Int. J. Nano Dimens., 6(5):525-537, (Special Issue) 2015

525

ABSTRACT: In this article, finite difference method (FDM) is used to solve sixth-order derivatives of differentialequations in buckling analysis of nanoplates due to coupled surface energy and non-local elasticity theories. Theuniform temperature change is used to study thermal effect. The small scale and surface energy effects are added into thegoverning equations using Eringen’s non-local elasticity and Gurtin-Murdoch’s theories, respectively. Two differentboundary conditions including simply-supported and clamped boundary conditions are investigated. The numericalresults are presented to demonstrate the difference between buckling obtained by considering the surface energy effectsand that obtained without the consideration of surface properties. The results show that the finite difference methodcan be used as a powerful method to determine the mechanical behavior of nanoplates. In addition, this method can beused to solve higher-order derivatives of differential equations with different types of boundary condition with littlecomputational effort. Moreover, it is observed that the effects of surface properties tend to increase in thinner and largernanoplates; and vice versa.

Keywords: Buckling analysis; Finite difference method; Nanoplate; Non-local elasticity theory; Surface energy theory.

Int. J. Nano Dimens., 6(5): 525-537, 2015, (Special Issue for NCNC, Dec. 2014, IRAN)

DOI: 10.7508/ijnd.2015.05.010

*Corresponding Author: Morteza KarimiEmail: morteza.karimi@me.iut.ac.irTel.: (+98) 311391-5237Fax: (+98) 311391-5216

Received 19 April 2015; revised 18 July 2015; accepted 5 August 2015; available online 01 December 2015

INTRODUCTIONNanostructures such as nanoplates, nanobeams and

nanotubes have attracted worldwide attention, becauseof their superior mechanical, thermal and electricalperformances. These small-scale nanostructures havewide-ranging applications in many areas such ascommunications, and in mechanical and biologicaltechnologies. In recent years, there has beensignificant interest in developing micro/nanomechanical and micro/nanoelectromechanicalsystems (MEMS/NEMS), such as capacitive sensor,switches, actuators, and so on. These devices can becontributed to novel technological developments inmany fields leading to industrial revolution [1, 2]. Thecontinuum mechanics approaches are widely preferreddue to their simplicity. The main feature of structuresis their high surface-to-volume ratio, which makeselastic response of their surface layers to be different

Finite difference method for sixth-order derivatives of differentialequations in buckling of nanoplates due to coupled surface energy and

non-local elasticity theories

M. Karimi*, A. R. Shahidi

Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

from macroscale structures. That is why Gurtin andMurdoch [3, 4] developed a theoretical framework basedon continuum mechanics concepts that included theeffects of surface and interfacial energies. In theirapproach, the surface was modeled as a mathematicallayer of zero thickness perfectly bonded to anunderlying bulk substance. The surface (interface) hadits own properties, which were different from those ofthe bulk material. These properties, such as surfaceenergies, affect on physical, mechanical, and electricalproperties as well as mechanical response of thenanostructure and can cause interesting behaviors. Forexample, Assadi [5] and Assadi et al. [6] investigatedforce and free vibration of nanoplates consideringsurface energy effects. Moreover, Assadi and Farshi[7, 8] studied surface energy effects on the vibrationand buckling of circular nanoplate. They showed thatsurface energy effects could have significant effectson the nanostructures. Moreover, increasing in thethickness causes reduction of the surface energyeffects. Ansari and Sahmani [9] analyzed bending and

Research Paper

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M. Karimi and A. R. Shahidi

buckling of nanobeams by considering surface stresseffects and normal stresses, using different beamtheories. They showed that the difference between thebehaviors of nanobeams predicted by with and withoutsurface effects would depend on the magnitudes ofthe surface elastic constants. In these works [5-9],because of using classical solutions, e.g., Navier’smethod as used for simply-supported boundaryconditions, the above-mentioned researchers were notable to study other boundary conditions.

Challamel and Elishakoff [10] analyzed buckling ofnanobeams, incorporating the surface stress effectsinto the Euler–Bernoulli and Timoshenko beam theories.They explained that the surface elasticity effects maysoften a nanostructure for some specific boundarycondition. Recently, Karimi et al. [11] studied size-dependent free vibration analysis of rectangularnanoplates with the consideration of surface energyeffects using finite difference method (FDM). Theyreported that the effects of surface properties tend todiminish in thicker nanoplates, and vice versa. Recently,Ansari et al. [12] investigated surface energies effectson the buckling, and maximum deflection of nanoplatesusing first order shear deformation plate theory andgeneralized differential quadrature method (GDQM).Moreover, they [13] analyzed forced vibration ofTimoshenko nanobeams based on the surface stresselasticity theory using GDQM. They reported that thesignificance of surface energy effects on the responseof nanoplate would rely on its size, type of edgesupports, and the selected surface constants. On theother hand, Mouloodi et al. [14, 15] analyzed the surfaceenergy effects on the bending and vibration ofmulticrystalline nanoplate. Furthermore, Wang andWang [16] studied the surface energy effects on thebending and vibration of Mindlin nanoplates. In theseworks [14-16], the finite element method (FEM) wasutlized to solve governing equations based on classicalbeam and plate theories. They explained that dependingon the boundary conditions, the deflections andfrequencies of nanoplates had a dramatic dependenceon the surface energy effects. In these works [5-16],the effect of non-local parameter was not consideredinto governing equations. On the other hand, the orderderivatives of differential equations in buckling andvibration were less or equal fourth-order.

In nanoscale, the small scale effects cannot beignored. That is why, some researchers investigatedthe effects of surface energy effects on the buckling

and vibration considering nonlocal elasticity theory.For example, Wang and Wang [17, 18] studied bucklingand vibration of rectangular nanoplates including bothsurface energy and non-local elasticity using Navier’smethod. They explained that by increasing the valueof non-local parameter, the surface energy effects coulddecrease. Farajpour et al. [19, 20] analyzed the surfaceenergy and non-local effects on the axisymmetricbuckling and vibration of circular graphene sheets inthermal environment using DQM. They [19, 20] studiedonly clamped boundary condition. They showed thatthe size effects would decrease with an increase in thevalue of surface residual stresses. In these works [17-20], the small scale effects were only considered forthe bulk of nanoplates.

On the other hand, some researchers investigatedsurface energy effects on the buckling and vibrationof nanobeams and nanoplates including non-localparameter into both nanoplate bulk and surface. Forexamples, Mahmoud et al. [21] studied static analysisof nanobeams including surface energy effects intogoverning equations using non-local finite elementmethod. Moreover, Eltaher et al. [22] combined effectsof non-local and surface energy on the vibration ofnanobeams. They used an efficiently finite elementmodel to descretize nanobeam domain and solve theequation of motion numerically. They showed that thesurface properties have significant effects when thethickness of beam structure approaches to its intrinsiclength. Recently, Karimi et al. [23] Combined surfaceenergy effects and non-local two variable refined platetheories on the buckling and vibration of rectangularnanoplates using DQM. They explained that the non-local effects on the shear buckling and vibration aremore important than that of biaxial buckling andvibration. Shokrani et al. [24] investigated bucklinganalysis of double-orthotropic nanoplates embeddedin elastic media based on non-local two-variable refinedplate theory using GDQM without considering surfaceenergy effects. They showed that the effects of non-local parameter for shear buckling are more noticeablethan that of biaxial buckling.

It should be noted that, with considering non-localparameter for only nanoplate bulk [17-20], in the finalequations displacement nanoplates sixth-orderderivatives appear. In this case, for solving equationswith numerical methods we need three boundaryconditions for any edge of nanoplates. But in the work[19, 20] two boundary conditions were considered. In

Int. J. Nano Dimens., 6(5):525-537, (Special Issue) 2015

527

this case, the answer of equations might be wrong. Butwhen the non-local effects considered for both bulkand surface [21-23], in the final equations displacementnanoplates fourth-order derivatives appear. In thiscase, for solving equations with numerical methodswe need two boundary conditions for any edge ofnanoplates. In this case, the numerical methods forsolving equations with various boundary conditionscould be done easily. On the other hand, the answer ofequations for two cases, consideration of non-localparameter for only bulk and for both bulk and surfaceof nanoplates are different.

In recent years some researchers studied influenceof magnetic field on the nanofluids. For example,Sheikholeslami and Rashidi [25] investigated effect ofspace dependent magnetic field on free convection ofFe3O4-water nanofluid. They indicated thatenhancement in heat transfer decrease with increasingof Rayleigh number, while for two other activeparameters different behavior was observed. Moreover,Sheikholeslami and Domiri Ganji [26] analyzednanofluid flow and heat transfer between parallel platesconsidering Brownian motion using DTM. Theyreported that skin friction coefficient would increasewith increasing of the squeeze number and Hartmannnumber but it was decreased with increasing ofnanofluid volume fraction. Recently, Sheikholeslami etal. [27] studied lattice Boltzmann method for simulationof magnetic field effect on hydrothermal behavior ofnanofluid in a cubic cavity. They showed thatenhancement in heat transfer has direct relationshipwith Hartmann number, while it had inverse relationshipwith Rayleigh number. In addition, they [28] investigatedmagnetic field effects on natural convection around ahorizontal circular cylinder inside a square enclosurefilled with nanofluid. They reveal that the averageNusselt number was an increasing function ofnanoparticle volume fraction as well as the Rayleighnumber, while it was a decreasing function of theHartmann number. Soleimani et al. [29] studied naturalconvection heat transfer in a nanofluid filled semi-annulus enclosure. They reported that there was anoptimum angle of turn in which the average Nusseltnumber is maximum for each Rayleigh number.

The main objective of this article is to numericallyinvestigation. In the present work, FDM is used tosolve sixth-order derivatives of differential equationsin buckling analysis of nanoplates due to coupledsurface energy and non-local elasticity theories. The

uniform temperature change is used to study thermaleffect. In this article, small-scale and surface effectsare introduced using the Eringen’s non-local elasticityand Gurtin-Murdoch’s theory, respectively. Twodifferent boundary conditions including simply-supported and clamped boundary conditions areinvestigated. First, the governing differential equationis introduced according to the literature, and then thisequation is solved using the FDM to obtain thebuckling for several combinations of boundaryconditions. To verify the accuracy of the resultsobtained by the FDM, these results are compared withthe results of the analytical approach.

EXPERIMENTALCoupling surface energy and non-local elasticity theories

By using non-local elasticity theory [30] anddisregarding body forces, the stress equilibriumequation for a linear homogeneous non-local elasticbody can be written as:

VxxdVxCxx klijklnlij )()(),( (1)

Here, ij

nl, ij and C

ijkl are the stress, strain and fourth-

order elasticity tensor, respectively. (|x-x’|, ) isregarded as a non-local modulus, %x-x’% represents aEuclidean distance and is a material constant =e

0 a

0

/l) depending on the internal characteristics length, aand external characteristic length, l. Parameter a is alattice parameter, granular size, or the distance betweenC–C bonds. Parameter e

0 is estimated such that

relations of non-local elasticity model could providesatisfactory atomic dispersion curves of plane wavesby using approximations from atomic lattice dynamics.Since a constitutive law of integral form is difficult toimplement, a simplified differential form of Eq. (1) isused as the basis:

klijklnlij Cg )1( 22

(2)

In the above equation, 2=(∂2/∂x2)+(∂2/∂y2) is the

Laplacian. g2=(e0 a

0)2 is the non-local parameter. The

force and moment resultants of the non-local elasticitycan be defined as [17, 18]:

dzNNNh

h

nlyx

nlyy

nlxxyxyyxx

2

2),,(),,(

dzzMMMh

h

nlyx

nlyy

nlxxyxyyxx

2

2),,(),,(

(3)

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528

M. Karimi and A. R. Shahidi

Here, h denotes the thickness of the nanoplates.σ

xxnl, σ

yynl, and σ

xynl are the non-local stresses, where the

superscript nl denotes “non-local”. The displacementcomponents in the x and y directions are obtained fromthe Kirchhoff’s plate model, as follows [11, 17, 18]:

,x

wzu

y

wzv

(4)

The resulting strain components in the Cartesiancoordinates can be derived by using the relations of Eq.(3), which are always considered to be the same for bothnanoplates bulk and surface [11, 17, 18]:

2

2

x

wzxx

, 2

2

y

wzyy

,yx

wzyx

2

2 (5)

As was discussed in the previous section, the externallayers of all the elastic solids have additional materialproperties. Based on the Gurtin-Murdoch theory of elasticsolid surfaces [3], the stress-strain relations of surfacenanoplates are expressed by [11, 17, 18, 23]:

)2

()())(( ,,,,

hzuuuu sssssss

)2

()())(( ,,,,

hzuuuu sssssss

(6)

)2

(h

zx

wss

where α=β==1,2. In Eq. (6), λs and μs are the surfaceLame constants. Substituting the strain components fromEq. (5) into Eq. (6) yields additional surface stresscomponents in terms of transverse displacement (w), asfollows[11, 17, 18, 23]:

2

2

2

2

,,

)()2(2

)()2(

y

w

x

wh

vu

ssss

yss

xsss

xx

s

2

2

2

2

,,

)()2(2

)()2(

x

w

y

wh

uv

ssss

xss

ysss

yy

s

(7)

yx

wh

vvu

ss

xs

xyxy

ss

2

,,,

)2(2

)(

x

wss

xz

,y

wss

yz

In the above relations, superscripts + and representthe upper and lower surfaces of nanoplate,respectively. The resultant stresses are obtained bythe following integral equations [11, 17, 18, 23]:

dzzh

Mh

h

nlxx

xx

s

xx

sxx

2

22)(

dzzh

Mh

h

nlyy

yy

s

yy

syy

2

22)(

dzzh

Mh

h

nlyx

yx

s

yx

syx

2

22)(

(8)

Using the principle of virtual work, the governingequations of nanoplates subjected to thermal effect canbe obtained as [11, 17, 18]:

0,, yyxxxx NN

0,, xyxyyy NN

0)(

)()(

)(2

,

,,

,,,,

xyTxyyx

yyTyyyyxx

Txxxx

z

s

z

syxyxyyyyxxxx

wNN

wNNwNN

MMM

(9)

where α=x, y. the non-local stress-strain relations ofbulk material subjected to thermal effect are written as[17-20, 23]:

0

)1(

)1(

00

0)1()1(

0)1()1(22

22

22

TE

TE

G

EE

EE

g

yx

yy

xx

nlyx

nlyy

nlxx

nlyx

nlyy

nlxx

(10)

Int. J. Nano Dimens., 6(5):525-537, (Special Issue) 2015

529

where E, v, and G denote the elastic modulus,Poisson’s ratio, and shear modulus of the nanoplate,respectively. where α and ΔT represent the thermalexpansion coefficient and temperature change,respectively. Using Eqs. (3), (5) and (10), we obtain[17-20, 23]:

yx

wy

wx

w

D

M

M

M

g

M

M

M

yx

yy

xx

yx

yy

xx

2

2

2

2

2

22

22

100

01

01

(11)

where D=Eh3/12(1-ν2) is the classical flexural rigidityof the nanoplate without the consideration of surfaceeffects. Thermal effect can cause bending and bucklingin the nanoplates. According to the theory of thermalelasticity, the thermal force can be expressed as [17-20]:

ThE

NN Tyy

Txx

)1(

(12)

0TyxN

Using Eqs. (7), (9), (11) and (12) and assuming thesame properties for the upper and lower surfaces, wecan obtain the following governing equation:

0))((

)2())1(

2(

)())1(

2(

)33(

2)2()

2(

2

22

2

2

2

2

2

2

22

22

2

2

4

4

22

4

4

4

22

2

2

2

6

6

42

6

24

6

6

6

22

4

4

22

4

4

42

y

wN

yx

wN

x

wN

y

w

x

w

gy

wN

yx

wN

x

wN

y

w

yx

w

x

wT

hE

gy

w

x

wT

hE

y

w

yx

w

yx

w

x

w

Ehg

y

w

yx

w

x

wEhD

yyxyxx

yyxyxx

s

s

ss

(13)

where Es=2μs+λs. For simplicity, the followingnomenclature is used for uniform biaxial compressionratios (see Fig.1)

NNxx , NN yy , 0xyN (14)

The finite difference method is a powerful methodfor solving differential equations. Recently, KaramoozRavari et al. [31, 32] studied non-local effect on thebuckling of rectangular, circular, and annular nanoplatesusing finite difference method. Moreover, Karimi et al.[11] investigated size-dependent free vibration analysisof rectangular nanoplates with the consideration ofsurface energy effects but without considering non-local effect using finite difference method. The finitedifference method replaces the nanoplate differentialequation and the expressions defining the boundaryconditions with equivalent differences equations. Thesolution of the bending problem thus reduces to thesimultaneous solution of a set of algebraic equationswritten for every nodal point within the nanoplate. Fig.2 shows a rectangular nanoplate and the grid pointswhich will be used in the finite difference method. Byusing this method, Eq. (15) can be used to estimate thederivative of the transverse displacement, w, for thei,j-th point as a function of its neighboring points.

)(2

1),1(),1( jiji

x

wwrdx

dw

)(2

1)1,()1,( jiji

y

wwrdy

dw

)2(1

),1(),(),1(22

2

jijijix

wwwrdx

wd

)2(1

)1,(),()1,(22

2

jijijiy

wwwrdy

wd (15)

)46

4(1

),2(),1(),(

),1(),2(44

4

jijiji

jijix

www

wwrdx

wd

)46

4(1

)2,()1,(),(

)1,()2,(44

4

jijiji

jijiy

www

wwrdy

wd

)(2

41

)1,(),1()1,(),1(

),()1,1()1,1(

)1,1()1,1(

2222

4

jijijiji

jijiji

jiji

yxwwww

www

ww

rrdydx

wd

Int. J. Nano Dimens., 6(5):525-537, (Special Issue) 2015

530

M. Karimi and A. R. Shahidi

),3(),2(),1(),(

),1(),2(),3(

66

6

61520

1561

jijijiji

jijiji

x wwww

www

rdx

wd

\

)3,()2,()1,(),(

)1,()2,()3,(

66

6

61520

1561

jijijiji

jijiji

y wwww

www

rdy

wd

)1,2(),2()1,2(

)1,1(),1()1,1(

)1,(),()1,(

)1,1(),1()1,1(

)1,2(),2()1,2(

2424

6

2

484

6126

484

2

1

jijiji

jijiji

jijiji

jijiji

jijiji

yx

www

www

www

www

www

rrdydx

wd

(15)

)2,1()2,()2,1(

)1,1()1,()1,1(

),1(),(),1(

)1,1()1,()1,1(

)2,1()2,()2,1(

4242

6

2

484

6126

484

2

1

jijiji

jijiji

jijiji

jijiji

jijiji

yx

www

www

www

www

www

rrdydx

wd

Here rx and r

y are the distance between two grid

points in the x and y directions, respectively.Substituting Eq. (15) into Eq. (13) and developing acomputer code in MATLAB the governing equationare solved.

(a)

(b)

h

Nanoplate

Surface layer

x

y

z

μs , s, τs

E, v Bulk

Surface

Length (a)

Width (b)TΔ

Fig. 1 (a): The geometry of rectangular nanoplate with surface layers and thermal loading, (b) biaxial loading

Int. J. Nano Dimens., 6(5):525-537, (Special Issue) 2015

531

4),2(),1(

),(),1(),2(4

222

14

641

)1(2

2

yjiji

jijijix

ss

rww

wwwr

gThE

gEh

D

)1,(),1()1,(),1(22

),()1,1()1,1()1,1()1,1(22

)2,()1,(),()1,()2,(

4

42

464

jijijijiyx

jijijijijiyx

jijijijiji

wwwwrr

wwwwwrr

wwwww

(16)

),3(),2(),1(),(

),1(),2(),3(

6

22

)1,(),()1,(2

),1(),(),1(2

61520

1561

2

21

21

)1(2

jijijiji

jijiji

x

s

jijijiy

jijijix

s

wwww

www

r

Ehg

wwwr

wwwr

ThE

)1,2(),2()1,2(

)1,1(),1()1,1(

)1,(),()1,(

)1,1(),1()1,1(

)1,2(),2()1,2(

24

)3,()2,()1,(

),()1,()2,()3,(

6

2

484

6126

484

2

3

615

201561

jijiji

jijiji

jijiji

jijiji

jijiji

yx

jijiji

jijijiji

y

www

www

www

www

www

rr

www

wwww

r

)1,1()1,1()1,1(22

),1(),(),1(2

)2,1()2,()2,1(

)1,1()1,()1,1(

),1(),(),1(

)1,1()1,()1,1(

)2,1()2,()2,1(

42

1

21

2

484

6126

484

2

3

jijijiyx

jijijix

xx

jijiji

jijiji

jijiji

jijiji

jijiji

yx

wwwrr

wwwr

N

www

www

www

www

www

rr

(16)

Fig. 2: The rectangular nanoplate and finite difference grid points

Int. J. Nano Dimens., 6(5):525-537, (Special Issue) 2015

532

M. Karimi and A. R. Shahidi

02

4

1

21

24

)1,(),1(

)1,(),1(

22),()1,1(

)1,1()1,1()1,1(22

)1,(),()1,(2

)1,(),1(

)1,(),1(

22),()1,1(

jiji

jiji

yxjiji

jijijiyx

jijijiy

yy

jiji

jiji

yxjiji

ww

ww

rrww

wwwrr

wwwr

N

ww

ww

rrww

(16)

Boundary conditions In this paper, simply-supported and clamped

boundary conditions are surveyed at all plate edges.The boundary conditions are written as:

Simply-supported boundary conditionsThe simply-supported boundary conditions at all

edges of nanoplate can be written as:

02

2

x

w

02

2

y

w(17)

0wThese conditions lead to the following expressions:

)1,()3,()1,()3,(

)2,(),()2,(),(

,

,,

jijijiji

jijijiji

wwww

wwww

),1(),3(),1(),3(

),2(),(),2(),(

,

,,

jijijiji

jijijiji

wwww

wwww

(18)

0),1()1,1()1,1()1,( jijijiji wwww

Clamped boundary conditionsThe clamped boundary conditions could be

expressed as follows:

0

x

w

0

y

w (19)

0w

These conditions lead to the following expressions:

)1,()3,()1,()3,(

)2,(),()2,(),(

,

,,

jijijiji

jijijiji

wwww

wwww

),1(),3(),1(),3(

),2(),(),2(),(

,

,,

jijijiji

jijijiji

wwww

wwww

(20)

0),1()1,1()1,1()1,( jijijiji wwww

RESULTS AND DISCUSSIONIn this section, it is attempted to demonstrate the

surface, thermal, and non-local effects (small scaleeffects) on the buckling of rectangular nanoplates. Thecritical buckling load ratio is (CBLR) defined in thefollowing from:

CBLR = (21)

CBLR= buckling load with both non-local and surface energy theoriesbuckling load without both non-local and surface energy theories

Table 1 shows the numerical results for bucklingratio of silver nanoplates with simply-supportedboundary conditions versus mode number, m. Thematerial properties of silver nanoplates in all examplesare taken as E=76 GPa and ν=0.3. Surface elasticmodulus, surface residual stress are Es=1.22 N/m,τs=0.89 N/m, respectively. From Table 1, goodagreements can be observed between the FDM resultsand the results in [17]. The results in [17] are based onan exact analytical solution. Thus, the obtained resultscan be trusted.

Finite difference method results are sensitive tolower grid points, a convergence test is performed todetermine the minimum number of grid points requiredto obtain stable and accurate results for Eq. (16). InFig. 3, non-dimensional buckling load (N a2/D) is plottedversus the number of grid points for various non-localparameters. According to Fig. 3, the present solutionis converging. From this figure, it is clearly seen that12 number of grid points (N=M=12) are sufficient toobtain the accurate solutions for the buckling analyses.N and M are the number of grid points in the x and ydirections, respectively.

Fig. 4(a and b) show the effects of non-localparameters on the critical buckling load ratio (CBLR)of nanoplates with simply-supported (SSSS) and

Int. J. Nano Dimens., 6(5):525-537, (Special Issue) 2015

533

clamped (CCCC) boundary conditions versus lengthand thickness nanoplates, respectively. This figureindicates that the CBLR would decrease withincreasing values of non-local parameter, and thisdecreasing of CBLR is more pronounced for thenanoplate with the clamped boundary conditions. It isobserved in Fig. 4(a) that when the nanoplate lengthr e a c h e s a c e r t a i n v a l u e ( e . g . 5 0 nm), the non-local effectdisappears. Also, it can be seen in Fig. 4(b) that byincreasing nanoplate thickness (h=1-2 nm), the CBLRwould diminish heavily. Therefore, the influence ofsurface effect on the CBLR is more significant whenthe length and thickness of nanoplates would increaseand decrease, respectively.

Fig. 5 shows the variation of load ratio versus non-l o c a l p a r a m e t e r ( g2) for different temperature changes( ΔT=0-120o K) and with simply-supported and clampedboundary conditions. The length and thickness ofnanoplates in Fig. 5 were considered as a=40 nm andh=3 nm, respectively. The influences of surface energy

and residual surface stress have not been taken intoaccount (Es=τs=0 N/m). Here, the temperature changeis considered at a low, or at room, temperature.Therefore, the thermal expansion is α=1.9×K-1 [33].According to Fig. 5, the CBLR would decrease withincreasing values of non-local parameter, and thisdecreasing of CBLR more pronounced for the nanoplatewith the clamped boundary conditions. On the otherhand, it is observed that as the value of temperaturechange increases, the CBLR would decrease. Thismeans that the small-scale effects increase withincreasing the temperature changes. This situation ismore pronounced for the nanoplate with simply-supported boundary conditions.

Figs. 6 (a and b) illustrate the variation of CBLRwith non-local parameter for various surface residualstresses and for simply-supported and clampedboundary conditions, respectively. The temperaturechange, length and the thickness of nanoplate in allthe examples were considered as ΔT=40o K, a=15 nm

Table 1: CBLR for silver nanoplates with simply-supported boundary conditions (h=5 nm, Es =1.22 N/m, τs=0.89 N/m, ΔT =0)

a=10 nm a=20 nmg2 nm2 m=1 2 3 4 m=1 2 3 4

0FDM 1.028 1.022 1.020 1.019 1.059 1.034 1.026 1.023[17] 1.028 1.022 1.020 1.019 1.059 1.034 1.026 1.023

1FDM 0.860 0.690 0.520 0.391 1.012 0.923 0.828 0.729[17] 0.863 0.691 0.523 0.392 1.012 0.924 0.828 0.727

2FDM 0.744 0.521 0.350 0.242 0.968 0.834 0.695 0.568[17] 0.745 0.525 0.356 0.248 0.969 0.836 0.695 0.566

(a) (b)

Fig. 3: Convergence study and minimum number of grid points required for obtaining accurate results for non-dimensionalbuckling load by finite difference method with various non-local parameter, (a) simply-support boundary conditions, (b)

clamped boundary conditions,( ΔT=0o K)

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Fig. 4: Changes of the critical buckling load ratio for different non-local parameters, (a) versus length nanoplates, (b) versusthickness nanoplates.

Fig. 5: Changes of the buckling load ratio with non-local parameters for different temperatures (Es=τs=0 N/m).

Fig. 6: Change of load ratio with non-local parameter for various values of surface residual stress, (a) simply-supportedboundary conditions, (b) clamped boundary conditions (Es=0 N/m)

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Fig. 7: Change of load ratio with non-local parameter for different values of surface elasticity modulus, (a) simply-supportedboundary conditions, (b) clamped boundary conditions ( τs=0 N/m).

and h=3 nm, respectively, unless noted otherwise. Theinfluences of surface energy have not been taken intoaccount (Es=0 N/m). Again, it is clearly seen that theCBLR decreases with increasing non-local parameter.Moreover, the CBLR increases with improving surfaceresidual stress. This observation means that the non-local effects diminish with increasing surface residualstress. In addition, by comparing Fig. 6(a) and Fig. 6(b),it can be found that the buckling load ratio in Fig. 6(a)is more sensitive to the surface residual stress.

Fig. 7 (a and b) show the variation of load ratiowith non-local parameter for different surface elasticmodulus and for simply-supported and clampedboundary conditions, respectively. The influencesof surface residual stress have not been taken intoconsideration (τs=0 N/m). It is seen that when thesurface elastic modulus increases, the CBLR wouldaugment. In other words, with increasing the valueof elastic modulus, the non-local effects coulddiminish.

Fig. 8: Surface energy effects on the higher buckling modes of rectangular nanoplates, (a) simply-supported boundaryconditions, (b) clamped boundary conditions; m=1:4 and n=1:4 are the half-wave number (buckling modes) along x and y

directions, respectively; (τs=0.1 N/m)

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To show the influence of surface energy on thehigher modes of buckling nanoplates, the variation ofnon-dimensional buckling load (N a2/D) versus modenumber, m, for various surface elastic modulus havebeen shown in Fig. 8(a and b), respectively. Simply-supported and clamped boundary conditions areinvestigated in two figures. In Fig. 8(a and b), m=1:4and n=1:4 are the half-wave number (buckling modes)along x and y directions, respectively. To focus on theeffect of surface elastic modulus, the non-localparameter is assumed to be zero (g2 (nm2)). It is foundthat by increasing the surface elastic modulus, the non-dimensional buckling load would increase. On the otherhand, the gap between the curves gradually widenswith increasing buckling mode number. This means thatthe surface effects are more significant in higherbuckling modes.

CONCLUSIONIn this study, FDM was used to solve sixth-order

derivatives of differential equations in bucklinganalysis of nanoplates due to coupled surface energyand non-local elasticity theories. The uniformtemperature change was used to study thermal effect.The small scale and surface energy effects are addedinto the governing equations using Eringen’s non-local elasticity and Gurtin-Murdoch’s theories,respectively. Two different boundary conditionsincluding simply-supported and clamped boundaryconditions are investigated. The governingdifferential equation is introduced according to theliterature, and then this equation is solved using theFDM to obtain the buckling for several combinationsof boundary conditions. To verify the accuracy ofthe results obtained by the FDM, these results arecompared with the results of the analytical approach.From the results of the present work, the followingconclusions were important:The finite difference method could be used as apowerful method to determine the mechanicalbehavior of nanoplates. Moreover, this method couldbe used to solve higher-order derivatives ofdifferential equations with different types of boundarycondition with little computational effort.The surface effects on the CBLR of nanoplatesincreased with increasing nanoplate length anddecreasing non-local parameter and nanoplate thickness.The non-local parameter decreased with increasingin the values of surface residual stress, surface elastic

modulus and decreasing in the degree of temperaturechange. This means that the surface effects were moresignificant when the values of surface residual stress,surface elastic modulus increased and the temperaturechange decreased.The non-dimensional buckling load increased withrising in the values of surface elastic modulus andmode numbers. This means that the surface effectsare more important in higher buckling modes.In all the findings, the CBLR values were alwayshigher for nanoplates with simply-supportedboundary conditions than those with clampedboundary conditions.

NOMENCLATURE

E Modulus of elasticity

Es Surface elastic modulus

Poisson’s ratioτs Surface residual stress

λs, μs Surface Lame constants

ij

Strain components

ij

s Surface stress components

ij

b Bulk stress components

Mij

Bending moment components

D Flexural rigidity

x,y,z Cartesian coordinates

u, v, w Displacements in the x, y and z directions,

respectively

a, b, h Length, width and thickness of nanoplate,

respectively

rx, r

yDistance between two grid points in the x

and y directions, respectively

Superscripts

( )+, ( )- Upper and lower surfaces of nanoplate,

respectively

( )s Surface of nanoplate

Subscripts

( ),x , ( )

,y Partial derivatives with respect to x and y,

respectively

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How to cite this article: (Vancouver style)

Karimi M, Shahidi A. R., (2015), Finite difference method for sixth-order derivatives of differential equations inbuckling of nanoplates due to coupled surface energy and non-local elasticity theories. Int. J. Nano Dimens. 6(5):525-537.DOI: 10.7508/ijnd.2015.05.010URL: http://ijnd.ir/article_17146_1117.html

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using finite difference method. Meccanica. 48: 135–144.

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