TWO NEW REFINED SHEAR DISPLACEMENT MODELS FOR FUNCTIONALLY GRADED SANDWICH PLATES
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
/ h z h ± = , 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
/ 1 / h h = − , 2 h , 3 h , / 4 / h h = , 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)
across the thickness, satisfying shear stress free
V =
, z ∈ [ h , h ] (9a)
1 2 surface conditions.
z ∈ [ h 2 , h 3 ] (9b)
2.1.1. Assumptions of the present plate theory (9b) Assumptions of the present plate theory are as
(i) The displacements are small in comparison
with the plate thickness and, therefore,
(n )
where V ,( n = 1 , 2 , 3 ) denotes the volume
strains involved are infinitesimal. fraction function of layer n ; k is the volume (ii) The transverse displacement w includes
fraction index ( 0 ≤k ≤ +∞ ), which indicates the two components of bending w b , and shear
material variation profile through the thickness. w s . These components are functions of
The effective material properties, like Young’s
modulus E , Poisson’s ratio ν , and thermal
coordinates x, y only.
expansion coefficient α then can be expressed w ( x , y , z ) = w b ( x , y ) + w s ( x , y ) (11) by the rule of mixture [31, 35] as
(iii) The transverse normal stress σ
is negligible
P ( z ) = P 2 + ( P 1 − P ) V ( n 2 ) (10)
in comparison with in-plane stresses σ x and
(iv) The displacements u in x-direction and v where in y- is the effective material property of direction consist of extension, bending, and
P (n )
FGM of layer n . P 2 and P 1 denote the property shear components.
of the bottom and top faces of layer 1
( h 1 ≤ z ≤ h 2 ), respectively, and vice versa for U = u + u + (12) 0 b u s , V = v 0 + v b + v s layer 3 ( h 3 ≤ z ≤ h 4 ) depending on the volume
V (n fraction ) ( n = 1 , 2 , 3 ). For simplicity, The bending components u b and v b are assumed to Poisson’s ratio of plate is assumed to be
be similar to the displacements given by the constant in this study for that the effect of classical plate theory. Therefore, the expr ession for
Poisson’s ratio on the deformation is much less u b and v b can be given as
than that of Young’s modulus [36].
2.1. present refined shear deformation theory The shear components u s and v s give rise, in Unlike the other theories, the number of conjunction with w s , to the parabolic variations unknown functions involved in the present of shear strains γ xz , γ yz and hence to shear refined shear deformation theory is only four,
stresses τ xz , τ yz through the thickness of the as against five in case of other shear plate in such a way that shear stresses τ xz , deformation theories [21 – 26]. The theory τ yz
presented is variationally consistent, does not are zero at the top and bottom faces of the require a shear correction factor, and gives rise plate. Consequently, the expression for u s and to transverse shear stress variation such that the v s can be given as transverse shear stresses vary parabolically
2.1.2. Displacement Field and Constitutive
df ( z )
Equations dz
In the present analysis, displacement field For elastic and isotropic FGMs, the constitutive models satisfying the condition of zero relations can be written as:
( n transverse shear stresses on the top and bottom ) σ
Q 11 Q 12 0 ε x
surface of the plate are considered. Based on σ y = Q 12 Q 22 0 ε y and
τ xy
0 Q 66 xy 0 γ
the assumptions made in preceding section, the
displacement field can be obtained using Eqs.
where ( σ x , σ y , τ xy , τ yz , τ yx ) and ( ε x , ε y ,
(15a)
γ , γ , γ ) are the stress and strain
components, respectively. Using the material where the function f (z ) is chosen in the form properties defined in Eq. (10), stiffness coefficients, RSDT1 and RSDT2 Q ij , can be expressed as
f ( z ) = z − sin
for the RSDT1 model and (15b) Q 11 = Q 22 =
f ( z ) = z + for the RSDT2 model (15c)
The strains associated with the displacements in Eq.
2.1.3. Equilibrium Equations
xy = γ xy + z k xy + f ( z ) k xy
The equilibrium equations are derived by
yz = g ( z ) γ yz
using the virtual work principle, which can
xz = g ( z ) γ xz
be written for the plate as
− h ∫∫ / 2 Ω [ σ x δ ε x + σ y δ ε y + τ xy δ γ xy + τ yz δ γ yz + τ xz δ γ xz ] d Ω dz − ∫ Ω q δ W d Ω = 0 where
2 where
is the top surface.
Substituting Eqs. (17) and (18) into Eq. (20)
ε and integrating through the thickness of the
plate, Eq (20) can be rewritten as
+ M s δ k s + M s δ k s + S s δ γ s + S s δ γ s d Ω − q ( δ w + δ w ) d Ω = 0 y (21) y xy xy yz yz xz xz ] ∫ Ω b b + M s δ k s + M s δ k s + S s δ γ s + S s δ γ s d Ω − q ( δ w + δ w ) d Ω = 0 y (21) y xy xy yz yz xz xz ] ∫ Ω b b
( n ) z dz , (22a)
D 11 D 12 0 (
yz ) ∑∫ xz yz
τ , τ ) g ( z ) dz . (22b)
where h n + 1 and h n are the top and bottom z-
11 H 12 0
coordinates of the nth layer.
H = H 12 H 22 0 ,
(25d)
The governing equations of equilibrium can be
0 0 H 66
derived from Eq. (21) by integrating the
displacement gradients by parts and setting the S = { S xz , S yz } , γ = { γ xz , γ yz } , A =
0 A 55
coefficients δ u 0 , δ v 0 , δ w b and δ w s zero
(25e) separately. Thus one can obtain the equilibrium
The stiffness coefficients A ij and B ij , etc., are equations associated with the present shear
deformation theory,
defined as
( A 22 , B 22 , D 22 , B 22 s , D 22 s , H s 22 )( = A 11 , B 11 , D 11 , B 11 s , D 11 s , H 11 s ) ,
Using Eq. (18) in Eq. (22), the stress resultants
of a sandwich plate made up of three layers can
44 = A 55 =
[ g ( z ) ] dz , ∑∫
be related to the total strains by (26c)
Substituting from Eq. (24) into Eq. (23), we obtain
M s B s D s H s k s the following equation
where A 11 d 11 u 0 + A 66 d 22 u 0 + ( A 12 + A 66 ) d 12 v 0 − B 11 d 111 w b − ( B 12 + 2 B 66 ) d 122 w b
− ( B 12 s + 2 B 66 s ) d 122 w s − B 11 s d 111 w s = 0 { ,
N x , N y , N xy } , M = { M x , M y , M xy } ,
(27a) M t s =
( 12 + 2 B 66 ) d 112 w s − B 22 d 222 w s = 0 ,
(27b)
{ ε x , ε y , γ xy } , k = { k x , k y , k xy } ,
b b b b ε= t
− D 22 d 2222 w b − D 11 s d 1111 w s − 2 ( D 12 s + 2 D 66 s ) d 12 w s − D 22 s d 2222 w s + q = 0 { ,
xy }
B 11 d 111 u 0 + ( B 12 + 2 B 66 ) d 122 u 0 + ( B 12 + 2 B 66 ) d 112 v 0 + B 22 d 222 v 0 − D 11 d 1111 w b − 2 ( D 12 + 2 D 66 ) d 12 w b
(25b)
(27c)
xy
Following the Navier solution procedure, we
(27e)
B s 11 d 111 u 0 + ( B 12 s + 2 B 66 s ) d 122 u 0 + ( B 12 s + 2 B 66 s ) d 112 v 0 + B 22 s d 222 v 0 − D s 11 d 1111 w b − 2 ( D 12 s + 2 D 66 s ) d 12 w b
− D 22 s d 2222 w b − H 11 s d 1111 w s − 2 ( H 12 s
+ 2 H 66 s ) d 12 w s − H 22 s d 2222 w s + A 55 s d 11 w s + A 44 s d 22 w s + q = 0 assume the following solution form for ( u 0 , v 0 , w b , w s ) that satisfies the boundary
where d ij , d ijl and d ijlm are the following conditions,
differential operators:
u 0 U mn cos( λ x ) sin( µ y )
2 ∂ 3 0 V mn sin( λ x ) cos( µ y )
, (32) d ij =
w b W sin( λ x ) sin( µ y ∂ ) x i ∂ x j ∂ x i ∂ x j ∂ x bmn l
, d ijl =
w s W smn sin( λ x ) sin( µ y )
d ijlm =
, ( i , j , l , m = 1 , 2 ).
where U mn , V mn , W bmn , and W smn are arbitrary parameters to be determined using Eqs. (27). One
3. NUMERICAL PROCEDURE
obtains the following operator equation,
Rectangular plates are generally classified in
[] (33) K {}{} ∆ = P ,
accordance with the type of support used. We are here concerned with the exact solution of where {} ∆ and {} F denotes the columns Eqs. (27) for a simply supported FGM plate. The following boundary conditions are T {}{ ∆ = U mn , V mn , W bmn , W smn } , and
imposed at the side edges: (34)
{}{ F T = 0 , 0 , − q , − q .
mn }
x = − a / 2 , a / 2 (29a)
a 11 a 12 a 13 a 14
= 0 , N y = 0 , and M y = M y = 0 [] K = 12 22 23 24
a 13 a 23 a 33 a 34
at y = − b / 2 , b / 2 (29b)
a 14 a 24 a 34 a 44
To solve this problem, Navier presented the
in which:
external force in the form of a double
66 trigonometric series: µ )
( A 11 λ + A
a 11 = −
a 12 = − λ µ ( A 12 + A 66 )
∑∑ mn sin( λ
12 B [ 2 B 11 λ + ( B + 2 66 ) µ ] m = 1 n = 1
x ) sin( µ y ) ,
22 = − ( A 66 λ + A 22 µ )
2 where 2 λ = m π / a and µ= n/ π b , and m and n a
are mode numbers. For the case of a a 23 = µ [( B 12 + 2 B 66 ) λ + B 22 µ ]
sinusoidally distributed load, we have 2
24 = µ [( B 12 + 2 B 66 ) λ + B 22 µ ]
4 2 2 a 4 33 = − ( D 11 λ + 2 ( D 12 + 2 D 66 ) λ µ + D 22 µ )
m =n = 1 , and q 11 = q 0 (31) a = − D s λ 4 + 2 ( D s + 2 D s ) λ 2 µ 2 s
34 ( 11 12 66 + D 22 µ
44 = − ( H 11 λ + 2 ( H 11 + 2 H 66 ) λ µ + H 22 µ + A 55 λ + A 44 µ )
where q 0 represents the intensity of the load at the plate center.
4. NUMERICAL RESULTS
E = 70 × 10 In this study, two new shear deformation 9 • Metal (Aluminium, Al): M
N/m 2 ; ν = 0 . 3 .
theories for FGM sandwich plates are • Ceramic (Alumina, Al 9
2 O 3 ): E C = 380 × 10 ; considered, and comparisons are made
N/m 2 ; ν = 0 . 3 .
with solutions obtained using other shear The various non-dimensional parameters used
are
deformation theories available in the
10 literature. Symmetric and non-symmetric hE
0 a b • center deflection
sandwich plates are examined. Note that the
core of the plate is fully ceramic while the • axial stress σ x = 2 σ x , , ,
10 2 h a b h
bottom and top surfaces of the plate are metal-
h rich. b • shear stress τ
xz =
τ xz 0 , , 0 , In the following, we note that several kinds of aq 0 2
sandwich plates are used: • thickness coordinate z = z / h .
• The (1-0-1) FGM sandwich plate: The plate is symmetric and made of only two E where the reference value is taken as 0 =1
equal-thickness FGM layers, i.e. there is no GPa. We also take the shear correction factor K
core layer. Thus, h 1 =h 2 = 0 .
= 5/6 in FSDPT. Numerical results are
• The (1-1-1) FGM sandwich plate: Here the presented in Tables 2 – 5 using different plate
theories. Additional results are plotted in Figs. plate is symmetric and made of three equal
3 to 5 using the present new shear deformation thickness layers. In this case, we have,
theory (RSDT1). It is assumed, unless
otherwise stated, that a / h = 10 and a / b = 1 . • The (1-2-1) FGM sandwich plate: The
Table 2 contains the dimensionless center
plate is symmetric and we have: h 1 = − h / 4 ,
deflection w for an FG sandwich plate
h 2 = h / 4 . subjected to a sinusoidally distributed load . The
• The (2-1-2) FGM sandwich plate: Here the deflections are considered for k = 0, 1, 2, 3, 4, plate is also symmetric and the thickness of and 5 and different types of sandwich plates. the core is half the face thickness. In this Table 2 shows that the effect of shear
deformation is to increase the deflection. The • The (2-2-1) FGM sandwich plate: In this difference between the shear deformation case the plate is not symmetric and the theories is insignificant for fully ceramic plates core thickness is the same as one face ( k = 0 ). It can be observed that the results while it is twice the other. Thus, obtained by the present refined theories RSDT1
case, we have, h 1 = − h / 10 , h 2 = h / 10 .
h 1 = − h / 10 , h 2 = 3 h / 10 .
and RSDT2 are identical to those of sinusoidal shear deformation plate theory (SSDPT) and
The FG plate is taken to be made of aluminum parabolic shear deformation plate theory and alumina with the following material (PSDPT), respectively. properties:
Table 3 compares the deflections of different types of the FGM rectangular sandwich plates
with
k = 2. The deflections decrease as the k = 2. The deflections decrease as the
GPa).
Table 4 lists values of axial stress σ x for k = 0, Fig 4 contains the plots of the axial stress σ x
1, 2, 3, 4, and 5 and different types of sandwich through-the-thickness of the FGM sandwich plates. All theories (RSDT1, RSDT2, PSDPT, plates. The stresses are tensile above the mid- SSDPT, ESDPT and FSDPT) give the same plane and compressive below the mid-plane axial stress σ x for a fully ceramic plate ( k = 0). except for the nonsymmetric (2-2-1) FGM In general, the axial stress increases with the plate. The axial stress is continuous through the volume fraction exponent k . However, the plate thickness. The results demonstrate a fully ceramic plates ( k = 0) give the largest nonlinear variation of the axial stress through axial stresses. It is to be noted that the CPT the plate thickness for FGM plates. It is yields identical axial stresses as the FSDPT and important to observe that the maximum stress so Table 4 lacks the results of CPT.
depends on the value of the volume fraction Table 5 shows similar results of transverse exponent k and the kind of the sandwich plate. shear stress τ xz for a FGM sandwich plate In Fig. 5 we have plotted the through-the- subjected to a sinusoidally distributed load . The thickness distributions of the transverse shear
results show that the transverse shear stresses stress τ xz : The maximum value occurs at a as per the FSDPT may be indistinguishable. As point on the mid-plane of the plate and its the volume fraction exponent increases for FG magnitude for a FG plate is larger than that for plates, the shear stress will increase and the
a homogeneous (ceramic or metal) plate. fully ceramic plates give the smallest shear Because of the non-symmetry of the (2-2-1) stresses.
FGM plate, the maximum value of the It can be observed that the results obtained by transverse shear stress, τ xz (Fig. 5d), occurs as the present two models RSDT1 and RSDT2 are discussed before at a point on the mid-plane of
identical to those of the sinusoidal shear the plate. deformation plate theory (SSDPT) and the It is important to observe that the stresses (Figs. parabolic shear deformation plate theory 4 and 5) for a fully ceramic plate are the same (PSDPT), respectively. In general, the fully as that for a fully metal plate. This is because ceramic plates give the smallest deflections and the plate for these two cases is fully shear stresses and the largest axial stresses. As homogeneous and the stresses do not depend on the volume fraction exponent increases for the modulus of elasticity. FGM sandwich plates, the deflection, axial stress and shear stress will increase.
5. CONCLUSION
Fig. 3 shows the variation of the center In this study, two new shear deformation deflection with side-to-thickness ratio for theories were proposed to analyse the static different types of FGM sandwich plates. The behaviour of FGM sandwich plates. Unlike any FGM plate deflection is between those of plates other theory, the theory presented gives rise to
made of ceramic (Al 2 O 3 ) and metal (Al). It can only four governing equations resulting in
be observed that, deflection of metal rich FGM considerably lower computational effort when plate is more when compared to ceramic rich compared with the other higher-order theories plate. This can be accounted to the Young’s reported in the literature having more number
modulus of ceramic (Al 2 O 3; 380 GPa) being of governing equations. Bending and stress analysis under transverse load were analysed modulus of ceramic (Al 2 O 3; 380 GPa) being of governing equations. Bending and stress analysis under transverse load were analysed
4. C. F. Lü, C. W. Lim, and W. Q. Chen, shear deformation theories. The developed
“Exact solutions for free vibrations of theories give parabolic distribution of the
functionally graded thick plates on transverse shear strains, and satisfy the zero
elastic foundations,” Mechanics of traction boundary conditions on the surfaces of
Advanced Materials and Structures. 16, the plate without using shear correction factors.
The accuracy and efficiency of the present
5. C. F. Lü, C. W. Lim, and W. Q. Chen, theories has been demonstrated for static
“Semi-analytical analysis for multi- behavior of symmetric and non-symmetric
directional functionally graded plates: functionally graded sandwich plates. All
3-D elasticity solutions,” Int. J. Numer. comparison studies demonstrated that the
Meth. Engng, 79, 25–44 (2009). deflections and stresses obtained using the
6. C.P. Wu, S.E. Huang, “Three- present two new shear deformation theories
dimensional solutions of functionally (with four unknowns) and other higher shear
graded piezothermo-elastic shells and deformation theories such as PSDPT and
plates using a modified Pagano SSDPT (with five unknowns) are almost
method.” Comput Mater Continua, 12, identical. The extension of the present theory is
also envisaged for general boundary conditions
7. S.S.Vel and R.C. Batra, “Three- and plates of a more general shape. In
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vibration of functionally graded theories RSDT1 and RSDT2 are accurate and
rectangular plates,” J. Sound Vib., 272, simple in solving the static behaviors of
symmetric and non-symmetric FGM sandwich
8. E. Reissner, “The effect of transverse plates.
shear deformation on the bending of elastic plates,” J. Appl. Mech., 12, 69-
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(1983) k=5
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metal
0,0 ceramic
(b)
a/h
(b) -2,5 -2,0 -1,5 -1,0
ceramic k=0.5 0,5 k=2
k=5 metal
(c) -2,5
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 metal (1-0-1) FGM sandwich plate. (b) The (1-1-1) 0,0
FGM sandwich plate. (c) The (1-2-1) FGM
sandwich plate. (d) The (2-1-2) FGM sandwich -0,3
plate, (e) The (2-2-1) FGM sandwich plate.
(d)
(e) -2,0 -1,5 -1,0 -0,5 0,0
Figure. 4: Variation of axial stress σ x through
ceramic
the plate thickness for various values of k and k=1
different types of sandwich plates: (a) The (1-0-
1) FGM sandwich plate. (b) The (1-1-1) FGM 0,0
sandwich plate. (c) The (1-2-1) FGM sandwich -0,2
plate. (d) The (2-1-2) FGM sandwich plate, (e)
The (2-2-1) FGM sandwich plate. -0,5
Figure. 5: Variation of transverse shear stress
τ xz through the plate thickness for various
values of k and different types of sandwich
-0,1 -0,2
plates: (a) The (1-0-1) FGM sandwich plate. (b)
The (1-1-1) FGM sandwich plate. (c) The (1-2-
-0,4 -0,5
1) FGM sandwich plate. (d) The (2-1-2) FGM
sandwich plate, (e) The (2-2-1) FGM sandwich
Table 1 Displacement models.
Unknown function
CPT
Classical plate theory
-0,1 -0,2
FSDPT
First shear deformation plate 5
Parabolic shear deformation 5
-0,5 (b) 0,00 0,05 0,10 0,15
plate theory [21, 22]
Sinusoidal shear deformation 5
plate theory [23]
HSDPT
Hyperbolic shear deformation 5
ceramic
plate theory [24]
Exponential shear
metal
deformation plate theory [25]
TSDPT
Trigonometric shear
deformation plate theory [26]
RSDT1
Refined shear deformation 4
plate theory 1 (Present)
RSDT2
Refined shear deformation 4
-0,5 (c) 0,00
plate theory 2 (Present)
Table 2 Effects of volume fraction exponent on the Table 3 Effect of aspect ratio a/ b on the dimensionless dimensionless center deflections w of the different deflection of the FGM sandwich plates ( k = 2).
sandwich square plates.
Table 4 Effects of volume fraction exponent on the
dimensionless axial stress σ x of the FGM square plate
Table 5 Effects of volume fraction exponent on the
dimensionless transverse shear stress τ xz of the FGM
sandwich square plates