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Buckling analysis of a defective annular graphene sheet in elastic medium
Accepted Manuscript
Buckling analysis of a defective annular graphene sheet in elastic medium M. Fadaee PII: DOI: Reference:
S0307-904X(15)00563-6 10.1016/j.apm.2015.09.029 APM 10728
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
12 February 2015 20 July 2015 23 September 2015
Please cite this article as: M. Fadaee , Buckling analysis of a defective annular graphene sheet in elastic medium, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.09.029
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ACCEPTED MANUSCRIPT Highlights > Static stability of a defective annular graphene sheet is considered. > Translational addition theorem is employed to analyze the problem. > The graphene sheet is elastically restrained at the bottom surface.
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> The eccentricity and size of defect have significant effects on the buckling loads.
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Buckling analysis of a defective annular graphene sheet in elastic medium
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M. Fadaee1 1
Department of Mechanical Engineering, College of Engineering, Qom University of Technology, Iran.
Abstract
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In this article, an analytical approach is presented to analyze static stability of defective annular graphene sheet. Due to production process and constrains conditions, graphene sheet may be opposed to structural defect. Some of the defects can be modelled as an eccentric hole. The graphene sheet is elastically restrained at the bottom surface.
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Nonlocal thin plate theory as well as the translational addition theorem are employed to
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solve the problem. The stability and accuracy of results are examined by the literature and the finite element analyses. Effects of eccentricity of defects, nonlocality, Winkler
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and Pasternak foundation parameters and various boundary conditions on the critical
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buckling load of an annular graphene sheet are investigated. It is observed that the eccentricity and size of defects have significant effect on the critical load.
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Keywords: Graphene sheet, nonlocal elasticity, defect, eccentric hole, translational addition theorem, buckling analysis.
1. Introduction
1
Corresponding author. E-mail address: [email protected] (Mohammad Fadaee).
ACCEPTED MANUSCRIPT Invention of graphene sheets initiated a new perspective in the mechanical devices. Graphene, a special monolayer of hexagon-lattice, is even stiffer and stronger than carbon nanotube (CNT). Their superior mechanical, thermal and electrical properties let to produce small devices that were impossible before. Therefore, understanding the mechanical behavior of the graphene sheets such as buckling phenomenon is an
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important issue.
Various methods are applied to analyze different mechanical, electrical and optical
properties of graphene sheets. These methods can be categorized in three groups as: atomistic methods, continuum methods and atomistic-continuum methods. Non-
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classical continuum methods which considering non continuity of nano scale structures, have lower precision respect to the two other methods but they can present exact phrases with low computational effort for various problems. Nonlocal theory of
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elasticity which was presented by Eringen [1] is one of the such continuum theories. A few studies have been performed in the field of buckling analysis of graphene sheets,
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especially for circular ones. Farajpour et al. [2] studied axisymmetric buckling of circular graphene sheet in a thermal environment based on nonlocal elasticity. They
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used differential quadrature method (DQM) and Galerkin method to solve the problem
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and showed that their results have good agreement with molecular dynamic simulation results. Ravari and Shahidi [3] applied finite difference method to nonlocal buckling
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equation of annular plate and investigated effect of small scale on the radial compressive critical load. Bedroud et al. [4] analyzed buckling of circular and annular nanoplate based on nonlocal first order shear deformation. They investigated the effect of small scales on buckling loads for different geometry, boundary conditions and axisymmetric or asymmetric mode numbers. Asemi et al. [5] investigated thermal effects on the stability of a circular graphene sheet using nonlocal plate model. Natsuki
ACCEPTED MANUSCRIPT et al. [6] studied the buckling properties of circular double-layered graphene sheets (DLGSs), using plate theory. An analytical solution of coupled governing equations is proposed for predicting the buckling properties of circular DLGSs. Neek-Amal and Peeters [7] investigated the stability of circular monolayer graphene subjected to a radial load using non-equilibrium molecular dynamics simulations. Zenkour and Sobhy [8]
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studied the thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium using nonlocal elasticity theory. Malekzadeh et al. [9] considered small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral
nanoplates embedded in an elastic medium. Wang et al. [10] reported thermal buckling
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properties of rectangular nanoplates with small-scale effects. They derived the critical temperatures for the nonlocal Kirchhoff and Mindlin plate theories by nonlocal continuum mechanics.
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According to this literature survey, it can be concluded that all researches in buckling
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analysis have focused on annular nano plates with a centric hole. But graphene sheets may be defected in different form in the production process (e.g. single or double
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vacancy and Stone-Wales defects). Also, graphene sheets can be constrained in an area except the edge, for example pinning in a point. These defects and pin holes can be
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considered as eccentric circular hole. In order to analytically model an eccentric hole in
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a circular graphene sheet, the translational addition theorem (TAT) can be employed. Watson [11] presented basic concepts of this theorem in 1992. Lin [12] used the TAT for vibration analysis of circular membranes and plates. Alsahlani and Mukherjee [13] analyzed dynamics of a circular membrane with an eccentric circular areal constraint under arbitrary initial conditions using the TAT. They studied the symmetric and antisymmetric modes of vibrations. Hasheminejad and Ghaheri [14] presented the method of separation of variables in elliptical coordinates in conjunction with the
ACCEPTED MANUSCRIPT translational addition theorems for Mathieu functions to investigate the free flexural vibrations of a fully clamped thin elastic panel of elliptical planform containing an elliptical cutout of arbitrary size, location, and orientation. The lack of study on the buckling of defective annular graphene sheets is clear. Therefore, this paper presents static stability analysis of an annular graphene sheet with
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a hole in an arbitrary point on its surface as a defect and resting on elastic foundation. Governing equation is derived based on nonlocal theory of elasticity. This equation is analytically solved using the translational addition theorem. Then, critical buckling
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loads are found for various geometry and scale parameters. Results are compared with literature and the finite element analyses and good agreements are achieved. Finally, effects of eccentricity, nonlocality and elastic foundation parameters on buckling loads are investigated.
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2. Mathematical formulations 2.1. Geometrical configuration
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Consider an eccentric annular graphene sheet resting on elastic foundation with Winkler
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and Pasternak parameters as KW and KG , respectively, as shown in Fig.1. The sheet has inside radius R2 , outside radius R1 , eccentricity , thickness h and is subjected to a
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radially compression load N . To interpret mathematical formulations, two polar
coordinates (r1 ,1 ) and (r2 , 2 ) are taken to coincide with the center of the outer and inner circles, respectively. 2.2. The equilibrium equation
ACCEPTED MANUSCRIPT Based on nonlocal theory presented by Eringen [1], the stress at a reference point in an elastic medium depends on the strain at every point of the body. According to this theory, the constitutive equation will be obtained as [1]
(1 2 2 )σ C : ε
(1)
where and are stress and strain tensors, C is fourth order elastic modulus tensor,
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2 is Laplacian operator and 2 (e0 a)2 . e0 is a constant correlated by material. The parameter e0 a is known as small scale. A certain value for small scale is not available and for any type of analysis, this will be found by comparing the results of continuum
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modeling with atomistic ones.
For buckling analysis, the displacement fields of an isotropic graphene sheet according
u (r2 , 2 , z ) z
W (r2 , 2 ) r2
v (r2 , 2 , z ) z
1 W (r2 , 2 ) r2 2
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to the classical plate theory, for example, in polar coordinate (r2 , 2 ) are considered as
(2)
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w (r2 , 2 , z ) W (r2 , 2 )
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respectively.
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where u , v and w are the displacement functions in the r2 , 2 and z directions,
Substituting Eq. (2) into the strain-displacement relations, it can be concluded that
2W r22
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r r z 2 2
z W 1 2W ( ) r2 r2 r2 22
r
1 W z 2W 2 ( ) , r2 r2 2 r2 2
2 2
2 2
r z 0
,
2
,
z 0 2
zz 0
The stress-strain relations may be expressed for a graphene sheet as
(3)
ACCEPTED MANUSCRIPT r2 r2 1 0 r2 r2 E( z) (1 2 2 ) 22 2 1 0 22 1 1 r 0 0 r 22 2 2 2
(4)
Using Eqs. (1) to (4), the equilibrium equation of thin annular graphene sheet resting on elastic Winkler and Pasternak foundations is obtained as [15] 2
2
4
G
2
W
K G N 2W KW W 0
where
2.3. Analytical solution procedure
(5)
(6)
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2 1 1 2 2 1 1 2 2 ( 2 ) 2 2 2 2 2 2 r2 r2 r2 r2 2 r2 r2 r2 r2 2 Eh3 D 12(1 2 )
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D N K W K
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To obtain the solution of Eq. (5), the separation of variables method is employed. Therefore, the transverse displacement is supposed as
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W ( r2 , 2 ) ( r2 ). ( 2 )
(7)
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Substituting Eq. (7) into (5), it can be concluded that (8)
2 4 4 4 r22 d 2 2 1 d 2 1 d 2 2 r ( ) ( ) n2 2 2 2 r2 dr2 d 2 ( ) 2 ( r2 ) dr22 2 2 2
(9)
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r22 d 2 1 1 d 1 2 4 4 4 1 d 2 1 r ( ) ( ) n2 1 2 2 r2 dr2 d 2 ( ) 1 ( r2 ) dr22 1 2 2
where
W (r2 , 2 ) 1 (r2 ).1 ( 2 ) 2 (r2 ).2 ( 2 )
(10)
ACCEPTED MANUSCRIPT Because the sheet is complete circular, the period of functions 1 ( 2 ) and 2 ( 2 ) must be 2π ( n 0,1, 2,... ) and consequently
1 ( 2 ) A n cos (n 2 ) B n sin (n 2 )
(11a)
2 ( 2 ) C n cos (n 2 ) D n sin (n 2 )
(11b)
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Then, according to Bessel differential equations [11], it is obtained from Eqs. (8) and (9) that
1 (r2 ) E n I n ( r2 ) Fn K n ( r2 )
(12)
2 (r2 ) G n J n ( r2 ) H n Y n ( r2 )
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(13)
where J n ( r2 ) , Y n ( r2 ) , I n ( r2 ) , K n ( r2 ) are Bessel functions of the first and second kinds and modified Bessel functions of the first and second kinds of order n ,
4 4 4 2 2
(14)
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4 4 4 2 , 2
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respectively. A n , B n ,…, H n are variable coefficients. Also,
,
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KW 4 2 D N 2K G
2 KW K G N D 2N 2K G
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Using Eqs. (11) to (13) into Eq. (10) as well as the superposition principle, the
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transverse deflection of the sheet can be rewritten as
W (r2 , 2 ) A1n J n ( r2 ) A2 n Yn ( r2 ) A3n I n ( r2 ) A4 n K n ( r2 ) cos (n 2 )
(15)
n 0
B1n J n ( r2 ) B2 n Yn ( r2 ) B3n I n ( r2 ) B4 n K n ( r2 ) sin(n 2 ) n 1
where A1n , A2 n , A3n , A4 n and B1n , B2 n , B3n , B4 n are the mode shape coefficients of buckling, which are determined by applying the boundary conditions. In Eq. (15), the cosine series is corresponding to symmetric buckling modes of graphene sheet and that
ACCEPTED MANUSCRIPT with sin(n 2 ) is for antisymmetric modes. For more information about the symmetric and antisymmetric buckling modes, section 3.2.4 will be presented. 2.3.1. Exact solution for symmetric buckling modes Since, the cosine function is symmetric, the part of transverse displacement which is
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multiplied by the cosine function, is identified as the symmetric solution. First, the symmetric buckling modes are considered. Therefore, the following solution of displacement should be considered as
W S (r2 , 2 ) A1n J n ( r2 ) A2 n Yn ( r2 ) A3n I n ( r2 ) A4 n K n ( r2 ) cos (n 2 )
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n 0
(16)
Herein, it is assumed that the graphene sheet is clamped in the outer and inner edges. Satisfying the clamped boundary conditions at the inner edge leads to
r2 R 2
0
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W S (r2 , 2 )
(17a)
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A1n J n ( R 2 ) A 2 n Y n ( R 2 ) A3n I n ( R 2 ) A 4 n K n ( R 2 ) 0 (17b)
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dW S (r2 , 2 ) 0 r2 R 2 dr2
A1n J n ( R 2 ) A 2 n Y n( R 2 ) A3n I n ( R 2 ) A 4 n K n ( R 2 ) 0
dJ n dY n dI dK n , Y n , I n n , K n dr2 dr2 dr2 dr2
(18)
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J n
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where
In order to satisfy the clamped boundary condition at the outer edge, the displacement
W S ( R2 ,2 ) should be expressed in terms of the polar coordinates of the outer circle (r1 ,1 ) . This can be carried out by employing the translational addition theorem for cylindrical vector wave functions [11-12], involving the regular and modified Bessel
ACCEPTED MANUSCRIPT functions. This theorem for the first and second regular and modified Bessel functions from the (r2 ,2 ) coordinate to (r1 ,1 ) coordinate may be presented as (19a)
Cos ( n 2 ) Cos ( n m)1 ) Yn ( r2 ) Yn m ( r1 ) J m ( ) Sin( n 2 ) m Sin( n m)1 )
(19b)
Cos ( n 2 ) Cos ( n m)1 ) m I n ( r2 ) ( 1) I n m ( r1 ) I m ( ) Sin( n 2 ) m Sin( n m)1 )
(19c)
Cos ( n 2 ) Cos ( n m)1 ) K n ( r2 ) K n m ( r1 ) I m ( ) Sin( n 2 ) m Sin( n m)1 )
(19d)
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Cos (n 2 ) Cos (n m)1 ) J n ( r2 ) J n m ( r1 ) J m ( ) Sin(n 2 ) m Sin( n m)1 )
Substituting Eqs. (19a-d) into Eqs. (17) and applying the clamped boundary condition at the outer edge, leads to
n 0 m
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r1 R1
0
(20b)
J m ( ) A1n J n m ( R1 ) A 2 n Y n m ( R1 ) Cos (n m )1 0 m I m ( ) (1) A3n I n m ( R1 ) A 4 n K n m ( R1 )
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dW S (r1 , 1 ) 0 r1 R1 dr1
(20a)
J m ( ) A1n J n m ( R1 ) A 2 n Y n m ( R1 ) Cos (n m )1 0 m I m ( ) (1) A3n I n m ( R1 ) A 4 n K n m ( R1 )
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W S (r1 ,1 )
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n 0 m
Applying the orthogonality relation for the cosine functions as 2
cos (m n)1 cos q1 d1 mn,q mn, q
(21)
0
where mn,q is the second Kronecker delta and using the Bessel function properties as
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J p (r ) (1) p J p (r ) , Y p (r ) (1) p Yp (r ) I p (r ) I p (r )
, K p (r ) K p (r )
Eqs. (20a) and (20b) can be rewritten as
n 0 q 0
n 0 q 0
J q n ( ) (1) n J q n ( ) A1n J q ( R1 ) A2 n Yq ( R1 ) 0 I q n ( ) I q n ( ) (1) q n A3n I q ( R1 ) A4 n K q ( R1 )
(23a)
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1 J q n ( ) (1)n J q n ( ) A1n J q ( R1 ) A2 n Yq( R1 ) 2 0 1 I ( ) I ( ) (1) q n A I ( R ) A K ( R ) 3n q 1 4n 1 qn qn q 2
(23b)
Eqs. (17a), (17b), (23a) and (23b) should be simultaneously satisfied to obtain the
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symmetric buckling loads of the eccentric annular graphene sheet. Expanding these equations from n=0 to N horizontally and from q=0 to N vertically, gives the matrix equation as 0 that the dimensions of matrix are (4 N 4) (4 N 4) . The
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determinant of matrix is the closed-form buckling equation for C-C eccentric annular graphene sheet. The vector X ( (4N 4) 1 ) is composed of A1n , A2 n , A3n , A4 n for n=0
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to N. Therefore, to calculate the symmetric mode shape coefficients as A1n , A2 n , A3n ,
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A4 n , null space of the matrix B should be determined.
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It is worth noting that eliminating A3n and A4 n from Eqs. (23) with using of Eqs. (17) leads to a decrease in the computational complexities. This is because of the order of
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matrix for finite n and q which would be reduced to half as (2 N 2) (2 N 2) . It
will be led to simplicity and quickness of computation in the calculations. 2.3.2. Exact solution for antisymmetric buckling modes The antisymmetric modes coincide with the sine term as sin(n 2 ) in Eq. (15). The
boundary conditions of antisymmetric modes in the inner edge are the same as Eqs.
ACCEPTED MANUSCRIPT (17). Following the same procedure as in the case of the symmetric modes and substituting the orthogonality relation for the sine functions as 2
(24)
sin (m n)1 sin q1 d1 mn,q mn, q 0
The outer edge boundary conditions in Eqs. (23a) and (23b) can be rewritten for
n 1 q 1
n 1 q 1
J q n ( ) (1) n J q n ( ) A1n J q ( R1 ) A2 n Yq ( R1 ) 0 I q n ( ) I q n ( ) (1) q n A3n I q ( R1 ) A4 n K q ( R1 )
(25a)
1 J q n ( ) (1)n J q n ( ) A1n J q ( R1 ) A2 n Yq( R1 ) 2 0 1 I ( ) I ( ) (1)q n A I ( R ) A K ( R ) 3n q 1 4n 1 qn qn q 2
(25b)
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antisymmetric modes as
For antisymmetric buckling modes, the obtaining of the buckling equation is exactly the
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3. Results and discussion
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same as that for symmetric modes, but the indices n and q vary from 1 to .
Based on the analytical solution mentioned in section 2, a computer code has been
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provided to obtain critical buckling loads of eccentric annular graphene sheets resting
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on elastic foundation. First, in order to examine validation of the present approach, three comparison studies are done. Then, effects of eccentricity, nonlocality, elastic
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foundation parameters and boundary conditions are investigated in some figures. Also, the following non-dimensional parameters are introduced as
K R2 K R4 R2 NR12 2 , , N , K G G 1 , KW W 1 , 2 R1 R1 D D D R1
(26)
It should be mentioned that the results in case I and case II of section 3.1 are obtained in macro size as circular and annular plates using the classical plate theory, whereas the
ACCEPTED MANUSCRIPT results of section 3.2 and case III in section 3.1 are calculated in nano size as the graphene sheet using the nonlocal theory. 3.1. Comparison study Case I: Table 1 shows comparison of critical buckling load parameter N NR12 / D for
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clamped and simply supported circular plates for various values of Winkler foundation parameter K W . This table reports excellent agreement between results of the present method and the literature obtained by a finite element method.
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Case II: Table 2 gives critical buckling load parameters N NR12 / D of CC and CS (the
clamped outer edge and the simply supported inner edge) eccentric ( 0, 0.1, 0.2 ) annular plates resting on Winkler foundation ( 0.2 , Kw 0,15.6,31.2,62.4 ) based on the present method and a 3D brick element modeling in traditional finite element software
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(ABAQUS) created on the basis the 3D elasticity. A mesh sensitivity analysis was
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carried out to ensure independency of finite element (FE) results from the number of elements. It is obvious that the present results capture finite element ones, precisely.
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Maximum value of error between two methods is 1.34%.
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Case III: In Fig. 2, behavior of the buckling load parameter N of CS and CC centric
annular graphene sheets versus nonlocal parameter is plotted based on the present
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analytical method and those obtained by Ref. [3]. Good agreement between results of the present method and the literature approves correctness of present method. 3.2. Parametric study 3.2.1. Effect of eccentricity
ACCEPTED MANUSCRIPT Fig. 3 shows effects of the eccentricity parameter on the non-dimensional buckling load N CC and CS graphene sheets for various values of Winkler modulus parameter K w ( 0.01, KG 0, 0.2 ). According to Fig. 3, increasing the eccentricity decreases
the critical buckling load parameter while increasing the Winkler modulus parameter increases the critical buckling load due to enhancing the stiffness of the graphene sheet
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for both boundary conditions. Also, it is evident that the critical buckling load parameter N is less affected by increasing the Winkler modulus parameter.
3.2.2. Effect of nonlocality
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Effects of the nonlocal parameter on the non-dimensional buckling load N of CS graphene sheets are considered in Fig. 4 for various values of eccentricity ratio ( Kw 100, KG 10, 0.4 ). These figures reveal that increasing the eccentricity ratio and
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the nonlocal parameter decrease the critical buckling load parameter. Also, it is
nonlocal parameter
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obvious that for higher values of the eccentricity ratio and the outer radius R 1 , the has less effects on the critical buckling load parameter than
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lower values of ones. It is because when the outer radius R 1 increases, size of the
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graphene sheet tends to micro one. Since the nonlocal parameter is defined for structures in nano size, therefore, its effect on the critical buckling load decreases.
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3.2.3. Effect of shear modulus K G Fig. 5 reports effects of the non-dimensional shear modulus K G on the non-dimensional
buckling load N of CS and CC graphene sheets for various values of eccentricity ratio
( Kw 100, 0.4, 0.01). It can be concluded from Fig. 5 that for all values of
eccentricity ratio, the critical buckling load parameter N varies linearly respect to K G .
ACCEPTED MANUSCRIPT Due to increasing the stiffness of graphene sheet, the buckling load N increases by enhancing the K G . 3.2.4. 3D view of buckling mode shapes 3-D plots of first four buckling mode shapes of a CC eccentric annular graphene sheet
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are illustrated in Fig. 6 to make the better physical sense of the symmetric and antisymmetric mode shapes. Also, this figure shows how boundary conditions can affect
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the buckling mode shapes of the eccentric graphene sheet.
4. Conclusion
The main objective of this study was to investigate buckling analysis of annular
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graphene sheet with an eccentric defect resting on Winkler and Pasternak foundations. Equilibrium equation was obtained based on nonlocal thin plate theory and solved using
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the translational addition theorem, analytically. Critical buckling loads were presented
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for various edge support, eccentricity, Winkler and Pasternak modulus parameters and nonlocal parameters. The results are validated by the literature and a finite element
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analysis. Buckling load of a specific eccentric annular graphene sheet is strongly depends on eccentricity and the nonlocal parameter. It is observed that increasing
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nonlocal parameter and eccentricity ratio decreases critical buckling loads for various boundary conditions. Enhancing the outer radius and eccentricity decreases nonlocal effect. Also, sensibility of critical buckling load of a graphene sheet to the eccentricity ratio is deceased by increasing the Winkler modulus parameter.
ACCEPTED MANUSCRIPT 5. References [1] A.C. Eringen, Nonlocal continuum field theories, Springer, (2002). [2] A. Farajpour, M. Dehghany, A.R. Shahidi, Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment,
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Compos. Part B 50(0) (2013) 333-343. [3] M.K. Ravari, A. Shahidi, Axisymmetric buckling of the circular annular nanoplates using finite difference method, Meccanica 48(1) (2013) 135-144.
[4] M. Bedroud, S. Hosseini-Hashemi, R. Nazemnezhad, Buckling of circular/annular
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Mindlin nanoplates via nonlocal elasticity, Acta mech 224(11) (2013) 26632676.
[5] S.R. Asemi, A. Farajpour, M. Borghei, A.H. Hassani, Thermal effects on the stability of circular graphene sheets via nonlocal continuum mechanics, Latin
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Amer. J. Solids Struct. 11(4) (2014).
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[6] T. Natsuki, J.X. Shi, Q.Q. Ni, Buckling instability of circular double-layered graphene sheets, J. Phys.: Condens. Matter 24 (2012) 135004 (5pp).
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[7] M. Neek-Amal, F. M. Peeters, Buckled circular monolayer graphene: a graphene nano-bowl, J. Phys.: Condens. Matter 23 (2011) 045002 (8pp).
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[8] A. Zenkour, M. Sobhy, Nonlocal elasticity theory for thermal buckling of nanoplates
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lying on Winkler–Pasternak elastic substrate medium, Physica E. 53 (2013) 251-259. [9] P. Malekzadeh, A. R. Setoodeh, A. Alibeygi Beni, Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an
elastic medium, Compos. struct. 93 (2011) 2083-2089. [10] Y. Z. Wang, H. T. Cui, F. M. Li, K. Kishimoto, Thermal buckling of a nanoplate with small-scale effects, Acta Mechanica 224 (2013) 1299–1307.
ACCEPTED MANUSCRIPT [11] G.N. Watson, A treatise on the theory of Bessel functions, Cambridge university press, (1922). [12] W. Lin, Free transverse vibrations of uniform circular plates and membranes with eccentric holes, J. Sound Vib. 81(3) (1982) 425-435. [13] A. Alsahlani, R. Mukherjee, Dynamics of a circular membrane with an eccentric
Practice Theory 31 (2013) 149–168.
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circular areal constraint: Analysis and accurate simulations, Simul. Model.
[14] S. M. Hasheminejad, A. Ghaheri, Exact solution for free vibration analysis of an eccentric elliptical plate, Arch. Appl. Mech. 84 (2014) 543–552.
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[15] S.C. Pradhan, T. Murmu, Small scale effect on the buckling analysis of single-
layered graphene sheet embedded in an elastic medium based on nonlocal plate theory, Physica E 42 (2010) 1293–1301.
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[16] K. K. Raju, G. V. Rao, Post-buckling of cylindrically orthotropic circular plates on elastic foundation with edges elastically restrained against rotation, Comput.
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Struct. 18(6) (1984) 1183-1187.
ACCEPTED MANUSCRIPT Figures Caption Fig. 1. Geometry of a radially loaded eccentric graphene sheet resting on elastic foundation Fig.2. Comparison of the buckling load parameter N of CS and CC annular graphene sheet for various values of nonlocal parameter when a) 0.333 b) 0.5 Fig.3. Effects of the eccentricity parameter on the non-dimensional buckling load ( N ) of a) CC and b) CS graphene sheets for various values of Winkler modulus parameter K w ( 0.01, KG 0, 0.2 )
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Fig.4. Effects of the nonlocal parameter on the non-dimensional buckling load ( N ) of CS graphene sheets with a) R1 10 nm and b) R1 5 nm for various values of eccentricity ratio ( Kw 100, KG 10, 0.4 )
Fig.5. Effects of the non-dimensional shear modulus K G on the non-dimensional buckling
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load ( N ) of a) CS and b) CC graphene sheets for various values of eccentricity ratio ( Kw 100, 0.4, 0.01 )
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Fig. 6. First four buckling mode shapes of a CC eccentric annular graphene sheet resting on elastic foundation with 0.01, KG 0, Kw 100, 0.2 and 0.6
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ACCEPTED MANUSCRIPT
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Fig. 1. Geometry of a radially loaded eccentric graphene sheet resting on elastic foundation
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a)
b)
160
90
Ref [3] (C-C) Present (C-C) Ref [3] (C-S) Present (C-S)
80
Ref [3] (C-C) Present (C-C) Ref [3] (C-S) Present (C-S)
140
70
60
80
50
60 40 0
0.5
1
1.5
2
0
0.5
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40
100
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120
1
1.5
2
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Fig.2. Comparison of the buckling load parameter N of CS and CC annular graphene sheet for various values of nonlocal parameter when a) 0.333 b) 0.5
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a)
b)
40
35
_ _Kw=100 _Kw=50 _Kw=10
30
Kw=0
25
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Kw=0
30
35
_ _Kw=100 _Kw=50 _Kw=10
25
20 20 15 0
0.2
0.4 _
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15 0.6
0.8
0
0.2
0.4 _
0.6
0.8
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Fig.3. Effects of the eccentricity parameter on the non-dimensional buckling load ( N ) of a) CC and b) CS graphene sheets for various values of Winkler modulus parameter K w ( 0.01, KG 0, 0.2)
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a)
b)
80
80
_ _ _
70
_ _ _
70
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60
60
50
50 40
40
30
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30 20
0
0.4
0.8
1.2
1.6
2
0
0.4
0.8
1.2
1.6
2
Fig.4. Effects of the nonlocal parameter on the non-dimensional buckling load ( N ) of CS graphene sheets with a)
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R1 10 nm and b) R1 5 nm for various values of eccentricity ratio ( Kw 100, KG 10, 0.4 )
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a)
b)
_ _ _
60
_ _ _
70
50
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60
50
40
30
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40
30
0
4
8
_
G
12
16
20
0
4
8
_
G
12
16
20
Fig.5. Effects of the non-dimensional shear modulus K G on the non-dimensional buckling load ( N ) of a) CS and b)
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CC graphene sheets for various values of eccentricity ratio ( Kw 100, 0.4, 0.01 )
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Mode II (Antisymmetric)
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Mode I (Symmetric)
Mode IV (Antisymmetric)
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Mode III (Symmetric)
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Fig. 6. First four buckling mode shapes of a CC eccentric annular graphene sheet resting on elastic foundation with 0.01, KG 0, Kw 100, 0.2 and 0.6
ACCEPTED MANUSCRIPT Tables Caption Table 1. Comparison of critical buckling load parameter N NR12 / D for clamped and simply supported circular plates resting on Winkler foundation with K w 0, 0.1, 0.2, 0.5,1, 2, 5,10 . Table 2. Comparison of critical buckling load parameter N NR12 / D for C‐C and C‐S eccentric annular plates resting on Winkler foundation with 0.2 , 0, 0.1, 0.2 and
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K w 0,15.6 , 31.2, 62.4 .
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ACCEPTED MANUSCRIPT Table 1. Comparison of critical buckling load parameter N NR12 / D for clamped and simply
0
0.1
0.2
0.5
FEM [16]
4.1978
4.2150
4.2322
4.2839
Present
4.1978
4.2150
4.2322
4.2839
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supported circular plates resting on Winkler foundation with K w 0, 0.1, 0.2, 0.5,1, 2, 5,10 .
FEM [16]
14.6826
14.6962
14.7098
Present
14.6820 14.6956
14.7092
B. C’s
Kw
Method
Simply Supported Clamped
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1
2
5
10
4.3700
4.5424
5.0591
5.9203
4.3700
4.5424
5.0591
5.9203
14.7507
14.8187
14.9548 15.3624 16.0396
14.7501
14.8181
14.9542 15.3618 16.0389
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ACCEPTED MANUSCRIPT Table 2. Comparison of critical buckling load parameter N NR12 / D for C‐C and C‐S eccentric annular plates resting on Winkler foundation with 0.2 , 0, 0.1, 0.2 and K w 0,15.6 , 31.2, 62.4 . B.C’s
Buckling load parameter ( N )
Method
C‐C
0
Kw 0
K w 15.6
K w 31.2
K w 62.4
ABAQUS
56.352
56.846
57.339
58.326
Present
56.287
56.780
57.274
58.260
Error (%)
0.115
0.116
0.113
0.113
0.1
ABAQUS
47.748
48.472
Present
47.626
48.360
Error (%)
0.256
0.2
ABAQUS
39.755
Present
39.416
Error (%)
0.853
C‐S
0
ABAQUS
41.472
42.210
42.947
44.420
Present
41.403
42.141
42.879
44.352
Error (%)
0.166
0.163
0.158
0.153
50.588
49.085
50.509
AN US 0.231
0.205
0.156
40.643
41.522
43.245
40.312
41.199
42.948
0.778
0.687
M
ED
PT
CE
49.186
0.814
0.1
ABAQUS
35.197
36.129
37.055
38.889
AC
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Present
35.024
35.952
36.876
38.707
Error (%)
0.492
0.490
0.483
0.468
0.2
ABAQUS
30.190
31.273
32.350
34.481
Present
29.786
30.867
31.934
34.048
Error (%)
1.338
1.298
1.286
1.256
Buckling analysis of a defective annular graphene sheet in elastic medium M. Fadaee PII: DOI: Reference:
S0307-904X(15)00563-6 10.1016/j.apm.2015.09.029 APM 10728
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
12 February 2015 20 July 2015 23 September 2015
Please cite this article as: M. Fadaee , Buckling analysis of a defective annular graphene sheet in elastic medium, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.09.029
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ACCEPTED MANUSCRIPT Highlights > Static stability of a defective annular graphene sheet is considered. > Translational addition theorem is employed to analyze the problem. > The graphene sheet is elastically restrained at the bottom surface.
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> The eccentricity and size of defect have significant effects on the buckling loads.
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Buckling analysis of a defective annular graphene sheet in elastic medium
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M. Fadaee1 1
Department of Mechanical Engineering, College of Engineering, Qom University of Technology, Iran.
Abstract
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In this article, an analytical approach is presented to analyze static stability of defective annular graphene sheet. Due to production process and constrains conditions, graphene sheet may be opposed to structural defect. Some of the defects can be modelled as an eccentric hole. The graphene sheet is elastically restrained at the bottom surface.
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Nonlocal thin plate theory as well as the translational addition theorem are employed to
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solve the problem. The stability and accuracy of results are examined by the literature and the finite element analyses. Effects of eccentricity of defects, nonlocality, Winkler
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and Pasternak foundation parameters and various boundary conditions on the critical
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buckling load of an annular graphene sheet are investigated. It is observed that the eccentricity and size of defects have significant effect on the critical load.
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Keywords: Graphene sheet, nonlocal elasticity, defect, eccentric hole, translational addition theorem, buckling analysis.
1. Introduction
1
Corresponding author. E-mail address: [email protected] (Mohammad Fadaee).
ACCEPTED MANUSCRIPT Invention of graphene sheets initiated a new perspective in the mechanical devices. Graphene, a special monolayer of hexagon-lattice, is even stiffer and stronger than carbon nanotube (CNT). Their superior mechanical, thermal and electrical properties let to produce small devices that were impossible before. Therefore, understanding the mechanical behavior of the graphene sheets such as buckling phenomenon is an
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important issue.
Various methods are applied to analyze different mechanical, electrical and optical
properties of graphene sheets. These methods can be categorized in three groups as: atomistic methods, continuum methods and atomistic-continuum methods. Non-
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classical continuum methods which considering non continuity of nano scale structures, have lower precision respect to the two other methods but they can present exact phrases with low computational effort for various problems. Nonlocal theory of
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elasticity which was presented by Eringen [1] is one of the such continuum theories. A few studies have been performed in the field of buckling analysis of graphene sheets,
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especially for circular ones. Farajpour et al. [2] studied axisymmetric buckling of circular graphene sheet in a thermal environment based on nonlocal elasticity. They
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used differential quadrature method (DQM) and Galerkin method to solve the problem
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and showed that their results have good agreement with molecular dynamic simulation results. Ravari and Shahidi [3] applied finite difference method to nonlocal buckling
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equation of annular plate and investigated effect of small scale on the radial compressive critical load. Bedroud et al. [4] analyzed buckling of circular and annular nanoplate based on nonlocal first order shear deformation. They investigated the effect of small scales on buckling loads for different geometry, boundary conditions and axisymmetric or asymmetric mode numbers. Asemi et al. [5] investigated thermal effects on the stability of a circular graphene sheet using nonlocal plate model. Natsuki
ACCEPTED MANUSCRIPT et al. [6] studied the buckling properties of circular double-layered graphene sheets (DLGSs), using plate theory. An analytical solution of coupled governing equations is proposed for predicting the buckling properties of circular DLGSs. Neek-Amal and Peeters [7] investigated the stability of circular monolayer graphene subjected to a radial load using non-equilibrium molecular dynamics simulations. Zenkour and Sobhy [8]
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studied the thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium using nonlocal elasticity theory. Malekzadeh et al. [9] considered small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral
nanoplates embedded in an elastic medium. Wang et al. [10] reported thermal buckling
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properties of rectangular nanoplates with small-scale effects. They derived the critical temperatures for the nonlocal Kirchhoff and Mindlin plate theories by nonlocal continuum mechanics.
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According to this literature survey, it can be concluded that all researches in buckling
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analysis have focused on annular nano plates with a centric hole. But graphene sheets may be defected in different form in the production process (e.g. single or double
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vacancy and Stone-Wales defects). Also, graphene sheets can be constrained in an area except the edge, for example pinning in a point. These defects and pin holes can be
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considered as eccentric circular hole. In order to analytically model an eccentric hole in
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a circular graphene sheet, the translational addition theorem (TAT) can be employed. Watson [11] presented basic concepts of this theorem in 1992. Lin [12] used the TAT for vibration analysis of circular membranes and plates. Alsahlani and Mukherjee [13] analyzed dynamics of a circular membrane with an eccentric circular areal constraint under arbitrary initial conditions using the TAT. They studied the symmetric and antisymmetric modes of vibrations. Hasheminejad and Ghaheri [14] presented the method of separation of variables in elliptical coordinates in conjunction with the
ACCEPTED MANUSCRIPT translational addition theorems for Mathieu functions to investigate the free flexural vibrations of a fully clamped thin elastic panel of elliptical planform containing an elliptical cutout of arbitrary size, location, and orientation. The lack of study on the buckling of defective annular graphene sheets is clear. Therefore, this paper presents static stability analysis of an annular graphene sheet with
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a hole in an arbitrary point on its surface as a defect and resting on elastic foundation. Governing equation is derived based on nonlocal theory of elasticity. This equation is analytically solved using the translational addition theorem. Then, critical buckling
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loads are found for various geometry and scale parameters. Results are compared with literature and the finite element analyses and good agreements are achieved. Finally, effects of eccentricity, nonlocality and elastic foundation parameters on buckling loads are investigated.
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2. Mathematical formulations 2.1. Geometrical configuration
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Consider an eccentric annular graphene sheet resting on elastic foundation with Winkler
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and Pasternak parameters as KW and KG , respectively, as shown in Fig.1. The sheet has inside radius R2 , outside radius R1 , eccentricity , thickness h and is subjected to a
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radially compression load N . To interpret mathematical formulations, two polar
coordinates (r1 ,1 ) and (r2 , 2 ) are taken to coincide with the center of the outer and inner circles, respectively. 2.2. The equilibrium equation
ACCEPTED MANUSCRIPT Based on nonlocal theory presented by Eringen [1], the stress at a reference point in an elastic medium depends on the strain at every point of the body. According to this theory, the constitutive equation will be obtained as [1]
(1 2 2 )σ C : ε
(1)
where and are stress and strain tensors, C is fourth order elastic modulus tensor,
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2 is Laplacian operator and 2 (e0 a)2 . e0 is a constant correlated by material. The parameter e0 a is known as small scale. A certain value for small scale is not available and for any type of analysis, this will be found by comparing the results of continuum
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modeling with atomistic ones.
For buckling analysis, the displacement fields of an isotropic graphene sheet according
u (r2 , 2 , z ) z
W (r2 , 2 ) r2
v (r2 , 2 , z ) z
1 W (r2 , 2 ) r2 2
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to the classical plate theory, for example, in polar coordinate (r2 , 2 ) are considered as
(2)
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w (r2 , 2 , z ) W (r2 , 2 )
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respectively.
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where u , v and w are the displacement functions in the r2 , 2 and z directions,
Substituting Eq. (2) into the strain-displacement relations, it can be concluded that
2W r22
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r r z 2 2
z W 1 2W ( ) r2 r2 r2 22
r
1 W z 2W 2 ( ) , r2 r2 2 r2 2
2 2
2 2
r z 0
,
2
,
z 0 2
zz 0
The stress-strain relations may be expressed for a graphene sheet as
(3)
ACCEPTED MANUSCRIPT r2 r2 1 0 r2 r2 E( z) (1 2 2 ) 22 2 1 0 22 1 1 r 0 0 r 22 2 2 2
(4)
Using Eqs. (1) to (4), the equilibrium equation of thin annular graphene sheet resting on elastic Winkler and Pasternak foundations is obtained as [15] 2
2
4
G
2
W
K G N 2W KW W 0
where
2.3. Analytical solution procedure
(5)
(6)
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2 1 1 2 2 1 1 2 2 ( 2 ) 2 2 2 2 2 2 r2 r2 r2 r2 2 r2 r2 r2 r2 2 Eh3 D 12(1 2 )
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D N K W K
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To obtain the solution of Eq. (5), the separation of variables method is employed. Therefore, the transverse displacement is supposed as
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W ( r2 , 2 ) ( r2 ). ( 2 )
(7)
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Substituting Eq. (7) into (5), it can be concluded that (8)
2 4 4 4 r22 d 2 2 1 d 2 1 d 2 2 r ( ) ( ) n2 2 2 2 r2 dr2 d 2 ( ) 2 ( r2 ) dr22 2 2 2
(9)
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r22 d 2 1 1 d 1 2 4 4 4 1 d 2 1 r ( ) ( ) n2 1 2 2 r2 dr2 d 2 ( ) 1 ( r2 ) dr22 1 2 2
where
W (r2 , 2 ) 1 (r2 ).1 ( 2 ) 2 (r2 ).2 ( 2 )
(10)
ACCEPTED MANUSCRIPT Because the sheet is complete circular, the period of functions 1 ( 2 ) and 2 ( 2 ) must be 2π ( n 0,1, 2,... ) and consequently
1 ( 2 ) A n cos (n 2 ) B n sin (n 2 )
(11a)
2 ( 2 ) C n cos (n 2 ) D n sin (n 2 )
(11b)
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Then, according to Bessel differential equations [11], it is obtained from Eqs. (8) and (9) that
1 (r2 ) E n I n ( r2 ) Fn K n ( r2 )
(12)
2 (r2 ) G n J n ( r2 ) H n Y n ( r2 )
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(13)
where J n ( r2 ) , Y n ( r2 ) , I n ( r2 ) , K n ( r2 ) are Bessel functions of the first and second kinds and modified Bessel functions of the first and second kinds of order n ,
4 4 4 2 2
(14)
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4 4 4 2 , 2
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respectively. A n , B n ,…, H n are variable coefficients. Also,
,
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KW 4 2 D N 2K G
2 KW K G N D 2N 2K G
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Using Eqs. (11) to (13) into Eq. (10) as well as the superposition principle, the
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transverse deflection of the sheet can be rewritten as
W (r2 , 2 ) A1n J n ( r2 ) A2 n Yn ( r2 ) A3n I n ( r2 ) A4 n K n ( r2 ) cos (n 2 )
(15)
n 0
B1n J n ( r2 ) B2 n Yn ( r2 ) B3n I n ( r2 ) B4 n K n ( r2 ) sin(n 2 ) n 1
where A1n , A2 n , A3n , A4 n and B1n , B2 n , B3n , B4 n are the mode shape coefficients of buckling, which are determined by applying the boundary conditions. In Eq. (15), the cosine series is corresponding to symmetric buckling modes of graphene sheet and that
ACCEPTED MANUSCRIPT with sin(n 2 ) is for antisymmetric modes. For more information about the symmetric and antisymmetric buckling modes, section 3.2.4 will be presented. 2.3.1. Exact solution for symmetric buckling modes Since, the cosine function is symmetric, the part of transverse displacement which is
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multiplied by the cosine function, is identified as the symmetric solution. First, the symmetric buckling modes are considered. Therefore, the following solution of displacement should be considered as
W S (r2 , 2 ) A1n J n ( r2 ) A2 n Yn ( r2 ) A3n I n ( r2 ) A4 n K n ( r2 ) cos (n 2 )
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n 0
(16)
Herein, it is assumed that the graphene sheet is clamped in the outer and inner edges. Satisfying the clamped boundary conditions at the inner edge leads to
r2 R 2
0
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W S (r2 , 2 )
(17a)
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A1n J n ( R 2 ) A 2 n Y n ( R 2 ) A3n I n ( R 2 ) A 4 n K n ( R 2 ) 0 (17b)
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dW S (r2 , 2 ) 0 r2 R 2 dr2
A1n J n ( R 2 ) A 2 n Y n( R 2 ) A3n I n ( R 2 ) A 4 n K n ( R 2 ) 0
dJ n dY n dI dK n , Y n , I n n , K n dr2 dr2 dr2 dr2
(18)
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J n
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where
In order to satisfy the clamped boundary condition at the outer edge, the displacement
W S ( R2 ,2 ) should be expressed in terms of the polar coordinates of the outer circle (r1 ,1 ) . This can be carried out by employing the translational addition theorem for cylindrical vector wave functions [11-12], involving the regular and modified Bessel
ACCEPTED MANUSCRIPT functions. This theorem for the first and second regular and modified Bessel functions from the (r2 ,2 ) coordinate to (r1 ,1 ) coordinate may be presented as (19a)
Cos ( n 2 ) Cos ( n m)1 ) Yn ( r2 ) Yn m ( r1 ) J m ( ) Sin( n 2 ) m Sin( n m)1 )
(19b)
Cos ( n 2 ) Cos ( n m)1 ) m I n ( r2 ) ( 1) I n m ( r1 ) I m ( ) Sin( n 2 ) m Sin( n m)1 )
(19c)
Cos ( n 2 ) Cos ( n m)1 ) K n ( r2 ) K n m ( r1 ) I m ( ) Sin( n 2 ) m Sin( n m)1 )
(19d)
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Cos (n 2 ) Cos (n m)1 ) J n ( r2 ) J n m ( r1 ) J m ( ) Sin(n 2 ) m Sin( n m)1 )
Substituting Eqs. (19a-d) into Eqs. (17) and applying the clamped boundary condition at the outer edge, leads to
n 0 m
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r1 R1
0
(20b)
J m ( ) A1n J n m ( R1 ) A 2 n Y n m ( R1 ) Cos (n m )1 0 m I m ( ) (1) A3n I n m ( R1 ) A 4 n K n m ( R1 )
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dW S (r1 , 1 ) 0 r1 R1 dr1
(20a)
J m ( ) A1n J n m ( R1 ) A 2 n Y n m ( R1 ) Cos (n m )1 0 m I m ( ) (1) A3n I n m ( R1 ) A 4 n K n m ( R1 )
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W S (r1 ,1 )
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n 0 m
Applying the orthogonality relation for the cosine functions as 2
cos (m n)1 cos q1 d1 mn,q mn, q
(21)
0
where mn,q is the second Kronecker delta and using the Bessel function properties as
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J p (r ) (1) p J p (r ) , Y p (r ) (1) p Yp (r ) I p (r ) I p (r )
, K p (r ) K p (r )
Eqs. (20a) and (20b) can be rewritten as
n 0 q 0
n 0 q 0
J q n ( ) (1) n J q n ( ) A1n J q ( R1 ) A2 n Yq ( R1 ) 0 I q n ( ) I q n ( ) (1) q n A3n I q ( R1 ) A4 n K q ( R1 )
(23a)
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1 J q n ( ) (1)n J q n ( ) A1n J q ( R1 ) A2 n Yq( R1 ) 2 0 1 I ( ) I ( ) (1) q n A I ( R ) A K ( R ) 3n q 1 4n 1 qn qn q 2
(23b)
Eqs. (17a), (17b), (23a) and (23b) should be simultaneously satisfied to obtain the
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symmetric buckling loads of the eccentric annular graphene sheet. Expanding these equations from n=0 to N horizontally and from q=0 to N vertically, gives the matrix equation as 0 that the dimensions of matrix are (4 N 4) (4 N 4) . The
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determinant of matrix is the closed-form buckling equation for C-C eccentric annular graphene sheet. The vector X ( (4N 4) 1 ) is composed of A1n , A2 n , A3n , A4 n for n=0
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to N. Therefore, to calculate the symmetric mode shape coefficients as A1n , A2 n , A3n ,
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A4 n , null space of the matrix B should be determined.
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It is worth noting that eliminating A3n and A4 n from Eqs. (23) with using of Eqs. (17) leads to a decrease in the computational complexities. This is because of the order of
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matrix for finite n and q which would be reduced to half as (2 N 2) (2 N 2) . It
will be led to simplicity and quickness of computation in the calculations. 2.3.2. Exact solution for antisymmetric buckling modes The antisymmetric modes coincide with the sine term as sin(n 2 ) in Eq. (15). The
boundary conditions of antisymmetric modes in the inner edge are the same as Eqs.
ACCEPTED MANUSCRIPT (17). Following the same procedure as in the case of the symmetric modes and substituting the orthogonality relation for the sine functions as 2
(24)
sin (m n)1 sin q1 d1 mn,q mn, q 0
The outer edge boundary conditions in Eqs. (23a) and (23b) can be rewritten for
n 1 q 1
n 1 q 1
J q n ( ) (1) n J q n ( ) A1n J q ( R1 ) A2 n Yq ( R1 ) 0 I q n ( ) I q n ( ) (1) q n A3n I q ( R1 ) A4 n K q ( R1 )
(25a)
1 J q n ( ) (1)n J q n ( ) A1n J q ( R1 ) A2 n Yq( R1 ) 2 0 1 I ( ) I ( ) (1)q n A I ( R ) A K ( R ) 3n q 1 4n 1 qn qn q 2
(25b)
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antisymmetric modes as
For antisymmetric buckling modes, the obtaining of the buckling equation is exactly the
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3. Results and discussion
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same as that for symmetric modes, but the indices n and q vary from 1 to .
Based on the analytical solution mentioned in section 2, a computer code has been
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provided to obtain critical buckling loads of eccentric annular graphene sheets resting
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on elastic foundation. First, in order to examine validation of the present approach, three comparison studies are done. Then, effects of eccentricity, nonlocality, elastic
AC
foundation parameters and boundary conditions are investigated in some figures. Also, the following non-dimensional parameters are introduced as
K R2 K R4 R2 NR12 2 , , N , K G G 1 , KW W 1 , 2 R1 R1 D D D R1
(26)
It should be mentioned that the results in case I and case II of section 3.1 are obtained in macro size as circular and annular plates using the classical plate theory, whereas the
ACCEPTED MANUSCRIPT results of section 3.2 and case III in section 3.1 are calculated in nano size as the graphene sheet using the nonlocal theory. 3.1. Comparison study Case I: Table 1 shows comparison of critical buckling load parameter N NR12 / D for
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clamped and simply supported circular plates for various values of Winkler foundation parameter K W . This table reports excellent agreement between results of the present method and the literature obtained by a finite element method.
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Case II: Table 2 gives critical buckling load parameters N NR12 / D of CC and CS (the
clamped outer edge and the simply supported inner edge) eccentric ( 0, 0.1, 0.2 ) annular plates resting on Winkler foundation ( 0.2 , Kw 0,15.6,31.2,62.4 ) based on the present method and a 3D brick element modeling in traditional finite element software
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(ABAQUS) created on the basis the 3D elasticity. A mesh sensitivity analysis was
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carried out to ensure independency of finite element (FE) results from the number of elements. It is obvious that the present results capture finite element ones, precisely.
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Maximum value of error between two methods is 1.34%.
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Case III: In Fig. 2, behavior of the buckling load parameter N of CS and CC centric
annular graphene sheets versus nonlocal parameter is plotted based on the present
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analytical method and those obtained by Ref. [3]. Good agreement between results of the present method and the literature approves correctness of present method. 3.2. Parametric study 3.2.1. Effect of eccentricity
ACCEPTED MANUSCRIPT Fig. 3 shows effects of the eccentricity parameter on the non-dimensional buckling load N CC and CS graphene sheets for various values of Winkler modulus parameter K w ( 0.01, KG 0, 0.2 ). According to Fig. 3, increasing the eccentricity decreases
the critical buckling load parameter while increasing the Winkler modulus parameter increases the critical buckling load due to enhancing the stiffness of the graphene sheet
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for both boundary conditions. Also, it is evident that the critical buckling load parameter N is less affected by increasing the Winkler modulus parameter.
3.2.2. Effect of nonlocality
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Effects of the nonlocal parameter on the non-dimensional buckling load N of CS graphene sheets are considered in Fig. 4 for various values of eccentricity ratio ( Kw 100, KG 10, 0.4 ). These figures reveal that increasing the eccentricity ratio and
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the nonlocal parameter decrease the critical buckling load parameter. Also, it is
nonlocal parameter
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obvious that for higher values of the eccentricity ratio and the outer radius R 1 , the has less effects on the critical buckling load parameter than
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lower values of ones. It is because when the outer radius R 1 increases, size of the
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graphene sheet tends to micro one. Since the nonlocal parameter is defined for structures in nano size, therefore, its effect on the critical buckling load decreases.
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3.2.3. Effect of shear modulus K G Fig. 5 reports effects of the non-dimensional shear modulus K G on the non-dimensional
buckling load N of CS and CC graphene sheets for various values of eccentricity ratio
( Kw 100, 0.4, 0.01). It can be concluded from Fig. 5 that for all values of
eccentricity ratio, the critical buckling load parameter N varies linearly respect to K G .
ACCEPTED MANUSCRIPT Due to increasing the stiffness of graphene sheet, the buckling load N increases by enhancing the K G . 3.2.4. 3D view of buckling mode shapes 3-D plots of first four buckling mode shapes of a CC eccentric annular graphene sheet
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are illustrated in Fig. 6 to make the better physical sense of the symmetric and antisymmetric mode shapes. Also, this figure shows how boundary conditions can affect
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the buckling mode shapes of the eccentric graphene sheet.
4. Conclusion
The main objective of this study was to investigate buckling analysis of annular
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graphene sheet with an eccentric defect resting on Winkler and Pasternak foundations. Equilibrium equation was obtained based on nonlocal thin plate theory and solved using
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the translational addition theorem, analytically. Critical buckling loads were presented
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for various edge support, eccentricity, Winkler and Pasternak modulus parameters and nonlocal parameters. The results are validated by the literature and a finite element
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analysis. Buckling load of a specific eccentric annular graphene sheet is strongly depends on eccentricity and the nonlocal parameter. It is observed that increasing
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nonlocal parameter and eccentricity ratio decreases critical buckling loads for various boundary conditions. Enhancing the outer radius and eccentricity decreases nonlocal effect. Also, sensibility of critical buckling load of a graphene sheet to the eccentricity ratio is deceased by increasing the Winkler modulus parameter.
ACCEPTED MANUSCRIPT 5. References [1] A.C. Eringen, Nonlocal continuum field theories, Springer, (2002). [2] A. Farajpour, M. Dehghany, A.R. Shahidi, Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment,
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Compos. Part B 50(0) (2013) 333-343. [3] M.K. Ravari, A. Shahidi, Axisymmetric buckling of the circular annular nanoplates using finite difference method, Meccanica 48(1) (2013) 135-144.
[4] M. Bedroud, S. Hosseini-Hashemi, R. Nazemnezhad, Buckling of circular/annular
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Mindlin nanoplates via nonlocal elasticity, Acta mech 224(11) (2013) 26632676.
[5] S.R. Asemi, A. Farajpour, M. Borghei, A.H. Hassani, Thermal effects on the stability of circular graphene sheets via nonlocal continuum mechanics, Latin
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Amer. J. Solids Struct. 11(4) (2014).
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[6] T. Natsuki, J.X. Shi, Q.Q. Ni, Buckling instability of circular double-layered graphene sheets, J. Phys.: Condens. Matter 24 (2012) 135004 (5pp).
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[7] M. Neek-Amal, F. M. Peeters, Buckled circular monolayer graphene: a graphene nano-bowl, J. Phys.: Condens. Matter 23 (2011) 045002 (8pp).
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[8] A. Zenkour, M. Sobhy, Nonlocal elasticity theory for thermal buckling of nanoplates
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lying on Winkler–Pasternak elastic substrate medium, Physica E. 53 (2013) 251-259. [9] P. Malekzadeh, A. R. Setoodeh, A. Alibeygi Beni, Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an
elastic medium, Compos. struct. 93 (2011) 2083-2089. [10] Y. Z. Wang, H. T. Cui, F. M. Li, K. Kishimoto, Thermal buckling of a nanoplate with small-scale effects, Acta Mechanica 224 (2013) 1299–1307.
ACCEPTED MANUSCRIPT [11] G.N. Watson, A treatise on the theory of Bessel functions, Cambridge university press, (1922). [12] W. Lin, Free transverse vibrations of uniform circular plates and membranes with eccentric holes, J. Sound Vib. 81(3) (1982) 425-435. [13] A. Alsahlani, R. Mukherjee, Dynamics of a circular membrane with an eccentric
Practice Theory 31 (2013) 149–168.
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circular areal constraint: Analysis and accurate simulations, Simul. Model.
[14] S. M. Hasheminejad, A. Ghaheri, Exact solution for free vibration analysis of an eccentric elliptical plate, Arch. Appl. Mech. 84 (2014) 543–552.
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[15] S.C. Pradhan, T. Murmu, Small scale effect on the buckling analysis of single-
layered graphene sheet embedded in an elastic medium based on nonlocal plate theory, Physica E 42 (2010) 1293–1301.
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[16] K. K. Raju, G. V. Rao, Post-buckling of cylindrically orthotropic circular plates on elastic foundation with edges elastically restrained against rotation, Comput.
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Struct. 18(6) (1984) 1183-1187.
ACCEPTED MANUSCRIPT Figures Caption Fig. 1. Geometry of a radially loaded eccentric graphene sheet resting on elastic foundation Fig.2. Comparison of the buckling load parameter N of CS and CC annular graphene sheet for various values of nonlocal parameter when a) 0.333 b) 0.5 Fig.3. Effects of the eccentricity parameter on the non-dimensional buckling load ( N ) of a) CC and b) CS graphene sheets for various values of Winkler modulus parameter K w ( 0.01, KG 0, 0.2 )
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Fig.4. Effects of the nonlocal parameter on the non-dimensional buckling load ( N ) of CS graphene sheets with a) R1 10 nm and b) R1 5 nm for various values of eccentricity ratio ( Kw 100, KG 10, 0.4 )
Fig.5. Effects of the non-dimensional shear modulus K G on the non-dimensional buckling
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load ( N ) of a) CS and b) CC graphene sheets for various values of eccentricity ratio ( Kw 100, 0.4, 0.01 )
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Fig. 6. First four buckling mode shapes of a CC eccentric annular graphene sheet resting on elastic foundation with 0.01, KG 0, Kw 100, 0.2 and 0.6
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Fig. 1. Geometry of a radially loaded eccentric graphene sheet resting on elastic foundation
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b)
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Ref [3] (C-C) Present (C-C) Ref [3] (C-S) Present (C-S)
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Ref [3] (C-C) Present (C-C) Ref [3] (C-S) Present (C-S)
140
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60
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60 40 0
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1.5
2
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100
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120
1
1.5
2
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Fig.2. Comparison of the buckling load parameter N of CS and CC annular graphene sheet for various values of nonlocal parameter when a) 0.333 b) 0.5
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b)
40
35
_ _Kw=100 _Kw=50 _Kw=10
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Kw=0
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_ _Kw=100 _Kw=50 _Kw=10
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20 20 15 0
0.2
0.4 _
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0.8
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0.4 _
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0.8
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Fig.3. Effects of the eccentricity parameter on the non-dimensional buckling load ( N ) of a) CC and b) CS graphene sheets for various values of Winkler modulus parameter K w ( 0.01, KG 0, 0.2)
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Fig.4. Effects of the nonlocal parameter on the non-dimensional buckling load ( N ) of CS graphene sheets with a)
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R1 10 nm and b) R1 5 nm for various values of eccentricity ratio ( Kw 100, KG 10, 0.4 )
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Fig.5. Effects of the non-dimensional shear modulus K G on the non-dimensional buckling load ( N ) of a) CS and b)
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CC graphene sheets for various values of eccentricity ratio ( Kw 100, 0.4, 0.01 )
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Mode II (Antisymmetric)
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Mode I (Symmetric)
Mode IV (Antisymmetric)
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Mode III (Symmetric)
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Fig. 6. First four buckling mode shapes of a CC eccentric annular graphene sheet resting on elastic foundation with 0.01, KG 0, Kw 100, 0.2 and 0.6
ACCEPTED MANUSCRIPT Tables Caption Table 1. Comparison of critical buckling load parameter N NR12 / D for clamped and simply supported circular plates resting on Winkler foundation with K w 0, 0.1, 0.2, 0.5,1, 2, 5,10 . Table 2. Comparison of critical buckling load parameter N NR12 / D for C‐C and C‐S eccentric annular plates resting on Winkler foundation with 0.2 , 0, 0.1, 0.2 and
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K w 0,15.6 , 31.2, 62.4 .
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ACCEPTED MANUSCRIPT Table 1. Comparison of critical buckling load parameter N NR12 / D for clamped and simply
0
0.1
0.2
0.5
FEM [16]
4.1978
4.2150
4.2322
4.2839
Present
4.1978
4.2150
4.2322
4.2839
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supported circular plates resting on Winkler foundation with K w 0, 0.1, 0.2, 0.5,1, 2, 5,10 .
FEM [16]
14.6826
14.6962
14.7098
Present
14.6820 14.6956
14.7092
B. C’s
Kw
Method
Simply Supported Clamped
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1
2
5
10
4.3700
4.5424
5.0591
5.9203
4.3700
4.5424
5.0591
5.9203
14.7507
14.8187
14.9548 15.3624 16.0396
14.7501
14.8181
14.9542 15.3618 16.0389
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Buckling load parameter ( N )
Method
C‐C
0
Kw 0
K w 15.6
K w 31.2
K w 62.4
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56.352
56.846
57.339
58.326
Present
56.287
56.780
57.274
58.260
Error (%)
0.115
0.116
0.113
0.113
0.1
ABAQUS
47.748
48.472
Present
47.626
48.360
Error (%)
0.256
0.2
ABAQUS
39.755
Present
39.416
Error (%)
0.853
C‐S
0
ABAQUS
41.472
42.210
42.947
44.420
Present
41.403
42.141
42.879
44.352
Error (%)
0.166
0.163
0.158
0.153
50.588
49.085
50.509
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0.205
0.156
40.643
41.522
43.245
40.312
41.199
42.948
0.778
0.687
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49.186
0.814
0.1
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35.197
36.129
37.055
38.889
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35.024
35.952
36.876
38.707
Error (%)
0.492
0.490
0.483
0.468
0.2
ABAQUS
30.190
31.273
32.350
34.481
Present
29.786
30.867
31.934
34.048
Error (%)
1.338
1.298
1.286
1.256