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

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

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.

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

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.

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.

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.

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.

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

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

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]

HOW FGM DIFFERS FROM

TRADITIONAL COMPOSITE

Schematic structure, elastic modulus (-) and thermal conductivity (---) of

an FGM (a) and a homogeneous material (b). (Cherradi,1994)

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.

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]

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):

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]

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.

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

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.

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

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.

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

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

Tem

pera

ture

(0C

)

Non-dimensional thickness, x/L

n=0

n=0.5

n=1

n=2

n=5

n=15

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

tem

per

ature

(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

Tem

per

ature

(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

Tem

per

ature

(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

Tem

per

ature

(0C

)

x/L

n=2

Present work

Reference :Nguyen et. al

max error= 4.5%

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

Tem

per

ature

(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

Tem

per

ature

(0C

)

x/L

n=15

Present work

Reference :Nguyen et. al

max error= -5.75%

VALIDATION (CONTD.)

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

2.15ZrO2 2.0

FGM 2[5]Ni 90.7

3.01Al2O3 30.1

FGM 3[30]Ti-6Al-4V 7.5

3.58ZrO2 2.09

FGM 4[16]Cr-Ni 11.4

5.18ZrO2 2.2

FGM 5[29]Al 204

97.6ZrO2 2.09

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

Ther

mal

co

nd

uct

ivit

y,W

/(m

K)

x/L

n=0.2FGM 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

Ther

mal

co

nd

uct

ivit

y,

W/(

mK

)

x/L

n=1.0FGM 1

FGM 2

FGM 3

FGM 4

FGM 5

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

Ther

mal

co

nd

uct

ivit

y,

W/(

mK

)

x/L

n=5FGM 1

FGM 2

FGM 3

FGM 4

FGM 5

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

ther

mal

co

nd

uct

ivit

y,

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

ther

mal

co

nd

uct

ivit

y,

W/(

mK

)

x/L

Ni/Al2O3 n=0.2

n=1

n=5

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

Ther

mal

co

nd

uct

ivit

y,

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

Ther

mal

co

nd

uct

ivit

y,

W/(

mK

)

x/L

n=0.2

n=1

Cr-Ni/ZrO2

Ti-6Al-4V/ZrO2

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

Ther

mal

co

nd

uct

ivit

y (

W/m

K)

x/L

Al/ZrO2 n=0.2

n=1

n=5

SIMULATION OF TEMPERATURE FOR

DIFFERENT FGMSEach 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

Tem

per

ature

, 0C

x/L

n=0.2

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

Tem

per

ature

, 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

Tem

per

ature

, 0C

x/L

n=5

FGM 1

FGM 2

FGM 3

FGM 4

FGM 5

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.

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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THANKYOU

39

QUESTIONS???