Dissertation

1. M. Tech Dissertation presentation
on
“THERMAL ANALYSIS OF FUNCTIONALLY
GRADED MATERIAL(FGM) PLATE USING FINITE
ELEMENT METHOD(FEM)”
DEENBANDHU CHHOTU RAM UNIVERSITY
OF SCIENCE & TECHNOLOGY, MURTHAL
SUBMITTED BY UNDER THE GUIDANCE OF:
RAJANI DALAL DR. RAJKUMAR
ROLL NO. 11001504028 (PROFESSOR)
MECHANICAL DEPARTMENT
DR. SURESH KUMAR VERMA
ASSOCIATE PROFESSOR
MECHANICAL DEPARTMENT

2. CONTENTS
 PROBLEM DEFINITION
 OBJECTIVES
 LITERATURE REVIEW
 INTRODUCTION TO FUNCTIONALLY GRADED
MATERIALS
 THEOROTICAL FORMULATION OF FGMs
 FINITE ELEMENT FORMULATION OF CONDUCTION IN
FGMs
 METHODOLOGY
 ANALYSIS AND RESULTS
 CONCLUSIONS
 FUTURE SCOPE OF WORK
 REFERENCES

3. PROBLEM DEFINITION
• Pure metals find little use in engineering applications.
• Alloys of quite dissimilar metals are difficult to combine.
• Composite materials undergo delamination under high
temperature applications.
• Hence FGMs were proposed in 1980s.
• FGM are non-homogenous and it is difficult to evaluate
characteristics analytically.
• An important numerical tool FEM is used to evaluate
thermal characteristics of FGMs.

4. OBJECTIVES
 Literature Review : Study the previous papers on thermal analysis
in functionally graded materials
 Mathematical modeling of the problem.
 Development of computer code for the FEM model.
 Validation of the developed computer code.
 Results and Discussion : Determination of temperature profile and
comparing it with conventional materials

5. LITERATURE REVIEW
 Cho et al[5] developed a numerical technique for finite element analysis of the thermal
characteristics of functionally graded materials and investigated the effect of significant
governing parameters such as variation function of material composition and relative
thickness of FGM layer inserted between metal and ceramic layers. Isoparametric bilinear
two-dimensional quadrilateral element was chosen for finite element mesh in time domain
and then Galerkin variational formulation in space coordinates was done. It was concluded
that considerable improvement is possible by inserting FGM layer between metal and
ceramic layers in classical biomaterial layered composites.
 Z.S. Shao [19] presented the solutions of temperature, displacements, and
thermal/mechanical stresses in a functionally graded circular hollow cylinder by using a
multi-layered approach in which it was assumed that the hollow cylinder is composed of 10
fictitious layers. It was shown that due to non homogeneity of the material properties the
variation of temperature is not linear through the thickness direction.
 Bao et. al. [20] established a solution method for the one-dimensional (1D) transient
temperature and thermal stress fields in FGMs. Finite-element method is used for space
discretization which results in a system of first-order differential equations. Transient
solutions of these equations were obtained using either finite-difference method or mode
superposition.

6.  Yin et. al.[21] proposed a multiscale modeling method to derive effective thermal
conductivity in two-phase graded particulate composites. In the particle-matrix zone, a
graded representative volume element is constructed to represent the random
microstructure at the neighborhood of a material point. At the steady state, the particle’s
averaged heat flux is solved by integrating the pairwise thermal interactions from all other
particles. The homogenized heat flux and temperature gradient are further derived,
through which the effective thermal conductivity of the graded medium is calculated. In
the transition zone, a transition function is introduced to make the homogenized thermal
fields continuous and differentiable. By means of temperature boundary conditions, the
temperature profile in the gradation direction was solved. Parametric analyses and
comparisons with other models and available experimental data were presented and
validated.
Wang et al.[28] proposed an efficient meshless method for transient heat transfer and
thermoelastic analysis of FGMs. The analog equation method is used to obtain an
equivalent homogenous system to the original non-homogenous governing equation, after
which radial basis functions and fundamental solutions are used to construct the related
approximated solutions of particular part and complementary part, respectively. Finally,
all unknowns are determined by satisfying the governing equations at interior points and
boundary conditions at boundary points. Numerical experiments showed that a good
agreement was achieved between the results obtained from the proposed meshless method
and available analytical solutions. The appropriate graded parameter can lead to different
temperature distribution, low stress concentration and little change in the distribution of
stress fields in the domain under consideration.

7.  Yangzin et. al.[29] discuss the steady heat conduction problem of a Ti-6Al-4V/ZrO2
composite FGM plate under heating boundary by the FEM. They showed that the
temperature distribution of the three-layered composite FGM plate is very gentle and
smooth Compared with the nongraded two-layered composite plate. Also, the variation of
temperature with the change in FGM layer thickness, composition and porosity were
shown with the help of FEM model.
 H. Nguyen-Xuan et al.[31] paper presented an improved finite element approach in which
a node-based strain smoothing is merged into shear-locking-free triangular plate elements.
The formulation used only linear approximations and its implementation into finite
element programs is quite simple and efficient. The method was then applied for static,
free vibration and mechanical/thermal buckling problems of functionally graded material
(FGM) plates. In the FGM plates, the material properties were assumed to vary across the
thickness direction by a simple power rule of the volume fractions of the constituents. The
behavior of FGM plates under mechanical and thermal loads was numerically analyzed in
detail through a list of benchmark problems. The numerical results showed high reliability
and accuracy of the present method compared with other published solutions in the
literature.

8.  Yang et. al.[37] focused on the finite element simulation on thermal stress for W/Cu FGM
with different graded layers, composition and thicknesses. In addition, the variance of
stresses for functionally graded coatings with the steady state heat flux were simulated by
finite element analysis (ANSYS Workbench). The results showed that the W/Cu FGM was
effectively beneficial for the stress relief of W coating. Meanwhile, the maximum von
mises stress decreased approximately by 52.8 % compared to monolithic W plasma facing
material. And the four-layer FGM with a compositional exponent of 2 was optimum for
1.5 mm W coating.
 Wu-Xiang[38] studied the thermal conduction behavior of the three-dimensional
axisymmetric functionally graded circular plate under thermal loads on its top and bottom
surfaces. A temperature function that satisfies thermal boundary conditions at the edges
and the variable separation method were used to reduce equation governing the steady
state heat conduction to an ordinary differential equation (ODE) in the thickness
coordinate which was solved analytically. Next, resulting variable coefficients ODE due to
arbitrary distribution of material properties along thickness coordinate was also solved by
the Peano-Baker series. The numerical results confirm that the influence of different
material distributions, gradient indices and thickness of plate to temperature field in plate
cannot be ignored.

9. INTRODUCTION
 2-phase composites
 Continuously varying volume
fractions
 Material properties vary with
location
 The matrix alloy (the metal), the
reinforcement material (the
ceramic), the volume, shape, and
location of the reinforcement,
and the fabrication method can
all be tailored to achieve
particular desired properties.
 Naturally occurring FGM :
bamboo, bone Fig 1: Functionally graded material

10. ORIGIN OF FGM
PROBLEM: EXTREME WORKING CONDITIONS
COMPOSITES
DELAMINATION
PROJECT: Niino of NAL INTERIOR
(1984) 2000 K 1000K OF SPACE
PLANE
10mm
Functionally Graded Materials 107/11/2012

11. STRUCTURE
 Composed of a ceramic
and a metal
 Material transitions from
0% at 1 end to 100% to
the other end
 FACE TEMP
Ceramic High
Metallic Lower
 The smooth transition of
material provide thermal
protection as well as
structural integrity
Continuously graded
microstructure of FGMs
[Photo courtesy of NASA]

12. HOW FGM DIFFERS FROM
TRADITIONAL COMPOSITE
Schematic structure, elastic modulus (-) and thermal conductivity (---) of
an FGM (a) and a homogeneous material (b). (Cherradi,1994)

13. ADVANTAGES OF FGM
(i) Thermal stresses can be reduced;
(ii) Thermal stresses at critical locations can be reduced;
(iii) Stress jumps at the interface can be avoided;
(iv) The driving force for crack extension, the stress intensity factor,
can be reduced; and
(v) The strength of the interfacial bond can be increased.

14. APPLICATIONS OF FGM
 FG Thermal Barrier
coatings(TBCs) for turbine
blades
 FG thermal protection systems
for spacecraft
 FG prosthesis joint increasing
adhesive strength and
reducing pain
 FG polyester-calcium
phosphate materials for bone
replacement
 FG layer between the Cr–MO
shank and ceramic tip of a
cutting tool improving the
thermal strength.
FGM application for a turbine blade
design[Birman,2007]
FGM application for relaxation of stress
concentration in lathe bits[Birman,2007]

15. TYPES
 Power law type (P-type):
Figure Illustration of FGM structure
Vc(x)= volume of ceramic at any point x throughout the thickness L
Pc = Property of ceramic, Pm= Property of metal, n= power law index
 Exponential type(E-type):

16. Effect of grading parameter “n”
on the volume fraction Vc [Naghdabadi,2005]
Variation of the effective material property vs
the non-dimensional thickness[Wang, 2005]

17. FINITE ELEMENT MODELLING FOR
FGMS
 How to model a material
with continuously varying
properties?
 The simplest approach is to
use homogeneous elements
each with different
properties, giving a
stepwise change in
properties in the direction
of the material gradient.

18. Assumptions :
 There are no heat sources within the plate.
 Material’s properties for each same ordinate x are
homogenous and isotropic.
 Creeps are neglected and perfect bonding.
 Temperature independent material constants.
 Initially stress free state.
 The width of the plate is assumed to be infinite

19. ELEMENT EQUATIONS
 Direct approach will be used
 1- D heat flow under steady conditions
 Discretization
 Fourier’s law:
q= heat flux(W)
kx = thermal conductivity of the material that varies along the thickness
direction, x (W/mK-1 ),
A = area normal to the heat flow
Kc & Km = thermal conductivity of ceramic and metal respectively.
Vc(x) = ceramic volume fraction along the thickness direction and
n= power law index.

20.  Nodal heat flow entering a typical node
 conservation of energy requires Q2=-Q1
 In matrix notation
 Or [Ke]{Te}={Qe}
where [Ke] = element thermal conduction stiffness matrix,
{Te} = element column vector of nodal temperatures and
{Qe} = element column vector of nodal heat fluxes

21. 7. METHODOLOGY
 FEM:Direct Approach
 Step 1 : Discretization: element chosen is 1-D, 2 node element
 Step 2: Constitutive relations for element stiffness matrix
Type of FGM chosen: P Type
Volume fraction equation
Rule of mixtures for effective properties
General 1-D heat conduction equation
 Step 3 : Assembly of element equations
 Step4 : Apply boundary conditions
 Step 5 : Solve for the unknowns i.e, the temperature at each node point.
Hence the temperature profile for a particular FGM can be simulated
 The FEM code is developed in Microsoft Visual C++ 6.0 environment.

22. 8. ANALYSIS & RESULTS
Properties Aluminium Zirconium
oxide(ZrO2 )
E (Gpa) 70 200
k (W/mK) 204 2.09
ρ (Kg/m3) 2707 5700
α 23x10-6/0C 10x10-6/0C
TABLE : MATERIAL PROPERTIES
Problem To determine the temperature distribution of the given plate subject
to constant temperatures at both ends.
10 mm

23. Temperature distribution through the thickness of Al/ZrO2 plate
0
50
100
150
200
250
300
350
0 0.2 0.4 0.6 0.8 1 1.2
Temperature(0C)
Non-dimensional thickness, x/L
n=0
n=0.5
n=1
n=2
n=5
n=15

24. VALIDATION
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
temperature(0C)
x/L
n=0
Present work
Reference :
Nguyen et. al
max error= 0%
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature(0C)
x/L
n=0.5
Reference :
Nguyen et. al
Present work
max error= -5.27%
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature(0C)
x/L
n=1
Present work
Reference :
Nguyen et. al
max error= 2.06%
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature(0C)
x/L
n=2
Present work
Reference :
Nguyen et. al
max error= 4.5%

25. 0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature(0C)
x/L
n=5
Present work
Reference :
Nguyen et. al
max error= -5.28%
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature(0C)
x/L
n=15
Present work
Reference :
Nguyen et. al
max error= -5.75%
VALIDATION(CONTD.)

26. Determination of thermal conductivity of different FGMs for
different “n”
Material property of FGMs
Different FGMs Constituents k(W/mK-1) Km/kc
FGM 1[20]
NiCoCrAlY 4.3
2.15
ZrO2 2.0
FGM 2[5]
Ni 90.7
3.01
Al2O3 30.1
FGM 3[30]
Ti-6Al-4V 7.5
3.58
ZrO2 2.09
FGM 4[16]
Cr-Ni 11.4
5.18
ZrO2 2.2
FGM 5[29]
Al 204
97.6
ZrO2 2.09

27. Plot of thermal conductivity of different FGMs versus x/L
for a particular power law index “n”
0
50
100
150
200
250
0 0.2 0.4 0.6 0.8 1 1.2
Thermalconductivity,W/(mK)
x/L
n=0.2
FGM 1
FGM 2
FGM 3
FGM 4
FGM 5
0
50
100
150
200
250
0 0.2 0.4 0.6 0.8 1 1.2
Thermalconductivity,W/(mK)
x/L
n=1.0
FGM 1
FGM 2
FGM 3
FGM 4
FGM 5

28. Plot of thermal conductivity of different FGMs versus x/L
for a particular power law index “n” (CONTD.)
0
50
100
150
200
250
0 0.2 0.4 0.6 0.8 1 1.2
Thermalconductivity,W/(mK)
x/L
n=5 FGM 1
FGM 2
FGM 3
FGM 4
FGM 5

29. Plot of thermal conductivity for different FGMs for different
power law index ‘n’
0
1
2
3
4
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
thermalconductivity,W/(mK)
x/L
NiCoCrAlY/ZrO2 n=0.2
n=1
n=5
0
10
20
30
40
50
60
70
80
90
100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
thermalconductivity,W/(mK)
x/L
Ni/Al2O3 n=0.2
n=1
n=5

30. Plot of thermal conductivity for different FGMs for different
power law index ‘n’(CONTD.)
0
1
2
3
4
5
6
7
8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Thermalconductivity,W/(mK)
x/L
n=1
n=0.2
n=5
0
2
4
6
8
10
12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Thermalconductivity,W/(mK)
x/L
n=0.
2
n=1
Cr-Ni/ZrO2
Ti-6Al-4V/ZrO2

31. Plot of thermal conductivity for different FGMs for
different power law index ‘n’(CONTD.)
0
50
100
150
200
250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Thermalconductivity(W/mK)
x/L
Al/ZrO2 n=0.2
n=1
n=5

32. SIMULATION OF TEMPERATURE FOR
DIFFERENT FGMS
Each of the FGM plate is then subjected to same constant temperature
conditions i.e, Thot = 300 0C and Tcold = 20 0C
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature,0C
x/L
n=0.2
FGM 1
FGM 2
FGM 3
FGM 4
FGM 5

33. 0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature,0C
x/L
n=1.0
FGM 1
FGM 2
FGM 3
FGM 4
FGM 5
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature,0C
x/L
n=5
FGM 1
FGM 2
FGM 3
FGM 4
FGM 5

34. CONCLUSIONS
 A very simple technique of finite element method, direct approach, has been used to
simulate the temperature profile of a rather complex and non-homogeneous
material.
 The temperature distribution of the FGM plate is always less and gradual as
compared to the homogenous materials.
 On comparison of thermal conductivities of different types of FGM for same value
of n it was observed that for n=0.2, thermal conductivity quickly tends towards that
of the ceramic thermal conductivity and that is why it is used for high temperature
applications. For n=5, the thermal conductivity does not change much over the
thickness and then suddenly drops to the ceramic thermal conductivity; hence the
FGM is metal rich and that is why it is used for structural applications.
 On simulation of temperature for each FGM, it was observed that Al/ZrO2 shows the
most smooth and gradual variation of temperature along its thickness as compared to
all other FGMs. Also we can conclude that as the ratio of km/kc increases the
temperature distribution becomes smoother and gradual, thus showing good
characteristics of FGM under high thermal loading where thermal stresses are likely
to occur. Hence greater the ratio of km/kc, better will be its thermo-mechanical
properties and response to loadings.

35. SCOPE OF FUTURE WORK
 The work can be further extended to determine the
thermal stresses and mechanical stresses under the
application of thermo-mechanical loads.
 A 3D heat transfer analysis can be performed that
will provide more accurate nodal temperatures.

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39. THANKYOU
39
QUESTIONS???