Variational Principles for Nonlocal Continuum Model of Orthotropic Graphene Sheets Embedded in An Elastic Medium

Acta Mathematica Scientia 2012,32B(1):325–338 http://actams.wipm.ac.cn

VARIATIONAL PRINCIPLES FOR NONLOCAL CONTINUUM MODEL OF ORTHOTROPIC GRAPHENE SHEETS EMBEDDED IN AN ELASTIC MEDIUM∗ Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday Sarp Adali School of Mechanical Engineering, University of KwaZulu-Natal, Durban, South Africa E-mail: [email protected]

Abstract Equations governing the vibrations and buckling of multilayered orthotropic graphene sheets can be expressed as a system of n partial differential equations where n refers to the number of sheets. This description is based on the continuum model of the graphene sheets which can also take the small scale effects into account by employing a nonlocal theory. In the present article a variational principle is derived for the nonlocal elastic theory of rectangular graphene sheets embedded in an elastic medium and undergoing transverse vibrations. Moreover the graphene sheets are subject to biaxial compression. Rayleigh quotients are obtained for the frequencies of freely vibrating graphene sheets and for the buckling load. The influence of small scale effects on the frequencies and the buckling load can be observed qualiatively from the expressions of the Rayleigh quotients. Elastic medium is modeled as a combination of Winkler and Pasternak foundations acting on the top and bottom layers of the mutilayered nano-structure. Natural boundary conditions of the problem are derived using the variational principle formulated in the study. It is observed that free boundaries lead to coupled boundary conditions due to nonlocal theory used in the continuum formulation while the local (classical) elasticity theory leads to uncoupled boundary conditions. The mathematical methods used in the study involve calculus of variations and the semi-inverse method for deriving the variational integrals. Key words variational formulation; multilayered graphene sheets; nonlocal theory; Rayleigh quotient; vibration; buckling; semi-inverse method 2000 MR Subject Classification ∗ Received

49S05; 74H45; 74K99

November 17, 2011. The research reported in this paper was supported by research grants from the University of KwaZulu-Natal (UKZN) and from National Research Foundation (NRF) of South Africa. The author gratefully acknowledge the support provided by UKZN and NRF.

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Introduction

Multilayered graphene sheets are a relatively new class of materials and offer a variety of physical properties with many applications in several fields [1]. Graphene sheets refer to nanostructures consisting of carbon atoms which form flat sheets with a regular hexagonal arrangement. The graphene sheet is a two-dimensional lattice structure and has many unique properties which cannot be matched by conventional materials. With a Young’s modulus around 1 TPa and higher, graphene sheet is one of the stiffest and strongest materials available [2–4]. It is already in use in many advanced technologies including nanocomposites [5, 6], electromechanical resonators [7], light emitting devices [8], electrochemical capacitors [9], sensing devices [10], lithium-ion batteries [11], desalination of sea water [12], electrooxidation [13], and as sensors for the detection of cancer cells [14, 15] to give a few examples. Review articles [16, 17] discuss further applications of graphene in many other fields. In studying the mechanical behavior of graphene sheets, available techniques include molecular dynamics simulations, experimental investigations and continuum mechanics based approaches. However the computations taking atomic lattice dynamics into account require extensive computational capacity in terms of computer memory and processing speed leading to unrealistic computational times even for relatively small nanostructures. Similarly experimental approaches are relatively difficult to implement due to the atomic scale of the phenomenon and require specialized and expensive equipment. These difficulties encountered in molecular and experimental studies led to the development of continuum mechanics models for the analysis of nanostructures [18–22]. However, the shortcomings of classical continuum mechanics in modeling a nano-scale phenomenon became apparent in experimental studies and in molecular dynamic simulations indicating that as the length scales become smaller, the effect of interatomic and intermolecular forces cannot be neglected [23–25]. Thus at nano scales the size effects become prominent which need to be taken into account in a continuum model. However, classical continuum mechanics is a scale free theory and as such its accuracy is of limited when applied to nanostructures. In particular, models of graphene sheets based on classical elastic constitutive relations lead to inaccurate results due to the nano-scale thickness of the sheets. This shortcoming of the classical elasticity led to the employment of nonlocal models in the study of nanostructures [22]. Nonlocal constitutive relations relate the stress at a point to the strains at all points of the domain as opposed to local elastic theory which relates the stress to the strain at the same point. Nonlocal theory was developed in early seventies and earlier works include Edelen and Laws [26], Eringen [27, 28] and more recently the book by Eringen [29]. Nonlocal continuum models of single-layered graphene sheets were used to study their vibration behaviour by Murmu and Pradhan [30], Shen et al. [31] and Narendar and Gopalakrishnan [32]. He et al. [33], Behfar and Naghdabadi [34], Liew et al. [35], and Jomehzadeh and Saidi [36] studied the vibrations of multilayered graphene sheets employing the classical elasticity theory. Nonlocal continuum models were used in [37] and [38] to study the buckling of multilayered graphene sheets and in [39–44] to study the vibrations of multilayered graphene sheets. Most of these studies considered the simply supported boundary conditions due to the complications which arise in the case of other boundary conditions. In this regard the derivation of variational

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principles and the applicable natural boundary conditions is helpful in formulating approximate methods of solution. The present study aims at deriving the variational principles and the natural boundary conditions for multilayered orthotropic graphene sheets of rectangular shape which are subject to biaxial compressive forces and embedded in an elastic medium. Constitutive modeling of the graphene sheets is based on a nonlocal continuum mechanics model and the elastic medium is modeled as a combination of Winkler and Pasternak foundations. The interaction between the layers is taken into account by introducing the van der Waals forces which cannot be neglected at the nano scale. The variational principle for buckling is obtained as a special case. As such the present study extends the results of Adali [45] which neglected the effects of in-plane loads and the elastic medium. Variational principles were derived previously for a number of cases involving nano-structures. These include multi-walled nanotubes under buckling loads [46], and undergoing linear and nonlinear vibrations [47, 48] which are based on nonlocal theory of EulerBernoulli beams. The corresponding results based on nonlocal Timoshenko theory are given in [49] for nanotubes undergoing vibrations and in [50] for nanotubes under buckling loads. In the present study Rayleigh quotients for the frequencies of freely vibrating graphene sheets and for the buckling load of graphene sheets subject to biaxial compressive loads are given. The variational formulation of the problem is obtained by the semi-inverse method developed by He [51, 52] which was applied to several problems of mathematical physics. Examples of the application of the semi-inverse method to derive variational principles can be found in [53–57].

2

Nonlocal Continuum Model

Multilayered graphene sheets consist of n-layers of graphene which interact with each other via van der Waals forces which are modeled as linear elastic springs. Fig. 1a shows multilayered sheets with the interaction between the adjacent layers indicated by arrows and the elastic mediums supporting the top and bottom layers. The thickness of each layer is h. Multilayered sheets are embedded in an elastic medium which acts on the top and bottom layers and is modeled as a Winkler foundation with a modulus of kw and a Pasternak foundation with a modulus of kp . The multilayered sheets are of rectangular shape of dimensions a and b with top and bottom layers labeled as i = 1 and i = n (Fig. 1a). Biaxial compressive forces N1 and N2 act on the rectangular sheets in x and y directions, respectively, as shown in Fig. 1b. The graphene, in general, is an orthotropic material with Young’s moduli E1 and E2 in the x and y directions, the shear modulus G12 and Poisson’s ratios v12 and v21 . The bending stiffnesses D11 , D12 , D22 and D66 are given by [40] D11 =

E1 h3 , D

D12 =

v12 E2 h3 , D

D22 =

E2 h3 , D

D66 =

G12 h3 , 12

(1)

where D = 12(1 − v12 v21 ). The nonlocal continuum model of the vibrations of multilayered graphene shells subject to biaxial compressive forces can be described by a system of partial differential equations (see [39, 41, 43]). Let wi (x, y, t) denote the transverse deflection of the ith layer and η the small scale parameter of the nonlocal elastic theory as defined in [27]. The deflection function is defined as a smooth function and sufficiently differentiable. The nonlocal differential equations governing

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the transverse vibrations in the time interval t1 ≤ t ≤ t2 can be expressed as [40] D1 (w1 , w2 ) = L(w1 ) + (1 − η 2 ∇2 )(H(w1 ) − c12 Δw1 + f (x, y, t)) = 0,

(2)

Di (wi−1 , wi , wi+1 ) = L(wi ) + (1 − η 2 ∇2 )(M (wi ) + N (wi ) + c(i−1)i Δwi−1 − c(i+1)i Δwi ) = 0,

for i = 1, 2, · · · , n − 1,

Dn (wn−1 , wn ) = L(wn ) + (1 − η 2 ∇2 )(H(wn ) + c(n−1)n Δwn−1 ) = 0,

(3) (4)

where f (x, y, t) is a transverse load acting on the topmost layer (i = 1) and the symbol Δwi is the difference operator defined as Δwi ≡ wi+1 − wi .

Fig. 1

Multilayered graphene sheets a) Side view, b) Top view

(5)

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In eqs. (2)–(4) L(wi ), H(wi ), K(wi ), M (wi ) and N (wi ) are differential operators defined as

2

L(wi ) = D11 wixxxx + 2(D12 + 2D66 )wixxyy + D22 wiyyyy ,

(6)

H(wi ) = K(wi ) + M (wi ) + N (wi ),

(7)

K(wi ) = kw wi − kp ∇2 wi ,

(8)

M (wi ) = (m0 − m2 ∇2 )witt ,

(9)

N (wi ) = N1 wixx + N2 wiyy ,

(10)

2

∂ ∂ with ∇2 = ∂x 2 + ∂y 2 . In eqs. (6)–(10), the subscripts x, y and t denote differentiations with respect to that variable and in eq. (9), m0 = ρh and m2 = ρh3 /12 with ρ denoting the mass density. The coefficient c(i−1)i is the interaction coefficient of van der Waals forces between the (i − 1)th and ith layers. The constant η = e0 α is a material parameter defining the small scale effect in nonlocal continuum mechanics where e0 is an experimentally determined constant and has to be determined for each material independently [58]. α is an internal characteristic length such as lattice parameter, size of grain, granular distance, etc. [32].

3

Variational Formulation

Following the semi-inverse method developed by He [51, 52], a trial variational functional V (w1 , w2 , · · · , wn ) is defined as V (w1 , w2 , · · · , wn ) = V1 (w1 , w2 ) +

n−1 

Vj (wj−1 , wj , wj+1 ) + Vn (wn−1 , wn ),

(11)

j=2

where V1 (w1 , w2 ) =

5 

Uj (w1 ) − Ta (w1 ) − Tb (w1 )

j=1



t2



b



a

+ t1

Vi (wi−1 , wi , wi+1 ) =

3 

0

0



t2



b



+ t1 5  j=1

(12)

Uj (wi ) − Ta (wi ) − Tb (wi )

j=1

Vn (wn−1 , wn ) =

[(f − η 2 ∇2 f )w1 + F1 (w1 , w2 )]dxdydt,

0

0

a

Fi (wi−1 , wi , wi+1 )dxdydt, for i = 2, 3, · · · , n − 1, (13) 

Uj (wn ) − Ta (wn ) − Tb (wn ) +

t2 t1

 0

b

 0

a

Fn (wn−1 , wn )dxdydt,

with the functionals Uj (wi ), Ta (wi ), and Tb (wi ) defined as    1 t2 b a 2 2 2 U1 (wi ) = (D11 wixx + 2D12 wixx wiyy + D22 wiyy + 4D66 wixy )dxdydt, 2 t1 0 0

(14)

(15)

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U2 (wi ) = −

1 2

U3 (wi ) = −

η2 2

1 2

U5 (wi ) =

η2 2



b



b

b

0



t1

a

b



0

2 2 )dxdydt, (N1 wix + N2 wiy a

0



0 t2



0



t1

a 0



t2

t1 t2



0

t1



U4 (wi ) =



t2

Vol.32 Ser.B

(16)

2 2 (N1 (wixx + wixx wiyy ) + N2 (wiyy + wixx wiyy ))dxdydt,

(kw wi2 + kp (∇wi )2 )dxdydt for i = 1, n, a

0



1 2

Tb (wi ) =

η2 2

2

t2



t1





0 t2

t1

2 wix

b



a

0 b

0

 0

2 2 2 (m0 wit + m2 (wixt + wiyt ))dxdydt,

a

(18)

2 2 (kw (∇wi )2 + kp (wixx + 2wixx wiyy + wiyy ))dxdydt

for i = 1, n, Ta (wi ) =

(17)

(19) (20)

2 2 2 [m0 witt (wixx + wiyy ) + m2 (wixxt + 2wixyt + wiyyt )]dxdydt, (21)

2 + wiy .

where (∇wi ) = In equations (12)–(14), Fi (wi−1 , wi , wi+1 ) are used to compute the variational forms of the coupling terms appearing in eqs. (2)–(4). Variational derivatives of Fi (wi−1 , wi , wi+1 ) are given by     2 2    ∂ ∂ δFj ∂Fj ∂Fj ∂Fj − , (22) = − δw1 ∂w1 ∂x ∂w1x ∂y ∂w1y j=1 j=1     i+1 i+1    δFj ∂Fj ∂Fj ∂Fj ∂ ∂ − , = − δwi ∂wi ∂x ∂wix ∂y ∂wiy j=i−1 j=i−1

(23)

     n n   δFj ∂Fj ∂Fj ∂Fj ∂ ∂ − , = − δwn ∂wn ∂x ∂wnx ∂y ∂wny j=n−1 j=n−1

(24)

where i = 2, 3, · · · , n − 1. The unknown functions Fi (wi−1 , wi , wi+1 ) are to be determined such that the Euler-Lagrange equations of the variational functional (11) should yield the coupling terms in the differential equations (2)–(4). Comparing equations (22)–(24) with equations (2)– (4), we need 2  δFj = −c12 (1 − η 2 ∇2 )Δw1 , δw 1 j=1

(25)

i+1  δFj = (1 − η 2 ∇2 )(c(i−1)i Δwi−1 − ci(i+1) Δwi ) for i = 2, 3, · · · , n − 1, δw i j=i−1

(26)

n  δFj = c(n−1)n (1 − η 2 ∇2 )Δwn−1 , δw n j=n−1

(27)

in order to obtain the coupling terms in eqs. (2)–(4) from the Euler-Lagrange equations of the variational functional (11). From equation (25), it follows that c12 [(Δw1 )2 + η 2 (∇(Δw1 ))2 ]. (28) F1 (w1 , w2 ) = 4

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Similarly from equations (26) and (27), we obtain i+1

Fi (wi−1 , wi , wi+1 ) = Fn (wn−1 , wn ) =

1 c(j−1)j [(Δwj−1 )2 + η 2 (∇(Δwj−1 ))2 ] for i = 2, 3, · · · , n − 1, (29) 4 j=i

c(n−1)n [(Δwn−1 )2 + η 2 (∇(Δwn−1 ))2 ]. 4

(30)

With Fi given by equations (28)–(30), we observe that the Euler-Lagrange equations of the variational functional (11) correspond to the governing eqs. (2)–(4). As such the functional V (w1 , w2 , · · · , wn ) given by eq. (11) constitutes a classical variational principle for the vibrating rectangular graphene sheets subject to compressive loads and embedded in an elastic medium.

4

Rayleigh Quotients for Vibration Frequencies and Buckling Load

In the present section Rayleigh quotients are derived for multilayered graphene sheets embedded in an elastic medium. Two cases under consideration involve sheets undergoing free vibrations subject to compressive loads and sheets subject to biaxial buckling loads. Rayleigh quotients are derived next for these two cases. 4.1 Freely Vibrating Graphene Sheets under Compressive Loads Let the harmonic motion of the ith layer be expressed as wi (x, y, t) = Wi (x, y)e

√

−1ωt

,

(31)

where ω is the vibration frequency. The equations governing the free vibrations are obtained by substituting equation (31) into equations (2)–(4) with f (x, y, t) = 0 and replacing the deflection wi (x, y, t) by Wi (x, y). The operator M (Wi ) now becomes M (Wi ) = ω 2 (−m0 + m2 ∇2 )Wi .

(32)

The other operators are time-independent and remain the same. The variational principle for free vibrations is the same as the one given by equations (11)–(14) with the deflection wi (x, y, t) replaced by Wi (x, y), and the triple integrals replaced by the double integrals with respect to ba x and y, i.e., 0 0 Fi (Wi−1 , Wi , Wi+1 )dxdy, and Uj (wi (x, y, t)) replaced by UjF V (Wi (x, y)). The functionals Ta (wi ) and Tb (wi ) now become TaF V (Wi ) and TbF V (Wi ) given by TaF V (Wi ) = ω 2 τa (Wi ),

(33)

TbF V (Wi ) = ω 2 η 2 τb (Wi ),

(34)

where 1 τa (Wi ) = 2 1 τb (Wi ) = 2



b



0

 0

b

a

0

 0

a

(m0 Wi2 + m2 (∇Wi )2 )dxdy,

(35)

2 2 [m0 (∇Wi )2 + m2 (Wixx + 2Wixx Wiyy + Wiyy )]dxdy.

(36)

The functions Fi (Wi−1 , Wi , Wi+1 ) are of the same form as given by equations (28)–(30) since the functions Fi (Wi−1 , Wi , Wi+1 ) are independent of time. Next the Rayleigh quotient is obtained

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for the vibration frequency ω from equations (11)–(14) and (33)–(34) as  n 5 n   b a UjF V (Wi ) + 0 0 Fi dxdy ω 2 = min

i=1

j=1

Wi

n

i=1

i=1

,

(τa (Wi ) + η 2 τb (Wi ))

(37)

where Fi (i = 1, 2, · · · , n) are given by equations (28)–(30) with wi (x, y, t) replaced by Wi (x, y). 4.2 Graphene Sheets under Buckling Loads In this case the multilayered graphene sheets are subject to the biaxial buckling loads N1 and N2 = λN1 only and the deflection is a function of x and y only, that is, wi = Wi (x, y). Thus the time derivatives of wi = Wi (x, y) are equal to zero and consequently the operator M (wi ) = 0. The triple integrals are replaced by the double integrals with respect to x and y, and Uj (wi (x, y, t)) (j = 1, 4, 5) are replaced by UjB (Wi (x, y)). The functionals U2 (wi ) and U3 (wi ) now become

where 1 ψ1 (Wi ) = 2 ψ2 (Wi ) =

1 2



b



0

 0

a 0

b



a 0

U2 (Wi ) = −N1 ψ1 (Wi ),

(38)

U3 (Wi ) = −N1 η 2 ψ2 (Wi ),

(39)

2 2 (Wix + λWiy )dxdy,

(40)

2 2 (Wixx + Wixx Wiyy + λ(Wiyy + Wixx Wiyy ))dxdy.

(41)

The Rayleigh quotient for the buckling load N1 can be expressed as n

N1 = min

i=1

(U1B (Wi ) + U4B (Wi ) + U5B (Wi )) + n

Wi

i=1

5

n   b a i=1

0

0

(ψ1 (Wi ) + η 2 ψ2 (Wi ))

Fi dxdy .

(42)

Boundary Conditions

Next we take the variations of the functional (11) using equations (12)–(14) with respect to wi in order to derive the natural and geometric boundary conditions. The first variations of V (w1 , w2 , · · · , wn ) with respect to wi , denoted by δwi V , are given by δw1 V (w1 , w2 , · · · , wn ) = δw1 V1 (w1 , w2 ) + δw1 V2 (w1 , w2 , w3 ),

(43)

δwi V (w1 , w2 , · · · , wn ) = δwi Vi−1 (wi−2 , wi−1 , wi ) + δwi Vi (wi−1 , wi , wi+1 ) +δwi Vi+1 (wi , wi+1 , wi+2 ) for i = 2, 3, · · · , n − 1, δwn V (w1 , w2 , · · · , wn ) = δwn Vn−1 (wn−2 , wn−1 , wn ) + δwn Vn (wn−1 , wn ). The first variation of Vi (wi−1 , wi , wi+1 ) with respect to wi is given by δwi Vi (wi−1 , wi , wi+1 =

5  j=1

δwi Uj (wi ) − δwi Ta (wi ) − δwi Tb (wi )

(44) (45)

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t2



b



a

+ t1

0

0

δwi Fi dxdydt for i = 1, 2, · · · , n − 1, n.

333

(46)

Let δwi denote the variation of wi satisfying the boundary conditions δwix (x, 0, t) = 0, δwix (x, b, t) = 0, δwiy (0, y, t) = 0, δwiy (a, y, t) = 0,

(47)

∂wi i where the following notation was used δ( ∂w ∂x ) = δwix , δ( ∂y ) = δwiy . Moreover the deflections wi (x, y, t) and their space derivatives vanish at the end points t = t1 and t = t2 , i.e., δwi (x, y, t1 ) = 0, δwi (x, y, t2 ) = 0, δwix (x, y, t1 ) = 0, δwix (x, y, t2 ) = 0, etc. t ba The first variations δwi Uj (wi ), δwi Ta (wi ), δwi Tb (wi ) and t12 0 0 δwi Fi dxdydt are obtained by integration by parts. These computations yield the following results:  t2  b  a δwi U1 (wi ) = (D11 wixxxx + 2(D12 + 2D66 )wixxyy + D22 wiyyyy )δwi dxdydt t1

0

0

+B1 (wi , δwi ),  t2  b  a δwi U2 (wi ) = (N1 wixx + N2 wiyy )δwi dxdydt + B2 (wi , δwi ), t1

0

δwi U3 (wi ) = −η 2  δwi U4 (wi ) =

 δwi Ta (wi ) =

t2



1 2 +





t2

t1

η2 2



0

t1

=

b



0 t2

t1

t2





b

t2

t1



0

 0

b

(N1 ∇2 wixx + N2 ∇2 wiyy )δwi dxdydt + B3 (wi , δwi ),

(kw wi − kp ∇2 wi )δwi dxdydt + B4 (wi , δwi ),



a

a

0

a

0



0



b

0



t2

a 0

t1

t1

δwi Tb (wi ) = η

b

b

0

0



2



t1

t1

(49)

0

t2



t2

δwi U5 (wi ) = η 2





(48)

[−kw ∇2 wi + kp ∇2 (∇2 wi )]δwi dxdydt + B5 (wi , δwi ),

(−m0 witt + m2 (∇2 witt ))δwi dxdydt + B6 (wi , δwi ),

 0

a

∇2 (m0 witt − m2 ∇2 witt )δwi dxdydt + B7 (wi , δwi ),

(50)

(51)

(52)

(53)

(54)

a

δwi (Fi (wi−1 , wi , wi+1 ))dxdydt 0 b a 

0 b

0

(c(i−1)i Δwi−1 − ci(i+1) Δwi )δwi dxdydt



a 0

(−c(i−1)i ∇2 (Δwi−1 ) + ci(i+1) ∇2 (Δwi ))δwi dxdydt + B8 (wi , δwi ), (55)

where Bj (wi , δwi ) are boundary terms which are given in the Appendix. Using the fundamental lemma of calculus of variations, the boundary conditions at x = 0, a and y = 0, b are obtained from equations (A1)–(A10), given in the appendix. The boundary conditions for i = 1, 2, · · · , n at x = 0, a are given by D11 wixx + D12 wiyy + η 2 (−N1 wixx − 0.5(N1 + N2 )wiyy +kp ∇2 wi − m0 witt + m2 wixxtt ) = 0,

(56)

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or wix = 0,

(57)

−D11 w1xxx − D12 w1xyy − 2D66 w1xyy − N1 wix + η 2 N1 wixxx +0.5η 2 (N1 + N2 )wixyy + kp wix + η 2 kw wix − η 2 kp (∇2 wi )x + m2 wixtt +η 2 m0 wixtt − η 2 m2 (w1xxx + w1xxy )tt + 0.5η 2 Cix = 0,

(58)

or wi = 0,

(59)

where C1z = c12 (Δw1z ),

(60)

Ciz = c(i−1)i Δw(i−1)z + ci(i+1) Δwiz for i = 2, · · · , n − 1,

(61)

Cnz = c(n−1)n Δw(n−1)z ,

(62)

and the boundary conditions for i = 1, 2, · · · , n at y = 0, b are given by D12 wixx + D22 wiyy + η 2 (−N2 wiyy − 0.5(N1 + N2 )wixx +kp ∇2 wi − m0 witt + m2 wiyytt ) = 0,

(63)

or wiy = 0,

(64)

−D22 w1yyy − D12 w1xxy − 2D66 w1xxy − N2 wiy + η 2 N2 wiyyy +0.5η 2 (N1 + N2 )wixxy + kp wiy + η 2 kw wiy − η 2 kp (∇2 wi )y + m2 wiytt +η 2 m0 wiytt − η 2 m2 (wiyyy + wixyy )tt + 0.5η 2 Ciy = 0,

(65)

or wi = 0.

(66)

It is observed that when η = 0, the natural boundary conditions are coupled through equations (59)–(61), that is, the nonlocal formulation of the problem leads to natural boundary conditions which contain derivatives of wi−1 and wi+1 in the expression for wi , e.g., equations (58) and (65).

6

Conclusions

The variational formulations for vibrating multilayered graphene sheets are derived based on a continuum model with nonlocal constitutive equations. The graphene sheets are embedded in an elastic medium modeled as a combination of Winkler and Pasternak foundations, of rectangular shape and subject to biaxial compressive forces. Rayleigh quotients are given for the frequencies of freely vibrating graphene sheets as well as for the buckling load. A semi-inverse approach was employed in the derivation of the variational principles and the Rayleigh quotient

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for free vibrations was obtained. The formulation was used to obtain the natural boundary conditions. The variational principles presented here form the basis of several approximate and numerical methods of solution and facilitates the implementation of complicated boundary conditions. The nonlocal theory used in the formulation allows the inclusion of small scale effects in the calculations.

Appendix The boundary terms appearing in eqs. (48)–(55) are given below. B1 (wi , δwi ) =

2 

B1j (wi , δwi ),

(A1)

j=1

where  B11 (wi , δwi ) =

t2



t1

b



0

(D11 wixx + D12 wiyy )δwix

−(D11 wixxx + D12 wixyy + 2D66 wixyy )δwi  B12 (wi , δwi ) =

t2



t1

a

0



t2



t1

b

0

x=0

dydt,

(D12 wixx + D22 wiyy )δwiy

−(D22 wiyyy + D12 wixxy + 2D66 wixxy )δwi B2 (wi , δwi ) = −

x=a

 x=a  N1 wix δwi  dydt − x=0

B3 (wi , δwi ) =

2 

t2

t1

 0

a

y=b y=0

dydt,

y=b  N2 wiy δwi  dxdt, y=0

B3j (wi , δwi ),

(A2)

(A3)

j=1

where B31 (wi , δwi ) = −

η2 2



t2



t1

b

0

(2N1 wixx + (N1 + N2 )wiyy )δwix

−(2N1 wixxx + (N1 + N2 )wixyy )δwi B32 (wi , δwi ) = −

η2 2



t2



t1

0

a

B4 (wi , δwi ) =

t2



t1

B5 (wi , δwi ) = η 2



0 t2

t1

+η 2

b



 x=a  kp wix δwi  dydt + x=0



t2

t1

b



0

 0

x=0

dydt,

(2N2 wiyy + (N1 + N2 )wixx )δwiy

−(2N2 wiyyy + (N1 + N2 )wixxy )δwi 

x=a

t2 t1

 0

a

y=b y=0

dydt,

y=b  kp wiy δwi  dxdt,

kp ∇2 wi δwix + (kw wix − kp (∇2 wi )x )δwi

a

(A4)

y=0

x=a x=0

dydt



y=b kp ∇2 wi δwiy + (kw wiy − kp (∇2 wi )y )δwi dxdt, x=0

(A5)

336

ACTA MATHEMATICA SCIENTIA

 B6 (wi , δwi ) = −m2



t2

 x=a  wixtt δwi  dydt − m2

b

x=0

0

t1

B7 (wi , δwi ) =

2 

t2

Vol.32 Ser.B



a

0

t1

y=b  wiytt δwi  dydt, y=0

B7j (wi , δwi ),

(A6)

(A7)

j=1

where B71 (wi , δwi ) = η 2





t2

0

t1



+η 2

B72 (wi , δwi ) = η 2 m2

x=a  (m0 witt − m2 wixxtt )δwix − m0 wixtt δwi ) dydt x=0



t2

a

0 t2 

t1

b

0



t1



2

η 2 2 

x=0

a

t2

y=b  ((wiyyytt + wixyytt )δwi ) dydt, y=0

0



t2

x=a  ((wixxxtt + wixxytt )δwi ) dydt



t2

t1



η2 2

y=b  (m0 witt − m2 wiyytt )δwiy − m0 wiytt δwi ) dydt, y=0

t1



+η 2 m2 B8 (w1 , δw1 ) =

b

x=a  c12 (Δw1x )δw1  dydt

b

x=0

0



a

+ η B8 (wi , δwi ) = 2

t1 2

+

η 2



t2



t2

b

0

x=a  (c(i−1)i Δw(i−1)x + ci(i+1) Δwix )δwi  dydt x=0



t1

(A8)

y=0

0

t1

y=b  c12 (Δw1y )δw1  dydt,

a

y=b  (c(i−1)i Δw(i−1)y + ci(i+1) Δwiy )δwi  dydt y=0

0

for i = 2, · · · , n − 1, B8 (wn , δwn ) =

η2 2



t1 2

+

t2

η 2



 t2

t1

b

0

(A9)

x=a  c(n−1)n Δw(n−1)x Δwn  dydt

 0

x=0

a

y=b  c(n−1)n Δw(n−1)y Δwn  dxdt. y=0

(A10)

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