author's personal copy - kgut.ac.ir · and vibration of nano-wires by using timoshenko beam...

Analysis of the buckling of rectangular nanoplatesby use of finite-difference method

M. R. Karamooz Ravari • S. Talebi •

A. R. Shahidi

Received: 10 September 2012 / Accepted: 26 February 2014 / Published online: 11 March 2014

� The Author(s) 2014

Abstract In recent years nanostructures have been

widely used in industry, for example in nanoelectro-

mechanical systems (NEMS); knowledge of the

mechanical behavior of nanostructured materials is

therefore important. In the work discussed in this

paper, the non-dimensional buckling load of rectan-

gular nano-plates was determined for general bound-

ary conditions. Non-local theory was used to derive

the governing equation, and this equation was then

solved, by use of the finite-difference method, by

applying different combinations of boundary condi-

tions. To verify the proposed method, the non-

dimensional buckling load determined for a simply

supported plate was compared with results obtained by

use of local theory and with results reported in the

literature. When the method was used to calculate the

buckling load of nano-beams, results were in good

agreement with literature results. As a novel contri-

bution of the work, non-symmetric boundary condi-

tions were also studied. The non-dimensional buckling

load was obtained for several values of aspect ratio,

non-local variables, and different types of boundary

condition. For better understanding, mode shapes are

also depicted. The finite-difference method could be a

powerful means of determination of the mechanical

behavior of nanostructures, with little computational

effort, and the results could be as reliable as those

obtained by use of other methods. The ability to deal

with a combination of boundary conditions illustrates

the advantages of this method compared with other

methods.

Keywords Nanoplate � Buckling load � Finite-

difference method � Non-local elasticity theory �Rectangular plates

1 Introduction

Nanostructures have a wide range of applications,

because of their superior mechanical, thermal, and

electrical properties. These properties enable produc-

tion of small devices that were, previously,

unobtainable.

Two approaches are used for mechanical analysis

of nano-scale structures—the molecular dynamic

model and continuum models. It has been shown in

many studies that continuum models can be used

effectively for analysis of nanostructures [1].

In recent years much effort has been devoted to

experimental and theoretical study of the mechanical

behavior of nanostructures. Because controlling the

experimental conditions is not easy for nanostructures,

theoretical methods are important.

M. R. Karamooz Ravari (&) � S. Talebi � A. R. Shahidi

Department of Mechanical Engineering, Isfahan

University of Technology, 84156-83111 Isfahan, Iran

e-mail: [email protected]

S. Talebi

e-mail: [email protected]

123

Meccanica (2014) 49:1443–1455

DOI 10.1007/s11012-014-9917-x

Author's personal copy

By using Gurtin’s continuum surface theory based

on the Euler–Bernoulli and Timoshenko beam models,

Liu and Rajapakse [2] developed a general model for

analysis of the behavior of nano-beams with arbitrary

cross sections. Wang and Feng [3] studied the stability

and vibration of nano-wires by using Timoshenko

beam theory. Farshi et al. [4] studied the size-

dependent vibration of nano-tubes by using a modified

beam theory based on the Timoshenko beam. Fu and

Zhang [5] used the analog equation method to solve

the problem of electrically actuated nano-beams by

consideration of the surface. Fu et al. [6] studied the

effects of surface energy on non-linear bending and

vibration of nano-beams. Phadikar et al. [7] reported

finite element formulations for non-local elastic

Euler–Bernoulli beams and Kirchoff plates. They

used non-local differential elasticity theory and the

Galerkin finite-element technique. Ansari et al. [8]

studied the axial buckling characteristics of single-

walled carbon nano-tubes, including the thermal

environment effect. They incorporated Eringen’s

non-local elasticity equations into the classical Don-

nell shell theory to establish a non-local elastic shell

model which takes small-scale effects into account.

The Rayleigh–Ritz technique in conjunction with the

set of beam functions as modal displacement functions

was used to consider the four commonly used

boundary conditions. In other research [9] they used

modified strain-gradient theory to investigate the

thermal post-buckling characteristics of micro-beams

made of functionally graded materials undergoing

thermal loading. Asgharifard Sharabiani and Haeri

Yazdi [10] studied non-linear free vibration of func-

tionally graded nano-beams within the framework of

the Euler–Bernoulli beam model including von Kar-

man geometric non-linearity. Lu et al. [11] studied the

effect of other surface properties and explained the

size-dependent mechanical behavior of nano-plates by

use of generalized Kirchhoff and Mindlin plate theory.

Sheng et al. [12] analyzed the three dimensional

elasticity of nano-plates, including their surface prop-

erties, by using the theory of laminated structures.

Assadi et al. [13] used the laminated plate theory to

study the effects of surface properties on the dynamic

behavior of nano-plates in thermal environments by

using the size-dependent Young’s elastic modulus.

Behfar and Naghdabadi [14] used classical continuum

modeling to study the vibration of a multilayer

graphene sheet embedded in an elastic medium.

Murmu and Pradhan [15, 16] studied the size-depen-

dent vibration of nano-plates by use of non-local

continuum theory. Ansari et al. [17] investigated

vibration analysis of single-layered graphene sheets

(SLGSs) by using the non-local continuum plate

model. They used the generalized differential quadra-

ture method (DQM) to obtain the frequencies of free

vibration of simply-supported and clamped SLGSs.

Jomehzadeh and Saidi [18] conducted three-dimen-

sional vibration analysis of nano-plates by decoupling

the field equations of Eringen theory. Liao-Liang Ke

et al. [19] developed a Mindlin micro-plate model

based on the modified couple stress theory for free

vibration analysis of micro-plates. Murmu et al. [20]

introduced an analytical method to determine the

natural frequencies of the non-local double-nanoplate

system. They derived explicit closed-form expressions

for natural frequencies for the case in which all four

ends are simply supported. Malekzadeh et al. [21]

investigated the free vibration of orthotropic arbitrary

straight-sided quadrilateral nano-plates by using the

non-local elasticity theory. They used the DQM as an

efficient and accurate numerical tool to solve the

governing equation. Kiani [22] investigated the

vibration of elastic thin nano-plates traversed by a

moving nano-particle involving Coulomb friction by

using the non-local continuum theory of Eringen.

Pouresmaeeli et al. [23] presented an analytical

approach for free vibration analysis of double-ortho-

tropic nano-plates with all edges simply-supported.

Aksencer and Aydogdu [24] studied forced vibration

of nano-plates by using the non-local elasticity theory.

The Navier-type solution method was used for simply

supported nano-plates. Gurses et al. [25] analyzed the

free vibration of nano-sized annular sector plates by

use of the non-local continuum theory. The method of

discrete singular convolution was used for numerical

computations. Alibeygi Beni and Malekzadeh [26]

studied the free vibration of orthotropic non-prismatic

skew nano-plates on the basis of first-order shear

deformation theory in conjunction with Eringen’s non-

local elasticity theory. Assadi [27] reported an

analytical method for study of the size-dependent

forced vibration of rectangular nano-plates under

general external loading by using a generalized form

of the Kirchhoff plate model. Satish et al. [28]

conducted thermal vibration analysis of orthotropic

nano-plates, for example graphene, by using the two-

variable refined-plate theory and non-local continuum

1444 Meccanica (2014) 49:1443–1455

123


mechanics for small-scale effects. Mohammadi et al.

[29] studied the free vibration behavior of circular and

annular graphene sheet by using the non-local elas-

ticity theory. Analytical frequency equations for

circular and annular graphene sheets were obtained

for different kinds of boundary condition. Malekzadeh

and Shojaee [30] extended the application of a two-

variable refined plate theory to the free vibration of

nano-plates. Huang [31] studied size-dependent bend-

ing, buckling, and vibration of nano-plates by using

the non-linear Kirchhoff plate theory and Von-

Karman non-linearity assumptions. Aksencer and

Aydogdu [1] reported use of the Levy-type solution

method for vibration and buckling of nano-plates

using non-local elasticity theory. Ansari et al. [32]

studied the applicability of an elastic plate theory

incorporating inter-atomic potentials to biaxial buck-

ling and vibration analysis of SLGSs and accounting

for small-scale effects. Babaei and Shahidi [33]

studied buckling of the quadrilateral nano-plates by

use of non-local plate theory. Pradhan and Murmu [34]

analyzed the buckling of rectangular SLGSs under

biaxial compression by use of the non-local elasticity.

Sakhaee-Pour [35] studied the buckling of graphene

nano-sheets by use of atomistic modeling. Farajpour

et al. [36] studied axisymmetric buckling of circular

graphene sheets by using the non-local continuum

plate model. Nabian et al. [37] studied the pull-in

instability of circular micro-scale plates under uniform

hydrostatic and non-uniform electrostatic pressure.

Murmu and Pradhan [38] studied the elastic buckling

behavior of orthotropic small scale plates under

biaxial compression. Samaei et al. [39] used non-local

Mindlin plate theory to investigate the effect of length

on the buckling behavior of a single-layer graphene

sheet embedded in a Pasternak elastic medium. They

obtained an explicit solution for the buckling loads of

graphene sheets. Hashemi and Samaei [40] proposed

an analytical solution based on the non-local Mindlin

plate theory and considering small-scale effects for

analysis of the buckling of rectangular nano-plates.

Narendar [41] analyzed the buckling of isotropic

nano-plates by using the two-variable refined-plate

theory and non-local small-scale effects. Ansari and

Rouhi [42] derived analytical expressions by use of the

Galerkin method, which enables rapid and accurate

calculation of the critical buckling loads of monolayer

graphene sheets with different boundary conditions

from static deflection under a uniformly distributed

load. Malekzadeh et al. [43] investigated small-scale

effects on the thermal buckling characteristics of

orthotropic arbitrary straight-sided quadrilateral nano-

plates embedded in an elastic medium. Ghorbanpour

Arani et al. [44] presented an analytical approach for

buckling analysis and smart control of an SLGS using

a coupled poly(vinylidene fluoride) nano-plate. Faraj-

pour et al. [45] investigated the buckling response of

orthotropic SLGS by using non-local elasticity theory.

They supposed two opposite edges of the plate were

subjected to linearly varying normal stresses. Naren-

dar and Gopalakrishnan [46] examined wave propa-

gation in a graphene sheet embedded in elastic

medium (polymer matrix) by incorporating non-local

theory into the classical plate model. They investi-

gated the effect of non-local scale effects in detail.

Murmu et al. [47] reported an analytical study of the

buckling of a double-nano-plate system (DNPS)

subjected to biaxial compression using non-local

elasticity theory. The two nano-plates of DNPS were

bonded by an elastic medium. Farajpour et al. [48]

investigated the non-linear buckling characteristics of

multi-layered graphene sheets subjected to a non-

uniformly distributed in-plane load through its thick-

ness. They modeled graphene sheet as an orthotropic

nano-plate with size-dependent material properties.

Karamooz Ravari and Shahidi [49] recently studied

the axisymmetric buckling of circular annular nano-

plates by using the finite-difference method. They

used a half-integrated grid to repel the singularity at

the center of the plate.

Because of the widespread use of nano-plates in

MEMS and/or NEMS components, buckling of them

is of interest. Because buckling of a nanostructure

initiates instability, knowledge of the buckling load is

important.

According to the literature, the buckling of rectan-

gular nano-plates has been studied only for particular

boundary conditions, because of limitations of the

methods used to solve such problems. As far as we are

aware, the method we describe below has not previ-

ously been used for studying of nano-plates and

graphene sheets.

In the work discussed in this paper the governing

differential equation was derived initially. This equa-

tion was then solved by use of the finite-difference

method to obtain the non-dimensional buckling load

for several combinations of boundary conditions. The

obtained results were compared with those from the

Meccanica (2014) 49:1443–1455 1445

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literature, obtained by use of different methods. Small-

scale effects on the buckling load of quadrilateral

graphene sheets under different conditions were then

examined. The obtained results show that the finite-

difference method can be used to solve complex

problems with non-symmetrical boundary conditions

without the need for much computational effort.

Accordingly, this paper suggests the finite-difference

method as a powerful method for solving a vast range

of problems involving nano-plates and nano-beams.

To do so, one needs to obtain the governing differential

equation and solve that by means of this method.

2 Non-local theory

According to the atomic theory of lattice dynamics and

experimental observations on phonon dispersion, for a

nano-plate the stress field at a reference point in an

elastic continuum depends not only on strain at that

point but also on strains at all other points in the

domain. So, the most general form of the constitutive

relationship for nanostructures involves an integral

over the whole body; this may be known as non-local

theory of elasticity. The stress equation for a linear

homogenous non-local elastic body, neglecting the

body force, using non-local elasticity theory [50], can

be written as:

rij xð Þ ¼Z

k x� x0j j; að Þcijklekl x0ð ÞdV x0ð Þ 8x 2 V

ð1Þ

where rij, eijand Cijkl are the stress, strain, and fourth-

order elasticity tensors, respectively, k(|x - x0|, a) is

the non-local modulus, |x - x0| is the Euclidean

distance, and a is a material constant which depends

on the internal (a) and external (‘) characteristic

lengths and is defined as a ¼ e0a‘ .

e0 is estimated such that the relationships of the

non-local elasticity model furnish a satisfactory

approximation of atomic dispersion curves of plane

waves and those of atomic lattice dynamics.

Because solving the constitutive integral is diffi-

cult, a simplified equation in the differential form of

Eq. (1) is used as a basis of all non-local constitutive

formulations [50]:

1� a2‘2r2� �

r ¼ C : e ð2Þ

where (:) represents the double dot product. r2 is

the Laplacian operator and is given by r2 ¼o2�ox2 þ o2

�oy2

� �. By use of the Laplacian, the

two-dimensional non-local constitutive relationship

can be expressed as [50]:

rxx � e0að Þ2r2rxx ¼E

1� m2ð Þ exx þ meyy

� �ð3aÞ

ryy � e0að Þ2r2ryy ¼E

1� m2ð Þ eyy þ mexx

� �ð3bÞ

sxy � e0að Þ2r2sxy ¼ Gcxy ð3cÞ

Stress resultants for the rectangular plate (Fig. 1)

under directional loads can be written as:

Nxx ¼Z h=2

�h=2

rxxdz Nyy ¼Z h=2

�h=2

ryydz Nxy

¼Z h=2

�h=2

rxydz; ð4aÞ

Mxx ¼Z h=2

�h=2

zrxxdz Myy ¼Z h=2

�h=2

zryydz Mxy

¼Z h=2

�h=2

zrxydz; ð4bÞ

By use of Eqs. (3a)–(3c) and (4) one can obtain the

following non-local constitutive relationships based

on classical plate theory:

Nxx � e0að Þ2 o2Nxx

ox2þ o2Nxx

oy2

� �¼ C

ou

oxþ m

ov

oy

� �

ð5aÞ

Nyy � e0að Þ2 o2Nyy

ox2þ o2Nyy

oy2

� �¼ C

ov

oyþ m

ou

ox

� �

ð5bÞ

Nxy � e0að Þ2 o2Nxy

ox2þ o2Nxy

oy2

� �

¼ C1� mð Þ

2

ou

oyþ ov

ox

� �ð5cÞ

Mxx � e0að Þ2 o2Mxx

ox2þ o2Mxx

oy2

� �

¼ �Do2w

ox2þ m

o2w

oy2

� �ð5dÞ

Myy � e0að Þ2 o2Myy

ox2þ o2Myy

oy2

� �

¼ �Do2w

oy2þ m

o2w

ox2

� �ð5eÞ

1446 Meccanica (2014) 49:1443–1455

123


Mxy � e0að Þ2 o2Mxy

ox2þ o2Mxy

oy2

� �

¼ �D 1� mð Þ o2w

oxoy

� �ð5fÞ

where

D ¼ Eh3

12 1� m2ð Þ ð6aÞ

C ¼ Eh

1� m2ð Þ ð6bÞ

E is the elastic modulus, m is Poisson’s ratio, h is

plate thickness, w is out-of-plane displacement, and

u and v are the in-plane displacements in the x and

y directions, respectively.

Note that the stress resultants displacement rela-

tionships given in Eq. (5) reduce to that of the classical

relationship when the scale coefficient e0a is set to

zero [36].

Because the principle of virtual work is indepen-

dent of constitutive relationships, it can be used to

derive the equilibrium equations of the non-local

plates. By use of the principle of virtual work, the

following equilibrium equations can be obtained [34]:

oNxx

oxþ oNxy

oy¼ m0

o2u

ot2ð7aÞ

oNyy

oyþ oNxy

ox¼ m0

o2v

ot2ð7bÞ

o2Mxx

ox2þ 2

o2Mxy

oxoyþ o2Myy

oy2þ qþ o

oxNxx

ow

ox

� �

þ o

oxNxy

ow

oy

� �þ o

oyNyy

ow

oy

� �þ o

oyNxy

ow

ox

� �

¼ 0

ð7cÞ

In this compression study we consider the

relationships:

Nxx ¼ N0 Nyy ¼ lN0 Nxy ¼ 0; ð8Þ

where l denotes the compression ratio (l = Nyy/Nxx).

Using Eqs. (5) and (6) and Eq. (7c) we obtain the non-

local governing differential equation for buckling of

graphene sheets under two directional compressions:

Do4w

ox4þ 2

o4w

ox2oy2þ o4w

oy4

� �

þ N0 e0að Þ2 o4w

ox4þ lþ 1ð Þ o4w

ox2oy2þ l

o4w

oy4

� ��

� o2w

ox2þ l

o2w

oy2

� ��¼ 0

ð9ÞEquation (9) yields:

r4wþ N w2 o4w

ox4þ 1þ lð Þ o4w

ox2oy2þ l

o4w

oy4

� ��

� 1

L2

o2w

ox2þ l

o2w

oy2

� ��¼ 0

ð10Þ

where w ¼ e0aL

and N ¼ N0L2

Dare the non-dimensional

non-local variable and the non-dimensional buckling

load, respectively.

In this paper we will use finite-difference method to

solve Eq. (10). This approach will be helpful for

nanostructures with different types of boundary con-

dition, including non-symmetrical boundary condi-

tions, with little computational effort.

3 Solution procedure

3.1 The finite-difference method

The finite-difference method is a powerful method for

solving differential equations. Figure 2 shows a rect-

angular plate and the grid points which will be used in

the finite-difference method. By use of this method

Eqs. (11)–(17) can be used to estimate the derivative

of the out-of-plane displacement for the kl-th point as a

function of its neighboring points.

owkl

ox¼ 1

2hx

wkðlþ1Þ � wkðl�1Þ� �

ð11Þ

Fig. 1 Graphene sheet modeled as a continuum nano-plate

Meccanica (2014) 49:1443–1455 1447

123


owkl

oy¼ 1

2hy

wðkþ1Þl � wðk�1Þl� �

ð12Þ

o2wkl

ox2¼ 1

h2x

wkðl�1Þ þ wkðlþ1Þ � 2wkl

� �ð13Þ

o2wkl

oy2¼ 1

h2y

wðk�1Þl þ wðkþ1Þl � 2wkl

� �ð14Þ

o4wkl

ox4

¼ 1

h4x

wkðl�2Þ � 4wkðl�1Þ þ 6wkl � 4wkðlþ1Þ þ wkðlþ2Þ� �

ð15Þ

o4w

oy4

¼ 1

h4y

wðk�2Þl � 4wðk�1Þl þ 6wkl � 4wðkþ1Þl þ wðkþ2Þl� �

ð16Þ

o4w

o2xo2y¼

1

h2xh2

y

wðk�1Þðl�1Þ þ wðk�1Þðlþ1Þ þ wðkþ1Þðl�1Þ�

þwðkþ1Þðlþ1Þ � 2wðk�1Þl � 2wðkþ1Þl � 2wkðl�1Þ

�2wkðlþ1Þ þ 4wkl

�ð17Þ

Substituting Eqs. (11)–(17) in Eq. (10) yields Eq.

(18) which should be valid for all points and will be

discussed in the section ‘‘Solution’’.

Aþ �N Bw2 � 1

L2C

� �¼ 0 ð18Þ

where:

Fig. 2 The rectangular

plate and finite difference

grid points

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3.2 Boundary conditions

Because clamped and simply supported boundary

conditions have been studied in this paper, in this sub-

section these boundary conditions are introduced.

Consider m and n grid points distributed in x and

y directions, respectively. Accordingly, there are

m 9 n grid points. On each edge of the plate, however,

two restrictions exist which correspond to the bound-

ary conditions as follows.

3.2.1 Clamped boundary condition

For the clamped edge, the out-of-plane displacement

and the slope are neglected. Equations (19) and (20)

show the corresponding formulas for the upper edge,

for example:

wjy¼0 ¼ 0) w2l ¼ 0 ð19Þ

ow

oyjy¼0 ¼ 0) w1l ¼ w3l ð20Þ

3.2.2 Simply supported boundary condition

For the simply supported edge of a rectangular plate,

the mathematical relationships are:

w ¼ 0;o2w

on2¼ 0 ð21Þ

where n is the direction normal to that of the

corresponding simply supported edge. Equation (21)

yields the following relationships for the upper edge,

for example:

wjy¼0 ¼ 0) w2l ¼ 0 ð22Þ

o2w

oy2jy¼0 ¼ 0) w1l ¼ �w3l ð23Þ

3.3 Solution

Because Eq. (18) should be valid for all internal grid

points, by applying this equation for all internal points

and by using the boundary conditions (m -

4) 9 (n - 4), equations are obtained which contain

(m - 4) 9 (n - 4) unknowns. This can be expressed

as an Eigenvalue problem, as follows:

Qw ¼ 0 ð24Þwhere Q[(m-4) 9 (n-4)] 9 [(m-4) 9 (n-4)] is a func-

tion of �Nand w is a vector containing the displacement

of the grid points. The evident solution of Eq. (24)

yields the non-dimensional buckling load.

4 Result and discussion

A MATLAB program was developed to obtain the

non-dimensional buckling load and displacement

vector. The graphene sheet was used for the analysis

and the maximum length of the plate was considered to

be L = 45.29 nm, as mentioned in the literature [35,

51]. Notice that the non-local variable is non-dimen-

sional here, but the maximum value of e0a should not

be larger than 2 nm. This value was chosen on the

basis of the discussion by Wang and Wang [52] and

because it has also been used for graphene sheets in the

A ¼ 1

h4y

wðk�2Þl þ wðkþ2Þl� �

þ 1

h4x

wkðl�2Þ þ wkðlþ2Þ� �

þ 2

h2xh2

y

wðk�1Þðl�1Þ þ wðkþ1Þðl�1Þ þ wðk�1Þðlþ1Þ þ wðkþ1Þðlþ1Þ� �

� 4

h4x

þ 4

h2xh2

y

!wkðl�1Þ þ wkðlþ1Þ� �

� 4

h4y

þ 4

h2xh2

y

!wðk�1Þl þ wðkþ1Þl� �

þ 6

h4x

þ 8

h2xh2

y

þ 6

h4y

!wkl

B ¼ lh4

y


þ 1

h4x


þ 1þ lh2

xh2y

wðk�1Þðl�1Þ þ wðkþ1Þðl�1Þ þ wðk�1Þðlþ1Þ þ wðkþ1Þðlþ1Þ� �

� 4

h4x

þ 2ð1þ lÞh2

xh2y

!wkðl�1Þ þ wkðlþ1Þ� �

� 4

h4y

þ 2ð1þ lÞh2

xh2y

!wðk�1Þl þ wðkþ1Þl� �

þ 6

h4x

þ 4ð1þ lÞh2

xh2y

þ 6lh4

y

!wkl

C ¼ lh2

y


þ 1

h2x


� 21

h2x

þ lh2

y

!wkl

Meccanica (2014) 49:1443–1455 1449

123


literature [51]. To verify the proposed method, two

methods were used. In the first method the results

obtained for e0a = 0 were compared with the exact

solution [53], because the governing equation has an

exact solution when e0a = 0. Notice that if e0a = 0

numerical methods are needed to solve the equation.

Figure 3a shows the change of non-dimensional

buckling load with aspect ratio (L/b) for different

numbers of grid points for a fully simply supported

plate and e0a = 0. The results are in good agreement

with the exact solution using only 15 grid points and

the maximum error is less than 2 %. To obtain

authentic results in each case, first, the non-dimen-

sional buckling load was calculated for different

numbers of grid points; the variation of the results is

slight. Nine grid points in the x and y direction yield

reliable results and have been used in all cases. In the

second method, the obtained non-dimensional buck-

ling load was verified by comparison with that

obtained for nano-beams, as proposed in Ref. [34].

In this method the non-dimensional buckling load of a

nano-plate with L/b = 0.001 is compared with that for

nano-beams reported by Wang et al. [54]. As shown in

Fig. 3b the results are in good agreement with those

obtained for nano-beams [54].

Notice that because all edges are simply supported

it is possible to find a solution for one quarter of the

plate only and the number of grid points decreases

from 81 to 15 for the example shown in Fig. 4.

Figure 5 shows the change of non-dimensional

buckling load with non-local variable for several

values of the aspect ratio and for the fully simply

supported and fully clamped boundary conditions. As

the non-local variable increases the non-dimensional

buckling load decreases for both clamped and simply

supported boundary conditions. Also, when the aspect

ratio is increased the non-dimensional buckling load

decreases in both cases.

Figure 6 shows the effects of increasing the com-

pression ratio (l) for L/b = 1 and L/b = 2 for fully

clamped and fully simply supported boundary condi-

tions. In all cases, as the compression ratio is increased

the non-dimensional buckling load decreases.

Fig. 3 Change of non-dimensional buckling load with (a) aspect ratio (L/b) for different numbers of grid points for a fully simply

supported plate and (e0a = 0), and (b) non-dimensional non-local variable for a nano-beam

Fig. 4 Reduction in the number of grid points when the plate

and its boundaries are symmetric

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Fig. 5 Change of non-dimensional buckling load with non-dimensional non-local variable for several values of aspect ratio for

(a) fully simply supported (b) fully clamped boundary conditions

Fig. 6 Change of non-dimensional buckling load with non-

dimensional non-local variable for several values of l for

(a) simply supported boundary conditions and L/b = 1,

(b) simply supported boundary conditions and L/b = 2,

(c) clamped boundary conditions and L/b = 1, and

(d) clamped boundary conditions and L/b = 2

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Fig. 7 Dependence of non-dimensional buckling load on non-dimensional non-local variable for different modes for (a) fully simply

supported and (b) fully clamped boundary conditions (l = 1, L/b = 1)

Fig. 8 Buckling modes for (a) SSSS and m = 1, (b) SSSS and m = 2, (c) SSSS and m = 3, (d) CCCC and m = 1, (e) CCCC and

m = 2, and (f) CCCC and m = 3

Fig. 9 Combinations of boundary conditions

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If P grid points are used, P buckling modes are

obtained. Figure 7 shows the dependence of non-

dimensional buckling load on non-local variable for

different modes; the results are in good agreement

with those from literature for fully simply supported

boundary conditions [34]. Figure 8 depicts several

buckling modes for fully clamped and simply sup-

ported boundary conditions.

Although simply supported boundary conditions

are physically more common, clamped boundary

conditions and the combination of simply supported

and clamped boundary conditions could be important

theoretically. Accordingly, as a novel contribution of

this paper several combinations of boundary condi-

tions were studied. Figure 9 shows some combinations

of boundary conditions. For better comparison, the

dependence of non-dimensional buckling load on non-

local variable for the cases given in Fig. 9 is plotted in

Fig. 10. As shown in Fig. 10, a fully clamped nano-

plate has the highest non-dimensional buckling load

and a fully simply supported nano-plate has the lowest.

These results also make sense.

Notice that the presented method can be used to

solve the buckling problem for general plates and nano

plates, because, as the non-local variable approaches

zero, Eq. (10) approaches the governing equation for a

general plate, which corresponds to classical contin-

uum model. The finite-difference method can also be

applied to the buckling and vibration of layered nano-

plates, nano-composites, multi layer composites, etc.

To achieve these objectives one must obtain the

governing equation of motion for the problem, choose

a suitable grid for the geometry of the structure, and

solve the obtained Eigenvalue problem.

5 Conclusion

The governing differential equation for buckling of a

graphene sheet was obtained by use of non-local theory

and solved by use of the finite-difference method. The

non-dimensional buckling load and out-of-plane dis-

placement vector were obtained. The obtained results

were verified by use of local theory, and a reliable

number of grid points were obtained. The effects of

changing the non-local variable, the aspect ratio, the

compression ratio, and the mode number on the value of

non-dimensional buckling load were studied for CCCC

and SSSS boundary conditions and the buckling mode

was depicted for some cases. As a novel contribution of

this paper, several combinations of clamped and simply

supported boundary conditions were studied. The

results show the finite-difference method could be used

as a powerful method to determine the mechanical

behavior of nano-plates. Also, in comparison with other

methods, for example the Galerkin method, the finite-

difference method can be used to solve a variety of

problems with different types of boundary condition

with little computational effort.

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