Composite Structures 111 (2014) 349–353

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Composite Structures

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Resonance behavior of FG rectangular micro/nano plate basedon nonlocal elasticity theory and strain gradient theorywith one gradient constant

0263-8223/$ - see front matter � 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2014.01.012

⇑ Corresponding author. Tel.: +98 9177014665.E-mail address: maziar.janghorban@gmail.com (M. Janghorban).

Mohammad Rahim Nami, Maziar Janghorban ⇑School of Mechanical Engineering, Shiraz University, Shiraz, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Available online 27 January 2014

Keywords:Resonance behaviorFunctionally graded materialRectangular micro/nano plateNonlocal elasticity theoryStrain gradient elasticity theory

In this article, for the first time, the resonance behaviors of functionally graded micro/nano plates are pre-sented using Kirchhoff plate theory. To consider the small scale effects, the nonlocal elasticity theory andstrain gradient theory with one gradient parameter are adopted. In this work, one can see the differentbehaviors of these two theories. To solve the governing equations, an analytical approach is used to inves-tigate simply supported functionally graded rectangular micro plates. To show the accuracy of presentmethodology, our results are compared with the results for isotropic gradient micro plate. The effectsof gradient parameter, aspect ratio and nonlocal parameter are also studied.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded micro/nano structures have been studiedfor several times in recent years [1]. Janghorban and Zare [2] con-sidered the free vibration of functionally graded nanotubes basedon Timoshenko beam theory using differential quadrature method.Janghorban and Zare [3] also presented the static analysis of func-tionally graded nanotubes on the basis Euler–Bernoulli beam mod-el using harmonic differential quadrature method. Nateghi et al. [4]proposed the buckling analysis of functionally graded micro beamsbased on modified couple stress theory. Three different beam the-ories, i.e. classical, first and third order shear deformation beamtheories, were considered to study the effect of shear deformations.Sadeghi et al. [5] presented the strain gradient elasticity formula-tion for analysis of Functionally Graded micro-cylinders. The mate-rial properties were assumed to obey a power law in radialdirection. The governing differential equation was derived as afourth order ODE. Akgöz and Civalek [6] investigated the bucklingbehavior of size-dependent micro beams made of functionallygraded materials for different boundary conditions on the basisof Bernoulli–Euler beam and modified strain gradient theory. Thehigher-order governing differential equation for buckling with allpossible classical and non-classical boundary conditions was ob-tained by a variational statement. Nabian et al. [7] studied the sta-bility of a functionally graded clamped–clamped micro-platesubjected to hydrostatic and electrostatic pressures. Equilibrium

positions of the micro-plate were determined and shown in thestate control space. To study the stability of the equilibrium posi-tions, the motion trajectories were given for different initial condi-tions in the phase plane. Uymaz [8] presented a forced vibrationanalysis of functionally graded nanobeams based on the nonlocalelasticity theory. The solution was obtained by using Navier meth-od for various shear deformation theories. It was shown that thedynamic deflections obtained by the local theory are smaller thanobtained by the nonlocal theory due to the nonlocal effects.Mohammadi et al. [9] investigated the buckling analysis of func-tionally graded rectangular micro plates based on the classicalplate theory and the strain gradient theory. The governing equa-tions were obtained for a rectangular micro plate which was sub-jected to the in plane loads. It was assumed that the micro platewas made of functionally graded materials and the material prop-erties vary through the thickness according to the power law dis-tribution. Hasanyan et al. [10] studied the pull-in instabilities ina functionally graded microelectromechanical system due to theheat produced by the electric current. Material properties of two-phase MEMS were assumed to vary continuously in the thicknessdirection. It was shown that the pull-in voltage strongly dependsupon the variation through the thickness of the volume fractionsof the two constituents. Asghari and Taati [11] proposed a size-dependent formulation for mechanical analyses of inhomogeneousmicro-plates based on the modified couple stress theory. The gov-erning differential equations of motion were derived for function-ally graded plates with arbitrary shapes utilizing a variationalapproach. Utilizing the derived formulation, the free-vibrationbehavior as well as the static response of a rectangular FG

350 M.R. Nami, M. Janghorban / Composite Structures 111 (2014) 349–353

micro-plate was proposed. Noori and Jomehzadeh [12] used themodified couple stress theory to study vibration analysis of func-tionally graded rectangular micro-plates. Considering classicaland first order plate theories, the couple governing equations ofmotion were obtained by using the Hamilton’s principle. It wasfound that the natural frequency parameter of micro-plates willdecrease as thickness-length ratio increases especially for lowerlength scale values. Mohammad-Alasti et al. [13] presented themechanical behavior of a functionally graded cantilever micro-beam subjected to a nonlinear electrostatic pressure and tempera-ture changes. It was assumed that the top surface is made of puremetal and the bottom surface from a metal–ceramic mixture. Keand Wang [14] proposed the dynamic stability of microbeamsmade of functionally graded materials (FGMs) based on the modi-fied couple stress theory and Timoshenko beam theory. The high-er-order governing equations and boundary conditions werederived by using the Hamilton’s principle. Ke et al. [15] also inves-tigated the nonlinear free vibration of micro beams made of func-tionally graded materials (FGMs) based on the modified couplestress theory and von Kármán geometric nonlinearity. The non-classical beam model was developed within the framework of Tim-oshenko beam theory which contains a material length scaleparameter. Jafari Mehrabadi et al. [16] studied the mechanicalbuckling of a functionally graded nanocomposite rectangular platereinforced by aligned and straight single-walled carbon nanotubes(SWCNTs) subjected to uniaxial and biaxial in-plane loadings. Theequilibrium and stability equations were derived using the Mindlinplate theory considering the first-order shear deformation (FSDT)effect and variational approach. Ke et al. [17] considered the non-linear free vibration of functionally graded nanocomposite beamsreinforced by single-walled carbon nanotubes based on Timo-shenko beam theory and von Kármán geometric nonlinearity.

In present research, the resonance behaviors of functionallygraded rectangular micro/nano plates are investigated on the basisof nonlocal elasticity and gradient elasticity theories using Kirch-hoff plate theory. Navier method is used to find the transversedeflections of simply supported functionally graded rectangularmicro/nano plate under external loading. The results of presentwork can be used as bench mark for future works.

2. Review of nonlocal elasticity theory

The nonlocal theory of elasticity, the points undergo transla-tional motion as in the classical case, but the stress at a point de-pends on the strain in a region near that point [19]. As forphysical interpretation, the nonlocal theory incorporates longrange interactions between points in a continuum model. Suchlong range interactions occur between charged atoms or moleculesin a solid [20]. Consider a single layer graphene sheet with as-sumed isotropic material in continuum model. The nonlocal consti-tutive behavior of a Hookean solid can be represented by thefollowing differential constitutive relation [21–23]:

ð1� lr2Þr ¼ t ð1Þ

where l is the nonlocal parameter, t is the macroscopic stress ten-sor at a point and r is the nonlocal stress tensor.

3. Review of strain gradient elasticity theory

Another famous theory which can be used for considering thesmall scale effects is strain gradient elasticity theory. For this the-ory, different formats are presented in the literature. In presentarticle, a simple strain gradient elastic theory with just one con-stant [18] is used to study the resonance behaviors of functionally

graded rectangular micro plates. In this theory, the stress–strainrelations can be defined as follow,

ðrijÞ ¼ Cijklðeij � leij;mmÞ ð2Þ

where l is the gradient parameter, eij are the strain components andCijkl are the elastic constants.

4. Governing equations

In this section, to study the resonance behaviors of functionallygraded rectangular micro/nano plates, the governing equationswith considering size effects are proposed.

4.1. Nonlocal elasticity theory

According to the classical plate theory for macro plates, the dis-placement components at an arbitrary material point of the platecan be expressed as [21],

uðx; y; zÞ ¼ uðx; yÞ � z@w@x

vðx; y; zÞ ¼ vðx; yÞ � z@w@y

wðx; y; zÞ ¼ wðx; yÞ ð3Þ

where u; v and w are the displacements in Cartesian coordinate.And the strain–displacement relations with considering the abovedisplacements can be defined as below,

ex ¼ �z@2w@x2

ey ¼ �z@2w@y2

cxy ¼ �2z@2w@x@y

ð4Þ

By substituting the above strains in Eq. (1), following relations areachieved,

rx � lr2rx ¼E

1� v2 �z@2w@x2 � vz

@2w@y2

!ð5Þ

ry � lr2ry ¼E

1� v2 �z@2w@y2 � vz

@2w@x2

!ð6Þ

sxy � lr2sxy ¼E

2ð1þ vÞ �2z@2w@x@y

!ð7Þ

where E is the Young’s modulus and v is the Poisson’s ratio. Now byintegrating and differentiating from above equations and by defin-ing the stress and moment resultants, following relations areobtained,

@2Mx

@x2 � lr2 @2Mx

@x2

!¼ D� � @

4w@x4 � v @4w

@x2@y2

!ð8Þ

@2My

@y2 � lr2 @2My

@y2

" #¼ D� � @

4w@y4 � v @4w

@x2@y2

!ð9Þ

@2Mxy

@x@y� lr2 @2Mxy

@x@y

" #¼ D�ð1� vÞ 2

@4w@x2@y2

!ð10Þ

where ðMx;My;MxyÞ ¼R h=2�h=2ðrx;ry;�sxyÞzdz and D� ¼

R h2

�h2

EðzÞ1�v2 z2dz.

According to the following equilibrium equation,

M.R. Nami, M. Janghorban / Composite Structures 111 (2014) 349–353 351

@2Mx

@x2 � 2@2Mxy

@x@yþ @

2My

@y2 þ qðx; yÞ ¼ qðzÞh @2w@t2 ð11Þ

where q is the density and q is the transverse loading. and with con-sidering Eqs. (8)–(10), the governing equation on the basis of non-local elasticity theory is expressed as,

�qðx; yÞ þ qðzÞh @2w@t2

!� lr2 �qðx; yÞ þ qðzÞh @

2w@t2

!

¼ D� � @4w@x4 �

@4w@y4 � 2

@4w@x2@y2

!ð12Þ

By considering the simply supported boundary condition for func-tionally graded micro plate, one can assume that

qðx; yÞ ¼X1m¼1

X1n¼1

Q mn sinax sinby sinXt

wðx; yÞ ¼X1m¼1

X1n¼1

wmn sinax sinby sinXt ð13Þ

where a ¼ mpa ; b ¼ np

b and X is the load frequency. Due to the factthat the nonlocal parameter loses its effect at the edges of the platein view of deflections being zero there, simply supported boundaryconditions for the nonlocal plate are the same as those of the localplate theory [24]. This assumption can be seen in several articles[23–25]. Now, by inserting the above series in Eq. (12), one can ob-tained a closed form solution for forced vibration of functionallygraded micro plates on the basis of nonlocal elasticity theory asfollow,

wmn ¼Q mn þ lðða2 þ b2ÞQmnÞ

�X2qðzÞh� lðða2 þ b2ÞX2qhÞ þ D�ða4 þ b4 þ 2a2b2Þð14Þ

4.2. Strain gradient elasticity theory

According to the constitutive Eq. (2) for strain gradient elastic-ity theory with one gradient parameter and the strain–displace-ment relations in Eq. (4), the stresses in terms of displacementscan be defined as follow,

-150

-100

-50

0

50

100

150

0 50 100 150 200 250 300 350 400 450

Load frequency

Def

lect

ion

ratio

Present[18]

Fig. 1. Comparison between the present results with the results for isotropicnanoplate.

rx ¼E

1� v2 �z@2w@x2 � vz

@2w@y2

!� E

1� v2 lr2 �z@2w@x2 � vz

@2w@y2

!

ð15Þ

ry ¼E

1� v2 �z@2w@y2 � vz

@2w@x2

!� E

1� v2 lr2 �z@2w@y2 � vz

@2w@x2

!

ð16Þ

sxy ¼E

2ð1þ vÞ �2z@2w@x@y

!� E

2ð1þ vÞ lr2 �2z

@2w@x@y

!ð17Þ

According to the procedure of finding the stress and moment resul-tants in nonlocal elasticity theory, following relations with consid-ering gradient parameter can be obtained,

@2Mx

@x2¼D� �@

4w@x4�v @4w

@x2@y2

!� lD� �@

6w@x6� @6w@x4@y2

�v @6w@x2@y4

�v @6w@x4@y2

!

ð18Þ

@2My

@y2 ¼ D� � @4w@y4 � v @4w

@x2@y2

!

� lD� � @6w@y6 �

@6w@y4@x2 � v @6w

@y2@x4 � v @6w@y4@x2

!ð19Þ

@2Mxy

@x@y¼ D�ð1� vÞ 2

@4w@x2@y2

!� lD�ð1� vÞ 2

@6w@y4@x2 þ 2

@6w@y2@x4

!

ð20Þ

By employing the equilibrium Eq. (11), the governing equation ofmotion for forced vibration of functionally graded rectangular microplate using one gradient parameter can be written as follow,

�qðx; yÞ þ qðzÞh @2w@t2

!¼ D� � @

4w@x4 �

@4w@y4 � 2

@4w@x2@y2

!

� lD� � @6w@x6 �

@6w@y6 � 3

@6w@x4@y2 þ

@6w@x2@y4

! !ð21Þ

The boundary conditions for simply supported gradient plate can beachieved from variational method as follow for all edges [26],

-20

-15

-10

-5

0

5

10

15

20

0 10 20 30 40 50 60Load frequency

Def

lect

ion

ratio

µ=1.0µ=1.5µ=2.0µ=2.5

Fig. 2. The effect of nonlocal parameter on resonance position ðp ¼ 1Þ.

0

5

10

15

20

ectio

n ra

tio

l=1

l=2

l=3

l=4

352 M.R. Nami, M. Janghorban / Composite Structures 111 (2014) 349–353

w ¼ 0;@2w@x2 ¼

@2w@y2 ¼ 0;

@4w@x4 ¼

@4w@y4 ¼

@4w@x2@y2 ¼ 0 ð22Þ

From above boundary conditions, it is found that the Navier methodcan satisfy the simply supported boundary condition for gradientplate. So by adopting the Eq. (13), one can express the closed formsolution for functionally graded rectangular micro plates as follow,

wmn ¼Q mn

�X2qðzÞhþ D�ða4 þ b4 þ 2a2b2Þ þ lD�ða6 þ b6 þ 3ða4b2 þ a2b4ÞÞð23Þ

-20

-15

-10

-5

400 20 60 80 100Load frequency

Def

l

Fig. 4. The effect of gradient parameter on resonance position ðp ¼ 1Þ.

5. Numerical results

In this section, the numerical results for resonance behavior offunctionally graded rectangular micro plates are presented. It isimportant to note that the material properties of functionallygraded micro plates are assumes as [11],

E ¼ ðEc � EmÞzhþ 0:5

� �p

þ Em

q ¼ ðqc � qmÞzhþ 0:5

� �p

þ qm ð24Þ

where Ec ¼ 380 GPa; qc ¼ 3960; Em ¼ 70 GPa; qm ¼ 2700 and theaverage value of Poisson’s ratio is assumed to be 0.3. In Fig. 1, ourresults are compared with the results presented by Papargyri-Bes-kou and Beskos [18] for isotropic nanoplates. From this comparison,it can be found that the amplitude of plate and the resonance posi-tion obtained by our work and Papargyri-Beskou and Beskos [18]

(a) =1.0

(c) =5.0

-40

-30

-20

-10

0

10

20

30

40

Load frequency

Def

lect

ion

ratio

-20

-15

-10

-5

0

5

10

15

20

Load frequency

Def

lect

ion

ratio

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

Def

lect

ion

ratio

Fig. 3. The effect of power law index

are the same. In this study, the deflection ratios are expressed asbelow,

Deflection ratio ¼ Deflection using nonlocal=gradient theoryDeflection using local theory

Deflection ratios can show us the importance of size effects instudying forced vibration of functionally graded micro/nano plates.

=3.0

(d) =7.0

-20

-15

-10

-5

0

5

10

15

20

20

Load frequency

Def

lect

ion

ratio

-20

-15

-10

-5

0

5

10

15

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

Load frequency

(b)

on resonance position ðl ¼ 1Þ.

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 5 10 15 20 25 30 35 40

Load frequency

Def

lect

ion

ratio

b/a=2b/a=3b/a=4

Fig. 5. The effect of aspect ratio on resonance position ðl ¼ 1; p ¼ 1Þ.

M.R. Nami, M. Janghorban / Composite Structures 111 (2014) 349–353 353

Next, the influences of power law index, nonlocal parameter, aspectratio and gradient parameter are studied on the resonance phenom-ena. Fig. 2 depicts the effects of nonlocal parameter on the forcedvibration of functionally graded rectangular micro plates. Fromnumerical results, it is observed that with the increase of nonlocalparameter, the resonance position will move to the lower load fre-quencies. In studying functionally graded structures, power law in-dexes usually play an important role. In Fig. 3, the influences ofpower law indexes on the resonance behavior of micro plates arepresented. In this figure the nonlocal parameter is assumed to be1 nm2. It can be found that increasing the power law indexes, theresonance position moves to the lower load frequencies. It seemsthat the power law indexes have more effect in comparison withnonlocal parameter. Above results are concluded from Eq. (14).

A review of literature shows that there is no study on forcedvibration of functionally graded rectangular micro/nano platesusing gradient elasticity theory. So as another example, the influ-ences of gradient parameter on the forced vibration of functionallygraded micro plates are proposed in Fig. 4. It is shown that with theincrease of gradient parameter, the resonance position will moveto higher load frequencies. This result is found from Eq. (23). FromFigs. 1 and 3, it is seen that the effects of nonlocal parameter andgradient parameter are not the same. This result can be found eas-ily from Eqs. (14) and (23). It is recommended to other researchersto compare the results of these two theories with experimental testto find out which of them are more accurate. As the final numericalexample, Fig. 5 expressed the influences of aspect ratio on theresonance phenomena. In this figure the gradient parameter isassumed to be 1 nm2 and the numerical results are calculated fromEq. (23). It is shown that increasing the aspect ratio will causedecreasing the resonance frequency. This phenomenon may beseen in forced vibration of functionally graded macro plates, too.From Figs. 4 and 5, one can understand that the geometricalparameter has more effect in comparison with gradient parameter.The presented new results for the forced vibration of functionallygraded micro plates on the basis of nonlocal elasticity theory andgradient elasticity theory with one gradient parameter can be usedas a benchmark solution for future researches.

6. Conclusion

In this work, the resonance behaviors of functionally graded mi-cro/nano rectangular plate were studied. In order to see the size ef-fects, nonlocal elasticity theory and strain gradient theory with one

gradient parameter were used to derive the governing equations.Closed form solution for both theories was also presented. Theinfluences of different parameters such as power law index, non-local parameter and gradient parameter were investigated, too. Itwas observed that with the increase of nonlocal parameter, the res-onance position will move to the lower load frequencies. It wasalso shown that with the increase of gradient parameter, the reso-nance position will move to higher load frequencies.

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