A Three-Dimensional Solution for Bending Analysis of Functionally Graded Ceramic–Metal Sandwich Plates with
Stretching Effect
Ahmed Hamidi1, Mohamed Zidour1, Abdelouahed Tounsi2and Adda Bedia El abbes2
1Civil Engineering Department. 2Civil Engineering Department.
University of Tiaret University of Sidi Bel Abbes
Tiaret14000, Algeria Sidi Bel Abbes22000, Algeria
Abstract- In this research, a simple but accurate sinusoidal plate theory for the thermomechanical bending analysis of functionally graded sandwich plates is presented. The main advantage of this approach is that, in addition to incorporating the thickness stretching effect, it deals with only 5 unknowns as the first order shear deformation theory (FSDT), instead of 6 as in the well-known conventional sinusoidal plate theory (SPT).
The material properties of the sandwich plate faces are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The core layer is made of an isotropic ceramic material. Comparison studies are performed to check the validity of the present results from which it can be concluded that the proposed theory is accurate and efficient in predicting the thermomechanical behavior of functionally graded sandwich plates. The effect of side-to-thickness ratio, aspect ratio, the volume fraction exponent, and the loading conditions on the thermomechanical response of functionally graded sandwich plates is also investigated and discussed..
Index Terms - Sandwich plate; thermomechanical; analytical modelling; functionally graded material; stretching effect.
I. INTRODUCTION
Sandwich structures are employed in a variety of engineering industries including aircraft, construction and transportation where strong, stiff and light structures are required. The advantages of these structures are that it provides high specific stiffness and strength for a low-weight consideration. Due to the mismatch of stiffness properties between the face sheets and the core, sandwich plates are susceptible to face sheet/core debonding, which is a major problem in sandwich construction, especially under impact loading. To increase the resistance of sandwich plates to this type of failure, the concept of a Functionally Graded Material (FGM) is being actively explored in sandwich structure design.
FGMs are achieved by gradually changing the composition of the constituent materials along one (or more) direction(s), usually in the thickness direction, to obtain smooth variation of material properties and optimum response to externally applied loading. Increased use of FGMs in various structural applications necessitates the development of accurate theoretical models to predict their response.
Most of these above-mentioned theories neglect the thickness stretching effect (i.e.,z 0) by assuming a constant transverse displacement through the thickness of the plate.
This assumption is appropriate for thin or moderately thick FGM plates, but is inadequate for thick FGM plates (Carrera et al. [12]; Bessaim et al. [08]; Houari et al. [16]; Hebali et al.
[15]; Fekrar et al. [13]; Belabed et al. [06]. The effect of the thickness stretching in FG plates was studied by Carrera et al.
[12] and Bessaim et al. [08], and it becomes significant in thick plates. Thus, it should be taken into consideration.
This research work aims to present a simple quasi-3D theory with only five unknowns for thermomechanical bending analysis of FGM sandwich plates. The beauty of the present formulation is that, in addition to including the thickness stretching effect (
z 0), the displacement field is modeled with only 5 unknowns as the FSDT, instead of 6 as in the well- known conventional sinusoidal plate theory (SPT). The sandwich plate faces are assumed to have isotropic, two- constituent (metal-ceramic) material distribution through the thickness, and the modulus of elasticity, Poisson’s ratio, and thermal expansion coefficient of the faces 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. The plate’s governing equations are obtained by using the principle of virtual work. Numerical results for deflections and stresses are investigated. The effects of temperature field on the dimensionless axial and transverse shear stresses of the FGM sandwich plate are studied.II. THEORETICAL FORMULATION
Consider a sandwich plate composed of three layers as shown in Fig. 1. Two FG face sheets are made from a mixture of a metal and a ceramic, while a core is made of an isotropic homogeneous material. The material properties of FG face sheets are assumed to vary continuously through the plate thickness by a power law distribution as
1 2
( )) 2
(n (z) P P P V n
P
where P(n) is the effective material property of FGM of layer n like Young’s modulusE , Poisson’s ratio , and thermal (1) (1)
expansion coefficient. P1 and P2 are the properties of the top and bottom faces of layer 1, respectively, and vice versa for layer 3 depending on the volume fractionV(n), (n1,2,3) defined by
] , [ for
] , [ for 1
] , [ for
3 3 2
2 ) 3 3 (
2 ) 1
2 (
1 0 0
1 ) 0 1 (
h h h z
h h V z
h h z V
h h h z
h h V z
p p
where p is the power law index (0 p), which dictates the material variation profile through the thickness.
Fig. 1 Geometry and coordinates of rectangular FGM sandwich plate.
. A. Kinematics
The displacement field of the present formulation is considered based on the following assumptions: (1) The transverse displacement is superposed into three parts, namely:
bending, shear and stretching components; (2) the axial displacement is divided into extension, bending and shear components; (3) the bending parts of the axial displacements are similar to those given by CPT; and (4) the shear parts of the axial displacements give rise to the sinusoidal variations of shear strains and hence to shear stresses through the thickness of the plate in such a way that the shear stresses vanish on the top and bottom surfaces of the plate. Based on these assumptions, the following displacement field relations can be obtained:
) , , ( ) ( ) , , ( ) , , ( ) , , , (
) ( )
, , ( ) , , , (
) ( )
, , ( ) , , , (
0 0
t y x z g t y x w t y x w t z y x w
y z w y f z w t y x v t z y x v
x z w x f z w t y x u t z y x u
s b
s b
s b
where u0 and v0 denote the displacements along the x and y coordinate directions of a point on the mid-plane of the plate;
wb and ws are the bending and shear components of the transverse displacement, respectively; and the additional displacement accounts for the effect of normal stress (stretching effect). The shape functions f(z) and g(z) are given as follows
h
z z h
z
f( ) sin
and g(z)1 f
zThe non-zero strains associated with the displacement field in Eq. (3) are:
xys ys xs
bxy by bx
xy y x
xy y x
k k k z f k
k k
z ( )
0 0 0
,
0
) 0
(
xz yz xz
yz g z
, z g('z)0z where
x v y u x
vx u
xy y x
0 0
0 0
0 0 0
,
y x
w y xw
w
k k k
b b b
bxy by bx
2 2 2
2 2
2
,
y x
w y xw
w
k k k
s s s
xys sy xs
2 2 2
2 2
2
,
x x wy y w
s s
xz
yz
0
0 , z0
and
dz z z dg g(' ) ( )
B. Constitutive relations
The linear constitutive relations are given as:
(2)
(3)
(4)
(5)
(6)
(7)
(8) (6)
(7)
(8)
xy xz yz z y x
xy xz yz z y x
T T T
Q Q Q Q Q Q
Q Q Q
Q Q Q
66 55 44 33 23 13
23 22 12
13 12 11
0 0 0 0 0
0 0
0 0 0
0 0 0
0 0
0 0 0
0 0 0
0 0 0
where (x, y, z,yz,xz,xy) and (x, y, z, yz,
xz, xy) are the stress and strain components, respectively.
where T TT0 in which T0 is the reference temperature.
The applied temperature distribution T(x,y,z) through the thickness are assumed to be
) , ( 1sin
) , ( ) , ( ) , ,
( 1 2 T3 x y
h y z
x hT y z x T z y x
T
Using the material properties defined in Eq. (1), stiffness coefficients, Qij, can be expressed as
1 , ) (
33 2 22
11 Q Q Ez Q
1 , ) (
23 2 13
12
Q Q E z
Q
1
,2 )
66 (
55
44 Q Q Ez Q
C. Governing equations
The governing equations of the present theory are derived using the principle of virtual work; the following expressions can be obtained:
0
2 /
2 /
wd q
dz d
h
h yz yz xz xz
xy xy z z y y x x
where is the top surface and q is the distributed transverse load.
Substituting Eqs. (3), (6) and (9) into Eq. (12) and integrating through the thickness of the plate, Eq (12) can be rewritten as
0 0
0
0 0
0 0
w d q S
S
k M k M k M
k M k M k M
N N
N U N
s xz xz s yz
yz
xys xys sy sy xs xs
bxy bxy by by bx bx
xy xy z z y y x x
where the stress resultants (N, Mb, Ms, Ss and Nz) are as follows:
3
1 1
, ,1 ,
,
n h
h b i b i i i
n
n
dz f z S
M
N ,
ix,y,xy
,
3
1 1
) (
n h
h s i i
n
n
dz z g
S ,
ixz,yz
and
3
1 1
) ('
n h
h z z
n
n
dz z g
N
The governing equations of equilibrium can be derived from Eq. (13) by integrating the displacement gradients by parts and setting the coefficients u0, v0, wb, ws and to zero separately. Thus one can obtain the equilibrium equations associated with the present simple quasi-3D theory,
0 :
0 2
:
0 2
:
0 :
0 :
2 2 2
2 2
2 2 2
2 2 0 0
z syz xzs
syz xzs sy xys
xs s
by xyb
bx b
y xy x xy
y N S x S
y q S x S y M y
x M x
w M
y q M y
x M x
w M
y N x v N
y N x u N
By substituting Eq. (6) into Eq. (9) and the subsequent results into Eq. (14), the stress resultants are readily obtained as:
sT bT T
a z s b s s s
s s
s b
M M N R
L L k k H D B
D D B
B B A M
M N
0
,
s A S ,
,0 0
Tz ys xs
by xb y a
a x z
N k k R
k k L L
R N
where
Nx Ny Nxy
N , , ,
b bxy
b y
b Mx M M
M , , ,
s xys
s y
s Mx M M
M , , ,
Tx, Ty,0
T N N
N , MbT
MbTx ,MybT,0
,
xsT, sTy ,0
,sT M M
M ,
(11b) (11a)
(11c) (8)
(8)
(12)
(14)
(13)
(14a)
(14b)
(15a) (13)
(15)
(16 a)
(16 b)
(17 a)
(17 b)
x0,0y,xy0
,
b bxy
b y
b kx k k
k , , ,
xs ys xys
s k k k
k , , ,
66 22 12
12 11
0 0
0 0
A A A
A A
A ,
66 22 12
12 11
0 0
0 0
B B B
B B
B ,
66 22 12
12 11
0 0
0 0
D D D
D D
D ,
s s s
s s s
B B B
B B B
66 22 12
12 11
0 0
0 0
,
s s s
s s s
D D D
D D D
66 22 12
12 11
0 0
0 0
,
s s s
s s s
H H H
H H H
66 22 12
12 11
0 0
0 0
,
s yzs
xz S S
S , ,
xz,yz
,
s s
s
A A A
55 44
0
0 ,
3
1 ( )
) (
) ( ) 11(
1
) (' ) ('
)
n (
h
h n
n n
n
a
a n
n
dz z g z g
z f Q z
R R L
L
Here the stiffness coefficients are defined as:
z z f z z f z f z
dzQ
H D B D B A
H D B D B A
H D B D B A
n n n
h
h n
s s s
s s s
s s s
n
n
2
1 1 ) ( ), ( ), ( , ,
,1 ( )
) 3 (
1
2 2
) 11(
66 66 66 66 66 66
12 12 12 12 12 12
11 11 11 11 11 11
1
and
s s s
s s s
H D B D B A
H D B D B A
11 11 11 11 11 11
22 22 22 22 22 22
, , , , ,
, , , , ,
,
2 )
11( 1 ) (
Ez Q n
( )
, 12 )
3 (
1 55 2 44
1
n h
h s
s n
n
dz z z g A E
A
The resultant efforts, NTx NTy , MxbT MybT, MsTx MsTy and NTz induced by the thermal effect are expressed by
3
1
) ( ) 2 (
) ( ) (
1 (' )
) ( 1 2
) 1 ( 1
) (
n h
h
n n n
n
Tz xsT xbT Tx
n
n
dz z g
z f T z z
E
N M M N
D. Governing equations in terms of displacements
Introducing Eq. (15) into Eq. (13), the governing equations can be expressed in terms of displacements (u0,
v0
, wb, ws, ) and the appropriate equations take the form:
2
,2
1 1 111
11 122
66 12
122 66 12 111
11
0 12 66 12 0 22 66 0 11 11
p Ld w d B w d B B
w d B B w d B
v d A A u d A u d A
s s s s
s
b b
2
,2
2 2 222
22 112
66 12
112 66 12 222
22
0 12 66 12 0 11 66 0 22 22
p Ld w d B w d B B
w d B B w d B
u d A A v d A v d A
s s s s
s
b b
,2 2
2 2
2 2
3 22 11 2222
22
1122 66 12 1111
11 2222
22
1122 66 12 1111
11 0 222 22
0 112 66 12 0 122 66 12 0 111 11
p d
d L w d D
w d D D w d D w d D
w d D D w d D v d B
v d B B u d B B u d B
s a s
s s s s
b s
b b
,
2 2
2 2
2 2
4 22 55 11 44
22 11 22
55 11 44 2222 22
1122 66 12 1111
11 2222
22
1122 66 12 1111
11 0 222 22
0 112 66 12 0 122 66 12 0 111 11
p d A d A
d d R w d A w d A w d H
w d H H w d H w d D
w d D D w d D v d B
v d B B u d B B u d B
s s
s s s s
s s
s s s s
b s s
s b b s
s s
s s s
s s
55
22 44 11 55 22 5,11 44 22
11 0
2 0 1
p d A d A R w d A R
w d A R w d w d L v d u d L
s s
s a s
s s b
a b
where dij , dijl and dijlm are the following differential operators:
j ij xi x d
2 ,
l j
ijl xi x x
d
3 ,
(15d)
(17e)
(17f)
(19)
(18b)
(18c)
(18a)
(18b)
(18c)
(18d)
(18e) (17 c)
(17 d)
(18a)
(20a)
(20b)
(20c)
(20d)
(20e)
(21)
m l j
ijlm xi x x x
d
4 ,
i xi
d
, (i, j,l,m1,2).
The components of the generalized force vector
p are given byx p NxT
1 ,
y p N
Ty
2 ,
2 2 2 2
3 y
M x
q M p
bTy bTx
,
2 2 2 2
4 y
M x
q M
p xsT ysT
, p5NTz
III. ANALYTICALSOLUTIONS
Consider a simply supported rectangular plate with length a and width b under transverse load q . To solve this problem, Navier presented the transverse mechanical and temperature loads q,T1,T2, and T3 in the form of a double trigonometric series as
) sin(
) sin(
3 2 1 0
3 2
1 x y
t t t q
T T T q
where q0,t1, t2 and t3 are constants, /a, /b. Based on Navier solution method, we assume the following solutions form for displacements (u0, v0, wb, ws,):
) sin(
) sin(
) sin(
) sin(
) sin(
) sin(
) cos(
) sin(
) sin(
) cos(
0 0
y x
y x
W
y x
W
y x
V
y x
U
w w v u
s b s b
whereU ,V ,Wb, Ws and unknown parameters must be determined. By considered equations (20) and (24), the following equation is obtained
C
Pwhere
U,V,Wb,Ws,
t and
C is the symmetric matrix expressed by
55 45 35 25 15
45 44 34 24 14
35 34 33 23 13
25 24 23 22 12
15 14 13 12 11
a a a a a
a a a a a
a a a a a
a a a a a
a a a a a
C
in which:
11 2 66 2
11 A A
a
12 66
12 A A
a
] ) 2 (
[ 11 2 12 66 2
13 B B B
a
] ) 2 (
[ 11 2 12 66 2
14 Bs Bs Bs
a
L a15
2 22 2
66
22 A A
a
] )
2
[( 12 66 2 22 2
23 B B B
a
] )
2
[( 12 66 2 22 2
24 Bs Bs Bs
a
L a25
11 4 12 66 2 2 22 4
33 D 2(D 2D ) D
a
11 4 12 66 2 2 22 4
34 Ds 2(Ds 2Ds ) Ds
a
2 2
35 La a
2
2 44 4 55
22
2 66 2 4 11
44 11 2( 2 )
s s
s
s s s
A A
H
H H a H
2 55 2 2 2
44
45 A A R
a s s
As As Ra
a55 442 552
and the components of the generalized force vector
P P1,P2,P3,P4,P5
t are expressed by
.-
. , ,
,
3 2 1 5
3 2 2 1
0 2 4
3 2 2 1
0 2 3
3 2 1 2
3 2 1 1
t R t L t L h P
t F t D t B h
q P
t D t D t B h
q P
t B t B t A P
t B t B t A P
T T a T
T s T s T s
T a T T T a T T
T a T T
where
3
1
) 2 ( ) ( 2 ) ( ) (
1
, ,1 2
) 1 ( 1
) ( , ,
n h
h
n n n
n T T T
n
n
dz z z z
E D B A
(22)
(23)
(24)
(25)
(26)
(25)
(26)
(27a) (27)
(27)
(28)
(29a)
3
1
) ( ) ( 2 ) ( ) (
1
,1 ) ( 2
) 1 ( 1
) ( ,
n h
h
n n n
n T a T a
n
n
dz z z z
E D B
3
1
) ( ) 2 (
) ( ) (
1
) ( , ,1 ) ( 2
) 1 ( 1
) ( , ,
n h
h
n n n
n T s T s T s
n
n
dz z z z z f
E F D B
1 2
(' )
,1 , ( )
, )( 1
) ( , ,
3
1
) ( ) ( 2 ) ( ) (
1
dz z z z z g
E R L L
n h
h
n n n
n T Ta T
n
n
with zz/h,f(z) f(z)/h and
h
z) 1sin z
(
.
IV. NUMERICAL RESULTS AND DISCUSSION
To assess the performance of present theory under mechanical and thermal loads, simply supported functionally graded sandwich plates are considered with following material properties:
Metal (Titanium, Ti-6Al-4V): P2 66.2 GPa;
3 /
2 1
; 2 10.3(106/K).
Ceramic (Zirconia, ZrO2): P1 117.0 GPa;1 1/3; )
/ 10 ( 11 .
7 6
1 K
.
The results are presented in the following normalized forms for displacements and stresses according to Saidi et al (2013) for the purpose of presentation in this article.
center deflection
,2 / 2
10 ) /(
10
2 2 3 0 0 3
0 4
3 w a b
h a t h
E a w q
,
axial stress
,2 ,2 / 2
10 /
10
2 2 2 0 2 0
0 2
h b a h a t E h a
q x
x
,
shear stress
,0
,2 ) 0 10 /(
/ 1
2 0 0 0
b h
a t E h a
q xz
xz
.
where the reference values are taken as E0 1 GPa and
0 106
/K.
It is assumed, unless otherwise stated, that a/h10 , 1
/b
a , t1 0, and q0 t2 t3 100. The shear correction factor of FSDT is fixed to be K5/6.
Fig. 2 shows the effect of the aspect ratio a/ on theb dimensionless center deflection w for FG sandwich plate. The effect of the mechanical and thermal loads is taken into consideration. It is found that the aspect ratio effect is more pronounced on the thermomechanical bending deflection w (q0t2t3100 and) of the FG sandwich plate.
.
0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0
-0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8
q0=t2=t3=100 q0=t3=0,t2=100 q0=100,t2=t3=0
a/b
Fig. 2 Effect of mechanical and temperature loads on the dimensionless center deflection of FG sandwich plate versus a/b (tFGM 0.6h, p2).
In Fig. 3 and 4, we have plotted the through-the-thickness distributions of the dimensionless axial stress x and the
transverse shear stress xz of the FG sandwich plate for 2
p and tFGM 0.6h, respectively. These figures show the great influence played by the different thermal and bending loads on the axial and transverse shear stresses.
Fig. 3 Effect of mechanical and temperature loads on the dimensionless axial stress of FG sandwich plate(tFGM 0.6h,p2).
(29b)
(29c)
(29d) w
