Accepted Manuscript

Bending, buckling and free vibration analysis of size-dependent functionally gradedcircular/annular microplates based on the modified strain gradient elasticity theory

R. Ansari, R. Gholami, M. Faghih Shojaei, V. Mohammadi, S. Sahmani

PII: S0997-7538(14)00110-7

DOI: 10.1016/j.euromechsol.2014.07.014

Reference: EJMSOL 3100

To appear in: European Journal of Mechanics / A Solids

Received Date: 10 January 2014

Revised Date: 17 May 2014

Accepted Date: 30 July 2014

Please cite this article as: Ansari, R., Gholami, R., Shojaei, M.F., Mohammadi, V., Sahmani, S.,Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annularmicroplates based on the modified strain gradient elasticity theory, European Journal of Mechanics / ASolids (2014), doi: 10.1016/j.euromechsol.2014.07.014.

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Bending, buckling and free vibration analysis of size-dependent functionally

graded circular/annular microplates based on the modified strain gradient elasticity

theory

R. Ansaria, R. Gholami*,b, M. Faghih Shojaeia, V. Mohammadia, S. Sahmania

a Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran b Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran

Abstract

A Mindlin microplate model based on the modified strain gradient elasticity theory is developed to

predict axisymmetric bending, buckling, and free vibration characteristics of circular/annular microplates

made of functionally graded materials (FGMs). The material properties of functionally graded (FG)

microplates are assumed to vary in the thickness direction. In the present non-classical plate model, the

size effects are captured through using three higher-order material constants. By using Hamilton’s

principle, the higher-order equations of motion and related boundary conditions are derived. Afterward,

the generalized differential quadrature (GDQ) method is employed to discretize the governing differential

equations along with various types of edge supports. Selected numerical results are given to indicate the

influences of dimensionless length scale parameter, material index and radius-to-thickness ratio on the

deflection, critical buckling load and natural frequency of FG circular/annular microplates.

Keywords: Circular/Annular microplates; Functionally graded materials; Bending; Buckling; Free

vibration; Modified strain gradient theory; Size effect.

1. Introduction

The use of structures that are made of functionally graded materials (FGMs) is increasing due to the

smooth variation of mechanical properties along some preferred direction which leads to continuous

stress distribution in these structures. Recently, FGMs have been concerned for their applications in

micro-structures such as micro-electro-mechanical systems (MEMS) and atomic force microscopes [1-4]

to achieve high sensitivity and desired performance.

The dependency of deformation behavior on the size effects has been experimentally observed in the

micro-bending test of the microbeams [5-7]. Therefore, it is essential to consider small scale effects in the

*Corresponding author. Tel. /fax: +98 1412222906.

E-mail address: gholami_r@liau.ac.ir (R. Gholami).

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analysis of the behavior of functionally graded (FG) microbeams. Conventional continuum mechanics

fails to predict the size-dependent response of the structures at micro- and nano-scale due to lacking

intrinsic length scales. In recent years, several higher-order elasticity theories have been introduced to

develop size-dependent continuum models [8-13].

By reformulating and extending the Mindlin’s theory, Fleck and Hutchinson [14] developed new type

of continuum theory namely as strain gradient theory in which the second-order deformation tensor

separated into the stretch gradient tensor and rotation gradient tensor which leads to additional higher-

order stress components compared to the couple stress theory. After that, Lam et al. [15] introduced

modified strain gradient theory (MSGT) with three material length scale parameters relevant to dilatation

gradient, deviatoric gradient and symmetric rotation gradient tensors. Several size-dependent beam and

plate models have been developed based on the MSGT to capture the size effects in the micro-scale

structures [16-25]. For example, Kong et al. [16] investigated the static and dynamic responses of Euler-

Bernoulli micro-beams using MSGT. They studied the effect of thickness to the material length scale

parameter ratio of the micro-beams on their static deformation and vibrational behavior. Wang et al. [17]

presented Timoshenko microbeams formulations based on the MSGT.

Another type of the higher-order continuum theories is the couple stress theory elaborated by Mindlin

and Tiersten [26] and Koiter [27] in which four material length scale parameters (two classical and two

additional) are used to incorporate micro-structure related size effect. Various researches have been

carried out in which size-dependent continuum models are developed based on couple stress theory [28-

35]. Yang et al. [36] first proposed the modified couple stress theory (MCST) in which the constitutive

equations contain only one additional material length scale parameter which causes to create symmetric

couple stress tensor and to use it more easily. This property has attracted some researchers to derive the

size-dependent governing equations and corresponding boundary conditions for the microbeams and

microplates [37-44]. For instance, utilizing the MCST, Asghari et al [37-38] proposed the size-dependent

beam models based on the Timoshenko and Euler-Bernoulli theories and investigated the static and

vibration behavior of FG microbeams. Furthermore, Ke et al. [39] developed a Mindlin microplate model

based on the MCST for the free vibration analysis of microplates.

In the present work, the bending, buckling and free vibration responses of FG circular/annular

microplates are studied based on the modified strain gradient elasticity theory and Mindlin plate theory.

The developed non-classical Mindlin plate model contains three material length scale parameters which

has the capability to interpret the size effects. To analyze the bending, buckling, and free vibration

characteristics of FG microplates, the generalized differential quadrature (GDQ) method is utilized to

discretize the governing differential equations along with different boundary conditions. The influences of

material index, dimensionless length scale parameter and radius-to-thickness ratio on the deflection,

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critical axial buckling loads and natural frequencies of FG circular/annular microplates are discussed in

detail. Furthermore, a comparison is made between the various plate models on the basis of the classical

theory (CT), MCST and MSGT.

2. Formulation of size-dependent equations of motion and corresponding boundary conditions

As it can be seen in Figure 1, an annular microplate composed of functionally graded materials through

the thickness with the inner radius a, outer radius b and thickness h is considered.

2.1. Functionally graded materials

The FG microplate is supposed to be made of ceramic and metal in a way that the materials at bottom

surface ( )2z h= − and top surface( )2z h= are metal-rich and ceramic-rich, respectively. The

effective Young’s modulus( )E , Poisson’s ratio( )ν and density( )ρ of the FG microplate can be defined

as

( ) ( ) ( ) ,c m f mE z E E V z E= − + (1a)

( ) ( ) ( ) ,c m f mz V zν ν ν ν= − + (1b)

( ) ( ) ( ) .c m f mz V zρ ρ ρ ρ= − + (1c)

the subscripts c and m are ceramic and metal phases, respectively. By defining k as the power-low index,

the volume fraction of the constituents ( )fV z can be defined by a simple power low function as follows

[45]

( ) 1.

2

k

f

zV z

h = +

(2)

2.2. Modified strain gradient theory

The governing equations of motion and corresponding boundary conditions are obtained based on the

first-order shear deformation plate theory and MSGT by implementing the Hamilton’s principle. The

principle can be presented in analytical form as

( )2

1

0,t

extT S

t

W dtδ Π − Π + =∫ (3)

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where TΠ , SΠ and extW denote the kinetic energy, the total strain energy and the work done by external

forces, respectively.

Based on the MSGT presented by Lam et al. [15], the strain energy in a continuum made of a linear

elastic material occupying region V undergoing infinitesimal deformations is stated as

( )

(1) (1)

V

1

2s s

S ij ij i i ijk ijk ij ijp m dVσ ε γ τ η χΠ = + + +∫ (4)

where the components of the strain tensor ijε , dilatation gradient tensor iγ , deviatoric stretch gradient

tensor (1)ijkη and symmetric rotation gradient tensor s

ijχ are defined by [15]

( )

( ) ( )

( ) ( )( )

, ,

,

(1) (1), , ,

, , ,

1,

2,

1;

51 1Θ Θ , Θ .

2 2

1;

3

ij i j j i

i mm i

ijk jik ijk ij mmk mmi ki mmj ijk jk i ki j ij k

sij i j j i i i

s s s s sjk

u u

curl

ε

γ ε

η η η δ η δ η δ η η ε ε ε

χ

= +

=

= = − + +

= + =

= + +

u

(5)

here iu denotes the components of the displacement vector u , Θi expresses the infinitesimal rotation

vector Θ and the symbol of δ represents the Kronecker delta.

For a linear isotropic elastic material, the constitutive equation can be expressed by the components of

kinematic parameters effective on the strain energy density as follows [15,46]

2 (1) 2 (1) 20 1 2 2 , 2 , 2 , 2s s

ij kk ij ij i i ijk ijk ij ijp l l m lσ λ ε δ µε µ γ τ µ η µ χ= + = = = (6)

where ijσ is the classical stress tensor and ip , (1)ijkτ and s

ijm are also called the higher-order stresses.

The parameters λ and µ appeared in the constitutive equation of the classical stress σ , denote the Lame

constants and are given as

2, .

1 2(1 )

E Eνλ µν ν

= =− +

(7)

where E and ν are Young’s modulus and Poisson’s ratio, respectively. Moreover, 0l , 1l , 2l appeared

in higher-order stresses are the additional independent material length scale parameters connected with

the dilatation gradients, deviatoric stretch gradients and symmetric rotation gradients, respectively.

According to the first-order shear deformation plate theory, in which the in-plane displacements are

expanded as linear functions of the plate thickness and the transverse deflection is constant through the

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plate thickness, the displacement field in a Mindlin plate corresponding to an axisymmetric problem can be

expressed as

( ) ( ) ( )1 2 3, , , 0, , .u U r t z r t u u W r t= + Ψ = = (8)

in which U andW represents the displacement components along the radial and axial directions,

respectively, and Ψ represents the small rotation of a transverse normal about θ - axes.

By introducing Eq. (8) into Eq. (5), the nonzero components of the strain–displacement relations can be

obtained as

1 , , .

2r rz

U U zz

r r r r

W

rθε ε ε∂ ∂Ψ ∂ = + = + Ψ = Ψ∂ + ∂ ∂

(9)

By using Eqs. (5) and (8), the non-zero components of the rotation vector, the dilatation gradient tensor

and the symmetric rotation gradient tensor can be obtained as, respectively

1Θ .

2

W

rθ∂ = Ψ − ∂

(10a)

2 2

2 2 2 2

11, .r z

U U Uz

r r r r r r r r r rγ γ

∂ ∂ ∂ Ψ ∂Ψ Ψ ∂Ψ Ψ= + − + + − = + ∂ ∂ ∂ ∂ ∂ (10b)

2

2

1 1.

4 4sr

WW

r r r rθχ ∂Ψ ∂ ∂ = − + − Ψ ∂ ∂ ∂ (10c)

Also, the non-zero components of deviatoric stretch gradient tensor can be derived as

( ) ( )2 2 2

2 2 2 2 2

2 2(1) (1) (1) (1) (1

1 1

) (1)2 2

(1)

1 1 12 , 2 ,

5 5

1 1 48 4 , 2 ,

15 15

12

5

rrz r

rrr zzz

zr zrr z z z

r

U U U z W

r r r r r r r r r r r

W W

r r r r r rθθ θ θ θθ

θθ

η

η η η η η η

η η

η ∂ ∂ ∂ Ψ ∂Ψ Ψ ∂ ∂Ψ Ψ− + + − + − + + ∂ ∂ ∂ ∂ ∂ ∂

∂Ψ ∂ Ψ Ψ ∂Ψ ∂= = = + − = = = − − ∂ ∂

= =

∂ ∂

=2 2

(1) (1)2 2 2 2

2 2(1) (1) (1)

2 2 2 2

1 4 4 43 3 ,

15 15

1 13

4

13 .

15 15

r r

zzr zrz rzz

U U U z

r r r r r r r r

U U U z

r r r r r r r r

θ θ θθη

η η η

∂ ∂ ∂ Ψ ∂Ψ Ψ= = − + − + − + − ∂ ∂ ∂ ∂

∂ ∂ ∂ Ψ ∂Ψ Ψ= = = − − + + − − + ∂ ∂ ∂ ∂

(11)

2.3. Derivation of governing equations and boundary conditions

If CΠ and 1 2 3NC s s sΠ = Π + Π + Π are used to illustrate the strain energies corresponding to the classical

elastic theory and MSGT, respectively, the total strain energy of FG microplate can be expressed as

S C NCΠ = Π + Π (12)

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where

1 .

2C r r

A

N MU WN M U dA

r r rQ

r rθ θ ∂ ∂Ψ ∂ Π = + + + Ψ + + Ψ ∂ ∂ ∂

∫

(13a)

2 2

1 2 2 2 2

1.

11

2p

zs r r

A

U U UP M dA

r r r r r r r r rP

r

∂ ∂ ∂ Ψ ∂Ψ Ψ ∂Ψ Ψ Π = + − + + − + + ∂ ∂ ∂ ∂ ∂ ∫

(13b)

2 2

2 2 2

2

2

2

2

2

211

22

1

h

s rr r rrz

h

rr

r

z r r

U U U WT

r r r r r r

T Mr r r r r

T T

M dA

θθ

θθ θθ

−

∂ ∂ ∂ ∂Ψ + − + ∂ ∂ ∂ ∂

Ψ ∂ Ψ ∂Ψ Ψ + + + − ∂ ∂

Π = +

∫

(13c)

2

3 2

1 1.

2 2r

s

A

Y WdA

r r r r r

Wθ ∂Ψ ∂ ∂ Ψ Π = − + − ∂ ∂ ∂ ∫

(13d)

in whichA stands for the area occupied by the mid-plane of the FG microplate. Introducing the stiffness

components as

{ } ( ) ( ){ }{ }

{ } ( ){ }

{ } ( ){ }

22

11 11 11

2

22

12 12 12

2

22

55 55 55

2

, , 2 1, , ,

, , 1, , ,

, , 1, , ,

h

h

h

h

h

h

A B D z z z z dz

A B D z z z dz

A B D z z z dz

λ µ

λ

µ

−

−

−

= +

=

=

∫

∫

∫

(14)

The normal resultant forces, shear forces, bending moments, and other higher-order resultants and

moments in a section caused by higher-order stresses are defined as follows

11 12 11 12 11 12 11 12

11 12 11 12 11 12 11 12

55

, ,

, ,

.

r

r

s

U U U UN A A B B N A A B B

r r r r r r r rU U U U

M B B D D M B B D Dr r r r rW

r r r

rAQ

θ

θ

κ

∂ ∂Ψ Ψ ∂ Ψ ∂Ψ= + + + = + + +∂ ∂ ∂ ∂∂ ∂Ψ Ψ ∂ Ψ ∂Ψ= + + + = + + +∂ ∂ ∂ ∂ = + Ψ

∂∂

(15a)

2 22 2 2

55 0 55 0 55 02 2 2 2

2 22 2

55 0 55 02 2 2 2

12 2 , 2 ,

1

1

12 2 .

r z

pr

U U UP A l B l P A l

r r r r r r r r r r

U U UM B l D l

r r r r r r r r

∂ ∂ ∂ Ψ ∂Ψ Ψ ∂Ψ Ψ = + − + + − = + ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ Ψ ∂Ψ Ψ= + − + + − ∂ ∂ ∂ ∂

(15b)

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2 22 21 55 1

2 255

2 2

22

5

21 12

5rrr

l B lU U U

r r r r r r r

AT

r

=

∂ ∂ ∂ Ψ ∂Ψ Ψ− + + − + ∂ ∂ ∂ ∂ 2 22 2

1 55 12 2 2 2

2 255

5

1

5

2

21 12

22

5,

5

28 4 ,

15

rr

rz

r

r

l D lU U U

r r r r r r r r

A l WT

r r

BM

r

∂ ∂ ∂ Ψ ∂Ψ Ψ− + + − + ∂ ∂ ∂ ∂

=

∂Ψ ∂ Ψ= + −

∂ ∂

(15c)

2 255 2

2

2 1

4r

A l WY

r r r r r

Wθ

∂Ψ ∂ ∂ Ψ= − + − ∂ ∂ ∂

(15d)

where sκ is a shear correction factor.

The kinetic energy of FG microplate can be expressed as:

2 22

2

2 2 2

1 2 3 1

1

2

12

2

h

T

A h

A

U Wz dzdA

t t t

U U WI I I I dA

t t t t t

ρ−

∂ ∂Ψ ∂ Π = + + ∂ ∂ ∂

∂ ∂ ∂Ψ ∂Ψ ∂ = + + + ∂ ∂ ∂ ∂ ∂

∫ ∫

∫

(16)

where

{ } ( ){ }2

21 2 3

2

, , 1, , .h

h

I I I z z z dzρ−

= ∫ (17)

If 0rN and 0q are used to denote the external radial force and external transverse load, respectively, the

work done by the external forces applied on the plate can be expressed as

2

0 0

1.

2ext

r

A

WW q W N dA

r

∂ = + ∂ ∫ (18)

Implementing Eq. (3) and taking the variation of U , W , Ψ , integrating by parts and setting the

coefficients of Uδ , Wδ andδΨ equal to zero, the following governing equations of motion (19a-19c)

and the associated boundary conditions (20a-20c) will be achieved as:

( ) ( ) ( )2 2 2

1 22 2 2 2

1 1 1 r r r rr r rrr

P TN P T UrN r P r T I I

r r r r r r r r t tθθθ θθ∂ + +∂ ∂ ∂ ∂ Ψ− − + + + = +

∂ ∂ ∂ ∂ ∂ (19a)

2 2

0 0 12 22

1 1 1

2r

r rrzrY w rY W

r r N rT q Ir r r r r t

Qr r

θ θ∂ ∂ ∂ ∂ + + − + = ∂

∂ ∂ ∂ ∂ +

∂ (19c)

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( )2 2 2

3 22 2 2 2

1 1

1

22

1

2p

r r

pp

z zr z r

r rr

rz r r

rrr

Mr M Y

r r r r

M M Ur M r M I I

r r r t

P TM r P rT M r

t

Y Qθθ

θθ

θθθθ θ

∂ − + − ∂

+∂ ∂ Ψ ∂− + +

+ ++ + + + +

= +∂ ∂ ∂

(19d)

Also, the corresponding boundary conditions are

( )10 0, 0 0.r r

r r rrr r rrr

P T UU or N rP rT or P T

r r r rθθδ δ+∂ ∂ = − + + = = + = ∂ ∂

(20a)

0

1 1 1 0 0, 0 0.

2 2 2r

rrz r r rrz r

YW or Q rT rY N or T Y

r r

w

rr

W

rθ

θ θδ δ∂ ∂ ∂ ∂

∂ = − − + + = = − = ∂

(20b)

( )0 0,

0

1 12

0.

2

pp r r

r rrr

pr r

r z rrz

rr

r

M MrM rM P Tor M

or M Mr

Yr r r

θθθδ

δ

Ψ = =

∂Ψ = + = ∂

+∂− + + + + +∂

(20c)

Therefore, based on the MSGT, the governing equations of motion of a size-dependent FG

circular/annular microplate are achieved. The presented model can be reduced to the size-dependent plate

model based on the MCST by setting 0 1 0l l= = and the classical plate model by putting the all material

length scale parameters equal to zero.

3. Non-dimensional form of equations of motion and boundary conditions in terms of displacements

Substituting Eqs. (15a)-(15d) into Eqs. (19a)-(19c) and by introducing the following non-dimensional

quantities

( ) ( ) ( )

( )

( )

23 01 2

1 2 3 0210 10 10 110

55 55 5511 11 1111 55 11 55 11 55 2 2

110 110 110 110 110 110

000 0 1 2

110

, , , , , , , , , , , ,

, , , , , , , , , , ,

, , ,rr

U W I q bI Ir bu w I I I q

b h h I I h I h A h

A B DA B Da a b b d d

A A A h A h A h A h

lNN

A

ξ η ψ

= = = = Ψ = =

=

= =l l l( )1 2 110

10

, , , .

l l At

h b Iτ =

(21a)

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2 2 2 2 2 2 2 21 0 1 2 0 1 3 1 2 4 1 2

2 2 2 2 2 2 2 2 25 1 2 6 0 1 2 7 0 1 8 0 1

2 29 1 2 10

4 4 8 1 16 12 , 2 , , ,

5 45 15 4 15 4

1 32 1 1 2, 2 , 2 , 2 ,

2 15 4 15 5

2,

2

1

15 4

c c c c

c c c c

c c

= + = − = = −

= − = + + = + = −

+

=

−

l l l l l l l l

l l l l l l l l l

l l2 2 2

0 1 2

4 12 .

15 4 = − −

l l l

(21b)

, the non-dimensional forms of equations of motion for FG annular/circular microplate based on the first-

order shear deformation theory and MSGT in terms of displacements can be expressed as

55 2 55 2 55 2 1 5511 11 112 2 2 2 2 2 2 2

55 2 1 5511 1 22

2 2 4 3

2 2 4 3

4 3 2 2

4 3 22 2 2

3 3 3

3 1

1 2

2

a c a c b c c au u u u ua a b

b c c b ub I I

ψη ξ ξ η ξ ξ ξ ξ η ξ ξ η ξ ξ ξ

ψ ψ ψ ψ ψη ξ ξ ξ ξ η ξ ξ ξ τ

+ + − − + + − +

+ − − − + = +

∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂∂ ∂

∂∂ ∂ 2τ∂

(22a)

2 2

2 2

4 3 3 2 2 23 4

02 4 3 3

255 2

55 2 2

55 55 55 502 2 21

1 1 1

4

2 1

s

r

w w

c cw w wq

a lw wk a

a a a c w wN I

ψ ψ ψ ψη η η ηξ ξ ξ ξ ξ η ξ ξ ξ ξ ξ ξ

ψ ψξ ξ ξ η ξ η ξ ξ ξ ξ ξη τ

∂ ∂ ∂ ∂+ + + − + ∂ ∂ ∂ ∂

− +

∂ ∂+ −∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂+ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − − + + =

(22b)

11 552

1 55 1 55 552 2

2 2

112 2 2

4 3 4

552

3 3 24

4 3 4 3 3 2

22 2 55

22 2

1

1

1

2 2 1

3 3

s

ud

c

c c du

u u wb a k

c b c d au u w w

b u u

ψ ψ ψ η ηψξ ξ ξ ξ ξ ξ ξ ξ ξ

ψ ψη ξ ξ ξ η ξ ξ ξ ξ ξ ξ

η ξ ξ ξ

η

ξ ξ

∂ ∂ ∂ ∂ ∂+ − ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ + − − +

∂ ∂− +∂ ∂

− + − + + +

+ +

2 2

2 255 1

2

2

55 255 6

2

2 2

3 22 2

2 2

2 2 2 2

2112

1

1

5 4

1a aw w ua c I I

ψ ψ ψη ξ ξ ξ ξ ξ

ψ ψ ψ ψ ψξ ξ η τ τξ ξ ξ ξ ξ ηξ ξ

− +

+ + − + + = +

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂+ ∂ ∂ ∂ ∂ ∂ ∂

l l

(22c)

It is observed from Eqs. (22a)-(22c) that for a linear homogeneous plate ( )11 55 2 0b b I= = = , the in-plane

displacement( )u is uncoupled with the transverse displacements ( ),w ψ .

The associated boundary conditions can be handled as a same way and expressed in terms of

displacements. The associated boundary condition for clamped (C) end can be written as

0.u w

u wψψ

ξ ξ ξ∂ ∂ ∂= = = = = =∂ ∂ ∂

(23)

and for simply supported (SS) edge, the boundary conditions can be expressed as

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2 255 55

1 55 55 82 20,

a bu u uu w c a b c

ψ ψ ψ ψξ ξ ξ ξ ξ ξ ξ ξ ξ

∂ ∂ ∂ ∂ ∂= = = + + − + − = ∂ ∂ ∂ ∂ ∂

2255 3 55 9 55 2

55 24 0,4

a c a c aw wca

ψ ψξ η ξ ξ η ξ ξ

∂ ∂ ∂+∂

− =∂ ∂

− l

3 2 35

2 21 55 1 55 45 55

67

113 2 3 2 2 2

255 7 55 2 55 10 55 7 55 712

11 2 2 3

552 2 2

122 2 3

21 1

2 2 20

4

c b c a c cu u wd

b c a a c d c b cdu wb u

d dc a

b

η ηψ ψ ψ

ξ ξ ξ ξ ξ ξ ξ ξ ξ

ψξ ξ ξ ξ ξ η ξ ξ ξ η

η η

η η ξ

∂ ∂ ∂ ∂ ∂ ∂− + − + + + + + ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂+ + + + − + + − = ∂ ∂

l

(24)

Also, the regular boundary condition ( )0ξ = can be stated as

2 255 55

1 55 5

4

5 82 2

3 2 255 3 55

3 2 2

22 255 2

1 2 5

3 2

552 5 2 2

0,

1

14 1 10,

15 2 4s

b du u u uu c b d c

a c aw w

aa wk a

c

ψ ψ ψψξ ξ ξ ξ ξ ξ ξ ξ

η ξ ξ ξ η ξ

η ξ ξ η ξ

ξ

ψ

ψ ψη ξ

∂ ∂ ∂ ∂ ∂= = = + + − + − = ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂− + − ∂ ∂ ∂

∂ ∂ − − + − + = ∂ ∂

ll l

(25)

4. Solution procedure

Different numerical techniques can be employed to solve the governing equations and associated

boundary conditions. The GDQ method [47] has shown a great potential in solving complicated partial

differential equations and has been successfully applied in many investigations. In this paper, this method

is employed to discretize the governing equations and associated boundary conditions. When discretizing

the problem, the grid points are located at the shifted Chebyshev–Gauss–Lobatto points as

1 11 cos ; 1, 2,3,..., .

2 1i

ii n

n

αξ α π− − = + − = − (26)

wheren is the total number of nodes distributed along the radial direction and a bα = . Also, the

column vectors ofu , w andψ can be described as follow

{ } { } { }1 2 1 2 1 2, ,..., , , ,..., , , ,..., .n n nu u u w w w ψ ψ ψ= = =u w ψ (27)

in which ( ) ( ), ,i i i iu u w wξ ξ= = and ( ) ( ); , 2,...,i i i i nψ ψ ξ= = . By applying the GDQ method, the

discrete counterpart of governing equations can be expressed as

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( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

155 2 55 2 55 211 11 112 2 2 2 2 2 2

4 3 1 4 31 55 55 2 1 5511 1 22 2 2 2 2

3 3 3

3 1 2,

1

2

a c a c b ca a b

c a b c c bb I I

η η η

η η η

+ + − − + +

− + + − − − + = +

2 2ξ ξ ξ

ξ ξ ξ ξ ξ

uD u D u D ψ

ξ ξ ξ ξ ξ

ψD u D u D ψ D ψ D ψ u ψ

ξ ξ ξ ξ ξ&&&&

(28a)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

21 1 1 155 2

55 2 2

4 3 3 155 553 5512

540

1 1 1

4

2,

1

s

r

a lk a

a aI

c c a cN

η η η ηη

η ηη

+ −

+

+ + + − +

− + − − + +

=

2 2ξ ξ ξ ξ ξ ξ

2 2ξ ξ ξ ξ 0ξ ξ

ψ ψD w D w D ψ D w D w D ψ

ξ ξ ξ ξ ξ

D w D w D ψ D ψ D w D qwξ ξ

wξ

&&

(28b)

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 111 552 2

4 3 4 3 31 55 1 55 552 2

1 1552 2 2 2 2

11

4

22

2 55

1

2 2 1

3 3

1

1 1

sb a k

c b c d a

b

d

c

c c d

η η

η η

η

η

η

+ + − − +

− + − +

+ −

+ +

+ + − +

− +

2 2ξ ξ ξ ξ ξ

2ξ ξ ξ ξ ξ ξ

2 2ξ ξ ξ ξ

ψD u D u D ψ D ψ D w ψ

ξ ξ ξ ξ

D u D u D ψ D ψ D

u

u

w D wξ ξ ξ

ψD u D u D ψ D ψ

ξ ξ ξ ξ ξ ξ

( ) ( ) ( ) ( )2 2

1 155 1 55 255 6 3 22 2

2 1.

112

15 4

a aa c I I

η η

+

+ + − + + = +

2 2ξ ξ ξ ξ

ψ ψD ψ D ψ D w D w ψ

ξ ξ ξu

ξ ξ

l l&& &&

(28c)

where

( ) ( )

( )( ) ( )

( ) ( )( )

( )

11 1

1,

; 0

; , 1,2,..., 1

; , 1,2,..., 2,3,..., 1

; , 1,2,..., 1,2,..., 1

x

i

i j j

r r rij r ij

ij iii j

nr

ijj j i

I r

xi j and i j n and r

x x x

w wr w w i j and i j n and r n

x x

w i j and i j n and r n

ξ−

−

= ≠

= Ρ ≠ = = − Ρ= =

− ≠ = = − − − = = = −∑

D

(29)

and �� is a N N× identity matrix and ( ) ( )1,

.n

i i jj j i

x x x= ≠

Ρ = −∏

The associated boundary conditions can be handled as the same way.

The governing equations (28a)-(28c) conjugating with the associated boundary condition yield the

following matrix form equation

( )0 .e rN+ + + =0gM K d 0qd K&& (30)

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where M , eK ,and gK denote the mass matrix, stiffness matrix and geometric stiffness matrix,

respectively. Also, 0q andd are the transverse load vector and displacement vector, respectively. The Eq.

(30) can be used for the static bending, static buckling and free vibration analysis of FG circular/annular

microplate. By neglecting the inertia term and setting 0 0rN = , Eq. (30) can be reduced to a static

bending problem of the FG circular/annular microplate as

.e + =0qK d 0 (31)

Also, by setting 0r crN P= − , the Eq. (30) can be reduced to a static buckling problem of FG

circular/annular microplate, if the inertia term and the transverse load vector neglected

( ) .e crP− =gK K d 0 (32)

in which crP denotes the dimensionless critical buckling load. Furthermore, by substituting

0 0rN = =0q into Eq. (29) and considering the dynamic displacement vector d as * ie ωτ=d d ( ω and

*d denote the dimensionless natural frequency and mode shape of FG circular/annular microplate,

respectively), Eq. (30) gives a free vibration problem

( )2 * .e ω− =K M d 0 (33)

By solving Eqs. (31), (32) and (33), respectively, the deflection, critical buckling load and natural

frequency of an FG circular/annular microplate can be obtained.

5. Results and Discussion

In this work, based on the MSGT, the static bending, buckling, and free vibration of the FGM

circular/annular microplates is analyzed. It is considered the FG microplates consist of aluminum (Al) and

ceramic (SiC) with the material properties 32702,3.0,70 mkgGPaE mmm === ρν for Al and

33100,17.0,427 mkgGPaE ccc === ρν for SiC [48]. Moreover, it is assumed that the top and the

bottom surfaces of the plate are metal-rich and ceramic-rich, respectively. It should be pointed out that the

experimental data is needed to evaluate the length scale parameters of a homogeneous epoxy or FG

microplate. The length scale parameter of an isotropic homogeneous microbeam has been experimentally

obtained as ml µ6.17= by Lam et al. [15]. However, so far, there is no available experimental data

relevant to the FG microplates in open literature. In order to quantitatively analyze the size effect of the

FG microplates, the values of length scale parameters for the FG microplates are approximately assumed

to be equal to ml µ15= in the following examples.

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5.1. Convergence and comparison studies

A critical part of the numerical methods is to show the convergence of their results. Tables 1 and 2

represents the convergence criteria of GDQ method used to evaluate the critical buckling loads and

natural frequencies of FG annular microplates with different types of boundary conditions. This pattern of

convergence of the numerical technique reflects its efficiency and reliability.

So as to ensure the validity and accuracy of the present model, the results obtained are compared with

those previously obtained based on modified couple stress model [49] in Table 3. As it was mentioned

before, the present modified strain gradient plate model can be easily reduced to the classical model

( )0 1 2 0l l l= = = and modified couple stress model ( )0 1 0l l= = . According to this table, good

agreement is observed between the generated results for FG annular microplates with C-C and SS-SS

boundary conditions and those of [49] based on the modified couple stress theory.

5.2. Bending analysis of FG circular/annular microplate under a uniformly distributed transverse load

On the basis of results given in Table 4, it is found that by increasing the b h ratio of annular FG

microplates, the maximum deflection will increase too and this behavior is the same for all kinds of edge

supports.

Figure 2 depicts the effect of material index on the maximum deflection of FG annular microplates

with various values of dimensionless length scale parameter and boundary conditions. It is found that

increasing of the value ofk leads to higher maximum deflection especially for FG microplates with

higher values of h l .

Illustrated in Figure 3 is a comparison between the maximum defections predicted by different plate

models corresponding to various boundary conditions. It is revealed that MSGT predicts the lowest value

of maximum deflection among various types of plate model and this prediction is the same for all kinds of

edge supports.

Plotted in Figure 4a are the dimensionless deflection curves of FG annular microplates predicted by

MSGT corresponding to various values of dimensionless length scale parameter and they are compared

with the dimensionless deflection curve predicted by CT. it can be seen that by increasing the value of

h l , the maximum deflection of FG annular microplate increases and it tends to the value of predicted by

CT. This pattern is the same for all types of boundary conditions. Figure 4b shows the same results

relevant to circular FG microplates and the previous manner can be found for these microplates too.

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Figure 5 illustrates the effect of inner-to-outer radius ratio on the dimensionless deflection curve of FG

annular microplates corresponding to different plate models. It is observed that among various types of

plate models, CT and MSGT predict the maximum and minimum values of deflection, respectively, and

this prediction is the same for all types of boundary conditions.

5.3. Buckling analysis of FG circular/annular microplates

Table 5 gives the first five dimensionless critical buckling loads of FG annular microplates corresponding

to different values of material property gradient index and various edge supports. It is indicated that by

increasing the value of k , the value of critical buckling load for the all first five modes and boundary

conditions decreases.

Depicted in Figure 6 is the effect of material index on the dimensionless critical buckling loads of FG

annular microplates with various values of dimensionless length scale parameter. It is found that for all

types of edge supports, increasing of the value ofh l leads to lower critical buckling load and tends to the

value predicted by CT and this behavior is more significant for FG microplates with lower values of

material property gradient index.

The results tabulated in Table 6 are the first five dimensionless critical buckling loads of FG annular

microplates with various values of dimensionless length scale parameter and boundary conditions. It can

be seen that increasing of h l leads to lower critical buckling load. In other words, by increasing the

influence of size effect, the stiffness of FG annular microplates decreases.

Figure 7 presents a comparison between the dimensionless critical buckling loads predicted by different

plate models for different boundary conditions. It is revealed that for all values of dimensionless length

scale parameter, especially for lower ones, MSGT and CT predict the maximum and minimum values of

critical buckling loads, respectively, among various plate models.

Figure 8 shows the effect of inner-to-outer radius ratio on the dimensionless critical buckling load and

corresponding mode-shape of FG annular microplates. It is found that higher value of inner-to-outer

radius ratio leads to the higher critical buckling load and increases the width of the buckling mode-shape.

This pattern is the same for all kinds of edge supports.

In Table 7, the effect of b h ratio on the critical buckling load of FG annular microplates with various

boundary conditions is demonstrated. It can be observed that the critical buckling load of annular

microplates decreases by increasing the value ofb h aspect ratio.

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5.4. Free vibration analysis of FG circular/annular microplates

Table 8 gives the first five dimensionless natural frequencies of FG annular microplates corresponding to

different values of material property gradient index and various boundary conditions. It is revealed that by

increasing the value of k , the value of natural frequency for the all first five modes and boundary

conditions decreases.

Illustrated in Figure 9 is the effect of material index on the dimensionless natural frequency of FG

annular microplates with various values of dimensionless length scale parameter. It is found that for all

types of boundary conditions, increasing of the value of h l leads to lower natural frequency and tends to

the frequency predicted by CT and this behavior is more significant for FG microplates with lower values

of k .

In Table 9 the first five dimensionless natural frequencies of FG annular microplates with various values

of dimensionless length scale parameter and boundary conditions are given. It can be observed that

increasing of h l leads to lower natural frequency. In other words, by increasing the influence of size

effect, the stiffness of FG annular microplates decreases.

Plotted in Figure 10 is a comparison between the dimensionless critical buckling loads predicted by

different plate models corresponding to different edge supports. It is found that for all values of

dimensionless length scale parameter, especially for lower ones, MSGT and CT predict the maximum and

minimum values of natural frequencies, respectively, among various plate models.

Shown in Figure 11 is the effect of inner-to-outer radius ratio on the dimensionless natural frequency

and corresponding mode-shape of FG annular microplates. It is found that higher value of inner-to-outer

radius ratio leads to the higher natural frequency and increases the width of the vibrational mode-shape.

This response is the same for all kinds of boundary conditions.

Tabulated in Table 10 is the results demonstrated the effect of b h ratio on the free vibration

characteristics of FG annular microplates with various edge supports. It can be seen that the fundamental

frequency of annular microplates decreases by increasing the value of b h aspect ratio.

6. Conclusion

In the present work, bending, buckling and free vibration responses of the FG circular/annular microplates

were studied based on the modified strain gradient elasticity theory and Mindlin plate theory. The

developed non-classical Mindlin plate model contains three material length scale parameters which has

the capability to interpret the size effects. To obtain the deflection, critical buckling loads and natural

frequencies of FG microplates, the generalized differential quadrature (GDQ) method was utilized to

discretize the governing differential equations along with different boundary conditions.

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It was observed that for all types of edge supports, increasing of the value of h l leads to lower critical

buckling load and natural frequency and this behavior is more significant for FG microplates with lower

values of material index. Furthermore, it was found that higher value of inner-to-outer radius ratio leads to

the higher natural frequency and critical buckling load. Moreover, it leads to increase the width of the

vibrational mode-shape. This response is the same for all kinds of boundary conditions. Also, it was

revealed that by increasing the value of dimensionless length scale parameter, the maximum deflection of

FG annular microplate increases and it tends to the value of predicted by CT. This pattern is for the

circular FG microplates.

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List of the Tables:

Table 1: Convergence of the first dimensionless natural frequency and critical buckling load of FG annular microplates predicted by MSGT corresponding to various numbers of nodes

( )2, 2, 0.5, 6k h l α η= = = =

Table 2: Comparisons of the first dimensionless natural frequencies and critical buckling load of SS-SS annular microplate for various gradient index and numbers of nodes predicted by MSGT

( )2, 0.5, 6h l α η= = =

Table 3: Comparisons of the dimensionless frequencies and critical buckling loads of SS-SS and C-C

annular microplate corresponding to the MCST and CT( )1.2, 0.5, 6k α η= = =

Table 4: Effectb h on the maximum deflection of FG annular microplates with 1.2, 2k h l= = and

0.5α = predicted by MSGT

Table 5: Effect of gradient index k on the first five dimensionless critical buckling loads for FG annular

microplates with 2, 0.5h l α= = and 6η = predicted by MSGT

Table 6: Size effect on the first five dimensionless critical buckling loads for FG annular microplates with

1.2, 2, 0.5k h l α= = = and 6η = predicted by MSGT

Table 7: Effectb h on the first five dimensionless critical buckling loads for FG annular microplates

with 1.2, 2k h l= = and 0.5α = predicted by MSGT

Table 8: Effect of gradient index k on the first five dimensionless natural frequencies for FG annular

microplates with 2, 0.5h l α= = and 6η = predicted by MSGT

Table 9: Size effect on the first five dimensionless natural frequencies for FG annular microplates with

1.2, 2, 0.5k h l α= = = and 6η = predicted by MSGT

Table 10: Effectb h on the first five dimensionless natural frequencies for FG annular microplates with

1.2, 2k h l= = and 0.5α = predicted by MSGT

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Table 1: Convergence of the first dimensionless natural frequency and critical buckling load of FG annular microplates predicted by MSGT corresponding to various numbers of nodes

( )2, 2, 0.5, 6k h l α η= = = =

BC SS-SS C-C SS-C C-SS n ω

crP ω crP ω

crP ω crP

9 3.63559 6.29685 4.99987 4.64933 0.36452 0.83675 0.58579 0.49948 12 3.61675 6.27879 4.96912 4.54749 0.36155 0.83463 0.5821 0.48495 15 3.61582 6.27881 4.97196 4.55877 0.36136 0.8346 0.5826 0.4863 18 3.61574 6.2788 4.9718 4.55799 0.36135 0.8346 0.58258 0.48619 21 3.61574 6.2788 4.97181 4.55801 0.36135 0.8346 0.58258 0.4862 24 3.61574 6.2788 4.97181 4.55801 0.36135 0.8346 0.58258 0.4862

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Table 2: Comparisons of the first dimensionless natural frequencies and critical buckling load of SS-SS annular microplate for various gradient index and numbers of nodes predicted by MSGT

( )2, 0.5, 6h l α η= = =

SiC k=0.6 k=1.2 k=2 Al n ω

crP ω crP ω

crP ω crP ω

crP

9 5.69878 4.08563 3.63559 3.33009 2.38105 0.9641 0.47138 0.36452 0.30064 0.14656 12 5.69303 4.06775 3.61675 3.3141 2.37949 0.96392 0.46828 0.36155 0.29839 0.14662 15 5.69268 4.06682 3.61582 3.31335 2.37928 0.96377 0.46806 0.36136 0.29825 0.14659 18 5.69267 4.06675 3.61574 3.31329 2.37928 0.96378 0.46804 0.36135 0.29825 0.1466 21 5.69267 4.06675 3.61574 3.31329 2.37928 0.96378 0.46804 0.36135 0.29825 0.1466 24 5.69267 4.06675 3.61574 3.31329 2.37928 0.96378 0.46804 0.36135 0.29825 0.1466

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Table 3: Comparisons of the dimensionless frequencies and critical buckling loads of SS-SS and C-C

annular microplates corresponding to the MCST and CT ( )1.2, 0.5, 6k α η= = =

Study BCs h/l = 1 h/l = 2 h/l = 3 CT

Present Ref. [49] Present Ref. [49] Present Ref. [49] Present Ref. [49]

Vibration

C-C

1ω 8.6259 8.6096 6.0651 6.0563 5.0823 5.0742 3.8664 3.8554

2ω 9.1419 9.1478 9.0698 9.0737 9.0127 9.0156 7.9155 7.8910

3ω 17.5934 17.6027 13.7467 13.7261 11.429

9 11.4110 9.3004 9.2992

SS-SS 1ω 5.2321 5.2254 3.5455 3.5431 3.0276 3.0256 2.4941 2.4922

2ω 8.8583 8.8630 8.598 8.6049 8.076 8.0798 6.6646 6.6575

3ω 15.6644 15.6229 11.1483 11.1268 9.869 9.8531 9.2559 9.2506

Buckling C-C crP 1.6615 1.6534 0.8284 0.8250 0.5918 0.5892 0.3718 0.3699

SS-SS crP 0.7523 0.7502 0.3529 0.3523 0.2598 0.2594 0.1784 0.1781

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Table 4: Effectb h on the maximum deflection of FG annular microplates with 1.2, 2k h l= = and

0.5α = predicted by MSGT

b/h=5 b/h=6 b/h=7 b/h=8 b/h=10

SS-SS microplate 0.0727 0.089 0.1062 0.1247 0.1668 C-C microplate 0.024 0.0306 0.0371 0.0434 0.0558 SS-C microplate 0.0386 0.0482 0.0576 0.067 0.0868 C-SS microplate 0.0454 0.0563 0.0671 0.0779 0.1005

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Table 5: Effect of gradient index k on the first five dimensionless critical buckling loads for FG annular

microplates with 2, 0.5h l α= = and 6η = predicted by MSGT

SiC k=0.1 k=0.2 k=0.6 k=1.2 k=2 k=5 k=10 Al

SS-SS microplate

1crP 0.9648 0.7624 0.6564 0.4678 0.3613 0.2983 0.2208 0.1876 0.1466

2crP 2.1708 1.7155 1.4747 1.0405 0.7977 0.6557 0.4855 0.4127 0.3248

3crP 3.8635 3.0536 2.625 1.8515 1.4175 1.164 0.8593 0.7301 0.5751

4crP 6.2261 4.9216 4.231 2.9828 2.2818 1.8722 1.3803 1.1721 0.9242

5crP 9.2872 7.3445 6.3154 4.452 3.4041 2.7911 2.0547 1.7439 1.3764

C-C microplate

1crP 2.2781 1.7989 1.5458 1.0896 0.8346 0.6861 0.5077 0.4316 0.3394

2crP 3.5872 2.8361 2.4385 1.7198 1.3162 1.0802 0.7968 0.6768 0.5337

3crP 6.3267 5.0039 4.3028 3.0338 2.3197 1.9019 1.4 1.1882 0.9381

4crP 9.0248 7.1401 6.1405 4.3299 3.31 2.7126 1.9951 1.6929 1.3374

5crP 13.1332 10.3951 8.9416 6.3056 4.8183 3.9462 2.8983 2.4583 1.9442

SS-C microplate

1crP 1.5819 1.2495 1.0742 0.759 0.5827 0.4796 0.3553 0.3019 0.2368

2crP 2.9319 2.3171 1.992 1.4049 1.0756 0.8833 0.6524 0.5544 0.4369

3crP 5.0035 3.9558 3.4012 2.3985 1.8351 1.5055 1.1098 0.9423 0.7429

4crP 7.6789 6.0731 5.2219 3.6815 2.8152 2.3082 1.6992 1.4423 1.1385

5crP 11.1191 8.7974 7.5659 5.3346 4.0776 3.3413 2.4566 2.0844 1.6468

C-SS microplate

1crP 1.3128 1.0383 0.893 0.6324 0.4863 0.4004 0.2964 0.2521 0.1981

2crP 2.8157 2.2252 1.9127 1.3488 1.0327 0.8483 0.6269 0.5327 0.4195

3crP 4.8254 3.8142 3.2797 2.313 1.7698 1.4523 1.0705 0.9091 0.7168

4crP 7.5532 5.9739 5.1363 3.6213 2.7691 2.2706 1.6718 1.4192 1.12

5crP 10.9556 8.6673 7.4539 5.2557 4.0175 3.2923 2.4208 2.0539 1.6227

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Table 6: Size effect on the first five dimensionless critical buckling loads for FG annular microplates with

1.2, 2, 0.5k h l α= = = and 6η = predicted by MSGT

h/l=1 h/l=2 h/l=3 h/l=4 h/l=5 h/l=7 h/l=15 h/l=30 CT

SS-SS microplate

1crP 0.7422 0.3613 0.2633 0.2203 0.1977 0.1761 0.1567 0.1524 0.1509

2crP 2.1407 0.7977 0.5372 0.4397 0.3917 0.3472 0.3077 0.2989 0.296

3crP 4.4646 1.4175 0.8501 0.6495 0.5555 0.4724 0.4029 0.3881 0.3831

4crP 7.834 2.2818 1.2529 0.8921 0.7248 0.5786 0.4586 0.4334 0.425

5crP 12.2645 3.4041 1.7632 1.1887 0.9227 0.6909 0.5018 0.4623 0.4491

C-C microplate

1crP 2.2067 0.8346 0.5668 0.4647 0.413 0.363 0.3143 0.301 0.2934

2crP 4.1123 1.3162 0.7937 0.6081 0.5206 0.4423 0.3743 0.3577 0.3491

3crP 7.951 2.3197 1.2756 0.909 0.7384 0.5885 0.463 0.4352 0.4242

4crP 11.9115 3.31 1.7168 1.159 0.9007 0.6756 0.4915 0.4524 0.4371

5crP 17.8659 4.8183 2.4017 1.5555 1.1636 0.822 0.5424 0.4832 0.4626

SS-C microplate

1crP 1.3958 0.5827 0.4117 0.3421 0.3057 0.27 0.2358 0.227 0.2225

2crP 3.1675 1.0756 0.6818 0.5399 0.472 0.4103 0.3557 0.3428 0.3371

3crP 6.0797 1.8351 1.0472 0.7701 0.6409 0.5272 0.4323 0.4115 0.4036

4crP 9.9343 2.8152 1.4962 1.0341 0.8199 0.633 0.4795 0.4469 0.4353

5crP 14.9347 4.0776 2.0668 1.3628 1.0368 0.7527 0.5206 0.4719 0.4551

C-SS microplate

1crP 1.1066 0.4863 0.3519 0.2963 0.267 0.2381 0.2099 0.2024 0.1983

2crP 3.0137 1.0327 0.6593 0.5245 0.46 0.4013 0.3493 0.3369 0.3311

3crP 5.8398 1.7698 1.0145 0.749 0.6253 0.5165 0.4258 0.4058 0.3978

4crP 9.7576 2.7691 1.4742 1.0206 0.8104 0.627 0.4767 0.4448 0.4331

5crP 14.704 4.0175 2.0385 1.3458 1.025 0.7456 0.5176 0.4698 0.453

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Table 7: Effectb h on the first five dimensionless critical buckling loads for FG annular microplates

with 1.2, 2k h l= = and 0.5α = predicted by MSGT

b/h=5 b/h=6 b/h=7 b/h=8 b/h=10

SS-SS microplate

1crP 0.4428 0.3613 0.3029 0.2581 0.1931

2crP 1.0234 0.7977 0.6563 0.5595 0.4337

3crP 1.9104 1.4175 1.1205 0.9262 0.6916

4crP 3.1644 2.2818 1.7537 1.4124 1.0103

5crP 4.794 3.4041 2.5727 2.0366 1.41

C-C microplate

1crP 1.0672 0.8346 0.6885 0.588 0.4565

2crP 1.7642 1.3162 1.0444 0.8657 0.6486

3crP 3.2041 2.3197 1.7883 1.4436 1.0359

4crP 4.649 3.31 2.5074 1.9891 1.3815

5crP 6.8277 4.8183 3.6129 2.8343 1.9236

SS-C microplate

1crP 0.7277 0.5827 0.4869 0.4178 0.3223

2crP 1.4163 1.0756 0.8673 0.7286 0.5562

3crP 2.51 1.8351 1.4291 1.1653 0.8511

4crP 3.9327 2.8152 2.1458 1.7133 1.2056

5crP 5.764 4.0776 3.0673 2.4154 1.6534

C-SS microplate

1crP 0.6008 0.4863 0.4097 0.3534 0.2743

2crP 1.3569 1.0327 0.8347 0.7026 0.538

3crP 2.4184 1.7698 1.3802 1.1268 0.825

4crP 3.8676 2.7691 2.1112 1.6864 1.1876

5crP 5.6786 4.0175 3.0226 2.3807 1.6305

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Table 8: Effect of gradient index k on the first five dimensionless natural frequencies for FG annular

microplates with 2, 0.5h l α= = and 6η = predicted by MSGT.

SiC k=0.1 k=0.2 k=0.6 k=1.2 k=2 k=5 k=10 Al

SS-SS microplate

1ω 5.6927 5.0921 4.7513 4.0667 3.6157 3.3133 2.8872 2.6743 2.3793

2ω 15.1571 13.5925 12.6448 10.7128 9.5022 8.7602 7.852 7.4624 6.9719

3ω 17.0742 15.3056 14.3146 12.3109 10.9577 10.0284 8.6901 8.0131 7.0804

4ω 23.1922 20.8139 19.4332 16.5735 14.6144 13.279 11.4405 10.5908 9.5632

5ω 34.2406 30.5361 28.3585 23.9411 21.1252 19.3417 16.974 15.8107 14.1592

C-C microplate

1ω 10.0072 8.9461 8.334 7.0933 6.2788 5.7394 4.9938 4.6294 4.1347

2ω 21.0965 18.855 17.49 14.7101 12.9677 11.9 10.6067 10.0581 9.3233

3ω 23.828 21.3534 19.9484 17.085 15.1607 13.8567 12.009 11.0892 9.8572

4ω 43.0956 38.5653 36.0848 31.1173 27.4038 25.0704 22.0179 20.5167 18.0397

5ω 44.5584 39.7539 36.92 31.1486 27.8337 25.5602 22.2503 20.5339 18.4079

SS-C microplate

1ω 7.9047 7.0664 6.5847 5.6109 4.9718 4.548 3.9603 3.6711 3.2747

2ω 17.8931 16.0389 14.9242 12.6504 11.2074 10.3081 9.1877 8.6993 8.0607

3ω 20.5521 18.4084 17.1909 14.7219 13.0697 11.9495 10.3603 9.5691 8.507

4ω 29.1252 26.1021 24.3826 20.8717 18.4934 16.8758 14.6119 13.5159 12.0763

5ω 39.4959 35.2247 32.7056 27.5744 24.2998 22.2387 19.5309 18.205 16.3233

C-SS microplate

1ω 7.2246 6.4612 6.0235 5.1394 4.558 4.1711 3.6324 3.367 3.0047

2ω 17.638 15.8085 14.7064 12.4593 11.0373 10.154 9.058 8.5825 7.9649

3ω 19.9999 17.9182 16.7398 14.3506 12.7476 11.6574 10.1049 9.3286 8.2814

4ω 28.0619 25.1634 23.5081 20.1198 17.8221 16.2605 14.0848 13.0422 11.6952

5ω 38.8921 34.69 32.2154 27.1774 23.9578 21.9253 19.2457 17.9335 16.0752

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Table 9: Size effect on the first five dimensionless natural frequencies for FG annular microplates with

1.2, 2, 0.5k h l α= = = and 6η = predicted by MSGT

h/l=1 h/l=2 h/l=3 h/l=4 h/l=5 h/l=7 h/l=15 h/l=30 CT

SS-SS microplate

1ω 5.2193 3.6157 3.0699 2.799 2.6461 2.4917 2.3456 2.3119 2.3

2ω 12.1588 9.5022 8.3506 7.6059 7.1694 6.7208 6.2846 6.1817 6.1449

3ω 15.3908 10.9577 9.5666 9.2023 9.0399 8.8863 8.7239 8.67 8.631

4ω 17.6817 14.6144 14.4614 14.2639 13.2111 12.1425 11.1549 10.9296 10.8524

5ω 35.7267 21.1252 16.4169 14.4723 14.3951 14.3693 14.3492 14.3446 14.3428

C-C microplate

1ω 10.263 6.2788 5.1406 4.6241 4.3327 4.02 3.6548 3.5237 3.4158

2ω 20.445 12.9677 10.8012 9.6631 8.9657 8.2223 7.4144 7.1488 6.9435

3ω 26.8049 15.1607 11.9192 10.7189 10.17 9.6593 9.1203 8.9261 8.754

4ω 46.2659 27.4038 20.4175 17.2335 15.506 13.7775 12.0323 11.5137 11.1547

5ω 49.8073 27.8337 22.8117 20.7289 19.645 18.4229 16.6519 15.8883 15.3862

SS-C microplate

1ω 7.748 4.9718 4.1529 3.7661 3.5445 3.3089 3.051 2.9701 2.9139

2ω 16.0568 11.2074 9.6858 8.7534 8.1851 7.5904 6.9699 6.7853 6.6632

3ω 22.2686 13.0697 10.6897 9.9306 9.5913 9.2692 8.9199 8.7937 8.6852

4ω 25.8668 18.4934 16.7574 15.6785 14.3541 12.9745 11.6183 11.2479 11.0293

5ω 42.7951 24.2998 18.4843 16.281 15.878 15.5741 15.2901 15.1871 15.0978

C-SS microplate

1ω 6.9761 4.558 3.8339 3.4909 3.2946 3.0861 2.8562 2.7822 2.7286

2ω 15.6266 11.0373 9.5103 8.5797 8.0252 7.4479 6.8458 6.6657 6.5452

3ω 21.4553 12.7476 10.5665 9.8803 9.563 9.2552 8.9151 8.7913 8.6852

4ω 24.1315 17.8221 16.3651 15.3898 14.1756 12.8529 11.5395 11.1783 10.9626

5ω 42.2163 23.9578 18.2862 16.1652 15.7488 15.4816 15.2529 15.1815 15.1221

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Table 10: Effectb h on the first five dimensionless natural frequencies for FG annular microplates with

1.2, 2k h l= = and 0.5α = predicted by MSGT

b/h=5 b/h=6 b/h=7 b/h=8 b/h=10

S-SSS microplate

1ω 4.0004 3.6157 3.3109 3.0563 2.6456

2ω 9.9581 9.5022 9.1003 8.6537 7.7695

3ω 12.236 10.9577 10.1333 9.6826 9.3131

4ω 12.3951 14.6144 16.9186 17.2233 14.9241

5ω 23.2817 21.1252 18.8985 19.2474 19.9831

C-C microplate

1ω 7.1032 6.2788 5.6995 5.2642 4.6341

2ω 14.1153 12.9677 12.1661 11.547 10.4129

3ω 17.4897 15.1607 13.5591 12.4308 11.2034

4ω 27.8867 27.4038 24.3124 21.9621 18.6971

5ω 31.5056 27.8337 27.9447 26.0994 23.7298

SS-C microplate

1ω 5.559 4.9718 4.5442 4.2097 3.6991

2ω 11.9645 11.2074 10.6564 10.1749 9.1647

3ω 14.8654 13.0697 11.8233 10.9937 10.2143

4ω 17.0539 18.4934 20.1741 19.5774 16.8313

5ω 27.6941 24.2998 21.7293 22.2151 21.7865

C-SS microplate

1ω 5.0725 4.558 4.1788 3.8793 3.4178

2ω 11.7689 11.0373 10.4886 9.9816 8.9558

3ω 14.4627 12.7476 11.5801 10.8352 10.147

4ω 16.2466 17.8221 19.5797 19.2654 16.6054

5ω 27.3501 23.9578 21.4405 21.7521 21.6985

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List of the Figures:

Figure 1: Schematic diagram of an FG annular microplate: kinematic parameters, coordinate system, and geometry

Figure 2: Effect of gradient index k on the maximum deflection of FG annular microplates with 0.5α =and 6η = corresponding to different dimensionless length scale parameter ( )h l predicted by MSGT

Figure3: Comparison of the maximum defection predicted by different plate models corresponding to

various boundary conditions( )1.2, 6, 0.5k η α= = =

Figure 4a: Size effect on the dimensionless deflection curve of FG annular microplates predicted by

MSGT and CT corresponding to various boundary conditions( )1.2, 6, 0.5k η α= = =

Figure 4b: Size effect on the dimensionless deflection curve of FG circular microplates predicted by

MSGT and CT corresponding to various boundary conditions( )1.2, 6, 0.5k η α= = =

Figure 5: Effect inner-to-outer radiusα on the dimensionless deflection curve of FG annular microplates

with 1.2, 3k h l= = and 6η = predicted by different plate models

Figure 6: Effect of gradient index k on the dimensionless critical buckling load of FG annular microplates

with 0.5α = and 6η = corresponding to different dimensionless length scale parameter ( )h l predicted

by MSGT

Figure 7: Comparison of the dimensionless critical buckling load predicted by different plate models

corresponding to various boundary conditions( )1.2, 6, 0.5k η α= = =

Figure 8: Effect inner-to-outer radiusα on the dimensionless critical buckling load and corresponding

mode shape of FG annular microplates with 1.2, 4k h l= = and 6η = predicted by MSGT

Figure 9: Effect of gradient index k on the dimensionless natural frequency of FG annular microplates

with 0.5α = and 6η = corresponding to different dimensionless length scale parameter ( )h l predicted

by MSGT

Figure 10: Comparison of the dimensionless natural frequency predicted by different plate models

corresponding to various boundary conditions( )1.2, 6, 0.5k η α= = =

Figure 11: Effect inner-to-outer radiusα on the dimensionless natural frequency and corresponding mode

shape of FG annular microplates with 1.2, 4k h l= = and 6η = predicted by MSGT

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Figure 1: Schematic diagram of an FG annular microplate: kinematic parameters, coordinate system, and geometry

0

Ceramic

�

Metal

r

a b

h

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Figure 2: Effect of gradient index k on the maximum deflection of FG annular microplates with 0.5α =and 6η = corresponding to different dimensionless length scale parameter ( )h l predicted by MSGT

0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35(a) SS-SS

Gradient index ( k )

Max

imu

m d

efle

ctio

n

h/l = 1h/l = 1.5h/l = 2h/l = 4 CT

0 2 4 6 8

0.04

0.08

0.12

0.16

(b) C-C

Gradient index ( k )

Max

imu

m d

efle

ctio

n

0 2 4 6 80

0.04

0.08

0.12

0.16

0.2

0.24 (c) SS-C

Gradient index ( k )

Max

imu

m d

efle

ctio

n

0 2 4 6 8

0.05

0.1

0.15

0.2

0.25(d) C-SS

Gradient index ( k )

Max

imu

m d

efle

ctio

n

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Figure3: Comparison of the maximum defection predicted by different plate models corresponding to

various boundary conditions( )1.2, 6, 0.5k η α= = =

1 2 4 6 8 10 12 14 16 18 20

0.06

0.1

0.14

0.18

0.22(a) SS-SS

Length scale (h/l)

Max

imu

m d

efle

ctio

n

CTMCSTMSGT

12 4 6 8 10 12 14 16 18 20

0.02

0.04

0.06

0.08

0.1

0.12(b) C-C

Length scale (h/l)M

axim

um

def

lect

ion

1 2 4 6 8 10 12 14 16 18 20

0.04

0.06

0.08

0.1

0.12

0.14

0.16(c) SS-C

Length scale (h/l)

Max

imu

m d

efle

ctio

n

1 2 4 6 8 10 12 14 16 18 20

0.04

0.06

0.08

0.1

0.12

0.14

0.16 (d) C-SS

Length scale (h/l)

Max

imu

m d

efle

ctio

n

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Figure4a: Size effect on the dimensionless deflection curve of FG annular microplates predicted by

MSGT and CT corresponding to various boundary conditions( )1.2, 6, 0.5k η α= = =

0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

ξ

Def

lect

ion

(a) SS-SS

h/l = 1.5h/l = 2h/l = 3h/l = 7h/l = 15CT

0.5 0.6 0.7 0.8 0.90

0.025

0.05

0.075

0.1

(b) C-C

ξD

efle

ctio

n

0.5 0.6 0.7 0.8 0.90

0.05

0.1

0.15

(c) SS-C

ξ

Def

lect

ion

0.5 0.6 0.7 0.8 0.90

0.05

0.1

0.15

(d) C-SS

ξ

Def

lect

ion

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Figure 4b: Size effect on the dimensionless deflection curve of FG circular microplates predicted by

MSGT and CT corresponding to various boundary conditions( )1.2, 6, 0.5k η α= = =

-1 -0.5 0 0.5 10

2

4

6

8

10

12

(a) SS circular microplate

ξ

Def

lect

ion

-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

(b) Clamped circular microplate

ξ

Def

lect

ion

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Figure 5: Effect inner-to-outer radiusα on the dimensionless deflection curve of FG annular microplates

with 1.2, 3k h l= = and 6η = predicted by different plate models

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

ξ

w

(a) SS-SS

MSGTMCSTCT

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

(b) C-C

ξw

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

(c) SS-C

ξ

w

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2(d) C-SS

ξ

w

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Figure 6: Effect of gradient index k on the dimensionless critical buckling load of FG annular microplates

with 0.5α = and 6η = corresponding to different dimensionless length scale parameter ( )h l predicted

by MSGT

0 2 4 6 8

0.5

1

1.5

2(a) SS-SS

Gradient index ( k )

Pcr

h/l = 1h/l = 1.5h/l = 2h/l = 4 CT

0 2 4 6 8

1

2

3

4

5

6(b) C-C

Gradient index ( k )

Pcr

0 2 4 6 8

0.5

1

1.5

2

2.5

3

3.5

(c) SS-C

Gradient index ( k )

Pcr

0 2 4 6 8

0.5

1

1.5

2

2.5

3(d) C-SS

Gradient index ( k )

Pcr

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Figure7: Comparison of the dimensionless critical buckling load predicted by different plate models

corresponding to various boundary conditions( )1.2, 6, 0.5k η α= = =

12 4 6 8 10 12 14 16 18 200.1

0.2

0.3

0.4

0.5

0.6

0.7(a) SS-SS

Length scale (h/l)

Pcr

CTMCSTMSGT

1 2 4 6 8 10 12 14 16 18 20

0.4

0.8

1.2

1.6

2

(b) C-C

Length scale (h/l)P

cr

12 4 6 8 10 12 14 16 18 20

0.2

0.4

0.6

0.8

1

1.2

(c) SS-C

Length scale (h/l)

Pcr

1 2 4 6 8 10 12 14 16 18 20

0.2

0.4

0.6

0.8

1

(d) C-SS

Length scale (h/l)

Pcr

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Figure 8: Effect inner-to-outer radiusα on the dimensionless critical buckling load and corresponding

mode shape of FG annular microplates with 1.2, 4k h l= = and 6η = predicted by MSGT

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3(a) SS-SS

ξ

w

α= 0.01, pcr= 0.11138

α= 0.1, pcr= 0.11482

α= 0.3, pcr= 0.14525

α= 0.5, pcr= 0.21885

α= 0.7, pcr= 0.37284

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3(b) C-C

ξw

α= 0.01, pcr= 0.21932

α= 0.1, pcr= 0.24889

α= 0.3, pcr= 0.33281

α= 0.5, pcr= 0.46469

α= 0.7, pcr= 0.76038

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3(c) SS-C

ξ

w

α= 0.01, p

cr= 0.21604

α= 0.1, pcr= 0.21594

α= 0.3, pcr= 0.25157

α= 0.5, pcr= 0.34204

α= 0.7, pcr= 0.50879

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3(d) C-SS

ξ

w

α= 0.01, p

cr= 0.11371

α= 0.1, pcr= 0.13466

α= 0.3, pcr= 0.19524

α= 0.5, pcr= 0.29867

α= 0.7, pcr= 0.48892

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Figure 9: Effect of gradient index k on the dimensionless natural frequency of FG annular microplates

with 0.5α = and 6η = corresponding to different dimensionless length scale parameter ( )h l predicted

by MSGT

0 2 4 6 82

4

6

8(a) SS-SS

Gradient index ( k )

ω

h/l = 1h/l = 1.5h/l = 2h/l = 4 CT

0 2 4 6 8

5

10

15

(b) C-C

Gradient index ( k )ω

0 2 4 6 8

4

6

8

10

12(c) SS-C

Gradient index ( k )

ω

0 2 4 6 8

4

6

8

10

(d) C-SS

Gradient index ( k )

ω

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Figure 10: Comparison of the dimensionless natural frequency predicted by different plate models

corresponding to various boundary conditions( )1.2, 6, 0.5k η α= = =

1 2 4 6 8 10 12 14 16 18 202

2.5

3

3.5

4

4.5

5(a) SS-SS

Length scale (h/l)

ω

CTMCSTMSGT

1 2 4 6 8 10 12 14 16 18 2023456789

10(b) C-C

Length scale (h/l)ω

1 2 4 6 8 10 12 14 16 18 202

3

4

5

6

7

(c) SS-C

Length scale (h/l)

ω

1 2 4 6 8 10 12 14 16 18 202

3

4

5

6

(d) C-SS

Length scale (h/l)

ω

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Figure 11: Effect inner-to-outer radiusα on the dimensionless natural frequency and corresponding mode

shape of FG annular microplates with 1.2, 4k h l= = and 6η = predicted by MSGT

0 0.2 0.4 0.6 0.8 10

1

2

3(a) SS-SS

ξ

w

α= 0.01, ω= 0.9951α= 0.1, ω= 1.1117α= 0.3, ω= 1.6192α= 0.5, ω= 2.799α= 0.7, ω= 6.125

0 0.2 0.4 0.6 0.8 10

1

2

3(b) C-C

ξw

α= 0.01, ω= 1.4838α= 0.1, ω= 1.8179α= 0.3, ω= 2.7715α= 0.5, ω= 4.6241α= 0.7, ω= 9.9326

0 0.2 0.4 0.6 0.8 10

1

2

3(c) SS-C

ξ

w

α= 0.01, ω= 1.4472α= 0.1, ω= 1.6187α= 0.3, ω= 2.2954α= 0.5, ω= 3.7661α= 0.7, ω= 7.8699

0 0.2 0.4 0.6 0.8 10

1

2

3

(d) C-SS

ξ

w

α= 0.01, ω= 1.0224α= 0.1, ω= 1.2752α= 0.3, ω= 2.0208α= 0.5, ω= 3.4909α= 0.7, ω= 7.5402

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Research Highlights:

• A size-dependent model for FG circular/annular microplates is developed.

• The bending, buckling and vibration of FG circular/annular microplates is investigated.

• The effects of length scale parameter, material index, radius-to-thickness ratio and BCs are studied.

• A comparison is made between the results of MSGT, MCST and CT.

• The size effect is prominent when the plate thickness is comparable with its length scale parameter.