icM

ic A, Iran

biaeditederausinha

type of planar loading, mode numbers and boundary conditions are discussed in details.

m andnly 0ndedh a un

scale effect must play an important role in the nanoscale struc-tures, but this small scale effect has been ignored when classicallocal continuum theory was adopted [11]. Really, we cannotneglect the van der Waals interaction between the atoms and itsinner distance in contrast with the main structure [12]. On theother hand, performing the experiment at nanoscale is very

ingen is the mostof view. Also, Suny betweeny solution

bending of nano-scale structures. Peddieson et al. [28] rthe nonlocal elasticity theory to develop a nonlocal Bernoulbeam model. After that, the nonlocal elasticity theory hawidely used due to its simplicity, high reliability and close agree-ment with MD simulations for mechanical analysis of carbon nano-tubes and graphene sheets [26,27,29]. However, on contrary to thehuge studies presented for mechanical analysis of one-dimensionalnanostructures such as nanobeams, nanorods and CNTs, only someworks are presented on two-dimensional ones such as graphenesheets. So, our understanding of the mechanical behaviors of

Corresponding author. Address: P.O.B. 9187144123, Mashhad, Iran.E-mail addresses:m.e.golmakani@mshdiau.ac.ir (M.E. Golmakani), rezatalabjavad@

mshdiau.ac.ir (J. Rezatalab).

Composite Structures 119 (2015) 238250

Contents lists availab

Composite S

sevical and chemical exfoliation of graphite, reduction of graphiteoxide, epitaxial growth on SiC, and chemical vapor deposition(CVD) on transition metals [10]. In order to study the mechanicalbehavior of nanostructures, it has been reported that the small

stress theory), the nonlocal elasticity theory of Erreasonable from the physical and atomic pointset al. [27] found that there exists an inconsistencistic simulation and the strain gradient elasticithttp://dx.doi.org/10.1016/j.compstruct.2014.08.0370263-8223/ 2014 Elsevier Ltd. All rights reserved.atom-for thest usedli/Eulers beendented structural, mechanical and electrical properties [2]. Thesuperior properties of these structures have led to its applicationsin many elds such as nano sensors, electrical batteries, superfastmicroelectronics, micro electromechanical systems (MEMS),nano-electromechanical systems (NEMS), biomedical, bioelectrical,reinforcement role at composites, etc. [39]. The most commonlyemployed methods for graphene manufacturing are micromechan-

theory [1721] and nonlocal elasticity theory [2225] are pro-posed. These theories are comprised of information about theinteratomic forces and internal lengths that is introduced as smallscale effect in nonlocal elasticity theory [25]. Chen [26] employinglattice dynamics and MD showed that among the size-dependentcontinuum theories (micromorphic theory, microstructure theory,micropolar theory, Cosserat theory, nonlocal theory and couple1. Introduction

Graphene, rst discovered by Geiis a monolayer (with a thickness of oized carbon atoms (covalently boarranged in a honeycomb lattice wit 2014 Elsevier Ltd. All rights reserved.

Novoselov [1] in 2004,.34 nm) of sp2 hybrid-to three other atoms)ique series of unprece-

difcult and expensive; also the atomistic simulation such asmolecular dynamics (MD) is highly computationally expensiveand cannot be applied for more number of atoms at surface. So,using some other methods is vital. In recent years, various size-dependent continuum theories such as couple stress theory [13],strain gradient elasticity theory [1416], modied couple stressNonuniform biaxial bucklingDifferential quadrature method

tions. The accuracy of the present results is validated by comparing the solutions with those reported bythe available literatures. Finally, inuences of small scale effect, aspect ratio, polymer matrix properties,Nonuniform biaxial buckling of orthotropin an elastic medium based on nonlocal

M.E. Golmakani a, J. Rezatalab a,b,aDepartment of Mechanical Engineering, College of Engineering, Mashhad Branch, IslambYoung Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad

a r t i c l e i n f o

Article history:Available online 6 September 2014

Keywords:Orthotropic nanoplateNonlocal Mindlin theory

a b s t r a c t

In this article, non-uniformded in a Pasternak elastic mgraphene sheet are subjecare derived in terms of genof orthotropic nanoplatesquadrature method (DQM)

journal homepage: www.elnanoplates embeddedindlin plate theory

zad University, Mashhad, Iran

xial buckling analysis of orthotropic single-layered graphene sheet embed-um is investigated using the nonlocal Mindlin plate theory. All edges of theto linearly varying normal stresses. The nanoplate equilibrium equationslized displacements based on rst-order shear deformation theory (FSDT)g the nonlocal differential constitutive relations of Eringen. Differentials been used to solve the governing equations for various boundary condi-

le at ScienceDirect

tructures

ier .com/locate /compstruct

results are compared with those of available references and molec-ular dynamics results. Excellent agreement between the results isobserved fortunately. Finally, inuences of many parameters suchas small scale effect, aspect ratio, polymer matrix properties, distri-bution of planar loading, mode numbers and boundary conditionsare discussed in details.

2. Formulation

The single-layered graphene sheet is simulated as a rectangularnano-plate and the elastic medium is modeled using an elasticfoundation, both Winkler-type and Pasternak-type elastic founda-tion. Fig. 1 shows the discrete model and continuummodel used inthis study for a single-layer graphene sheet resting on two-param-eter foundation with length lx, width ly and thickness h. As seen inFig. 1, a Cartesian coordinate system is placed at one corner of thegraphene sheet with the x,y and z axes along the length, width andthickness, respectively. Also, the linear variations of in-plane com-pressive loadings along the x and y axes denoted in Fig. 2 by Py(x)and Px(y), respectively, are dened by:

Pxy P1 1 k1 y

; Pyx P2 1 k2 x

posite Structures 119 (2015) 238250 239graphene sheets such as buckling is essential for their engineeringdesign and manufacture. Pradhan [30] employed higher-ordershear deformation theory using the nonlocal differential constitu-tive relations of Eringen in order to study buckling behavior of iso-tropic single-layered graphene sheet and obtained an analyticalsolution for critical buckling load of these nanoplates. Murmuand Pradhan [31] carried out biaxial buckling study of orthotropicgraphene sheets based on nonlocal Kirchhoff model and obtainedexplicit expression for modied buckling load. They [32] alsoimplemented nonlocal elasticity theory to investigate the bucklingbehavior of single-layered graphene sheet (SLGS) embedded in anelastic medium. Their results show that the buckling loads of SLGSare strongly dependent on the small scale coefcients and the stiff-ness of the surrounding elastic medium. Narendar [33] presentedthe buckling analysis of isotropic graphene sheets using the twovariable rened plate theory and nonlocal small scale effects. Heconcluded that the present theory, which does not require shearcorrection factor, is not only simple but also comparable to therst-order and higher order shear deformable theory. Samaeiet al. [34] investigated the effect of length scale on buckling behav-ior of an isotropic single-layer graphene sheet embedded in a Pas-ternak elastic medium using a nonlocal Mindlin plate theory andextracted explicit solution for the buckling loads of graphene sheet.Farajpour et al. [35] studied uniaxial buckling response of orthotro-pic nanoscale plates under linearly varying in-plane load via non-local Kirchhoff theory. They found that for the case of pure in-plane bending, the nonlocal effects are relatively more than othercases. Using Levys method, Pradhan [36] investigated bucklingbehavior of biaxially compressed graphene sheets based on non-local elasticity theory. He found that nonlocal parameter andboundary conditions signicantly inuence the critical bucklingloads of the small size graphene sheets. Ansari and Shamani [37]studied the biaxial buckling behavior of single-layered graphenesheets based on nonlocal plate models and molecular dynamicsimulations. They extracted the appropriate values of nonlocalparameter relevant to each type of nonlocal elastic plate modeland chirality. They also showed that the present nonlocal platemodels with their proposed proper values of nonlocal parameterhave an excellent capability to predict the biaxial bucklingresponse of SLGSs. Analooei et al. [38] used nonlocal continuummechanics and spline nite strip method due to elastic bucklingand vibration analysis of orthotropic nanoplates. Their resultsrevealed that small scale effect plays considerable role in the anal-ysis of small sizes plates. Murmu et al. [39] reported an analyticalstudy on the buckling of double-nanoplate-system (DNPS) sub-jected to biaxial compression using nonlocal elasticity theory. Sar-rami-Foroushani and Azhari [40] using the nonlocal classical platetheory and nite strip method studied vibration and buckling ofsingle and multi-layered graphene sheets. They observed that inthe nanoscale structures the critical buckling load and natural fre-quency are highly dependent on nonlocal parameter.

As far as knowledge of authors is concerned, there is no litera-ture considering the non-uniform biaxial buckling analysis oforthotropic nanoplate embedded in a Pasternak elastic mediumbased on the nonlocal Mindlin plate theory. Thus, this study is pre-sented considering the non-uniform biaxial buckling of embeddedgraphene sheet under various distribution of linearly planar loadalong the edges. Governing equations are derived based on Mindlintheory with considering orthotropic property and nonlocal theoryof Eringen in order to consider the size effects. Both Winkler-typeand Pasternak-type foundation models are employed to simulatethe interaction between the graphene sheet and the surroundingelastic medium. The created eigenvalue problem is solved using

M.E. Golmakani, J. Rezatalab / Comthe differential quadrature method for simply-supported boundarycondition, clamped boundary condition and combination of them.To verify the accuracy of the present consequences, simpliedly lx

k0P1 1 k2 xlx

1

where P1 is normal stress along the x direction at origin point andk0, k1, k2 are optional parameters dened to express the loading dis-tribution. According to the rst-order shear deformation theory(FSDT), the following displacement eld can be expressed as:

ux; y; z; t u0x; y; t zuxx; y; tvx; y; z; t v0x; y; t zuyx; y; twx; y; z; t w0x; y; t

8>: 2where u, v and w are the displacement components of point (x,y)along x, y and z directions, respectively at time t. Also, u0, v0 andw0 are the displacement functions of the middle surface of thegraphene sheet. Moreover, ux and uy are the local rotations for xand y directions, respectively. The general strain relations areexpressed as:Fig. 1. Graphene sheet in a Pasternak medium under biaxial buckling load.

positexxeyycxycyzcxz

26666664

37777775

@u0@x z @ux@x@v0@y z

@uy@

@u0@y @v0@x z @ux@y

@uy@x

@w0@y uy@w0@x ux

26666666664

37777777775

3

where eii and cij (i = x,y and j = x,y) are normal and shear strains,respectively.

Nonlocal continuum theory states that the stress at a referencepoint x in an elastic continuum depends not only on the strain at xbut also on the strains at all other points in the body [2225].According to Eringen [24] the nonlocal constitutive behavior of aHookean solid is represented by the following differential constitu-tive relation

sij 1 lr2rij; l e0a2; r2 @2=@x2 @2=@y2 4In which rij and sij are nonlocal and local stresses, respectively.Also, e0 is a material constant, a is the internal characteristic length(like CC bond length for carbon nanotube) and l is a scale coef-cient that describes the small scale effect for mechanical behavior

Fig. 2. Linear distribution of planar load along edges of graphene sheet.

240 M.E. Golmakani, J. Rezatalab / Comof nanostructures. So, based on Eqs. (3) and (4), the stressstrainequations of a rectangular orthotropic nanoplate are expressed asfollows:

rxxryyrxyryzrxz

26666664

37777775 lr2

rxxryyrxyryzrxz

26666664

37777775

Q11 Q12 0 0 0Q12 Q22 0 0 00 0 Q66 0 00 0 0 C44 00 0 0 0 C55

26666664

37777775

exxeyycxycyzcxz

26666664

377777755

where the coefcients of Qij and Cij are stiffness of the orthotropiclayer dened by:

Q11 E1

1 v12v21 ; Q21 Q12 v12E2

1 v12v21 ;

Q22 E2

1 v12v21 ; Q66 G12 C44 G23; C55 G13 6

Here, E1 and E2 are Youngs moduli in directions x and y, respec-tively, G12, G13, G23 are shear modulus and v12 and v21 denotePoissons ratios. Moreover, Ni (i = x,y,xy),Mi (i = x,y,xy) and Qi(i = x,y) are in-plane, moment and shear stress resultants of nonlocalelasticity, respectively, which are dened as:Nx;Ny;Nxy;Qx;Qy R h

2

h2rx;ry;rxy; k rxz; k ryzdz

Mx;My;Mxy R h

2

h2rx;ry;rxyzdz

8>: 7In which k = 5/6 is the transverse shear correction coefcient. Bysubstituting Eqs. (3) and (5) into Eq. (7), the nonlocal constitutiverelations in terms of displacements are obtained

NxxNyyNxyMxxMyyMxyQyQx

266666666666664

377777777777775lr2

NxxNyyNxyMxxMyyMxyQyQx

266666666666664

377777777777775

A11 A12 0 0 0 0 0 0A12 A22 0 0 0 0 0 00 0 A66 0 0 0 0 00 0 0 D11 D12 0 0 00 0 0 D12 D22 0 0 00 0 0 0 0 D66 0 00 0 0 0 0 0 H44 00 0 0 0 0 0 0 H55

266666666666664

377777777777775

@u0@x@v0@y

@v0@x @u0@y

@ux@x@uy@y

@ux@y

@uy@x

@w0@y uy@w0@x ux

2666666666666666664

3777777777777777775

8where Aij (i,j = 1,2,6), Dij (i,j = 1,2,6) and H44, H55 are the extensional,bending and shear stiffness of the graphene sheet dened by:

Aij;Dij h; h3

12

!Qiji;j1;2;6 ; H44 k C44h; H55 k C55h

9Using the principle of virtual displacements, the following govern-ing equations can be obtained [41]:

@Nxx@x

@Nxy@y

0@Nxy@x

@Nyy@y

0

@Qx@x

@Qy@y

qeff Nix@2w0@x2

2Niy@2w0@x@y

Nixy@2w0@y2

" # 0

@Mx@x

@Mxy@y

Qx 0@Mxy@x

@My@y

Qy 0 10

where Ni is pre-buckling in-plane stress resultant which is com-puted for this problem as follows:

Nix P1 1 k1yly

; Niy k0P1 1 k2

xlx

; Nixy 0 11

Furthermore, the effective transverse load qeff is dened as follows:

qeff q kww kpr2w 12In which kw, kp are Winkler and shear coefcients of foundationparameter, respectively. Also, q is external transverse load that van-ishes in this study and has been deleted from next relations. There-fore, the following governing equations can be obtained in terms ofthe displacements:

H55@ux@x

lkw kp H55 @2w0@x2

! H44

@uy@y

lkw kp H44 @2w0@y2

! kww0

P1 1 k1 yly

@2w0@x2

l @4w0@x4

l @4w0

@x2@y2

" #(

k0 1 k2 xlx

@2w0@y2

l @4w0@y4

l @4w0

@x2@y2

" # !)

e Structures 119 (2015) 2382502l k1ly @

3w0@x2@y

k0k2lx

@3w0

@x@y2 lkpr4w0 0

13For nuis immainparam

W

D 12 21 11

positD 1 A11 2 A11 A11

k1 m12E22E11 ; k2 E22E11

; k3 G12DE11 ; k4 kG23DE11

; k5 k G13DE11Eq. (13) can be rewritten as the following normalized form:

k5@ux@X

k4b@uy@Y

k5 gc1 c2@2W

@X2 k4 gc1 c2b2

@2W

@Y2

c1W gc2@4W

@X4 2b2 @

4W

@X2@Y2 b4 @

4W

@Y4

" #

P 1 k1Y @2W

@X2 g @

4W

@X4 gb2 @

4w0@X2@Y2

" #(

k0b21 k2X @2W

@Y2 gb2 @

4W

@Y4 g @

4W

@X2@Y2

" #

2gb2 k1 @3W

@X2@Y k0k2 @

3W

@X@Y2

!) 0

@2ux@X2

k1 k3b@2uy@X@Y

k3b2 @2ux@Y2

12 k5a2ux

12 k5a2 @W@X

k3@2uy@X2

k1 k3b @2ux

@X@Y k2b2

@2uy@Y2

12 k4a2uy

12 k4ba2 @W@Y

15

In order to complete the formulation, the governing equation (15)should be accompanied by a set of boundary conditions. The follow-ing cases of boundary conditions are used in this study:

(a) For simply supported boundary condition edges:

X 0;1 : W uy @ux@X 0Y 0;1 : W ux @uy@Y 0

(16

(b) For clamped boundary condition edges:

X 0;1 : W uy ux 0Y 0;1 : W ux uy 0

(17

3. Solution procedure

In this paper, in order to solve the equilibrium equations thedifferential quadrature method (DQM) is applied. This methodhas shown superb accuracy, efciency, convenience and greatpotential in solving complicated partial differential equations

[42]. Tmulatmerical solution method especially for nanoscale problems, itportant to use non-dimensional equations instead ofrelations. By introducing the following non-dimensionaleters

w0lx; X xlx ; Y

yly; a lxh ; b lxly ; g

e0alx

21 m m ; A E11h ; c kwl2x ; c kp ; P P1 14D11@2ux@x2

D12 D66@2uy@x@y

D66 @2ux@y2

H55 ux @w0@x

0

D66@2uy@x2

D12 D66 @2ux

@x@y D22

@2uy@y2

H44 uy @w0@y

0

M.E. Golmakani, J. Rezatalab / Comhe differential quadrature (DQ) method provides simple for-ion and low computational costs in contrast with the othernumerical methods such as dynamic relaxation (DR), nite differ-ence (FD), nite element (FE), etc. The DQ method was introducedby Bellman and Casti [43,44]. Many researchers have recently sug-gested the application of the DQM to the analysis of nanostructures[4548]. The basic idea of the differential quadrature method isbased on the approximation of partial derivative of a function withrespect to a space variable at a discrete point as a weighted linearsum of the function values at all discrete points in the wholedomain. Its weighting coefcients are only depending on the gridspacing. Therefore, every partial differential equation can be sim-plied to a set of algebraic equations using these coefcients[49]. DQM can be subdivided into several subsets with respect tothe applied function and satised types of boundary conditions.In this paper, polynomial function and direct substitutiontechnique are used to this end. Thus, the rst order derivativesof a function F(x,y) for point (xi,yj) from a rectangular sheetcan be expressed as a following linear sum of the function values[49]:

@Fxi; yj@x

XNk1

axikFxk; yj; i 1;2; . . . ;N 18

@Fxi; yj@y

XMr1

ayjrFxi; yr; j 1;2; . . . ;M 19

In which N andM are the number of grid points along x and y direc-tions, respectively. Also, ax and ay are obtained as follows [49]:

axij RxixixjRxj for i j

axii XNj1;i

axij i; j 1;2; . . . ;N

20

ayij PyiyiyjPyj for i j

ayii XMj1;i

ayij i; j 1;2; . . . ;M

21

where R(x) and P(y) are dened as:

Rxi YN

j1;ixi xj

Pyi YM

j1;iyi yj

22

Also, higher order partial derivatives are expressed as:

@nFxi; yj@xn

XNk1

cnik Fxk; yj 23

@mFxi; yj@ym

XMr1

cmjr Fxi; yr 24

@abFxi; yj@xa@yb

XNk1

XMr1

caik cbjr Fxk; yr 25

In which superscripts (m,n,a,b) are order of derivative. Also, c and care the weighing coefcients along x and y directions, respectively,which are dened as follows:

c1 ax; c1 ay 26

Cnij n axijcn1ii cn1ij

xixj

for i j

XN 27

e Structures 119 (2015) 238250 241Cnii j1;i

cnij i; j 1;2; . . . ;N

respectively. Moreover, the load value for all edges is equal. As

assumed as: Nix P 1 k1 yly ; Niy Nixy 0.

mentioned reference are attained by means of Kirchhoff nonlocal

positCmij m ayijcm1ii cm1ij

yiyj

for i j

Cmii Xmj1;i

cmij i; j 1;2; . . . ;M

28

With implementation of DQM into Eq. (15), the following equationscan be obtained:

k5XNk1

c1ik uxXk;Yj k4bXMr1

c1jr uyXi;Yr k5 gc1 c2

XNk1

c2ik Wk;j k4 gc1 c2 b2 XMr1

c2jr Wi;r c1Wi;j

gc2XNk1

c4ik Wk;j 2b2 XNk1

XMr1

c2ik c2jr Wk;r b4

XMr1

c4jr Wi;r

" #

P 1 k1YXNk1

c2ik Wkj gXNk1

c4ik Wkj gb2XNk1

XMr1

c2ik c2jr Wkr

" #(

k0b21 k2XXMr1

c2jr Wir gb2XMr1

c4jr Wir gXNk1

XMr1

c2ik c2jr Wkr

" #

2gb2 k1XNk1

XMr1

c2ik c1jr Wkr k0k2

XNk1

XMr1

c1ik c2jr Wkr

!) 0 29

XNk1

c2ik uxXk;Yj k1 k3 bXNk1

XMr1

c1ik c1jr uyXk;Yr

k3b2 XMr1

c2jr uxXi;Yr 12 k25uxXi;Yj

12k5a2XNk1

c2ik WXk;Yj 30

k3XNk1

c2ik uyXk;Yjk1k3bXNk1

XMr1

c1ik c1jr uxXk;Yrk2b2

XMr1

c2jr uyXi;Yr12k24uyXi;Yj12k4ba2XMr1

c1jr WXi;Yr 31

In order to obtain a better mesh point distribution, GaussCheby-shevLobatto technique has been dened as follows:

xi lx2 1 cos i1N1p

i 1;2; . . . ;Nyj ly2 1 cos j1M1p

j 1;2; . . . ;M

32In fact, this distribution leads to higher stability of equations set andincreases the convergence speed of solution.

4. Results and discussion

In this section the obtained results of biaxial buckling analysis ofthe orthotropic single-layered graphene sheet (SLGS) embedded inan elastic matrix is investigated based on nonlocal Mindlin platetheory. First, in order to show the efciency and accuracy of thepresent numerical analysis, the current results are compared withsome simpler ones obtained by MD and other nonlocal solutions.After that, the parametric study is presented for considering theeffects of different parameters such as small scale effect, aspectratio, polymer matrix properties, type of planar loading, modenumbers and boundary conditions on the buckling behavior of theSLGS.

4.1. Comparison study

242 M.E. Golmakani, J. Rezatalab / ComThe accuracy and validity of the present method concerning thenonlocal buckling loads of square/rectangular SLGS under uniaxialtheory and DQ method. It is obviously concluded that the resultsare of high accuracy and strongly conrm each other.

4.2. Parametric study

It is of importance to state that the material properties appliedin this study are dened in Table 5. Additionally, the exact value ofsmall scale effect is not known, so it is assumed varyingly between0 to 2 nm [50]. To conduct a thorough analysis of the bucklingresults, the variables P1, NP1, NP2 and PR, are taken into account.P1 (Pa m) represents critical buckling load; NP1 and NP2 denotenon-dimensional critical buckling loads and PR shows the criticalbuckling load ratio which are dened as:Also, in Table 3 the present results for the critical buckling loadratio of simply supported SLGS subjected to linear uniaxialcompression (e0a = 2 nm,k1 = 1) are compared with those of non-local Kirchhoff model reported by Farajpour et al. [35] for differentmodes and the mentioned geometry and material properties.

The comparative results in Tables 2 and 3 imply the highaccuracy and strong agreement between the results. It should benoted that the results reported by Farajpour et al. [35] are obtainedbased on Kirchhoff nonlocal model and numerical DQ method andthe slight difference between the results can be attributed to thisfact.

Sample 3. In the third example, uniaxial buckling of isotropic SLGSembedded in a Pasternak elastic medium with simply supportedboundary condition is examined. Material properties and geometryof the graphene sheet are dened as E = 1.06 TPa, m = 0.25,h = 0.34 nm and lx = 15 nm. The dimensionless Winkler and Paster-nak elastic foundation are assumed as kwl

4x

D 10 and kpl2x

D 2, respec-tively. As indicated in Table 4, the present results for the criticalbuckling load ratio (dened as nonlocal buckling load to localbuckling load) are compared with those reported by Pradhan andMurmu [32]. Similar to the previous example, the results of theindicated in Table 1, it is obvious that the present results are in goodagreement with those of the reported solutions by MD [37].

Sample 2. In the second example, the critical buckling load oforthotropic SLGS without elastic foundation is investigated underlinearly varying in-plane load. The ultimate results on the basisof dimensionless critical buckling load for different values of loadfactor and small scale effect are compared to the results of Faraj-pour et al. [35] in Table 2. The material properties applied in thisanalysis are comprised of E1 = 1765 GPa, E2 = 1588 GPa, m12 = 0.3,m21 = 0.27. Also, the length, width and thickness of the graphenesheet are 10, 20 and 0.34 nm, respectively. Furthermore, accordingto the model reported by [35], distribution of the in-plane load is and biaxial compression are examined by three comparison studiesas follows;

Sample 1. In the rst step of validation, the present results for theuniform nonlocal biaxial buckling load of isotropic square graphenesheet with a simply supported boundary condition are comparedwith those of molecular dynamic (MD) simulations as reported byAnsari and Sahmani [37] in Table 1. It is notable that the materialproperties of SLGS are taken as E = 1 TPa and m = 0.16. Also, thethickness, h, and the nonlocal effect, l, are 0.34 nm and 1.81 nm2,

e Structures 119 (2015) 238250NP1 P0A11 ; NP2 P0l

2x

D11; PR PNonLocal

PLocal33

gle-

positTable 1Comparison of present results with those of MD [37] simulation for orthotropic sin

M.E. Golmakani, J. Rezatalab / ComMoreover, it is noticed that POF and AOF in the following resultsintroduce the presence and absence of the elastic foundation,respectively.

boundary condition.

lx = ly (nm) Critical buckling load (Pa m)

MD results [37] Pre

4.990 1.0837 1.08.080 0.6536 0.610.77 0.4331 0.414.65 0.2609 0.218.51 0.1714 0.122.35 0.1191 0.126.22 0.0889 0.030.04 0.0691 0.033.85 0.0554 0.037.81 0.0449 0.041.78 0.0372 0.045.66 0.0315 0.0

Table 2Comparison of present results of Pl

2x

D11with those of nonlocal Kirchhoff model [35] for linea

Load factor (k1) e0a (nm)

0.0 0

0.0 Present 15.195 1Ref. [35] 15.246 1

0.5 Present 20.068 1Ref. [35] 20.136 1

1.0 Present 28.183 2Ref. [35] 28.283 2

1.5 Present 41.369 3Ref. [35] 41.536 4

2.0 Present 60.937 5Ref. [35] 61.225 5

Table 3Comparison between the present results with the ones reported by [35] for the critical buckcompression (e0a = 2 nm, k1 = 1).

Mode number 1 2 3

Present results 0.654 (N =M = 21) 0.353 (N =M = 21) 0.329 (N =MRef. [35] 0.654 0.353 0.328

Table 4Comparison of present results with those of Kirchhoff nonlocal model [32] foruniform uniaxial buckling of embedded nanoplate with simply supported boundarycondition.

ly (nm) e0a (nm) Buckling load ratio

Ref. [32] Present results

15 2 0.769 0.7691.5 0.854 0.8541 0.923 0.9280.5 0.980 0.981

25 2 0.840 0.8401.5 0.918 0.9021 0.954 0.9530.5 0.987 0.988

75 2 0.880 0.8801.5 0.920 0.9281 0.967 0.9660.5 0.992 0.991layered graphene sheet under uniform biaxial compression with simply supported

Percent of relative error (%)

sent results (N =M = 11)

749 0.8165523 0.1952356 0.5825645 1.3702751 2.1815239 3.9886917 3.1317707 2.3032561 1.3245453 0.8162372 0.1051313 0.6903

r uniaxial buckling and simply supported boundary condition.

e Structures 119 (2015) 238250 243Fig. 3(a) and (b) illustrate the non-dimensional critical bucklingload in terms of different values of small scale effect and the loadfactor k0 for simply supported and clamped boundary conditions,respectively. As seen, the critical buckling load decreases as a con-sequence of increasing the small scale effect. This reduction isobserved slight in the beginning but along with increase of non-local effect the slope of the reduction becomes steep. Moreover,the small scale parameter has more effect on the results forclamped boundary condition compared to simply supported one.It is also found that the slope remains rather unchanging consider-ing the cases in which the small scale parameter is assumedbeyond 0.75 nm in simply supported boundary condition. How-ever, this event happens for clamped boundary condition for thevalues greater than 0.5 nm.

Fig. 4(a) and (b) show the buckling load ratio in terms of non-local effect for different values of load factor k0 in simply supportedand clamped boundary conditions, respectively. It is observablethat with increase of small scale effect, the values of buckling loadratio decrease intensively and it consequently makes a huge differ-ence between nonlocal and local models. Analyzing the diagrams,the important point to focus on is that variations of the load factormake almost no effect on the results.

.5 1.0 1.5 2.0

4.740 13.526 11.893 10.1744.789 13.571 11.933 10.208

9.462 17.845 15.673 13.3909.528 17.905 15.726 13.435

7.286 24.904 21.732 18.4337.383 24.993 21.809 18.497

9.873 35.962 30.893 25.7840.034 36.107 31.014 25.882

8.339 51.720 43.483 31.3448.616 51.961 43.681 29.400

ling load ratio of different modes of simply supported SLGS subjected to linear uniaxial

4 5 6

= 21) 0.219 (N =M = 21) 0.203 (N =M = 21) 0.136 (N =M = 29)0.218 0.201 0.132

In Figs. 5 and 6, the same analysis is repeated as Figs. 3 and 4with only difference that elastic foundation does not exist here.The diagrams reveal that the conclusions in previous case are againvalid in the present one so that regardless of variations of the slopeof the diagrams, absence of elastic medium noticeably affects thereduction of critical buckling load. Furthermore, it is observed thatthe reduction effect of small scale effect on the buckling loaddecreases in the absence of elastic foundation. Also, the effect ofsmall scale parameter on the results decreases with raising the val-ues of k0 for both with and without elastic foundation. In addition,the differences between the local and nonlocal results have thegreater values in clamped case compared to simply supported one.

Figs. 7 and 8 illustrate the small scale effect parameter on non-dimensional buckling load (NP2) for different boundary conditionsand rst-to-fourth mode shapes of a graphene sheet imposed touniform and non-uniform biaxial loadings, respectively. Propertiesand geometry of the applied graphene sheet in this diagram are

brought in Table 5 for kp = 0, kw = 1.13e16 Pa. The results pointout that in local state (e0a = 0), a signicant difference existsbetween the results obtained from various buckling mode shapes.But, by enlarging the small scale parameter, the aforementioneddifference sharply decreases. So that, the decreasing aspect of thesmall scale parameter in higher buckling modes is more consider-able and the curvature of these curves is sharper than the oneswith lower buckling modes. As a result, by increasing the smallscale parameter, the curves converge to a constant value. Clearly,changing the boundary condition of the edges from simply sup-ported to the clamped accelerates the convergence of the dia-grams. Hence, it can be concluded that when graphene sheetswith large nonlocal parameters are involved there is no urge to cal-culate the buckling loads of high modes. The point which is of cer-tain concern is that when the curves become closer to each other invarious modes, transition phenomenon takes place very rapidlyfrom one mode to another by slightly increasing the compressive

Table 5Geometrical and material properties of graphene nanoplate, elastic foundation properties and shear correction factor used in the parametric study.

E1 (Gpa) E2 (Gpa) G12 (Gpa) m12 lx (nm) ly (nm) h (nm) kw Gpanm

kp (Pa m) k

1765 1588 694.88 0.3 10.2 10.2 0.34 1.13 1.13 0.8333

wit

244 M.E. Golmakani, J. Rezatalab / Composite Structures 119 (2015) 238250Fig. 3. Non-dimensional buckling load (NP2) based on small scale parameter (e0a)boundary condition and (b) clamped boundary condition.Fig. 4. Buckling load ratio (PR) based on the small scale parameter (e0a) and various lcondition and (b) clamped boundary condition.h various load factor k0 and presence of elastic medium for (a) simply-supportedoad factor k0 for presence of elastic medium with (a) simply-supported boundary

Fig. 5. Non-dimensional buckling load (NP2) based on small scale parameter (e0a) with various load factor k0 and absence of elastic medium for (a) simply-supportedboundary condition and (b) clamped boundary condition.

Fig. 6. Buckling load ratio (PR) based on the small scale parameter (e0a) and various load factor k0 for absence of elastic medium with (a) simply-supported boundarycondition and (b) clamped boundary condition.

Fig. 7. Effect of small scale parameter on the non-dimensional buckling load (NP2) for different boundary conditions and rst-to-fourth mode shapes of a graphene sheetimposed to uniform biaxial loadings (kp = 0,kw = 1.13e16 Pa, k0 = k1 = 1, k2 = 0).

M.E. Golmakani, J. Rezatalab / Composite Structures 119 (2015) 238250 245

posit246 M.E. Golmakani, J. Rezatalab / Comload and therefore, the process of calculations demands a morecrucial accuracy. What is more to conclude is that owing to sharp-ness of the curvatures in higher modes, it is still possible to employlinear approximation in order to model the curves at small scaleparameter values beyond 1 nm. Also, a concurrent analysis ofFigs. 7 and 8 reveals that the type of loading plays no role in theobtained conclusions of this section and increase of k1 does notnecessarily lead to the growth of critical buckling load value.

In Fig. 9(a) and (b), effect of aspect ratio (b) on non-dimensionalcritical buckling load is investigated for different values of the loadfactor in simply supported boundary condition and width of 20 nmwhen no elastic foundation exists. According to Ref. [37] value ofnonlocal parameter is deemed to be 1.81 nm2. The same analysisis similarly carried out in case of clamped boundary condition inFig. 10(a) and (b). All the diagrams unanimously demonstrate thatincreasing the aspect ratio at constant width will consequently

Fig. 8. Effect of small scale parameter on the non-dimensional buckling load (NP2) for dimposed to non-uniform biaxial loadings (kp = 0, kw = 1.13e16 Pa, k0 = k1 = 1, k2 = 0).

Fig. 9. Effect of aspect ratio (b) on the non-dimensional buckling load (NP1) in simply-supfor (a) k0 = 1 and (b) k0 = 2.e Structures 119 (2015) 238250cause a reduction to critical buckling load. It is also evident thatas the aspect ratio grows increasingly slope of the curve tendsdownwards to zero meaning that effect of length variations canbe looked over for extensive values of length. It is inferred throughanalysis of diagrams that increase of coefcients k1 and k2 imposesan eventual increase on critical buckling load and also slope of thecurves. Analogy of the curves suggests this conclusion that unlikethe two load factors k1 and k2, increasing k0 results in decrease ofcritical buckling load. Concerning the load factors, it should be keptin mind that increase of k1 and k2 reduce the resultant of compres-sive load whereas increase of k0 signies growth of it. Besides, Ana-lytic view of the diagrams determines that critical buckling load isobserved greater in clamped boundary condition rather than insimply supported one.

Fig. 11(a) and (b) shows the amount of critical buckling load interms of aspect ratio at width of 20 nm with various load factor (k1

ifferent boundary conditions and rst-to-fourth mode shapes of a graphene sheet

ported boundary condition and various load factor (k2,k1) without elastic foundation

Fig. 10. Effect of aspect ratio (b) on the non-dimensional buckling load (NP1) in clamped boundary condition and various load factor (k2,k1) without elastic foundation for (a)k0 = 1 and (b) k0 = 2.

Fig. 11. Non-dimensional buckling load (NP1) based on the aspect ratio (b) for various load factor (k1,k2) with k0 = 2 and presence of elastic medium: (a) simply-supportedboundary condition and (b) clamped boundary condition.

Fig. 12. Variation of critical buckling load (P1) based on the square nanoplate length (lx = ly) for different types of planar load distribution and absence of elastic medium: (a)simply-supported boundary condition and (b) clamped boundary condition.

M.E. Golmakani, J. Rezatalab / Composite Structures 119 (2015) 238250 247

posit248 M.E. Golmakani, J. Rezatalab / Comand k2) and presence of elastic foundation for simply supportedand clamped boundary conditions, respectively. It is obvious thatthe conclusion at hand for Figs. 9 and 10 replicates its validityfor this case as well. As indicated, increasing the variations of theload distribution (k1 and k2) raises the effect of aspect ratio onthe results. Also, by making a comparison between Figs. 911 itcan be understood that presence of elastic foundation brings downthe slope and values of the curves, thus the diagrams have becomeindependent of length at smaller aspect ratios.

In Fig. 12(a) and (b), effect of growth in length and width(length of square nanoplate) is discussed for ve different kindsof loading on critical buckling load in simply supported andclamped boundary conditions, respectively. As expected, increaseof length has led to decrease of critical buckling load and so theplane is faced with buckling phenomenon in lower compressive

Fig. 13. Non-dimensional buckling load (NP2) based on Winkler module (kw) for variocondition.

Fig. 14. Non-dimensional buckling load (NP2) variations of rst-to-fourth mode shapes odifferent small scale parameters (k0 = 1,k1 = k2 = 0,kp = 1.13 Pa m).e Structures 119 (2015) 238250loads. It is clear that decreasing slope of the diagrams is very steepin the beginning and tends downwards zero in the following.Hence, controlled variation of dimensions for large graphenesheets makes no certain impact on the results and it can beignored. It is also concluded that resultant of compressive loadand how it is distributed both affect the critical buckling load butin square graphene sheets what is of importance is actually thevalue of load not the distribution.

Fig. 13(a) and (b) indicate the effect of Winkler elastic founda-tion on critical buckling load for different values of nonlocalparameter in simply supported and clamped boundary conditions,respectively. Furthermore, as it is noticed, growth of Winkler mod-ule raises the critical buckling load with an unchanging slope. Byincreasing the nonlocal parameter, a decrease of critical bucklingload occurs for different values of Winkler elastic foundation. In

us nonlocal parameter (l) with (a) simply-supported and (b) clamped boundary

f a simply supported graphene sheet in terms of dimensionless Winkler module and

es o

positaddition, with increase of Winkler elastic foundation, the differ-ence of obtained critical buckling load becomes greater betweennonlocal and local theories.

Fig. 14 shows non-dimensional buckling load (NP2) variations ofrst-to-fourth mode shapes of a simply supported graphene sheetin terms of dimensionless Winkler module for different small scaleparameters. In this study, a uniform biaxial type of loading hasbeen utilized and the geometrical features are in accordance withTable 5. As observed, the curves at hand show linear behavior

Fig. 15. Non-dimensional buckling load (NP2) variations of rst-to-fourth mode shapfor different small scale parameters (k0 = 1,k1 = k2 = 0,kw = 2.26e16 Pa).M.E. Golmakani, J. Rezatalab / Comand posses constant slopes. Thus, by determining the slope of eachdiagram, it is possible to identify the critical buckling load valuewithout any complicated calculations involved. Furthermore, itcan be observed that small scale parameter does not have anynoticeable effect on the slope of the diagrams. In other words,the small scale parameter directs the curves toward each otherand does not inuence their slopes. Obviously, increasing the Win-kler module leads to convergence of the results for differentmodes. In addition, unlike the increase of e0a which results inreduction of the critical buckling load, increase of the Winklermodule makes it grow.

In Fig. 15, non-dimensional buckling load (NP2) variations of asimply supported graphene sheet subjected to uniform biaxialloading is depicted for different values of shear modules of elasticfoundation along with different modes and small scale parameters.It is seen that kp. adds to the critical buckling load in a linear trend.Another point to bear in mind is that the slope of all the diagramsare the same for different values of small scale parameter exceptfor the fact that their starting points are different from each other.

5. Conclusions

In this study, biaxial buckling of a single layered rectangulargraphene sheet in a medium elastic is studied with linear load dis-tribution along the edges. To full this objective, equilibrium equa-tions are obtained through Mindlin orthotropic plate models, andEringen nonlocal elasticity theory has been applied to considerthe small scale effect parameter. Governing equation for differentboundary conditions are solved using DQ method. In order tovalidate the results, some comparison studies are carried outbetween the present results and those reported by MD and otherliteratures. Then, the effect of some important parameters is inves-tigated and some of the most signicant results are presented asfollows:

The study is representative of this fact that increase of smallscale parameter leads to reduction of critical buckling load. Thisreduction shows nonlinear behavior at rst and then it turns to

f a simply supported graphene sheet in terms of shear modules of elastic foundation

e Structures 119 (2015) 238250 249linear. The reduction effect of small scale effect on the buckling loaddecreases in the absence of elastic foundation.

The effect of small scale parameter on the results decreaseswith raising the values of k0 for both with and without elasticfoundation

Rise of length-to-width ratio at a constant width creates areduction in critical buckling load at the beginning, however,as the increase of length continues the slope tends downwardszero.

The fact that how the loading is distributed along the edgesmakes a crucial effect on creation of mode shapes and this iswhile the critical buckling load is mostly affected by the resul-tant in-plane loads rather than its distribution.

The extension of length of the square graphene sheet and a shiftfrom clamped to simply supported boundary condition bringsabout a reduction in critical buckling load. Besides, at high val-ues of length, the critical buckling load is independent of thelength variation.

Critical buckling load increases at a constant slope as a result ofincreasingWinkler and shear modulus of the elastic foundation.

By increasing of shear elastic foundation, difference betweenthe obtained critical buckling loads based on nonlocal and localtheories does not vary.

Increasing the variations of the load distribution (k1 and k2)raises the effect of aspect ratio on the results.

The decreasing effect of the small scale parameter on highermodes is more prominent than lower ones, thus, the signi-cance of taking nonlocal effect into analysis of high modes goesup.

The small scale parameter if increased and shifting the bound-ary conditions from simply supported to clamped cause criticalload values of various modes to become close to each other anddirect the results to a constant value.

With increase of Winkler module the obtained results for differ-ent modes converge to each other at the beginning and if thisincrease is continued the mode shapes might change with thesame number of modes.

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Nonuniform biaxial buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal Mindlin plate theory1 Introduction2 Formulation3 Solution procedure4 Results and discussion4.1 Comparison study4.2 Parametric study

5 ConclusionsReferences