The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory

Composites Part B 98 (2016) 78e87

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Composites Part B journal homepage: www.elsevier.com/locate/compositesb

The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory Shahriar Dastjerdi a, *, Masoumeh Lotfi b, Mehrdad Jabbarzadeh b a b

Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 March 2016 Received in revised form 15 April 2016 Accepted 1 May 2016 Available online 10 May 2016

In the paper, effects of eccentric vacant defect on bending analysis of circular graphene sheets have been studied based on nonlocal elasticity theory considering the Mindlin theory of the plates. The governing equations have been derived for circular graphene sheet including eccentric vacant defect. Because of existence of an eccentric vacant defect on the plate, the anti-symmetric problem has been obtained. So, the constitutive equations are partial differential equations system. Considering this fact that there is not any analytical solution available for solving the nonlinear constitutive equations based on the first order shear deformation theory, a new semi-analytical polynomial method is applied which has been presented by the authors recently. By applying the mentioned method, the governing equations have been solved. Regarding this fact that no study has been done heretofore in case of the effects of eccentric vacant defect on bending analysis of graphene sheets, the obtained results of local analysis have been compared with the results of Abaqus software. At the following, the effects of nonlocal parameters, size and location of vacant defect on the results have been surveyed. © 2016 Elsevier Ltd. All rights reserved.

Keywords: A. Nano-structures A. Plates B. Defects B. Elasticity C. Micro-mechanics

1. Introduction Graphene, the latest discovered carbon nanostructure, is an allotrope of carbon in the form of a two-dimensional, atomic-scale, honey-comb lattice in which one atom forms each vertex. Graphene has many extraordinary properties. It is about 100 times stronger than the strongest steel and is even stiffer and stronger than carbon nanotube (CNT). Nowadays Graphene becomes one of the most important nano-sized structural elements which has been generally used. Its amazing properties as the lightest and strongest material, compared with its ability to conduct heat and electricity better than anything else, mean that it can be integrated into a huge number of applications which let to produce small devices that were impossible before. Therefore, understanding the mechanical behaviors of the graphene sheets such as vibration, buckling and bending phenomenon are important issues. Several methods have been applied to analyze different mechanical properties of graphene sheets. These methods can be categorized in three groups as: atomistic methods, continuum methods and atomistic-continuum methods. Classical theories in

* Corresponding author. P.O.B. 9185734943, Mashhad, Iran. E-mail address: [email protected] (S. Dastjerdi). http://dx.doi.org/10.1016/j.compositesb.2016.05.009 1359-8368/© 2016 Elsevier Ltd. All rights reserved.

description of the proper mechanical behaviors of nanostructures are with some shortcomings, therefore the researchers have attempted to provide suitable models and theories for this aim. Non-classical continuum methods can present accurate phrases with low computational try for different problems. Nonlocal theory of elasticity which was presented by Eringen [1,2] is one of the continuum theories. In recent years, several studies on the mechanical behavior of graphene sheets (GSs) have been carried out, such as their buckling instability and vibrational characteristics. Wang et al. [3] developed a nonlinear continuum model for the vibrational analysis of multilayer GSs, in which there are nonlinear van der Waals (vdW) interactions between the two adjacent layers. Arash and Wang [4] investigated free vibration of single-layer (SLGSs) and double-layer graphene sheets (DLGSs) by applying nonlocal continuum theory and molecular dynamic (MD) simulations. Eltaher et al. [5] aims at directing the light to research work concerned with bending, buckling, vibrations, and wave propagation of nanobeams modeled according to the nonlocal elasticity theory of Eringen and studied on static and dynamic behavior of nanoscale beam structures. Kumar et al. [6] studied the thermal vibration analysis of DLGSs embedded in polymer elastic medium, using the plate theory and €z and nonlocal continuum mechanics for small scale effects. Akgo

S. Dastjerdi et al. / Composites Part B 98 (2016) 78e87

Civalek [7] studied on static bending response of single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium based on higher-order shear deformation microbeam models in conjunction with modified strain gradient theory and simulated interactions between SWCNTs and surrounding elastic medium by Winkler elastic foundation model and investigated that the bending behavior of SWCNTs is dependent on the small-size, stiffness of the elastic foundation and also effects of shear deformation, especially for smaller slenderness ratios. Mohammadi et al. [8] studied free transverse vibration analysis of circular and annular graphene sheets with different boundary conditions using the nonlocal continuum plate model. They derived governing equations by using the nonlocal elasticity theory for SLGS. Analytical frequency equations for circular and annular graphene sheets were obtained based on different cases of boundary conditions. Hashemi et al. [9] developed an exact solution for free vibration of coupled double viscoelastic graphene sheets by viscopasternak medium. Mohammadi et al. [10] investigated the shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment. Farajpour et al. [11] studied axisymmetric buckling of circular graphene sheet in a thermal environment based on nonlocal elasticity. Bedroud et al. [12] analyzed buckling of circular and annular nanoplate based on nonlocal first order shear deformation. Asemi et al. [13] investigated thermal effects on the stability of a circular graphene sheet using nonlocal plate model. Natsuki et al. [14] studied the buckling properties of circular double-layered graphene sheets (DLGSs), using plate theory. Zenkour and Sobhy [15] studied the thermal buckling of nanoplates lying on WinklerePasternak elastic substrate medium using nonlocal elasticity theory. Wang et al. [16] reported thermal buckling 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. Dastjerdi et al. [17,18] studied the nonlinear bending behavior of single layer sector and bilayer rectangular graphene sheets which is subjected to uniform transverse loads resting on a WinklerPasternak elastic foundation based on nonlocal elasticity theory. Dastjerdi et al. [19] derived the constitutive equations of graphene sheets based on nonlocal elasticity theory considering the first and higher order shear deformation theories in Cartesian and Cylindrical coordinates systems. In Continue, Dastjerdi et al. [20] worked on static analysis of single layer annular/circular graphene sheets based on nonlocal theory of Eringen. They presented a new semianalytical polynomial method (SAPM) for solving the ordinary and partial differential equations system [17,18,20]. According to above, it can be resulted that the most of researches have focused on annular nanoplates with a centric hole and especially for vibration and buckling analyses considering the classical plate theory. But graphene sheets may be defected in different form in the production process (e.g. single or double 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 considered as eccentric circular hole. The defects in graphene sheets can be categorized in three types and based on the genesis mechanism of the defects: removing carbon atoms (vacancies), adding carbon atoms or other impurities (adatoms), and rearrangement of existing carbon atoms (different ring sizes like Stone-Wales defects made by rotating a carbon bond). In recent years, vacancies and Stone-Wales defects and their effect on properties of GSs have been studied. Liu et al. [21,22] calculated the arrangement of pentagon/heptagon defects and related out-of-plane deformations up to about 3 A in grain boundaries based on the chirality and wry angles of grains. Yazyev and Louie [23], worked on a series of hybrid molecular dynamics simulations to study the structures, energies, and

79

structural transformations of grain boundaries of graphene sheets and presented a general approach for constructing the dislocations in graphene grain boundaries using ab initio calculations and investigated strong tendency toward out-of-plane deformation. They showed that defects in grain boundaries have important effects on physical properties of graphene and mentioned that these effects may be used for engineering graphene-based nano-materials and functional devices. Lusk and Carr [24,25] suggested a method for engineering defects in graphene by introducing defect domains and presented a set of stable defects. Zhang et al. [26] investigated the possibility of controlling the shape of graphene using defects based on both continuum and atomistic simulations. Watson [27] presented basic concepts of this theorem in 1992. Lin [28] employed the TAT for vibration analysis of circular membranes and plates. Alsahlani and Mukherjee [29] analyzed dynamics of a circular membrane with an eccentric circular areal constraint under arbitrary initial conditions using the TAT. They studied the symmetric and anti-symmetric modes of vibrations. Hasheminejad and Ghaheri [30] presented the method of separation of variables in elliptical coordinates compounded with the translational addition theorems for Mathieu functions to investigate the free flexural vibrations of a fully clamped thin elastic panel of elliptical plan-form containing an elliptical cutout of arbitrary size, location, and orientation. M. Fadaee et al. [31] present an analytical approach to analyze static stability of defective annular graphene sheet and investigated that the eccentricity and size of defects have significant effect on the critical load. E. Allahyari, M. Fadaee et al. [32] studied on effects of a single vacancy defect or a pin hole on free vibration behavior of a double layer graphene sheet and investigated that the fundamental natural frequency of an annular double-layer graphene sheet with a free eccentric circular defect is less affected by the size and location of the defect and the surface effect parameters have more significant effects on the in-phase vibration modes than the anti-phase ones. E. Jomehzadeh et al. [33] studied on bending of curved monolayer graphene and obtained governing nonlinear nonlocal elastic equations for a monolayer graphene with an initial curvature and found that the bending stiffness of graphene strongly depends on the initial configuration. Again E. Jomehzadeh et al. [34] studied on the effects of rippling on the bending stiffness of a monolayer graphene and calculated for a rippled graphene and the effects of rippling, material discreteness and quantify how the rippling strongly increases the effective bending stiffness of graphene and interacts with the discrete nature of the material. The lack of study on the bending analysis of defective annular graphene sheet is clear. Therefore, this paper presents bending analysis of an annular graphene sheet which is resting on elastic foundation with a hole in an arbitrary point on its surface as a defect. Governing equations are derived based on first order nonlocal elasticity theory. The constitutive equations are analytically solved applying an innovated semi-analytical polynomial method (SAPM) [17,18,20]. Because there is not any available numerical method in this case, the maximum deflection has been calculated for various types of boundary condition and location of defect and compared with Abaqus software. Following this calculations, the suitable agreements between the results have been achieved. Finally, the effects of eccentricity and nonlocal parameters on deflection have been investigated. 2. Governing equations The vacant defect is usually available in graphene structures. Consequently, studying the mechanical behavior of graphene sheets including the vacant defect could be an interesting issue. A circular graphene sheet including an eccentric hole is considered as

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Fig. 1(a). And also Fig. 1(b) shows the position of the plate on Winkler-Pasternak elastic medium. ro is the outer radius of the plate, ri defines the size of defect. According to Fig. 1(a) the inner hole is not on the center of the plate. In this study, the vacant defect is considered to be eccentric which is covering the whole geometry of the plate. The distance between the center of the plate and inner hole is assumed e. And S is the distance between the center of the inner hole and the outer boundary. The plate is embedded on a two parameters Winkler-Pasternak elastic foundation thoroughly. An uniform transverse load q is imposed on the surface of the plate. As the result of existing an eccentric hole, the problem is supposed to be axisymmetric and must be formulated in both r and q directions. Consequently, the system of partial differential equations (PDE's) would be obtained. The parameter S can be calculated considering the geometry of plate in Fig. 1(a) as follow:

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 o 2 S¼e  sin ðqÞ þ cosðqÞ e

(1)

Many researchers studied the mechanical behavior of the nanostructures based on the classical plate theory (CLPT) which is the simplest theory of the plates. This theory is only valuable for the thin plates and assumes some suppositions; for example, neglecting the effects of transverse shear deformations. When the thickness to length ratio is considerable, the effects of transverse shear deformations are remarkable and must be considered. Therefore in this paper, the governing equations are derived based on the firstorder shear deformation theory (FSDT) which assumes the neglected assumptions in classical plate theory. Considering the first-order shear deformation theory, the displacement field can be regarded as:

Uðr; q; zÞ ¼ uðr; qÞ þ zfðr; qÞ

(2)

Vðr; q; zÞ ¼ vðr; qÞ þ zjðr; qÞ

(3)

Wðr; qÞ ¼ wðr; qÞ

(4)

u, v and w are the displacement components of the mid-plane along the r, q and z directions, respectively. f and j define the rotation functions of the transverse normal about circumferential and radial directions. The plate is considered to have large deformation. Therefore, the von-Karman assumptions are considered with following strain field:

εr ¼

  vu vf 1 vw 2 þz þ vr vr 2 vr

(5)

εq ¼

      1 vv z vj 1 1 vw 2 þu þ þf þ r vq r vq 2 r vq

(6)

grz ¼

vw þf vr

(7)

gqz ¼

1 vw þj r vq

(8)

grq ¼

  1 vu vf vv vj 1 vw vw þ þz  v  zj þ þ z r vq vq vr vr r vr vq

(9)

Size effects or nonlocal effects are the changes that are caused by the decrease in the size of a body at small scale. The effects of small scale and atomic forces directly enter into the calculations as the material parameters [2]. In this theory, the stress at a reference point is a function of strain field in every points on whole domain of the plate. For the first time, Eringen demonstrated a differential form of the nonlocal constitutive equation from nonlocal balance law as follow [2]:

  1  mV2 sNL ¼ sL ; m ¼ ðe0 aÞ2

Fig. 1. (a) Geometry of circular graphene sheet including an eccentric defect, (b) Schematic view of Winkler-Pasternak elastic foundations.

(10)

In Eq. (10) a is internal characteristic length, and e0 is material constant which is defined by experimental tests. The parameter eoa is also called the small-scale parameter which is revealing the small-scale effect on the responses of nano-size structures. The value of the small-scale parameter depends on several circumstances such as the boundary conditions, chirality, number of walls, and the nature of motions. Wang [35] proved that the value of e0 a must be between 0 and 2 nm in nonlocal continuum mechanics.

S. Dastjerdi et al. / Composites Part B 98 (2016) 78e87

Similarly, in this paper, the range of nonlocal coefficient is taken between 0 and 2 nm. Applying Eq. (10), the nonlocal stresses can be defined in cylindrical coordinates system as follows [2]: (the letter L shows the local and NL nonlocal form of the stresses) 2 NL sNL r  m V sr 

 4 vsNL 2 NL rq  2 sNL r  sq 2 v q r r

2 NL sNL q  m V sq þ

 4 vsNL 2 NL rq þ 2 sNL r  sq 2 r vq r



2 NL sNL rz  m V srz 



1 NL 2 vsNL 2 NL rz sNL qz  m V sqz  r 2 sqz þ r 2 vq

¼ sLr

(11)

¼ sLq

(12)



¼ sLrq

¼ sLrz 

¼ sLqz

v2 1 v 1 v2 þ þ vr 2 r vr r 2 vq2

ðMr ; Mq ; Mrq Þ ¼

QrL ¼

ks Eh vw þf 2ð1 þ nÞ vr

QqL ¼

ks Eh 1 vw þj 2ð1 þ nÞ r vq

MrL ¼

   Eh3 vf n vj   þ þ f vr r vq 12 1  n2

(27)

MqL ¼

   Eh3 vf 1 vj   n þ þf 2 vr r vq 12 1  n

(28)

(16)

ðsr ; sq ; srq Þzdz





(24)

 (25)  (26)

(15)

h

Z2

  Eh vv v 1 vu 1 vw vw  þ þ 2ð1 þ nÞ vr r r vq r vr vq

(14)

The nonlocal force, moment and shear force components NiNL ; MiNL ði ¼ r; q; r qÞ, and QiNL ði ¼ r; qÞ can be obtained from the equations below:

NL

  !   ! vu 1 vw 2 u 1 vv 1 1 vw 2 þ þ þ þ vr 2 vr r r vq 2 r vq

NrLq ¼

(13)

V2 is the Laplacian operator in cylindrical coordinates system which is expressed as follow:

V2 ¼

n

(23)

!

!

1 NL 2 vsNL srz  2 qz 2 r r vq

Eh 1  n2

!

 4 NL 2 v  NL NL s þ s  s r q q r r2 r 2 vq

2 NL sNL r q  m V sr q 

NqL ¼

81

(17)

MrLq ¼

  Eh3 vj 1 vf j þ  24ð1 þ nÞ vr r vq r

(29)

In this study, by using the principle of stationary total potential energy, the constitutive equations in combination with the related boundary conditions along the edges of circular graphene plate including an eccentric hole can be derived. The variational equations of the total potential energy in nonlocal form can be presented as follow:

h 2

dP ¼ dU þ dU ¼ 0 ðNr ; Nq ; Nrq ; Qr ; Qq ÞNL ¼

Z

h 2

ðsr ; sq ; srq ; ks srz ; ks sqz ÞNL dz

(18)

h 2

 



1  mV2 NiNL ¼ NiL ; i ¼ ðr; q; r qÞ 

(19)

(20)

 1  mV2 QiNL ¼ QiL ; i ¼ ðr; qÞ

(21)

In Eqs. (19)e(21), NiL ; MiL ði ¼ r; q; r qÞ and QiL ði ¼ r; qÞ are the local in-plane force, moment and the shear force resultants which can be resulted from Eqs. (17) and (18) in local form for isotropic material as follows:

Eh 1  n2

 þ ks sNL qz dgqz dV 

(31) 

dU ¼ ∭ q  kw w þ kp V2 w dV

1  mV2 MiNL ¼ MiL ; i ¼ ðr; q; r qÞ

NrL ¼



NL NL NL dU ¼ ∭ sNL rr dεrr þ sqq dεqq þ srq dgr q þ ks srz dgrz V

Applying Eq. (10), the formulation between the local and nonlocal force, moment and shear force components can be presented below:



(30)

   !! vu 1 vw 2 u 1 vv 1 1 vw 2 þ þ þn þ vr 2 vr r r vq 2 r vq 

(22)

(32)

V

In Eq. (30) and (31), U and U are the strain energy and potential of applied forces, respectively. ks is the transverse shear correction coefficient. Also, kw and kp are the Winkler and Pasternak stiffness coefficients of elastic foundation. Applying the principle of stationary total potential energy, the nonlocal governing equations would be obtained in cylindrical coordinates system in terms of nonlocal force, moment and the shear force resultants as:

NL du : Nr;r þ

dv : NrNL q;r þ

 1  NL Nrq;q þ NrNL  NqNL ¼ 0 r

(33)

 1  NL Nq;q þ 2NrNL q ¼0 r

(34)

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S. Dastjerdi et al. / Composites Part B 98 (2016) 78e87

dw :

NL Qr;r

 1  NL Qq;q þ QrNL þ þ r

3.1. Semi-analytical polynomial method (SAPM) 2

q  kw w þ kp V w þ

v2 w NrNL 2 vr

! !! 2 1 vw 1 v2 w 1 vw NL 1 v w þ  þ 2Nrq r vr r 2 vq2 r vrvq r 2 vq

þ NqNL ¼0

(35) NL dj1 : Mr;r þ

 1  NL Mrq;q þ MrNL  MqNL  QrNL ¼ 0 r

dj2 : MrNL q;r þ

 1  NL Mq;q þ 2MrNL  QqNL ¼ 0 q r

(36)

(37)

In this method, every function in differential equations is estimated by a polynomial in general form depended on the grid point distribution. Contrary to collocation method, there is no need for the introduced polynomial functions to satisfy the boundary conditions. Every PDE or set of PDE's system would be solved conveniently and quickly considering different types of boundary conditions. A partial differential equation is considered as follow:

vn f ðr; qÞ vðn1Þ f ðr; qÞ vf ðr; qÞ vn f ðr; qÞ vðn1Þ f ðr; qÞ þ þ þ þ ::: þ n ðn1Þ vr v r vn q r v vðn1Þ q n n n vf ðr; qÞ v f ðr; qÞ v f ðr; qÞ v f ðr; qÞ þ ::: þ þ þ þ ::: þ ðn1Þ vq vr vq vrvðn1Þ q vr 2 vðn2Þ q ¼0 (43)

Substituting Eqs. (11)e(15) into Eqs. (16) and (17) and then applying the obtained nonlocal form of force, moment and shear force resultants into Eqs. (33)e(37) and neglecting some insignificant terms gives the five constitutive equations which can be expressed in local form as follows: L du : Nr;r þ

 1 L Nrq;q þ NrL  NqL ¼ 0 r

dv : NrLq;r þ

1 r



NqL;q þ 2NrLq ¼ 0

(38)

(39)

   1 L Qq;q þ QrL þ 1  mV2 r ! v2 w 1 vw 1 v2 w þ 2  q  kw w þ kp V2 w þ NrL 2 þ NqL r vr r vq2 vr !! 1 v2 w 1 vw  ¼0 þ 2NrLq r vrvq r 2 vq

L dw : Qr;r þ

(40) L dj1 : Mr;r þ

 1 L Mrq;q þ MrL  MqL  QrL ¼ 0 r

dj2 : MrLq;r þ

 1 L Mq;q þ 2MrLq  QqL ¼ 0 r

(41)

The function f(r,q) is presented as follow:

f ðr; qÞ ¼

N X M X

aðiþjð1ði1ÞðM1ÞÞÞ r ði1Þ qðj1Þ

(44)

i¼1 j¼1

In Eq. (44), N is the number of grid points in r and similarly M in

q directions. The grid points are shown in Fig. 2. By substituting Eq. (44) into Eq. (43), the partial differential equation is transformed to the algebraic equation. According to Fig. 2, number of 2M equations would be derived from boundary conditions (dark points in boundaries) and MðN  2Þ ¼ MN  2M equations from Eq. (44) (bright points). So, there are M$N algebraic equations and M$N unknown ai totally. By substituting the obtained ai ; i ¼ 1::M$N into Eq. (44), the function f(r,q) will be determined. For the set of partial differential equations the similar procedure can be applied too. If there are nonlinear differential equations, the nonlinear algebraic equations will be obtained. Hence, a numerical method must be applied to solve the set of nonlinear equations. In this paper, the Newton-Raphson numerical method is applied to solve the nonlinear algebraic equations system. According to the mentioned explanations for SAPM to achieve solution of the governing equations (Eqs. (38)e(42)), the below displacements and rotations can be introduced as follows [17]:

(42)

The constitutive equilibrium equations can be derived in terms of displacements and rotations by substituting Eqs. (22)e(29) into Eqs. (38)e(42).

3. Numerical solution According to the governing equations, it can be seen that a system of nonlinear partial differential equations has been obtained which cannot be solved by last analytical methods. Solving the partial differential equations is one of the time consuming and challenging issues which researchers apply different methods to solve the problems as well accurately and rapidly. Each method has its own advantages and disadvantages. In this paper because of eccentricity and axisymmetric nonlinear problem, it could not be possible to apply common analytical and numerical methods, so a new semi-analytical polynomial method (SAPM) which presented before by the authors, is applied.

Fig. 2. The schematic picture of grid points distribution for an eccentric annular/circular plate.

S. Dastjerdi et al. / Composites Part B 98 (2016) 78e87

N X M X

u¼

ðj1Þ

aðiþjð1ði1ÞðM1ÞÞÞ r ði1Þ q

(45)

i¼1 j¼1

83

ri ¼ 0:1m; ro ¼ 0:5m; h ¼ 0:01m; E ¼ 1:9  1011 N=m2 ;

n ¼ 0:29; q ¼ 104 Pa (51)

v¼

N X M X

aðiþjð1ði1ÞðM1ÞÞþM$NÞ r

ði1Þ ðj1Þ

q

(46)

In continue, a circular graphene sheet by an eccentric defect and boundary conditions of CC (clamped inner and outer edges) is considered with below conditions:

(47)

ri ¼ 1nm; ro ¼ 5nm; h ¼ 0:34nm; e ¼ 1nm; E ¼ 1:06TPa;

i¼1 j¼1

w¼

N X M X

aðiþjð1ði1ÞðM1ÞÞþ2M$NÞ r ði1Þ q

ðj1Þ

n ¼ 0:3; q ¼ 0:1GPa; e0 a ¼ 1nm

i¼1 j¼1

f¼

N X M X

(52) ði1Þ ðj1Þ

(48)

aðiþjð1ði1ÞðM1ÞÞþ4M$NÞ r ði1Þ qðj1Þ

(49)

aðiþjð1ði1ÞðM1ÞÞþ3M$NÞ r

q

i¼1 j¼1

j¼

N X M X i¼1 j¼1

Substituting above polynomial functions (Eqs. (45)e(49)) into the governing equations, the PDE's can be transformed to the algebraic equations.

4. Boundary conditions In this paper, all types of boundary conditions are considered in category of the simply supported (S), clamped (C) and free edges (F) which can be defined as follows:

S : u ¼ v ¼ w ¼ j ¼ Mr ¼ 0 C: u¼v¼w¼j¼f¼0 F : Nr ¼ Nrq ¼ Qr ¼ Mr ¼ Mrq ¼ 0

r ¼ ri ; ro r ¼ ri ; ro r ¼ ri ; ro

(50)

5. Numerical analysis and discussion Regarding to this fact that there is not any research has already been done in field of effect of eccentric defects on bending analysis of graphene sheets, in this paper the obtained results of local theory for different distances of defect from center of the sheet and various boundary conditions are compared with Abaqus software. As per results of Table 1, achieved values of the research are in acceptable compliance with Abaqus software. Therefore, applied method for solving the constitutive equations of eccentric defect for different types of boundary conditions is acceptable. Specifications of mentioned plate are as following:

Table 1 Comparison between the results of present paper and Abaqus software. e (m)

CC

FC

w (mm)

0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2

Abacus

Paper

0.03862 0.05972 0.08747 0.1225 0.1646 0.5717 0.6077 0.6164 0.5987 0.5626

0.03936 0.05757 0.08611 0.1121 0.1507 0.5711 0.5895 0.6012 0.5773 0.5467

The number of grid points in r direction is considered N ¼ 7 and in q direction is M ¼ 9. In Table 2, the maximum deflections of the plate are compared in two cases of internal defect and effect of Winkler-Pasternak elastic medium. First, Pasternak medium is considered to be omitted, and the value of Winkler foundation varies. It is observed that by increasing the value of Winkler elastic medium, the maximum deflection is decreased. Also, by changing the location of defect and moving from center to edges, the maximum deflection will be increased. It is observed that the raising rate of deflection will be more depended on the situation of defect from center. However, the raising rate will be reduced by increasing the defect eccentricity and moving toward the edges. More over in next try, by eliminating the Winkler elastic medium and changes in value of Pasternak medium, the similar results have been achieved considering this conclusion that the effect of Pasternak elastic medium is more than Winkler, and the maximum deflection is decreased more. When the defect is moving toward edges, the effect of Winkler elastic medium is more, and it is resulted that Winkler elastic medium has more effect during the movement of the eccentric defect. In Table 3, the effect of FSDT (first order shear deformation theory) to CLPT (classical plate theory) analysis is shown for a circular graphene sheet which is including eccentric defect with following specifications. The boundary conditions are considered as clamped-clamped (CC).

ri ¼ 0:142nm; ro ¼ 5nm; E ¼ 1:06TPa; n ¼ 0:3; q ¼ 0:1GPa; e0 a ¼ 1nm; kw ¼ 1:13GPa=nm; kp ¼ 1:13Pa:m (53) The dimensionless thickness is presented as follow:

h* ¼

h 0:34nm

(54)

One carbon atom is deleted in the mentioned sheet, so the radius of internal defect is going to be 0.142 nm which is equal to the length of covalent bond of carbon atoms. (CeC). As it is observed, in CLPT and FSDT analyses, the maximum deflection is decreased by increasing the dimensionless thickness h* , and it is increased by raising eccentricity of defect. In governing conditions for this issue, it is shown that similar results of FSDT analysis are more than CLPT, furthermore, it is observed that the ratio of results for CLPT to FSDT are reduced by growing of thickness. On the other hand, by raising thickness, the differences of results of two CLPT and FSDT theories are increasing considerably. Therefore, for high value of thicknesses, the results of CLPT analysis do not have suitable accuracy, and FSDT theory must be applied. Taking a look on Table 3, it is considered that by taking the dimensionless thickness h* ¼ 1, which is equal to the thickness of one carbon atom, the results of mentioned theories are approximately in 10% deviation, thus, it is more recommended to use FSDT

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S. Dastjerdi et al. / Composites Part B 98 (2016) 78e87

Table 2 The effect of eccentric defect and elastic medium on the maximum deflection. W (nm)

kw ðGPa=nmÞ kp ¼ 0

0 1 2 3 4 5 10 0 1 2 3 4 5 10

kp ðPa:mÞ kw ¼ 0

e1 ¼ 0

e2 ¼ 0:5 nm

e3 ¼ 1 nm

e4 ¼ 1:5 nm

e2 =e1

e3 =e2

e4 =e3

0.01934 0.01532 0.01267 0.01079 0.00939 0.00831 0.00526 0.01934 0.01408 0.01107 0.00913 0.00777 0.00676 0.00410

0.02821 0.02120 0.01693 0.01405 0.01198 0.01044 0.00631 0.02821 0.02037 0.01597 0.01314 0.01116 0.00970 0.00584

0.04126 0.02788 0.02452 0.02259 0.02103 0.01965 0.01461 0.04126 0.02658 0.02151 0.01854 0.01641 0.01472 0.00957

0.08682 0.04257 0.03528 0.03555 0.03695 0.03815 0.03883 0.10287 0.05276 0.03574 0.02665 0.02155 0.01854 0.01370

0.45863 0.38381 0.33622 0.30213 0.27582 0.25631 0.19962 0.45863 0.44673 0.44263 0.43921 0.43629 0.43491 0.42439

0.46260 0.31509 0.44831 0.60782 0.75542 0.88218 1.31537 0.46260 0.30486 0.34690 0.41095 0.47043 0.51752 0.63869

1.10422 0.52690 0.43882 0.57370 0.75701 0.94147 1.65776 1.10422 0.98495 0.66155 0.43743 0.31322 0.25951 0.43155

Table 3 The comparison between the results of CLPT and FSDT analyses. e (nm)

w (nm) FSDT h*

CLPT h*

0 0.05 0.1 0.2 0.5

1

2

4

1

2

4

1

2

4

0.01811 0.01864 0.01915 0.02019 0.02495

0.00401 0.00416 0.00430 0.00463 0.00682

0.000555 0.000576 0.000597 0.000645 0.001709

0.01989 0.02046 0.02105 0.02229 0.02592

0.00537 0.00553 0.00571 0.00613 0.00799

0.00133 0.00135 0.00139 0.00149 0.00200

0.91050 0.91104 0.90974 0.90579 0.96258

0.74674 0.75226 0.75306 0.75530 0.85356

0.41721 0.42689 0.42921 0.43315 0.85450

theory which has higher accuracy. Finally, it is observed that by increasing eccentricity of defect, the results of two CLPT and FSDT theories have been approached to each other. Now a circular graphene sheet is considered with eccentric defect as below conditions:

ri ¼ 1nm; ro ¼ 5nm; h ¼ 0:34nm; e ¼ 1nm; E ¼ 1:06TPa;

n ¼ 0:3; q ¼ 0:1GPa; e0 a ¼ 1nm; kw ¼ 1:13GPa=nm; kp ¼ 1:13Pa:m (55) To study the effect of nodes quantity on each direction on accuracy of solution, Fig. 3 is presented. It is clear that acceptable

0.1

result is obtained by only 7 no. grid points in r and q direction. Fig. 3 is obtained for different boundary conditions and it is observed that there is not any difference in accuracy for various types of boundary conditions, and in purpose to gain an accurate solution, quantity of nodes is independent from boundary conditions. Also selecting more nodes will be caused to more calculations considerably, therefore by choosing optimum and suitable grid point quantity, calculations could be reduced, with the same accuracy. The effect of increasing nonlocal parameter on maximum deflection is shown in Fig. 4. It is observed that the same conclusion is obtained in last studies [19] which by increasing the nonlocal parameter, the maximum deflection is decreased. And whatever the plate is more constrained, the effect of nonlocal parameter is

0.1

CC

0.09

CC

0.09

FS

0.08

FC

0.08

FC

0.07

0.07

0.06

0.06

w (nm)

w (nm)

CLPT/FSDT h*

0.05

FS

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01 0

0 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Node number in each direction (N×N) Fig. 3. The effect of number of grid points on the results.

0

1

2

3

μ (nm^2) Fig. 4. Variation of the maximum deflection due to nonlocal parameter.

4

S. Dastjerdi et al. / Composites Part B 98 (2016) 78e87

0.14

0.1

0.8 0.7 0.6 0.5 0.4 0.3 0

0.5

1

1.5

2

e (nm) Fig. 6. Variation of Rm versus the eccentricity.

ignorable. The same results can be obtained for FC boundary conditions, but the effect of nonlocal analysis by increasing the eccentricity is more than the case with CC boundary conditions. In FC boundary conditions by heightening e0a ¼ 1 nm to e0a ¼ 2 nm it is shown that decreasing rate of Rm is tend to zero, also by increase of the eccentricity, the nonlocal and local analyses are going to approach to each other. The existence of eccentric defect in circular graphene sheet is studied in Fig. 7 for two different types of boundary conditions. Parameter Rv is presented below:

Rv ¼

wwith defect wwithout defect

(56)

The eccentric defect is set e ¼ 1 nm. According to Fig. 7, it is observed that in CC boundary conditions by increase of the nonlocal parameter, Rv is decreased, and variation of nonlocal parameter has reducing effect on existence of defect. However, the procedure for FC boundary conditions is vice versa. Comparing two curves in Fig. 7, it is shown that the existence of defect in CC boundary conditions has more effect about 70% variations in Rv along the increase of nonlocal parameter rather than FC boundary

CC

2.4 2.2

FC

2 1.8 1.6

0.08

Rv

w (nm)

0.9

CC (e0a = 1 nm) CC (e0a = 2 nm) FC (e0a = 2 nm) FC (e0a = 1 nm)

0.12

CC (e0a = 1 nm) CC (e0a = 2 nm) FC (e0a = 2 nm) FC (e0a = 1 nm)

1

Rm

decreased. The results are obtained for a specific location of eccentric defect, in addition, for other distances from the center of the plate could be obtained too. To survey the effects of different places of eccentricity defect, and nonlocal parameter on the results, Fig. 5 is drawn. Fig. 5 is presented for different values of nonlocal parameter and types of boundary conditions. According to mentioned figure, it is clear that in one specified nonlocal parameter and CC boundary conditions, by increasing the eccentricity of defect, the maximum deflection is grown. At first, the curve slope is decreased gently, but along the raising of eccentricity, the curve slope is increased considerably. In these conditions, it is seen that by raising the nonlocal parameter in CC boundary conditions, slope of the diagram is decreased or in other word, by increasing the small scale effects in CC boundary conditions, variations of the maximum deflection is reduced by heightening the eccentricity. The slope of deflection curve must be zero at the edges in CC boundary conditions. This issue causes considerable decline of maximum deflection. When the eccentricity (e) is increased, the distance between two edges will be increased too. Consequently, the effect of boundary conditions will be decreased on the deflection. Therefore, as the eccentricity is more than a particular value, the maximum deflection is increased with a steep slope. Obtained results for FC boundary conditions are against the results of CC boundary conditions. By increasing the eccentricity (e), the maximum deflection is decreased, but its slope is extremely gradual, also with increasing eccentricity (e), the slope of diagram tends to zero and no considerable change in results is obtained. It can be noted that unlike the CC boundary conditions which by increase of nonlocal parameter along the variations of e value, two diagrams are getting far from each other, the two diagrams are approaching to each other in FC boundary conditions, therefore, obtained results for nonlocal parameters of e0a ¼ 1 nm, e0a ¼ 2 nm, by increasing e are going to tend to each other. Consequently, it is concluded that in FC boundary conditions, as well as raise of eccentricity e, the small scale effects is become smaller. To study the effect of nonlocal analysis, Fig. 6 is drawn by using mentioned specifications of the plate in Fig. 6. Rm is presented as ratio of nonlocal deflection to local [19]. It is observed that in CC boundary conditions, the nonlocal analysis has more effect on decreasing of deflection. By increasing the eccentricity (e) in CC boundary conditions, it is observed that the effect of nonlocal analysis is increased. But the growing value is very low and is

85

1.4 1.2

0.06

1 0.04

0.8 0.6

0.02

0.4 0.2

0 0

0.5

1

1.5

e (nm) Fig. 5. The effect of eccentricity on the maximum deflection.

2

0

0.5

1

1.5

2

2.5

3

μ (nm^2) Fig. 7. Variation of Rv versus the nonlocal parameter.

3.5

4

S. Dastjerdi et al. / Composites Part B 98 (2016) 78e87

conditions by 20% variations. Therefore, as a result, it is investigated that in FC boundary conditions and by small size of defect in comparison with outside diameter of plate, defect could be ignored and then utilize from the results of whole non-defected plate, which has more convenience calculations. One of the other items which is supposed to be studied in analysis of defected graphene plate, is effects of size of the defect. In this regard, Fig. 8 is drawn for variations of maximum deflection and different nonlocal parameters based on changes in size of eccentric defect. It is shown that in all different boundary conditions and nonlocal parameters, by increasing size of defect, maximum deflection is reduced. It is observed that by growing the size of defect, deflections approach for both cases of different nonlocal parameters for CC and FC boundary conditions. Therefore, it is concluded that by raising the defect size, the nonlocal effects become less, on the other hand, by increasing the defect size, the effect of nonlocal analysis can be ignorable. This effect in FC boundary condition is more than CC. However, in small size of defect, due to small scale effects, nonlocal effects are more, which is shown in Fig. 8. Finally, the size of defect is considered to be constant, and the outer radius of plate varies. In Fig. 9, the effect of increasing in outer radius of plate is shown. By increasing ro, maximum deflection will be increased for all types of boundary conditions and nonlocal parameters. Also, it can be noted that the raising slope is reductive and slope of the curve is tended to zero, in addition, approach to each other dependent form type of boundary conditions. Fixing and approaching of curve's slope to zero in FC boundary condition is clearly more than CC boundary condition. Also, it is observed that in different types of boundary conditions by increasing outside diameter and nonlocal parameter the curves approach to each other, consequently, the effect of nonlocal analysis become smaller. It is resulted from the fact that the effects of small scales in small dimensions are more considerable. According to the diagram, it is observed that by growing the size of plate, the effect of nonlocal analysis becomes more unworthy of attention.

6. Conclusion In this paper, for the first time, the effect of eccentric defect has been investigated on bending analysis of circular graphene sheet based on nonlocal elasticity theory and considering the Mindlin assumptions. The governing equations has been derived with

0.05

CC (e0a = 2 nm) FC (e0a = 2 nm) FC (e0a = 1 nm)

w (nm)

0.03

0.02

0.01

0 1

1.5

2

2.5

3

ri (nm) Fig. 8. The effect of defect size on maximum deflection.

CC (e0a = 2 nm)

0.09

FC (e0a = 2 nm)

0.08

FC (e0a = 1 nm)

0.07 0.06 0.05 0.04 0.03 0.02 0.01 5

6

7

8

9

10

ro (nm) Fig. 9. Maximum deflection versus the variation of outer radius ro.

respect to nonlocal parameter, and solved by a new semi-analytical polynomial method which called SAPM. The main obtained results can be categorized as summery below:  Although the new presented method SAPM is extremely simple in formulation, but the obtained results are accurate. Consequently, in this case, bending analysis of circular graphene sheet including an eccentric defect, SAPM is so useful and recommended for further studies.  FSDT analysis gives more accurate results in comparison with CLPT. So, in this paper, FSD theory has been applied.  Both Winkler and Pasternak foundations affect the results. However, when the defect is moving toward the boundaries, the effect of Winkler elastic medium is more.  The eccentric defect has different effect on the results depending on boundary conditions. The defect and nonlocal parameter have considerable effect on CC boundary conditions but in FC is smaller.  The effect of presence of defect is highly depends on the types of boundary conditions. For example, defect affects the results in CC boundary conditions extremely more than FC.  By increasing the size of defect, the effect of nonlocal parameter becomes smaller. References

CC (e0a = 1 nm)

0.04

CC (e0a = 1 nm)

0.1

w (nm)

86

3.5

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