Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium

Physica E 56 (2014) 400–409

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Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium Mohammed Sobhy n Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt

H I G H L I G H T S







The sinusoidal shear deformation plate theory is used to analyze the bending and vibration of the nanoplates. The nanoplates are assumed to be embedded in two-parameter elastic foundations and subjected to mechanical and thermal loads. The governing equations are solved analytically for various boundary conditions. A detailed parametric study is carried out to highlight the influences of the different parameters on the bending and the frequency of the nanoplates.

art ic l e i nf o

a b s t r a c t

Article history: Received 22 August 2013 Received in revised form 30 September 2013 Accepted 16 October 2013 Available online 26 October 2013

In the present paper, the sinusoidal shear deformation plate theory (SDPT) is reformulated using the nonlocal differential constitutive relations of Eringen to analyze the bending and vibration of the nanoplates, such as single-layered graphene sheets, resting on two-parameter elastic foundations. The present SDPT is compared with other plate theories. The nanoplates are assumed to be subjected to mechanical and thermal loads. The equations of motion of the nonlocal model are derived including the plate foundation interaction and thermal effects. The governing equations are solved analytically for various boundary conditions. Nonlocal theory is employed to bring out the effect of the nonlocal parameter on the bending and natural frequencies of the nanoplates. The influences of nonlocal parameter, side-to-thickness ratio and elastic foundation moduli on the displacements and vibration frequencies are investigated. & 2013 Elsevier B.V. All rights reserved.

Keywords: Graphene Nonlocal plate theory Elastic foundations Boundary conditions Bending Vibration

1. Introduction There exists a large class of problems in classical physics and continuum mechanics (classical field theories) that fall outside their domain of applications. Fracture of solids, stress fields at the dislocation core and at the tips of cracks, singularities present at the point of application of concentrated loads (forces, couples, heat, etc.), sharp corners and discontinuities in bodies, and the failure in the prediction of short wavelength behavior of elastic waves are some major anomalies that defy classical treatment [1]. On the other hand, the classical continuum theory cannot predict the size effect. At nanometer scales, size effects often become prominent [2]. To overcome this weakness, the nonlocal continuum theory developed by Eringen [3,4] has been used in the continuum models for accurate prediction of nanostructures mechanical behaviors. Unlike the local theories which assume

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that the stress at a point is a function of strain at that point, the nonlocal elasticity theory assumes that the stress at a point is a function of strains at all points in the continuum. Eringen's nonlocal elasticity [3,4] allows one to account for the small scale effect that becomes significant when dealing with micro- and nanostructures. The small scale parameter (Eringen's nonlocal elasticity parameter) e0 of carbon nanotubes is calibrated by using molecular dynamics [5] and lattice dynamics [6]. It has been found that the calibrated values of the parameter e0 depend on the material, geometry and boundary conditions of the nanotubes. Based on the nonlocal elasticity of Eringen, a number of paper have been published attempting to develop nonlocal structures and apply them to analyze the bending, vibration and buckling. On the basis of the nonlocal elasticity theory, the buckling and vibration of nanoplates were studied by Aksencer and Aydogdu [7] using Navier and Levy type solutions. Ansari et al. [8] studied the free vibration of single-layered graphene sheets based on the first-order shear deformation theory using the generalized differential quadrature method. Vodenitcharova and Zhang [9] introduced the pure bending and bending-induced local

M. Sobhy / Physica E 56 (2014) 400–409

buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube. Murmu and Pradhan [10] employed nonlocal continuum mechanics to investigate small-scale effects on the free in-plane vibration of nanoplates. Roque et al. [11] studied the bending, buckling and free vibration of Timoshenko nanobeams based on a meshless method. Li et al. [12] investigated the natural frequency, steady-state resonance and stability for the transverse vibrations of a nanobeam subjected to a variable initial axial force, including axial tension and axial compression, based on nonlocal elasticity theory. In Pradhan and Phadikar [13], the equations of motion of the nonlocal theories were derived using the classical plate theory and first-order shear deformation theory and then solved using Navier's approach to obtain the vibration of the graphene sheets. Also, Pradhan and Phadikar [14] presented the vibration of embedded multi-layered graphene sheets considering the small scale effects and employing the nonlocal classical plate theory. Ansari et al. [15] employed the finite element method to study the vibration of embedded multi-layered graphene sheets with different boundary conditions embedded in an elastic medium. A non-classical solution has been proposed to analyze bending and buckling responses of nanobeams including surface stress effects by Ansari and Sahmani [16]. Recently, Zenkour and Sobhy [17] studied the thermal buckling of single-layered graphene sheets lying on an elastic medium. Other investigations on the nanomaterials are presented in the open literature [18,19]. Plates resting on an elastic foundation is a classical topic in civil, mechanical and aeronautical engineering [20–25]. On the other hand, nanostructural elements such as nanobeams, nanomembranes and nanoplates are commonly used as components in nanoelectromechanical devices. The importance of these structures grows due to modern technology involving graphene sheets [26–28] or carbon nanotube [29–31] embedded in an elastic medium. A nonlocal higher-order shear deformation theories are used in some studies to illustrate the behavior of nanostructures. Thai [32] investigated the bending, buckling and vibration of simply supported nanobeams using the third-order shear deformation beam theory and the nonlocal differential constitutive relations of Eringen. Aghababaei and Reddy [33] investigated the vibration of simply supported isotropic nanoplate based on nonlocal elasticity theory using the third-order shear deformation theory. Aydogdu [34] proposed a generalized nonlocal beam theory to study bending, buckling and free vibration of nanobeams. In the present work the sinusoidal shear deformation plate theory (SDPT) in conjunction with nonlocal continuum mechanics has been employed to study the nonlocal effects on the bending and vibration of single-layered graphene sheet on elastic foundations in thermal environment. Five different boundary conditions (namely simply supported, simply supported-clamped, clamped– clamped, clamped-free and free–free) are used to study the bending and vibration of nanoplates. The results obtained by using SDPT are compared with those obtained by using the higher- and first-order shear deformation plate theories (HDPT and FDPT) and the classical plate one (CPT). The elastic foundations is considered as Pasternak's foundation model. A detailed parametric study is carried out to highlight the influences of the nonlocal parameter, thickness-to-length ratio, the boundary conditions and other parameters on the bending and the frequency of the nanoplates.

2. Formulation Consider a single-layered graphene sheet with length Lx, width Ly and constant thickness h that rests on two-parameter elastic foundations. Let U, V and W be the plate displacements parallel to a right-hand set of axes (x, y, z), where x is the longitudinal axis

401

and z is perpendicular to the plate. The origin of the coordinate system is located at the corner of the middle plane of the plate. The plate is assumed to be relatively thin and exposed to elevated temperature and subjected to a transverse sinusoidal or uniform load qðx; y; tÞ. 2.1. Nonlocal elasticity According to Eringen [1,3,4], the stress field at a point x in an elastic continuum not only depends on the strain field at the point (hyperelastic case) but also on strains at all other points x′ of the body. Eringen attributed this fact to the atomic theory of lattice dynamics and experimental observations. Thus, the nonlocal stress tensor components sNL ij at point x are expressed as Z sNL λðjx′  xj; τÞsLij ðx′Þ dx′; i; j ¼ x; y; z; ð1Þ ij ðxÞ ¼ V

sLij ðxÞ

where are the components of the classical macroscopic stress tensor at point x and the kernel function λðjx′  xj; τÞ represents the nonlocal modulus, jx′  xj being the distance in Euclidean norm and τ ¼ e0 a=Lx is a material constant that depends on the internal characteristic length a (e.g. distance between carbon–carbon bonds, granular size, lattice parameter) and external characteristic length Lx. The parameter e0 is Eringen's nonlocal elasticity constant suitable to each material. The macroscopic stress sL at a point x in a Hookean solid is related to the strain ɛ at the point by the generalized Hooke's law sL ðxÞ ¼ C : ɛðxÞ;

ð2Þ

where C is the fourth-order elasticity tensor and: denotes the double-dot product. The constitutive Eqs. (1) and (2) together define the nonlocal constitutive behavior of a Hookean solid. Eq. (1) represents the weighted average of the contributions of the strain field of all points in the body to the stress field at a point. Nevertheless, the integral constitutive relation in the elasticity problems (1) is difficult to solve, in addition to possible lack of determinism. However, for the case where the thermal effect is taken into account, it is possible to represent the integral constitutive relations in an equivalent differential form as in Eringen [4] ð1  τ2 L2x ∇2 ÞsNL ij ¼ C ijkl ɛ kl  C ijkl αΔT δkl ;

ð3Þ

where ∇2 is the Laplacian operator which is defined by ∇2 ¼ ð∂2 =∂x2 þ∂2 =∂y2 Þ, α is the coefficient of thermal expansion, δij is Kronecker's delta and ΔT is the temperature difference and it is given as

ΔT ¼ T  T 0 ¼ Applied temperature–reference temperature; The applied temperature distribution Tðx; y; z; tÞ through the thickness is assumed to be [35] z ΘðzÞ Tðx; y; z; tÞ ¼ T 1 ðx; y; tÞ þ T 2 ðx; y; tÞ: h h

ð4Þ

From Eq. (3), the plane stress nonlocal constitutive relations can be rewritten as  2 NL  ∂ sxx ∂2 sNL E xx þ ðɛ xx þ νɛyy  ð1 þ νÞαΔTÞ; ¼ sNL xx  μ 2 2 1  ν2 ∂x ∂y ! ∂2 sNL ∂2 sNL E yy yy þ ðɛyy þ νɛ xx ð1 þ νÞαΔTÞ; sNL ¼ yy  μ 1  ν2 ∂x2 ∂y2 ! ∂2 sNL ∂2 sNL E xy xy ɛxy ;  μ þ sNL ¼ xy 2ð1 þ νÞ ∂x2 ∂y2 ! ∂2 sNL ∂2 sNL E yz yz ɛyz ;  μ þ sNL ¼ yz 2ð1 þ νÞ ∂x2 ∂y2

402

sNL xz  μ

M. Sobhy / Physica E 56 (2014) 400–409

 2 NL  ∂ sxz ∂2 sNL E xz ɛxz ; ¼ þ 2ð1 þ νÞ ∂x2 ∂y2

μ ¼ e20 a2 :

ð5Þ

M xy  μ

2.2. The generalized displacement field In this study, the generalized displacement field, taking into account the shear deformation effect, is presented as [35–37] U ¼ u  zw;x þ ΘðzÞφ;

V ¼ v  zw;y þ ΘðzÞψ ;

W ¼ w;

ð6Þ

where u, v and w are the displacement components of the material point (x, y) in the mid-plane along the x, y and z directions, respectively; φ and ψ are the flexural rotations about the y- and x-axis, respectively. The function ΘðzÞ represents a shape function determining the distribution of the transverse shear strains and stresses along the thickness. The displacement fields of CPT, FDPT, 2 HDPT and SDPT are obtained by setting Θ ¼ 0; z; zð1  4z2 =3h Þ and h=π sin ðπ z=hÞ, respectively. All of the displacements u; v; w; φ; ψ are independent of z. The six strain components compatible with the displacement field in Eq. (6) are 9 9 8 2 9 8 8 ∂ w > > ∂u ∂φ > > > > > > > > > > > > > > > > > > 8 9 > ∂x > > > > > > ∂x2 > ∂x > > > > > > > > > ɛ xx > > > > > > > > > = < ∂2 w = = < ∂ψ < = < ∂v ɛyy ¼ þ  z ; Θ ðzÞ 2 ∂y ∂y ∂y > > > > > > > > :ɛ > ; > > > > > > ∂v ∂u > > ∂ ψ ∂φ > > 2 > xy > > > > > > > > > > > > > > > > > ∂ w > > > > > ; > ; : ∂x þ ∂y > : ∂x þ ∂y > >2 : ∂x∂y ; ( ) ( ) ɛ yz dΘðzÞ ψ ɛzz ¼ 0; ¼ ; ð7Þ φ ɛxz dz

  2   ∂ M xy ∂2 M xy E ∂v ∂u ∂2 w þ A2  2 A4 þ ¼ 2 2 2ð1 þ νÞ ∂x ∂y ∂x∂y ∂x ∂y    ∂ψ ∂φ þ A5 ; þ ∂x ∂y

  2    2  ∂ Sxx ∂2 Sxx E ∂u ∂v ∂ w ∂2 w þ A þ ν  þ ν ¼ A5 3 ∂y 1  ν2 ∂x ∂x2 ∂y2 ∂x2 ∂y2    ∂φ ∂ψ þν A6  ð1 þ νÞαST ; þ ∂x ∂y   2    2  ∂ Syy ∂2 Syy E ∂v ∂u ∂ w ∂2 w þν A3  Syy  μ ¼ þ þ ν 2 A5 2 2 2 2 ∂y ∂x 1ν ∂y ∂x ∂x ∂y    ∂ψ ∂φ T þν A6  ð1 þ νÞαS ; þ ∂y ∂x    2   ∂ Sxy ∂2 Sxy E ∂v ∂u ∂2 w þ A3  2 A5 Sxy  μ þ ¼ 2 2 2ð1 þ νÞ ∂x ∂y ∂x∂y ∂x ∂y    ∂ψ ∂φ þ A6 ; ð11Þ þ ∂x ∂y

Sxx  μ

! ∂2 Q yz ∂2 Q yz Eψ B þ ¼ 2ð1 þ νÞ ∂x2 ∂y2  2  ∂ Q xz ∂2 Q xz EφB ¼ þ Q xz  μ 2 2 2ð1 þ νÞ ∂x ∂y

Q yz  μ

The nonlocal stress resultants N i ; M i ; Si and Qj are expressed as Z h=2   fN i ; M i ; S i g ¼ sNL 1; z; Θ dz; i Q j ¼ K^

Z

 h=2

h=2  h=2

sNL j

dΘ dz; dz

Z fA1 ; A2 ; A3 ; A4 ; A5 ; A6 g ¼ B ¼ K^

Z

h=2



Z

  2    2  ∂ M xx ∂2 M xx E ∂u ∂v ∂ w ∂2 w þ A M xx  μ þ ν  þ ν ¼ A4 2 ∂y ∂x2 ∂y2 1  ν2 ∂x ∂x2 ∂y2    ∂φ ∂ψ þν A5  ð1 þ νÞαM T ; þ ∂x ∂y   2    2  ∂ M yy ∂2 M yy E ∂v ∂u ∂ w ∂2 w þν A2  M yy  μ þ þ ν 2 A4 ¼ 2 2 2 2 ∂y ∂x 1ν ∂y ∂x ∂x ∂y    ∂ψ ∂φ þν A5  ð1 þ νÞαM T ; þ ∂y ∂x

2

 h=2

n

1; z; Θ; z2 ; zΘ; Θ

2

o

dz; ð13Þ

dz;

 h=2 h=2

 h=2

ð8Þ

where K^ is the shear correction factor of FDPT. Substituting Eqs. (5) and (7) into Eq. (8) gives the constitutive relations as   2    2  ∂ Nxx ∂2 N xx E ∂u ∂v ∂ w ∂2 w N xx  μ þν A1  þ þ ν 2 A2 ¼ 2 2 2 2 ∂x ∂y 1ν ∂x ∂y ∂x ∂y    ∂φ ∂ψ þν A3  ð1 þ νÞαNT ; þ ∂x ∂y   2    2  ∂ N yy ∂2 N yy E ∂v ∂u ∂ w ∂2 w þν A1  N yy  μ þ þ ν 2 A2 ¼ 2 2 2 2 ∂y ∂x 1ν ∂y ∂x ∂x ∂y    ∂ψ ∂φ þν A3  ð1 þ νÞαNT ; þ ∂y ∂x   2   ∂ N xy ∂2 N xy E ∂v ∂u ∂2 w N xy  μ þ A1  2 A2 þ ¼ 2ð1 þ νÞ ∂x ∂y ∂x∂y ∂x2 ∂y2    ∂ψ ∂φ þ A3 ; þ ð9Þ ∂x ∂y

dΘ dz

h=2

and fN T g, fM T g and fST g are the thermal force and moment resultants Z h=2 Z h=2 ΔT dz; fMT g ¼ ΔTz dz; fN T g ¼ fP T g ¼

ði ¼ xx; yy; xy; j ¼ yz; xzÞ;

ð12Þ

where

 h=2

2.3. Nonlocal stress resultants

ð10Þ

 h=2

ΔT Θ dz:

ð14Þ

3. Equations of motion As is customary [20–28], the foundation is assumed to be a compliant foundation, which means that no part of the plate lifts off the foundation in the large deflection region. The load– displacement relationship of the foundation is assumed to be R ¼ K 1 w  K 2 ∇2 w, where R is the force per unit area, K1 is Winkler's foundation stiffness and K2 is the shearing layer stiffness of the foundation. The equations of motion can be obtained in a systematic manner by using the Hamilton's principle; the results of which are € ∂Nxx ∂N xy ∂w € ¼ 0; I 13 φ þ I 11 u€ þ I 12 ∂x ∂x ∂y € ∂N ∂N ∂w δv : xy þ yy  I11 v€ þ I12  I13 ψ€ ¼ 0; ∂y ∂x ∂y

δu :

δw :

  ∂2 M xy ∂2 M yy ∂2 M xx ∂u€ ∂v€ €  I 12 þ þ w þ 2  R þ q  I 11 ∂x ∂y ∂x∂y ∂x2 ∂y2    2  2 € ∂ w € € ∂ψ€ ∂ w ∂φ þ I 22 þ ¼ 0;  I 23 þ ∂x ∂y ∂x2 ∂y2

€ ∂Sxx ∂Sxy ∂w € ¼ 0;  I 33 φ þ Q xz  I 13 u€ þI 23 ∂x ∂x ∂y € ∂S ∂S ∂w δψ : xy þ yy  Q yz  I13 v€ þ I23  I33 ψ€ ¼ 0; ∂y ∂x ∂y

δφ :

ð15Þ

M. Sobhy / Physica E 56 (2014) 400–409

where the 2 I 11 I 12 6I 4 12 I 22 I 13 I 23

inertias Iij are defined by 3 2 3 I 13 1 z Θ Z h=2 7 I 23 7 ρ6 5¼ 4 z z2 zΘ 5dz:  h=2 2 I 33 Θ zΘ Θ

403

where ω ¼ ωmn denotes the eigenfrequency, I ¼ functions Xm(x) and Yn(y) are given as ð16Þ

pffiffiffiffiffiffiffiffi  1 and the

X m ðxÞ ¼ sin ϖ m x þ ζ m cos ϖ m x þ ηm sinh ϖ m x þ ξm cosh ϖ m x; Y n ðyÞ ¼ sin ϖ n y þ ζ n cos ϖ n y þ ηn sinh ϖ n y þ ξn cosh ϖ n y:

In order to derive the nonlocal equations of motion in terms of the displacement and rotation components, one should use the constitutive relations (9)–(12), noting that the coupling stiffness A2 and A3 and the inertias I12 and I13 are all equal to zero. For this case, in-plane displacements are uncoupled from the bending deflections. Thus one can obtain     E αð1 þ νÞ 2 4 2 ∂φ ∂ψ ∇ þ þ δw : A ∇ w A ∇ ð A T þ A T Þ 5 5 4 4 1 2 h ∂x ∂y 1  ν2    € ∂ψ€ ∂φ €  I 22 ∇2 w € þ I 23 þ ¼ 0; þð1  μ∇2 Þ R q þ I 11 w ∂x ∂y    E ∂2 φ ∂2 φ 2 ∂w þ2A  2A þ ð1 þ νÞ δφ : ∇ þ ð1  ν Þ A  B φ 5 6 6 ∂x 2ð1  ν2 Þ ∂x2 ∂y2     € ∂ 2 ψ 2α ∂ ∂w € ¼ 0;  I 33 φ  ðA5 T 1 þA6 T 2 Þ þ ð1  μ∇2 Þ I 23  A6 ∂x ∂x∂y h ∂x    E ∂2 ψ ∂2 ψ 2 ∂w þ2A 2A þ ð1 þ νÞ δψ : ∇ þ ð1  ν Þ A  B ψ 5 6 6 ∂y 2ð1  ν2 Þ ∂y2 ∂x2     € ∂ 2 φ 2α ∂ ∂w  I 33 ψ€ ¼ 0:  ðA5 T 1 þ A6 T 2 Þ þ ð1  μ∇2 Þ I 23  A6 ∂y ∂x∂y h ∂y ð17Þ

ð21Þ

All constants in the former equation for five different kinds of boundary conditions, i.e. SS, SC, CC, CF and FF, are listed in Appendix A. The external force and the applied temperature are presented in the following trigonometric series form: fqðx; y; tÞ; T i ðx; y; tÞg ¼

1

∑

m;n ¼ 1;3;5;…

fqmn ; T imn g sin ϖ nm x sin ϖ nn y eðIωtÞ ;

i ¼ 1; 2;

ð22Þ

where ϖ nm ¼ mπ =Lx , ϖ nn ¼ nπ =Ly and the coefficients qmn and Timn for the case of uniformly distributed load are defined as follows:     16 ; qmn ; T imn ¼ q0 ; T ni mnπ 2

ð23Þ

while for the sinusoidal load, qmn ¼ q0 , T imn ¼ T ni and m ¼ n ¼ 1, where q0 represents the intensity of the load at the plate center and T ni are constants and may be called temperature parameters. With the help of Eqs. (19) and (22), Eq. (17) become 9 8 9 2 38 P 11  ω2 G11 P 12  ω2 G12 P 13  ω2 G13 > < A mn > = > < F1 > = 6 P  ω2 G 7 2 P 22  ω G22 P 23 4 21 5 B mn ¼ F 2 ; 21 > > : ; > :F > ; P 31  ω2 G31 P 32 P 33  ω2 G33 C mn 3 ð24Þ

4. Exact solutions The exact solution of Eq. (17) for the single-layered graphene sheet under various boundary conditions can be constructed using the analytical solution method. In this method, the displacements are represented by functions that satisfy at least the different geometric boundary conditions, and represent approximate shapes of the deflected surface of the plate. The nanoplate is assumed to have simply supported (S), clamped (C) or free (F) edges or have combinations of these boundary conditions. Then, these boundary conditions on the edges perpendicular to x-axis take the form S : w ¼ ψ ¼ M xx ¼ Sxx ¼ T ¼ 0; C : w ¼ φ ¼ ψ ¼ T ¼ 0;

F : M xx ¼ M xy ¼ Q xz ¼ T ¼ 0:

ð18Þ

The following representation for the displacement quantities, that satisfy the above boundary conditions, is appropriate in the case of our problem:

Fig. 1. Comparison of the frequency ratios of the SSSS square nanoplate for various values of the nonlocal parameter μ (m ¼n¼ 1, k1 ¼k2 ¼0).

wðx; y; tÞ ¼ ∑∑A mn Ω mn ðx; y; tÞ; m n

φðx; y; tÞ ¼ ∑∑B mn Φ mn ðx; y; tÞ; m n

ψ ðx; y; tÞ ¼ ∑∑C mn Ψ mn ðx; y; tÞ;

ð19Þ

m n

where A mn ; B mn and C mn are the unknown coefficients, and m and n are the mode numbers. In the present case, the admissible functions Ω mn ðx; y; tÞ; Φ mn ðx; y; tÞ and Ψ mn ðx; y; tÞ are selected as beam eigenfunction and can be expressed as [38]

Ω mn ðx; y; tÞ ¼ X m ðxÞY n ðyÞeðIωtÞ ; ∂X m ðxÞ Y n ðyÞeðIωtÞ ; ∂x ∂Y ðyÞ Ψ mn ðx; y; tÞ ¼ X m ðxÞ n eðIωtÞ ; ∂y

Φ mn ðx; y; tÞ ¼

ð20Þ

Fig. 2. Comparison of the frequency ratios of the SSSS square nanoplate for various values of the nonlocal parameter μ (m ¼n¼ 2, k1 ¼ k2 ¼ 0).

404

M. Sobhy / Physica E 56 (2014) 400–409

where the coefficients Pij, Gij and Fi (i, j¼1, 2, 3) are given in Appendix B. 4.1. Bending solution For static thermomechanical bending problem, we consider A mn ; B mn and C mn to be time-independent, consequently, the time derivative terms in Eq. (17) and therefore, the terms containing ω are omitted. Eq. (24) has been solved for each pair of integers (m, n) to determine the magnitude of A mn ; B mn and C mn as A mn ¼  B mn ¼  Fig. 3. Comparison of the frequency ratios of the SSSS rectangular nanoplate (Ly ¼ 0.5 Lx) for various values of the nonlocal parameter μ (m¼ n¼ 1, k1 ¼ k2 ¼ 0).

C mn ¼ 

ðP 22 P 33  P 23 P 32 ÞF 1 þ ðP 13 P 32  P 12 P 33 ÞF 2 þ ðP 12 P 23  P 22 P 13 ÞF 3

Γ

ðP 23 P 31  P 21 P 33 ÞF 1 þ ðP 11 P 33  P 13 P 31 ÞF 2 þ ðP 13 P 21  P 11 P 23 ÞF 3

Γ

ðP 32 P 21  P 22 P 31 ÞF 1 þ ðP 12 P 31  P 11 P 32 ÞF 2 þ ðP 11 P 22  P 12 P 21 ÞF 3

Γ

Fig. 4. Percentage differences D1, D2 and D3 against the side-to-thickness ratio Lx/h for different values of the nonlocal parameter μ.

; ; ;

ð25Þ

M. Sobhy / Physica E 56 (2014) 400–409

405

Fig. 5. Percentage differences Dn1 , Dn2 and Dn3 against the side-to-thickness ratio Lx =h for different values of the nonlocal parameter μ.

Table 1 The deflection w of single-layered graphene sheets without or resting on elastic foundations for various values of nonlocal parameter (μ) ðT n1 ¼ 0; T n2 ¼ 100Þ. k1

k2

0

0

100

0

0

20

100

20

μ

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

Boundary conditions SSSS

SSSC

SCSC

SSCC

SCCF

CCCC

CCFF

CFFF

7.46088 7.46094 7.46099 7.46104 7.46110 7.46115 5.87874 4.31597 3.40960 2.81784 2.40112 2.09177 3.61743 1.92454 1.31102 0.99411 0.80059 0.67014 3.19987 1.62004 1.08457 0.81514 0.65294 0.54458

5.16952 5.16956 5.16960 5.16964 5.16967 5.16971 4.34536 3.35579 2.73333 2.30566 1.99372 1.75613 2.85434 1.41378 0.93958 0.70359 0.56235 0.46833 2.58375 1.23171 0.80859 0.60184 0.47929 0.39821

3.90487 3.90490 3.90493 3.90496 3.90499 3.90502 3.40819 2.71927 2.26204 1.93645 1.69279 1.50360 2.33669 1.10362 0.72241 0.53694 0.42725 0.35478 2.14925 0.98255 0.63684 0.47109 0.37380 0.30982

3.88863 3.88866 3.88869 3.88872 3.88874 3.88877 3.43667 2.81023 2.37696 2.05944 1.81676 1.62525 2.45621 1.11223 0.71888 0.53106 0.42106 0.34881 2.26782 1.00223 0.64325 0.47361 0.37478 0.31007

4.11319 4.11321 4.11324 4.11327 4.11330 4.11333 2.82641 2.19386 1.79268 1.51553 1.31261 1.15761 2.08780 1.08692 0.73471 0.55490 0.44580 0.37255 1.69589 0.88282 0.59673 0.45068 0.36207 0.30258

2.73170 2.73172 2.73174 2.73176 2.73179 2.73181 2.51375 2.14886 1.87649 1.66539 1.49698 1.35952 1.91476 0.80763 0.51174 0.37453 0.29534 0.24379 1.80507 0.74768 0.47149 0.34430 0.27116 0.22365

0.86880 0.86880 0.86881 0.86882 0.86882 0.86883 0.80168 0.68849 0.60330 0.53688 0.48363 0.43999 0.61531 0.26324 0.16744 0.12276 0.09690 0.08004 0.58087 0.24389 0.15435 0.11290 0.08900 0.07345

0.67664 0.67665 0.67665 0.67666 0.67666 0.67667 0.56014 0.47856 0.41773 0.37062 0.33306 0.30241 0.45840 0.20847 0.13491 0.09972 0.07909 0.06554 0.40179 0.18489 0.12007 0.08890 0.07058 0.05852

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M. Sobhy / Physica E 56 (2014) 400–409

Table 2 Free vibration Ω of single-layered graphene sheets without or resting on elastic foundations for various values of nonlocal parameter (μ). k1

k2

0

0

100

0

0

20

100

20

μ

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

Boundary conditions SSSS

SSSC

SCSC

SSCC

SCCF

CCCC

CCCF

CFCF

1.93861 1.17816 0.92261 0.78347 0.69279 0.62772 2.18396 1.54903 1.36479 1.27485 1.22122 1.18552 2.78410 2.31969 2.20092 2.14629 2.11486 2.09443 2.96017 2.52831 2.41979 2.37020 2.34177 2.32332

2.30839 1.36590 1.06339 0.90069 0.79530 0.71994 2.51780 1.69530 1.46242 1.34867 1.28064 1.23522 3.10655 2.61182 2.49421 2.44127 2.41113 2.39166 3.26517 2.79819 2.68863 2.63954 2.61165 2.59365

2.63247 1.51868 1.17619 0.99399 0.87658 0.79288 2.81776 1.81988 1.54535 1.41150 1.33135 1.27774 3.40301 2.85656 2.73435 2.68025 2.64967 2.63000 3.54829 3.02743 2.91223 2.86139 2.83271 2.81426

2.77248 1.61759 1.25565 1.06219 0.93723 0.84803 2.94915 1.90281 1.60602 1.45955 1.37117 1.31173 3.48844 3.02444 2.92003 2.87374 2.84755 2.83066 3.63044 3.18608 3.08687 3.04298 3.01815 3.00213

1.49564 1.07678 0.88460 0.76847 0.68863 0.62941 1.80426 1.47439 1.33996 1.26601 1.21901 1.18644 2.09929 2.09467 2.09304 2.09219 2.09167 2.09131 2.32926 2.32422 2.32243 2.32150 2.32092 2.32053

3.41369 1.91809 1.47795 1.24630 1.09778 0.99219 3.55859 2.16260 1.78318 1.59614 1.48287 1.40637 4.07737 3.52717 3.41380 3.36452 3.33680 3.31893 4.19943 3.66581 3.55646 3.50895 3.48221 3.46494

2.13624 1.50620 1.22863 1.06352 0.95096 0.86794 2.36266 1.81006 1.58542 1.46056 1.38037 1.32426 2.62583 2.78131 2.83071 2.85490 2.86920 2.87860 2.81311 2.95689 3.00275 3.02522 3.03848 3.04719

0.51783 0.56062 0.61614 0.69228 0.80609 1.00432 1.13854 1.15777 1.18446 1.22403 1.28895 1.41589 0.00000 0.66842 1.09517 1.53199 2.07634 2.90595 0.97124 1.21364 1.49087 1.83466 2.30711 3.07254

Fig. 6. Dimensionless center deflection w versus the side-to-thickness ratio Lx =h of SSSS single-layered graphene sheets subjected to sinusoidal and uniform loads for various values of the nonlocal parameter μ.

By solving the former equation (27), one can get the natural frequencies of the single-layered graphene sheet.

where

Γ ¼ P 11 P 22 P 33  P 11 P 23 P 32  P 12 P 21 P 33 þ P 32 P 21 P 13  P 22 P 31 P 13 þP 31 P 12 P 23 :

Fig. 7. Dimensionless center deflection w versus the nonlocal parameter μ for various values of the elastic foundation parameters.

ð26Þ

4.2. Free vibration solution To obtain the free vibration solution for a nanoplates using the nonlocal sinusoidal theory, the external thermal and mechanical loads are considered to be eliminated. The following eigen value problem for the natural frequencies can be obtained as    P 11  ω2 G11 P 12  ω2 G12 P 13  ω2 G13      P 23 ð27Þ  P 21  ω2 G21 P 22  ω2 G22  ¼ 0:   2 2  P 31  ω G31 P 32 P 33  ω G33 

5. Numerical results In this section, numerical results are given for analytical solutions given in the above section. The thermomechanical bending and free vibration of single-layered graphene plates are explained when these plates are resting on two-parameter elastic foundations and subjected to thermal and mechanical loads with various cases of the boundary conditions. In the present work, the plate is considered to be a square plate. The material properties are Young's modulus E ¼1 TPa, Poisson's ratio ν ¼0.19, the coefficient of thermal expansion α ¼ 1:6  10  6 K  1 , and the mass density

M. Sobhy / Physica E 56 (2014) 400–409

ρ ¼ 2300 kg/m3 [29]. The used non-dimensional parameters are 10  3 Eh

3

w¼

k1 ¼

q0 L4x K 1 L4x ; D

w

  Lx Ly ; ; 2 2

k2 ¼

K 2 L2x ; D

Ω¼

ωL2y qffiffiffiffiffiffiffiffiffiffiffiffi ρh=D; π2

pffiffiffiffiffiffiffiffiffi

Ωn ¼ ωL2y ρ=D;

3

D¼

h E : 12ð1  ν2 Þ

The results in this paper are demonstrated in tabulated form, as shown in Tables 1 and 2, and graphical form, as shown in Figs. 1–12, using the following fixed data (unless otherwise stated) Lx =h ¼ 10; m ¼ n ¼ 1; k1 ¼ 100; k2 ¼ 10; q0 ¼ 10; T 0 ¼ 298 K (room temperature), T n1 ¼ T n2 ¼ 100; h ¼ 0:34 nm, μ ¼2 nm2. The SDPT has been employed in the present numerical examples. The results of this theory are compared with those of HDPT and FDPT as well as CPT by estimating the percentage differences of central deflections as   w CPT  w SDPT    100; D1 ¼   w SDPT

  w FDPT  w SDPT    100; D2 ¼   w SDPT

  w HDPT  w SDPT    100; D3 ¼   w SDPT

Fig. 8. Free vibration Ω versus the nonlocal parameter μ for various values of the elastic foundation parameters.

407

and the percentage differences of free vibration as     ΩCPT  ΩSDPT      100; Dn ¼ ΩFDPT  ΩSDPT   100; Dn1 ¼  2    ΩSDPT ΩSDPT   ΩHDPT  ΩSDPT    100: Dn3 ¼  

ΩSDPT

In order to verify the accuracy of the present formulations, the frequency ratio, which is defined as Frequency ratio natural frequency Ω calculated using nonlocal theory n natural frequency Ω calculated using local theory n

¼

is compared with the available results in literature for simply supported nanoplates. In Figs. 1–3, the frequency ratios are presented and compared with those reported by Aksencer and Aydogdu [7]. It can be observed that the results of the present study are in excellent agreement with the published results. The frequency ratios increase as the length of the nanoplate Lx increases and the nonlocal parameter μ decreases. The percentage differences of central deflection D1, D2 and D3 and free vibration Dn1 , Dn2 and Dn3 vs the side-to-thickness ratio Lx =h for different values of the nonlocal parameter μ are plotted in Figs. 4 and 5. It can be observed that the difference of w between the SDPT and the CPT increases with the ratio Lx =h increasing while the difference between the deflection obtained by the SDPT and that obtained by the FDPT decreases with the ratio Lx =h increasing. Furthermore, for the classical model (i.e. μ ¼0 nm2), the percentage difference D3 has the same behavior of D2 whereas, for the nonlocal continuum model (i.e. μ a0 nm2 ), it is similar to the behavior of D1 (see Fig. 4). From Fig. 5, it can be seen that the differences between the frequencies obtained by the SDPT and those obtained by the other theories decrease directly as Lx =h increases for μ ¼0 nm2 but, the variation of these differences become quite different for μ ¼2, 5 nm2, especially, with the ratio Lx =h becoming smaller. Generally, the classical model (i.e. μ ¼0 nm2) will give a higher estimation, especially, for thick plates. Also, the SDPT is more agreement with HDPT than FDPT or CPT. Also, the difference between the nonlocal theory and the local one is larger for a thick nanoplate than that for a thin one, i.e., the difference between the nonlocal theory and the local one decreases as Lx =h increases. Tables 1 and 2 exhibit the effects of the elastic foundation parameters k1 and k2 and nonlocal parameter μ on the deflection and free vibration of the nanoplates under various boundary

Fig. 9. Dimensionless center deflection w versus the length of the square nanoplate for various values of the temperature parameters T n1 and T n2 (h is varied).

408

M. Sobhy / Physica E 56 (2014) 400–409

conditions. As it is expected, with the presence of the elastic foundation, the deflection decreases and the vibration increases. It is to be noted that the deflection and the vibration of plates resting on elastic foundations will decrease with the nonlocal parameter increasing except the vibration of CCCF and CFCF plates. However, the deflection of the foundationless nanoplate slowly increases with the increasing of the nonlocal parameter μ as shown in Table 1. Whereas the vibration of the foundationless nanoplate rapidly decreases with the increasing of the nonlocal parameter μ except that of the CFCF plate as shown in Table 2. The central deflection w of nanoplates resting on twoparameter elastic foundations under sinusoidal and uniform loading against the side-to-thickness ratio Lx =h for various values of the nonlocal parameter is illustrated in Fig. 6. Compared with the deflection of plates subjected to different loads, similar nonlocal parameter effect can be observed, the deflection decreases as μ increases. Note that the deflection can be changed not only by the nonlocal parameter, but also by the external load type change. The central deflection of the nanoplates calculated under uniformly distributed load are usually overpredicted. The

Fig. 10. Dimensionless center deflection w versus the side-to-thickness ratio Lx =h of single-layered graphene sheets under various boundary conditions (μ¼ 0.5 nm2).

Fig. 12. Free vibration Ω versus the length of the square nanoplate for various values of the mode numbers m and n (μ ¼1 nm2, h is varied).

differences between the curves decrease as the plate become thinner. Figs. 7 and 8 display the variations of the deflection w and the frequency Ω, respectively, versus the nonlocal parameter μ for different values of the elastic foundation parameters k1 and k2. It can be seen that w and Ω decrease monotonically as μ increases. Also, the higher the elastic foundation parameters change are, the smaller the deflection becomes. However, this phenomena becomes the opposite for the frequency Ω. The relation between the deflection w and the length of the square nanoplate for T n1 ; T n2 ¼ 100; 200; 300; 400; 500 are presented in Fig. 9. It can be observed that all of the natural frequencies become larger with the temperature parameters increasing. Furthermore, the differences between these temperature changes are obvious for a length greater than 10 nm. Regardless of T n1 and T n2 , w increases with the length of the plate increasing. For various cases of the boundary conditions, the deflection w and the frequency Ω of the nanoplates resting on elastic foundations are displayed in Figs. 10 and 11, respectively. It is noted that the simply supported boundary condition gives results for deflection greater than the free boundary condition. However, the clamped boundary condition has intermediate values (see Fig. 10). The clamped boundary condition always overpredicts the vibration frequencies magnitude. It can be also noticed that the frequency of CFCF plate has an exception behavior with the variation of Lx =h. Natural frequency for various lengths of the plate and various values of vibration modes is plotted in Fig. 12. The value of nonlocal parameter (μ) is assumed to be 1 nm2. It can be observed that as length and vibration modes increase, frequency increases. Further, it is important to observe that the effect of the vibration modes is more obviousness in the case of large nanolengths.

6. Conclusions

Fig. 11. Free vibration Ω versus the side-to-thickness ratio Lx =h of single-layered graphene sheets under various boundary conditions.

In this paper, based on the nonlocal continuum model, the thermomechanical bending and free vibration of the singlelayered nanoplates are studied. The present nanoplates are assumed to be resting on Pasternak's elastic foundations with various boundary conditions. The governing equations are derived by using the SDPT and compared with other shear deformation plate theories as well as classical one. The disagreement between STPT and CPT is very higher than that between the first and the other shear deformation theories, indicating the shear

M. Sobhy / Physica E 56 (2014) 400–409

1 2 2  ð1 þ νÞðϖ nm þ ϖ nn ÞðT 1mn A4 þT 2mn A5 ÞαE1 κ 14 ; h 1 F 2 ¼ ð1 þ νÞðT 1mn A5 þ T 2mn A6 Þϖ nm κ 15 αE1 ; h 1 F 3 ¼ ð1 þ νÞðT 1mn A5 þ T 2mn A6 Þϖ nn κ 16 αE1 ; h

Table A.1 Coefficients of beam functions. B.C.

ϖi

ζi

ηi

ξi

SS

0

0

0

SC

ϖ 1 ¼ π=Lx ϖ 2 ¼ 2π=Lx ϖ i ¼ iπ=Lx ði 4 2Þ ϖ 1 ¼ 3:9266=Lx

0

sin ϖ i Lx  sinh ϖ i Lx

0

CC

ϖ 2 ¼ 7:0685=Lx ϖ i ¼ ð4i þ 1Þπ=4Lx ði 4 2Þ ϖ 1 ¼ 4:7300=Lx

1

sin ϖ i Lx  sinh ϖ i Lx cos ϖ i Lx  cosh ϖ i Lx

CF

ϖ 2 ¼ 7:8532=Lx ϖ i ¼ ð2i þ 1Þπ=2Lx ði 4 2Þ ϖ 1 ¼ 1:8751=Lx

FF

ϖ 2 ¼ 4:6941=Lx ϖ i ¼ ð2i  1Þπ=4Lx ði 4 2Þ ϖ 1 ¼ 4:7300=Lx

sin  cos



ϖ i Lx  sinh ϖ i Lx ϖ i Lx  cosh ϖ i Lx

sin ϖ i Lx þ sinh ϖ i Lx cos ϖ i Lx þ cosh ϖ i Lx

1

in which

sin ϖ i Lx þ sinh ϖ i Lx cos ϖ i Lx þ cosh ϖ i Lx

E1 ¼ E=ð1  ν2 Þ; E2 ¼ E=2ð1 þ νÞ; Z Ly Z Lx ðX m Y n ; X m Y ″n ; X m Y ⁗ ðκ 1 ; κ 3 ; κ 5 Þ ¼ n ÞX m Y n dx dy; 0

ðκ 9 ; κ 11 ; κ 13 Þ ¼ ðκ 6 ; κ 8 ; κ 12 Þ ¼



sin ϖ i Lx  sinh ϖ i Lx cos ϖ i Lx  cosh ϖ i Lx

1



sin ϖ i Lx  sinh ϖ i Lx cos ϖ i Lx  cosh ϖ i Lx

ϖ 2 ¼ 7:85321=Lx ϖ i ¼ ð2i þ 1Þπ=2Lx ði 4 2Þ

ðκ 2 ; κ 4 ; κ 10 Þ ¼ ðκ 14 ; κ 15 Þ ¼

deformation effect. The influences of (i) nonlocal parameter, (ii) elastic foundation stiffnesses, (iii) boundary condition type, (iv) loading type, (v) temperature parameters, (vi) length and (vii) side-to-thickness ratio on the outlined results are discussed. The deflection decreases as the nonlocal parameter, elastic foundation stiffnesses and side-to-thickness ratio increase while it is proportional to the length of the plates and the temperature parameters. The uniform load distribution overpredicts the deflection magnitude. The free vibrations of the nanoplates increase as the nonlocal parameter decreases and elastic foundation stiffnesses increase. As the size of the nanoplate increases the effect of vibration modes becomes more significant and predicts smaller frequencies.

Appendix A Coefficients of beam functions are given in Table A.1. Appendix B P 11 ¼  ðκ 5 þ 2κ 11 þ κ 13 ÞðE1 A4 þ μK 2 Þ þ ðκ 3 þ κ 9 ÞðK 2 þ μK 1 Þ  K 1 κ 1 ; P 12 ¼ ðκ 11 þ κ 13 ÞE1 A5 ; P 13 ¼ ðκ 5 þ κ 11 ÞE1 A5 ; P 21 ¼  ðκ 8 þ κ 12 ÞE1 A5 ; P 22 ¼ ðκ 8 E2 þ κ 12 E1 ÞA6  κ 6 E2 B; 1þν κ 8 E1 A6 ; P 31 ¼  ðκ 4 þ κ 10 ÞE1 A5 ; P 23 ¼ 2 1þν κ 10 E1 A6 ; P 33 ¼ ðκ 4 E1 þ κ 10 E2 ÞA6  κ 2 E2 B; P 32 ¼ 2 G11 ¼  ðκ 5 þ 2κ 11 þ κ 13 ÞμI 22 þ ðκ 3 þ κ 9 ÞðI 22 þ μI 11 Þ  κ 1 I 11 ; G12 ¼  ½κ 9  μðκ 11 þ κ 13 ÞI 23 ; G13 ¼  ½κ 3  μðκ 5 þ κ 11 ÞI 23 ; G21 ¼ ½κ 6  μðκ 8 þ κ 12 ÞI 23 ; G22 ¼  ½κ 6  μðκ 8 þ κ 12 ÞI 33 ; G31 ¼ ½κ 2  μðκ 4 þ κ 10 ÞI 23 ; G33 ¼  ½κ 2  μðκ 4 þ κ 10 ÞI 33 ; F 1 ¼ ½1 þ μðϖ nm þ ϖ nn Þκ 14 qmn 2

2

409

κ 16 ¼

Z

Ly

Z 0 Lx

Z

0

0

0

Z

0 Ly

Z Z

Ly

0 Ly 0

Ly

Z

0 Lx

Z Z

0 Lx 0

Z

Lx 0

Lx

ðX ″m Y n ; X ″m Y ″n ; X ⁗ m Y n ÞX m Y n dx dy;

ðX ′m Y n ; X ′m Y ″n ; X ‴m Y n ÞX ′m Y n dx dy; ðX m Y ′n ; X m Y ‴n ; X ″m Y ′n ÞX m Y ′n dx dy;

ðX m sin ϖ nm x; X ′m cos ϖ nm xÞY n sin ϖ nn y dx dy;

X m Y ′n sin ϖ nm x cos ϖ nn y dx dy;

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