[Doi 10.1016%2Fj.finel.2004.07.001] Bao-Lin Wang; Zhen-Hui Tian -- Application of Finite...

8/11/2019 [Doi 10.1016%2Fj.finel.2004.07.001] Bao-Lin Wang; Zhen-Hui Tian -- Application of Finite ElementFinite Differenc

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Finite Elements in Analysis and Design41 (2005) 335349

www.elsevier.com/locate/finel

Application of finite elementfinite difference method to thedetermination of transient temperature field in functionally graded

materials

Bao-Lin Wanga,b,, Zhen-Hui Tianb

aSchool of Aerospace, Mechanical and Mechatronic Engineering, Mechanical Engineering Building J07, The University of

Sydney, Sydney, NSW 2006, AustraliabCenter for Composite Materials, Harbin Institute for Technology, Harbin 150001, China

Received 7 January 2004; accepted 29 July 2004

Abstract

A finite element/finite difference method (FEM/FDM) is developed to solve the time-dependent temperature field

in non-homogeneous materials such as functionally graded materials. The method uses the finite element space

discretization to obtain a first-order system of differential equations, which is solved by employing finite differencescheme to resolve the time-dependent response. A computation code is developed in the programming environment

MATLAB. Temperature-dependent material properties are taken into account in the program. 2004 Elsevier B.V. All rights reserved.

Keywords:Transient temperature; Functionally graded materials; Finite element method; Finite difference method

1. Introduction

A functionallygradedmaterial (FGM) is usually a combination of two materialphases that has a gradualtransition from onematerial at one surface to another material at theopposite surface.This gradual changeof properties can be tailored to different applications and service environments.

School of Aerospace, Mechanical and Mechatronic Engineering, Mechanical Engineering Building J07, The University of

Sydney, Sydney, NSW 2006, Australia. Tel.: +61-2-935-17618;fax: +61-2-935-14841.

E-mail addresses:[email protected],[email protected](B.-L. Wang).

0168-874X/$- see front matter 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.finel.2004.07.001
http://www.elsevier.com/locate/finelmailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/finel


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336 B.-L. Wang, Z.-H. Tian / Finite Elements in Analysis and Design 41 (2005) 335 349

Functionally graded materials are expected to be used for high temperature and high heating rate envi-ronments.Accordingly, determination of transient temperature field is an important topic for those advan-tage materials. However, it is rather difficult to do so due to the arbitrarily distributed and continuously

varied material properties of FGMs. Since most of FGMs show one-dimensional non-homogeneity, thetemperature field canbe readily obtained by composite laminated plate model for simple one-dimensionalheat conduction. Taking into account the effect of temperature dependency of material properties, Tani-

gawa et al.[1]investigated a one-dimensional transient heat conduction problem of a FGM plate. Alsothe associate thermal stress distributions for an infinite long FGM plate was formulated. Jin and Paulino

proposed a multi-layered material model to treat the transient thermal conduction problem for an FGMstrip[2].They gave a feasible solution approach by means of Laplace transform technique and obtainedan analytical asymptotic solution of temperature for short times. Elperin and Rudin studied temperature

field and the associated thermal stresses in a functionally graded material caused by a laser thermal shock[3]. Abd-Alla et al. investigated the transient temperature field in a non-homogeneous orthotropic multi-

layered cylinder by means of Laplace transform technique[4].Noda and Obata obtained a perturbationsolution for the one-dimensional transient temperature field in FGMs[5].

A key feature that distinguishes FGMs from homogeneous materials is that the thermal properties

of the former vary spatially. Thus, thermal analysis of functionally graded materials is considerablymore complex than in the corresponding homogeneous case. The above-referred papers consideredonly one-dimensional temperature distribution in FGMs. For FGMs with complex geometries and/or

complex thermal boundary conditions, a numerical method such as finite element method is of greatimportance. In this paper, a finite element method in conjunction with the finite difference methodis introduced to solve the system of time-dependent equations that govern the transient temperature

distribution.

2. Thermal conductivity equation

Suppose in a coordinatesystemx (whose components are xi , where i =1, 2, 3) there isa solid occupyinga space, which is surrounded by a surfaceS. The temperature inside the solid may vary from point to

point, and with time. LetT (x, t)denote this temperature, which is assumed to be a continuous functionsof the coordinatesxiand timet. A basic law of heat conduction may be stated as

qi = kijT

xj(1)

in whichqiare the components of the heat flux vectorqandT /xjthe temperature gradients. Hereafter,the summations over the indices iand jwill be assumed when appearing twice in an equation. Thecomponents of the thermal conductivity tensor k(x)are denoted by kijand are generally considered to

be symmetric, i.e., kij =kj i . If the solid is anisotropic, heat will not necessarily flow in the direction ofthe temperature gradient.

The heat flow is controlled by the following conduction equation:

qi,i +Q= cT

t, (2)


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B.-L. Wang, Z.-H. Tian / Finite Elements in Analysis and Design 41 (2005) 335 349 337

where(x)is the mass density,c(x)the specific heat, and Qis the internal heat generation rate per unitvolume. Substitution of Eq. (2) into (1) gives the governing equation in terms of temperature:

kij T

xj

,i

= Q+c Tt

. (3)

In Eqs. (1)(3), material properties ,candkijare considered to be functions of space coordinates xi .

The thermal conduction equation(3)must be solved forprescribed boundary and initial conditions. Theinitial condition specifies the temperature distribution at time zero. This is, T (xj, 0)= T0(xj, 0). Heatconduction boundary conditions take several forms. The frequently encountered conditions are specified

surface temperature and specified surface eat flow:

qi ni =h, on boundary Sq , (4a)

T =T , on boundary ST, (4b)

whereSq + ST = S, and the over bar represents the known value, niare components of the unit vector n

normal to the exterior ofS. Eq. (4a) indicates that on the boundary Sqthe thermal flux is prescribed (his

positive if it is directed towards the exterior of the body). Conversely, Eq. (4b) describes the temperatureboundary condition onST.

3. Finite elementfinite difference method (FEM/FDM)

For functionally graded materials, the material properties k, and care complex functions of spatial

positionx, and the diffusion equation (3) is not amenable to analytical solutions. Numerical technique hasgained wide acceptance in engineering applications. Here, we develop a finite elementfinite differencemethod (FEM/FDM) to solve Eq. (3).

Suppose the medium undergoes a virtual temperature change T, multiply Eq. (3) by Tand thenintegrate in the entire space domaingiving

c

T

t

kij

T

xj

,i

Q

TdV =0. (5)

The above equation becomes, after the Green formula,

c Tt

TdV +

s

qi niTdV

qiT,idV

QTdV =0. (6)

Since we assume Tis zero on the boundary ST. Then from boundary (4a) expression (6) becomes

cT

tTdV +

sq

hTdV

qiT,idV

QTdV =0. (7)

The finite element/finite difference method involves two essential procedures: (a) using finite elementspace discretization to obtain a first-order system of differential equations, and (b) finding transient

response via finite difference method.


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(a) Finite element formulation. Let the continuum be divided into a finite number of elements inter-connected only at nodal points. For each element occupying space e, the temperature at any point canbe expressed in terms of their values at their nodal points by

T (x1, x2, x3, t )= [N]{T} (8)

in which[N]is known as the shape function matrix and is a function of spatial positions,{T}is a vector,which contains the temperature values at the nodal points of the element. It follows from Eq. (8) thattemperature gradients T /xjat any point in region ecan be written as

{T} = {T /x1, T /x2, T /x3}T = [B]{T}, (9)

where[B] = [L][N]and[L]denotes a differential operator matrix. Substituting Eq. (9) into Eq. (1), we

obtain the heat fluxesqiin the element

{q} = {q1, q2, q3}T

= [k][B]{T}, (10)

where[k] = [kij]is a matrix containing the thermal conductivities of the medium.Finally, by substituting Eqs. (8)(10) into Eq. (7), the finite element approximation of the heat equation

can be obtained as (after assemblage)

[C]{T} + [K]{T} = {p}, (11)

where the dot represents differentiation with respect to time. The element matrices and external heat loadvector are given by

[C] =

c[N]T[N] d, (12a)

[K] =

[B]T[k][B] d, (12b)

{p} =

S

[N]Th ds+

Q[N]T d. (12c)

In Eqs. (12), material properties ,cand[k]are functions of spatial coordinatesxi , and{T}and{p}arefunctions of timet. If the space is divided into a large number of elements, the material can be treated ashomogeneous in each element. If the applied thermal loads are independent of time, then {p} is a constant

vector. The numerical integration scheme, such as GaussLegendre integration, is used to evaluate the

integrals involved in Eqs. (12).The problem now is to solve the matrix differential equation (11). There are many general methods

and several techniques for solving first-order matrix differential equations. Among many numerical tech-niques, we consider the method of direct numerical difference in time that has been proven popular in

finite element analysis.(b)Finding transient response via finite difference. Since we can not determine the nodal temperature

{T}from Eq. (11) for all values of timetin an interval[0, t0], we will have to be satisfied with computing

approximations {T}mof{T (tm)}for some points(tm)Mm=0in the interval. We assume that the points are

equidistant, i.e., thattm = m(t), m = 0, . . . , M , where the step length tis defined as t= t0/Mfor an

integerM. To make the derivation somewhat more general, we assume that we have known{T}m, which


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is the approximation of{T (tm)}, and that we want to compute{T}m+1. The differential equation (11) atthe pointt=tmis

[C]{T (tm)} + [K]{T}m= {p(tm)}. (13)

Replacing the derivative by a difference quotient

{T (tm)} {T}m+1 {T}m

t, (14)

the differential equation would then become

[C]{T}m+1 {T}m

t+ [K]{T}m= {pm}, (15)

where{pm} = {p(tm)}. Similarly, if the differential equation (13) is satisfied at the pointt=tm+1, we

obtain

[C]{T}m+1 {T}m

t+ [K]{T}m+1= {p(m+1)}. (16)

Both Eqs. (15) and (16) can be used to solve {T}m+1. In order to improve numerical precise, thearithmetic

average of them is taken. This yields[C]

t+[K]

{T}m+1=

[C]

t(1 )[K]

{T}m+ (1){pm} + {p(m+1)}, (17)

where m= 0 M, {T}m+1on the left-hand side of Eq. (17) are unknowns, and all of the terms on

the right-hand side are known. Eq. (17) represents a general family of recurrence relations; a particular

algorithm depends on the value ofselected. If=0, the algorithm is the forward difference method(Euler method); if = 0.5, the algorithm is the CrankNicolson method; if =23 , the algorithm is known

as the Galerkin method; if =1, the algorithm is the backward difference method. The time-marchingalgorithm (17) is unconditionally stable for 0.5[6].Further discussion on solving the discretizedtime-dependent equation (13) can be found in[6].

In the numerical examples in this paper, Eq. (17) is solved with =0.5. According to the theory ofdifferential equation (e.g.,[7];), Eq. (17) for the time portion is unconditionally stable and converges

with the truncation error of order(t)2 for =0.5. This is a distinct improvement over to previous twoapproximations (15) and (16) because both of them lead to a truncation error of order (t). The numberof time steps in Eq. (17) can be chosen such that the required precise can be achieved.

3.1. Consideration of position-dependent material properties

In Eqs. (12), material properties , cand [k]can be functions of spatial coordinates xi . In order to

consider the non-homogeneity of the material properties inside an element, it is more convenient tospecify the material properties at the nodal points of the element. The material properties at any point inthe element can be approximated as

(x1, x2, x3)= [N]{}, (18a)

c(x1, x2, x3)= [N]{c}, (18b)


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[k(x1, x2, x3)] = [N]{[k]} (18c)

in which the quantities in parentheses {} are composed of the corresponding values at the nodal points of

the element. The element matrices [C] and [K] can be evaluated by substituting Eqs. (18) into Eqs. (12).

3.2. Consideration of temperature-dependent material properties

In high temperature environments, material properties (density , specific heatcand thermal conduc-

tivityk, etc.) may become temperature-dependent. Finite element/finite difference equations for such aproblem can be derived using a similar procedure outlined before. The matrix equations obtained havethe same forms as those given there, provided that the coefficient matrices [C] and [K] are functions of

temperatureTand/or its gradients, i.e.,

[C(T)] = (T )c(T )[N]

T

[N] d, (19a)

[K(T )] =

[B]T[k(T )][B] d. (19b)

Substituting of Eq. (8) into Eqs. (19) leads to

[C(T)] =

([N]{T})c([N]{T})[N]T[N] d, (20a)

[K(T )] = [B]T[k([N]{T})][B] d. (20b)

Assembly of the element equations to form the global system of equations is same as Eqs. (11) and (17),provided that the global matrices [C] and [K] are functions of{T}. The equation system will then becomenon-linear and can be solved iteratively. To avoid iterative operation, one can simply assume that material

properties (and then [C] and [K]) at the time interval(tm, tm+1)are functions of temperature vector{T}mat timetm, which has already been known in each step.

4. Illustrative examples

In this study, a finite element code is developed in theprogrammingenvironmentMATLAB. Eightkinds

of elements are constructed as shown inTable 1,in which TROD2, TROD2N, TROD3 and TROD3N areone-dimensional elements, and TQUAD4, TQUAT4N, TQUAD9 and TQUAD9N are two-dimensionaliso-parametric elements. Further, element types TROD2N, TROD3N, TQUAT4N and TQUAT9N havingspatially varied material properties inside the elements (i.e., Eqs. (13) are used to calculate the material

properties of the elements). For one-dimensional thermal conductivity problems, TROD2 and TQUAD4,TROD2N and TQUAD4N, TROD3 and TQUAD9, and TROD3N and TQUAD9N give the same results.Also given in Table 1is the number of Gauss integration points used to numerically evaluate the integrals

involved in Eqs. (12).To demonstrate the applicability of the numerical procedure, some simple examples are discussed

below.


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Table 1

Element types

Element name Element shape Description Material properties GaussLegendrewithin the element points

TROD2 1 2 One-dimensional 2-node linear element H 2

TROD2N 1 2 Same as TROD2 N 3

TROD3

1 3 2 One-dimensional 3-node quadratic element H 3

TROD3N

1 3 2 Same as TROD3 N 3

TQUAD4 1

4 3

2

Two-dimensional 4-node linear element H 22

TQUAD4N 1

4 3

2

Same as TQUAD4 N 33

TQUAD9 1

8 6

4 37

5

3

2

9

Two-dimensional 49 node quadratic element

(Any one or more of nodes 59 can be vacant) H 33

TQUAD9N 1

8 6

4 37

5

3

2

9

Same as TQUAD9 N 33

H: Homogeneous element (the properties inside the elements are constants and are their values at the center of the element).

N: Non-homogeneous (the properties inside the element are calculated from their values at the nodes of the element).

4.1. One-dimensional heat conduction in a homogeneous rod

Consider the temperature rise in a one-dimensional homogeneous rod whose length is l. The rod isinitially at a zero temperature environment, and is suddenly heated to T0at its two ends at x =0 andl.The series solution for this problem is

T ( x , t )

T0= 1+

4

n=1,3,5

1

nexp

(n)2

t

0

sin

n

x

l

, (21)

where0= cl2/kis a characteristic time parameter.Table 2shows the temperature at the center of the

rod for att= 0.050, 0.10, 0.20, and 0.50. The time interval[0, 0.50]is divided into 5000 time steps


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Table 2

Transient temperature at the center of a one-dimensional rod (the time interval 00.5t0is divided into 5000 time steps, which

has leaded to convergent results)

Element type Time Number of nodes in the entire rod Exact solution

3 Nodes 5 Nodes 7 Nodes 11 Nodes

TROD2 t /t0 =0.05 0.176 0.205 0.216 0.223 0.228

t /t0 =0.1 0.548 0.526 0.525 0.525 0.526

1 2 t /t0 =0.2 0.864 0.832 0.821 0.825 0.823

t /t0 =0.5 0.996 0.993 0.992 0.991 0.991

TROD3 t /t0 =0.05 0.242 0.215 0.226 0.227 0.228

t /t0 =0.1 0.540 0.522 0.525 0.525 0.526

1 3 2

t /t0 =0.2 0.831 0.833 0.823 0.823 0.823t /t0 =0.5 0.992 0.991 0.991 0.991 0.991

(i.e., M =5000). It has been found that further increasing ofMdoes not change the calculated tem-perature distribution for this problem. The convergence of the results with the increasing number of

elements (nodes) is evident. The FEM/FDM result obtained by TROD3 with 11 nodes is the most ac-curate solution when comparing with the series solution. This suggests that TROD3 is more efficientthan TROD2. This is partially due to the fact that TROD3 uses a quadratic interpolation function in the

elements.

4.2. A two-dimensional heat conduction in a homogeneous square plate

Consider the conduction heat transfer in a homogeneous square plate of dimension x = (0, 2l) by

y= (0, 2l), conductivityk, density, specificc. The plate is subjected to a sudden uniform internal heatgeneration ofQ0. The edges of the plate are maintained at a temperature ofT = 0. We wish to determinethe steady temperature distribution inside the plate using the FEM/FDM method.

The problem has symmetries that can be exploited in the finite element analysis. Only 14of the plate is

needed to consider. The solutions obtained by two different meshes of rectangular elements are comparedat the center of the plate for varying time, as shown inFig. 1,which gives the normalized results obtainedby 44and20 20 TQUAD4 elements. The convergence of the results with increasing element numbersto stable values is significant. By comparing the steady solution (tapproaches infinity) with the series

solution, we know that results obtained from 20 20 TQUAD4 elements is the most accurate solution.For example, the steady values of Tat (x,y) = (0, 0), (0, 0.25), (0, 0.5)and (0, 0.75)are 0.2948,0.2790, 0.2295 and 0.1398, respectively, calculated by 2020 TQUAD4 Elements. Those results are

almost identical to the series solutions given in[8].The steady values ofTcalculated by 4 4 TQUAD4Elements at (x,y)= (0, 0), (0, 0.25), (0, 0.5)and (0, 0.75)are 0.2984, 0.2824, 0.2322 and 0.1414,

respectively, which are same as those in[8].


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T/T

0

2020 TQUAD4 Elements

44 TQUAD4 Elements

t/t0

0.0

0.30

0.25

0.20

0.15

0.10

0.00

0.05

0.2 0.4 0.6 0.8 1.0

Fig. 1. Transient temperature at the center of a homogeneous square plate under a sudden internal heat generation (t0

= cl2/ k,

T0= Q0l2/k).

4.3. One-dimensional heat conduction in a FGM strip

The main purpose of this article is to develop a numerical model applicable to the determination of the

temperature distribution in FGMs with arbitrarily distributed properties. In this sub-section, temperaturehistories for a one-dimensional FGM material strip are obtained using the developed FEM/FDM analysismodel. The FGM strip has a length land is made from a PSZ/Ti-6Al-4V composition system[9].Both

phases are regarded as isotropic materials. Their properties are kTi= 18.1 W/(m K),Ti= 4420 Kg/m3,

cTi=808.3 J/(kg K),kp =2.036W/(m K), p =5600 Kg/m3

andcp =615.6 J/(kg K), wherekis thecoefficient of thermal conductivity, cis the specific heat and is the mass density. The subscripts p and

Ti denote PSZ and Ti-6Al-4V, respectively. The volume fraction of Ti-6Al-4V in the FGM is varied from100% on the top surface(x = 0)to 0% on the bottom surface (x = l)of the strip. Therefore, the FGM is

pure Ti-6Al-4V on its top surface (x =0)and pure PSZ on its bottom surface(x =l). Inside the strip,the material properties are expressed as an exponential function ofxas many authors have done so [e.g.,[1012]]

fFGM= fTiexp[(x/l)], =ln(fP/fTi), (22)

wherefrepresents the density, the specific, or the coefficient of thermal conductivity.

Considered is the case that the temperature at x =0 is suddenly raised to T0, which is maintainedthereafter. The temperature is kept zero atx= l. In the FEM/FDM analysis, the strip is divided into some

elements of equal length. The total numbers of nodes in the strip are fixed as 5, 9, and 17, respectively.Five thousands time steps in the time interval [0, 0.5t0] are used, which have leaded to convergent results,where t0 =TicTil

2/kTi = 19739 l2 is the characteristic time obtained for Ti-6Al-4V. There is no closed-

form solution for the transient temperature field in such a non-homogeneous strip. The exact solution forthe steady temperature field for this problem can be obtained as

T(x)=T0

1

1exp(x/l)

1exp()

, =ln(kP/kTi). (23)
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Table 3

Transient temperatures in a FGMstrip (Element types:TROD3 andTROD3N; total 17 nodes; the time interval 00.5t0is divided

into 5000 time steps; t0 =19739l2, numbers in parentheses denote solutions obtained from TROD3)

Positions(x/ l) T (x)/T 0 Steady values

t/t0=0.05 t/t0=0.1 t/t0=0.2 t/t0=0.5 t /t0 =infinity Exact

solution

0.25 0.412 0.586 0.724 0.849 0.9080.908

(0.413) (0.586) (0.725) (0.849) (0.908)

0.5 0.0468 0.178 0.372 0.618 0.7490.749

(0.0471) (0.178) (0.373) (0.619) (0.749)

0.75 0.000472 0.0146 0.0983 0.325 0.4740.474

(0.000481) (0.0147) (0.0989) (0.326) (0.474)

Table 4Transient temperatures in a FGM strip with 9 nodes (Element types: TROD3 and TROD3N, the time interval 00.5t0is divided

into 5000 time steps, t0 =19739l2, numbers in parentheses denote solutions obtained from TROD3)



solution

0.25 0.413 0.586 0.724 0.849 0.9080.908

(0.416) (0.589) (0.726) (0.850) (0.908)

0.5 0.0447 0.177 0.372 0.618 0.7490.749

(0.0455) (0.180) (0.376) (0.621) (0.749)

0.75 0.00182 0.0142 0.0977 0.325 0.4740.474

(0.00192) (0.0147) (0.0996) (0.329) (0.474)

Table 5

Transient temperatures in a FGM strip with 5 nodes (Element types: TROD3 and TROD3N, the time interval 00.5t0is divided

into 5000 time steps, t0 =19739l2, numbers in parentheses denote solutions obtained from TROD3)



solution

0.25 0.420 0.591 0.729 0.849 0.9070.908

(0.310) (0.469) (0.634) (0.802) (0.874)

0.5 0.0177 0.169 0.373 0.617 0.749 0.749(0.0267) (0.171) (0.380) (0.632) (0.749)

0.75 0.00152 0.00768 0.0956 0.328 0.4730.474

(0.00142) (0.00313) (0.0664) (0.258) (0.374)

Tables 35summarize the FEM/FDM results for positions x =0.25l, 0.5land 0.75l. If we considerthe results from the element-type TROD3N with total 17 nodes in the strip as the exact solution, then

we find that using of the non-homogeneous element (i.e., TROD3N) has a considerable improvement ofresults. The FEM/FDM solutions obtained by the non-homogeneous element, TROD3N, with 9 nodes

show negligible errors. Conversely, the results obtained by the homogeneous element, TROD3, with 9
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0.0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25

T

/T

0

44 TQUAD4N Elements

2020 TQUAD4N Elements

22 TQUAD4N Elements

22 TQUAD4 Elements

t/t0

Fig. 2. Transient temperature at the center of a non-homogeneous square plate under a sudden internal heat generation (andt0 = cl

2/k0,T0= Q0l2/ k0).

nodes still have significant errors, especially at the beginning of heating (i.e, for short-time response).

These facts indicate that the efficiency of the element-type TROD3N is higher than TROD3 for theanalysis of heat conduction in non-homogeneous materials. To further demonstrate the efficiency of thenon-homogeneouselements, in the followingwe considera two-dimensional transient thermalconduction

in a non-homogeneous square plate.

4.4. A two-dimensional heat conduction in a FGM square plate

Consider again the 2l 2lplate discussed in Section 4.2. The only difference between this problemand that in Section 4.2 is that the thermal conductivity kis non-homogeneous such that

k(x,y)=k0(1+(x/l)2 +(y/l)2]. (24)

The temperature histories at the center of the plate obtained by 2 2 TQUAD4, 2 2 TQUAD4N, 4 4

TQUAD4N and 2020 TQUAD4N are shown inFig. 2.The solutionsobtained by 44 TQUAD4N and 2020 TQUAD4N elementsare very close.Compared

with the solution obtained by 2 2 homogeneous elements, TQUAD4, the solution obtained by 2 2

non-homogeneous elements, TQUAD4N, is closer to that obtained by 20

20 TQUAD4N elements.Those facts again indicate that the non-homogeneous elements can provides an ideal solution without

dividing the FGMs into too many elements.

4.5. A FGM strip with temperature-dependent material properties

Investigate a FGM strip of thickness 10 mm. The material phases inside the FGM change from pure

ZrO2at x =0 linearly to Ti-6Al-4V at x =10 mm. The temperature at the x =0 side of the strip issuddenly heated toT0, which is maintained thereafter. The temperature at the x= 10 mm side of the strip

is kept zero. The properties of ZrO2and Ti-6Al-4V are[13]:


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0. 000 0. 002 0. 004 0. 006 0. 008 0. 0100

200

400

600

800

1000

Transient temperature att=10

seconds

Steady temperature

x(m)

T

Fig. 3. Temperature distribution in a FGM strip (T0 =1000K. The length of the strip is 1 cm. Solid lines consider tempera-

ture-dependent material properties; dash-dot lines do not consider temperature-dependent material properties.).

[ZrO2]:

k=1.71+0.21103T +0.116106T2 [W/(mK)],

c=274+0.795T 6.19104T2 +1.71107T3 [J/(kgK)],

=3657/{1.0+ (T 300.0)}3 [Kg/(m3)]

[Ti-6Al-4V]:

k=1.1+0.017T [W/(mK)],

c=350+0.878T 9.74104T2 +4.43107T3 [J/(kgK)],

=4420.0/{1.0+(T 300.0)}3 [Kg/(m3)].

The simple Rule-of-Mixture is applied to evaluate the overall property distribution of the FGM strip.The entire strip is divided into one hundred linear TROD2 elements of equal length.Figs. 3and4plotted

the temperature distribution forT0=1000 and 100 K, respectively. It can be shown that the influence oftemperature-dependent material properties is more significant for higher temperature field than for lowertemperature field.

5. Conclusion remarks

Because of the non-homogeneous material properties, it is very difficult to obtain the exact solution of

thermal conductivity equations for non-homogeneous materials, such as functionally graded materials.Therefore, numerical solution is required. In this paper, the finite element/finite difference method forthe thermal conductivity problems in non-homogeneous materials was discussed. A computer code using

commercial software MATLAB was developed. Although only numerical examples for one-dimensionaland two-dimensional elements are given, the extension of the method to the three-dimensional problems

is obvious and straightforward.


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0.000 0.002 0.004 0.006 0.008 0.0100

20

40

60

80

100

Transient temperature att=10

seconds

Steady temperature

x(m)

T

Fig. 4. Temperature distribution in a FGM strip (T0 =100K. The length of the strip is 1 cm. Solid lines consider tempera-ture-dependent material properties; dash dot lines do not consider temperature-dependent material properties.).

The discussion of transient temperature field in this paper considered the most commonly encounteredsituations. The formula and the analysis program are general enough so that they can be easily used in

engineering problems.A number of practical situations arise where thegoverning equations are simplified.For example, one-dimensional transient thermal conduction is of great importance for FGMs application

in high temperature environments. The model developed in this paper can be easily extended to such aproblem. We discuss this below.

Suppose a material layer undergone a transient thermal conduction along the xdirection. The layer is

divided into a number of 2-node elements along the xdirection (e.g.,Nelements andN+ 1 nodes). Theproperties are considered to be constant inside each of the elements. We start from theith element whoselength isli . Denote the material properties of theith layer with a subscripti. If the temperature inside the

element is assumed as a linear function of positionx, wherex [0, li ], then the shape function [N]forthei th element can be chosen as

[N] = [(1x/ li ),x/li ]. (25)

Since the material properties in each element are constants, then the element matrices and load vector

can be directly obtained as follows:

[B]i = [1/ li , 1/ li ], (26)

[C]i =Si

ll

c[N]T[N] dr =

C

(i)11 C

(i)12

C(i)21 C

(i)22

=

(c)i li

6

2 11 2

Si , (27a)

[K]i =Si

li

[B]Tk[B] dr =

ki

li

1 11 1

Si , (27b)

{p}i =

pi1pi2

=

hi Si

hi+1Si+1

+Si

li

Q[N]T dr, (27c)

whereSiis the area of the layer at theith element. This area is in the plane perpendicular to the x-axis.


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The assembly of element matrices to form the global matrices in Eq. (11) gives

(1) {T} = {T1, T2, . . . , T N+1}T.

(2) The non-zeroith row andjth column elementsCijin global matrix [C] are:

The first row: C11= C(1)11, C12= C

(1)12.

Theith row(N+1 > i > 1):Ci(i 1)= C(i1)21 , Cii =C

(i1)22 +C

(i)11 , Ci(i +1)= C

(i)12 .

The last row:C(N+1)N =C(N)21 , C(N+1)(N+1)= C

(N)22 .

(3) The assembly of the global matrix [K] is the same as that of the global matrix [C].(4) Theith elementspiin the global load vector {p}are:

p1= p(1)1 , pi =p

(i1)2 +p

(i)1 , pN+1= p

(N )2 .

Eqs. (27) provide the element matrices for the one-dimensional plates. Similarly, the element matrices

for one-dimensional heat conduction in cylinder and sphere can be obtained as follows.(i)Cylinder. The volume integral dfor an axially symmetric cylinder is d = 2rHidr , whereHiis

the length (along the axi-symmetric axis) of the cylinder at theith element. Hence,

[C]i =2Hi

ll

c[N]T[N]rdr =

(c)i li

6

ri+1+3ri ri+1+ riri+1+ ri 3ri+1+ ri

Hi , (28a)

[K]i =2Hi li

[B]T[k][B]rdr =ki

ri+1+ ri

li 1 1

1 1Hi , (28b)

{p}i =

2hi Hi ri

2hi+1Hi+1ri+1

+2Hi

li

Q[N]Trdr. (28c)

(ii)Sphere. The volume integral dfor a rotationally symmetric sphere is d=4r2 dr. Then,

[C]i =4

ll

c[N]T[N]r

2 dr

=(c)i li

15

2r2i+1+6ri+1ri +12r

2i 3r

2i+1+4ri+1ri +3r

2i

3r2i+1+4ri+1ri +3r2i 12r

2i+1+6ri+1ri +2r

2i

, (29a)

[K]i =4

li

[B]T[k][B]r

2 dr =4

3

r2i+1+ ri ri+1+ r2i

liki

1 11 1

, (29b)

{p}i =

4hi r

2i

4hi+1r2i+1

+4

li

Q[N]Tr2 dr. (29c)

In expressions (28) and (29), rb =rN+1> rN> > r2> r1 = ra , li =ri+1 ri , whereraandrbare

the inner radius and the outer radius of the cylinder (or sphere), respectively.Now that the global matrices [K] and [C] are known, one can make a computation program to find

the transient solution of Eq. (11), using finite difference scheme given in Eq. (17). Although the material


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properties in each element are assigned as constants, one can improve the computation accuracy byincreasing thenumber ofelements.Theabovediscretizationmethod likes thelaminated plate model,whichhas been used by Tanigawa et al.[1]and Jin and Paulino[2],for solving the one-dimensional transient

heat conduction equations. In their analysis, the FGMs were divided into a number of homogeneoussub-layers.

Acknowledgements

BLW acknowledges the awards of an ARC Australian Research Fellowship and a Discovery Project

(#DP0346037) by the Australian Research Council (ARC). Part of this research was supported by theNSFC (#10102004) and SRF for ROCS, SEM.

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plate with temperature-dependent material properties, J. Thermal Stresses 19 (1996) 77102.

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107 (1) (2001) 7398.

[3] T. Elperin, G. Rudin,Thermal stresses in functionallygraded materialscaused by a laser thermal shock,Heat Mass Transfer

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[4] A.M. Abd-Alla, A.N. Abd-Alla, N.A. Zeidan, Thermal stresses in a nonhomogeneous orthotropic elastic multi-layered

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[5] Y. Obata, N. Noda, Transient thermal stresses in a plate of functionally graded material, Ceramic Trans. 34 (1993) 403.

[6] Kenneth H. Huberner, Earl A. Thornton, Ted G. Byrom, The Finite Element Method for Engineers, Wiley-Interscience

Publication, Wiley, NewYork, 1995.[7] Sadik Kaka, Yaman Yener, Heat Conduction, Taylor & Francis, Washington, DC, 1993.

[8] J.N.Reddy, D.K.Gartling, in:The FiniteElementMethodinHeatTransfer andFluid Dynamics,CRC PressInc.,BocaRaton,

FL, 1994, p. 67.

[9] T. Fujimoto,N. Noda, Crack Propagation in a Functionally GradedPlateUnderThermal Shock,vol. 70,2000, pp. 377386.

[10] F. Erdogan, B.H. Wu, Crack problem in FGM layers under thermal stresses, J. Thermal Stresses 19 (1996) 237265.

[11] F. Erdogan, M. Ozturk, Periodic cracking of functionally graded coatings, Int. J. Eng. Sci. 33 (1995) 21792195.

[12] N. Shbeeb, W.K. Binienda, K. Kreider, Analysis of the driving force for a generally oriented crack in a functionally graded

strip sandwiched between two homogeneous half planes, Int. J. Fract. 14 (2000) 2350.

[13] Y. Tanigawa, M. Matsumoto, T. Akai, Optimization of material composition to minimize thermal stresses in

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