a new class of composites functionally graded material.docx
TRANSCRIPT
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Analysis of Functionally Graded Materials
1. Introduction
A new class of composites functionally graded material (FGM) is a two component composite
characterized by a ratio that is continuously varying from 100% of one component through to
100% of another component that can be defined by a function. FGM may possess a number of
advantages over traditional composites including reduction in in-plane and transversal stresses
along its thickness, improved thermal properties and reduced stress concentration. Due to its
favorable properties in engineering applications FGM has drawn a considerable attention from
numerous researchers. A comprehensive overview on applications and processing technique of
FGM is presented by Mahamood et al.[1] . Similarly B. Kieback [2] et al. discussed different
types of FGM manufacturing methods.
2. Analysis of FGM
A new beam element is proposed to study the thermo- elastic behavior of functionally graded
beam structures by Chakraborty et al. [3]. J. Murin andV. Kutis [4] presented thermo-elastic
analysis of a multilayered FGM beam in which continuous variation of effective material
properties is taken in both longitudinal and transversal directions. Butcher et al. [5] prepared and
investigated the material properties of particulate FGM structure experimentally. A finiteelement method based micromechanical analysis is proposed in by Lee et al.[6] to understand the
fracture behavior of functionally graded foams. Sina et al. [7] presented a new beam theorywhich is used to analyze free vibration of functionally graded beams. The beam properties are
assumed to be varied through the thickness following a simple power law distribution in terms of
volume fraction of material constituents. An analytical method is developed for temperature-
dependent free vibration analysis of functionally graded beams by Mahi et al. [8]. The
formulation used is based on a unified higher order shear deformation theory. Becker et al. [9]
proposed an approximate solution for residual stress calculation in functionally graded materials.
Giunta1 et al. [10] proposed several axiomatic refined theories for the linear static analysis of
beams made of functionally graded materials. The analysis of static deformations of functionally
graded plates is performed by using the collocation method, the radial basis functions and a
higher-order shear deformation theory by Ferreira et al. [11]. Mechanical behavior of rectangular
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and circular plates made of functionally graded materials (FGMs) is investigated by Momennia
and Akbarzadeh [12]. The analysis is based on the finite element approach using Abaqus and the
results are validated using analytical solutions reported in the literature. Komeili et al. [13] made
attempt to investigate bending of functionally graded piezoelectric beams based on the Euler-
Bernoulli beam theory under mechanical loads.
3. Effective Properties of FGM
Effective properties of FGM may be obtained using basic three laws i.e. Power Law (P-FGM),
Exponential Law (E-FGM) and Sigmoid Law (S-FGM). Khorshidv and Eslami [14] compared
solutions of all three laws for thermal buckling solution of circular FGM plate.
In Power Law (P-FGM), a model is created that describes the function of composition
throughout the material. In Figure 2, the volume fraction Vc, describes the volume of ceramic at
any point zacross, the thickness h according to a parametern which controls the shape of the
function [15].
Vc (z) = (1)
z Vc (h/2) = 1
h/2
h/2
Vc (-h/2) = 0
Figure 1 Ceramic Volume Fractions across the FGM Layer
It follows that the volume fraction of metal, Vm(z), in the FGM is 1-Vc(z).
() ( )() (2)Pc is the material property of the pure ceramic and Pm is the material property of the pure metal.
0
Vc (z)
Graded Layer
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Figure 2 Effect of Power Law Index (n) on the Volume Fraction Vc
Exponential Law (E-FGM) is also used to describe the material properties of FGM. Cited paper
[16] studied the influence of an exponential volume fraction law on the vibration frequencies of
thin functionally graded cylindrical shells. It directly correlates the properties of FGM with
ceramic and metal properties
() () (3)Where
and
Chung and Chi[17] defined the Sigmoid Law (S-FGM) in which volume fraction is calculated
using two power-law functions to ensure smooth distribution of stresses among all the interfaces.
The two power law functions are defined by:
() For (4)() For (5)By using rule of mixture, the Youngs modulus of the S-FGM can be calculated by:
() () () for (6)() () () for (7)Farhatnia et al.[18] presented an an alytical solution which is based on simple Euler-Bernoulli
type beam theory for long, slender beam. The principle of stationary potential function is used to
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
z/h
Vm
n=2
n=1
n=5
n=0.2n=0.3
n=0.4
n=3
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obtain the static finite element equations for the FGM composite beam. The Farhatnia model
consists following basic assumptions
1. The beam is assumed to be in a state of plane strain, it is normal to the xz plane.2. Euler-Bernoulli type beam theory is applied.3. There is no variation in thickness along the length of beam.4. Poissons ratio is to be held constant along FG layer.5. Material properties are independent of temperature gradient.
For a cantilever beam, the displacement field can be written as:
w (x, z) = w(x)
u (x, z) = u0(x)z
()
(8)
In above equations, u and w are denoted as horizontal and vertical displacement of beam across
the thickness. It may be noted that u0 denotes displacement of points on the middle surface of the
beam along the x direction. It is assumed that zz is negligible. Then the stress-strain relations
take the form:
() () () () (9)Where the plane strain Young modulus is given by:
The expressions for axial strain and stress can be derived as:
() () () , () () () () (10)
(11)
[] [ ] , , ()
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Here,[] and [] both are stiffness matrices and , k are axial strain in the middle surfaceand the beam curvature. According to Euler-Bernoulli beam theory, the axial force and bending
moment, N and M, are defined
(N, M) = [() ()]() (12)Since the axial force resultant is zero, the expressions for the deformation take the form (13)C0, C1, and C2 are the coefficients of mid-plane strain and curvature. Using Equation (9), the
axial stresses in ceramic, metal and FGM section across the thickness of proposed model are
obtained.
Yaghoobi[19] investigated the effect of varying material parameters on position of neutral
surface and it is suggested that position of neutral surface is shifted. The position of neutral
surface for functionally graded beam is obtained by considering FGM beam under uniformly
distributed load. The position of neutral surface is given by following expression
()
()(14)
Comparison with Fatemeh Farhatina Model (Mechanical Loading)
Comparing the axial stress obtained in present work with the results obtained by Farhatina et al.
for a three-layered beam model where the transition is made thin FGM layer. The material
considered and their properties are given in Table 1. A functionally graded cantilever composite
beam subjected to uniformly distributed load is considered as shown in Figure 2. An FGM
interlayer is placed in between these layers. Material properties vary according to the power law.
Table 2 shows the modulus of elasticity obtained for three-layer FGM beam
The stress distribution for two layered and three-layer composite is presented in Figure 3. The
observations are summarized below:
1. In the absence of FGM layer between the ceramic and metal layer, the stress distributionsare discontinuous at the interface.
2. The introduction of a small FGM layer smoothens the stresses.
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3. Axial stresses vary linearly with z within the metallic ceramic layer, and approximatelyparabolically within the functionally graded layer.
The stress distribution reasonably matches with that of Fatemeh Farhatina model [18] (Figure.3).
Table 1 Thermo-Elastic Properties for Metallic (Steel) and Ceramic (Al2O3) Phases
Material ( c-
) E (GPa)
Al2O3(ceramic) 6.9 10-
0.25 390
Steel(metal) 14 10-
0.25 210
CERAMIC (Al2O3)
METAL (Steel)
FGM
x
z
h = 0.02m
I = 0.5m
Figure 2 Three Layered Composite FGM Beam
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Table 2Youngs Modulus across the Thickness of Three Layered Composite FGM Beam.
Thickness
(z) m
Modulus of Elasticity
(E) GPa
0.01 390
0.005 390
0.00375 360.96
0.0025 334.08
0.00125 309.21
0 286.18
-0.00125 264.87
-0.0025 245.15
-0.00375 226.90
-0.005 210
-0.01 210
Table 3 Axial Stress of Three Layered Composite FGM Beam across the Thickness
Thickness (z)m Ceramic (x)
Pa
FGM (x)
Pa
Metal (x)
Pa
0.01 2.230910
---- ----
0.005 0.934610 ---- ----
0.00375 ---- 0.565110 ----
0.0025 ---- 0.245410 ----
0.00125 ---- -0.029810 ----
0 ---- -0.265410 ----
-0.00125 ---- -0.465710 ----
-0.0025 ---- -0.634710 ----
-0.00375 ---- -0.776010 ----
-0.005 ---- ---- -0.892710
-0.01 ---- ---- -1.590710
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(a)
(b)
Figure 3 (a) Axial Stress Distribution with FGM Beam, (b) Reprinted From ref. [18]
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 108
-0.01
-0.005
0
0.005
0.01
Axial Stress(Pa)
depth(m)
steel
FGM
Alumina
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Neutral Surface Positioning
Yaghoobi[19] investigated the effect of varying material parameters on position of neutral
surface and it is suggested that position of neutral surface is shifted. The position of neutral
surface for functionally graded beam is obtained by considering FGM beam under uniformly
distributed load. The position of neutral surface is given by following expression
()
()(14)
-6 -5 -4 -3 -2 -1 0 1 2 3 4
x 107
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
Axial Stress (Pa)
Depth(m)
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Consider a functionally graded cantilever beam of length l, width b, thickness h, with co-ordinate
system oxyzhaving the origin o as shown in Figure 4. The beam is subjected to a uniformly
distributed load qand the Youngs modulus E varies continuously in the thickness direction by
equation (2).
() ( ) (15)
Figure 4 Shifting of neutral surface in FGM beam
To determine the position of neutral surface construct a new co-ordinate system such that the
new axis is placed at the neutral axis, which will be determined below. Then we have where
, (16) is the distance of the neutral surface from the mid-plane of the beam. Similar to the usualtreatment in the Euler-Bernoulli beam theory the strain and stress can be expressed as
(17)
() (18)Here the small deformation assumption has been employed. The position of the neutral surface
can be determined by choosing h0such that the total axial force at cross-section vanishes.
() (19)
()
()
Where w is the deflection of the functionally graded beam and is the curvature radius of the
neutral surface.
By changing the interval of integral
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()
Then
() ()
The position of neutral surface can be determined by below equation
()
()(20)
Substituting Equation 15 into Equation 20 and integrating, gives neutral surface non-dimensional
shift (h0/h) in the proposed model.
()()()
()() (21)By using this expression for proposed model value ofh0 is obtained as 0.0003614 m for n=.4.
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0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
n
h0/h
Ec/Em=10
Ec/Em=8
Ec/Em=6
Ec/Em=2
Ec/Em=4
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( ())(())
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 108
-0.01
-0.005
0
0.005
0.01
Stresses given by exponential law
Stresses given by power law
Ceramic layer
FGM layer
Metal layer
-0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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1 1 1 1 1 1 1 1 1 1 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1