Bending Response of Composite Material Plates with Specific Properties, Case of a Typical FGM "Ceramic/Metal" in Thermal Environments
Abdelrahmane Bekaddour Benyamina
1*, Bachir Bouderba
2, Abdelkader Saoula
1Received 30 December 2017; Revised 27 March 2018; Accepted 09 May 2018
1 Department of Civil Engineering, Faculty of Applied Sciences, Ibn Khaldoun University,
BP 78 Zaaroura, P.B. 14000, Tiaret, Algeria 2 Department of Science and Technology,
Institute of Science and Technology, El-Wancharissi University Center, Tissemsilt,
Route de BOUGARA, Ben Hamouda, P.B. 38004, Tissemsilt, Algeria
* Corresponding author, email: beny104@gmail.com
62(4), pp. 930–938, 2018 https://doi.org/10.3311/PPci.11891 Creative Commons Attribution b research article
PP Periodica Polytechnica Civil Engineering
Abstract
The rapid development of composite materials and structures in recent years has attracted the increased attention of many engineers and researchers. These materials are widely used in aerospace, military, mechanical, nuclear, marine, optical, electronic, chemical, biomedical, energy sources, automotive fields, ship building and structural engineering industries. In conventional laminate composite structures, homogeneous elastic plate are bonded together to obtain improved mechani- cal and thermal properties. However, the abrupt change in material properties across the interface between the different materials can cause strong inter-laminar stresses leading to delamination, cracking, and other damage mechanisms at the interface between the layers. To remedy these defects, func- tionally graded materials (FGM) are used, in which the prop- erties of materials vary constantly. The purpose of this paper is to analyze the thermomechanical bending behavior of func- tionally graded thick plates (FGM) made in ceramic/metal.
This work presents a model that employed a new transverse shear function. The numerical results obtained by the present analysis are presented and compared with those available in the literature (classical, first-order, and other higher-order theories). It can be concluded that this theory is effective and simple for the static analysis of composite material plates with specific properties "Case of a typical FGM (ceramic/metal)"
in thermal environments.
Keywords
FGM, shear function, mechanical loads, thermal behavior
1 Introduction
For the first time, in 1984–1985, a group of Japanese sci- entist proposed the concept of FGM. Five years later, the first international conference [1] was held at Sendai-City in Japan.
The interest was such that a rapid progress of FGM research in Japan was noticed from 1984 to 1996. At the beginning FGM was designed as a thermal barrier material for aero- space and fusion reactors applications [2]. Later, FGMs were developed for the military, automotive, biomedical and semi- conductor industries, and as a general structural element in high thermal environments temperature-resistant materials.
Functionally gradient materials (FGMs) are inhomogeneous at the microscopic scale, in which the mechanical properties vary regularly and continuously from one surface to another.
This is done by gradually varying the volume fraction of the constituent materials; these materials are made from a mixture of ceramic and metal or a combination of different materials.
The ceramic constituent provides a high temperature resis- tant material due to its low thermal conductivity and protects the metal from oxidation. The ductile metal component, on the other hand, prevents fracture caused by high temperature gradient stresses in a very short period of time. In addition, a mixture of a ceramic and a metal with a continuously varying volume fraction can be easily manufactured Surash et al [3], Plindera et al [4], [5] and Markworth et al [6]. A number of journals dealing with various aspects of the FGM have been published in recent years [7], [8], and [9]. Proceedings of inter- national seminar on FGM also shed light on the latest research in these materials, their manufacture, Biest et al [10], studies of FGM plates on mechanics, thermal properties and ther- mo-mechanical response of FGM plates; Reddy [11], Reddy and Chin [12], Vel and Batra [13, 14], Cheng and Batra [15].
A recent critical review of thermal analysis of FGM plates has been published by Swaminathan et al [16].
In this paper a new exponential refined shear deformation plate theory is employed. The exponential function in terms of thickness coordinates is used in the displacement field to con- sider the shear deformations. The originality of the theory is that it does not require a shear correction factor, satisfying the
Fig. 1 Geometry and coordinate system of the FGM rectangular plate
nullity of shear stress at the upper and lower surfaces of the plate, the unknown functions number is only four, while five or more in the case of other shear deformation theories [16]. Numerical examples are presented to illustrate the accuracy and efficiency of this theory by comparing the results obtained with those cal- culated using various other theories. When Aydogdu [17] com- pared various HSDPT’s (Higher-order Shear Deformation Plate Theories) with available 3D analysis, it has been pointed out that, while the parabolic shear deformation proposed by Reddy [18] and hyperbolic shear deformation theory proposed by Sol- datos [19] plate theories yields more accurate predictions for natural frequencies and buckling loads, the transverse displace- ments and stresses are best predicted by ESDT (the exponential shear deformation plate theory) proposed by Karama et al [20].
2 Theoretical formulation
Consider a rectangular plate in FGM of thickness h, a length a according to the x-direction, and has a width b following the y-direction, as shown in the Fig. 1. The plate material proper- ties varying gradually across the thickness (h), from the bot- tom surface in metal (z = –h/2) to the top surface in ceramic (z = h/2).
The present theory function is based on (ESDPT) the expo- nential shear deformation plate theory proposed by Karama et al [20], it is employed in FGM beams context by Osfero et al [21]. This function Eq. (1) is used as an exponential distribu- tion of the shear transverse stresses, it satisfies the nullity of these stresses on the upper and lower plate surfaces without using shear correction factors. The unknown functions num- ber is only four, while five or more in the case of other shear deformation theories [16], Table 1.
The proposed present theory based on the assumption that the axial and transverse displacements consist of one bend- ing and one shear part so that the bending component does not contribute to the shear forces and so, the shear component does not contribute to the bending moments [22].
2.1 Basic assumptions
For the present theory the following assumptions are used:
• The displacements are small in comparison with the plate thickness and, therefore, strains involved are infintesimal.
Table 1 Plate theories displacement models.
Model Transverse shear
function Unknown functions
CPT (classical plate theory) ψ(z) = 0 3
FSDPT (Reissner [23]) ψ(z) = z 5
PSDPT (Reddy [18]) 5
SSDPT(Touratier [24],[25] ) 5
ESDPT (Karama et al [20]) 5
Present (RESDPT) 4
• The transverse displacement w includes two components of bending wb and shear ws . These components are functions of coordinates x, y only.
• The transverse normal stress σz is negligible in comparison with in-plane stresses σx and σy.
• The displacement u in x-direction and v in y-direction con- sist of extension, bending and shear components.
2.2 Kinematics
In this paper a new exponential refined shear deformation plate theory is employed. The exponential function in terms of thickness coordinates is used in the displacement field to con- sider the shear deformations. The originality of the theory is that it does not require a shear correction factor (Table 1), sat- isfying the nullity of shear stress at the upper and lower sur- faces of the plate, the proposed shear function is [21]:
Based on the above assumptions, the displacement field is obtained as follows:
The kinematic relations can be obtained as follows:
And :
ψ z z z
( ) h
= 1−4 2 3 2 ψ z πh πz
( ) h
= sin
ψ z ze hz ( )
= −2 2
ψ z z ze hz ( )
= − −2 2
ψ( ) = −z z ze− 2zh
2
U x y z u x y z wx( , , )= ( , )− ∂∂ −b ( )z wx∂ s
∂
0 ψ
W x y z w x y w x y( , , )= b( , )+ s( , ) V x y z v x y z wy( , , )= ( , )− ∂∂ −b ( )z wy∂ s
∂
0 ψ
ε ε γ
ε ε γ
x y xy
x y xy
xb yb
z k k
= +
0 0
0 kk
z k k
xyb k
xs ys xys
+ψ( ) ,
γ
γ ξ γ
γ
yz xz
yzs xzs
z
= ( )
(1)
(2)
(3) (4)
(5)
(6)
Where :
Where ξ(z) is the considred warping funtion derivative (1), its variation along the thickness is dipcted in Fig. 2, it is defined as follows:
2.3 Constitutive equations
P-FGM is one of the most favorable models for FGMs.
Effective material properties such as Young’s modulus E, shear modulus G, mass density ρ and thermal expansions α are assumed to vary continuously in the depth direction accord- ing to power-low. Poisson’s ratio ν is assumed to be constant through the thickness of the plate, Dalale et al [26] states that the effect of this coefficient is not important as that of Youg modulus in deformations.
The effective material properties of FG plate with two kind of porosities that distributed identical in two phases of ceramic and metal can be expressed by using the modified rule of mix- ture as [27], [28] :
In which PC and PM are the corresponding properties of ceramics and metal, respectively, and α' is the volume fraction of porosities (α' << 1), for perfect FGM α' is set to zero, VC and VM are the volume fraction of ceramic and metal that are attached as [29], [30] and [31]:
In this paper the material properties P(z) are assumed to vary continuously through the depth of the plate by a power mixture law [3], [32], [33] and [34], as follows:
Where p is the volume fraction exponent or power indice in P-FGMs, which takes values higher or equal to zero.
Fig. 2 Warping funtion derivative variation along the non-dimensional width z/h.
Using the material properties (12), the linear constitutive relations are:
Where (σx, σy, τxy, τyz, τyx) and (εx, εy, γxy, γyz, γzx) are the terms of the stresses and deformations, respectively. The stiffness coefficients can be expressed by:
A polynomial temperature distribution applied through the thickness z is considred here [26], it is a combanation of three parts, a constant in T1 , a linear in T2 and sinusoidal in T3 , as follows:
In Eq. (13), ∆T = T – T0 in which T0 is the reference tem- perature.
2.4 Governing equations
The governing equations of equilibrium can be derived by using the principale of virtual work, which is expressed in this case as follows:
Where Ω is the top surface and q is a distributed mechanical loads on it.
ε ε γ
x y xy
ux vx uy v
x
0 0 0
0
0
0 0
=
∂∂
∂∂
∂∂ +∂
∂
=
−∂∂
−∂
, k k k
xw
x w
b yb xyb
b
b 2
2 2
∂∂
− ∂∂ ∂
y x ywb
2
2 2
γ γ
yzs xzs
s
s xs ys xys
wy wx
k k k
=
∂∂
∂∂
=
−∂∂
−∂∂
− ∂∂ ∂
2 2 2
2
2 2
xw yw
x yw
s
s
s
ξ ψ
( )z d z( )
= −1 dz
P z( )=P VC C− ′ +P VM M− ′
α α
2 2
V z
h V V
C
p
C M
= + + =
1
2 , 1
P z P( )= M +(P P VC− M) C
σ σ τ
x ε α y xy
Q Q Q Q
Q
x
=
11 12 −
12 22
66
0 0
0 0
∆∆
∆ T
y T
xy ε α
γ
−
τ τ
γ yz γ
zx Q
Q yz zx
= 44
55
0 0
Q Q E z Q E z
11= 22=1 2 12 1 2
− =
−
( ), ( )
ν ν ,
ν
Q Q Q E z
44 = 55= 66=2 1
(
( )+ν)
,T x y z T x y z
h T x y z
h T x y ( , , )= ( , )+ ( , )+ sin ( , ),
1 2 3
π1 π
σ δε σ δε τ δγ τ δγ τ δγ
x x y y xy xy yz yz xz xz d
h h
. . .
. .
/
/ ∫ + +
∫
+ +
−
Ω
Ω
2 2
dz− ∫q wd. =
Ω
δ Ω 0 (7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15) (16)
(17)
(18)
Fig. 3 Stresses resultants of the FGM plate
Substituting Eqs. (5–6) and (13–14) into Eq. (18) and inte- grating through the thickness of the plate, Eq. (18) can be rewritten as [32], [35]:
Stresse resultants N, M, and S in Fig. 3 are defined by:
Substituting Eqs. (13–14) into Eqs. (20–21) and integrating through the thickness of the plate, the stress resultants are in Eqs. (22–23) respectively, as follows:
Where :
Extensional and bending stiffness A, B and transverse shear stiffness coupling BS, DS and HS, are defined as follows:
And the shear stress resultants are :
With Aij, Bij, etc, are the stifness matrix terms of the plate, defined by:
And
Membrane resultants, flexural moments and additional shear moments, NxT = NyT, MxbT = MybTand MxsT = MysT, Eq. (15), due to thermal loads are defined respectively by:
The governing equations of equilibrium can derived from equation Eq. (19) by integration the displacement gradients by parts and setting the coefficients δu0, δν0 and δw0 to zero. Thus one can obtain the equilibrium equations associated with the present shear deformation theory:
Nx N y N
M k M k M k M k
x y xy xy
xb xb yb yb xyb xyb xs
∫∫ + +
+ + + +
δε δε δε
δ δ δ δ
0 0 0
xxs ys ys xys xys yxs yzs
xzs xzs
M k M k S S dxdy q wb w
+ + +
+ − +
δ δ δγ
δγ ∫ .(δ δ ss)dxdy=0
N N N
M M M
M M M
x y xy
xb yb xyb xs ys xys
x y
, ,
, ,
, ,
, ,
= σ σ ττ ξ
h xy
h z
z
( )
dz
−∫
1
2 2
( ) ,
/ /
S Sxzs yzs xz yz z dz
h h
, , ( ) .
/
( )
/( )
= ∫ τ τ ξ− 2
2
N M M
A B B B D D B D H
k k
b s
s s
s s s
b s
=
ε
− N M M
T bT sT
S S
A A
yz xz
xzs yzs
s s
= 44
55
0 0
γ γ
A B D
A B D
A B D
Q z z
h 11 11 11 h
12 12 12
66 66 66
11 2
2
1
( )
−
= , ,
/
//2 1
1 2
∫
−
ν ν
dz
B D H
B D H
B D H
Q
s s s
s s s
s s s
11 11 11
12 12 12
66 66 66
11
= ψψ ψ ν
ν ( ) , , ( )
/
/ z z z dz
h h
1
1 1
2
2
2
( )
−
∫
−
A B D A B D B D Hs s s B D Hs s
22 22 22 11 11 11
22 22 22 11 11 1
, , , , ,
( , , ) ( , ,
( ) (
=)
= 11
44 55
2 2
2
2 1
s
s s
hh
A A E z z dz
)
( ) ( ) ,
/
= = /
∫−
(
+ν)
ξ N M M
E z z T z z
xT xbT xsT
= −( )
( ) ( ) 1
1
ν α ψ
−∫ dz
h h
/ /
,
2
N N N N M M M M 2
M M M M N N
x y xy t b
xb yb xyb t
s xs ys xys t T
x
= =
= =
{ } { }
{ }
, , , , , ,
, , , TT Ty t
bT xbT ybT t sT
xsT ysT t
b xb
N
M M M M M M
k k k
, , ,
, , , , , ,
,
0
0 0
{ }
{ } { }
= =
=
{
yyb,kxyb t}
, ks ={
k k kxs, ys, xys t}
, ε={
ε ε γx y xy0, 0, 0}
t,A
A A A A
A B
B B B B
B
= =
11 12
12 22
66
11 12
12 22
66
0 0
0 0
0 0
0 0
,
= =
,
D ,
D D
D D D
B
B B
s B
s s
11 12
12 22
66
11 12
0 0
0 0
0
1
12 22
66
11 12
12 22
66
0
0 0
0 0
0 0
s s
s
s
s s
s s
B B
D
D D
D D D
=
,
ss s
s s
s s
s
H
H H
H H
H
, =
11 12
12 22
66
0 0
0 0
,
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
Substituting Eqs. (22–23) into Eq. (28), we obtain the fol- lowing equations:
where {f} = {f1, f2, f3, f4}t is a generalized force vector, dij, dijl and dijlm are the following differential operators:
The components of the generalized force vector {f} are given by:
3 The exact solution for FGM plates
Generally, rectangular plates are classified according to used support type [32], [35] and [36]. The simple support boundary conditions are:
It is considered here the exact solution of Eq. (28) for a sim- ply supported FGM plate. To solve this problem, Navier assumes that the transverse mechanical and temperature loads, q and Ti are given as a double Fourier series as follows:
Where λ = π/a, μ = π/b, q0 and ti are constants.
Following the procedure of Navier solution, we assume the following solution:
Where Umn, Vmn, Wb and Ws are arbitrary parameters to determine under condition which the solution of the equation (34) satisfied equilibrium equations Eqs. (29–32). The follow- ing operator’s equation is obtained:
Where {∆} = {Umn, Vmn, Wb Ws} and [K] is the symmetrical matrix, in which:
The components of the generalized forces vector are given by:
A d u A d u A A d v B d wb B B d w
11 11 0 66 22 0 12 66 12 0
11 111 12 2 66 122
+ +
(
+)
− −
(
+)
bbs s
s s
B B d w B d ws f
−
(
12+2 66)
122 − 11 111 = 1,A d v A d v A A d u B d wb B B d w
22 22 0 66 11 0 12 66 12 0
22 222 12 2 66 112
+ +
(
+)
− −
(
+)
bbs s
s s
B B d w B d ws f
−
(
12+2 66)
112 − 22 222 = 2,B d u B B d u B B d v B d v D d
11 111 0 12 66 122 0
12 66 112 0 22 222 0 11
2 2
+
(
+)
+
(
+)
+− 11111 12 66 1122
22 2222 11 1111 12
2 2
2 2
w D D d w
D d w D d w
D D
b b
b s
s s
−
(
+)
− −
−
(
+ 666s)
d1122w D ds− 22 2222s ws = f3 B d u B B d u B B d vB d v
s s s s s
s
11 111 0 12 66 122 0 12 66 112 0
22 222
2 2
+
(
+)
+(
+)
+ 00 11 1111 12 66 1122
22 2222 11 1111
2 2
− −
(
+)
− −
D d w D D d w
D d w H d w
s b s s
b
s b s
ss s s
s
s s s
s s
s
H H d w
H d w A d w A d w f
−
(
+)
− + + =
2 12 2 66 1122
22 2222 55 11 44 22 4
dij = ∂∂ ∂x x di j ijl = x x xi ∂j l
∂ ∂ ∂
2 3
, ,
dijlm x x x x d x i j l m
i j l m i i
= ∂
∂ ∂ ∂ ∂4 , = ∂∂ , ( , , , =1 2, ).
f N
x f N
y f q M
x
M y
f q M
x
xT yT
xbT ybT
xsT
1 2 3
2 2
2
4
2 2
= ∂∂ =∂ 2
∂ = − ∂
∂ −∂
∂
= − ∂
∂
, ,
−−∂
∂
2 2
M yy
sT
.
q T
q
t x y i
i i
= 0 =
1 sin(λ )sin(µ ), ( , 2, 3)
u v w w
U x y
V x
b s
mn mn 0
0
=
cos( )sin( ) sin( ) c
λ µ
λ oos( ) sin( )sin( ) sin( )sin( ) µ
λ µ
λ µ
y
W x y
W x y
b s
,
Κ ∆{ }=
{ }
f ,Κ Κ Κ
11 11 2
66 2
12 12 66
13 11 2
12 2 66
= − +
= − +
= + +
( )
( )
A A
A A
B B B
λ µ
λ µ
λ λ
[ ( )) ]
[ ( ) ]
µ
λ λ µ
λ µ
2
14 11 2
12 66 2
22 66 2
22 2 23
Κ 2 Κ Κ
= + +
= − +
=
(
AB AB)
Bs s s
µµ λ µ
µ λ µ
[( ) ]
[( ) ]
B B B
Bs Bs Bs
12 66
2 22
2
24 12 66 2
22 2 33
2 2
+ +
= + +
= Κ
Κ −− + + +
= − + +
(
D D D D)
Ds Ds Ds
11 4
12 66 2 2
22 4
34 11
4
12 66
2
2 2
2 2
λ λ µ µ
λ λ
( )
( )
Κ µµ µ
λ λ µ
µ λ
2 22
4
44 11 4
11 66 2 2
22 4
55 2
2 2
+
= − + +
− +
( )
( )
D
H H H
H A
s
s s s
s s
Κ ( )
( ++A44s µ2) δ
δ δ
u N
x N y v N
x N y
w M
x
x xy
xy y
b xb
0 0
2 2
0 0 :
: :
∂∂ +∂
∂ =
∂
∂ +∂
∂ =
∂
∂ +22 0
2
2 2
2 2
2
2 2
∂
∂ ∂ +∂
∂ + =
∂
∂ + ∂
∂ ∂ +∂ x yM M
y q
w M
x
x yM M
xyb yb
s xs xys ys
δ :
∂∂ + ∂∂ +∂
∂ + =
y S
x S y q
xzs yzs
2 0
v w w w
y w
y N M M at x a
u w w w
x w
b s b s
x xb
xs
b s b
0 0
0 0
= = = ∂∂ =∂
∂ = = = = =
= = = ∂∂ =∂
, ,
ss yb
ys
x N y M M at y b
∂ = = = =0 =0, ,
f A t B t aB t f A t B t aB t
T T T
T T T
1 1 2 3
2 1 2 3
= + +
= + +
( )
( )
λ µ
, , f q h B t D t D t
f q h B t D
T T a T
s T s
3 0 2 2
1 2 3
4 0 2 2
1
= − − + + +
= − − + +
( ) ( )
( )
λ µ λ µ
,
TTt2+s TF t3
( )
.(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
Where:
In which: z z h= / , α( ) ( )ν α
z = −E z1 ( )z , Ψ( )z =ψ( ) /z h and ξ( )z πsin π z
= h
1 .
4 Analytical validation and numerical results
For the plate in FGM, the material properties used in the present study are:
• Metal (Aluminum):
EM = 70 (GPa), v = 0.3; αM = 23 × (10–6/C°).
• Ceramics (Zirconia):
EC = 151 (GPa), v = 0.3; αC = 10 × (10–6/C°).
Numerical results are presented in terms of non-dimen- sional stresses and deflection. The different non-dimensional parameters used are:
• The central deflection:w D a q w a b
=
10
0 2 2
2
4 ,
• Axial stress: σx= q σxa b h
1
102 0 2 2 2, ,
• The transverse shear stress: τxz = − q τxz b
1
10 0 0
20 , ,
• The coordinate thickness: z z h D= = h EC
( )
−/ , .
3
12 1ν2
The numerical results are given and represented in Fig. 4–9 by using the present refined exponential shear deformation plate theory (RESDT), it does not require a shear factor of cor- rection and the number of unknown functions for the present theory of a high order is only four. Noted that the factor of shear correction is taken k = 5/6 in the first order shear deformation plate theory (FSDPT).
In Figures 8–9, it is important to observe that the stresses for a plate entirely in ceramics are not the same as for a plate entirely in metal. This is because the plate is subjected to a field of temperature.
The relation between the present theory and the various theories of high order shear deformations and first order and the classical plate theory (ESDPT, PSDPT, SSDPT, FSDPT, CPT) is illustrated in Fig. 4–9. These figures also give the effects ofthe variation of the volume fraction exponent values p on the dimensionless center deflection and stresses of the FGM rectangular plate. It is clear that the deflection decreases as much as the side-to-thickness ratio (a/h) increases.
Figures 4–6 shows an excellent agreement of this theory with the other high order shear deformation theories for a square FGM plate subjected to a mechanical load.
Fig. 4 Variation of the dimensionless center deflection (W) through the thickness of a square FGM plate (p = 2) for various theories and different
side-to-thickness ratio (a/h) with (q0 = 100, ti = 0).
Fig. 5 Variation of dimensionless axial stress (σx) through the thickness of a square FGM plate (p = 2) for various theories with with (q0 = 100, ti = 0).
Fig. 6 Variation of dimensionless shear stress (τxz) through the thickness of a square FGM plate (p = 2) for various theories with (q0 = 100, ti = 0).
Dimensionless axial stress (σx), is represented in Fig. 5 and Fig. 8. One can see that the maximum compressive stresses occur at a point near the upper surface while maximum tensile stresses are at a point close to the lower surface of the FGM plate.
Fig. 9 illustrate the dimensionless shear stress (τxz) dis- tributions through the thickness of a square and rectangular FGM plate under thermal loads.
A B DT T T z z z dz
h
{ }
h{ }
= ∫−
, , ( ) , ,
/
/ α 1 2
2 2
a T a T h
B D h z z z dz
{ } {
}= ∫−
, ( ) ( ) ,
/
/ α Ψ 1
2 2
s T s T s T h
B D F h z z z z dz
{ } { }
= ∫−
, , ( ) ( ) , , ( )
/
/ α Ψ 1 ξ
2 2
(39) (40)
(41)
Fig. 7 Effect of the volume fraction exponent p on the dimensionless center deflection (W) of a rectangular FGM plate for different side-to-thickness ratio
(a/h) with (q0 = 100, ti = 10, b = 2a).
Fig. 8 Effect of the volume fraction exponent p on dimensionless axial stress (σx) through-the-thickness of a rectangular FGM plate with
(q0 = 100, ti = 10, b = 2a).
Fig. 9 Effect of the volume fraction exponent p on the dimensionless shear stress (τxz) through-the-thickness of a rectangular FGM plate with
(q0 = 100, ti = 10, b = 2a).
Maximum values of (τxz) occur with (z≅0 1. ) of the FGM plate, not in the center of the plate as in the homogeneous case.
The deflection and two axial stresses and shear stresses increase with as the thermal load increases.
The effect of the mechanical and thermal loads is taken into consideration. The deflection is larger for plates subjected to thermal load only whereas it is smaller for plates subjected to mechanical load only. With the inclusion of all loads (q0 = 100, t1 = t2 = t3 = 10), the deflection decreases as a/h increase.
Fig. 10 Effect of the thermal field on the dimensionless center deflection (W) of a rectangular FGM plate for different side-to-thickness ratio (a/h) with
(p = 2, q0 = 100, b = 2a).
Fig. 11 Effect of the thermal field on dimensionless axial stress (σx) through- the-thickness of a rectangular FGM plate with (p = 2, q0 = 100, b = 2a).
Fig. 12 Effect of the thermal field on the dimensionless shear stress (τxz) through-the-thickness of a rectangular FGM plate with (p = 2, q0 = 100, b = 2a).
Finally, Figures 10–12 show the effect of the thermal field on the deflection and stresses. For FGM plates subjected to thermal load, the deflection may be stable for all values of a/h ≥ 5.
The deflection and both axial stresses and shear stresses increase with the increase of the thermal load t3.
The figures emphasize the great influence played by the dif- ferent thermal and bending loads on the analyzed axial and transverse shear stresses.
