach for the structure andenergy of covalent systemsPhys. Rev.B 37 6991

[43] Tersoff J 1989Modeling solid-state chemistry: interatomicpotentials formulticomponent systems Phys. Rev.B 39 5566

[44] Stuart S J, Tutein AB andHarrison J A 2000A reactivepotential for hydrocarbonswith intermolecular interactionsJ. Chem. Phys. 112 6472–86

[45] Los JH and Fasolino A 2003 Intrinsic long-range bond-orderpotential for carbon: performance inMonte Carlo simulationsof graphitizationPhys. Rev.B 68 024107

[46] BrennerDW, ShenderovaOA,Harrison JA, Stuart S J,Ni B and Sinnott S B 2002A second-generation reactiveempirical bond order (REBO) potential energy expression forhydrocarbons J. Phys.: Condens.Matter 14 783–802

[47] Koukaras EN,KalosakasG, Galiotis C and Papagelis K 2015Phonon properties of graphene derived frommoleculardynamics simulations Sci. Rep. 5 12923

[48] HumphreyW,Dalke A and SchultenK 1996VMD: visualmolecular dynamics J.Mol. Graph. 14 33–8

[49] KrausD 2014Consolidated data analysis and presentationusing an open-source add-in for theMicrosoft Excel®

spreadsheet softwareMed.Writ. 23 25–8[50] Girifalce LA and LadRA 1956 Energy of cohesion,

compressibility, and the potential energy functions of thegraphite system J. Chem. Phys. 25 693

[51] Wagner P, IvanovskayaVV, RaysonM J, Briddon PR andEwels CP 2013Mechanical properties of nanosheets andnanotubes investigated using a new geometry independentvolume definition J. Phys.: Condens.Matter 25 155302

[52] PopovAM, Lebedeva I V, Knizhnik AA, Lozovik Y E andPotapkin BV 2012 Barriers tomotion and rotation of graphenelayers based onmeasurements of shearmode frequenciesChem. Phys. Lett. 536 82–6

[53] Han SM,BenaroyaH andWei T 1999Dynamics oftransversely vibrating beams using four engineering theoriesJ. SoundVib. 225 935–88

[54] KalosakasG, LathiotakisNN,Galiotis C and Papagelis K 2013In-plane forcefields and elastic properties of graphene J. Appl.Phys. 113 134307

[55] Gupta S S andBatra RC2010 Elastic properties andfrequencies of free vibrations of single-layer graphene sheetsJ. Comput. Theor. Nanosci. 7 2151–64

[56] Liu F,Ming P and Li J 2007Ab initio calculation of idealstrength and phonon instability of graphene under tensionPhys. Rev.B 76 064120

[57] WeiD andWang F 2014Graphene: a partially ordered non-periodic solid J. Chem. Phys. 141 144701

[58] Berinskii I E andKrivtsov AM2010Onusingmany-particleinteratomic potentials to compute elastic properties ofgraphene and diamondMech. Solids 45 815–34

[59] Androulidakis C, Koukaras EN, FrankO, Tsoukleri G,Sfyris D, Parthenios J, PugnoN, Papagelis K,NovoselovK S andGaliotis C 2014 Failure processes inembeddedmonolayer graphene under axial compression Sci.Rep. 4 5271

[60] ZhangDB, Akatyeva E andDumitrica T 2011 Bendingultrathin graphene at themargins of continuummechanicsPhys. Rev. Lett. 106 255503

[61] Shibuta Y and Elliott J A 2011 Interaction between twographene sheets with a turbostratic orientational relationshipChem. Phys. Lett. 512 146–50

10

2DMater. 3 (2016) 025033 AP Sgouros et al