two new refined shear displacement models for …...prosiding seminar nasional aplikasi teknologi...

Prosiding Seminar Nasional Aplikasi Teknologi Prasarana Wilayah (ATPW), Surabaya, 11 Juli 2012, ISSN 2301-6752

Manajemen dan Rekayasa Struktur C-75

TWO NEW REFINED SHEAR DISPLACEMENT MODELS FOR FUNCTIONALLY GRADED SANDWICH PLATES

ADDA BEDIA El Abbas(1) , MERDACI Slimane(1), ZIDI Mohamed(1), HEBALI Habib(1),

BACHIR BOUIADJRA Rabbab(2), TOUNSI Abdelouahed(1), BENYOUCEF Samir(1), BOURADA Mohamed

1. INTRODUCTION

(1)

(1) Laboratoire des Matériaux et Hydrologie, Université de Sidi Bel Abbes, BP 89 Cité Ben

M’hidi 22000 Sidi Bel Abbes, Algérie. [email protected]

(2) Université de Sciences et Technologie d’Oran (USTO), Algérie. Abstract— Two refined displacement models, RSDT1 and RSDT2, are developed for a bending analysis of functionally graded sandwich plates. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The developed models are variationally consistent, have strong similarity with classical plate theory in many aspects, doe not require shear correction factor, and give rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The accuracy of the analysis presented is demonstrated by comparing the results with solutions derived from other higher-order models. The functionally graded layers are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity, Poisson’s ratio of the faces, and thermal expansion coefficients are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated. It can be concluded that the proposed models are accurate and simple in solving the bending behavior of functionally graded plates.

Keywords— functionally graded plates, shear deformation, higher-order theories

In recent decades, a new class of plates/shells made up of functionally graded materials (FGM), in which the material properties continuously vary through the thickness has become popular in various engineering applications. Because of the feature of continuously distributed material properties in FGM plates/shells, some drawbacks of conventional multilayered composite

plates/shells resulting from the abrupt change of material properties at the interfaces between adjacent layers have been overcome, such as residual stress concentration, delamination, and matrix cracking. Consequently, this class of FGM plates/shells can provide more stable working performance than the conventional multilayered composite plates/shells usually achieve, and have been successfully applied in various advanced industries. Therefore, developing theoretical methodologies and



numerical modeling for the analysis of this class of FGM plates and shells has attracted considerable attention from researchers [1 – 7].

xw

zuzyxu∂∂

−= 00),,( ,

yw

zvzyxv∂∂

−= 00),,( ,

),(),,( 0 yxwzyxw = where u, v, and w are displacements along the x, y, and z coordinate directions, respectively, and u0, v0, and w0

xzuzyxu θ+= 0),,(

are the midplane displacements. However, in thick and moderately thick plates, the transverse shear strains have to be taken into account. There are numerous plate theories that include these strains. The first-order shear deformation plate theory (FSDPT), which is known as the Mindlin plate theory [8 – 10], considers the displacement field as linear variations of midplane displacements:

, yzvzyxv θ+= 0),,( , ),(),,( 0 yxwzyxw =

In this theory, the relation between the resultant shear forces and the shear strains depends on shear correction factors [11 – 13]. This theory is also used for FGM plates as is described by Sladek et al. [14]. Some other plate theories, e.g., higher-order shear deformation theories (HSDT), which include the effect of transverse shear strains, are reported in the literature.

Higher-order theories based on series

expansions were developed by Donnel

[15], Reissner [16], and Lo et al. [17, 18]

and were modified by Levinson [19],

Murthy [20], and Reddy [21]. The

displacement field in these theories is

xxx zzzuzyxu ξψθ 320),,( +++= ,

yyy zzzvzyxv ξψθ 320),,( +++= ,

zz zzwzyxw ψθ 20),,( ++=

Reddy [21, 22] put forward a parabolic shear deformation plate theory (PSDPT) which considers not only the transverse shear strains, but also their parabolic variation across the plate thickness. As a result, there is no need to use shear correction coefficients in computing the shear stresses. In this theory,

xhzz

xw

zuzyxu θ

−+

∂∂

−=2

20

0 341 ),,( ,

yhzz

yw

zvzyxv θ

−+

∂∂

−=2

20

0 341 ),,( ,

),(),,( 0 yxwzyxw = Touratier [23] used sinusoidal shear deformation plate theory (SSDPT) for describing the parabolic distribution of transverse shear strains across the plate thickness and took the displacement field in the form

xhzh

xw

zuzyxu θππ

+

∂∂

−= sin),,( 0

0 ,

yhzh

yw

zvzyxv θππ

+

∂∂

−= sin),,( 0

0 ,

),(),,( 0 yxwzyxw = Soldatos [24] employed hyperbolic shear deformation plate theory (HSDPT) for this purpose:

xzhzh

xw

zuzyxu θ

−

+

∂∂

−=21coshsinh),,( 0

0 ,

yzhzh

yw

zvzyxv θ

−

+

∂∂

−=21coshsinh),,( 0

0 ,

),(),,( 0 yxwzyxw =

Karama et al. [25] used an exponential

shear deformation plate theory (ESDPT):

(1)

(2)

(3)

(4)

(4)

(5)

(5)

(6)

(6)



( )x

hzzex

wzuzyxu θ

2/200),,( −+

∂∂

−= ,

( )y

hzzey

wzvzyxv θ

2/200),,( −+

∂∂

−= ,

),(),,( 0 yxwzyxw =

Ferreira et al. [26] assumed the

displacement field in the form

xhz

xw

zuzyxu θπ

+

∂∂

−= sin),,( 0

0 ,

yhz

yw

zvzyxv θπ

+

∂∂

−= sin),,( 0

0 ,

),(),,( 0 yxwzyxw = The description of various plate theories is

given in Table 1. There are also many

comparison studies on the behavior of

transverse shear stresses in composite

plates [27 – 31].

In this study, two new displacement

models for an analysis of simply supported

FGM sandwich plates are proposed. The plates are made of an isotropic material with material properties varying in the thickness

direction only. Analytical solutions for

bending deflections of FGM plates are

obtained. The governing equations are

derived from the principle of minimum total

potential energy. Numerical results for

displacements and stresses are presented for a metal–ceramic FG plate. To make the study reasonable, displacements and stresses are given for different homogenization schemes and exponents in the power-law that describes the variation of the constituents.

2. PROBLEM FORMULATION Consider the case of a uniform thickness, rectangular FGM sandwich plate composed of three microscopically heterogeneous layers as shown in Fig. 1. The top and bottom faces of the plate are at 2/hz ±= , and the edges of the plate are parallel to axes x and y. The sandwich plate is composed of three elastic layers, namely, ‘‘Layer 1’’, ‘‘Layer 2’’, and ‘‘Layer 3’’ from bottom to top of the plate (Fig. 2). The vertical ordinates of the bottom, the two interfaces, and the top are denoted by

2/1 hh −= , 2h , 3h , 2/4 hh = , respectively. Two homogenization techniques are used to find the effective properties at each point in FGM layer. The rule of mixtures is the conventional and simple technique which is widely used in composite materials. In this technique, the effective property of FGM can be approximated based on an assumption that a composite property is the volume weighted average of the properties of the constituents. Another widely used approach for characterization of the material gradation is the micromechanics technique. In this technique, the effective elastic moduli of an FGM are determined from the volume fractions and shapes of the constituents. The Mori–Tanaka method [32] and self-consistent method [33] are two popular schemes of micromechanics technique. Recently, Chehel Amirani et al. [34] studied the free vibration of sandwich beam with FG core and they showed that there is insignificant difference between the results obtained by these two techniques (micromechanics technique and the rule of mixtures technique). Hence, in the following sections, the rule of mixtures technique is used for its simplicity. The volume fraction of the FGMs is assumed to obey a power-law function along the thickness direction:

(8)

(7)



k

hhhz

V

−−

=12

1)1( ,

],[ 21 hhz∈

1)2( =V ,

],[ 32 hhz∈

k

hhhz

V

−−

=43

4)3( ,

],[ 43 hhz∈

where )(nV , ( 3,2,1=n ) denotes the volume fraction function of layer n ; k is the volume fraction index ( +∞≤≤ k0 ), which indicates the material variation profile through the thickness. The effective material properties, like Young’s modulus E , Poisson’s ratioν , and thermal expansion coefficient α then can be expressed by the rule of mixture [31, 35] as

( ) )(212

)( )( nn VPPPzP −+= where )(nP is the effective material property of FGM of layer n . 2P and 1P denote the property of the bottom and top faces of layer 1 ( 21 hzh ≤≤ ), respectively, and vice versa for layer 3 ( 43 hzh ≤≤ ) depending on the volume fraction )(nV ( 3,2,1=n ). For simplicity, Poisson’s ratio of plate is assumed to be constant in this study for that the effect of Poisson’s ratio on the deformation is much less than that of Young’s modulus [36]. 2.1. present refined shear deformation theory Unlike the other theories, the number of unknown functions involved in the present refined shear deformation theory is only four, as against five in case of other shear deformation theories [21 – 26]. The theory presented is variationally consistent, does not require a shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically

across the thickness, satisfying shear stress free surface conditions. 2.1.1. Assumptions of the present plate theory Assumptions of the present plate theory are as follows: (i) The displacements are small in comparison

with the plate thickness and, therefore, strains involved are infinitesimal.

(ii) The transverse displacement w includes two components of bending bw , and shear

sw . These components are functions of coordinates x, y only.

),(),(),,( yxwyxwzyxw sb +=

(iii) The transverse normal stress zσ is negligible

in comparison with in-plane stresses xσ and

yσ .

(iv) The displacements u in x-direction and v in y-direction consist of extension, bending, and shear components.

sb uuuU ++= 0 , sb vvvV ++= 0

The bending components bu and bv are assumed to be similar to the displacements given by the classical plate theory. Therefore, the expression for

bu and bv can be given as

xw

zu bb ∂

∂−= ,

yw

zv bb ∂

∂−=

The shear components su and sv give rise, in conjunction with sw , to the parabolic variations of shear strains xzγ , yzγ and hence to shear

stresses xzτ , yzτ through the thickness of the

plate in such a way that shear stresses xzτ , yzτ

are zero at the top and bottom faces of the plate. Consequently, the expression for su and

sv can be given as

(9b)

(9a)

(9b)

(9c)

(10)

(11)

(12)

(13)

(12)

(13)

(14)



xw

zfu ss ∂

∂−= )( ,

yw

zfv ss ∂

∂−= )(

2.1.2. Displacement Field and Constitutive Equations In the present analysis, displacement field models satisfying the condition of zero transverse shear stresses on the top and bottom surface of the plate are considered. Based on the assumptions made in preceding section, the displacement field can be obtained using Eqs. (11) – (14) as

),(),(),,(

)(),(),,(

)(),(),,(

0

0

yxwyxwzyxwy

wzf

yw

zyxvzyxvx

wzf

xw

zyxuzyxu

sb

sb

sb

+=∂∂

−∂∂

−=

∂∂

−∂∂

−=

where the function )(zf is chosen in the form RSDT1 and RSDT2

−=

hzhzzf sin)( π

π for the RSDT1 model and

+

−=

2

35

41 )(

hzzzf for the RSDT2 model

The strains associated with the displacements in Eq. (15) are

0 )(

)(

)( )( )(

0

0

0

=

=

=

++=

++=++=

z

sxzxz

syzyz

sxy

bxyxyxy

sy

byyy

sx

bxxx

zg

zg

kzfkzkzfkzkzfkz

εγγ

γγ

γγεεεε

where

xu

x ∂∂

= 00ε , 2

2

xw

k bbx

∂

∂−= ,

2

2

xw

k ssx

∂

∂−=

yv

y ∂∂

= 00ε , 2

2

yw

k bby

∂

∂−= ,

2

2

yw

k ssy

∂

∂−=

xv

yu

xy ∂∂

+∂∂

= 000γ , yx

wk bb

xy ∂∂∂

−=2

2 , yx

wk ss

xy ∂∂∂

−=2

2

yw ss

yz ∂∂

=γ , x

w ssxz ∂

∂=γ , )('1)( zfzg −= and

dzzdfzf )()(' =

For elastic and isotropic FGMs, the constitutive relations can be written as:

)()(

66

2212

1211)(

0000

n

xy

y

xnn

xy

y

x

QQQQQ

=

γεε

τσσ

and

)()(

55

44)(

00 n

zx

yznn

zx

yz

QQ

=

γγ

ττ

where ( xσ , yσ , xyτ , yzτ , yxτ ) and ( xε , yε ,

xyγ , yzγ , yxγ ) are the stress and strain

components, respectively. Using the material properties defined in Eq. (10), stiffness coefficients, ijQ , can be expressed as

,1

)(22211

ν−==

zEQQ

,1

)( 212

νν−

=zEQ

( ) ,12

)(665544 ν+

===zEQQQ

2.1.3. Equilibrium Equations

The equilibrium equations are derived by

using the virtual work principle, which can

be written for the plate as

[ ] 0 2/

2/

=Ω−Ω++++ ∫∫ ∫ Ω−

ΩdWqdzd

h

hxzxzyzyzxyxyyyxx δγδτγδτγδτεδσεδσ

where Ω is the top surface. Substituting Eqs. (17) and (18) into Eq. (20) and integrating through the thickness of the plate, Eq (20) can be rewritten as

[] ( ) 0

000

=Ω+−Ω++++

++++++

∫∫

Ω

Ω

dwwqdSSkMkM

kMkMkMkMNNN

bbsxz

sxz

syz

syz

sxy

sxy

sy

sy

sx

sx

bxy

bxy

by

by

bx

bxxyxyyyxx

δδγδγδδδ

δδδδεδεδεδ

(15c)

(15a)

(15b)

(16)

(17)

(18)

(19b)

(19a)

(19c)

(20)

(21)

(20)

(21)



where

( ) ,)(

1,,

,,,,,, 3

1

)(1

∑ ∫=

+

=

n

h

h

nxyyx

sxy

sy

sx

bxy

by

bx

xyyx n

n

dzzf

zMMMMMMNNN

τσσ

( ) ( )∑ ∫=

+

=3

1

)(1

.)(,,n

h

h

nyzxz

syz

sxz

n

n

dzzgSS ττ

where 1+nh and nh are the top and bottom z-coordinates of the nth layer. The governing equations of equilibrium can be derived from Eq. (21) by integrating the displacement gradients by parts and setting the coefficients 0 uδ , 0 vδ , bw δ and sw δ zero separately. Thus one can obtain the equilibrium equations associated with the present shear deformation theory,

02 :

02 :

0 :

0 :

2

22

2

2

2

22

2

2

=+∂

∂+

∂∂

+∂

∂+

∂∂

∂+

∂

∂

=+∂

∂+

∂∂

∂+

∂

∂

=∂

∂+

∂

∂

=∂

∂+

∂∂

qy

Sx

Sy

Myx

M

xM

w

qy

Myx

M

xM

w

yN

xN

v

yN

xN

u

syz

sxz

sy

sxy

sx

s

by

bxy

bx

b

yxy

xyx

δ

δ

δ

δ

Using Eq. (18) in Eq. (22), the stress resultants of a sandwich plate made up of three layers can be related to the total strains by

=

s

b

sss

s

s

s

b

kk

HDBDDABBA

MMN ε

, γsAS = ,

where

txyyx NNNN ,,= , tb

xyby

bx

b MMMM ,,= ,

tsxy

sy

sx

s MMMM ,,= ,

txyyx000 ,, γεεε = , tb

xyby

bx

b kkkk ,,= ,

tsxy

sy

sx

s kkkk ,,= ,

=

66

2212

1211

0000

AAAAA

A ,

=

66

2212

1211

0000

BBBBB

B ,

=

66

2212

1211

0000

DDDDD

D ,

=s

ss

ss

s

BBBBB

B

66

2212

1211

0000

,

=s

ss

ss

s

DDDDD

D

66

2212

1211

0000

,

=s

ss

ss

s

HHHHH

H

66

2212

1211

0000

,

tsyz

sxz SSS ,= , t

yzxz γγγ ,= ,

= s

ss

AAA

55

44

00 ,

The stiffness coefficients ijA and ijB , etc., are

defined as

( ) dzzfzfzzfzzQHDBDBAHDBDBAHDBDBA

n

n

n

h

h

n

sss

sss

sssn

n

−=

∑ ∫=

+

21

1)(),( ),(,,,1

)(

)(3

1

22)(11

666666666666

121212121212

111111111111 1

νν

, and ( ) ( )ssssss HDBDBAHDBDBA 111111111111222222222222 ,,,,,,,,,, = ,

2)(

11 1)(

ν−=

zEQ n

( ) [ ] ,)(12

)(3

1

25544

1

∑ ∫=

+

+==

n

h

h

ssn

n

dzzgzEAAν

Substituting from Eq. (24) into Eq. (23), we obtain the following equation

( ) ( )( ) ,02

2

111111226612

12266121111101266120226601111

=−+−

+−−+++

ss

sss

bb

wdBwdBB

wdBBwdBvdAAudAudA

( ) ( )

( ) ,02

2

222221126612

11266122222201266120116602222

=−+−

+−−+++

ss

sss

bb

wdBwdBB

wdBBwdBudAAvdAvdA

( ) ( ) ( )( ) ,022

2222

22222211226612111111222222

112266121111110222220112661201226612011111

=+−+−−−

+−−+++++

qwdDwdDDwdDwdD

wdDDwdDvdBvdBBudBBudB

ss

sss

ss

b

bb

(22a)

(22b)

(23)

(24)

(25a)

(25b)

(25c)

(25d)

(25e)

(26a)

(27a)

(27b)

(27c)

(26b)

(26c)

(27b)

(27c)



( ) ( ) ( )

( ) 022

2222

2244115522222211226612111111222222

112266121111110222220112661201226612011111

=+++−+−−−

+−−+++++

qwdAwdAwdHwdHHwdHwdD

wdDDwdDvdBvdBBudBBudB

ss

ss

ss

sss

ss

bs

bss

bsssssss

where ijd , ijld and ijlmd are the following

differential operators:

jiij xx

d∂∂∂

=2

, lji

ijl xxxd

∂∂∂∂

=3

,

mljiijlm xxxx

d∂∂∂∂

∂=

4, ).2,1,,,( =mlji

3. NUMERICAL PROCEDURE Rectangular plates are generally classified in accordance with the type of support used. We are here concerned with the exact solution of Eqs. (27) for a simply supported FGM plate. The following boundary conditions are imposed at the side edges:

0 and ,0 ,0 ,00 ====∂∂

=∂∂

=== sx

bxx

sbsb MMN

yw

yw

wwv at

2/ ,2/ aax −=

0 and ,0 ,0 ,00 ====∂∂

=∂∂

=== sy

byy

sbsb MMN

xw

xw

wwu

at 2/ ,2/ bby −= To solve this problem, Navier presented the external force in the form of a double trigonometric series:

∑∑∞

=

∞

=

=1 1

) sin() sin(),(m n

mn yxqyxq µλ ,

where am /πλ = and bn /πµ = , and m and n are mode numbers. For the case of a sinusoidally distributed load, we have

1== nm , and 011 qq = where 0q represents the intensity of the load at the plate center.

Following the Navier solution procedure, we assume the following solution form for ( sb wwvu ,,, 00 ) that satisfies the boundary conditions,

,

) sin() sin() sin() sin() cos() sin() sin() cos(

0

0

=

yxWyxWyxVyxU

wwvu

smn

bmn

mn

mn

s

b

µλµλµλµλ

where mnU , mnV , bmnW , and smnW are arbitrary parameters to be determined using Eqs. (27). One obtains the following operator equation, [ ] ,PK =∆ where ∆ and F denotes the columns ,,,, smnbmnmnmn

T WWVU=∆ and

.,,0,0 mnmnT qqF −−=

and

[ ]

=

44342414

34332313

24232212

14131211

aaaaaaaaaaaaaaaa

K ,

in which:

( )266

21111 µλ AAa +−= ( )661212 AAa +−= µλ

] )2([ 26612

21113 µλλ BBBa ++=

] )2([ 26612

21114 µλλ sss BBBa ++=

( )222

26622 µλ AAa +−=

] )2[( 222

2661223 µλµ BBBa ++=

] )2[( 222

2661224 µλµ sss BBBa ++=

( )422

226612

41133 )2(2 µµλλ DDDDa +++−=

( )4 22

2 26612

41134 )2(2 µµλλ ssss DDDDa +++−=

( )244

255

422

226611

41144 )2(2 µλµµλλ ssssss AAHHHHa +++++−=

4. NUMERICAL RESULTS

(27e)

(28)

(29a)

(29b)

(30)

(31)

(32)

(33)

(35)

(34)

(36)



In this study, two new shear deformation

theories for FGM sandwich plates are

considered, and comparisons are made

with solutions obtained using other shear

deformation theories available in the

literature. Symmetric and non-symmetric sandwich plates are examined. Note that the core of the plate is fully ceramic while the bottom and top surfaces of the plate are metal-rich. In the following, we note that several kinds of sandwich plates are used: • The (1-0-1) FGM sandwich plate: The

plate is symmetric and made of only two equal-thickness FGM layers, i.e. there is no core layer. Thus, 021 == hh .

• The (1-1-1) FGM sandwich plate: Here the plate is symmetric and made of three equal thickness layers. In this case, we have,

6/1 hh −= , 6/2 hh = . • The (1-2-1) FGM sandwich plate: The

plate is symmetric and we have: 4/1 hh −= , 4/2 hh = .

• The (2-1-2) FGM sandwich plate: Here the plate is also symmetric and the thickness of the core is half the face thickness. In this case, we have, 10/1 hh −= , 10/2 hh = .

• The (2-2-1) FGM sandwich plate: In this case the plate is not symmetric and the core thickness is the same as one face while it is twice the other. Thus,

10/1 hh −= , 10/32 hh = . The FG plate is taken to be made of aluminum and alumina with the following material properties:

• Metal (Aluminium, Al): 91070×=ME N/m2 3.0=ν; .

• Ceramic (Alumina, Al2O3910380×=CE): ;

N/m2 3.0=ν; . The various non-dimensional parameters used are

• center deflection

=

2,

210

02

0 bawqa

hEw ,

• axial stress

=

2,

2,

210

02

2 hbaqah

xx σσ ,

• shear stress

= 0,

2,0

0

baq

hxzxz ττ ,

• thickness coordinate hzz /= .

where the reference value is taken as 0E = 1 GPa. We also take the shear correction factor K = 5/6 in FSDPT. Numerical results are presented in Tables 2 – 5 using different plate theories. Additional results are plotted in Figs. 3 to 5 using the present new shear deformation theory (RSDT1). It is assumed, unless otherwise stated, that 10/ =ha and 1/ =ba . Table 2 contains the dimensionless center deflection w for an FG sandwich plate subjected to a sinusoidally distributed load. The deflections are considered for k = 0, 1, 2, 3, 4, and 5 and different types of sandwich plates. Table 2 shows that the effect of shear deformation is to increase the deflection. The difference between the shear deformation theories is insignificant for fully ceramic plates ( 0=k ). It can be observed that the results obtained by the present refined theories RSDT1 and RSDT2 are identical to those of sinusoidal shear deformation plate theory (SSDPT) and parabolic shear deformation plate theory (PSDPT), respectively. Table 3 compares the deflections of different types of the FGM rectangular sandwich plates with k = 2. The deflections decrease as the



aspect ratio ba / increases and this irrespective of the type of the sandwich plate. Table 4 lists values of axial stress xσ for k = 0, 1, 2, 3, 4, and 5 and different types of sandwich plates. All theories (RSDT1, RSDT2, PSDPT, SSDPT, ESDPT and FSDPT) give the same axial stress xσ for a fully ceramic plate ( k = 0). In general, the axial stress increases with the volume fraction exponent k . However, the fully ceramic plates ( k = 0) give the largest axial stresses. It is to be noted that the CPT yields identical axial stresses as the FSDPT and so Table 4 lacks the results of CPT. Table 5 shows similar results of transverse shear stress xzτ for a FGM sandwich plate subjected to a sinusoidally distributed load. The results show that the transverse shear stresses as per the FSDPT may be indistinguishable. As the volume fraction exponent increases for FG plates, the shear stress will increase and the fully ceramic plates give the smallest shear stresses. It can be observed that the results obtained by the present two models RSDT1 and RSDT2 are identical to those of the sinusoidal shear deformation plate theory (SSDPT) and the parabolic shear deformation plate theory (PSDPT), respectively. In general, the fully ceramic plates give the smallest deflections and shear stresses and the largest axial stresses. As the volume fraction exponent increases for FGM sandwich plates, the deflection, axial stress and shear stress will increase. Fig. 3 shows the variation of the center deflection with side-to-thickness ratio for different types of FGM sandwich plates. The FGM plate deflection is between those of plates made of ceramic (Al2O3) and metal (Al). It can be observed that, deflection of metal rich FGM plate is more when compared to ceramic rich plate. This can be accounted to the Young’s modulus of ceramic (Al2O3;

Fig 4 contains the plots of the axial stress

380 GPa) being

high when compared to that of metal (Al; 70 GPa).

xσ through-the-thickness of the FGM sandwich plates. The stresses are tensile above the mid-plane and compressive below the mid-plane except for the nonsymmetric (2-2-1) FGM plate. The axial stress is continuous through the plate thickness. The results demonstrate a nonlinear variation of the axial stress through the plate thickness for FGM plates. It is important to observe that the maximum stress depends on the value of the volume fraction exponent k and the kind of the sandwich plate. In Fig. 5 we have plotted the through-the-thickness distributions of the transverse shear stress xzτ : The maximum value occurs at a point on the mid-plane of the plate and its magnitude for a FG plate is larger than that for a homogeneous (ceramic or metal) plate. Because of the non-symmetry of the (2-2-1) FGM plate, the maximum value of the transverse shear stress, xzτ (Fig. 5d), occurs as discussed before at a point on the mid-plane of the plate. It is important to observe that the stresses (Figs. 4 and 5) for a fully ceramic plate are the same as that for a fully metal plate. This is because the plate for these two cases is fully homogeneous and the stresses do not depend on the modulus of elasticity. 5. CONCLUSION In this study, two new shear deformation theories were proposed to analyse the static behaviour of FGM sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations resulting in considerably lower computational effort when compared with the other higher-order theories reported in the literature having more number of governing equations. Bending and stress analysis under transverse load were analysed



and results were compared with previous other shear deformation theories. The developed theories give parabolic distribution of the transverse shear strains, and satisfy the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The accuracy and efficiency of the present theories has been demonstrated for static behavior of symmetric and non-symmetric functionally graded sandwich plates. All comparison studies demonstrated that the deflections and stresses obtained using the present two new shear deformation theories (with four unknowns) and other higher shear deformation theories such as PSDPT and SSDPT (with five unknowns) are almost identical. The extension of the present theory is also envisaged for general boundary conditions and plates of a more general shape. In conclusion, it can be said that the proposed theories RSDT1 and RSDT2 are accurate and simple in solving the static behaviors of symmetric and non-symmetric FGM sandwich plates. REFERENCES

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Figure. 1: Geometry of rectangular FGM sandwich plate with uniform thickness in rectangular Cartesian coordinates.

Figure. 2: The material variation along the thickness of the FGM sandwich plate

2 4 6 8 10 12 140,0

0,5

1,0

1,5

2,0

2,5

(a)

ceramic

metal

ceramic k=0.5 k=2 k=5 metal

a/h

2 4 6 8 10 12 140,0

0,5

1,0

1,5

2,0

2,5

(b)

ceramic

metal


a/h



2 4 6 8 10 12 140,0

0,5

1,0

1,5

2,0

2,5

ceramic

metal

(c)


a/h

2 4 6 8 10 12 140,0

0,5

1,0

1,5

2,0

2,5

(d)

ceramic

metal


a/h

2 4 6 8 10 12 140,0

0,5

1,0

1,5

2,0

2,5

ceramic

metal

(e)


a/h

Figure. 3: Dimensionless center deflection ( w ) as a function of side-to-thickness ratio (a/h) of an FGM sandwich plate for various values of k and different types of sandwich plates. (a) The (1-0-1) FGM sandwich plate. (b) The (1-1-1) FGM sandwich plate. (c) The (1-2-1) FGM sandwich plate. (d) The (2-1-2) FGM sandwich plate, (e) The (2-2-1) FGM sandwich plate.

-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

-2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0(a)


-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

-2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5(c)


-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

-2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0(d)


-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

-2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5(b)




-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

-2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0(e)


Figure. 4: Variation of axial stress xσ through the plate thickness for various values of k and different types of sandwich plates: (a) The (1-0-1) FGM sandwich plate. (b) The (1-1-1) FGM sandwich plate. (c) The (1-2-1) FGM sandwich plate. (d) The (2-1-2) FGM sandwich plate, (e) The (2-2-1) FGM sandwich plate.

-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45(a)

ceramic k=1 k=2 metal

-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35(b)


-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,00 0,05 0,10 0,15 0,20 0,25 0,30(c)


-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35(d)


-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35(e)


Figure. 5: Variation of transverse shear stress

xzτ through the plate thickness for various values of k and different types of sandwich plates: (a) The (1-0-1) FGM sandwich plate. (b) The (1-1-1) FGM sandwich plate. (c) The (1-2-1) FGM sandwich plate. (d) The (2-1-2) FGM sandwich plate, (e) The (2-2-1) FGM sandwich plate. Table 1 Displacement models.

Model Theory Unknown function CPT Classical plate theory 3

FSDPT First shear deformation plate theory [8 – 10]

5

PSDPT Parabolic shear deformation plate theory [21, 22]

5

SSDPT Sinusoidal shear deformation plate theory [23]

5

HSDPT Hyperbolic shear deformation plate theory [24]

5

ESDPT Exponential shear deformation plate theory [25]

5

TSDPT Trigonometric shear deformation plate theory [26]

5

RSDT1 Refined shear deformation plate theory 1 (Present)

4

RSDT2 Refined shear deformation plate theory 2 (Present)

4



Table 2 Effects of volume fraction exponent on the dimensionless center deflections w of the different sandwich square plates.

Table 3 Effect of aspect ratio ba / on the dimensionless deflection of the FGM sandwich plates ( =k 2).

Table 4 Effects of volume fraction exponent on the dimensionless axial stress xσ of the FGM square plate



Table 5 Effects of volume fraction exponent on the dimensionless transverse shear stress xzτ of the FGM sandwich square plates

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