Computational Materials Science 46 (2009) 151–161

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Computational Materials Science

journal homepage: www.elsevier .com/locate/commatsci

The statistical second-order two-scale analysis method for heat conductionperformances of the composite structure with inconsistent random distribution

Yan Yu a,*, Junzhi Cui b, Fei Han a

a Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Dongda Town, Chang’an District, Xi’an 710000, Chinab Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China

a r t i c l e i n f o

Article history:Received 26 September 2008Received in revised form 22 December 2008Accepted 10 February 2009Available online 19 March 2009

PACS:44.05.+e44.10.+i44.35.+c

Keywords:The SSOTS methodComposite structure with inconsistentrandom distributionMeso-structureHeat conduction propertiesFunctional gradient materialsNumerical technique

0927-0256/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.commatsci.2009.02.016

* Corresponding author. Tel.: +86 10 6256 8715; faE-mail address: xiaoxiaoyuyan@163.com (Y. Yu).

a b s t r a c t

In this paper, a statistical second-order two-scale (SSOTS) method is presented for predicting heat con-duction performances of the structures of composite materials with inconsistent random distribution.Firstly, the meso-scopic configuration for the structure with inconsistent random distribution is charac-terized. Secondly, the SSOTS asymptotic formulation for predicting the heat conduction parameters andthe temperature fields of the heat conduction problems of the structure is given by means of constructionway. Then the heat conduction properties for the functional gradient materials (FGM) with varying vol-ume fraction are predicted by the SSOTS method. The numerical results are compared with experimentaldata and theoretical results. Finally, macroscopic heat conduction properties for the structures with vary-ing probability distribution models including size, location and orientation distributions of grains are cal-culated. The numerical results show that the SSOTS method in this paper is valid to predict heatconduction performances of composite structures with inconsistent random distribution.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Random composites have been widely used in a variety of engi-neering and industrial products. According to the arrangementcharacteristics of inclusions, random composites are classified intothe materials with consistent random distribution and those withinconsistent random distribution (Fig. 1). For the former both thevolume fraction and probability distribution models of inclusionsare same at anywhere, such as normal concretes, and metal, cera-mic and polymer matrix composites. For the latter, however, theychange with position, such as functionally graded materials withvarying grain’s volume fractions [1], the cancellous bone withvarying pore’s sizes [2] and polymer with varying fiber’s orienta-tions in gas-assisted injection molded part [3]. For compositestructures with consistent random distribution, authors havedeveloped a computer generation method of meso-structure [4],and proposed a statistical second-order two-scale method [5] topredict their heat conduction properties. So, in this paper a strong

ll rights reserved.

x: +86 10 6254 2285.

emphasis will be laid on the investigation of heat conductionproperties of composite structures with inconsistent randomdistribution.

In recent years FGMs (Fig. 2) have attracted wide interest in thefields of material sciences as well as thermal and mechanical engi-neering, and they are made into the composite structures withinconsistent random distribution. In the fields of experimental[6–8] and theoretical studies [9–14] scientists have done a lot ofworks and made many valuable results. However, most studieswere devoted to the FGMs where the volume fraction of grains(pores) varies continuously along a single direction. And theoreti-cal prediction models can be just applied to predicting the macro-scopic effective properties of composite materials, but can not beemployed to calculate the temperature and heat flux fields inmeso-scopic level. Moreover, no model has taken into account allrandom parameters together, including the shape, orientationand spatial location, and the volume fraction and so on. So, in thispaper general structures with inconsistent random distributionwill be investigated, and for predicting their heat conductionproperties and calculating temperature and heat flux fields inmeso-scopic level a new statistical second-order two-scale (SSOTS)method are brought forward.

Fig. 3. (a) Composite structure with inconsistent random distribution and (b) 10parameters of an ellipsoid.

Fig. 1. The schematic representation for the meso-configuration of randomcomposites (a) composite structures with consistent random distribution and (b)composite structures with inconsistent random distribution.

Fig. 2. Functional gradient materials.

152 Y. Yu et al. / Computational Materials Science 46 (2009) 151–161

In this paper, to predict the macroscopic heat conductionbehaviors of the structures with inconsistent random distribu-tion, the following step is adopted: Firstly, the finite discretepoints inside the structure are chosen, and an appropriatee – cuboid domain centered at each point is taken as localmeso-scopic scale cell, and the characteristics of random distri-bution inside it will be represented. Further the heat conductionbehavior corresponding to the point can be evaluated by theSSOTS method. Finally, the effective heat conduction parametersinside the global structure are obtained by interpolation techni-que from discrete points.

The remainder of this paper is outlined as follows. In the follow-ing section, the meso-scopic configurations for composite struc-tures with inconsistent random distribution are represented. InSection 3, for the structures with inconsistent random distributionthe statistical second-order two-scale analysis formulation foreffective heat conduction parameters, temperature field and heatflux densities are derived. In Section 4, the numerical simulationfor heat conduction performances of various structures with incon-sistent random distribution are performed. Finally, the conclusionsare given in Section 5.

2. Representation of meso-scopic configuration of compositestructures with inconsistent random distribution

Suppose that the structure X with inconsistent random distri-bution is made from the grain phase A and the matrix phase B,and all grains are randomly arranged in the matrix. Refer to Ref.[4], every grain can be considered as an ellipsoid with ten randomparameters (Fig. 3): the location parameters, namely the centralpoints (x0; y0; z0) of the ellipsoid, and the shape parameters, namelythe long axis a, the middle axis b and the short axis c of the ellip-soid, and the orientation parameters, namely the direction b1 of thea-axis with respect to the z-axis, the direction m1 of the projectionof the a-axis on the xoy-plane with respect to the x-axis, the direc-tion b2 of the b-axis with respect to the z-axis, the direction m2 ofthe projection of the b-axis on the xoy-plane with respect to thex-axis. Then the composite structure X with inconsistent randomdistribution can be represented as follows:

(1) At every point x0 in the structure X there exists a much smal-ler constant ex0 than the size of the macroscopic structure Xso that all ellipsoids in the cuboid-shape cell Cx0 centered atx0 and with side length ex0 subject to specified probabilitydistribution model Pðx0Þ, and their volume fraction is notedas V(x0).

(2) PðxÞ and VðxÞ are continuous and differentiable in X.(3) Suppose there are N ellipsoids in the cell Cx0 , then define

�

xx0 ¼ x101; x201; x301; a1; b1; c1; b11; m11;b21; m21; � � �;x10N ; x20N; x30N; aN; bN; cN;b1N ; m1N ;b2N; m2N

�: ð2:1Þ

Obviously, xx0 is a sample of Pðx0Þ and the volume fraction of Nellipsoids is Vðx0Þ. Let x ¼ fxx0 ; x0 2 Cx0 � Xg, then x represents thedistribution characteristic of ellipsoids in the whole structure X.

In this paper the following material parameters for thecomposite structures with inconsistent random distribution areconsidered

kijðx;xÞ ¼kA

ij; x 2 XA;

kBij; x 2 XB;

(ð2:2Þ

where XA denotes the domain of grain phase, XB is the one of matrixphase. kA

ij and kBij are the constants and satisfy that maxfkA

ij;���

kBijgj < M1;M1 > 0.

Y. Yu et al. / Computational Materials Science 46 (2009) 151–161 153

3. The SSOTS analysis method for composite structures withinconsistent random distribution

3.1. The SSOTS asymptotic expansion of temperature solution for thestructures with inconsistent random distribution

Consider the following stationary heat conduction problem ofcomposite structures with inconsistent random distribution:

Leðheðx;xÞÞ ¼ � @@xi

keijðx;xÞ

@heðx;xÞ@xj

� �¼ f ðxÞ; x 2 X;

heðx;xÞ ¼ �hðxÞ; x 2 C1;

mikeijðx;xÞ

@heðx;xÞ@xj¼ pðxÞ; x 2 C2;

8>>><>>>:

ð3:1Þ

where heðx;xÞ denotes the increment of the temperature, �h is theone of the boundary temperature and f ðxÞ is the internal heatsource.

At first, suppose

(1) If gðxÞ is a function of a bounded random variable, its math-ematical expectation EgðxÞ exists.

(2) keijðx;xÞ 2 C1ðxÞ and f ðxÞ 2 C1ðXÞ. If ke

ijðx;xÞ 2 L2ðXÞ, onecan construct a smooth operator d: ke

ijðx;xÞ ! dkeijðx;xÞ

2 C1ðXÞ satisfying

dkeijð�;xÞ � ke

ijð�;xÞ������ ! 0 as d! 0; ð3:2Þ

(3) (keijðx;xÞÞn�n are symmetric and there exist two positive con-

stants c1; c2 (not depending on x) such that

c1gigi 6 keijðx;xÞ 6 c2gigi 8x 2 X: ð3:3Þ

By Lax–Milgram theorem, Poincaré’s inequality and supposition3, for a given x 2 P the Eq. (3.1) has one unique solution. In thissection, we will derive a SSOTS asymptotic expansion of tempera-ture field for heat conduction problems of composite structureswith inconsistent random distribution by means of constructionway, and establish a SSOTS formulation to evaluate the heat con-duction parameters.

It is well-known that the temperature increments of compositestructures with inconsistent random distribution do not only de-pend on its global behaviors, but also on random meso-scopic con-figurations, so in the cell Cx0 the increment of temperaturecorresponding to x0 can be expressed as heðx;xÞ ¼ hðx; n;xx0 Þ,where n ¼ x

ex0, x denotes the macroscopic coordinate and n is the lo-

cal one. By construction way we assume that in the cell Cx0 heðx;xÞ

can be expanded into a series in the following form:

heðx;xÞ ¼ hðx; n;xx0 Þ

¼ h0ðxÞ þ ex0Ha1 ðx; n;xx0 Þ@h0ðxÞ@xa1

þ e2x0 Ha1a2 ðx; n;xx0 Þ

@2h0ðxÞ@xa1@xa2

þ ba1ðx; n;xx0 Þ

@h0ðxÞ@xa1

!

þ e3x0P1ðx; n;xx0 Þ;

ð3:4Þwhere h0ðxÞ only depends on the macroscopic behaviors of thestructure, and does not reflect the meso-scopic configurations,and then it is called as the homogeneous solution, Ha1 ðx; n;xx0 Þ,ba1ðx; n;xx0 Þ and Ha1a2 ðx; n;xx0 Þ are called as local solutions, depen-

dent on ðx; n;xx0 Þ. They will be determined below.Considering

@

@xi¼ @

@xiþ 1

ex0

@

@ni; ð3:5Þ

(3.4) is substituted into (3.1) in Cx0 , which leads to the followingidentity:

@

@xike

ijðx;xx0 Þ@he x;xx0ð Þ

@xj

�

¼ e�1x0 �

@

@nikijðn;xx0 Þ

@Ha1 ðx; n;xx0 Þ@nj

� � @

@nikia1 ðn;xx0 Þ� ��

@h0

@xa1

þ e0x0 �

@

@nikijðn;xx0 Þ

@Ha1a2 ðx; n;xx0 Þ@nj

� �@2h0ðxÞ@xa1@xa2

� @

@nikijðn;xx0 Þ

@ba1ðx; n;xx0 Þ@nj

� @h0ðxÞ@xa1

� @

@nikijðn;xx0 Þ

@Ha1 ðx; n;xx0 Þ@xj

� @h0ðxÞ@xa1

� @

@nikijðn;xx0 ÞHa1 ðx; n;xx0 Þ� � @2h0ðxÞ

@xa1@xj

� @

@xikijðn;xx0 Þ

@Ha1 ðx; n;xx0 Þ@nj

� @h0ðxÞ@xa1

� kijðn;xx0 Þ@Ha1 ðx; n;xx0 Þ

@nj

� @2h0ðxÞ@xa1@xi

� @kijðn;xx0 Þ@xi

@h0ðxÞ@xj

� kijðn;xx0 Þ@2h0ðxÞ@xi@xj

!þ Oðex0 Þ ¼ f ðxÞ: ð3:6Þ

First, considering the coefficients of e�1x0 one obtains that

� @

@nikijðn;xx0 Þ

@Ha1 ðx; n;xx0 Þ@nj

� � @

@nikia1 ðn;xx0 Þ� ��

� @h0

@xa1

¼ 0; n 2 Q x0 ; ð3:7Þ

where Qx0 is the 1-normalized cell of Cx0 . Since @h0@xa1

arises from themacroscopic behavior of the structure, it is not identical to zero.Thus the following equation must be satisfied

� @

@nikijðn;xx0 Þ

@Ha1 ðx; n;xx0 Þ@nj

� ¼ @

@nikia1 ðn;xx0 Þ� �

; n 2 Q x0 :

ð3:8aÞ

Attach the following boundary condition on @Qx0

Ha1 ðx; n;xx0 Þ ¼ 0; n 2 @Q x0 : ð3:8bÞ

From the Eq. (3.8) it is easy to see that Ha1 ðx; n;xx0 Þ is independentof x, so it can be written as Ha1 ðn;xx0 Þ. Further, for any sample xx0 inQx0 Ha1 ðn;xx0 Þ is the solution of the following elliptic partial differ-ential equation

� @@ni

kijðn;xx0 Þ@Ha1 ðn;xx0 Þ

@nj

� �¼ @

@niðkia1 ðn;xx0 ÞÞ; n 2 Q x0 ;

Ha1 ðn;xx0 Þ ¼ 0; n 2 @Q x0 :

(ð3:9Þ

From Lax–Milgram Lemma, Poincare’s inequality and the supposi-tion 3, (3.9) has a unique solution.

Refer to Refs. [5,15], for any sample xx0 the homogeneousparameters are defined as

k̂a1a2 ðxx0 Þ ¼Z

Qx0

ka1a2 ðn;xx0 Þ þ ka2 jðn;xx0 Þ@Ha1 ðn;xx0 Þ

@nj

� dn:

ð3:10aÞ

From supposition 3 and theorem 2.1 in Ref. [15], it follows thatk̂a1a2 ðxx0 Þ is a bounded function of random variable. Further, Fromsupposition 1 the mathematical expectation of k̂a1a2 ðxx0 Þ exists, sothe expected homogeneous coefficients are defined as�ka1a2 ðx0Þ ¼ Ek̂a1a2 ðxx0 Þ. According to the Kolmogoov’s classical stronglaw of large number one can evaluate the expected homogenizationcoefficients

�ka1a2 ðx0Þ ¼ limM!1

PMm¼1k̂a1a2 ðxm

x0 ÞM

ð3:10bÞ

and �ka1a2 ðx0Þ is a function defined in the structure X .

154 Y. Yu et al. / Computational Materials Science 46 (2009) 151–161

Further, one can define the homogenized problem as follows:

� @

@xj

�ka1 jðxÞ@h0ðxÞ@xa1

� ¼ f ðxÞ; x 2 X;

h0ðxÞ ¼ �hðxÞ; x 2 C1;

mi�kijðxÞ

@h0ðxÞ@xj

¼ pðxÞ; x 2 C2:

ð3:11Þ

Similar to Ref. [15], from supposition 3 it is easy to prove that�ka1a2 ðxÞ is symmetric, positive and definite. So the homogenizedproblem (3.11) has uniquely homogenization solution h0ðxÞ. Solving(3.11) one can obtain the homogenization solution h0ðxÞ.

Next, compare the coefficient of e0x0 on both sides of Eq. (3.6). It

can be noted that the right of the equation is independent of ex0 , sothe coefficient of e0

x0 on left side of the equation is equal to f ðxÞ

� @

@nikijðn;xx0 Þ

@Ha1a2 ðx; n;xx0 Þ@nj

� @2h0ðxÞ@xa1@xa2

� @

@nikijðn;xx0 Þ

@ba1ðx; n;xx0 Þ@nj

� @h0ðxÞ@xa1

� @

@nikijðn;xx0 Þ

@Ha1 ðn;xx0 Þ@xj

� @h0ðxÞ@xa1

� @

@nikijðn;xx0 ÞHa1 ðn;xx0 Þ� � @2h0ðxÞ

@xa1@xj

� @

@xikijðn;xx0 Þ

@Ha1 ðn;xx0 Þ@nj

� @h0ðxÞ@xa1

� kijðn;xx0 Þ@Ha1 ðn;xx0 Þ

@nj

� @2h0ðxÞ@xa1@xi

� @kij n;xx0ð Þ@xi

@h0ðxÞ@xj

� kijðn;xx0 Þ@2h0ðxÞ@xi@xj

¼ f ðxÞ: ð3:12Þ

Considering that Ha1 ðn;xx0 Þ and kijðn;xx0 Þ are independent of x andf ðxÞ can be substituted by the Eq. (3.11), so the Eq. (3.12) is writtenas

� @

@nikijðn;xx0 Þ

@Ha1a2 ðx; n;xx0 Þ@nj

� � @

@nikia2 ðn;xx0 ÞHa1 ðn;xx0 Þ� ��

� ka2 jðn;xx0 Þ@Ha1 ðx; n;xx0 Þ

@nj

� � ka1a2 ðn;xx0 Þ þ �ka1a2 ðxÞ

@2h0ðxÞ@xa1@xa2

� @

@nikijðn;xx0 Þ

@ba1ðx; n;xx0 Þ@nj

� � @

�ka1 iðxÞ@xi

!@h0ðxÞ@xa1

¼ 0:

ð3:13Þ

Since @h0ðxÞ@xa1

and @2h0ðxÞ@xa1 @xa2

are not identical to zero, so for any x 2 Cx0

the two following equations must be satisfied:

� @

@nikijðn;xx0 Þ

@ba1ðx; n;xx0 Þ@nj

� ¼ � @

�ka1 iðxÞ@xi

; n 2 Q x0 ; ð3:14aÞ

� @

@nikijðn;xx0 Þ

@Ha1a2 ðx; n;xx0 Þ@nj

�

¼ @

@nikia2 ðn;xx0 ÞHa1 ðn;xx0 Þ� �

þ ka2 jðn;xx0 Þ@Ha1 ðn;xx0 Þ

@nj

� þ ka1a2 ðn;xx0 Þ � �ka1a2 ðxÞ; n 2 Q x0 : ð3:14bÞ

Further, the appropriate boundary conditions are imposed to(3.14a), (3.14b) to determine uniquely ba1

ðx; n;xx0 Þ and Ha1a2 ðx; n;xx0 Þ. Let them satisfy the following elliptic partial differentialequations

� @@ni

kijðn;xx0 Þ@ba1

ðx;n;xx0 Þ@nj

� �¼ � @�ka1a2 ðxÞ

@xa2; n 2 Q x0 ;

ba1x; n;xx0ð Þ ¼ 0; n 2 @Q x0 ;

8<: ð3:15Þ

� @@ni

kijðn;xx0 Þ@Ha1a2 ðx;n;xx0 Þ

@nj

� �¼ @

@nikia2 ðn;xx0 ÞHa1 ðn;xx0 Þ� �

þ ka2 jðn;xx0 Þ@Ha1 ðn;xx0 Þ

@nj

� �þ ka1a2 ðn;xx0 Þ � �ka1a2 ðxÞ; n 2 Q x0 ;

Ha1a2 ðx; n;xx0 Þ ¼ 0; n 2 @Q x0 :

8>>><>>>:

ð3:16Þ

From Lax–Milgram Lemma, Poincare’s inequality and supposition 3,ba1ðx; n;xx0 Þ and Ha1a2 ðx; n;xx0 Þ are determined uniquely.To sum up, one obtains the following theorem:

Theorem. The increment of temperature for heat conduction problem(3.1) of composite structures with inconsistent random distributionformally has a SSOTS asymptotic expansion as follows in the cellCðx0Þ � X

heðx;xÞ ¼ h0ðxÞ þ ex0Ha1 ðn;xx0 Þ@h0ðxÞ@xa1

þ e2x0 Ha1a2 ðx; n;xx0 Þ

@2h0ðxÞ@xa1@xa2

þ ba1ðx; n;xx0 Þ

@h0ðxÞ@xa1

!

þ e3x0P1ðx; n;xx0 Þ;

ð3:17Þ

where h0ðxÞ is the solution of the homogenized Eq. (3.11) with theheat conduction parameters (3.10b). Ha1 ðn;xx0 Þ, ba1

ðx; n;xx0 Þ andHa1a2 ðx; n;xx0 Þ are the local solutions satisfying (3.9), (3.15), and(3.16), respectively.

In practical computation of engineering only the sum of forethree terms in (3.17) are evaluated

heðx;xÞ ¼ h0ðxÞ þ ex0Ha1 ðn;xx0 Þ@h0ðxÞ@xa1

þ e2x0 Ha1a2 ðx; n;xx0 Þ

@2h0ðxÞ@xa1@xa2

þ ba1ðx; n;xx0 Þ

@h0ðxÞ@xa1

!:

ð3:18Þ

Further, from (3.18) the expansion formulation of heat flux densi-ties are given

qiðx;xx0 Þ ¼ �kijðx;xx0 Þ@heðx;xx0 Þ

@xj¼ �kijðn;xx0 Þ

@h0ðxÞ@xj

� kijðn;xx0 Þ@Ha1 ðn;xx0 Þ

@nj

@h0ðxÞ@xa1

� ex0kijðn;xx0 Þ

� Ha1 ðn;xx0 Þ@2h0ðxÞ@xj@xa1

þ @Ha1a2 ðx; n;xx0 Þ@nj

@2h0ðxÞ@xa1@xa2

þ@ba1ðx; n;xx0 Þ@nj

@h0ðxÞ@xa1

!

� e2x0kijðn;xx0 Þ

@Ha1a2 ðx; n;xx0 Þ@xj

@2h0ðxÞ@xa1@xa2

þ@ba1ðx; n;xx0 Þ@xj

@h0ðxÞ@xa1

!

� e2x0kijðn;xx0 Þ Ha1a2 ðx; n;xx0 Þ

@3h0ðxÞ@xj@xa1@xa2

þ ba1ðx; n;xx0 Þ

@2h0ðxÞ@xj@xa1

!: ð3:19Þ

3.2. Algorithm procedure

(1) Choose n points x1; x2; . . . ; xn, those are dispersed in wholestructure X. Let i ¼ 1 .

Y. Yu et al. / Computational Materials Science 46 (2009) 151–161 155

(2) Generate a sample xsxi and its corresponding cell Cs

xi accord-ing to the probability distribution model PðxiÞ and the vol-ume fraction VðxiÞ at the point xi. Further, partition Cs

xi intoFE mesh.

(3) Solve Ha1 ðn;xsxi Þ according to the problem (3.9) by finite ele-

ment method. Further, the homogenization coefficientsk̂a1a2 ðxs

xi Þ are evaluated by using the formula (3.10a).(4) Repeat the steps (2)–(3) for M samples xs

xi ðs ¼ 1;2; . . . ;MÞ,where M is the number of samples. Then the expected heatconduction parameters �ka1a2 ðxiÞ at the point xi can be givenby the following formula

Fig. 5.imental

�ka1a2 ðxiÞ _¼PM

s¼1k̂a1a2 ðxsxi Þ

M: ð3:20Þ

(5) Let i ¼ iþ 1. If i < n, go to (2), else go to (6).(6) The function of heat conduction parameters �ka1a2 ðxÞ can be

approximated by interpolation technique on conductionparameters of the n points x1; x2; . . . ; xn, spline or Lagrangeinterpolation [16]. Further, the homogenization solutionh0ðxÞ is obtained by solving problem (3.11).

(7) For a given point x0 in X, obtain Ha1 ðn;xsx0 Þ, ba1

ðx; n;xsx0 Þ and

Ha1a2 ðx; n;xsxi Þ corresponding to a sample xs

x0 by solving theEqs. (3.9), (3.15), and (3.16).

(8) From (3.18) and (3.19) the temperature and heat flux densitydistributions corresponding to the sample xs

x0 are evaluated.

Fig. 4. The structures with inconsistent random distribution.

Comparison of heat conduction parameters by SSOTS method with exper-data [18] for Al/Al2O3 FGM.

4. Numerical simulations

First, simple composite structures with inconsistent randomdistribution are considered, namely the whole structure X is thecuboid-shape domain and the probability distribution model PðxÞ

Fig. 6. Heat flux density distributions in a local cells around two material points (a)x3 ¼ 0:06 and (b) x3 ¼ 0:22 for Al/Al2O3 FGM.

Fig. 7. Heat flux density distributions in the same sample with 6% of Al grains forstructure with consistent and uniform random distribution.

Fig. 9. Comparison of heat conduction parameter between results evaluated bySSOTS method and experimental data [13].

Fig. 8. FGM with a transition zone.

156 Y. Yu et al. / Computational Materials Science 46 (2009) 151–161

and the volume fraction VðxÞ only depend on x3, shown in theFig. 4. Let the following mixed boundary conditions be satisfied

heðx;xÞ ¼ h1; x3 ¼ x03;

heðx;xÞ ¼ h2; x3 ¼ x003;

mikeijðx;xÞ

@heðx;xÞ@xj¼ 0; x1 ¼ x01; x1 ¼ x001; x2 ¼ x02; x2 ¼ x002:

ð4:1Þ

If the macroscopic heat conduction parameters are orthotropic,then the macroscopic heat conduction Eq. (3.11) can be transformedinto the following one-dimension problem:

Fig. 10. (a) Heat conduction parameters, (b) macroscopic temperatures, (c) macroscopic hdistributions. The curves denote the prediction results by the SSOTS method; the symbo

� @@x3

�k33ðx3Þ @h0ðx3Þ@x3

� �¼ f ðx3Þ;

h0ðx3Þ ¼ h1; x3 ¼ x03;

h0ðx3Þ ¼ h2; x3 ¼ x003:

8>><>>: ð4:2Þ

For the convenience of computation, the internal heat sourcef ðx3Þ is taken as zero and the boundary temperatures in thex3-direction are set as h1 ¼ 100; h2 ¼ 0. Macroscopically homogeni-zation solution is obtained by FEM, and its derivatives up to three

eat flux densities, for inconsistent random structures with different volume fractionls denote the theoretical those [13].

Y. Yu et al. / Computational Materials Science 46 (2009) 151–161 157

order are evaluated by the average technique of relative elements[17]. The generation of meso-scopic configuration at every materialpoint is based on the computer generation method [4]. The localsolutions Ha1 ðn;xx0 Þ, ba1

ðx; n;xx0 Þ, Ha1a2 ðx; n;xx0 Þ are evaluated bythe tetrahedron finite element.

4.1. Numerical computation for typical FGM structure

In this section, composite structure with inconsistent randomdistribution, where volume fraction of grains varies continuouslyalong the x3-direction, will be investigated to demonstrate thecapability of the SSOTS method for predicting heat conductionproperties, and the evaluated results are compared with experi-mental data and theoretical results.

(1) Ordinary FGMs. In the Al/Al2O3 FGM [18] the thermal con-ductivity of Al is 211.07 W/mK and the value for Al2O3 is38.27 W/mK. The spheres are taken as reinforcing grains. Thesegrains are uniformly distributed in the matrix and their volumefraction linearly varies from 0% to 25% in the x3-direction. Here,the macroscopically effective heat conduction parameter at anypoint is almost isotropic and their variation curve is shown inthe Fig. 5. It can be seen that the heat conduction parameter in-crease with the increment of volume fraction, and SSOTS methodprovides a smooth curve of conduction parameter which is a goodagreement with experimental data and falls into the Hashin–Shtrikman bounds, between super and low bounds.

Fig. 11. Heat flux density distributions for: (a) 25% ZrO2 grains, (b) 25% stainlesssteel grains in the case of 1-powder volume fraction distribution.

Fig. 6 shows heat flux density distribution around two materialpoints with 6% and 22% of Al grains. One can see that heat flux den-sities in the two local cells have a marked fluctuation. And localheat flux density distributions with a high volume fraction are rel-atively high. However, macroscopic heat flux density should beconstant because the internal heat source f ðxÞ is taken as zero.For the structure with consistent random distribution of grains,the heat flux density distribution is different from that of the struc-ture with inconsistent random distribution. Fig. 7 takes on heatflux density distribution of the same sample with 6% of Al grains,the heat flux density distribution is lower and has a smaller spanthan FGM globally.

Fig. 12. Heat flux density distribution for 25% ZrO2 grains in the case of 0.5-powdervolume fraction distribution.

Fig. 13. Composite structures with varying probability distribution of grainorientations along x3-direction.

158 Y. Yu et al. / Computational Materials Science 46 (2009) 151–161

(2) FGM with a transition zone. Fig. 8 shows that the material iscomposed of three material zones: in zone X1 discrete particles ofphase A are uniformly distributed in the continuous matrix phaseB; in transition zone X2 the particle and matrix phases cannot bewell defined because the two phases are interpenetrated into eachother as a connected network; in zone X3 discrete particles ofphase B are uniformly distributed in the continuous matrix ofphase A. The heat conduction parameters are evaluated by SSOTSmethod point by point in zone X1 and X3, however those in zoneX2 are obtained by spline interpolation. Set d as thick of the struc-ture (0 6 d 6 1).

At first, the heat conduction parameters for ZrO2/NiCoCrAlYFGM are calculated and compared with experimental results [13].

The thermal conductivity of NiCoCrAlY and ZrO2 are 4.3 W/mKand 2 W/mK, respectively. And the values for their density are7.324 g/cm3 and 6 g/cm3, respectively. The transition zone isbounded as the volume fraction from 40% to 60%. Spherical grainsare adopted in zone X1 and zone X3. Suppose the volume fractionof ZrO2 is linear change with the spatial position, then the thermalconduction parameter varying with the weight fraction of ZrO2 isshown in Fig. 9. The 100% ZrO2 coating has a lower value of thermalconductivity than that of bulk ZrO2 material, it is due to the exis-tence of lamellae structure, defects, cracks and fine interlamellaepores in the sprayed coatings. Otherwise, SSOTS method providesa good agreement with the experimental data.

Changing the volume fraction distribution greatly also affectsthe heat conduction property of composite structures with incon-sistent random distribution.

Fig. 14. (a) Heat conduction parameter, (b) temperature, (c) heat

Next, the heat conduction property for FGM consisting of stain-less steel and ZrO2 is analyzed and the results evaluated by SSOTSmethod are compared with the theoretical those [13].

Thermal conductivities of stainless steel and ZrO2 are 14.4 W/mK and 2.16 W/mK, respectively. Set the transition zone X2 withvarying ZrO2 volume fraction from 40% to 60%. The spheres ofZrO2 are taken as grain phase in zone X1, conversely, the spheresof stainless steel are grain phase in zone X3. The thermal responsesfor FGM of the distributions with different volume fraction areshown in Fig. 10. It can be seen that volume fraction has a great ef-fect on heat conduction properties. The higher the power of distri-bution curve is, the larger the heat conduction parameter,macroscopic temperature and heat flux density are. Especially,macroscopic heat flux density has a very large difference by 78%.All results evaluated by SSOTS method are compared with the the-oretical results [13], and it shows a good agreement.

Fig. 11 shows heat flux density distributions around two mate-rial points with 25% of ZrO2 grains and with 25% stainless steelgrains. To make the difference clear at first glance, the central slicein the x3-direction in the same sample is taken on. It can be seenthat values of heat flux densities at x3 ¼ 0:75 are relatively higherthan those at x3 ¼ 0:25, but most of them are around values ofhomogeneous heat flux density. When the power of volume frac-tion distribution is 0.5, the heat flux densities in the same slicewith 25% ZrO2 grains are illustrated in Fig. 12. The distribution issimilar to that of 1-powder, but the structure of 1-powder is withhigher values, it accords with the variation trend of macroscopicheat flux density.

flux density distributions, for different variation curves of r.

Y. Yu et al. / Computational Materials Science 46 (2009) 151–161 159

From the above numerical results, it follows that the heat con-duction properties for composite structures with inconsistent ran-dom distribution can be effectively predicted by the SSOTSmethod. And the phase volume fraction distribution has a great ef-fect on the heat conduction property of the structures.

4.2. Some extension of SSOTS’s applications

Above the heat conduction properties for composite structureswith inconsistent random distribution are analyzed as the volumefraction of grains Vðx3Þ changes with the spatial position. The heatconduction properties of composite structures with varying proba-bility distribution model Pðx3Þ will be studied below.

(1) Composite structures with varying probability distributionof grain orientations

The SiC prolate ellipsoidal grains, whose long axes are two twiceof middle and short axes, are selected as the grain phase, and C asthe matrix phase. Their thermal conductivities are 135 W/mK and9.5 W/mK, respectively. The volume fraction of grains is 19.6%, andall of grains are uniformly distributed in the matrix; The orienta-tion parameter b1 of long axis with respect to x3-axis obeys a nor-mal distribution, and the other orientation parameters all satisfyuniform distributions. Let the variance r of b1 change continuouslyfrom 0 to 5, and the sketch map of x3-section is shown in Fig. 13.From the statistical computation results the heat conductionparameters for this kind of composite structures are almost ortho-tropic, so the macroscopic heat conduction equation can be con-verted into the one-dimensional equation (4.2). For theconvenience of drawing, the transformation r ¼ r=5 is made andrange value of r is [0,1].

The smaller r is, the more grains are oriented toward x3-direc-tion and the more easily the composite material forms the conduc-tive pathway along x3-direction. With the increment of r theprobability distribution of grains trend to the same toward eachdirection due to the characteristic curve of normal distributions,so the heat conduction parameters along x3-direction are graduallylowered till the even variation, see Fig. 14a. For different variation

Fig. 15. Composite structures with varying probability distribution of grain sizesalong x3-direction.

curve of r versus x3-coordinate, at a given location heat conductionparameters have obvious difference. It is noted that the higher thepower is, the bigger the heat conduction parameters are. However,the temperature distributions are almost linear and the same. Va-lue of heat flux density for high power is 2.4% more than that forlow power.

(2) Structures with varying probability distribution of grainsizes

The composite structure is composed of BN spherical grains andHDPE matrix. Their thermal conductivities are 33 W/mK and0.26 W/mK, respectively. The volume fraction of grains is 19.6%.Suppose that the radius of spherical grain is r. Let r change contin-uously from 0.04 to 0.08, and the sketch map along x3-direction isshown in Fig. 15. Similarly, for the convenience, the transformationr ¼ r�0:04

0:04 is made and range value of r is changed into [0,1].Fig. 16 shows the variation trends of heat conduction parameter

and macroscopic temperature distributions in different grain-sizedistributions. The smaller the size of BN grains is, the higher sur-face energy the grain possesses and more easily the conductivepathways are formed, so heat conduction parameters decreasewith increment of radius of grains, it accords with the experimen-tal results of Ref. [19]. For different size distributions the heat con-duction parameters increase with the increment of power. Thetemperature distributions are all almost linear.

(3) Structures with varying probability distribution of grainpositions

The structure is still composed of BN spherical grains and HDPEmatrix. The volume fraction of grains is 8%; The center coordinatesof grains inside ex0-cell subject to a normal distribution with the

Fig. 16. (a) Heat conduction parameter, (b) temperature distributions, underdifferent r-varying curves.

Fig. 17. Structures with varying probability distribution of grain positions in x3-direction.

Fig. 18. (a) Heat conduction parameter, (b) temperature, (c) heat fl

160 Y. Yu et al. / Computational Materials Science 46 (2009) 151–161

variance r. Let r changed continuously from 0.2 to 0.6, and thesketch map of x3-direction is shown in Fig. 17. From SSOTS’s resultsthe heat conduction parameters for this kind of structures are al-most isotropic since the probability distribution models for the ori-entations of long axis and middle axis of grains are uniform, so themacroscopic heat conduction equation can be converted into theone-dimensional equation (4.2) by making use of the transforma-tion r ¼ r�0:2

0:6 .The smaller r is, the more grains are gathered around the cen-

tric point of ex0-cell and more easily the thermal insulation layersare formed due to low thermal conductivity of HDPE. However,the larger r is, the more grains are far from the center of ex0-celland are dispersed, and it leads to stronger heat transmission. Itcan be noted from Fig. 18 that in the beginning of x3-directionthe heat conduction parameter increases faster than that in theend of x3-direction. In a given location, the heat conduction param-eter, macroscopic temperature and heat flux density distributionsare strongly dependent on the distribution of grain location. Andthe heat flux densities have 13.1% of deviation between the highand low powder.

From the above numerical results it follows that the variation ofprobability distribution model of grains has a great effect on heatconduction properties. In this section, the heat conduction proper-ties for composite structures with varying size, orientation andlocation distribution of grains have been investigated. It is notedthat location and orientation distributions of grains have a greatereffect than size distribution of grains on macroscopic heat conduc-tion properties.

ux density distributions under different variation curves of r.

Y. Yu et al. / Computational Materials Science 46 (2009) 151–161 161

5. Conclusions

In this paper, SSOTS analysis method is developed to predict theheat conduction behaviors for composite structure with inconsis-tent random distribution. Using SSOTS method one can predictnot only heat conduction properties of inconsistent random struc-tures with varying volume fraction distribution of grains, but alsothose with varying probability model distribution:

(1) By the comparisons with experimental data, it is shown thatthe SSOTS method is feasible to predict the heat conductionperformances of FGM’s structure.

(2) From SSOTS’s numerical results it follows that the volumefraction distributions of grains have a greater effect thanthe location and orientation distribution of grains on macro-scopically effective heat conduction coefficients.

(3) The SSOTS method developed in this paper is helpful to thedesign and optimization of composite structures with incon-sistent random distribution. Before the manufacture of thesestructures, the expected macroscopic heat conduction prop-erty can be obtained by changing distribution situation ofmeso-structure.

In this paper, the heat conduction properties for simplestructures, where the volume fraction or probability distributionmodel of grains varies continuously along a direction are investi-gated in detail. However, the SSOTS method can also be appliedto predict heat conduction behaviors of those models with com-plex variation of volume fraction or probability distribution modelof grains.

Acknowledgements

This work is supported by National Natural Science Foundationof China (10590353), (90405016) and Center for High PerformanceComputing of Northwestern Polytechnical University.

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