An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity

Composite Structures 100 (2013) 290–299

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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity Shahrokh Hosseini-Hashemi a,b,⇑, Mojtaba Zare a, Reza Nazemnezhad a a b

School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16842-13114 Tehran, Iran Center of Excellence in Railway Transportation, Iran University of Science and Technology, Narmak, 16842-13114 Tehran, Iran

a r t i c l e

i n f o

Article history: Available online 4 January 2013 Keywords: Exact analytical solution Free vibration Nonlocal elasticity theory Mindlin plate theory Nano-plate

a b s t r a c t Eringen nonlocal theory is employed in Mindlin plate theory to consider small scale effects on free vibration of rectangular nano-plates. Introducing some auxiliary and potential functions, an exact analytical procedure is applied on the governing equations to decouple the displacement variables. It is believed that this method is new for solving vibration of nano-plates. The solution of natural frequencies is obtained for Levy-type boundary conditions (two opposite edges simply supported and the others arbitrary). In order to confirm the reliability of the method considered, the results are compared with several reported literature. The effect of nonlocal parameter is investigated on natural frequency of the nanoplate for different boundary conditions. Finally the influence of aspect ratio and thickness to length ratio on natural frequency is studied in detail. It is expected that results obtained in this paper serve as an accurate reference in future nano-structures issues. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Among different nano-structures, one can introduce nanoplates which concern with small scale fields. Due to the rapid development of technology, especially in micro and nano-scale fields, nano-plates are used in micro- or nano-electromechanical systems (MEMS or NEMS) for their superior mechanical, thermal and electrical properties. Dynamic behavior of nano-plates used as thin film elements [1], two-dimensional suspended nano-structures [2,3] nano-sheet and paddle-like resonators [4,5] requires a two-dimensional nano-structure analysis. Hence, one must consider small scale effects in order to refine classical theories to derive the governing equations for these structures. The scale effects are accounted by considering internal size as a material parameter. Experimental results show that as length scales of a material are reduced, the influences of long-range interatomic and intermolecular cohesive forces on the mechanical properties become prominent and cannot be neglected. The local (classic) continuum theory neglects the effects of long-range load on the motion of the body and long range inter atomic interactions. Therefore, the internal scale is neglected. Some methods like molecular dynamics [6] are presented in recent years which consider size effects and atomic lengths. Molecular dynamics models are limited to the small number of atoms and relatively short times. Therefore, the simulation time (cost) increases enormously ⇑ Corresponding author at: School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16842-13114 Tehran, Iran. Tel.: +98 2177240540. E-mail address: [email protected] (Sh. Hosseini-Hashemi). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.11.035

if we increase the length and the number of atoms. Nonlocal linear theory, which has both features of lattice parameter and classical elasticity, could be considered a superior theory for modeling nano-materials. Nonlocal theory of Eringen [7] is one of the wellknown continuum mechanics theories to account the small scale effect by specifying the stress at a reference point as a functional of the strain field at every point in the body. Hence, many papers dealt with analyzing nano-structures have been published on this topic. Buckling and vibration analyses of carbon nano-tubes with the help of beam and shell theories [8,9], application of nonlocal theory for beam vibration [10] and vibration analysis of graphite sheets using the plate theories [11] are some of the wide application of nonlocal theory. Study of the vibration and buckling analysis of nano-plates and graphene sheets can be seen in bending and vibration of plates via nonlocal Reddy plate theory [12], CPT and Mindlin nonlocal theory for plate vibration [13,14], free vibrations of single-layered graphene sheets [15], buckling of graphene sheets [16,17], vibration and buckling of nano-plates [18] and 3D vibration analysis of nano-plates [19]. But as reported in many of these literature the solution of the governing equation are based on numerical methods (e.g., finite element method [20], finite difference method [21], differential quadrature method [22]) and approximate analytical methods like Navier type solution method that assumes the variation of displacement variables harmonically [12,13]. Furthermore, many of these solutions are concerned with Navier boundary condition, i.e. all edges are simply supported and a few of them consider combinations of clamed and simply supported boundaries [18,22]. Hence, no exact closed-form

Sh. Hosseini-Hashemi et al. / Composite Structures 100 (2013) 290–299

291

Nomenclature a, b t l e0

q h E

m D x, y, z X, Y

eij tij

rij Nij Mij pi L

r2 ~2 r

ui u0r

plate length and width time internal length material constant mass density nano-plate thickness Young modulus of elasticity Poisson’s ratio flexural rigidity rectangular Cartesian coordinates non-dimensional rectangular Cartesian coordinates strain components nonlocal stress tensor local stress tensor nonlocal force resultants nonlocal moment resultants components of body force linear differential operator two-dimensional Laplacian operator non-dimensional Laplacian operator

components of displacement mid-plane displacements in x- and y-directions ur rotational displacements about the x- and y-axes ~r u non-dimensional displacements about the x- and y-axes w lateral displacement in z-direction ~ w non-dimensional lateral displacement in z-direction j, l nonlocal parameters f non-dimensional nonlocal parameter nl nonlocal term ks shear correction factor inertia terms Ik d thickness to length ratio g aspect ratio (length to width ratio) b non-dimensional frequency parameter x frequency parameter a(|x0  x|) nonlocal kernel function f(X, Y) auxiliary function Wt(X, Y) potential functions m, n number of half waves in x- and y-directions

solution is available in the literature for the free vibration analysis of nano-plates and various Boundary conditions (BCs). According to Hosseini-Hashemi et al. [23,24] an exact closed form solution procedure is established for vibration of single-layered and functionally graded plates based on some auxiliary and potential functions. This method has just been considered for local theory and can be applied to Levy-type support conditions and yield well convergence and accurate results without any approximations. Therefore, the main purpose of this article is to apply this exact method to solve the governing equations of motion of nano-plate for Mindlin theory base on nonlocal elasticity. In this regard, the rectangular plate equations of motion for Mindlin theory are derived via equations of momentum balance and base on nonlocal continuum model. The equations of the problem are coupled through displacement components. Introducing a set of auxiliary and potential functions, the governing equations are decoupled for transverse vibration analysis. By transforming the displacement variables into known functions the problem leads to a soluble form without any approximations. Two opposite edges are held simply supported and the other two edges may be given any combination of free (F), simply supported (S) and clamped (C). Applying the boundary conditions lead to characteristic equations which result natural frequencies accurately and analytically. In order to confirm the reliability of the method considered, the results are compared with several reported literature. Also, the effects of nonlocal parameter, aspect ratio and thickness to length ratio of the plate and different boundary conditions on non-dimensional vibration frequencies are investigated.

2.1. Summary of nonlocal continuum theory As mentioned earlier in nonlocal theory the stress in a material body point is a function of strain field of the same point and all other ones in material domain, so the stress tensor plays the essential role in this continuum theory which is defined as [7]:

Z

aðjx0  xjÞrij ðx0 ÞdV 0

Laðjx0  xjÞ ¼ dðjx0  xjÞ

ð2Þ

Which after applying Eq. (2) on Eq. (1) the integral forms of nonlocal stress tensor reduces to differential one:

Ltij ¼ rij

ð3Þ

The linear operator is an approximate model of the kernel obtained by matching the Fourier transforms of the kernel in the wave number space with the dispersion curves of lattice dynamics. For curve-fitting at low wave numbers relevant to the small internal length scale Eq. (2) is written as:

ð1  j2 r2 þ l4 r4  . . .Þt ij ¼ rij So the linear operator becomes:

L ¼ ð1  j2 r2 þ l4 r4  . . .Þ

ð4Þ

where j and l are small parameters proportional to the internal length scale. If first order approximation is to be considered, just the Laplacian form of the operator in Eq. (4) is maintained [27]. So for the two-dimensional case:

L ¼ 1  ðe0 lÞ2 r2

ð5Þ

In which l is internal length and e0 is material constant which is de@2 @2 fined by the experiment and r2 ¼ @x 2 þ @y2 is the two-dimensional Laplacian operator. Equations of motion for nonlocal linear elastic solids are obtained from nonlocal balance law as:

2. Problem formulation

t ij ¼

any other point in the body, i, j = x, y, z, for three dimensional Cartesian coordinate, rij is the local stress tensor and a(|x0  x|) is nonlocal kernel function depends on internal characteristic length. Eringen proposed a(|x0  x|) as a Green function of a linear differential operator L as:

ð1Þ

V

where the volume integral is taken over the body region V. x is the reference point in body which the stress tensor is calculated at, x0

€i tij;j þ pi ¼ qu

ð6Þ

pi and ui are the components of the body force and displacement vector respectively and q is mass density. Using Eq. (3) in Eq. (6) the nonlocal equations of motion in differential form become:

rij;j þ Lðpi  u€i Þ ¼ 0

ð7Þ

It should be noted that the boundary conditions here are based on nonlocal stress tensors tij rather than local ones rij [13].

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2.2. Mindlin plate equations via nonlocal elasticity The global Mindlin plate equations can be derived by integrating the equations of motion (6) through the thickness [28]. To this end, the nonlocal resultant forces N and nonlocal resultant moments M are introduced as follow:

Nij ¼

Z

þh2

2h

tij dz;

M ij ¼

Z

þ2h

2h

t ij zdz

ð8a; bÞ

Now, multiplying Eq. (6) by dz, then integrating through the thickness and attending to the fact that there are no shear forces applied to the faces of the plate and using the plane stress assumption and making use of Eq. (8a) we have:

Nir;r þ Pr ¼

Z

þh2

2h

qu€i dz

ð9Þ

where r takes the symbols x and y and P i ¼

R þ2h 2h

pi dz. Also multiply-

ing Eq. (6) by zdz then integrating through the thickness and noting that the body force is independent of z and using the knowledge in derivation of Eq. (9) we write:

M rs;s  Nrz ¼

Z

þ2h

h2

qu€ r zdz

ð10Þ

Since Eq. (10) does not have physical application for i = z, it has already been omitted through the derivation. s also takes the symbols x and y. Eqs. (9) and (10) are in nonlocal form. To have general equations of motion for the nonlocal theory in terms of known force and moment resultants we apply the definition of Eqs. (8) into Eq. (3) and using linear differential operator L from Eq. (5). Therefore the nonlocal force and moments can be obtained in terms of local ones: 2

½1  j2 r Nij ¼

NLij ;

2

½1  j2 r M ij ¼

M Lij

where j = e0l and the local (classical) resultant forces cal resultant moments MLij are defined by:

NLij

¼

Z

þh2

2h

rij dz;

M Lij

¼

Z

þ2h

2h

rij zdz

ð11a; bÞ N Lij

and the lo-

NLir;r

2

2

¼ ½1  j r  pi 

M Lrs;s  NLrz ¼ ½1  j2 r2 

h 2

ð12a; bÞ

Z

h 2

2h

!

qu€ i dz

ð13Þ

qu€ r zdz

ð14Þ

2h

uz ðx; y; z; tÞ ¼ wðx; y; tÞ

ð15Þ

where r = x, y and u0r are the mid-plane displacement components and ur are independent variables and denote the rotational displacements about the x and y axes considered positioning at the mid-surface of the plate and t is time. In order to obtain motion equations in terms of displacement variables we need the local force and moment resultants in terms of displacement fields. The strain–displacement relations are defined as:

1 2

eij ¼ ðui;j þ uj;i Þ

ð17Þ where E is the Young modulus of elasticity and m is the Poisson’s Ratio. Using Eqs. (12) and (17) simultaneously then inserting in Eqs. (13) and (14) and considering free vibration, the equations of motion of Mindlin plate theory in nonlocal continuum model are obtained:

12D h

2

ðu0x;xx þ mu0y;xy Þ þ

6D h

2

€ 0x ð1  mÞðu0x;yy þ u0y;xy Þ ¼ ½1  j2 r2 I0 u ð18aÞ

12D h

2

ðu0y;yy þ mu0x;xy Þ þ

6D h

2

€ 0y ð1  mÞðu0x;xy þ u0y;xx Þ ¼ ½1  j2 r2 I0 u ð18bÞ

6ks D h

2

€ ð1  mÞðux;x þ w;xx þ uy;y þ w;yy Þ ¼ ½1  j2 r2 I0 w

Dðux;xx þ muy;xy Þ þ

D 6ks D ð1  mÞðux;yy þ uy;xy Þ  2 ð1  mÞðux þ w;x Þ 2 h

€x ¼ ½1  j2 r2 I2 u Dðuy;yy þ mux;xy Þ þ 6ks D h

2

ð18cÞ

ð18dÞ D ð1  mÞðux;xy þ uy;xx Þ 2

€y ð1  mÞðuy þ w;y Þ ¼ ½1  j2 r2 I2 u

ð18eÞ

where the shear correction factor ks was introduced to consider that Eh3 the transverse shear strains, D ¼ 12ð1 m2 Þ is the flexural rigidity and the inertia term Ik is defined as:

Ik ¼

Z

h 2

qzk dz k ¼ 0; 2

ð19Þ

2h

As we see the in-plane and the out-of-plane displacement variables are uncoupled. Therefore for simplicity we just consider equations of the flexural deformations i.e. Eqs. (18c)–(18e) 3. Solution procedure

Now it’s time to apply Mindlin displacement field into equations of motion. Displacement variables in Mindlin theory are considered as [26]:

ur ðx; y; z; tÞ ¼ u0r ðx; y; tÞ þ zur ðx; y; tÞ

9 08 0 8 91 u þ mu0y;y > > ux;x þ muy;y > > > > > > x;x > > > > C B> > > > > 0 0 > > > > C B> u þ m u u þ m u > > > > y;y x;x y;y x;x < < = = C B E B 1m 0 C 0 1m rxy ¼ þ z ðu þ u Þ C B ð u þ u Þ x;y y;x x;y y;x 2 2 2 > > > > > C 1  m B> > > > > > > > > > > > > C B 1 m r > > > > 0 xz > ðux þ w;x Þ > > > > > > A @> 2 > > > > > > : : > ; ; > : ; 1 m ryz 0 ð u þ w Þ ;y y 2 8 9 rxx > > > > > > > > > > > < ryy > =



Finally applying the operator L to equations of motion (9) and (10) and making use of Eqs. (11) the general equations of motion for the nonlocal plate model become:

Z

Furthermore for an isotropic solid with plane stress assumption (rzz = 0) using Eqs. (15) and (16) the stress–displacement relations are defined based on Hooke’s law as:

ð16Þ

For generalizing solution the following non-dimensional terms are introduced:

x y h b w j ~2 ~ ¼ ;f ¼ ;r ~ r ¼ ur ; w X ¼ ;Y ¼ ;d ¼ ;g ¼ ;u ¼ a2 r2 a a a a a a rffiffiffiffiffiffi qh b ¼ xa D 2

ð20Þ

f and b are called non-dimensional nonlocal and frequency parameters respectively and the non-dimensional Laplacian operator is ~ 2 ¼ @22 þ @ 22 . Assume harmonic motion with respect defined as r @X @Y to time as:

~ y; tÞ ¼ wðx; ~ yÞeixt ~ r ðx; y; tÞ ¼ u ~ r ðx; yÞeixt ; wðx; u

ð21Þ

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Inserting non-dimensional parameters and using Eqs. (19) and (21) in equation of motions (18c)–(18e) the non-dimensional equations become:

Employing Eqs. (27) and (30) in (22b) and (22c) the general solutions to the dimensionless rotations in terms of potentials give [23]:

~ x ¼ C 1 W 1;X þ C 2 W 2;X þ W 3;Y u

ð32aÞ

~ y ¼ C 1 W 1;Y þ C 2 W 2;Y  W 3;X u

ð32bÞ

2

~ 2w ~ þ b2 d2 w ~ þ 6ks ð1  mÞðu ~ x;X þ u ~ y;Y Þ ¼ 0 ½6ks ð1  mÞ  nl r ð22aÞ

where:

" # " #  2 1m nl ~ 2 b2 d2 6ks ð1  mÞ ~x þ ~x  r u u  2 12 12 d2 þ

Cv ¼

1þm 6k ð1  mÞ ~ ;X ¼ 0 ~ y;Y Þ;X  s 2 ~ x;X þ u w ðu 2 d

ð22bÞ

" # " #  2 1m nl ~ 2 b2 d2 6ks ð1  mÞ ~y þ ~y r u u   2 12 12 d2 þ

ð22cÞ

where nl2 = f2b2d2 is substituted to distinguish the nonlocal term. ~ x;X þ u ~ y;Y ; differIntroducing an auxiliary function as f ðX; YÞ ¼ u entiating Eqs. (22b) and (22c) with respect to X and Y respectively and adding them together we obtain a new equation based on f ~ Solving Eq. (22a) and the new obtained one simultaand variable w. ~ is neously and collecting terms a differential equation in terms of w achieved:

~ 4w ~ 2w ~ þ a2 r ~ þ a3 w ~ ¼0 a1 r

ð23Þ

where: 2

a1 ¼  1 

nl 12

! þ

a3 ¼

2

ð24Þ

2





2 2

mÞ þ b12d  6ks ð1 d2

; v ¼ 1; 2

ð33Þ

6ks ð1  mÞ d2



1þm bv ; v ¼ 1; 2 2

ð34Þ

The dimensionless potential W3(X, Y) is the homogenous solution of Eqs. (22b) and (22c) satisfying:

~ 2 W 3 þ s2 W 3 ¼ 0 r 3

ð35Þ

where:

h 2 3

s ¼

mÞ  6ks ð1 d2 h i 2 ð12 mÞ  nl12

1 2 2 b d 12

i ð36Þ

The general solutions to the potential functions i.e. Eqs. (28) and (35) can be achieved by virtue of the separation variables principle. Rewriting these equations to the index form:

ð37Þ

Assume the solutions to be of the type Wt(X, Y) = qt(X)gt(Y) the independent functions q and g become:

6ks ð1  mÞ

ð6 þ nl Þb2 d4 þ 3ks ð12nl þ b2 d4 Þð1 þ mÞ

2 2 b2 d2 b12d



~ 2 W t þ s2 W t ¼ 0 t ¼ 1; 2; 3 r t

  2 2 1  nl12 nl

2

a2 ¼ 

s

ð12 mÞ

And:

dv ¼

1þm 6k ð1  mÞ ~ y;Y Þ;Y  s 2 ~ ;Y ¼ 0 ~ x;X þ u w ðu 2 d

dv 2

2 ½nl v 12

36ks d ð1 þ mÞ 

q00t  c2t qt ¼ 0

ð38aÞ

g 00t  k2t g t ¼ 0

6ks ð1mÞ d2

2 t

6ks ð1 þ mÞ

Eq. (23) can be written as:

~ 2  e1 Þðr ~ 2  e2 Þw ~ ¼0 ðr

ð25Þ

where e1 and e2 are the roots of auxiliary equation of Eq. (23) i.e.:

a1 e2 þ a2 e þ a3 ¼ 0

ð26Þ

ð38bÞ k2t

where c and are separation constants to be determined and t = 1, 2, 3. From now on the Levy-type BCs is developed where two opposite edges of the plate are simply supported and the remaining can have any of the conditions simply supported (S), clamed (C) and Free (F). The dimensionless boundary conditions in nonlocal field will be given below for an edge parallel to Y – normalized axes, i.e. X = 0 or X = 1:

According to superposition principle one can write the solution to Eq. (25) as:

e xx ¼ u ~ 0 ¼ 0 For a simply supportedðSÞedge; ~y ¼ w M

~ ¼ W1 þ W2 w

e xx ¼ M ~ xy ¼ N ~ xz ¼ 0 For a freeðFÞedge; M

ð27Þ

where W1(X, Y) and W2(X, Y) are dimensionless potentials satisfying the differential equations:

~ 2 W 1 þ s2 W 1 ¼ 0; r ~ 2 W 2 þ s2 W 2 ¼ 0 r 1 2

ð28a; bÞ

And:

s21 ¼ e1 ¼

a2 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða22  4a1 a3 Þ 2a1

; s22 ¼ e2 ¼

a2 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða22  4a1 a3 Þ 2a1 ð29a; bÞ

~0 ¼ u ~x ¼ u ~ y ¼ 0 For a clampedðCÞedge; w In the view of Eqs. (11) the natural boundary conditions can be changed to their local forms. The corresponding boundary conditions for the edges at Y = 0 and Y = g are obtained by interchanging subscripts x and y. The non-dimensional force and moment resultants may be obtained by use of Eqs. (12) and (17) and non-dimensional parameters as:

eL ¼u ~ x;X þ mu ~ y;Y M xx

eL ¼u ~ y;Y þ mu ~ x;X M yy

eL ¼u ~ x;Y þ mu ~ y;X M xy

Using Eqs. (27) and (28) in Eq. (22a) the solution to auxiliary function f yields:

eL ¼ u ~x þ w ~ ;X N xz

f ¼ b1 W 1 þ b2 W 2

On the assumption of a simply-supported edge at both X = 0 and 1 a simple test in boundary conditions reveals that the solutions of the equations q00t  c2t qt ¼ 0 are not suitable for satisfying this BC [25] and finally applying these conditions the solutions to Eqs. (38) may be selected:

ð30Þ

where: 2

bv ¼

½6ks ð1  mÞ  nl s2v  b2 d2 ; v ¼ 1; 2 6ks ð1  mÞ

ð31Þ

eL ¼ u ~y þ w ~ ;Y N yz

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Table 1 Comparison study of frequency ratio for SSSS nonlocal plate. f

0

(n, m) (1, 1)

a b

(1, 2)

(2, 1)

(2, 2)

(3, 2)

(3, 3)

Navier HSDT [12] Navier FSDT [13] Present Navier HSDT [12] Navier FSDT [13] Present Navier HSDT [12] Navier FSDT [13] Present Navier HSDT [12] Navier FSDT [13] Present Navier HSDT [12] Navier FSDT [13] Present Navier HSDT [12] Navier FSDT [13] Present

0.2

0.4

0.6

0.7477 0.7475 0.7475 0.5801 0.5799 0.5799 0.5801 0.5799 0.5799 0.4906 0.4904 0.4904 0.4040 0.4038 0.4038 0.3514 0.3512 0.3512

0.4904 0.4904 0.4904 0.3353 0.3353 0.3533 0.3353 0.3353 0.3533 0.2708 0.2708 0.2708 0.2155 0.2155 0.2155 0.1844 0.1844 0.1844

0.3512 0.3512 0.3512 0.2309 0.2308 0.2308 0.2309 0.2308 0.2308 0.1844 0.1844 0.1843 0.1456 0.1456 0.1456 0.1241 0.1241 0.1241

¼ 0:5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.8183 0.8183 0.7475 0.7475 0.6111 0.6111 0.5799 0.5799 0.4496 0.4496 0.4287 0.4287

0.5799 0.5798 0.4904 0.4904 0.3601 0.3601 0.3353 0.3353 0.2440 0.2440 0.2309 0.2309

0.4287 0.4287 0.3512 0.3512 0.2492 0.2492 0.2309 0.2309 0.1655 0.1655 0.1562 0.1562

a ¼ 0:4 b 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.8282 0.8279 0.7719 0.7631 0.6152 0.6152 0.5942 0.5942 0.4562 0.4562 0.4419 0.4419

0.5942 0.5941 0.5278 0.5170 0.3635 0.3635 0.3465 0.3465 0.2483 0.2483 0.2391 0.2391

0.4419 0.4117 0.3827 0.3749 0.2517 0.2518 0.2391 0.2391 0.1684 0.1684 0.1620 0.1620

¼1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 a b

(1, 1) (1, 2) (2, 1) (2, 2) (3, 2) (3, 3)

(1, 1) (1, 2) (2, 1) (2, 2) (3, 2) (3, 3)

Navier HSDT Present Navier HSDT Present Navier HSDT Present Navier HSDT Present Navier HSDT Present Navier HSDT Present

[12]

Navier FSDT Present Navier FSDT Present Navier FSDT Present Navier FSDT Present Navier FSDT Present Navier FSDT Present

[13]

[12] [12] [12] [12] [12]

[13] [13] [13] [13] [13]

Fig. 1. Comparison of frequency ratio variations with plate side length for different nonlocal parameter (SSSS, ba ¼ 1, m = n = 2).

Fig. 2. Comparison of frequency ratio variations with plate side length for different nonlocal parameter (SSSS, ab ¼ 2, m = n = 1).

Fig. 3. Comparison of frequency ratio variations with plate side length for different nonlocal parameter (SSSS, ab ¼ 2, m = n = 2).

Fig. 4. Comparison of frequency ratio variations with plate side length for different nonlocal parameter (SCSC, ba ¼ 1, m = n = 2).

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Fig. 5. Comparison of frequency ratio variations with plate side length for different nonlocal parameter (ab ¼ 2, m = n = 1).

Fig. 6. Comparison of frequency ratio variations with plate side length for different nonlocal parameter (ab ¼ 2, m = n = 2).

Table 2



Variations of non-dimensional frequency

b ¼ xa 2

qffiffiffiffi qh D

and frequency ratio (FR) for

different boundary conditions (m = 1, n = 1). 0

f BC SCSF SFSF SFSS SSSS SCSS SCSC

g = 0.6

0.2

0.4

0.6

0.8

bNL 17.5196 9.3561 13.9320 35.0643 45.0923 56.8967

FR 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

FR 0.8779 0.8759 0.8547 0.6335 0.6159 0.6080

FR 0.6820 0.6712 0.6328 0.3789 0.3631 0.3560

FR 0.5351 0.5166 0.4769 0.2633 0.2513 0.2458

FR 0.4354 0.4121 0.3762 0.2005 0.1910 0.1867

13.8996 9.4047 12.2549 24.2330 29.8086 36.7592

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.8667 0.8718 0.8548 0.7050 0.6914 0.6846

0.6559 0.6635 0.6330 0.4451 0.4308 0.4237

0.5014 0.5085 0.4771 0.3146 0.3030 0.2973

0.3988 0.4048 0.3764 0.2412 0.2319 0.2273

12.2606 9.4458 11.3810 19.0840 22.4260 26.7369

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.8616 0.8683 0.8548 0.7475 0.7374 0.7319

0.6458 0.6578 0.6331 0.4904 0.4785 0.4721

0.4904 0.5019 0.4772 0.3512 0.3412 0.3359

0.3886 0.3988 0.3765 0.2708 0.2626 0.2583

g = 0.8 Fig. 7. Change of Frequency ratio for six different boundary condition and nonlocal parameter (j2) [22].

qv ðXÞ ¼ Av sinðcv XÞ;

v ¼ 1; 2;

q3 ðXÞ ¼ A3 cosðc3 XÞ

8   2 2 2 > < g t ðYÞ ¼ At sinðkt YÞ þ Bt cosðkt YÞ; st ¼ ct þ kt g t ðYÞ ¼ At sinhðkt YÞ þ Bt coshðkt YÞ; s2t ¼ c2t  k2t > :

ð39Þ

SCSF SFSF SFSS SSSS SCSS SCSC

g=1

ð40Þ

where ct ¼ mp; m ¼ 1; 2; 3; . . . and At ; At ; Bt ; At and Bt are arbitrary constants.

SCSF SFSF SFSS SSSS SCSS SCSC

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Table 3



Variations of non-dimensional frequency

b ¼ xa 2

qffiffiffiffi qh D

and frequency ratio (FR) for

different boundary conditions (m = 2, n = 1). 0

f BC SCSF SFSF SFSS SSSS SCSS SCSC

g = 0.6

0.2

0.4

0.6

0.8

bNL 42.7132 36.3434 41.1144 60.2869 66.4133 73.8831

FR 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

FR 0.7324 0.6505 0.6308 0.5216 0.5111 0.5048

FR 0.4382 0.3929 0.3749 0.2923 0.2848 0.2804

FR 0.3045 0.2737 0.2599 0.1997 0.1942 0.1911

FR 0.2319 0.2087 0.1978 0.1511 0.1469 0.1445

39.9452 36.4958 39.2655 50.3100 53.3086 57.0503

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.6345 0.6438 0.6310 0.5594 0.5530 0.5486

0.3784 0.3869 0.3752 0.3197 0.3148 0.3116

0.2627 0.2691 0.2602 0.2194 0.2159 0.2135

0.2000 0.2051 0.1980 0.1664 0.1636 0.1617

38.7128 36.4246 38.3610 45.5845 47.2245 49.2606

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.6328 0.6396 0.6308 0.5799 0.5759 0.5732

0.3769 0.3832 0.3751 0.3353 0.3321 0.3300

0.2615 0.2663 0.2601 0.2308 0.2285 0.2269

0.1990 0.2028 0.1979 0.1752 0.1734 0.1721

g = 0.8 SCSF SFSF SFSS SSSS SCSS SCSC

g=1 SCSF SFSF SFSS SSSS SCSS SCSC

Table 4



Variations of non-dimensional frequency

b ¼ xa 2

qffiffiffiffi qh D

Fig. 8. Variations of ðg ¼ 0:5; m ¼ 1; n ¼ 2Þ.

frequency

ratio

with

thickness

to

length

ratio

and frequency ratio (FR) for

different boundary conditions (m = 2, n = 2). 0

f BC SCSF SFSF SFSS SSSS SCSS SCSC

g = 0.6

0.2

0.4

0.6

0.8

bNL 85.6022 51.6274 77.8972 121.7700 132.6770 143.6230

FR 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

FR 0.4963 0.6290 0.5052 0.3789 0.3692 0.3600

FR 0.2742 0.3731 0.2816 0.2006 0.1949 0.1893

FR 0.1866 0.2586 0.1922 0.1352 0.1313 0.1274

FR 0.1410 0.1967 0.1454 0.1018 0.0989 0.0959

65.9053 45.9143 61.6058 87.2357 94.0851 101.3480

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.5415 0.6298 0.5454 0.4451 0.4364 0.4286

0.3062 0.3739 0.3093 0.2412 0.2357 0.2306

0.2096 0.2591 0.2119 0.1635 0.1596 0.1560

0.1587 0.1971 0.1605 0.1233 0.1204 0.1176

55.9736 42.8870 53.3852 70.0219 74.4019 79.1951

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.5666 0.6305 0.5688 0.4904 0.4833 0.4770

0.3248 0.3745 0.3264 0.2708 0.2660 0.2617

0.2231 0.2596 0.2243 0.1844 0.1809 0.1778

0.1691 0.1975 0.1701 0.1393 0.1366 0.1343

g = 0.8 SCSF SFSF SFSS SSSS SCSS SCSC

g=1 SCSF SFSF SFSS SSSS SCSS SCSC

Fig. 9a. Variations of the fundamental frequency ratio with plate side length for different nonlocal parameter (g = 1).

fines the relation between nonlocal and local theory in free vibration as follow: The selection of functions gt depends on their conditions related to the values of st which also depend on non-dimensional frequency parameters b. Using correct forms of Eq. (40) and substituting into appropriate boundary conditions i.e. along the edges Y = 0 and Y = g leads to a characteristic determinant of the sixth order for each m. Solving the eigenvalue equations yields n non-dimensional frequency parameters b. In vibration m and n are called half-wave numbers and the set (m, n) (or simply mn) refers to the mode number.

Frequency RatioðFRÞ ¼ ¼

b calculated by nonlocal theory b calculated by local theory bNL bL

ð41Þ

In the forgoing analysis the nano-plate is considered to be simply supported along edges parallel to Y – normalized axes, i.e. X = 0 or X = 1 as mentioned earlier and the sequence of the boundaries on Y is according to the name of boundary condition. For example SCSF refers to clamed (C) boundary along edge Y = 0 and free (F) boundary along Y = g.

4. Results and discussion 4.1. Comparison studies For numerical results, the Poisson’s ratio m = 0.3 and shear correction factor ks = 0.86667 are used throughout the investigation. In the following discussions, we consider a parameter which de-

To verify the reliability and the high accuracy of the aforementioned method, a comparison study of the natural frequency ratios

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297

Fig. 9b. Variations of the fundamental frequency ratio with plate side length for different nonlocal parameter (g = 1).

Fig. 9d. Variations of the fundamental frequency ratio with plate side length for different boundary conditions (j = 2 nm, g = 1).

Fig. 9c. Variations of the fundamental frequency ratio with plate side length for different nonlocal parameter (g = 1).

Fig. 10a. Variations of fundamental frequency ratio with aspect ratio.

of the present method is carried out with the results reported in Refs. [12,13,18] for a nano-plate with all edges simply supported, and Refs. [18,22] for a nano-plate with two opposite edges simply supported and the other ones clamped. Table 1 represents some comparison numerical results for the frequency ratio of several modes of the nano-plate with simplysupported boundary condition, different values of nonlocal parameter ðf ¼ 0; 0:2; 0:4; 0:6Þ, specified values of inverse of aspect ratio ð1=g ¼ 1; 0:5; 0:4Þ and an arbitrary value of thickness to length ratio (d). As seen, the results obtained by the present solution are in an excellent compatibility with Navier solution of Reddy plate theory [12] and first-order shear deformation theory [13]. Also, all the results have good convergence to classical continuum (local) theory when f = 0. Figs. 1–6 represent comparison of variations of the frequency ratio versus the nano-plate side length for different values of nonlocal parameter and two different BCs, i.e. SSSS and SCSC, between the present exact method with Navier and Levy solutions of the classical plate theory [18]. In Figs. 1–4, it is observed that the re-

sults obtained on the basis of the present method have good agreements with the available data. But in Figs. 5 and 6, however, it can be seen considerable differences between the presented results for a nano-plate with SCSC support condition. According to Ref. [22] in which the vibration of graphene sheets is considered using differential quadrature method, the frequency ratio for the nano-plate with SCSC boundary condition is just a little less than the one with SSSS boundary condition and this difference approaches to zero as the nonlocal parameter decreases (Fig. 7). Hence, the results of the nano-plate with SSSS and SCSC boundary conditions obtained with the present exact method and shown in Figs. 5 and 6 are in an agreement with the Pradhan and Kumar [22] and they seem to be much more acceptable. 4.2. Benchmark results In this section, some new results for the free vibration analysis of the nano-plate are expressed as benchmark solutions for validating new computational techniques in the future.

298

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Fig. 10b. Variations of fundamental frequency ratio with aspect ratio.

Frequency ratios of the nano-plate with different boundary conditions, specified values of nonlocal parameter, selected values of aspect ratio and the thickness to length ratio d = 0.1 are listed in Tables 2–4 for the first three frequency modes. The non-dimensional frequencies are given for zero value of nonlocal parameter. As it’s seen for this value of nonlocal parameter the local and nonlocal frequencies are equal according to the definition of Eq. (41). The non-dimensional frequency for other nonlocal parameters can be obtained by multiplying their frequency ratio by their own bL. It’s evident through these tables that by increasing the nonlocal parameter, increasing of small scale effects, the natural frequencies of the nano-plate decrease. This implies that the nonlocal effects soften the structures and make them more flexible. Hence the frequency properties predicted using the local plate theories are considerably over-estimated and there is no exception for frequency mode and specific boundary condition. Furthermore, the lowest frequency ratio is related to the SCSC boundary condition which is the stiffest one and the highest frequency ratio may be contributed to one of the SFSF or SCSF boundary conditions according to the aspect ratio and frequency mode. Therefore different boundaries may affect more or less by increasing the nonlocal

parameter. Another considerable result is that the nonlocal effects have more significant influences on the higher order frequencies for all support conditions. Fig. 8 shows variations of the frequency ratio versus thickness to length ratio for selected values of nonlocal parameter, different boundary conditions and the (1, 2) mode number when aspect ratio is g = 0.5. It is seen from this figure that the frequency ratio is not basically influenced by thickness to length ratio. Hence for a fixed value of aspect ratio and nonlocal parameter, the frequency ratio of a specific mode and a determined boundary condition remains unchanged throughout variations of thickness to length ratio. This result may be reliable for thin and moderately thick nano-plates as the Mindlin theory concerns. But the reliability of this conclusion should be investigated for thick nano-plates by the respective theories. Figs. 9a–9c display variations of the fundamental frequency ratio of a square nano-plate with the external length for specified values of nonlocal parameter and different boundary conditions. In these figures, it is observed that as the external length increases, the frequency ratio increases and approaches the classical solution, which signifies a decrease in the small scale effects. Furthermore, increasing the nonlocal parameter leads to a decrease in frequency ratio. The curves in Figs. 9a–9c show that variations of the frequency ratio have the same trend as the scale effects become considerable for all boundary conditions. It’s interesting to know that convergence of the nonlocal natural frequency to local one is smoother for SSSS, SCSS and SCSC boundary conditions which are stiffer, as external lengths gets larger than internal ones, or the nonlocal effect becomes less significant. For the nano-plates with at least one free edge, this behavior is not as smooth as the ones with SSSS, SCSS and SCSC boundary conditions. This can be justified by the fact that the nonlocal theory softens the structures and causes more effect on stiffer boundaries rather than softer ones (i.e. boundaries with at least one free edge) as seen in Fig. 9d. To investigate the effects of the aspect ratio on frequency ratio, Figs. 10a–10b are plotted. These figures depict variations of the fundamental frequency ratio with aspect ratio for two specified values of nonlocal parameter ðf ¼ 0:2; 0:6Þ and six different boundary conditions. Two different behaviors can be observed from these figures for variation of the frequency ratio with aspect ratio. By increasing the aspect ratio, the frequency ratio of a nano-plate with SSSS, SCSS and SCSC boundary conditions, shown in Fig. 10a, increases, whereas this result is completely different for SFSF and SCSF cases, shown in Fig. 10b. For the case of SFSS boundary

Fig. 11. Variations of non-dimensional frequency with thickness to length ratio (g = 0.8, (m, n) = (3, 1)).

Sh. Hosseini-Hashemi et al. / Composite Structures 100 (2013) 290–299

condition, variation of the frequency ratio is independent of the aspect ratio. Generally the aspect ratio has more influences on the frequency ratio in the case of stiff boundary conditions (i.e. SCSC, SCSS and SSSS). Fig. 11pillustrates variations of the non-dimensional frequency ffiffiffiffiffiffiffiffiffiffiffiffi ðb ¼ xa2 qh=DÞ versus thickness to length ratio for six different boundary conditions, two selected nonlocal parameters (f = 0.25, 0.5), aspect ratio of g = 0.8 and mode number of (3, 1). From Fig. 11, it can be seen that the non-dimensional frequency decreases by an increase in the thickness to length ratio as it’s the case in local elasticity. Also, an increase in the nonlocal parameter decreases the non-dimensional frequency for all six boundary conditions. Furthermore, the value of non-dimensional natural frequency of the SCSC boundary condition is the highest while for the SFSF one is the lowest for all values of thickness to length ratio. 5. Concluding remarks In this paper, exact analytical solutions for free vibration analysis of Levy-type rectangular nano-plate are studied using nonlocal first-order shear deformation theory. The nonlocal plate theory accounts for small scale effect, transverse shear deformation and rotary inertia. Three coupled governing partial differential equations of motion for freely vibrating Levy-type rectangular nano-plates are exactly solved by introducing the potential functions and using the method of separation of variables. Comparison cases by those reported in the literature, for simply supported and clamed rectangular nano-plates, demonstrate highly stability and accuracy of the present exact procedure. Presented herein shows the effects of boundary conditions, variations of nonlocal parameter, thickness to length ratio and aspect ratio on the frequency values of a nano-plate. It’s shown that the frequency ratio is independent of thickness to length ratio and decreases with increasing the mode number and the value of nonlocal parameter for all boundary conditions. Furthermore, it’s seen that the difference between nonlocal theories and local theories is significant for high value of the nonlocal parameter. The nonlocal theory has the least and the most influence on free and clamped boundary conditions, respectively. All analytical results presented here can provide other research groups with a reliable source to check out their analytical and numerical solutions. References [1] Freund LB, Suresh S. Thin film materials. Cambridge, UK: Cambridge University Press; 2003. [2] Zalalutdinov MK, Baldwin JW, Marcus MH, Reichenbach RB, Parpia JM, Houston BH. Two-dimensional array of coupled nanomechanical resonators. Appl Phys Lett 2006;88:14350. [3] Tighe TS, Worlock JM, Roukes ML. Direct thermal conductance measurements on suspended monocrystalline nanostructures. Appl Phys Lett 1997;70: 2687–9.

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