JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

42, 4, pp. 755-770, Warsaw 2004

AN AVERAGED ISOTROPIC MODEL OF NONSTATIONARY

HEAT TRANSFER IN ANISOTROPIC HEXAGONAL-TYPE

CONDUCTORS

Ewaryst Wierzbicki
Urszula Siedlecka

Institute of Mathematics and Computer Science, Częstochowa University of Technology

e-mail: ewrystwierzbicki@interia.pl

Anewmacroscopicmodel for non-stationaryheat transfer processes in a
periodic hexagonal-type anisotropic rigid conductor is formulated. The
main aimof this contribution is to show that themacroscopic properties
of such a conductor are transversally isotropic. The tolerance averaging
technique as a tool of macroscopicmodelling is taken into account.

Keywords:macroscopicmodelling, heat conduction, periodic conductors

1. Introduction

The problem of examination of isotropic macroscopic models for mate-
rials having micro-periodic structures is not new in the recent literature. A
continuum isotropic model for a certain class of frames was formulated by
Woinowski-Krieger (1970). This model was generalized in a series of papers
and summarized byWoźniak (1970) to the case of hexagonal lattices with ri-
gid joints. An alternative approach to this problemwas proposed by Lewiński
(1984-1988). The anisotropic heat transfer effective modulus for chessboard
structures was determined by Jikov et al. (1994). Within the framework of
dynamic problems, a continuum isotropic model for hexagonal frames was
analyzed by Cielecka et al. (2000). An isotropic averaged model of dynamic
problems for periodically ribbedplateswas formulatedbyWoźniak andWierz-
bicki (2000b), for linear elastic composites byNagórko andWągrowska (2002),
and for grids by Cielecka and Jędrysiak (2003). Averaged isotropic properties
of linear elastic chessboard-type dynamic models were investigated by Augu-
stowska andWierzbicki (2000), Wierzbicki (2002).



756 E.Wierzbicki, U.Siedlecka

Fig. 1. Fragment of hexagonal-type periodic conductor

In this paper, a new averaged model for non-stationary heat transfer pro-
cesses in a hexagonal-type rigid conductor is proposed. A fragment of such
conductor is shown in Fig.1. The main result of this contribution is to show
that, under a certain condition, the overall properties of the considered con-
ductor are transversally isotropic even if its constituents are anisotropic. The
tolerance averaging technique as a tool of macroscopic modelling is taken into
account. The general form of the averaged equations, which can be found in
Woźniak and Wierzbicki (2000a), will be specified in order to describe the
hexagonal-type periodic rigid conductor, whosematerial and geometrical pro-
perties are invariant for rotation by 2π/3 with respect to the center of an
arbitrary hexagonal cell. All considerations will be carried out within the fra-
mework of the parabolic heat transfer equations for nonstationary problems.
The scope of the paper is restricted to the formulation of 3D-nonstationary
heat transfer model equations; some applications of these equations will be
presented in a separate paper.
Throughout the paper, we use superscripts A,B which run over 1, . . . ,N,

superscripts a,b which run over 1, . . . ,n and subscripts r,s which run over
1,2,3. Summation convention with respect to these indices holds. Tensor pro-
duct related to vectors p, q is denoted by p⊗q.

2. Preliminaries

In order to make the paper self-consistent, we outline in this section some
concepts which were explained inWoźniak andWierzbicki (2000a).



An averaged isotropic model of ... 757

Throughout the paper, we shall use the concept of the tolerance system.
The definition of the tolerance system is based on the concept of the tolerance
introduced by Zeeman (1965), and from the formal viewpoint it coincides
with a pair T = (F,ε(·)), where F will be identified in this paper with a set
of functions defined for every x=(x1,x2)∈ R2 endowed with pertinent unit
measures, and ε(·) is a mapping which assigns to every f(·) ∈ F a positive
real number εf which has the same unit measure as f. The positive number
εf is called the tolerance parameter and is treated as a certain admissible
accuracy related to computations of f.The concept of the tolerance parameter
coincideswith that introduced byFichera (1992) and called an upper bound of
negligibles. For every f(·)∈ F and x′,x′′ ∈ R2 we shall write f(x′)∼= f(x′′)
if and only if |f(x′)− f(x′′)| < εf, and say that the values of f(·) at x′

and x′′ we shall not discern. It can be seen that every tolerance is a certain
symmetric and reflexive but not transitive binary relation. For the sake of
simplicity, every tolerance relation will be denoted by the same symbol ∼.
Now, followingWoźniak andWierzbicki (2000a), we shall recall the concept

of the slowly-varying function. To this end, denote by ∆ a representative
hexagonal periodicity cell on the Ox1x2-plane. A continuous function ψ ∈ F
will be called slowly varying (with respect to a certain tolerance system T and
a periodicity cell ∆) and written as ψ(·) ∈ SL∆(T) if and only if for every
x′,x′′ ∈ R2, such that x′−x′ ∈ ∆, condition |ψ(x′)−ψ(x′′)| < εψ holds; it
will be assumed that this condition should be satisfied for all derivatives of ψ.
Let Oz be an axis normal to the periodicity Ox1x2-plane. In the direction of
the z-axis, the conductor will be treated as homogeneous. At the same time
every plane z = const is assumed to be a plane of the material symmetry.
The symbol ∂ stands for a partial derivative with respect to the z-coordinate
and ∇ is a gradient with respect to x=(x1,x2). For an arbitrary integrable
function f(·) defined in R2, we shall introduce the averaging operator

〈f〉(x)=
1

area∆

∫

∆(x)

f(y) dy

where ∆(x) = x+∆ for x ∈ R2. If f depends exclusively on z and is ∆-
periodic function then the above definition determinates a constant whichwill
be denoted by 〈f〉.
In order to describe theheat transfer in the hexagonal-type rigid conductor

we shall assume the heat conductivity ∆-periodic tensor field A(·) in the form

A(x)=

[
A 0

0 A33

]
x∈ R2 (2.1)



758 E.Wierzbicki, U.Siedlecka

where A33 = const. Moreover, according to the periodicity of the hexagonal
structure under consideration, we assume that this field, together with the
specificheatfield c(·), are both ∆-periodicfields and that theyattain constant
values in every constituent of the conductor. The problem of heat transfer in
a honeycomb-type periodic conductor is described by the parabolic equation

∇·q(x,z,t)+∂q3(x,z,t)− c(x,z)θ̇(x,z,t)= f(x,z,t) (2.2)

where the total flow q=(q,q3) is determined by constitutive equations

q(x,z,t)=−A(x) ·∇θ(x,z,t)
(2.3)

q3 =−A33(x)∂θ(x,z,t)

andwhere θ(·,z,t), z ∈ R, t ∈ (t0, t1) is a temperature field. The temperature
field will be represented here in a characteristic for the tolerance averaging
technique description form

θ(y,z,t) = θo(y,z,t)+gA(y)WA(x,z,t) y∈ (x) (2.4)

where θo(·,z,t) = 〈c〉−1〈cθ〉(·,z,t) is an averaged temperature field and
WA(·,z,t), A = 1, . . . ,N, represent the fluctuation of the temperature field
caused by nonhomogeneous periodic structure of the conductor and that is
why will be called fluctuation variables. Moreover, gA(·), A = 1, . . . ,N,
are postulated a priori ∆-periodic functions, usually called shape functions
and satisfying conditions 〈gA〉 = 0, 〈cgA〉 = 0 and 〈AgA〉 = 0. Fields
θo(·,z,t) and WA(·,z,t) are assumed to be slowly varying functions for every
(z,t) ∈ R× (t0, t1) and represent new basic unknowns. For these unknowns,
the following system of equations

∇·〈A〉 ·∇θ0+ 〈A33〉∂
2θ−〈c〉θ̇0+∇·〈A ·∇gA〉WA = 〈f〉

(2.5)

〈cgAgB〉ẆB + 〈∇gA ·A ·∇gB〉WB −〈A33g
AgB〉∂2WB +

+〈∇gA ·A〉 ·∇θ0 =−〈fgA〉

has been formulated inWoźniak andWierzbicki (2000a). Equations (2.5) have
constant coefficients. Let us observe that the shape functions depend on the
period length l and satisfy conditions gA(·) ∈ O(l), l∇gA(·) ∈ O(1), where
O(·) is the known ordering symbol. The coefficients 〈AgAgB〉 and 〈cgAgB〉 in
Eqs (2.5) are of the order l2 and, hence, model equations (2.4) describe the
effect of the microstructure size on the averaged properties of the conductor.



An averaged isotropic model of ... 759

It is worth noting that the known homogenized models of a periodic solid
are not able to describe the above effect. The choice of shape functions gA,
A = 1, . . . ,N, depends on the finite element discretization of the periodicity
cell by using a procedure similar to that given in Żmijewski (1987).
Amodel of the heat transfer in a hexagonal-type rigid conductor, given by

Eqs (2.5), is the starting point for the subsequent considerations. For detailed
discussion of model equations (2.4) the reader is referred to Woźniak and
Wierzbicki (2000a,b).

3. Modelling of hexagonal-type conductors

In order tu discuss isotropic properties of model equations (2.5) we shall
restrict considerations to the case of hexagonal-type composites satisfying the
following two assumptions.

Assumption 1. The material structure of the conductor is inva-

riant over rotation by 2π/3 with respect to the center of an arbi-
trary hexagonal periodicity cell.

Since we deal with a ∆-periodic conductor, the above assumption can be
written as

A(Qy)=QA(y)Q> c(Qy)= c(y) y∈ ∆ (3.1)

where Q represents rotationby 2π/3 related to the center of the representative
periodicity cell ∆. In the subsequent considerations, we shall use the following
representation of the conductivity tensor A(y,z) at (y,z)∈ ∆(x)×R

A(y,z)= a(1)(y)d1(y)⊗d1(y)+a(2)(y)d2(y)⊗d2(y)+A33e
3⊗e3 (3.2)

in which d1(y), d2(y) denote a pair of unit vectors determining the princi-
pal directions of the conductivity tensor at a point (y,z) ∈ R3. The scalars
a(1)(y), a(2)(y) denote conductivitymoduli related to these directions, respec-
tively, and e3 is the 0x3-axis unit vector. Bearing in mind Assumption 1
we conclude that the fields a(1)(·), a(2)(·) are invariant over rotation Q, i.e.
formulae

a(1)(Qy)= a(1)(y) a(2)(Qy)= a(2)(y) y∈ ∆ (3.3)



760 E.Wierzbicki, U.Siedlecka

describe a transformation procedure over rotation Q for these fields in the
representative periodicity cell. Bearing in mind (3.2) and (3.3), we conclude
that

A+QAQ>+Q>AQ=
3
2
1trA (3.4)

Similarly, bearing in mind Assumption 1 we conclude that the unit vector
fields d1(·), d2(·) are invariant over rotation Q, i.e. formulae

d
1(Qy)=Qd1(y) d2(Qy)=Qd2(y) y∈ ∆ (3.5)

describe a transformation procedure over the rotation Q for these fields in the
periodicity cell.
Passing to the formulation of the second assumption,we shall denote by G

the set of all shape functions taken into account in every special problem
analyzed in the framework of model equations (2.5). Since the set G has to
conform with the material structure of the conductor, we shall also assume
that

Assumption 2.The set G is invariant over rotation by 2π/3with
respect to the center of an arbitrary hexagonal periodicity cell.

Having used the denotations GQ ≡ {gQ(·) : g(·) ∈ G}, where gQ(y) ≡
g(Qy) for y∈ ∆, this condition can be written as

GQ ⊂ G (3.6)

At the sametime, let us recall the transformation formula (over rotation Q) for
shape function gradients. Namely, bearing inmind the above stated definition
of the symbol gQ, we obtain

∇gQ(y)=Q∇g(Q
>
y) y∈ ∆ (3.7)

for every g(·)∈ G.
Now, we shall outline the approach leading from model equations (2.5)

to the isotropic averaged model equations. This approach is implied by the
assumptions formulated above andwill be divided into five steps. To this end,
we shall introduce a new enumerations of shape functions and fluctuation
variables.



An averaged isotropic model of ... 761

Step 1. Introduction of new enumeration of the shape functions

Letus observe that thenatural consequence ofAssumption2 is that the set
G of all shape functions canbe represented thedisjoint sum G = G1∪G2∪. . .∪
Gn of classes Ga, a = 1, . . . ,n. Every class Ga consists exclusively of one or
three elements. Indeed, every shape function g ∈ G uniquely determines two
other shape functions gQ and gQ>, defined by gQ(y)= g(Qy) and gQ>(y)=
g(Q>y) for every y ∈ ∆. The class of shape functions determinined by the
shape function g will be denoted by G(g). Hence, g,gQ,gQ> ∈ G(g), and
G(g) includes no other shape functions. It must be emphasized that in many
cases, in which a certain shape function g ∈ G is invariant over the 2π/3
rotation Q (i.e. gQ = gQ> = g), the set G(g) consists of exclusively one
element g and then G(g) = {g}. Denoting by n3 the number of three-element
classes, we conclude that N = n+2n3. Hence, it is possible to introduce a
new enumeration of the shape functions. Namely, in every class Ga, shape
functions will be denoted by ga1, ga2, ga3 and related with each other by
formulas ga2 = (ga1)Q, g

a
3 = (ga2)Q. It must be emphasized that in the case

of one element class Ga, we have ga1 = ga2 = ga3 and we admiss a situation
inwhich one shape function posseses three different indices. At the same time,
formula (2.4), for every z ∈ R, t ∈ (t0, t1), will be rewritten in the form

θ(x,z,t) = θo(x,z,t)+ga1(y)W
a
1(x,z,t)+g

a
2(y)W

a
2(x,z,t)+

(3.8)

+ga3(y)W
a
3(x,z,t) y∈ ∆(x)

where the fluctuation variable field related to the shape functions ga1, ga2,
ga3 will be denoted by Wa1, Wa2, Wa3, respectively.

Step 2. Introduction of new fluctuation variables

Let us introduce in every cell ∆ a local coordinate system uniquely deter-
minedby theunitvectors e1,e2 related to the 0x1- and 0x2-axes, respectively.
Moreover, let t1 = e1, t2 = −1

2
e1 +

√
3
2
e2, t3 = −1

2
e1 −

√
3
2
e2 and t̃1 = �t1,

t̃2 = �t2, t̃3 = �t3, where � denote the Ricci-type tensor. The set of just
introduced vectors is shown in Fig.2.
Now we can define new variables

Ua = Wa1+W
a
2+W

a
3

(3.9)

V
a = t1Wa1+ t

2Wa2+ t
2Wa3 a =1, . . . ,n



762 E.Wierzbicki, U.Siedlecka

Fig. 2. Geometrical andmaterial structure of hexagonal-type periodicity cell
invariant over rotation by 2π/3

strictly related to introduced above formulated two invariant assumptions and
to the geometry of the hexagonal cell describedby the vectors t1, t2, t3. It can
be proved that relation (3.9) is an invertible transformation, and the formula

Wa1 =
1
3
Ua + t1V a Wa2 =

1
3
Ua + t2V a

Wa3 =
1
3
Ua + t3V a a =1, . . . ,n

(3.10)

determines new fluctuation variable fields Ua(·), V a(·) by the old fluctuation
variable fields Wa1(·), Wa2(·), Wa3(·). Nowwe are to introduce the just defi-
ned new fluctuation variable Ua, V a into model equations (2.5). To this end
we define new coefficients

A
ab
2 = 〈∇g

a
rA(∇gbs)>〉tr ⊗ ts A

ab
3 = 〈A33g

a
rg
b
s〉t

r ⊗ ts

[Ba] = 〈A ·∇gar〉⊗ tr [B
a]> = tr ⊗〈A ·∇gar〉

C
ab
2 = 〈cg

a
rg
b
s〉t

r ⊗ ts

(3.11)

aab2 = 〈∇(g
a
1+g

a
2+g

a
3) ·A ·∇(g

b
1+g

b
2+g

b
3)〉

aab3 = 〈A33(g
a
1+g

a
2+g

a
3)(g

b
1+g

b
2+g

b
3)〉

cab2 = 〈c(g
a
1+g

a
2+g

a
3)(g

b
1+g

b
2+g

b
3)〉

Moreover, let

[fa] = 〈f(ga1+g
a
2+g

a
3)〉 f

a = 〈garf〉t
r (3.12)



An averaged isotropic model of ... 763

After the formal introducing of (3.10) to model equations (2.5), bearing in
mind (3.11), (3.12) and

〈A ·∇(ga1+g
a
2+g

a
3)〉=0

〈∇(ga1+g
a
2+g

a
3) ·A ·∇g

a
1〉t
1+ 〈∇(ga1+g

a
2+g

a
3) ·A ·∇g

a
2〉t
2+

+〈∇(ga1+g
a
2+g

a
3) ·A ·∇g

a
3〉t
3 =0

we conclude the following alternative form of model equations

∇·〈A〉 ·∇θo +[Ba] :∇V a + 〈A33〉∂
2θo −〈c〉θ̇o = 〈f〉

cab2 U̇
b +aab2 U

b +aab3 ∂
2Ub = [fa] (3.13)

C
ab
2 V̇

b +Aab2 V
b +[Ba]>∇θo +Aab3 ∂

2
V
ab =−fa

In the subsequent analysis, term [Ba] :∇V a placed in thefirst fromEqs (3.13)
will be called the fluctuation term.

Step 3. Investigation of isotropic coefficients and their isotropic representa-

tions

Now we are going to show that the coefficients Aab2 , C
ab
2 and A

ab
3 are

isotropic. Indeed, formulas (3.1) and (3.7) yield

〈∇ga1A∇g
b
1〉= 〈∇g

a
2A(∇g

b
2)
>〉= 〈∇ga3A(∇g

b
3)
>〉

〈∇ga1A∇g
b
2〉= 〈∇g

a
2A(∇g

b
3)
>〉= 〈∇ga3A(∇g

b
1)
>〉

Hence, we denote

âab2 =
2
9
〈∇ga1A(∇g

b
1)
>+∇ga2A(∇g

b
2)
>+∇ga3A(∇g

b
3)
>〉
(3.14)

ãab2 =
2
9

[
〈∇ga1A(∇g

b
2)
>+∇ga2A(∇g

b
3)
>+∇ga3A(∇g

b
1)
>〉−

−〈∇gb1A(∇g
a
2)
>+∇gb2A(∇g

a
3)
>+∇gb3A(∇g

a
1)
>〉
]

Moreover, cf.Woźniak andWierzbicki (1999)

t
1⊗ t1+ t2⊗ t2+ t3⊗ t3 = �(t1⊗ t2+ t2⊗ t3+ t3⊗ t1)=

3
2
1 (3.15)

Bearing in mind (3.14) and (3.15), we arrive at the following form of the
coefficient Aab2

A
ab
2 = â

ab
2 1+ ã

ab
2 �
> (3.16)



764 E.Wierzbicki, U.Siedlecka

On the similar way it can be proved that

A
ab
3 = â

ab
3 1+ ã

ab
3 �
>

C
ab
2 = ĉ

ab
2 1+ c̃

ab
2 �
> (3.17)

where

âab3 =
2
9
〈A33(g

a
1g
b
1+g

a
2g
b
2+g

a
3g
b
3)〉

ĉab2 =
2
9
〈c(ga1g

b
1+g

a
2g
b
2+g

a
3g
b
3)〉

(3.18)

ãab3 =
2
9

[
〈A33(g

a
1g
b
2+g

a
2g
b
3+g

a
3g
b
1)〉−〈A33(g

b
1g
a
2+g

b
2g
a
3+g

b
3g
a
1)〉
]

c̃ab2 =
2
9

[
〈c(ga1g

b
2+g

a
2g
b
3+g

a
3g
b
1)〉−〈c(g

b
1g
a
2+g

b
2g
a
3+g

b
3g
a
1)〉
]

Analogously, we conclude that the averaged conductivity tensor 〈A〉 is trans-
versally isotropic. Indeed, denoting

a =
1
2
〈trA〉 (3.19)

one can prove that
〈A〉= a1 (3.20)

It must be emphasized that the representations of coefficients 〈A〉, Aab2 , A
ab
3

and Cab2 , given by Eqs (3.16), (3.17), (3.20) are isotropic.

Step 4. Investigation of the isotropic representation of fluctuation term [Ba] :
∇V a

In order to investigate the isotropic form of the fluctuation term [Ba] :
∇V a we introduce the following vector field

δV a ≡∇V a +Q∇V aQ>+Q>∇V aQ (3.21)

It is easy to verify that
δV a =Q · δV a ·Q> (3.22)

and hence

[Ba] :∇V a = [Ba] : δV a =Q[Ba]Q> : δV a =Q>[Ba]Q : δV a (3.23)



An averaged isotropic model of ... 765

From the above representations of the fluctuation term, bearing inmind (3.4)
and the representation of [Ba] given by the third formula from (3.11), we
conclude that

[Ba] :∇V a =
1
3

(
[Ba]+Q[Ba]Q>+Q>[Ba]Q

)
: δV a =

(3.24)

=
1
2

(
〈∇ga1 trA〉⊗ t

1 : δV a + 〈∇ga2 trA〉⊗ t
2 : δV a + 〈∇ga3 trA〉⊗ t

3 : δV a
)

Now,we shall apply the decompositions of the shape function gradients ∇ga1,
∇ga2, ∇ga3 with respect to the three vector bases (t1, t̃

1
), (t2, t̃

2
), (t3, t̃

3
),

respectively. To this end, we shall apply the following formulas

〈A ·∇ga1〉 · t
1 = 〈A ·∇ga2〉 · t

2 = 〈A ·∇ga3〉 · t
3

(3.25)

〈A ·∇ga1〉 · t̃
2
= 〈A ·∇ga2〉 · t̃

3
= 〈A ·∇ga3〉 · t̃

1

which have to hold for every a =1, . . . ,n. Moreover, cf. Woźniak andWierz-
bicki (2000b), we recall that

t
1⊗ t1+ t2⊗ t2+ t3⊗ t3 = t̃

1
⊗ t̃
1
+ t̃
2
⊗ t̃
2
+ t̃
3
⊗ t̃
3
=
3
2
1 (3.26)

Bearing in mind formulas (3.18) and denoting

[̂ba] =
3
4
〈∇gar · t

r trA〉 [b̃a] =
3
4
〈∇gar · t̃

r
trA〉 (3.27)

we arrive at the final isotropic form of the fluctuation term

[Ba] :∇V a =([b̂a]1+[b̃a]�) :∇V a (3.28)

being final result of Step 4.

Step 5. Formulation of governing equations

Bearing inmind the results obtained in the framework of Steps 1-4, we can
rewrite model equations (2.5) in an equivallent isotropic form

a∇
2
θo +[̂ba]∇·V a +[̃ba]∇· (� ·V a)+ 〈A33〉∂

2θo −〈c〉θ̇o = 〈f〉

cab2 U̇
b +aab2 U

b +aab3 U
b,33= [f

a] (3.29)

(ĉab2 1+ c̃
ab
2 �
>) · V̇ a +(âab2 1+ ã

ab
2 �
>) ·V a +([b̂a]1+[b̃a]�>) ·∇θo +

+(âab3 1+ ã
ab
3 �
>) ·∂2V a =−fa



766 E.Wierzbicki, U.Siedlecka

Moreover, if we assume that the periodicity cell of the hexagonal type compo-
site has threefold symmetry axes, then model equations (3.29) reduce to the
form

a∇
2
θo +[̂ba]∇·V a + 〈A33〉∂

2θo −〈c〉θ̇o = 〈f〉

cab2 U̇
b +aab2 U

b +aab3 U
b,33= [f

a] (3.30)

ĉab2 V̇
a + âab2 V

a +[̂ba]∇θo + âab3 ∂
2
V
a =−fa

in which the first and third from equations (3.30) have exclusively scalar co-
efficients in contrast to (3.29)1,3.
Model equations (3.29) represent an averaged model of the parabolic heat

transfer in hexagonal-type rigid conductors and are equvallent to equations
(2.5). The basic unknowns are: the averaged temperature field θo and new
fluctuation variables V a, Ua, a = 1, . . . ,n. These new fluctuation variables
are interrelated with the fluctuation variables W A, A = 1, . . . ,N, by means
of formula (3.9). The inverse relation is given by (3.10). Thus, we conclude
that the initial boundaryvalue problemsdiscussedbyWoźniak andWierzbicki
(2000a), and related toEqs (2.5) can also be applied to theproblems related to
above obtainedmodel equations (3.29). It has to be emphasized that equations
(3.29) have a physical sense only if basic unknowns θo, va, Ua are slowly
varying functions of x≡ (x1,x2)∈ R2.
The characteristic features of Eqs (3.29) can be listed as follows.

• Model equations (3.29) are transversally isotropic, and hence we ha-
ve proved the main thesis of this contribution that the averaged heat
transfer response of hexagonal-type rigid conductors can be transversal-
ly isotropic.

• It has to be underlined that the components of the periodic conductor
under consideration, in the general case, can be anisotropic in the pla-
ne z = const. We have to recall that every such plane is a material
symmetry plane for every component.

• Obtained equations (3.29), in a special case reduced to (3.30), concern
hexagonal-type conductors for which every cell has a threefold axis of
symmetry in every plane z = const. A similar result for elastodynamics
was obtained by Nagórko andWągrowska (2002).

In the next section, a few special cases of the obtained model equations
will be examined.



An averaged isotropic model of ... 767

4. Special cases

Asthefirst special case,we shall discussheatpropagation in theperiodicity
plane. Hence we shall assume that θo = θo(x1,x2, t), Ua = Ua(x1,x2, t) and
V a =V a(x1,x2, t). In this case, model equations (3.29) yield

a∇
2
θo +[̂ba]∇·V a +[̃ba]∇· (� ·V a)−〈c〉θ̇o = 〈f〉

cab2 U̇
b +aab2 U

b = [fa] (4.1)

(ĉab2 1+ c̃
ab
2 �
>) · V̇ b +(âab2 1+ ã

ab
2 �
>) ·V b +([b̂a]1+[b̃a]�>) ·∇θo =−fa

It can be observed that the coefficient (ĉab2 1+ c̃
ab
2 �
>) in Eq. (4.1) is of the

order l2, and hence represents the effect of the periodicity cell size on the
macroscopic heat transfer behaviour.Applying the formal limit passage l → 0,
we arrive at algebraic equations for the fluctuation variables V a

(âab2 1+ ã
ab
2 �
>) ·V b +([b̂a]1+[b̃a]�>) ·∇θo =0 (4.2)

It can be shown that this system has a unique solution for V a given by

V
a =−Mab([̂bb]1+[b̃b]�>) ·∇θo (4.3)

where Mab is defined by

M
ab · (âac2 1+ ã

ac
2 �
>)= δac1 (4.4)

Thus, after substituting the right-hand sides of formula (4.3) into the first
fromequations (4.1), we obtain a single equation for the averaged temperature
given by

A
eff :∇⊗∇θo −〈c〉θ̇o = 〈f〉 (4.5)

where

A
eff = a1−[b̂a][̂bb]Mab+[̂ba][̃bb](Mab·�>)−[b̃a][̂bb](�·Mab)−[b̃a][̃bb](�·Mab·�>)

(4.6)
If the periodicity cell of a hexagonal type composite has the threefold symme-
try axis, i.e. if themodel equations has a special form of (3.30), then equation
(4.5) for the averaged temperature reduces to the well known form of the
parabolic heat transfer equation

aeff∇
2
θo −〈c〉θ̇o = 〈f〉 (4.7)



768 E.Wierzbicki, U.Siedlecka

in which
aeff = a− [b̂a][̂bb]Mab (4.8)

and Mab is defined by
Mab · âac2 = δ

ac (4.9)

where δac denotes the Kronecker delta symbol.
In the second special case, let us consider heat propagation in the direction

normal to the periodicity plane, i.e. Ua = Ua(z,t), V a = V a(z,t), θo =
θo(z,t). In this case, we arrive at [b̂a] = [b̃a] = 0 and governing equations
(3.29) reduce to the form

〈A33〉∂
2θo −〈c〉θ̇o = 〈f〉

cab2 U̇
b +aab2 U

b +aab3 U
b,33= [f

a] (4.10)

(ĉab2 1+ c̃
ab
2 �
>) · V̇ a +(âab2 1+ ã

ab
2 �
>) ·V a +(âab3 1+ ã

ab
3 �
>) ·∂2V a =−fa

It can be observed that above equations are separated, i.e. every equation
describes independent evolution of exclusively one basic unknown.
In the third special case, let us assume that the set G of all shape func-

tions consists of only one shape function which will be denoted here by g, i.e.
G = {g}. Since in this case n = 1, the superscript a attain the num-
ber a = 1 and it can be omitted. Moreover, we arrive at [b̂] = [b̃] = 0,
â3 = ã3 = ĉ2 = c̃2 =0 and, sinilarly to the previous case of heat propagation
in the direction normal to the periodicity plane,model equations (3.29) reduce
to the form

a∇
2
θo + 〈A33〉∂

2θo −〈c〉θ̇o = 〈f〉

ca2U̇
a +aa2U

a +aa3U
a,33= [f

a] (4.11)

ĉab2 V̇
a + âab2 V

a + âab3 ∂
2
V
a =−fa

in which every equation contains only one variable.

5. Conclusions and summary of new results

At the end of this paper let us summarize themain new results and infor-
mations on the heat transfer in 3D-hexagonal-type conductors.



An averaged isotropic model of ... 769

• A newmacroscopic model for heat transfer in 3D rigid conductors with
a transversally honeycomb-type periodic structure is formulated. The
main characteristic feature of thismodel is that the governing equations
are transverally isotropic. Hence, the considered conductors have trans-
versally isotropic properties on the macroscopic level.

• It is shown that the isotropic behaviour of the conductor takes place on
themacroscopic level provided that thematerial properties of the hexa-
gonal cell are invariant over rotation by 2π/3 with respect to the center
of the cell. In a special case, this situation occurs if there exist threefold
axes of the cell material symmetry, which is a stronger condition.

• The obtained isotropic model also describes situations in which a
hexagonal-type conductor is made of anisotropic constituents. It means
that the isotropic properties of the constituents are not a necessary con-
dition for the isotropic behaviour of the conductor on the macroscopic
level.

The scope of this paper is restricted to the formulation of model equ-
ations. The application of these equations for finding a solution to some initial
boundary-value problems will be given in a separate paper.

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Makroskopowe modelowanie problemów niestacjonarnego przepływu

ciepła w przewodnikach heksagonalnych

Streszczenie

Celem pracy jest sformułowaniemakroskopowegomodelu niestacjonarnego prze-
pływu ciepław lokalnie anizotropowychheksagonalnychprzewodnikach prostych. Ja-
ko narzędzie modelowania zastosowano technikę tolerancyjnego uśredniania. Sformu-
łowano warunki wystarczające na to, by otrzymane równania były izotropowe. Za-
sadniczym rezultatem pracy jest wykazanie, że własnościmakroskopowe lokalnie ani-
zotropowych przewodnikówmogą być transwersalnie izotropowe.

Manuscript received February 23, 2004; accepted for print March 23, 2004