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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
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Manajemen dan Rekayasa Struktur C-76
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)
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( )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)
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Manajemen dan Rekayasa Struktur C-78
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)
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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
=
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)
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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)
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( ) ( ) ( )
( ) 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)
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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
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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
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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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30. M.A. Benatta, I. Mechab, A. Tounsi, E.A. Adda Bedia, “Static analysis of functionally graded short beams including warping and shear deformation effects,“ Computational Materials Science, 44, 765–773 (2008).
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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
ceramic k=0.5 k=2 k=5 metal
a/h
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2 4 6 8 10 12 140,0
0,5
1,0
1,5
2,0
2,5
ceramic
metal
(c)
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
(d)
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
ceramic
metal
(e)
ceramic k=0.5 k=2 k=5 metal
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)
ceramic k=0.5 k=2 k=5 metal
-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)
ceramic k=0.5 k=2 k=5 metal
-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)
ceramic k=0.5 k=2 k=5 metal
-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)
ceramic k=0.5 k=2 k=5 metal
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-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)
ceramic k=0.5 k=2 k=5 metal
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)
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(c)
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(d)
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(e)
ceramic k=1 k=2 metal
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
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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
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Table 5 Effects of volume fraction exponent on the dimensionless transverse shear stress xzτ of the FGM sandwich square plates