Micro temperature-dependent FG porous plate: Free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT

Accepted Manuscript

Micro temperature-dependent FG porous plate: Free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT Mohammad Hasan Shojaeefard , Hamed Saeidi Googarchin , Majid Ghadiri , Mohammad Mahinzare PII: DOI: Reference:

S0307-904X(17)30414-6 10.1016/j.apm.2017.06.022 APM 11826

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

21 August 2016 6 June 2017 14 June 2017

Please cite this article as: Mohammad Hasan Shojaeefard , Hamed Saeidi Googarchin , Majid Ghadiri , Mohammad Mahinzare , Micro temperature-dependent FG porous plate: Free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.06.022

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Highlights ● Free vibration and buckling analysis of micro temperature-dependent FG porous plates ●Utilizing CPT and FSDT in conjunction with MCST to derive the governing equations

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●Considerable porosity effects on the free vibration of the hinged plates

* Corresponding Author: E-mail address: [email protected] Λ E-mail address: [email protected]

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Micro temperature-dependent FG porous plate: Free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT Mohammad Hasan Shojaeefard1, Hamed Saeidi Googarchin2*, Majid Ghadiri3,

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Mohammad Mahinzare2Λ 1-Mechanical Engineering Department, Iran University of Science and Technology, Tehran, Iran, P.O. Box 16846-13114 2-School of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran, P.O. Box 16846-13114

3- Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran, P.O. Box 34149-16818

Abstract

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For the first time in this paper, free vibration and thermal buckling of micro temperature-

dependent FG porous circular plate subjected to a nonlinear thermal load are numerically studied. The governing equations are derived based on Hamilton's principal and using both classical and the first-order shear deformation theories in conjunction with the modified couple

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stress theory. Generalized Differential Quadrature method is applied to solve the equations with associated boundary conditions. The results reveal that the increase of size dependency

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and the temperature-change would lead to the increase of differences between the first natural

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frequencies predicted based on the two theories. In contrast, the porosity and the FG power index do have not any effect on that. While the effect of porosity on free vibration of clamped

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and free plates are negligible, but the effect of porosity for hinged ones is considerable as the temperature-change increase. Moreover, the critical conditions of the plates which are

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expressed by porosity, FG power index, size dependency, temperature-change and geometrical dimensions are presented, as well. Numerical results are in good agreement with those available in literature in some special cases. Keywords: micro FGM porous circular plate, Modified couple stress, thermal loading, Generalized Differential Quadrature.

* Corresponding Author: E-mail address: [email protected] Λ E-mail address: [email protected]

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1. Introduction Increasing the demand for the materials with conflicting properties (such as the lightness and the high strength, the hardness and the ductility) would lead to the construction of the

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composite materials. Since the delamination has known as a frequent failure in composites, functionally graded materials (FGM) have been introduced in order to overcome this problem. Due to existence of more advantages in the FGMs, there is an accelerating growth on

application of FGM in industry such as aerospace, automotive and MEMS. Consequently, there

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are many investigations on mechanical behavior analysis (such as vibration and buckling) of materials categorized with respected to their shape, size or loading of the FGM. Circular, annular, sectorial, rectangular and elliptic are the most famous geometries for FGM plates which have been studied. About circular and annular ones, Ansari et al. [1] and Reddy

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and Berry [2] studied FG micro-plates using the modified strain gradient elasticity and the

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modified couple stress theories respectively. Moreover, Kiani and Eslami [3] presented an analytical solution for thermal buckling of FGM plates. Sectorial plates, Hosseini Hashemi et

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al. [4] presented a solution for the vibration of radially FGM plates with variable thickness on an elastic foundation. Thermal post-buckling of FGM plates was presented by Kiani and

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Eslami [5]. Li et al. [6] presented relations between buckling load, frequency, and deflection of the thin homogeneous FGM plates. Kim and Reddy [7] investigated exact solutions for

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buckling, bending and vibration problems of the FGM plates using couple stress theory. Ansari et al.[8] investigated nonlinear vibration of micro-plates using the modified couple stress theory. Elliptic geometries, Cheng and Batra [9] studied thermo-elastic deformation of FGM plates. There are some differences in analysis of macro and micro-size FGMs. Although in micro-size

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structures the presence of a distributed moment is common, this is not the case in macro-size applications. This would lead to consider some terms known as couple stress [10–16] in equilibrium equations of micro-size FGM plates. Increasing the thickness to cross section characteristic ratio, high order theories are used in order to improve the results in the both sizes

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of plates.[7,17–21]

In practice, FGMs are subjected to mechanical, thermal, electrical and Magnetic loads or their combinations. Kiani and Eslami [3] and Lanhe [22] derived analytical solutions for buckling of FGM plates. Considering linear variation of temperature in terms of thickness, Dung and Dao

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[23] and Ma and Wang [24] investigated post-buckling of FGMs. Moreover, Tung [25] and Duc et al. [26] analytically studied post-buckling and vibration of FGMs with temperaturedependent properties respectively. Thermal effects on vibration of FGM plates were studied by

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Gupta et al [27,28]. Behavior of FGMs in magnetic field, Hu and Wang [29] , Dai and Dai [30] and Ghorbanpour Arani et al. [31] developed analysis for nonlinear response of a rotating plate

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to free vibration and stress variation in functionally graded piezoelectric materials respectively. While aforementioned studies were conducted assuming a perfect state for the materials,

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observations indicate that there is porosity in FGMs which leads to change in those mechanical

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behaviors predicted in literature. For instance, analysis by Ebrahimi [32] reveals that increasing the volume fraction of porosity leads to a little increase in frequency. Jabbari [33,34]

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developed a buckling analysis of plates subjected to equally radial compressive load to study the effects of the porosity distribution. Moreover, Wattanasakulpong et al. [35,36] predicted rotational and flexural vibrations of FGMs. Behravan Rad and Shariyat [37] proposed an analytical solution for porous FGM plates subjected to magneto-elastic loads. In this study, free vibration and buckling of a micro-size FGM porous circular plate subjected

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to a nonlinear thermal load are investigated. The governing equations of the plate are represented considering the modified couple stress as well as porosity and thermal effects. The classical and the first-order shear deformation theories are used in solution procedure with Generalized Differential Quadrature (GDQ) method (A complete historical background of

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GDQ method, as well as the main aspects of these techniques and a rich bibliography was presented in [38,39]). The results are presented for three boundary conditions: free, hinged and clamped. There is an agreement between results obtained in this study and those available in literature for special cases. Finally, comparative results are prepared to make a discussion on

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the impacts of the parameters like porosity and thermal load distribution on mechanical behavior of the micro-size plates. Novelties of the present research may be summarized as: 

Investigation on free vibration and thermal buckling of the continuous model of the micro-size

responses, for the first time.

Thermal effect is taken into account and material properties are considered to be temperature

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

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FGM plates based on FSDT and CPT, Considering the effects of the porosity on the vibrational

dependent. The nonlinear temperature distribution is considered through the thickness. The computational time for achieving the solution is significantly lower than other numerical

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

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methods. Moreover, the procedure would facilitate a parametric study.

2. Theoretical background

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In this section, the theoretical background of the problem is presented. Elaborating assumptions in the problem, the governing equation of the micro FGM porous circular plate subjected to a nonlinear thermal load is extracted.

2.1.Material It is assumed that there is an FG circular plate with a fixed thickness amount, h, and radius R,

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which is composed from metal and ceramic. The combination changes from the top to the bottom surface, i.e. as can be seen in Fig. 1, the top surface (z=h/2) of the plate is metal-rich

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whilst the bottom surface (z= -h/2) is ceramic-rich.

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Fig 1. The schematic view of the micro-size FGM porous circular plate

An arbitrary material characteristic, P, is defined, representing Young’s modulus E, mass

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density, ρ, Poisson ratio ν, and thermal expansion, β of the plate and is assumed to vary along

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the thickness with respect to volume fraction of the constituent components, Vi, in FGM as follows [40]:

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P( z )  Pt Vc (z)  Pb Vm ( z)

(1)

where, subscripts b and t denote the bottom and top surfaces of the plate which are enriched with ceramic and metal respectively. The subscripts c and m address the last two materials. Since Vc and Vm are metal and ceramic volume fractions in FGM respectively, it is obvious that:

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Vc ( z )  Vm ( z )  1

(2)

Therefore, equation (1) is rewritten as follows: P( z )  (Pt  Pb ) Vc ( z )  Pb

(3)

The temperature-dependent properties of components are defined as follows [40]:

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Pi (T )  P0 (P1T1  1  P1T1  P2T 2  P3T3 )

(4)

where, i is replaced with t and b to introduce the top and bottom surfaces respectively. P-1, P0, P1, P2, and P3 are the material coefficients and T is the temperature in material points of the

summarized in table 1 [40].

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plate. For instance, the coefficients for popular silicon nitride-stainless steel FGM are

Table 1. The material coefficients for temperature-dependent properties of the Si3N4 and SUS304 [40]. material

P-1

Si3N4

0

SUS304

0

P2

P3

348.43e+9

-3.070e-4

2.160e-7

-8.946e-11

201.04e+9

3.079e-4

-6.534e-7

0

0

2370

0

0

0

SUS304

0

8166

0

0

0

Si3N4

0

0.24

0

0

0

SUS304

0

0.3262

-2.002e-4

3.797e-7

0

Si3N4

0

5.8723e-6

6.095e-4

0

0

SUS304

0

12.330e-6

8.086e-4

0

0

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ν

P1

Si3N4

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ρ

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E

P0

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property

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β

Moreover, it is assumed that there is a linear reduction proportional to the volume fraction of porosity, α, in material properties. Thereafter variation of material characteristic along thickness is represented as follows [41]:

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P( z, T )  (Pt (T )  Pb (T )) Vc (z)  Pb (T )  (

Pt (T )  Pb (T ) ) 2

(5)

Assuming the coordination origin in the center of the plate (as shown in Fig. 1), the volume fraction of the ceramic, Vc, would be defined in power low form as follows:

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1 z Vc (z)  (  ) n 2 h where, n introduces the power of the volume fraction.

(6)

Consequently, one would represent the Young’s modulus, mass density, Poisson ratio and

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thermal expansion of the FGM porous material as follows:

E (T )  Em (T ) 1 z E (z, T )  ( Ec (T )  Em (T ))(  ) n  Em (T )  ( c ) 2 h 2 1 2

z h

1 2

z h

 (z, T )  ( c (T )   m (T ))(  ) n   m (T )  (

2

c (T )  m (T )

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 (z, T )  ( c (T )  m (T ))(  ) n  m (T )  (

c (T )  m (T )

2

)

(8)

)

(9)

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 (T )  m (T ) 1 z  (z, T )  ( c (T )  m (T ))(  ) n  m (T )  ( c ) 2

h

(7)

(10)

2

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where, Ei(T), βi(T), ρi(T) and νi(T) are defined in equation (4).

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In this study, it is assumed that there is nonlinear temperature distribution in thickness as follows[42]:

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1 z T (z)  Tm  (Tc  Tm )(  ) 2 h

while in cases of γ=0 and 1, there are uniform and linear temperature distribution along thickness respectively; for γ between zero and one, this would lead to nonlinearity in temperature distribution.

(11)

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2.2.The modified couple stress theory The strain energy, U, in an isotropic linear elastic material is obtained using the modified couple stress theory as follows [2,12,43]: 1 ( :   m :  )dV 2 V

(12)

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U=

ε and σ are linear strain and Cauchy stress tensors respectively. χ and m are symmetric curvature strain and deviatoric part of the couple stress tensors. Neglecting the modified couple

tensors are defined as follows [2,12,43]:

  tr ( ) I  2 1

  u  (u )T  2

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1 2

    ()T 

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m  2l 2 

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stresses, the second term in the right-hand side of equation (12) is omitted. The last four

(13) (14) (15) (16)

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where, λ and μ are Lame’s constant, u and l are displacement vector and material length scale parameter respectively, and Λ is a rotation vector defined by:

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1   curl u 2

(17)

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3. Formulation

In this section, the governing equation of the micro FGM porous circular plate subjected to the nonlinear thermal loading based on classical and the first-order shear deformation theories are presented.

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3.1.Classical theory In three-dimensional classical plate theory, displacement components (ur , uθ and uz in cylindrical coordinate) of the circular plates would be written as follows [44]:

u   z

w(r , , z, t ) r

(18a)

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ur  u ( r ,  , z , t )  z

w(r , , z, t ) 

uz  w(r , , t )

(18b)

(18c)

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where, u and w are radial and transverse displacements, ∂w/∂r and ∂w/∂θ represent rotations of a transverse normal line about r and θ coordinates respectively. For an axisymmetric problem, the circumferential displacement, uθ, is omitted and ur and uz would be independent of circumferential axis. Hence, displacement field is rewriten as fallows [44]:

u (r , z, t )  0

(19a) (19b) (19c)

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uz (r , z, t )  w(r , t )

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w r

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ur ( r , z , t )  u ( r , t )  z

The strain components would be extracted in terms of transverse displacement in the middle

u 2w z 2 r r

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 rr 

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surface of the plate as follows [44]:

u r

  

z w r r

 rz   r   z  0 As aforementioned, the rotation vector, Λ, would be defined as below:

(20a) (20b)

(20c)

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er 1   2r r u

re   rv

ez  z w

(21)

Substituting equation (21) into equation (16) would lead to only one non-zero component of

1 4

 r  (

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the symmetric curvature tensor:  2 w 1 w  ) r 2 r r

(22)

The governing equation of the motion would be extracted using the Hamilton’s principle as

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follows [45]: t

  (U  T  H )  0 0

(23)

forces acting on the plate.

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where, U and T are strain and kinetic energy respectively. H is the work due to the external

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The strain energy of the circular plate in axisymmetric condition is obtained by substituting the non-zero strain and the symmetric curvature strain components into equation (12) which leads

h 2

 u 2w u z w  2 w 1 w   (  z )   (  )  m (  h  rr r r 2  r r r r r 2  r r ) dzdA 

(24)

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1 U=  2A

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to:

2

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The constitutive equation for an isotropic material by considering the in-plan stress condition is defined as follows [44] :

 rr D11 D12    D21 D22  r D31 D32

in which,

D13  rr D23  D33  r

(25a)

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D11 =D22 =D33 = +2

(25b)

D12 =D21 =D31 =D13 =D32 =D23 =

The Lame’s constant is represented in the form of engineering constant (i.e. Young’s modulus,



vE (1  v)(1  2v)



E 2(1  v)

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E, and Poisson’s ratio, ν,) as follows: (26a)

(26b)

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Substituting the components of the stress and deviatoric part of the couple stress tensors into equation (24) would lead to: U=

1  u u 2w 1 w  2 w 1 w  M ( )  N ( )  M  M   (   ) dA rr  rr  r 2 A  r 2 r 2 r r r 2 r r 

(27)

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where, A is the cross-sectional area of the plate. The force, Nrr and Nθθ , the moment, Mrr and

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Mθθ , and the couple moment, Ωrθ , are defined as follows: h 2

u u  2 w B12 w N rr     rr dz  ( A11  A12 )  ( B11 2  ) r r r r r h 2

h 2

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N      dz  ( A12 

h 2

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h 2

u u 1 w 2w  A11 )  ( B11  B12 2 ) r r r r r

M rr     rr zdz  ( B11 h  2

h 2

u u  2 w D w  B12 )  ( D11 2  12 ) r r r r r

M       zdz  ( B12 

(28a)

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

h 2

D w u u 2w  B11 )  ( 11  D12 2 ) r r r r r

(28b)

(28c)

(28d)

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r 

h 2

 

mr dz  A55l 2 (

h 2

1 w  2 w  ) r r r 2

(28e)

where the constants are defined as follows: h 2

 A11 , B11 , D11   ( 2 ) 1, z, z 2  dz h 1  v (z)

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

E (z)

2

h 2

 A12 , B12 , D12    ( 2 ) 1, z, z 2  dz h 1  v (z)

A55 

h 2

2

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

E (z)v(z)

E (z)

 ( 2(1  v(z)) )dz

h  2

h/2

I1 , I2 , I3    (z) 1, z1 , z 2  dz

(29b)

(29c)

(29d)

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h /2

(29a)

1 v w   u   z  ( )2  ( ) 2  ( ) 2  z dA  2V t t   t

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T=

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In general, the kinetic energy of the plate is presented as [45]: (30)

And the work due to the external forces on the circular plates is calculated as follows:

1 w N T ( )2 dA   FwdA  2A r A

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H= 

(31)

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where, F represents the out-of-plane load acting on the top surface of the plate and NT is radial force that is caused due to the functionally graded thermal behavior of the material and the prescribed mechanical boundary condition for the plate which is subjected to thermal load. NT would be obtained as follows [41]:

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h 2

NT    

h 2

E (z)  (z)(T  T0 ) dz 1  v(z)

(32)

where, T and T0 are temperature distribution along thickness of the plate and the reference

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temperature, respectively. Since thermal effects cause compressive stresses, a negative sign is literally assumed in equations.

The integral form of equations of motion obtained by Hamilton’s principal and the differential

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form with associated boundary conditions are rewritten as follows:

1   2u ( (rN rr )  N )  I1 2 r r t

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1  1  2w T w ( (rM rr  rr ))  ( (rN  M  ))  I1 2 r r r r r t

R

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   2u w ( (rM rr  rr )  M   r  I 2 2  rN T ) w  0 r t r 0  w  (rM rr  rr ) 0 r  0

 rNrr   w 0

0

(33b)

(33c)

(33d)

(33e)

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R

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R

(33a)

where, the first two equations are known as equations of motion and the three last ones are the

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associated boundary condition. Substituting equations (28a) - (28e) into equations (33a)-(33e) leads to the equations of motion and the associated boundary conditions in terms of displacement as follows:  B11 (

 3 w 1  2 w 1 w  2u 1 u u  2u   )  A (   )  I 11 1 r 3 r r 2 r 2 r r 2 r r r 2 t 2

(34a)

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1   3u  2u 1 u u  2u T w ( D11  A55l )( ) w  (rN )  ( B11 )( 3  2 2   )  I1 2 r r r r r r r r 2 t 2

(r ( D11 (

2 2

3 3 w 1  2 w 1 w 1  2 w 1 w w 2  w   )  A l (   2 )  NT )) w  0 55 3 2 2 3 2 r r r r r r r r r r r

(34b)

at r  0, R

(r ( D11

2w 1 w  2 w 1 w  w 2  D  A l (   ))) 0 12 55 r 2 r r r 2 r r r

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(34c)

at r  0, R

 2 w B12 w u u (r (( B11 2  )  ( A11  A12 )))δw  0       at     r  0, R r r r r r

(34d) (34e)

2 

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where, the Laplacian operator in polar coordinate is defined as follows: 2 1   r 2 r r

(35)

3.2.The first-order shear deformation theory

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In the first three-dimensional first-order shear deformation plate theory, the displacement

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components (ur , uθ and uz in cylindrical coordinates) of circular plates are written as follows [44]:

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ur (r , , z, t )  u(r , , t )  z(r , , t )

(36a) (36b)

uz (r , , z, t )  w(r, , t )

(36c)

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u (r , , z, t )  z(r , , t )

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where, u and w are radial and transverse deflections, Φ and Ψ represent rotations of a transverse normal line about r and θ coordinate respectively. For the axisymmetric problem, circumferential displacement of uθ is omitted and Φ and w would be independent of circumferential axis. Hence, displacement filed is rewritten as follows [44]: ur (r , z, t )  u(r, t )  z(r, t )

(37a)

ACCEPTED MANUSCRIPT

u (r , z, t )  0

(37b)

uz (r , z, t )  w(r , t )

(37c)

The linear strain components are written again as follows [44]:

u  z r r

  

z r r

1 u 2 r

z r

u r

(38a)

CR IP T

 rr 

 rz  (  )

AN US

 r   z  0

(38b)

(38c)

(38d)

From equation (21), the components of vector Λ are [44]: r   z  0

M

1 w   (  ) 2 r

(39a) (39b)

ED

Substituting equation (21) into equation (16) leads to only one non-zero component of the symmetric curvature tensor:

 2 w  w  )  (  )) 2 r r r

(40)

CE

PT

1 4

 r  ((

The strain energy of the circular plate in axisymmetric condition would be obtained by

AC

substituting the non-zero strain and the symmetric curvature strain components into equation (12) which leads to:

u  u z 1 u z    rr (  z )    (  )  2 rz ( (  ))   1 r r r r r 2 r r U  dzdA 2 2A h 1  w  1 w   2mr ( ( 2  )  (  )) 2  4 r r 4 r  h 2

(41)

Substituting the components of stress and deviatoric part of the couple stress tensors into

ACCEPTED MANUSCRIPT

equation (41) leads to:

u u   w   N rr  N  M rr  N  N rz (   )   1 r r r r r U   dA 2 2 A 1  w  1 w    2 r (( r 2  r )  r (  r )) 

(42)

moment are defined as follows:

h 2

h 2

12

h 2

h 2

 

h 2

12

h 2

h 2



rz

dz  ks A55 (

h 2

CE 

(43b)

u u    B12 )  ( A11  B12 ) r r r r

   zdz  ( B 

(43c)

u u    B11 )  ( D11  D12 ) r r r r

(43d)

w  ) r

(43e)

h 2

1 2   2 w 1 w m dz  l A (( h r 2 55 r  r 2 )  r (  r ))

AC

r 

zdz  ( B11

h 2

M  

M rz 

rr

(43a)

u u    A11 )  ( B12  B11 ) r r r r

   dz  ( A 

M rr 

u u    A12 )  ( B11  B12 ) r r r r

PT

N 

dz  ( A11

AN US



rr

M



N rr 

ED

h 2

CR IP T

where, A is the cross-sectional area of the plate. The forces, the moments and the couple



2

(43f)

where, ks=π2/12 is known as the shear correction factor and the constants are defined in equations (29a)-(29d). Substituting the displacement components into equation (30) results following relation:

ACCEPTED MANUSCRIPT

T

1  u 2 u   w  I1 ( )  2 I 2 ( )( )  I 3 ( )2  I1 ( ) 2 dA   2 A  r r r r r 

(44)

where, the constants are defined in equation (29e). Substituting equations (42), (44), and (31) into equation (23) gives the equations of motion and

CR IP T

associated boundary conditions as follows: 1   2u  2 rN  N  I  I  r   1 2 2 2 r r t t

(45a)

(45b)

r 1  M 1 2  2u  2 ( r  )   ( rM )   N  I  I r r rz 2 3 2r r 2 2r r r r t 2 t 2

(45c)

( Nr ) u  0

(45d)

rr  ( w)  0 2r r r )( )  0 2

at r  0, R

(44e)

(45f)

(45g)

PT

( M rr 

at r  0, R

at r  0, R

M

 1  w (rr )  r  N rz  N r ) w  0 2r r 2r r

ED

(

at r  0, R

AN US

1 2 1  1  1  2w T w ( r  )  (  )  ( rN )  (  N r )  F  I r r rz 1 2r r 2 2r r 2r r r r r t 2

CE

Substituting equations (43a) - (43f) into equations (45a) - (45g) leads to the governing equations in terms of displacement components:  2u 1 u u  2 1    2u  2   )  B (   )  I  I 11 1 2 r 2 r r r 2 r 2 r r r 2 t 2 t 2

AC

A11 (

(46a)

ACCEPTED MANUSCRIPT

CR IP T

 l 4  3 2  2 1    4 w 2  3 w 1  2 w 1 w   A55 4 ( r 3  r r 2  r 2 r  r 3  r 4  r r 3  r 2 r 2  r 3 r )     2 w 1 w   N T w 2w    k A (    )   I  s 55 r 2 r r r r  1 t 2 r r   2  T  w 1 w    N ( r 2  r r )  F   

AN US

   2u 1 u u l 2  3 w 1  2 w 1 w w  B11 ( r 2  r r  r 2 )  A55 4 ( r 2  r r 2  r 2 r )  ks A55 ( r   )    l 2  2 1      ( D11  A55 4 )( r 2  r r  r 2 )   2u  2  I 2 2  I3 2 t t u    ( A11 (r r )  A12u )  ( B11 (r r )  B12)   u  0

(46c)

at r  0, R

(46d)

at r  0, R

M

 l 4  2  3 w 1  1  2 w  1 w   A55 4 ( r 2  r 3  r r  r r 2  r 2  r 2 r )    w  0   k A ( w  )  N T ( w )   s 55 r  r

( A55

l 4 1   2 w 1 w w ((  2 )  (  ))) ( )  0 4 r r r r r r

( D11

Φ Φ l 2 Φ  2 w 1 w  D12  A55 ((  2 )  (Φ  ))) Φ  0      at   r  0, R r r 4 r r r r

at r  0, R

(46e)

(46f)

(46g)

CE

PT

ED

(46b)

4. Solution method

AC

GDQ method is used in order to solve the governing equations with associated boundary conditions for buckling and free vibration of FGM circular micro plates. Bellman et al. presented GDQ method in early 1970s [46,47] . GDQ method resolution is affected by the accuracy of weight coefficients which are influenced themselves by the selection of grid points (see [38,39] for more details). Discretization rule of the method enables the

ACCEPTED MANUSCRIPT

approximation of m-th derivatives of displacement functions, w and Φ, with respect to r that is defined as linear sum of function values [48]:

 m u  r  , w  r  ,   r  m r  rp

m

uk , w k ,  k 

k 1

(47)

CR IP T

r

n

 Cik 

where:

AN US

uk  u  rk , t  wk  w  rk , t 

M

 k    rk , t 

(48a)

(48b)

(48c)

ED

n represents the total number of nodes distributed along r-axis and Cik(m) are weighting coefficients whose recursive formula can be found as follows [45]:

PT

ik

ik

AC

CE

Cik

1

M  ri      ri  rk  M  rk   n   Cik 1  k  1, i  k 

(49)

where, k=1,2,3,…,n. In order to improve mesh points distribution, the cosine pattern of Chebyshev-Gauss-Lobatto technique is used to generate the GDQ points as follows:

ACCEPTED MANUSCRIPT

1 (i 1) ri  (1  cos(  )) 2 (n  1)

(50)

where, i=1,2,3,…,n. The analysis indicates that above distribution would accelerate the

M r  

CR IP T

convergence of the solution. M(r) is defined as follows [45]: n

 r  r  i

k

k 1, k  i

(51)

The associated weighting coefficient for r-th derivative of displacement functions is written as

i  k and 2  m  n  1

i  k and 1  m  n  1

(52)

M

Cik

m

   m 1 1 Cik  m 1  Cik   m Cik   ri  rk      n   Cik  m     k 1,i  k

AN US

follows [45]:

ED

Applying GDQ method to the governing equations of the micro FGM porous circular plate extracted with the assumption of either classical theory, equations (34a) and (34b), or the first-

PT

order shear deformation theory, equations (46a) - (46c), would lead to the equations (53a)-(53b)

CE

and equations (54a)-(54c) respectively:

AC

ni  2 ni 3 1 ni  2 1 ni 1   4 2 ( D  ( A l ))(  C w  C w  C w  C wk )     11 55 ik k ik k k 2  ik 3  ik r r r k  1 k  1 k  1 k  1 i i i   ni ni   1  2 1  ( N T )( Cik wk   Cik wk )    2 I1wi ri k 1 k 1     ni ni ni  ( B11 )( Cik3uk  2 Cik 2uk  1  Cik1uk  u2i )  ri k 1 ri k 1 k 1  

(53a)

ACCEPTED MANUSCRIPT

ni  1 ni  2 1 ni 1   3 ( D11 )( Cik wk   Cik wk  2  Cik wk )  ri k 1 ri k 1 k 1    2I u 1 i ni ni   ui 1  2  2  ( A11 )( Cik uk   Cik uk  2 )  ri k 1 ri   k 1

(54a)

ED

M

AN US

  ni 3 2 ni  2 1 ni 1   C   C   Cik  k     ik k r 2    ik k ri k 1 i k 1   k 1   ni ni 2      l 2 ( A55 )   3i   Cik 4 wk   Cik3 wk   4  ri ri k 1  k 1     1 ni   ni    2  Cik 2 wk  13  Cik1 wk    r i k 1  r i k 1     2 I1wi   ni ni i  1 ni 1   2 1 Cik wk   Cik wk   Cik  k  2 )    (ks A55 )( ri k 1 ri k 1 k 1   ni ni     1  2 1 T   ( N )   Cik wk   Cik wk   ri k 1  k 1     F   

CR IP T

ni   u 1 ni 1  2 ( A )( C u  Cik uk  2i )    11  ik k ri k 1 ri k 1     2 (I u  I  ) 1 i 2 i n n i i  i  1  2 1  ( B11 )( Cik  k   Cik  k  2 )  ri k 1 ri  k 1 

(53b)

AC

CE

PT

ni   l2 1 ni  2 1 ni 1  3 ( A )(  C w  C w  C wk )    55  ik k ik k 2  ik 4 ri k 1 ri k 1 k 1   ni ni ni 2    l 1 1  2 1  (ks A55  A55 )( Cik wk   i )  ( D11 )( Cik  k   Cik  k  2i )    2 I 2 i 4 k 1 ri k 1 ri  k 1    ni ni  ( B11 )( Cik 2uk  1  Cik1uk  u2i )  ri k 1 ri k 1  

(54b)

(54c)

Now, governing equations of motion in matrix form are written as follows:

M

2d   Ke  N T K g  d  F  0 t 2

(55)

where, M, Ke, and Kg are mass matrix, stiffness matrix and geometric stiffness matrix, respectively. The matrixes dimensions are 2N×2N and 3N×3N in classical and the first-order

ACCEPTED MANUSCRIPT

shear deformation theories respectively, where, the unknown displacement vector, d, in last theories are defined respectively as follows:

u 

T

i



, wi 

T



(56a)

d  ui  , wi  , i  T

T

T



(56b)

CR IP T

d

Equation (55) is used for buckling and free vibration analysis of the FGM circular micro plate as it has been summarized.

The solution of the last equation leads to the deflection of the plate. Static buckling

AN US

-

Neglecting the inertia term and transverse load vector, q, and setting Nrr=-Pcr (in which Pcr represents the critical buckling load), equation (55) reduces to a static buckling eigen-value problem of the FGM circular micro plate as follows: e

 Pcr K g  d  0

M

K

(57)

Free vibration

PT

-

ED

in which the critical buckling load of the micro plate is obtained.

Substituting dynamic displacement vector, d, in the form of d*=deiω and dropping the radial

follows: 2

 Ke  d *  0

AC

 M

CE

load Nrr and transverse load F, equation (55) results to an eigenvalue system of equations as

(58)

5. Numerical Results In this section, the results of numerical solutions for the free vibration and temperature buckling of a FG circular micro plate are presented. The results are compared with available

ACCEPTED MANUSCRIPT

results in the literature. In order to elaborate the impact of size dependency (h/l), temperaturechange (ΔT), FG power index (n) and porosity factor (α) parameters on mechanical behaviors such as the first and the second non-dimensional natural frequencies (ω1 and ω2) and critical temperature buckling (ΔTcr), several figures and tables are presented.

i2 

I10 4 2 R i D110

at T  0,   0

In which ̅ represents natural frequencies and:

AN US

D110

Ec h3  12 1  vc 2 

CR IP T

Non-dimensional natural frequencies, ωi, are defined as follows:

I10  c h

(59)

(60)

(61)

5.1. Convergency and Validation

M

where, the subscript c addresses the ceramic material.

ED

In order to evaluate the convergency of the results with respect to grid-point-numbers, the non-

PT

dimensional natural frequencies of circular plates obtained by classical plate theory (CPT) and the first-order shear deformation theory (FSDT) are presented in Fig. 2. The convergency’s

CE

studies are conducted considering different boundary conditions. It can be found that the results are independent of the grid-point-numbers.

AC

It is noteworthy that the mixed boundary conditions (both displacements and stresses) in the formulation would generally lead to numerical instabilities in solution procedures due to higher condition numbers in the stiffness matrix. On the contrary this never happens with boundary conditions only on displacements.

ACCEPTED MANUSCRIPT

M

AN US

CR IP T

a)

AC

CE

PT

ED

b)

ACCEPTED MANUSCRIPT

M

AN US

CR IP T

c)

AC

CE

PT

ED

d)

ACCEPTED MANUSCRIPT

Fig. 2. The convergency of the first non-dimensional natural frequency results in respect to grid-point-numbers utilized in GDQ method: a) hinged circular plates obtained by classical plate theory (CPT), b) hinged circular plates obtained by the first-order shear deformation theory (FSDT), c) clamped circular plates obtained by classical plate theory (CPT), d) clamped circular plates obtained by the first-order shear deformation theory (FSDT), with α=0.1, R/h=20, γ=0.9, n=0.1, ΔT=20 and h/l=1.

CR IP T

In order to verify numerical results, comparisons between natural frequencies obtained with GDQM and those achieved from literature [44,49,50] are presented based on both classical and first-order shear deformation theories. Moreover, the critical buckling load obtained with GDQM and those results obtained in the literature [18] based on classical plate theory are

AN US

compared.

In table 2, the first five non-dimensional natural frequencies of free vibration of circular plates obtained in this study with those presented in literature [44,49,50] based on classical theory are compared. The comparison between results are presented for hinged, clamped and free

M

boundary conditions. Moreover, the convergency of the above results odtained by GDQM for

ED

the first and second non-dimensional natural frequencies of the plates results are presented in Fig. 3.

PT

The analysis indicates that there is a very good agreement between results. In table 3, the

CE

critical buckling load of clamped circular plates with different R/h are presented and compared with results obtained in [18] . Obviously, there is a very good agreement between results. In

AC

table 4, the first five non-dimensional natural frequencies of free vibration of circular plates subjected to different boundary conditions obtained in this study with those presented in the literature [51,52] based on the first-order shear deformation theory are presented. The analysis indicates that numerical results are consisted of those results available in the literature. The comparison between results are prepared for different boundary conditions, hinged, clamped and free. The analysis indicates that there is very good agreement between results.

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Table 2. The comparison of the first five non-dimensional natural frequencies, ω, of free vibration of circular plates obtained in this study with those presented in literature based classical theory. (R/h=20).

Clamped

Free

Present study (GDQM)

1 2 3 4 5 1 2 3 4 5 1

4.9351 29.7199 74.1559 138.3179 222.2145 10.2158 39.7711 89.1040 158.1839 247.0059 9.0031

2

38.4428

3 4

87.7508 156.8164

5

245.6357

Leissa (Exact) [44] 4.997 29.76 74.2 138.34 --10.2158 39.771 89.104 158.183 247.005 ---

CE

PT

ED

M

a)

AC

Yalcin (DTM) [49] 4.9351 29.72 74.1561 138.3181 222.215 10.2158 39.7711 89.1041 158.1842 247.0064 9.0031

Wang (GGDQ) [50] 4.935 29.72 74.156 138.318 222.215 10.216 39.771 89.104 158.184 247.006 9.003

---

38.4432

38.443

-----

87.7052 156.8183

87.75 156.818

---

245.6335

245.634

CR IP T

Hinged

Mode

AN US

B.C

ACCEPTED MANUSCRIPT

AN US

CR IP T

b)

AC

CE

PT

ED

M

c)

ACCEPTED MANUSCRIPT

AN US

CR IP T

d)

AC

CE

PT

ED

M

Fig. 3. The convergency of results in respect to grid-point-numbers utilized in GDQ method: a) first non-dimensional natural frequency of hinged circular plates obtained by classical plate theory (CPT), b) second non-dimensional natural frequency of hinged circular plates obtained by classical plate theory (CPT), c) first non-dimensional natural frequency of clamped circular plates obtained by classical plate theory (CPT), d) second non-dimensional natural frequency of clamped circular plates obtained by classical plate theory (CPT), with R/h=20.

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Table 3. The comparison of the critical buckling loads (N/m) of FG plate under radial compression for clamped edges obtained in this study with those presented in literature based classical theory. Present study (GDQM)

Najafizade et al [18]

0.01

510 885

510 842

0.02

4 087 087

4 086 739

0.03

13 793 919

0.04

32 696 698

0.05

63 860 738

0.06

11 0351 356

0.07

175 233 867

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0.08 0.09

ED PT CE

13 792 747 32 693 919 63 855 311

110 341 978 175 218 974

261 573 586

261 551 355

372 435 829

372 404 175

510 885 911

510 842 490

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0.1

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h/R

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Table 4. The comparison of the first five non-dimensional natural frequencies, ω, of free vibration of circular plates obtained in this study with those presented in literature based on the first-order shear deformation theory. Present Liew et

Irie et

study

al [51]

Mode

al [52]

(GDQM)

4.9351

4.935

al [52]

4.6963

4.6963

4.696

2

29.720

29.720

29.720

23.254

23.254

23.254

3

74.155

74.155

74.156

46.774

46.775

46.775

4

138.31

138.31

138.318

71.602

71.603

71.603

5

222.21

222.21

----

96.608

96.609

----

1

10.216

10.216

10.216

8.8068

8.8068

8.807

2

39.771

39.771

39.771

27.253

27.253

27.253

3

89.102

89.102

89.102

49.420

49.420

49.420

4

158.18

158.18

158.184

73.054

73.054

73.054

5

246.99

246.99

----

97.198

97.198

----

1

9.0029

9.0031

9.003

8.2674

8.2674

8.267

2

38.443

38.443

38.443

28.605

28.605

28.605

3

87.747

87.749

87.750

52.584

52.584

52.584

156.80

156.81

156.818

76.935

76.936

76.936

245.61

245.62

----

99.543

99.545

----

4

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CE

5

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4.9353

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Free

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1

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Clamped

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(GDQM) R/h=1000

Hinged

Liew et study

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B.C.

Present

5.2.Parametric results In this section, contributions of physical and geometrical parameters on free vibration of the micro FGM porous circular plate subjected to a nonlinear thermal load are evaluated. As aforementioned, classical and the first-order shear deformation theories in conjunction with modified couple stress theory are used to extract the governing equations.

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It has been known that while the application of the classical theory is efficient in analysis of thin-walled plates, where shear deformations are negligible, in the analysis of relatively thickwalled plates, the application of the first-order shear deformation theory would result in more accuracy. The fact is that, the simplified assumption about deformations in classical theory

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would lead to the prediction of a stiffer structure. As can be seen in Table 5, the first natural frequencies of the micro FGM porous circular plate which are obtained based on classical theory are bigger than those obtained based on the first-order shear deformation theory, in different size dependencies and temperature changes. Differences between the results obtained

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for the first natural frequency are increased as temperature-change and size dependency increase.

Table 5. The comparison of results obtained by classical theory (CPT) and the first-order shear

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deformation theory (FSDT) for the first non-dimensional natural frequency, ω1, of free vibration of hinged circular plates with α=0.1, R/h=20, γ=0.9, n=2 and different ΔT in terms of h/l.

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ω1 (ΔT=20)

h/l

0.5

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4 5

CPT

FSDT

13.582

11.252

13.735

11.271

7.5398

6.9441

7.8106

6.6422

4.9942

4.0814

5.3936

3.5392

4.1189

2.9550

4.5949

2.1447

4.0010

2.7896

4.4895

1.9101

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1 2

FSDT

PT

CPT

ω1 (ΔT=40)

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In table 6 and 7, the impact of FG power index and porosity on prediction of natural frequencies based on last two theories are seen. Unlike size dependency, FG power index or porosity has no effect on the difference between the results obtained with classical and the

constant in terms of last two parameters.

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first-order shear deformation theories. The fact is that the differences between results are

As can be seen above, the trend of numerical results in terms of different parameters are the same for both of the governing equations, therefore, the representation of the results are confined to those obtained from the solution of governing equations derived based on classical

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theory in conjunction with modified couple stress theory.

Table 6. FG power index effects on the first and the second non-dimensional natural frequencies, ω1 and ω2, of free vibration of clamped circular plates with α=0.2, γ=0.9, ΔT=60,

CPT

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5 6

FSDT

14.057

57.838

54.575

13.353

12.206

50.652

47.585

12.584

11.425

47.639

44.643

12.158

10.989

45.965

42.005

11.885

10.709

44.893

41.954

11.695

10.512

44.144

41.219

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3 4

CPT

15.190

2

ω2

FSDT

PT

1

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ω1 n

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R/h=20 and h/l=1.

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Table 7. FG power index effects on the first two non-dimensional natural frequencies, ω, of free vibration of hinged circular plates with ΔT=30, γ=0.9, R/h=20, h/l=1 and different α. ω1 (α=0)

ω1 (α=0.1) FSDT

CPT

FSDT

8.6549

7.8022

7.6759

6.7946

7.2634

6.3652

7.0339

6.1243

6.8869

5.9688 5.8596

1

8.6507

7.6965

2

7.7266

6.7304

3

7.3349

6.3149

4

7.1164

6.0808

5

6.9760

5.9296

6

6.8779

5.8233

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CPT

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n

6.7843

In Fig. 4, variation of the first non-dimensional natural frequency, ω1, of free vibration of circular

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plates with α=0.2, R/h=20, γ=0.9, h/l=1 in terms of FG power indexes and ΔT for clamped and free boundary conditions are presented. The results reveal that the first non-dimensional natural

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frequency of the both clamped and free plates are decreased with the increase of FG power index and temperature change. There is an asymptotic limit for the first natural frequency in terms of FG

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power index. Moreover, the first non-dimensional natural frequency of the clamped one is more

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than those obtained for the free one.

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Fig. 4. Variation of the first non-dimensional natural frequency, ω1, of free vibration of circular plates with α=0.2, R/h=20, γ=0.9, h/l=1 in terms of FG power index and ΔT: a) Clamped b) Free boundary conditions.

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In Fig. 5, variation of the second non-dimensional natural frequency, ω2, of free vibration of

circular plates with α=0.2, R/h=20, γ=0.9, h/l=1 in terms of the FG power index and ΔT, for clamped, hinged, free boundary conditions are presented. Reduction of the second natural

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frequency in terms of FG power index and temperature-change are the same as the variation of the first natural frequency in terms of those parameters. While the second non-dimensional natural frequency of the free plate is less than the clamped one, the second non-dimensional natural frequency of the hinged plate is less than other ones. Like variation of the first natural frequency,

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an asymptotic limit for the second natural frequency with increasing the FG power index is seen. In Fig 6, variation of the first non-dimensional natural frequency, ω1, of free vibration of

hinged circular plates with R/h=20, γ=0.9 and h/l=1 in terms of the FG power index, n, and ΔT for different porosity factors are compared. As it can be observed, increasing porosity leads to

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increase of the first natural frequency. This is because that mass reduction is more than

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stiffness reduction in plates due to the existence of void in the material.

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Fig. 5. Variation of the second non-dimensional natural frequency, ω2, of free vibration of

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circular plates with α=0.2, R/h=20, γ=0.9, h/l=1 in terms of FG power index and ΔT: a)

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Clamped b) Hinged c) Free boundary conditions.

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b)

Fig. 6. Variation of the first non-dimensional natural frequency, ω1, of free vibration of hinged circular plates with R/h=20, γ=0.9 and h/l=1 in terms of the FG power index, n, and ΔT for different porosity factors, α,: a) α=0, b) α=0.2.

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While the effect of porosity on the first natural frequency of the hinged plate is not negligible, it can be found from Fig. 7 that there is no porosity effect on the first and the second natural frequencies of the clamped and free boundary conditions. In Fig. 8, variation of the first non-

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dimensional natural frequency, ω1, in free vibration of hinged circular plates with R/h=20, γ=0.9 and h/l=1 in terms of FG power index, n, and porosity factor, α, in different temperature changes are present. The results illustrate that the increase of the temperature-change would

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lead to the increase of porosity-dependency of natural frequencies.

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Fig. 7. The variation of the first and the second non-dimensional natural frequencies, ω1 and

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ω2, in free vibration of hinged circular plates with R/h=20, h/l=1, ΔT=90 and γ=0.9 in terms of the FG power index, n, and porosity factor, α,: a) ω1 for clamped boundary condition, b) ω2 for

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clamped boundary condition, c) ω1 for free boundary condition, d) ω2 for free boundary condition.

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d)

Fig. 8. The variation of the first non-dimensional natural frequency, ω1, in free vibration of hinged circular plates with R/h=20, γ=0.9 and h/l=1 in terms of the FG power index, n, and porosity factor, α, in different temperature changes: a) ΔT=0, b) ΔT=30, c) ΔT=60, d) ΔT=90.

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In Fig. 9, variation of the first and the second non-dimensional natural frequencies, ω1 and ω2, in free vibration of circular plates with R/h=20, γ=0.9, n=2 and ΔT=40 in terms of size dependency, h/l, and porosity factor, α, for clamped and free boundary conditions are presented. The results show that the increase of size dependency in materials would lead the

second natural frequencies in terms of size dependency.

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decrease of natural frequencies of plates. There are asymptotic values for the first and the

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Fig. 9. Variation of the first and the second non-dimensional natural frequencies, ω1 and ω2, in free vibration of circular plates with R/h=20, n=2, ΔT=40 and γ=0.9 in terms of size dependency, h/l, and porosity factor, α,: a) ω1 for clamped boundary condition, b) ω2 for clamped boundary condition, c) ω1 for free boundary condition, d) ω2 for free boundary condition.

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Variation of the second non-dimensional natural frequency, ω2, in free vibration of hinged

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circular plates in terms of size dependency, h/l, and porosity factor are summarized in Fig. 10. The pattern of these variations is the same as what has been elaborated about the first non-

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dimensional natural frequencies in terms of different parameters.

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b)

Fig. 10. Variation of the second non-dimensional natural frequency, ω2, of free vibration of hinged circular plates with R/h=20 and γ=0.9: a) with n=1, ΔT=40 in terms of size dependency, h/l, and porosity factor, α,: b) with h/l=1, ΔT=30 in terms of FG power index, n, and porosity factor, α.

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In Fig. 119, variation of the first non-dimensional natural frequency, ω1, in free vibration of hinged circular plates with R/h=20 and γ=0.9 in terms of size dependency, and h/l, for different porosities and temperature changes are presented. Phenomenon of critical condition of the plate is illustrated in Fig. 11. This corresponds to the conditions of a plate at which the natural

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frequency curve has suddenly experienced a fall. At this semi-inter-section in this study, an unstable phenomenon possibly appears which is known as thermal buckling. The fact is that the nonuniform radial deformation of the plate due to the thermal loads leads to the creation of a reaction force on the boundaries. In critical conditions, the reaction force would lead to

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thermal buckling of the plates. As it is seen in Fig. 11, the thermal buckling depends on porosity factor, size dependency, FG power index and temperature change. In practice, the critical temperature-change for a plate with known geometrical and physical parameters would

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be definable.

In order to elaborate that, variation of critical temperature change, ΔTcr, of circular plates with

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R/h=20, γ=0.9 and n=1.2 in terms of size dependency, h/l, and porosity factor, α, for clamped and hinged boundary conditions are presented in Fig. 12. The analysis indicates that the critical

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temperature change, ΔTcr, decreases in terms of size dependency and increases in terms of the

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porosity. The latter is due to variation of flexural stiffness in plates. Moreover, critical temperature changes of clamped plates are more than those of hinged ones in the same

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geometrical and physical conditions.

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Fig. 11. The variation of the first non-dimensional natural frequency, ω1, in free vibration of hinged circular plates with R/h=20 and γ=0.9 in terms of size dependency, h/l, : a) n=0 and ΔT=20, b) n=0 and ΔT=40, c) n=1 and ΔT=20, d) n=1 and ΔT=40, e) n=2 and ΔT=20, f) n=2 and ΔT=40.

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Fig. 12. Variation of critical temperature buckling, ΔTcr, of circular plates with R/h=20, γ=0.9 and n=1.2 in terms of size dependency, h/l, and porosity factor, α, for different boundary conditions: a) clamped, b) hinged.

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6. Conclusion In this study, free vibration and thermal buckling of the micro FGM porous circular plates are numerically investigated. Classical and the first-order shear deformation theories in conjunction with the modified couple stress theory are used to derive the governing equations

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of motion. The impact of size dependency, temperature change, FG power index and porosity factor on the first and the second non-dimensional natural frequencies and critical temperaturechange are numerically presented. The results are in good agreement with those available in the

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literature. References

Ansari R, Gholami R, Faghih Shojaei M, Mohammadi V, Sahmani S. Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. Eur J Mech - A/Solids 2015;49:251–67. doi:10.1016/j.euromechsol.2014.07.014.

[2]

Reddy JN, Berry J. Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress. Compos Struct 2012;94:3664–8. doi:10.1016/j.compstruct.2012.04.019.

[3]

Kiani Y, Eslami MR. An exact solution for thermal buckling of annular FGM plates on an elastic medium. Compos Part Ba Eng 2013;45:101–10. doi:10.1016/j.compositesb.2012.09.034.

[4]

Hosseini-Hashemi S, Rokni Damavandi Taher H, Akhavan H. Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations. Compos Struct 2010;92:1734–43. doi:http://dx.doi.org/10.1016/j.compstruct.2009.12.016.

[5]

Kiani Y, Eslami MR. Thermal Postbuckling of Imperfect Circular Functionally Graded Material Plates: Examination of Voigt, Mori–Tanaka, and Self-Consistent Schemes. J Press Vessel Technol 2014;137:021201. doi:10.1115/1.4026993.

AC

CE

PT

ED

M

[1]

[6]

Li S-R, Wang X, Batra RC. Correspondence Relations Between Deflection, Buckling Load, and Frequencies of Thin Functionally Graded Material Plates and Those of Corresponding Homogeneous Plates. J Appl Mech 2015;82:111006. doi:10.1115/1.4031186.

[7]

Kim J, Reddy JN. Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory. Compos Struct 2013;103:86–98. doi:10.1016/j.compstruct.2013.03.007.

[8]

Ansari R, Faghih Shojaei M, Mohammadi V, Gholami R, Darabi MA. Nonlinear vibrations of functionally graded Mindlin microplates based on the modified couple stress

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theory. Compos Struct 2014;114:124–34. doi:10.1016/j.compstruct.2014.04.013. [9]

Cheng Z-Q, Batra RC. Three-dimensional thermoelastic deformations of a functionally graded elliptic plate. Compos Part B Eng 2000;31:97–106. doi:10.1016/S13598368(99)00069-4.

[10] Ghadiri M, Mahinzare M, Shafiei N, Ghorbani K. On size-dependent thermal buckling and free vibration of circular FG Microplates in thermal environments. Microsyst Technol 2017. doi:10.1007/s00542-017-3308-x.

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[11] Jomehzadaeh E, Noori HR, Saidi AR. The size-dependent vibration analysis of microplates based on a modified couple stress theory. Phys E Low-Dimensional Syst Nanostructures 2011;43:877–83. doi:10.1016/j.physe.2010.11.005. [12] Ke LL, Yang J, Kitipornchai S, Bradford MA. Bending, buckling and vibration of sizedependent functionally graded annular microplates. Compos Struct 2012;94:3250–7. doi:10.1016/j.compstruct.2012.04.037.

AN US

[13] Nateghi A, Salamat-talab M. Thermal effect on size dependent behavior of functionally graded microbeams based on modified couple stress theory. Compos Struct 2013;96:97– 110. doi:http://dx.doi.org/10.1016/j.compstruct.2012.08.048. [14] Wang Y-G, Lin W-H, Liu N. Large amplitude free vibration of size-dependent circular microplates based on the modified couple stress theory. Int J Mech Sci 2013;71:51–7. doi:10.1016/j.ijmecsci.2013.03.008.

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[15] Wang Y-G, Lin W-H, Zhou C-L. Nonlinear bending of size-dependent circular microplates based on the modified couple stress theory. Arch Appl Mech 2014;84:391– 400. doi:10.1007/s00419-013-0807-9.

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[16] Mohammadi K, Mahinzare M, Rajabpour A, Ghadiri M. Comparison of modeling a conical nanotube resting on the Winkler elastic foundation based on the modified couple stress theory and molecular dynamics simulation. Eur Phys J Plus 2017;132:115. doi:10.1140/epjp/i2017-11395-x.

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[17] Najafizadeh MM, Heydari HR. Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory. Eur J Mech - A/Solids 2004;23:1085–100. doi:10.1016/j.euromechsol.2004.08.004.

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[18] Najafizadeh MM, Heydari HR. An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression. Int J Mech Sci 2008;50:603–12. doi:10.1016/j.ijmecsci.2007.07.010.

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[19] Tornabene F, Fantuzzi N, Bacciocchi M. Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories. Compos Part B Eng 2014;67:490–509. doi:10.1016/j.compositesb.2014.08.012. [20] Mohammad-Abadi M, Daneshmehr AR. Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions. Int J Eng Sci 2014;74:1–14. doi:10.1016/j.ijengsci.2013.08.010. [21] Salamat-talab M, Nateghi A, Torabi J. Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. Int J Mech Sci 2012;57:63–73.

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[22] Lanhe W. Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos Struct 2004;64:211–8. doi:10.1016/j.compstruct.2003.08.004. [23] Dung D Van, Hoa LK. Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment. Compos Paart B Eng 2015;69:378–88. doi:10.1016/j.compositesb.2014.10.018.

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[24] Ma LS, Wang TJ. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings. Int J Solids Struct 2003;40:3311–30. doi:10.1016/S0020-7683(03)00118-5. [25] Tung H Van. Thermal and thermomechanical postbuckling of FGM sandwich plates resting on elastic foundations with tangential edge constraints and temperature dependent properties. Compos Struct 2015. doi:10.1016/j.compstruct.2015.06.043.

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[26] Duc ND, Tuan ND, Tran P, Dao NT, Dat NT. Nonlinear dynamic analysis of Sigmoid functionally graded circular cylindrical shells on elastic foundations using the third order shear deformation theory in thermal environments. Int J Mech Sci 2015;101-102:338–48. doi:10.1016/j.ijmecsci.2015.08.018. [27] Gupta a. K, Kumar L. Thermal effect on vibration of non-homogenous visco-elastic rectangular plate of linearly varying thickness. Meccanica 2007;43:47–54. doi:10.1007/s11012-007-9093-3. [28] Gupta AK, Tripti Johri, Vats RP. Study of thermal gradient effect on vibrations of a nonhomogeneous orthotropic rectangular plate having bi-direction linearly thickness variations. Meccanica 2010;45:393–400. doi:10.1007/s11012-009-9258-3.

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[29] Hu Y, Wang T. Nonlinear resonance of the rotating circular plate under static loads in magnetic field. Chinese J Mech Eng 2015;28:1277–84. doi:10.3901/CJME.2015.0720.097. [30] Dai H-L, Dai T, Yang L. Free vibration of a FGPM circular plate placed in a uniform magnetic field. Meccanica 2013;48:2339–47. doi:10.1007/s11012-013-9752-5.

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[31] Ghorbanpour Arani a., Khoddami Maraghi Z, Mozdianfard MR, Shajari a. R. Thermopiezo-magneto-mechanical stresses analysis of FGPM hollow rotating thin disk. Int J Mech Mater Des 2011;6:341–9. doi:10.1007/s10999-010-9141-3.

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[32] Ebrahimi F, Mokhtari M. Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method. J Brazilian Soc Mech Sci Eng 2014. doi:10.1007/s40430-014-0255-7.

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[33] Jabbari M, Joubaneh EF, Khorshidvand AR, Eslami MR. Buckling analysis of porous circular plate with piezoelectric actuator layers under uniform radial compression. Int J Mech Sci 2013;70:50–6. doi:http://dx.doi.org/10.1016/j.ijmecsci.2013.01.031. [34] Jabbari M, Mojahedin A, Joubaneh EF. Thermal Buckling Analysis of Circular Plates Made of Piezoelectric and Saturated Porous Functionally Graded Material Layers. J Eng Mech 2014:04014148. doi:10.1061/(ASCE)EM.1943-7889.0000872. [35] Wattanasakulpong N, Ungbhakorn V. Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerosp Sci Technol 2014;32:111– 20. doi:10.1016/j.ast.2013.12.002.

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[36] Wattanasakulpong N, Chaikittiratana A. Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method. Meccanica 2015;50:1331–42. doi:10.1007/s11012-014-0094-8. [37] Behravan Rad A, Shariyat M. Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations. Compos Struct 2015;125:558–74. doi:http://dx.doi.org/10.1016/j.compstruct.2015.02.049.

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[38] Tornabene F, Fantuzzi N, Ubertini F, Viola E. Strong Formulation Finite Element Method Based on Differential Quadrature : A Survey Strong Formulation Finite Element Method Based on Differential Quadrature : A Survey 2015. doi:10.1115/1.4028859. [39] Tornabene F, Fantuzzi N, Bacciocchi M. The strong formulation finite element method: stability and accuracy. Frat Ed Integrità Strutt 2014:251. [40] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stress 1998;21:593–626.

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[41] Ebrahimi F, Ghasemi F, Salari E. Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities. Meccanica 2015. doi:10.1007/s11012-015-0208-y. [42] Fazzolari FA, Carrera E. Thermal stability of FGM sandwich plates under various through-the-thickness temperature distributions. J Therm Stress 2014;37:1449–81. doi:10.1080/01495739.2014.937251.

M

[43] Eshraghi I, Dag S, Soltani N. Bending and free vibrations of functionally graded annular and circular micro-plates under thermal loading. Compos Struct 2016;137:196–207. doi:10.1016/j.compstruct.2015.11.024.

ED

[44] Leissa AW. Vibration of plates. DTIC Document; 1969. [45] Reddy JN. Energy and variational methods. John Wiley, New York; 1984.

PT

[46] Bellman R, Casti J. Differential quadrature and long-term integration. J Math Anal Appl 1971;34:235–8. [47] Bellman R, Kashef BG, Casti J. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J Comput Phys 1972;10:40–52.

CE

[48] Shu C. Differential quadrature and its application in engineering. Springer Science & Business Media; 2012.

AC

[49] Yalcin HS, Arikoglu A, Ozkol I. Free vibration analysis of circular plates by differential transformation method. Appl Math Comput 2009;212:377–86. doi:10.1016/j.amc.2009.02.032. [50] Wu T., Wang Y., Liu G. Free vibration analysis of circular plates using generalized differential quadrature rule. Comput Methods Appl Mech Eng 2002;191:5365–80. doi:10.1016/S0045-7825(02)00463-2. [51] Liew KM, Han J-B, Xiao ZM. Vibration analysis of circular Mindlin plates using the differential quadrature method. J Sound Vib 1997;205:617–30. [52] Irie T, Yamada G, Aomura S. Natural frequencies of Mindlin circular plates. J Appl Mech 1980;47:652–5.