·J

Functionally Graded Material: A Parametric Study onThermal Stress Characteristics using the

Crank- Nicolson-Galerkin Scheme

J. R. Cho *

School of Mechanical Engineering, Pusan National University

Jangjeon-Dong, Kumjung-Ku, Pusan 609-735, KOREA

J. Tinsley Oden t

Texas Institute for Computational and Applied MathematicsThe University of Texas at Austin, Austin, TX 78712

April 24, 1999

Abstract

In this paper. we analyze thermal stress characteristics of functionally graded ma-terials(FGM), the newly-introduced layered composite materials with great potentialas next generation composites. Among the several material parameters governing itscharacteristics, we study the effects of the material variation through the thicknessand the size of the FGM layer inserted between metal and ceramic layers using thefinite element method. Through a representative model problem, we observe differentthermal stress characteristics for different material variations and sizes of FGM. Thisbasic study provides insight into the concept of FGM and lays the foundation of FGMoptimization to control thermal stresses.

1 Introduction

Layered composite materials, due to their thermal and mechanical merits compared tosingle-composed materials, have been widely used for a variety of engineering applications.

•Assistant Professor.tDirector of TICAM, and Cockrell Faculty Regents' Chair in Engineering #2 .

1

However, owing to the sharp discontinuity in the material properties at interfaces betweentwo different materials, there may exist stress concentrations which result in severe materialfailure [27, 34].

To remedy such defects, in 1989, the NKK Corporation in Japan initiated studieson functionally graded materials(FGM)[19, 28]. At that time, applications to thermal-resistant structures of space shuttles were targeted[35]. However, recently, intensive andactive research for various engineering applications have been under way in many countries[19].

Thermal Stress

t

(I) I (2)

(CLC)

Thermal Stress

t!TTl

(I) ; Metal Layer

(2) ; Ceramic Layer

~I(3) ; FGM Layer

(FGM)

Figure 1: Illustration of thermal characteristics between CLC and FGM composites.

As indicated from the term functionally-graded, FGM has continuously varying ma-terial composition variation through the thickness such that it can satisfy the desired goals.The comparison between classical and FGM layered composites is illustrated in Fig. 1. Re-ferring to Fig. 1, material composition is 100% metal at the interface between layers (1) and(3), while it becomes 100% ceramic at the interface between layers (2) and (3). Therefore,since there is no sharp discontinuity in material composition through the thickness, jumpsin resulting thermal stresses are minimized. From the illustration, we see that FGM exhibitssmoother thermal stress distribution compared to classical layered composites(CLC).

In FGM, major concerns are the effects of the material composition variation andthe relative thickness ratio of the FGM layer (3) inserted between metal and ceramic layers,because its thermal and mechanical behavior is significantly affected by these two parameters(for example, the maximum value and the smoothness of the thermal-stress distributionare different for different choices of the two parameters). In this study, we hierarchicallyinvestigate the effects of such parameters on thermomechanical characteristics for plane-stress two-dimensional Ni - Al203 FGMs subject to uniform heating through the numericalsimulations using the finite element method.

2

2 Functionally Graded Materials

For the purpose of theoretical investigation, we consider a two-dimensional symmetric FGMbeam, as shown in Fig. 2, where 2d and 2dm represent, respectively, the thickness of thebeam and the middle FGM layer. The coordinate z is directed upwards normal to the mid-surface of the beam. Throughout this paper, we indicate the entire FGM composite by FGM,while the inserted middle layer by the FGM layer. Furthermore, we assume that ceramicand metal layers are homogeneous and isotropic, and the FGM layer is inhomogeneous butmacroscopically isotropic.

In this section, we hierarchically analyze the two parameters, and summarize thematerial-property variations through the thickness for the analysis ill the following sections.Here, by the term hierarchically (or hiera·rchica0 in this paper is meant that, by continuouslyvarying a specific parameter, the corresponding behavior of FGMs, when measured alongthe specifically defined basis, shows sort of sequential variations (but, not monotonically).In this paper, with the basis defined in the next section, we examine the thermal stresseswith the two parameters, the variation of material composition and the relative size of theFGM lalyer.

z'" Top surface

100% Ceramic

2d

BuUom surface I2L ~

Figure 2: A two-dimensional symmetric functionally graded composite beam.

2.1 Definition of Two Paralneters

Referring to Fig. 3, we define the composition volume fraction functions, "\1;;, ~~ E CO( -d, d)for ceramic and metal, respectively, such that

Vc(z) + Vm(z) = 1, Vc(z) ~ 1, I z I~ d

Vm(z) = [dm-Zr I z l~ dm2dm ' ~ (1)

Vm(z) = 1, -d ~ Z ~ -dm

Vm(z) = 0, dm~z~d

3

Z

Vm(Z)

"N->~

o

-d m-d

d

d m ...-- ~. '-' :..: '..: :...:-..:-..:....:.,:- ..

Figure 3: Composition volume fraction of metal through the FG'\;I thickness.

where N are non-negative real numbers (N ~ 0). By changing the value of N, we canconstruct an infinite number of such functions. Here, the special cases are for N -t aand for N -t 00, because these cases correspond to CLCs. We note that the former limitapproaches the metal layer-extended CLG(m-CLC), while the latter limit approaches theceramic layer-extended CLG( c-CLC), as depicted in Fig. 4. Next, referring to Fig. 2, wedefine the relative thickness rat1:0 X as

X = dm/d, 0::; X ::; 1 (2)

which is proportional to the relative region of the FGM layer with respect to the compositebody. As X tends to 0, FGMs approach CLCs, while they approach the fully FGMs(f-FGM)asX-tl.

According to the two defined parameters, we can construct two different hierarchicalfamilies Fft and FiI for layered composites to examine the variations in thermal stress char-acteristics. For the detailed mathematical abstraction and physical concept of hierarchicalmodeling, see Cho and Oden [7] for isotropic elastic structures and Oden and Zohdi [22] forheterogeneous elastic structures.

100% Ceramic

(a) c-CLC

IGraded

layer

100% Ceramic

(b) m-CLC

Figure 4: Ceramic layer- and metallayer-extencled classical layered composites; (a) c-CLCand (b) m-CLC.

4

(3)

... ,

First, for a given power index N(O < N < +(0), the x-hiera?'chical family :Ffi isdefined as a set of infinite layered composite materials which are sequentially distinguishedby the choice of relative thickness ratios X such that

F"Ji = {MXr : MRr = CLC, M}" = f-FGM, 0:::; X:::; I}

Here, the lowest model is CLC and the highest model is the f-FGM.

Along the similar pattern to the definition of F"Ji, the N-hierarchical family for afixed relative thickness X(O < X < 1) is defined as

Ffj = {M~ : M~ = m-CLC, M~ = c-CLC, 0:::; N :::;oo} (4)

In particulal', FGMs in this family approach m-CLCs as N -+ 0 while they approach c-CLCsas IV -+ 00. Therefore, in this family, the two lowest models are c- and m-CLCs, but thehighest model is not clear.

The corresponding basis for the parametric investigation of the hierarchical familiesis defined in the following section dealing with the thermal stress in FGMs.

2.2 Material Properties of Graded Composites

In order to compute the temperature and the overall strain/stress distributions in FGMs,one needs the appropriate estimates for properties of the graded layer, such as the thermalconductivity, the coefficient of thermal expansion, the Young's modulus, Poisson's ratio andso on. A large number of papers on the thermomechanical response and the material prop-erties of compositionally graded materials have been published, and they can be classifiedlargely into theoretical and experimental categories. The progress in theoretical methodsfor the overall thermomechanical properties can be seen in a book by Christensen [9], andthe discussion on experimental approaches in a paper by Suresh et al. [18]. In addition,the thermomechanical response by the theoretical approach can be modeled at microscopic(predicted upon the interaction between matrix and inclusions) and continuum levels (byaccounting for only the volume fraction and individual properties of the constituent phases).

The prediction methods of the overall therlTlomechanical properties within the fram-work of single continuum mechanics are generally classified into three; (a) direct, (b) varia-tional and (c) approximation approaches [30]. As is well described in a paper by Hill [15],the direct approach seeks the closed-form analytic solutions to the overall properties of idealcomposites, wherein geometrical and other simplifications are made for mathemat.ical deriva-tion. As an alternative approach of rigourous mathematical treatment in the direct met.hod,the variational approach provides the upper and lower bounds for the overall properties ex-pressed in terms of the phase volume fraction [14]. Besides the bounded-form estimation, as

5

..

described in [30], this approach does not specify the details in the phase geometry, one thusrequires some reasonable approximations to obtain corresponding closed-form estimates.

In the approximation approach, the self-consistent models by Hill [16], Walpole [32]and Budiansky [4], the mean-field micromechanics models by Mori and Tanaka [20] andWakashima and Tsukamoto [30, 31], the modified rule of mixtures by Tamura et al. [29]and the unit cell model by Ravichandran [23] are representative homogenized (or averag-ing) estimates for the overall material properties. These a.veraging methods are simple andconvenient to predict the overall thermomechanical response and properties, however, ow-ing to the assumed simplifications, the validity in real FGMs is affected by the detailedmicrostructure and other conditions. We here briefly describe each of these together withits assessment by comparing with the Hashin-Shtrikman's bounds, the experimental resultsand the detailed finite element analysis of the discrete models [11, 24, 25, 33].

The self-consistent model is based on the Eshelby's equivalent inclusion method, andhave been reported to be generally suitable for relatively simple microstructures of lowvolume fractions. According to the work by Grujicic and Zhang [11] with the VCFEM(a Voronoi cell finite element method proposed orginally by Ghosh et al. [13] for the cal-culation of material properties), the self-consistent method is not suitable for the FGMsof microstructures varying continuously, particularly consisting of interwined clusters ofphases. Furthermore Dvorak et al. [24] reported, from the numerical results obta.ined usingthe planar hexagonal-arrayed discrete model subjected to uniform and continuously vary-ing traction/essential boundary conditions, that the self-consistent models are better forskeleton microstructures in a wide transition zone.

The Wakashima- Tsukamoto mean-field micromechanics model is to exploit the Es-helby's equivalent inclusion method by incorporating the Mori- Tanaka concept [8, 20] forthe average (overall or regularized) stress, wherein the overall macromechanical propertiesE are defined by

(o-)n = E : (C:)n (5)

(-)n = I~lin < . > dn (6)

Where, it denotes a representative volume element(RVE) [30, 31]. Wakashima and Tsukamotoderived the expressions for the overall properties and presented the numerical results showingthe comparison with the Hashin-Shtrikman's bounds for two representative binary systems,the macroscopically isotropic particulate composite with randomly oriented and uniformlydispersed spheroidal particles, and the macroscopically tra.nsversely isotropic composite withunidirectionally continuous filaments of uniform cross section. We record the formula forthe former case in Appendix.

6

..

cr ceramictwo-phase composite

000~

a h

OCr. h h

00--0 hunit cell

...!!!.. CcFGM

(b)(a)

Figure 5: Schematic representation of two averaging approaches for stress-strain data ofFGMs; (a) the modified rule of mixtures and (b) the unit cell approximation.

As is well established in [20, 24], the Mori-Tanaka estimatel:i are derived with ran-domly oriented and uniformly dispersed ellipsoidals in an infinite continuous matrix. There-fore the Mori- Tanaka and Wakashima- Tsukamoto estima.tes produce genera.lly stiffer me-chanical response [33], and further they become increasingly more inaccurate as a macrostruc-ture tends to the microstructural size. In addition, they show good agreement with Hashin-Shtrikman's bounds only when the difference in elastic moduli of two constituent phases islarge, according to the work by Dvorak et al. [24, 25].

For the Young's modulus of two-phase graded materials, Tamura et al. [29] proposedthe modified rule of mixtures, and which has subsequently adapted by vVilliamson et al.[34], Suresh et al. [10, 18] and so on. According to this approach, each sublayer in thegraded layer is treated as an isotropic composite for which the uniaxial stress G" and strain C

are expressed in terms of the average uniaxial stress (ji and Ci (i denoted by each constituentphase) and the volume fraction such that

(7)

(8)

together with the ratio of stress to strain transfer q given by

q=CA - CB

Figure 5(a) schematically illustrates this approach, and the value q is determined by exper-iments. The value q of 4.5GPa chosen from the experiments has been justified by Sureshet al. [10, 18] and Williamson et al. [34] for the Ni - AL203 FGMs with a wide range ofvolume fraction and applied tractions.

Figure 5(b) shows a schematic idea of the unit cell approximation proposed byRavichandran [23]. In this approach, a two-phase composite is modelled by one with pe-riodically located cubic cells. With this model, the formula for the Young's modulus and

7

,

the Poisson's ratio of composites are derived. However, this method does not account forinteraction between two phases by assuming perfect bonding. Additionally the Young'smodulus is derived upon the assumption of the same Poisson's ratio for two phases. As-sessment through the comparison with experimental data is addressed in [23], where theYoung's modulus is reported to coincide with experimental data, but Poisson's ratio is not.

Although the aforementioned single homogenized continuum models have been ad-equate for static and quasi-static problems, they lose completely any characterization indynamical processes. This is because the single homogenized continuum model can notcapture dispersion and dissipation caused by structural effects in the dynamical process.The reader may refer to Bedford and Stern [21 for the superimposed continua model whichaccounts for thermal and mechanical interactions.

As pointed out in the recent paper by Suresh et al. [33], the averaging methods re-quire more detailed analysis for the constituent phases of nonuniform geometries with sharpcorners and for adequate capturing large fluctuations in microscopic thermomechanical fieldsand strain localization due to the non-uniform/non-periodic distributions of the constituent.phases. However, referring to the recent work by Dvorak et al. [25] on the assessment oft.he homogenized methods by the detailed finite element analysis with the discrete model,the averaging methods may be selectively applied to FGMs subjected to both uniform andnon-uniform overall loads with a reasonable degree of confidence.

According to comparison of the approaches discussed so far and the choice of metaland ceramic for our study, we employ the modified rule of mixtures for the Young's modulusof the graded layer. For the specific heat, the density, the thermal conductvity, the thermalexpansion coefficient and Poisson's ratio, we simply use the linear rule of mixtures. For thedual-phase composite materials for which the difference in thermal expansion coefficientsof two constituent phases is less than 1.0 x 10-70 [{-I, the justification of the linear rule ofmixtures for the thermal expansion coefficient has been provided by Schapery [26].

3 Thermal Stresses in FGM

For a mathematical characterization of thermal stresses in general FGMs, we consider athree-dimensional symmetric composite plate in which each layer is of uniform thickness,as shown in Fig 6. \\There, n is an open bounded Lipschitzian domain in fR3 with piecewisesmooth boundary an and w denotes a mid-surface (i.e., the plane with z = 0) in HP withpiecewise smooth line boundary ow.

We assume that the composition fractions are functions of z only, the essential bound-ary region anD is restricted to the top and bottom surfaces f± = {~: (x,y) E w,Z = ±d}

8

x boundary of ro

z N Mxyx

N~ / If

y

Figure 6: A three-dimensional symmetric FGM la.yered composite plate of uniform thicknessand in-plane thermal stress resultants.

while the natura.l boundary region anN is assigned to the lateral surface re = aw x (-d, d).Furthermore, we make the following additional assumptions for our study:

• The layers are perfectly bonded.

• The strains are small.

• The Kirchhoff-Love hypothesis holds.

• The body is initially stress-free state and has uniform room temperature To.

3.1 Basic Theories

The governing equations for time-dependent temperature field T( x; t) are composed of ageneral heat-diffusion equation based on Fourier'8 law, and the initial and boundary condi-tions (i.e. initial-boundary value problem):

To, at t = 0

To, for t E (0, t*]

f(x, y; t), for t E (0, t*]

19(x; t), for t E (0, t*]

aTpc- - Y' . (kY'T)at

T(x; t)

T(x, y, -d; t)

T(x, y, +d; t)

(-kY'T· n) I(anN;t)

aqat' in 0., t E (0, r]

(9)

Here, c and K indicate the specific heat and the thermal conductivity, respectively, whileq E L2(n) and t* denote the internal heat source anel the time interval under consideration.

9

Thermal and mechanical quantities are constants in the metal a.nd ceramic layers, butfunctions of z in the FGM layer. It is obvious all material properties are CO(O) functions.In addition, constant room temperature To is applied to the bottom surface f_while varyingtemperature f(x,y;t) E L2(aD.D) is applied to the top surface f+, and external heat fluxes13E U(aD.N) are applied to fe.

After obtaining the temperature field T( x; t), we compute thermal stress distribu-tions. In order to calculate them, we must derive the suitable equations. First, accordingto assumptions made so far, the non-vanishing strain components are expressed by

1Cx )Cy

,xy (X;t) 1cO )

x

cD

,;y (x,y;t) 111,0 )

x

",0+ Z· n,y

K,~y (x.y:t)

(10)

where {c~, c~, ,~y} are in-plane strains and {K,~, K,~} , K,~y are in-plane curvatures and torsion,respectively, of the mid-surface. Using the stress-strain relations of plane-stress problems,we have the non-vanishing stress components given by

(11 )+ z· K,~ ) 16T )+ z. K,~ - ( oE ) 6T

1 - v (z)

+ z'K,~ 0

o

]\

cOx

o c~

(1 - v) (z) I~ylax ) [1 1.1E

ay = ( 2) V 11 - 1.1

Txy (X;t) 0 0

where 6T is (T(x; t) - To).

In order to compute the thermal stresses due to the temperature gradient, the ap-plied external force Fext and moment Mext (thickness-integrated/unit area in w), we mustdetermine three in-plane strains, two in-plane curvatures and a torsion. First, we denotesix across-the-thickness thermal stress resultants, for which a notation convention is shownill Fig. 6, by

(12)

(13)(i = 0,1,2)Ii(x, y)

Hi(x, y)

Ji(x, y; t)

and we introduce moment-of-inertia-like quantities {Hi, Ii, Ji, Li}

Jd vE idz z-d (1 - 1.12)

Jd E '(

_, ZZ dz-d 1- II

Jd aE6T 'z~dz

-d(l-v)

Jd E '

zldz-d(l-v)

10

Since the thermal stress resultants should be statically equivalent to the applied forceand moment, it is not difficult to obtain the following system of simultaneous equations forthe four unknowns

and the relations for two unknowns given by

(14)

°Ixy [- L Fext + L A1ext] /£ }2 xy 1 xy

[L Fext - L Mext] /£1 xy ° xy

(15 )

Here, £ is defined as (Lt - LoL2).

We then compute thermal stress field with Eq. (11) at any time and for any appli-cation of external forces and moments.

Remark 3.1 From the -relations -in Eq. (15), we know thai {,~y,K~y} a-re time-independent.fUTthe-rmo-re they are identically zero if no external f01'ce and moment are applied. •

Here, we consider a simpler case of {cx = Cy} and {,XY = O} which corresponds touniform heating function f( x, y; t) = f( t) and an external forces and moments satisfying{Fext = Fext = Fext} {At/ext = Mext = l\1ext} and {Fext = Mext = O} Then using the factx y 'x y xy xy . ,

of {ax = ay = a} and {Txy = OJ, we have

E(z) [ ]a(z; t) = I () cO(z) + Z . KO(Z) - a(z)6T(z; t)1 - v z )

where two unknowns are

(16)

(17)cO = [-12(JO-Fext)+11(Jl+Mext)]/i}

KO = [11(Jo-Fext)+10(Jl+Mext)]ji

Here, the Ii are defined by substituting Ej(1 - v2) in Eq. (13) with E/(1 - v) while t isdefined as (ii - 1012),

The symmetric matrix in Eq. (13) containing {Hi, Ii} should be positive definite fornon-trivial solutions. However, Eqs. (15) and (17) do not become singular because the twodeterminants £ and i do not vanish for v,# -1, as proved in Lemma 3.1.

11

Lemma 3.1 For 1/ =F -1, the two determinants £: and i defined above aTe always negativereal numbers (i. e. £:, i E lR- ).

PROOF. Let denote zE(z)j(l -1/) and zE(z)j(l - 1/2) by zE(z), then

{

llC.zE(z) = 0,

1R+ ,

z<Oz=Oz>O

(18)

and which implies (1'. zE(z) dz) 2 < (1'. I zE(z) I dZ) 2 . Furthermore, from Cauchy-Schwartz inequality, we have

(19)

Then, the following inequality holds

Finally, we have £:, I E lR- .

3.2 Paranletric Investigation

(20)

•We now define two quantities for examining the thermal-stress characteristics of FGM mem-ber X in the previously defined hierarchical families. In fact, t.he relative thickness of heat-resisting layered composites used in most engineering applications is thin, and furthermoremajor concerns focus on the distributions of axial stress components (7x and O'y through thethickness.

By denoting the thermal-stress distribution of axial stress component O'a(a = X, y)of FGM members.:\It by O'a(./\It), we define two quantities as the basis for parametric inves-tigation on the effects of the two parameters.

The first quantity, the local sha'rpness 6(M) of the thickness-wise distributions ofaxial thermal-stress components O'a(M), is defined as

1) de! • I6(M = 21 Zo - Z

(21 )

Referring to Fig. 7, Zo is a vertical location in the FGM layer at which thermal stress reachest.he steepest value convex or concave, while z* is defined as its closest neighborhood location

12

Thermal stress, O'a

-dz

Figure 7: A definition of local sharpness .6.(}vl).

with a value of O'~ = O'~ - .6.0'0' Here, .6.0'0 is defined as a relative difference between O'~

and 0'~/..;2. When the steepest side intersects with the bottom or the top side of FGMs, weextrapolate the thermal stress distri bu t ion to calculate z*. The thickness-averaged L2-norm.IIO'a(M)llu(-dm,dm)l the other quantity, is defined by

0' = x,y (22)

The thermomechanical behavior exhibits the edge effect near the boundary when thedimension of the mid-surface of FGMs is finite. In addition, it varies along the mid-surfaceunder general thermal and mechanical loadings. However, since we are interested in itsvariations through the thickness with the two defined parameters N and ox, we restrict tothe xy-invariant thermomechanical distributions for our parametric analysis. Then, for thefunctionally graded heat-resisting composites with the composition volume fraction definedin Eq. (1), their thermal-stress distributions satisfy the following properties with respect tothe two defined quantities.

Property 3.1 FO'r a given N, the x-hierarchical fam.ily FJi has the following properties(except fOT -relatively small 01' large N ):

.6. ( .1\.1 ~ ) d ecreas es as ox. -t 1

II00Q(J\.1~)IIL2(-dm,dm) increases as X -t 0, 0' = x, Y

•13

Property 3.2 The N -hierarchical family :Fif exhibits the following property:

6(M~) -+ incTeases as N -+ 0 OT 00

•4 Formulation of the Approximations

An approximation of T(x; t) in Eq. (9) involves discretizations both in time and spacedomains. In this paper, we employ a finite difference method in time and a finite elemcntapproximation in space.

4.1 Time Discretization and Variational Formulation

Let us make finite N uniform partitions (or subinterval-wise uniform partitions) in timedomain (0, t~], then we have uniform (or subinterval-wise uniform) time intervals 6t = t* jNand (N + 1) time stages tn = n6t, (n = 0,1,' . " N).

For a time discretization, we use the Cmnk-Nicolson method. a first order timeapproximation with relatively higher local truncation error O(.6.tl, in which Tn+1/2 and(aTjat)71+1/2 at the mid of two time stages in and tn+! arc

(~~) n+1/2

(23)

Substituting Eq. (23), (aqjat)n+1/2' fn+1/2 and t9n+1/2 into the initial-boundary-value problem (9), we have a sequence of semi-discrete converted boundary value problemsfor N time stages tn (n = 0,1,2" . " N - 1):

~ (1' - l' ) - V . { kV (Tn+1 + Tn)} = (aq) . 06t n+1 'n 2 at' lI1n+1/2

Tn+1/2(X) - To, on y = -d ~ (24)

Tn+1/2(X) = fn+1/2(x, y), on y = +d

(-kVT·n)n+1/2(x) = ~n+1/2(X), on 80N

It is well known that the Crank-Nicolson scheme, while unconditionally convergent, pro-duces oscillatory results unless the time step is appropriately selected.

14

The converted boundary value problem (24) characterizes successively the tempera-ture distribution T(~; tn+d at time tn+1 with the previously obtained solution T(~; tn) attime tn and the known boundary data of To, fn+l/2 and ()n+l/2'

Now~we establish a weighted residual variational formulation for the solution T(~; tn+1)

at the (n, + 1)th time stage. First, we define the space \/(0) of admissible test temperaturefields such that every function v in \/(0) has finite thermal strains and its trace on aDDvanishes

V(o) = {v(~): v(~) E H1(0)",ov = o} (25)

with /0 defined as a trace operator, /0 : H1(0) ~ Hl/2(aoD). On the other hand, the trialfunctLon space Vn+do) for the (n + l)th time stage is defined as a linear manifold of V(O)

(26)

where {w*} are extended H1(0) functions satisfying w*1 = To and w*1 =fn+l/2' Wer_ f+note that the trial function space is time-stage-dependent owing to different trace data onr+ at different time stages, while the test function space is time-stage-dependent.

As usual, multiplying the converted partial differential equation by a test function Qand integrating over the domain 0, we arrive at a sequence of N abstract boundary valueproblems:

Given To E "~(O), find Tn+l E V(O) + {W*}n+l such that(27)

a(Tn+l' Q) = £(Q), VQ E V(o), (n = 0,1,2,' . " N - 1)

Here, a bilinear functional a(·,·) : V(o) x V(f2) ~ IR and a linear functional £(.) : V(o) ~ lRare, respectively, defined by (dx = dxdydz)

£(Q)

in pcTn+1Q dx + ~t in k(\1Tn+1 . \1Q) dx

{ pcTnQ dx - 6t ( k(\1Tn . \1Q) dxin 2 in

+ { qn+l/2Q dx - 6t { f}n+l/2Q dxin ianN

(28)

It is well known, from the Lax-Milgram theorem, t.hat the problem (27) is well posedand the solution depends continllosly on data.

4.2 Finite Element Approximations

Since finite element approximations of the problem (27) are standard, we do not describethe details on the procedure. Along the standard procedure, one arrives at the full-discrete

15

finite element approximation, the well-known Crank-Nicolson-Galerkin scheme. For thedetailed analysis on the stability and consistency of this scheme, see Johson [17] and Odenand Carey [21].

As is well known, this scheme produce oscillatory results when the time step is notsmall enough, particularly for the problem showing sudden changes in its time response byimpulsive heating loads or abrupt changes in the boundary condition. This is because onetries to reach directly at the steady state with a larger time step, and such a phenomenonprevails as the largest eigenmode (i.e., one of the shortest wavelength) dominates, such asexponential decay for increase in the transient response. One of these situations in the heatdiffusion problem is the problem subjected to thermal shock.

In order to determine the critical time step for our study, we examine finite-elementeigenvalues in the free response of the problem (9). The free response is expressed by

(29)

with eigenvalues ..\mn (assuming constant p,c and 1\:) given by

(30)

where, Lx and Ly denote the characteristic lengths of n E mN (when N is 2). Then, for auniform finite element mesh with varying material constants, the largest eigenvalue is

..\max ~ N [5f (;) max(31)

where d denotes a distance between two adjacent nodes. On the other hand, for a non-uniform mesh with varying material constants, it is

(32)

In order to prevent oscillations in such a largest eigenmode, it has been suggested that oneuse the time step satisfying

(33)

with a correction parameter Cf of 2 '" 3 [5]. The critical time step depends on the materialproperties and the size of finite elements. In particular, it has a proportional relation to thesquare of minimum size of finite elements.

For the entire time interval, use of the uniform time-stepping method with the timestep corresponding to the critical eigenmode may lead to a very expensive numerical com-putation. In fact, the time-response is characterized by dominant several eigenmodes, and

16

Table 1: Numerical data of thermal and mechanical properties of Ni and A1203.

[I Properties ~ Costituents ~Ni I Ah03

Density (kgJm3) 8900.0 3970.0Young's modulus (Gpa) 199.5 393.0Poisson's ratio 0.3 0.25Specific heat (J j kg . °K) 444.0 775.0Thermal conductivity (Wjm . °1\') 90.7 30.1Thermal expansion coefficient (OC x 10-6) 13.3 8.8

f(t) (oK)

~I I II I~~~

Tl = 1290 ' A

To= 290

3(a)

III

20

t (sec)

AI20) -1(uniform)

graded r(Unif:~) f(uniform) i

(b)2L

p

2d

Figure 8: A heating function f(t) and a FE mesh pattern.

the range of eigenmodes dominating the time-response varies along the time duration inmost cases. Therefore, it would be advisable to adopt the graded time-stepping method(similar to an irregular mesh construction in spatial approximations) or to divide the entiretime interval into several subintervals with different uniform time steps.

5 Numerical Experiments

To investigate the thermo-elastic characteristics of FGMs, we simulate a symmetric two-dimensional functionally graded composite beam of uniform thickness 2d = lOmm, 2L =Lmm, and unit depth. As for the metal and ceramic layers, we select Ni and A1203,

respectively, and their thermal and mechanical properties are summarized in Table 1.

Referring to Fig. 2, a room temperature To of 2900 K is kept on the bottom surfacewhile a heating temperature f(t) shown in Fig. 8(a) is applied on the top surface. Fur-thermore, the condition of no temperature gradient is assigned at the left and right sidesin order to exclude the boundary effect (i.e., for the x-invariant temperature field). With

17

this FGM model, we analyze the transient and the steady-state response in temperatureand thermal-stress fields. As for the values of the two parameters, X and N, we take N of0.3,0.7,1,5,10 and 100 for each X of 0.2,0 ..5,0.7 and 1.

For finite element approximations, we make six uniform partitions in the horizontaldirection and three layer-wise uniform partitions (i.e. 3x20 partitions) in the thicknessdirection so that total of 360(6x60) almost uniform biquadratic tensor-pro duct-type elementsare assigned regardless of the combination of the two defined parameters.

For a given heating condition, numerical and geometry data, we examine the timestep according to the previous discussion. For the x-invariant temperature distributionthrough the thickness and the layer-wise uniform mesh using biquadratic elements, we obtain6tcrit ~ 0.0001 sec for the largest eigenmode from Eqs.(3l) and (33) (with Cf of 3). On theother hand, for the smallest eigenmode, we obtain 6t ~ Isec (with Cf of 1). However, sincethe suitable time step depends definitely on the dominant eigenmode in the temperatureresponse, we first examine the temperature-time histories of the problem under considerationwith different uniform time steps 6t of 0.1, 1 and 5 sec. For this preliminary test, we use Xof 0.2 and N of 0.3 because this combination is expected to show the representative behaviorto determine the time step for our numerical experiments.

We compute and plot the temperature-time histories at the upper interface of thegraded-layer (i.e. at the point p in Fig. 8), which are depicted in Fig. 9. From the numericalinformation together with the 6tcrih we see that the system response is not dominated by thelargest eigenmode, and furthermore it is characterized by different eigenmodes for differentsubintervals. We thus divide the entire time interval into three subintervals 1(0 - 5 sec),II(5 -lOsec) and 111(10- 20sec), as shown in Fig. 8. For the three subintervals, we usedifferent uniform time steps of 0.05, 0.1 and 1 sec, respectively.

For the N -hierarchical families with X of 0.5 and 1.0, we plot the computed tem-perature distributions at time tA and ta in Figs. 10 and 11. Where we include the classicallayered composite (denoted by CLC in the figures) with a kink at the interface in its tem-perature distribution. By comparing the transient (at time tA) and the steady-state (attime ta) temperature fields, we observe the unsaturated and slightly curved distributions inthe Al203-dominated region at time tA owing to the relatively smaller termal conductivityof Al203. This tendency at time tA prevails as the relative thicknesH ratio increases.

When we compare the distributions in Fig. 10 with respect to the parameter N, weobserve that the temperature gradient at the FGM-AI203 interface becomes steeper as Ntends to O. But, as N approaches 100, such a steep gradient prevails at the other interfaceas N approaches 100. This follows from the previous parametric investigation. In otherwords, an FGM becomes m-CLC or c-CLC, respectively, as N -+ 0 or N approaches 00.

Ilowever, as shown in Fig. 11 for the f-FGM, the temperature distributions are smoother

18

700

650 I "'.

600

550 ~1/·i:' ;' 0,\ sec -: .': , 1.0 sec ....

g ./ f/ 3,Osec ----

~ 500::l.....& 450E~

400

350

300

2500 3 5 10 15 20

Time(sec)

Figure 9: Temperature-time histories at the point p for three different uniform time steps(X of 0.2 and N of 0.3).

and such kinks are not observed, due to the disapperance of the interface with sharp materialvariation.

For the N -hierarchical families with X of 0.2,0.5, 0.7 and 1.0, the computed thermal-stress distributions at time tA are presented in Figs. 14-16, while those at time tB are givenin Figs. 12, 13, 15 and 17. Where the bimaterial composite case is included in order fora comparison purpose. The transient response in thermal stress, when compared to thesteady-state response, shows the slightly curved pattern in the Al203-dominated regionaccording to the unsaturated curved-distribution of the temperature field. This tendencyprevails as N increases. Because the difference in termal-stress distributions at time tA andtB become more considerable as X increases, due to the extension of Al20a to the entirecomposite region, we exclude the trasient distributions for the cases with X of 0.2 and 0.5.

By examining Fig. 12 of the N -hierarchical family with X of 0.2, we see thermal-stress concentrations at the FGM-A120a interface for relatively small N and at the FGM-Ni interface for relatively large N, respectively. The concentration at the right interfaceis relaxed as N increases, while the other concentration near the left interface looses as Ndecreases. This figure represents well the effect of the volume fraction on the thermal-stressbehavior near the interface with steep material property varia.tions.

Through the next Figs. 13-17, we observe the effect of the relative thickness ratioall thermal-stress of the N -hierarchical families. Because the difference in thermal stressdistributions at time tA and tB is considerable when the relative thickness ratio is large,we include the trasient results for X of 0.7 and 1.0. When we compare the trasient and

19

1400 1400CLC -' CLC -

N:::OJ -: (al time A) N:::OJ -' (allime B)1200 ~ 0.7 1200 0.7 ..

1.0--: 1.0-5.0 ---- ' 5.0 -- ..100-: 100-

Q 1000 '" 1000~~ 0( (

'" '"0 0~ ~~ 800 ~ 800~ ~~ ~0 8.Q.E EI)

~I-< 600 • 600•

400 400

(loo%Ni) (FGM Layer) ~ (Ioo'k A1203)200 I , , ! , I 200

-5 -3 ·1 1 3 5 ·5 -3 -1 I 3Coordinate in the thickness direction (mm) Coordinate in the thickness direction (mm)

Figure 10: Temperature distributions of the lV-hierarchical members )'v(;;5'

1400 1400CLC- CLC -

N:::OJ - (at time A) N:::OJ- (al time B)

1200 ~ 0.7'" 12000,7 ......

1.0 -- 1.0 --5.0 ---- 5,0 ----100 ....,.., 100 _m

Q 1000 '" 1000~c 0( (

'" v

~ 0..., 800 a 800',1 <l~ ~8. &E E~

0

600 f- 600

400

200 200,5 ·3 .[ I 3 5 ·5 -3 ·1 1 3

Coordinate in the thickness direction (mm) Coordinate in the thickness direction (rom)

Figure 11: Temperature distributions of the N -hierarchical members .I\..1~o·

20

steady-state thermal-stress behavior, the difference becomes considerable near the FGM-Niinterface (near the bottom surface for the f-FGM) as N increases. On the other hand, whenwe examine the thermal-stress distributions with respect to the relative thickness ratio (i.e.,for the x-hierarchical families), we first see that the stress concentration near the FGM-Niinterface becomes monotonically relaxed as X increases. We note here that, the f-FGM withN of 100 displays a convex concentration in Fig. 16 and a sudden drop from the almost zerostress level in Fig. 17, respectively, near the bottom surface. The stress concentration atFGM-Ni interface for the relatively large N seems to be relaxed, but not for considerablysmall N ~ 0.7. For the x-hierarchical members M~.3 and .I\;f~,7' the stress concentrationsremain unrelaxed even when X is 1.0.

Figure 18 shows the variations of the estimated thickness-averaged U-norm of thermal-stress distributions in the middle FGM layer at time tA and ta, respectively. As the relativethickness ratio l/X increases, the values estimated at both time points increases mono-tonically for every x-hierarchical member, which implies that the resulting thermal stresstends to concentrate in the middle FGM layer as the FGM layer becomes thinner. This isconsistent with Property 3.1 in Section 3.2.

The last figures Figs. 19 and 20 show the numerical results concerned with the localsharpness L:.(M) for the x- and N -hierarchical families at time tA and ta, respectively.The estimated local sharpnesses of the N -hierarchical families are presented in Fig. 19,where the estimated values, except for j\;f~,3' M~.7 and /\;f~oo' decrease uniformly as Xincreases. For those FGM members, it is not relaxed but father increases significantly(except for the M&.7 at time tA). This is because the stress concentrations at the FGM-Al203 interface (for N ~ 0.7) and the FGM-Ni interface (for N 2:: 100) do not disappearas X increases, and which illustrates the previously described Property 3.1. The estimatedresults for the N - hierarchical families are shown in Fig. 20 (where, RT R refers to Relat£veThickness Ratio X). As declared in Property 3.2, the local sharpness of the N-hierarchicalfamily increases as N ~ 0 or N ~ 00. However, the rate of its increase for N ~ 00 is muchlower than for N ~ 0, and this slowness becomes more critical at the transient stage.

6 Conclusions

From the numerical results, we observe that thermomechanical characteristics, such as tem-perature and thermal stress distribution, the local sharpness and the t.hickness-a.veragedL2-norm, are quite different for various combinations of the two parameters, N and x·However, the temperature and thermal-stress distributions vary in a sequentia.l mannerwith respect to the two parameters, and the estimated quantities IIO'x(.A.1)11L2(_dm.dm) andL:.(M) of thermal stresses follow from Properties 3.1 and 3.2. Furthermore, the difference

21

'. I

500

400

~ 300

6>< 200><,.1

100a011

'0;

'" 0'"g'" -1000;§ -200"~

-300

-400

-500-5

CLC -N=O.3 -

0,71,0""5,0 ----10 .. ,..100-

-3 -I I 3 5Coordinate in the thickness direction (mm)

Figure 12: Thermal stress(O'x) distributions of the N-hierarchical members Mii2 at timetB.

500 r CLC -;N=O,3 - i

400 0,7 ........ ;1.0 ,... - i

-;- 300 5,0 ---- :c..~ 10 ........ :~ 200 100-:><><,.\

100 (N=IOOj~

'0;0'"'"§

-100'"-;a§ -200"~

-300

-400

-500-5 -3 -1 I 3 5

Coordinate in the thickness direction (mm)

Figure 13: Thermal stress(o-x) distributions of the N -hierarchical members A1~5 at timetB·

22

' ..

500

400

-. 300..c..6 200

><>< 100.. I

8 aC(j'Vi<Il -100<Il

~'" -200

01E ·300'"~ -400

-500

-600L

-5

'CLC-N=O,3 -, 0,7'

1.0---5,0 --,--10 ........

100 -

-3 -I 1 3 5Coordinate in the thickness direction (mm)

Figure 14: Thermal stress(ax) distributions of N-hierarchical members /\/t[;',7 at time tAo

500

400

"2 3000..

6~ 200.. I

~100

'Vi<Il 0<Il

~'" -100

01

5 -200~

·300

-400

-500-5

;CLC -N=OJ -; 0,7 ....... ,

1.0 .-.. -5.0 .--,-10 .." ....

100 -

-3 -I I 3 5Coordinate in the thickness direction (mm)

Figure 15: Thermal stress(ax) distributions of the N-hierarchical members M[;',7 at timelB.

23

'. .

500 CLC -400 N=O,3 -

0,7 ........

-- 300 ].0 ----cd 5,0 ----Q..

6 200 10 '>< 100 ->< 100 ,~ ..<~::.:,"',.,", ..,"~'\, (N=O,3)cd

~ 0"Vi

'" -100'"~on -200...§ -300.,~ -400

-500

-600L

-5 -3 -I I 3 5Coordinate in the thickness direction (mm)

Figure 16: Thermal stress(O'x) distributions of the N-hierarchical members Mf.a at timetAo

500

400

'2300Q..

6>< 200><,...~ 100'inonon

0~'"...

-100~~ -200

-300

-400-5

CLC -N=OJ -

0.7 ........1.0 .,----5,0 ----10 " .. ,..

100-

_.~-'-'''''-'-

.';:J~r". "~,,~-~",-.~::-:--c::.-, __...

(N=IOO)

-3 -J I 3 5Coordinate in the thickness direction (0101)

Figure 17: Thermal stress( 0'x) distri bu t ions of the N- hierarchical members /\..1 f.a at timelB·

24

'. ,

250 250

2 3 4Inverse of relative thickness ralio of FGM layer

5

.........

(at lime B)

234Inverse of relative thickness ratio of FGM layer

N::OJ -+-0.1 +-1.0 ,iI··

5.0 -J-

10-100 -J , ..

o1

150

200

5

(at lime A)N::O,3 -+-

0.1 +.

1.0-1)'5,0 -»--10 -....

100 .J.-

oI

200

150

100

Figure 18: A variation of "ax(M~)ilL2(-dm.dm) along to the relative thickness ratio X.

...-... -'

""1'

..,~·~-·:..~~~~;.~~·,~·~:·_~_::-:.:.~~~uUD ..:.J::m/l.".~,~'r>."r,

......' "' , , -..__ :'~..

o0,2 0.5 0.7

Relative thickness ratio of FGM layer

4

8

I2

10

16

14

18

(at lime A)

0.5 0.1Relative thickness ratio of FGM layer

Figure 19: A variation of .6.(M~) along to the relative thickness ratio X.

25

'. .20 18

100

RTR::O,2 -+--0.5 +..0.7 ·a·1.0 -l(-

(at lime B)

'\'.

\q\....\'\,

\L\I\','

.: ..,.-;:> •.~:::~':~~'t=-.f./:'..._..-, '"

I 10Power index N of metal I'olume fracuon

6

4

o0,1

16

12

14

10

100

RlR::O,2 -+--0.5 +..

O.7~'"1.0 -~-

(at lime A)

I 10Power index N of metal volume fraction

o0,1

6

2

18

16

10

14

12

Figure 20: A variation of 6(M~) along to the metal volume fraction parameter N.

ill thermal characteristics between the transient and the steady-state stages is noticeable,especially for large X and N.

Except for the relatively small or large N, the temperature and thermal-stress distri-butions become smoother as the relative thickness approaches unity. In the N -hierarchicalfamilies, the stress concentration prevails at the FGM-AI203 interface as N tends to zerowhile near the FGM-Ni interface as N goes up. Compared to the classical layered compos-ite, FGMs except for the three limiting cases (N ---+ 0, +00 and X ---+ 0) exhibit considerableimprovement in the temperature and thermal-stress distributions. This is due primarily tothe enforcement of continuity in material properties through the thickness

Based on our theoretical and numerical results, the selection of optimal (at least,quasi-optimal) combinations of the two parameters satisfying the predefined specificationfor heat-resisting FGM composites under given thermal circumstances would be worthwhile,and which represents a topic that deserves the future work.

Acknowledgements: The financial support given to the first author(JRC) by Pusan Na-tional University in Korea under the Grant-in-Scientijic-Assistance (May, 1997 - April,2001) and that to the second author(JTO) by the Office of Naval R.esearch under GrantNo. N00014-95-1-0401 are gratefully acknowledged. The authors would like to thank theanonymous referees for their valuable comments.

26

'. .

7 Appendix

According to the modified rule of mixtures, the Young's modulus of graded two-phasecomposites is given by

By denoting two phases in Fig. 5 by m and p together with the parameter c = h/a.,we have the volume fraction of the second phase p

c = V-1/3 1p - (35)

Then, according to the unit cell approximation, we calculate the Young's modulus of two-phase composites using

(36)

aBel the Poisson's ratio by

(37)

(38)

According to the Wakashima- Tsukamoto expressions for the macroscopically isotropicparticulate composites, the overall bulk modulus K and shear modulus p are obtained by

Y _ r aVcI<m(l( - I<m)\ - \m + T r F + V l'

m l\c a c1i.m

where a and bare

µ (39)

aKc(3I<m + 4µm)I<m(3Kc + 4µc)

(40)

(41)b (1 + e)µc (}' 8 )/( I' . )= ( )' e = 9 \m + µm 6 \m + l2µmµm + eµc

The overall mechanical properties are immediately obtained from basic parametric relations

1/

9I?p/(3I? + µ)

(3J{ - 2µ)/2(3K + p)

27

(42)

(43)

'. .

400

380

360

-;- 340c..~ 320

<J>::I

300"5'8E 280<J>

"bJ)

260t:::I0>- 240

220

200

]80-0,5

N=03(a) -(b) 'd

(c) - ---(d)

-0,25 0 0,25 0.5Coordinate in the thickness direction (mm)

Figure 21: Distributions of the Young's modulus calculated by the unit cell approach(a),the Wakashima- Tsukamoto expression(b), the modified rule of mixtures( c) and the linearrule of mixtures( d).

And the overall coefficient of thermal expansion & is related to the bulk moduli throughLevin's relation:

(44)

Furthermore, the overall specific heat c (neglecting the higher order term) and ther-mal conductivity k are expressed, respectively, by

(45 )

(46)

A comparison between the distributions of the Young's modulus calculated by the fourdifferent approaches is shown in Fig. 21, for which we used the material data given in TableL For each volume fraction N, the modified rule produces smaller values than those by theother approaches, while the linear rule of mixtures predicts the stiffest values.

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31