Dedicated to Professor Apolodor Aristotel R adut as70thAnniversary [610109]

Dedicated to Professor Apolodor Aristotel R ˘adut ¸˘a’s70thAnniversary
MULTISCALE MODELING OF HEAT TRANSFER IN COMPOSITE
MATERIALS
C. TIMOFTE
Faculty of Physics,
University of Bucharest,
P.O. Box MG-11, M ˘agurele-Bucharest, Romania, EU
E-mail : [anonimizat]
Received June 2, 2013
The goal of this paper is to analyze, using homogenization techniques, the ef-
fective thermal transfer in a periodic composite material formed by two constituents,
separated by an imperfect interface. The imperfect contact between the constituents
generates a contact resistance and, depending on the magnitude of this resistance, a
threshold phenomenon arises.
Key words : homogenization, composite materials, imperfect interfaces.
PACS : 44.05.+e, 44.10.+i, 44.30.+v, 44.35.+c, 81.05.Rm..
1. INTRODUCTION
In the last decades, the problem of thermal transfer in heterogeneous media
has been a subject of huge interest for a broad category of researchers: engineers,
mathematicians, physicists (see [2] and [12]). Also, addressing contact problems for
multiphase composites is important, since it is known that the macroscopic properties
of a composite can be affected by the imperfect bonding between its constitutive
components (for a review of the literature on imperfect interfaces in heterogeneous
media, we refer to [13] and [15]).
The main goal of this paper is to describe the macroscopic behavior of a system
of coupled partial differential equations arising in the modeling of thermal transport
in a two-component composite. We deal, at the microscale, with a periodic struc-
ture formed by two connected media with different thermal properties, separated by
an imperfect interface. We assume that we have nonlinear sources acting in each
media and that at the interface between the two constituents the flux is continuous,
but the temperature field has a jump. We are interested in describing the asymptotic
behavior, as the small parameter which characterizes the sizes of the two constituents
tends to zero, of the temperature field in the periodic composite. The imperfect con-
tact between the constituents generates a contact resistance and, depending on the
magnitude of this resistance, a threshold phenomenon arises. So, depending on the
rate exchange between the two phases, three important cases are considered and three
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2 Multiscale modeling of heat transfer in composite materials 1419
different types of limit problems are obtained from the same type of micromodel (see
Section 3).
For simplicity, we deal here only with the stationary case, but we mention that
the dynamic one can be treated in a similar manner (see [8] and [14]). Our setting can
be also relevant for studying electrical conduction in biological tissues (see [1], [17]
and [19]).
Our approach is based on the periodic unfolding method, recently introduced
by D. Cioranescu, A. Damlamian, G. Griso, P. Donato and R. Zaki (see [6] and [7]).
An advantage offered by our approach is that we can avoid the use of extension
operators and, therefore, we can deal rigorously with media with less regularity than
those usually considered in the literature.
Similar problems have been addressed, using different techniques, formal or
not, in [2], [3], [13] and [11]. Our approach, as already mentioned, is based on a
different method, the periodic unfolding method, which allows us to deal with more
general media. The results presented in this paper also constitute a generalization
of those obtained in [11, 14, 18, 19]. Corrector results and results for the case of
nonsymmetric matrices will be presented in a future paper.
For heat conduction problems in a periodic material with a different geometry,
we refer to [9] and [16] and the references therein.
The plan of the paper is as follows: in the second section, we formulate the
microscopic problem. In the third section, we give our main results, while the last
section is devoted to the proof of the convergence results. The paper ends with a few
conclusions and some references.
2. PROBLEM SETTING
Let
be an open bounded material body in Rn(n3), with a Lipschitz-
continuous boundary @
. We assume that
is formed by two constituents,
"
1and

"
2, representing two materials with different thermal characteristics, separated by an
imperfect interface ". We also assume that both phases
"
1and
"
2=
n
"
1are
connected, but only
"
1reaches the external fixed boundary @
. Here,"represents
a small parameter related to the characteristic size of the two constituents. Let Y1be
a Lipschitz open connected subset of the unit cell Y= (0;1)nandY2=YnY1. We
assume thatY2has a locally Lipschitz boundary and the intersections of the boundary
ofY2with the boundary of Yare identically reproduced on opposite faces of the cell.
Also, we suppose that, repeating Yby periodicity, the union of all the sets Y1is
connected and has a locally C2boundary (see [11]).
Let
Z"=fk2Znj"k+"Y
g;
K"=fk2Z"j"k"ei+"Y
;8i=1;ng;
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1420 C. Timofte 3
whereeiare the elements of the canonical basis of Rn.
We define

"
2=int([
k2K"("k+"Y2));
"
1=
n
"
2
and we set=YnY2.
For1;12R, with 0< 1< 1, letM(1;1;Y)be the set of all the
square matrices A2(L1(Y))nnsuch that, for any 2Rn,(A(y);)1jj2,
jA(y)j1jj, almost everywhere in Y. LetA"(x) =A(x=")defined on
,
whereA2M (1;1;Y)is a symmetric smooth Y- periodic matrix. We shall de-
note the matrix AbyA1inY1and byA2, respectively, in Y2.
Our goal is to describe the effective behavior of the solution (u";v")of the
following coupled system of equations:
8
>>>><
>>>>:div(A"
1ru") +(u") =fin
"
1;
div(A"
2rv") +(v") =fin
"
2;
A"
1ru"
=A"
2rv"
on";
A"
1ru"
="
h(u";v")on";
u"= 0 on@
:(1)
Here,is the unit outward normal to
"
1andf2L2(
).
Thus, we consider that the flux is continuous across the boundary ", but, since
the interface between the two phases is not perfect, the continuity of temperatures is
replaced by a Biot boundary condition.
We assume that the functions =(r)and=(r)are continuous, monoto-
nously non-decreasing with respect to rand such that (0) = 0 and(0) = 0 . More-
over, we suppose that there exist C0and an exponent q, with 0q<n= (n2),
such that
j(r)jC(1 +jrjq) (2)
and
j(r)jC(1 +jrjq): (3)
We also assume that
h(u";v") =h"
0(x)(v"u"); (4)
whereh"
0(x) =h0x
"
andh0(y)is aY-periodic, smooth real function with h0(y)
>0. Moreover, we consider that
H=Z
h0(y) d6= 0:
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4 Multiscale modeling of heat transfer in composite materials 1421
Let us notice that, following [14], we can treat is a similar manner the case in
which the function his of the following relevant form:
h(r;s) =h(r;s)(sr);
with 0<hminh(r;s)hmax<1. Also, we can deal with the more general
case in which the nonlinear functions andare multi-valued maximal monotone
graphs, as in [8].
For results concerning the well posedness of problem (1), we refer to [1], [14]
and [17–19].
Since it is impossible to solve this system, the microstructure must be homo-
genized in order to obtain a new model that describes its macroscopic properties.
Using the periodic unfolding method introduced by D. Cioranescu, A. Damlamian,
G. Griso, P. Donato and R. Zaki (see [6] and [7]), we can describe the asymptotic
behavior of the solution of system (1). This behavior depends on the values of the
parameter
,i.e.on the contact resistance between the two constituents. There are
three interesting cases to be considered: a)
= 1; b)
= 0; c)
=1.
In the most interesting case,
= 1, we obtain at the limit a new nonlinear sys-
tem (see (5)). At a macroscopic scale, the composite medium can be represented by a
continuous model, which conceives it as the superimposition of two interpenetrating
continuous media, coexisting at every point of the domain. For the other two cases,
we obtain at the limit only one equation (see (6) and (7)).
3. THE MAIN RESULTS
In this section, we shall describe the effective behavior of the solutions of the
microscopic model (1) for the above mentioned three cases.
a) Let us consider first the case
= 1.
Theorem 1. For
= 1, the solution (u";v")of system (1) converges, as
"!0, to the unique solution (u;v), withu;v2H1
0(
), of the following macroscopic
problem:
(
div(A1ru) +(u)H(vu) =fin
;
div(A2rv) + (1)(v) +H(vu) = (1)fin
:(5)
In (5),A1andA2are the homogenized matrices, defined by:
A1
ij=Z
Y1
aij+aik@1j
@yk
dy;
A2
ij=Z
Y2
aij+aik@2j
@yk
dy
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1422 C. Timofte 5
and1k2H1
per(Y1)=R;2k2H1
per(Y2)=R,k= 1;:::;n , are the weak solutions of
the cell problems
8
<
:ry
((A1(y)ry1k) =ryA1(y)ek; y2Y1;
(A1(y)ry1k)
=A1(y)ek
; y2;
8
<
:ry
((A2(y)ry2k) =ryA2(y)ek; y2Y2;
(A2(y)ry2k)
=A2(y)ek
; y2:
So, at a macroscopic scale, the composite medium, despite of its discrete struc-
ture, can be represented by a continuous model, which is similar to the so-called bi-
domain model , arising in the context of diffusion in partially fissured media (see [4]
and [11]) or in the case of electrical activity of the heart (see [1] and [17]).
b) For
= 0,i.e.for high contact resistance, we get, at the macroscale, only
one temperature field. So, u=v=u02H1
0(
)andu0satisfies:
div(A0ru0) +(u0) + (1)(u0) =fin
: (6)
Here, the effective matrix A0is given by:
A0
ij=Z
Y1
aij+aik@1j
@yk
dy+Z
Y2
aij+aik@2j
@yk
dy;
in terms of the functions 1k2H1
per(Y1)=R;2k2H1
per(Y2)=R,k= 1;:::;n; weak
solutions of the cell problems
8
<
:ry
((A1(y)ry1k) =ryA1(y)ek; y2Y1;
(A1(y)ry1k)
=A1(y)ek
; y2;
8
<
:ry
((A2(y)ry2k) =ryA2(y)ek; y2Y2;
(A2(y)ry2k)
=A2(y)ek
; y2:
Let us notice that in this case, the insulation provided by the interface is suffi-
cient to modify the limiting diffusion matrix, but it is not strong enough to force the
existence of two different limit phases.
c) For the case
=1,i.e.for weak contact resistance, we also get, at the limit,
u=v=u02H1
0(
)and, in this case, the effective temperature field u0satisfies:
div(A0ru0) +(u0) + (1)(u0) =fin
: (7)
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6 Multiscale modeling of heat transfer in composite materials 1423
The macroscopic coefficients are given by:
A0;ij=Z
Y1
aij+aik@w1j
@yk
dy+Z
Y2
aij+aik@w2j
@yk
dy;
wherew1k2H1
per(Y1)=R;w2k2H1
per(Y2)=R,k= 1;:::;n; are the weak solutions
of the cell problems
8
>><
>>:ry
(A1(y)ryw1k) =ryA1(y)ek; y2Y1;
ry
(A2(y)ryw2k) =ryA2(y)ek; y2Y2;
(A1(y)ryw1k)
= (A2(y)ryw2k)
 y2;
(A1(y)ryw1k)
+h0(y)(w1kw2k) =A1(y)ek
 y2:
We remark that in this case, the effective coefficients depend on h0.
4. PROOF OF THE MAIN RESULTS
We shall sketch now the proof for the main case, i.e.
= 1. Let
H1
@
(
"
1) =fv2H1(
"
1)jv= 0on@
\@
"
1g;
endowed with the norm kvkH1
@
(
"
1)=krvkL2(
"
1).
LetH"=H1
@
(
"
1)H1(
"
2), endowed with the scalar product
(u;')H"=Z

"
1ru1
r'1dx+Z

"
2ru2
r'2dx+
"Z
"(u1u2)('1'2) d;8u= (u1;u2);'= ('1;'2)2H":(8)
The norm associated to the scalar product (8) is equivalent to the standard norm in
H1
@
(
"
1)H1(
"
2), with constants independent of "(see [9] and [11]).
Let us give now the variational formulation of problem (1).
Find (u";v")2H"such that
Z

"
1A"
1ru"
r'1dx+Z

"
2A"
2rv"
r'2dx+Z

"
1(u")'1dx+Z

"
2(v")'2dx
+"Z
"h(u";v")('1'2) d=Z

"
1f'1dx+Z

"
2f'2dx;(9)
for any'= ('1;'2)2H".
The problem (9) has a unique weak solution (u";v")2H"(see, for instance,
[9], [11], [14] and [18]).
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1424 C. Timofte 7
Using the hypotheses we imposed on the data, we can obtain suitable energy
estimates, independent of ", for our solution (see [8], [18] and [19]). We get:Z

"
1A"
1ru"
ru"dx+Z

"
2A"
2rv"
rv"dx+"Z
"h(u";v")(v"u") dC;
whereCis independent of ". So,(u";v")is bounded in H"and there exists a constant
C>0, independent of ", such that
ku"kH1
@
(
"
1)C;
kv"kH1(
"
2)C
and
kv"u"kL2(")C"1
2:
For obtaining the macroscopic problem (5), we shall make use of the unfolding
operatorsT"
1andT"
2introduced in [6] and [9]. Since these operators map functions
defined on the oscillating domains
"
1and
"
2into functions defined on the fixed
domains
Y1and
Y2, respectively, we can avoid the use of extension operators.
Using our a priori estimates and the properties of the above mentioned un-
folding operators, we can prove that there exist u;v2H1
0(
),bu2L2(
;H1
per(Y1)),
bv2L2(
;H1
per(Y2))such that, up to a subsequence, for "!0, we have:
T"
1(u")*u weakly inL2(
;H1(Y1));
T"
1(ru")*ru+rybuweakly inL2(
Y1);
T"
2(v")*v weakly inL2(
;H1(Y2));
T"
2(rv")*rv+rybvweakly inL2(
Y2):
Let us take now 1;22C1
0(
) =D(
)as test functions in (9). We have:
Z

"
1A"
1ru"
r1dx+Z

"
2A"
2rv"
r2dx+
Z

"
1(u")1dx+Z

"
2(v")2dx+"Z
"h(u";v")(21) d=
Z

"
1f1dx+Z

"
2f2dx: (10)
We intent now to apply the corresponding unfolding operators in (10) and to pass to
the limit, with "!0. For passing to the limit in the terms involving the functions ,
andh, let us notice that, exactly like in [8] and [19], one can prove that, assuming (2)-
(4) and using the properties of the unfolding operators T"
1andT"
2(see, for instance,
[7]), one gets:Z

"
1(u")1dx!Z

Y1(u)1dxdy;
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8 Multiscale modeling of heat transfer in composite materials 1425
Z

"
1(v")2dx!Z

Y2(v)2dxdy
and
"Z
"h(u";v")(21)!HZ

(vu)(21) dx:
Thus, applying the unfolding operators and passing to the limit in (10), we get:
Z

Y1A1(ru+rybu)
r1dxdy+Z

Y2A2(rv+rybv)
r2dxdy+
Z

Y1(u)1dxdy+Z

Y2(v)2dxdy+
HZ

(vu)(21) dx=Z

Y1f1dxdy+Z

Y2f2dxdy: (11)
Now, we take in (9) the test functions w"
i="i(x)'ix
"
, withi= 1;2, where
i2D(
),'i2H1
per(Yi). Obviously,T"
i(w"
i)!0, strongly in L2(
Yi)and
T"
i(rw"
i)!iry'i, strongly in L2(
Yi). Hence, we can pass to the limit and
we obtain: Z

Y1A1(ru+rybu)
ry'11dxdy+
Z

Y2A2(rv+rybv)
ry'22dxdy= 0: (12)
Putting together (11) and (12) and using standard density arguments, we get:
Z

Y1A1(ru+rybu)
(rx1+rye'1) dxdy+
Z

Y2A2(rv+rybv)
(rx2+rye'2) dxdy+
Z

Y1(u)1dxdy+Z

Y2(v)2dxdy+HZ

(vu)(21) dx=
Z

Y1f1dxdy+Z

Y2f2dxdy; (13)
for1;22H1
0(
),e'12L2(
;H1
per(Y1))ande'22L2(
;H1
per(Y2)).
So, we have the variational formulation of the limit problem (5). Since uand
vare uniquely determined (see [14] and [18]), the whole sequences of microscopic
solutions converge to a solution of the unfolded limit problem and this completes the
proof of Theorem 1.
Let us discuss now briefly the other two interesting cases. We notice that
kT"
1(u")T"
2(v")kL2(
)C"1

2:
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1426 C. Timofte 9
Therefore, for the case
= 0, we have, at the macroscale, u=v=u0and, by un-
folding, we get
Z

Y1A1(ru0+rybu)
(rx +rye'1) dxdy+
Z

Y2A2(ru0+rybv)
(rx +rye'2) dxdy+
Z

Y1(u0) dxdy+Z

Y2(u0) dxdy=
Z

Y1f dxdy+Z

Y2f dxdy;(14)
for2H1
0(
),e'12L2(
;H1
per(Y1))ande'22L2(
;H1
per(Y2)), which leads im-
mediately to the macroscopic problem (6).
For the case
=1, we also obtain at the macroscale u=v=u0. Moreover,
we can prove that
T"
1(u")T"
2(v")
"*bubvweakly in L2(
):
Hence, by unfolding, we get
Z

Y1A1(ru0+rybu)
(rx +rye'1) dxdy+
Z

Y2A2(ru0+rybv)
(rx +rye'2) dxdy+
Z

Y1(u0) dxdy+Z

Y2(u0) dxdy+Z

h0(bvbu)(e'2e'1) dxd=
Z

Y1f dxdy+Z

Y2f dxdy; (15)
which gives exactly the limit problem (7).
5. CONCLUSIONS
Using the recently developed periodic unfolding method, the macroscopic be-
havior of the solution of a problem describing the heat transfer in a periodic com-
posite material formed by two constituents, separated by an imperfect interface, was
analyzed. One advantage offered by our approach is that it allows us to avoid the use
of extension operators and, as a result, to deal with much more general media. Our
setting is also relevant for studying the electrical conduction in biological tissues.
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10 Multiscale modeling of heat transfer in composite materials 1427
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RJP 58(Nos. 9-10), 1418–1427 (2013) (c) 2013-2013