# Stress and Strain Analysis of Functionally Graded Rectangular Plate with Exponentially Varying Properties

Authors: Amin Hadi; Abbas Rastgoo; A. R. Daneshmehr; Farshad Ehsani
Journal: Indian Journal of Materials Science
 (2013)

---

## Abstract

The bending of rectangular plate made of functionally graded material (FGM) is investigated by using three-dimensional elasticity theory. The governing equations obtained here are solved with static analysis considering the types of plates, which properties varying exponentially alongz direction. The value of Poisson’s ratio has been taken as a constant. The influence of different functionally graded variation on the stress and displacement fields was studied through a numerical example. The exact solution shows that the graded material properties have significant effects on the mechanical behavior of the plate.


---

## Body

## 1. Introduction

Recently, a new category of composite materials known as functionally graded materials (FGMs) has attracted the interest of many researchers. The FGMs are heterogeneous composite materials in which the mechanical properties vary continuously in certain direction. FGMs are used in many engineering applications such as aviation, rocketry, missiles, chemical, aerospace, and mechanical industries. Therefore, composites that are made of FGMs were considerably attractive in recent years.Several studies have been performed to analyze the behavior of functionally graded beam, plates, and shells. Hadi et al. [1, 2] studied an Euler-Bernoulli and Timoshenko beam made of functionally graded material subjected to a transverse loading at which Young’s modulus of the beam varies by specific function. Reddy [3] has analyzed the static behavior of functionally graded rectangular plates based on his third-order shear deformation plate theory. Cheng and Batra [4] have related the deflections of a simple supported functionally graded polygonal plate given by the first-order shear deformation theory and a third-order shear deformation theory to an equivalent homogeneous Kirchhoff plate. Cheng and Batra [5] also presented results for the buckling and steady state vibrations of a simple supported functionally graded polygonal plate based on Reddy’s plate theory. Loy et al. [6] studied the vibration of functionally graded cylindrical shells by using Love’s shell theory.Analytical 3D solutions for plates are useful because provided benchmark results to assess the accuracy of various 2D plate theories and finite element formulations. Cheng and Batra [7] used the method of asymptotic expansion to study the 3D thermoelastic deformations of a functionally graded elliptic plate. Recently, Vel and Batra [8] have presented an exact 3D solution for the thermoelastic deformation of functionally graded simple supported plates of finite dimensions. Reiter et al. [9] performed detailed finite element studies of discrete models containing simulated particulate and skeletal microstructures and compared the results with those that are computed from homogenized models in which effective properties were derived by the Mori-Tanaka and the self-consistent methods.Tanigawa [10] used a layerwise model to solve a one-dimensional transient heat conduction problem and the associated thermal stress problem of an inhomogeneous plate. He further formulated the optimization problem of the material composition to reduce the thermal stress distribution. Tanaka et al. [11, 12] have designed FGM property profiles by using sensitivity and optimization methods based on the reduction of thermal stresses. Jin and Noda [13] used the minimization of thermal stress intensity for a crack in a metal-ceramic functionally gradient material as a criterion for optimizing material property variation. In the same context, also were studied both the steady state [14] and the transient [15] heat conduction problems by them, but neglected the thermomechanical coupling (see also [16, 17]).The response of functionally graded ceramic-metal plates has been investigated by Praveen and Reddy [18] with using a plate finite element that accounts for the transversal shear strains, rotatory inertia, and moderately large rotations in von Kármán sense. Reddy and Chin [19] have studied the dynamic thermoelastic response of functionally graded cylinders and plates. Najafizadeh and Eslami [20] presented the buckling analysis of radially loaded solid circular plate that is made of functionally graded material.In this paper, an exact solution bending of rectangular plate made of functionally graded material (exponential form alongz direction) subjected to top and bottom pressures PT and PB, respectively, is investigated by using three-dimensional elasticity theory.


## 2. Analysis

The equilibrium of a weightless homogeneous transversally isotropic elastic FGM plate is considered. The geometry of the elastic FGM plate in relation to the coordinate axes is shown in Figure1.Figure 1
Geometry and boundary conditions of the plate.The plate is assumed under the action of top and bottom pressuresPT and PB, respectively,
(1)PT=PToSinπxaSinπyb,PB=PBoSinπxaSinπyb,
where PTo, and PBo are top and bottom pressures in x=a/2, y=b/2.In order to account for the changing material properties along thez direction, an exponential relationship is used as follows:
(2)E(z)=Eien(z/h).
Here Ei is the module of elasticity at z=0, and n is the inhomogeneity constants determined empirically. Equilibrium equations in three dimensions are defined as follows:
(3)∂σx∂x+∂τxy∂y+∂τxz∂z=0,∂τxy∂x+∂σy∂y+∂τyz∂z=0,∂τxz∂x+∂τyz∂y+∂σz∂z=0,
where σx, σy, and σz are normal stress components in x, y, and z direction, respectively. τxy, τyz, and τxz are shear stresses components.The displacement in thex, y, and z directions is denoted by u, v, and w, respectively, six strain components can be expressed as
(4)[εxλxyλxzλxyεyλyzλxzλyzεz]=[dudx12(dudy+dvdx)12(dudz+dwdx)12(dudy+dvdx)dvdy12(dvdz+dwdy)12(dudz+dwdx)12(dvdz+dwdy)dwdz],
where εx, εy, and εz are normal strain components in x, y, and z directions, respectively. λxy, λyz, and λxz are shear strain components.The stress-strain relations are(5)σx=E(z)(1-2υ)(1+υ)[(1-υ)∂u∂x+υ(∂v∂y+∂w∂z)],σy=E(z)(1-2υ)(1+υ)[(1-υ)∂v∂y+υ(∂u∂x+∂w∂z)],σz=E(z)(1-2υ)(1+υ)[(1-υ)∂w∂z+υ(∂u∂x+∂v∂y)],(6)τxy=G(z)[∂u∂y+∂v∂x],τxz=G(z)[∂u∂z+∂w∂x],τyz=G(z)[∂v∂z+∂w∂y],
where G(z)=E(z)/2(1+υ) is shear modulus, with E(z) the modulus of elasticity and υ Poisson’s ratio. The value of Poisson’s ratio has been taken as constant.The boundary conditions of the problem are the following. Along the sides of the plate, we have(7)x=0,a⟶w=0,σx=0,x=0,b⟶w=0,σy=0
the boundary conditions on the plate faces are as below:
(8)z=h2⟶σz=PT,τxz=τyz=0,z=-h2⟶σz=PB,τxz=τyz=0.
By the displacement field below, the boundary conditions (7) are satisfied:
(9)u=u(x,y,z)=U(z)cosπxasinπyb,v=v(x,y,z)=V(z)sinπxacosπyb,w=w(x,y,z)=W(z)sinπxasinπyb.
Substituting (9) into (5) and the resulting expressions for stress components into equilibrium equations (3), we see that the equilibrium equation is expressed as system of second-order ordinary differential equations:
(10)d2Udz2+nhdUdz+J1U+J3V-bπJ3dWdz+πnahW=0,d2Vdz2+nhdVdz+J2V+J3U-aπJ3dWdz+πnbhW=0,bπJ3dUdz+J4bU+aπJ3dVdz+J4aV+J5d2Wdz2+J5nhdWdz+J6W=0,
where
(11)[J1J2J3J4J5J6]=[-π2(1b2+2(1-υ)a2(1-2υ))-π2(1a2+2(1-υ)b2(1-2υ))-π2ab(11-2υ)-2nυπabh(1-2υ)2(1-υ)1-2υ-π2a2b2(a2+b2)].
Therefore
(12)(S2+nhS+J1)U+J3V+(-bπJ3S+πnah)W=0,J3U+(S2+nhS+J2)V+(-aπJ3S+πnbh)W=0,(bπJ3S+J4b)U+(aπJ3S+J4a)V+(J5S2+J5nhS+J6)W=0,S=ddz.
The general solution of (10) is as follows:
(13)U=C1es1z+C2es2z+C3es3z+C4es4z+C5es5z+C6es6z,V=C7es1z+C8es2z+C9es3z+C10es4z+C11es5z+C12es6z,W=C13es1z+C14es2z+C15es3z+C16es4z+C17es5z+C18es6z,
where s1,s2,…,s6 are the roots of the equation below:
(14)det[S2+nhS+J1J3-bπJ3S+πnahJ3S2+nhS+J2-aπJ3S+πnbhbπJ3S+J4baπJ3S+J4aJ5S2+J5nhS+J6]=0,
where C1,C2,…,C18 are arbitrary integration constants.The resulting displacement field is defined as follows:(15)u(x,y,z)=[C1es1z+C2es2z+C3es3z+C4es4z+C5es5z+C6es6z]CosπxaSinπyb,v(x,y,z)=[C7es1z+C8es2z+C9es3z+C10es4z+C11es5z+C12es6z]SinπxaCosπyb,w(x,y,z)=[C13es1z+C14es2z+C15es3z+C16es4z+C17es5z+C18es6z]SinπxaSinπyb.
Substituting (15) into (5), (6) stress component is calculated as follows:(16a)
σ
x
=
E
i
e
n
z
/
h
Sin
(
π
x
/
a
)
Sin
(
π
y
/
b
)
(
1
-
2
υ
)
(
1
+
υ
)
×
(
-
(
1
-
υ
)
π
(
C
1
e
s
1
z
+
C
2
e
s
2
z
+
C
3
e
s
3
z
+
C
4
e
s
4
z
+
C
5
e
s
5
z
+
C
6
e
s
6
z
)
a
-
υ
(
C
7
e
s
1
z
+
C
8
e
s
2
z
+
C
9
e
s
3
z
+
C
10
e
s
4
z
+
C
11
e
s
5
z
+
C
12
e
s
6
z
)
π
b
+
υ
(
C
13
e
s
1
z
s
1
+
C
14
e
s
2
z
s
2
+
C
15
e
s
3
z
s
3
+
C
16
e
s
4
z
s
4
+
C
17
e
s
5
z
s
5
+
C
18
e
s
6
z
s
6
)
)
,
(16b)
σ
y
=
E
i
e
n
z
/
h
Sin
(
π
x
/
a
)
Sin
(
π
y
/
b
)
(
1
-
2
υ
)
(
1
+
υ
)
×
(
-
(
1
-
υ
)
π
(
C
7
e
s
1
z
+
C
8
e
s
2
z
+
C
9
e
s
3
z
+
C
10
e
s
4
z
+
C
11
e
s
5
z
+
C
12
e
s
6
z
)
b
-
υ
(
C
1
e
s
1
z
+
C
2
e
s
2
z
+
C
3
e
s
3
z
+
C
4
e
s
4
z
+
C
5
e
s
5
z
+
C
6
e
s
6
z
)
π
a
+
υ
(
C
13
e
s
1
z
s
1
+
C
14
e
s
2
z
s
2
+
C
15
e
s
3
z
s
3
+
C
16
e
s
4
z
s
4
+
C
17
e
s
5
z
s
5
+
C
18
e
s
6
z
s
6
)
)
,
(16c)
σ
z
=
E
i
e
n
z
/
h
Sin
(
π
x
/
a
)
Sin
(
π
y
/
b
)
(
1
-
2
υ
)
(
1
+
υ
)
×
(
-
υ
π
(
C
7
e
s
1
z
+
C
8
e
s
2
z
+
C
9
e
s
3
z
+
C
10
e
s
4
z
+
C
11
e
s
5
z
+
C
12
e
s
6
z
)
b
-
υ
π
(
C
1
e
s
1
z
+
C
2
e
s
2
z
+
C
3
e
s
3
z
+
C
4
e
s
4
z
+
C
5
e
s
5
z
+
C
6
e
s
6
z
)
a
+
(
1
-
υ
)
(
C
13
e
s
1
z
s
1
+
C
14
e
s
2
z
s
2
+
C
15
e
s
3
z
s
3
+
C
16
e
s
4
z
s
4
+
C
17
e
s
5
z
s
5
+
C
18
e
s
6
z
s
6
)
)
,
(16d)
τ
x
y
=
E
i
e
n
z
/
h
π
Cos
(
π
x
/
a
)
Cos
(
π
y
/
b
)
2
(
1
+
υ
)
(
C
7
e
s
1
z
+
C
8
e
s
2
z
+
C
9
e
s
3
z
+
C
10
e
s
4
z
+
C
11
e
s
5
z
+
C
12
e
s
6
z
a
+
C
1
e
s
1
z
+
C
2
e
s
2
z
+
C
3
e
s
3
z
+
C
4
e
s
4
z
+
C
5
e
s
5
z
+
C
6
e
s
6
z
b
)
,
(16e)
τ
x
z
=
E
i
e
n
z
/
h
π
Cos
(
π
x
/
a
)
Sin
(
π
y
/
b
)
2
(
1
+
υ
)
×
(
C
13
e
s
1
z
+
C
14
e
s
2
z
+
C
15
e
s
3
z
+
C
16
e
s
4
z
+
C
17
e
s
5
z
+
C
18
e
s
6
z
a
+
C
1
s
1
e
s
1
z
+
C
2
s
2
e
s
2
z
+
C
3
s
3
e
s
3
z
+
C
4
s
4
e
s
4
z
+
C
5
s
5
e
s
5
z
+
C
6
s
6
e
s
6
z
π
)
,
(16f)
τ
y
z
=
E
i
e
n
z
/
h
π
Sin
(
π
x
/
a
)
Cos
(
π
y
/
b
)
2
(
1
+
υ
)
×
(
C
13
e
s
1
z
+
C
14
e
s
2
z
+
C
15
e
s
3
z
+
C
16
e
s
4
z
+
C
17
e
s
5
z
+
C
18
e
s
6
z
b
+
C
7
s
1
e
s
1
z
+
C
8
s
2
e
s
2
z
+
C
9
s
3
e
s
3
z
+
C
10
s
4
e
s
4
z
+
C
11
s
5
e
s
5
z
+
C
12
s
6
e
s
6
z
π
)
.The resultant moments on a unit of length are obtained by using relations (16a)–(16f):(17a)
M
x
=
E
i
Sin
(
π
x
/
a
)
Sin
(
π
y
/
b
)
(
1
-
2
υ
)
(
1
+
υ
)
×
[
∑
i
=
1
6
(
a
b
(
n
+
h
s
i
)
2
)
[
e
(
h
/
2
)
(
n
/
h
+
s
i
)
h
(
h
n
2
+
h
(
-
1
+
h
s
i
2
)
)
(
-
a
C
i
+
6
υ
π
+
b
(
C
i
(
-
1
+
υ
)
π
+
a
C
i
+
12
υ
s
i
)
)
-
e
-
(
h
/
2
)
(
n
/
h
+
s
i
)
h
(
-
h
n
2
+
h
(
-
1
-
h
s
i
2
)
)
(
-
a
C
i
+
6
υ
π
+
b
(
C
i
(
-
1
+
υ
)
π
+
a
C
i
+
12
υ
s
i
)
)
]
]
,
(17b)
M
y
=
E
i
Sin
(
π
x
/
a
)
Sin
(
π
y
/
b
)
(
1
-
2
υ
)
(
1
+
υ
)
×
[
∑
i
=
1
6
1
a
b
(
n
+
h
s
i
)
2
[
e
(
h
/
2
)
(
n
/
h
+
s
i
)
h
(
h
n
2
+
h
(
-
1
+
h
s
i
2
)
)
(
-
b
C
i
υ
π
+
a
(
C
i
+
6
(
-
1
+
υ
)
π
+
b
C
i
+
12
υ
s
i
)
)
-
e
-
(
h
/
2
)
(
n
/
h
+
s
i
)
h
(
-
h
n
2
+
h
(
-
1
-
h
s
i
2
)
)
(
-
b
C
i
υ
π
+
a
(
C
i
+
6
(
-
1
+
υ
)
π
+
b
C
i
+
12
υ
s
i
)
)
]
]
,
(17c)
M
x
y
=
E
i
π
Cos
(
π
x
/
a
)
Cos
(
π
y
/
b
)
2
(
1
+
υ
)
×
[
∑
i
=
1
6
h
(
a
C
i
+
b
C
i
+
6
)
(
n
+
h
s
i
)
2
[
e
(
h
/
2
)
(
n
/
h
+
s
i
)
(
h
n
2
+
h
(
-
1
+
h
s
i
2
)
)
-
e
-
(
h
/
2
)
(
n
/
h
+
s
i
)
(
-
h
n
2
+
h
(
-
1
+
h
s
i
2
)
)
]
]
.Using (16a), (16b), (16c), (16d), (16e), and (16f) the transverse shearing forces on a unit of length are by definition(18a)
Q
x
=
E
i
Cos
(
π
x
/
a
)
Sin
(
π
y
/
b
)
2
a
(
1
+
υ
)
[
∑
i
=
1
6
h
(
C
i
+
12
π
+
a
C
i
s
i
)
n
+
h
s
i
[
e
(
h
/
2
)
(
n
/
h
+
s
i
)
-
e
-
(
h
/
2
)
(
n
/
h
+
s
i
)
]
]
,
(18b)
Q
y
=
E
i
Sin
(
π
x
/
a
)
Cos
(
π
y
/
b
)
2
b
(
1
+
υ
)
[
∑
i
=
1
6
h
(
C
i
+
12
π
+
b
C
i
+
6
s
i
)
n
+
h
s
i
[
e
(
h
/
2
)
(
n
/
h
+
s
i
)
-
e
-
(
h
/
2
)
(
n
/
h
+
s
i
)
]
]
.


## 3. Results and Discussion

In the following, the obtained solution will be employed to analyze the effect of material inhomogeneity on the elastic field in the rectangular plate. Consider a rectangular plate with lengtha=1.5m, width b=1m, and thickness h=0.2m, with material property Ei=70GPa, subjected to top and bottom pressures PTo=-3×106Pa and PBo=-1.5×106Pa, respectively. It is assumed the Poisson’s ratio υ has a constant value of 0.3. Dimensionless and normalized variables are used. For different values of n, dimensionless modulus of elasticity along the z direction is plotted in Figure 2. According to this figure, at the same position -0.5≤z/h≤0, dimensionless modulus of elasticity is increasing as the parameter nisdecreasing, while for 0≤z/h≤0.5, dimensionless modulus of elasticity is increasing as the parameter nisincreasing.Figure 2
Distribution of modulus of elasticity.Figure3 displays the nondimensional displacement of the plate in z direction for different values of parameter n. This plot displays that the magnitude of changing w by z is low, so for, n<0 assuming plane strain is true and reasonable, but, for n>0, the amount of changing w by z is high.Figure 3
Distribution of displacement inz direction of the plate versus z/h at x=a/2, y=b/2.In Figure4 is shown the nondimensional displacement in the x direction versus x/a for z=h/3, y=b/2. In this plot displacement is decreasing as the parameter n is increasing. In Figure 5 is shown the nondimensional displacement in the y direction versus y/b for z=h/3, x=a/2.Figure 4
Distribution of displacement inx direction of the plate versus x/a at y=b/2, z=h/3.Figure 5
Distribution of displacement inx direction of the plate versus y/b at x=a/2, z=h/3.In Figure6 is shown the variation of nondimensional stress in the z direction versus nondimensional thickness at the x=a/2, y=b/2. This plot shows that the boundary conditions at the up and down surfaces are satisfied. Also at the constant z, by increasing the parameter n, it is observed the stress is decreased.Figure 6
Distribution of nondimensional stress in thez direction versus z/h at x=a/2, y=b/2.Figure7 shows the (τxz/PTo)×1018 according to the z/h for the variable amount of n: this figure probes that the stress’s component can be deleted proportionally to other components.Figure 7
Distribution ofτxz/PTo versus z/h at x=a/2, y=b/2.


## 4. Conclusion

It is apparent that close form solutions are important to simplified kinds of real engineering problems. In this paper is studied the rectangular plate that is made of functionally graded material with the variable properties (exponential form) by using 3D elasticity theory. Then some exact solution packages for stresses, displacements are presented.To show the effect of inhomogeneity on the stress distributions, different values were considered for material inhomogeneity parametern. The presented results show that the material inhomogeneity has a significant influence on the mechanical behaviors of the solid rectangular plate made of exponentially FG.



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*Source: 206239-2013-08-18.xml*