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
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DOI 10.1007/s11012-014-9917-x
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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
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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
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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
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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
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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
wðk�2Þl þ wðkþ2Þl� �
þ 1
h4x
wkðl�2Þ þ wkðlþ2Þ� �
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
wðk�1Þl þ wðkþ1Þl� �
þ 1
h2x
wkðl�1Þ þ wkðlþ1Þ� �
� 21
h2x
þ lh2
y
!wkl
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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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