Vol. 00, No. 00, January —-, 1–17
Effective Conductivity of a Composite Material
with Non-ideal Contact Conditions
L. P. Castro,a E. Pesetskayaa,b and S. V. Rogosinb∗
a
b
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Mechanical-Mathematical Faculty, Belarusian State University, 220030 Minsk, Belarus
(Received 00 Month 200x; in final form 00 Month 200x)
The effective conductivity of 2D doubly periodic composite materials with circular disjoint
inclusions under non-ideal contact conditions on the boundary between material components
is found. The obtained explicit formula for the effective conductivity contains all parameters
of the considered model such as the conductivities of matrix and inclusions, resistance coefficients, radii and centers of the inclusions, and also the values of special Eisenstein functions.
The method of functional equations is used to analyze the conjugation problem for analytic
functions which is equivalently derived from the initial problem. Existence and uniqueness
for the problem solution is obtained by using a reduction to certain mixed boundary value
problem for analytic functions in special functional spaces.
Keywords: 2D composite material; steady-state conductivity problem; effective
conductivity; non-ideal contact conditions; functional equations
AMS Subject Classification: 30E25; 35B27; 74Q05
1.
Introduction
This paper is devoted to the study of analytical solutions of mixed boundary value
problems for multiply connected domains and their application to determining a
formula for the effective properties of 2D composite materials.
We consider fibrous composite materials with disjoint parallel inclusions which
can be geometrically represented as doubly periodic multiply connected domains.
The steady-state conductivity problem for such materials is studied. The problem
of determining the effective properties of the composites turns out to be equivalent
to the problem of finding a potential function satisfying the Laplace equation in
each inner point of a considered domain and satisfying certain conditions on the
boundary of the domain. Ideal or non-ideal contact conditions can be incorporated
as boundary conditions. The derived mixed boundary value problem can be reduced
to an equivalent conjugation problem for analytic functions which is solved by the
method of functional equations (such method is extensively developed in [12]).
As for other works using the method of functional equations for analogous problems, we point out that a mixed boundary value problem to determining the effective conductivity of unbounded composite materials with infinitely many equal
disjoint inclusions in the case of ideal contact conditions is investigated in [2] and
[13]. Power series formula for periodic distributions of inclusions under ideal contact conditions are applied in [3] for the study of the effect of polydispersity in the
∗ Corresponding
author. Email: rogosin@gmail.com
ISSN: 0278-1077 print/ISSN 1563-5066 online
c —- Taylor & Francis
°
DOI: 10.1080/0278107YYxxxxxxxx
http://www.informaworld.com
random distributed inclusions case. In the case of non-ideal contact conditions, a
mixed boundary value problem for a bounded region with a finite number of inclusions was solved in [4]. Analytical and numerical study of the effective conductivity
problem under non-ideal contact conditions is performed in [5]. Variational methods are applied in the context of non-ideal contact to identify size effects and critical
dimensions in [9].
In some cases the homogenization method (cf., e.g. [1], [8]) allow us to reduce the
problem for composites with a rich microstructure to a doubly periodic problem
(e.g. based on the idea of the so-called representative cell [11]). Doubly periodic
composites are investigated in the framework of different physical models. Thus
in [14] the homogenization method is used in the study of elasticity problems,
diffusion problems, and thermoelasticity problems. The dominated terms of the
exact solution are obtained in [14] as solutions to the problem for doubly periodic
composites. In [6] elasticity problems for doubly periodic composites are solved by
means of the integral equation method.
In this paper, we provide an analytic investigation of the steady-state potential
conductivity problem under non-ideal contact conditions on the boundary between
the material components. Here, it is considered 2D unbounded doubly periodic
composite materials with a large quantity of circular disjoint inclusions of different
radii. The conductivity problem is written in the form of a mixed boundary value
problem for the Laplace equation in a multiply connected domain. Such a problem
is equivalently rewritten as a system of functional-differential equations, and an
explicit formula for the effective conductivity is obtained. This formula contains all
parameters of the considered model such as the conductivity of the matrix and inclusions, resistance coefficients, radii, centers of the inclusions, and also the values
of special Eisenstein functions. The solvability of the problem is performed in certain spaces of analytic functions. The compactness of the corresponding operators
plays a central role in the proofs of the results. Conditions are found under which
the method of successive approximations is successfully applied to the problem
under consideration.
2.
Problem formulation and consequent structural relations
In this section we provide a mathematical formulation of the problem in the form
of a boundary value problem, and start to analyse some of the consequences of the
stated conditions.
In here, it is important to have in mind that according to the theory of homogenization (cf. [1] and [8]) it is sufficient to study properties of a composite material
for one of its periodic cells which is called the minimal representative cell [11].
We consider a lattice L which is defined by the two fundamental translation
vectors “1” and “ı” (where ı2 = −1) in the complex plane C ∼
= R2 (with the
standard notation z = x + ıy). Here, the representative cell (see Fig. 1) is the
square
Q(0,0)
Let E :=
S
m1 ,m2
½
¾
1
1
:= z = t1 + ıt2 ∈ C : − < tp < , p = 1, 2 .
2
2
(1)
{m1 + ım2 } be the set of the lattice points, where m1 , m2 ∈ Z. The
cells corresponding to the points of the lattice E are denoted by
ª
©
Q(m1 ,m2 ) = Q(0,0) + m1 + ım2 := z ∈ C : z − m1 − ım2 ∈ Q(0,0) .
(2)
Figure 1. The representative cell Q(0,0) , and its disjoint inclusions.
It is considered the situation when mutually disjoint disks (inclusions) of different
radii Dk := {z ∈ C : |z − ak | < rk } with boundaries Tk := {z ∈ C : |z − ak | =
rk } (k = 1, 2, . . . , N ) are located inside the cell Q(0,0) and periodically repeated in
all cells Q(m1 ,m2 ) . We denote by
D0 := Q(0,0) \
Ã
N
[
D k ∪ Tk
k=1
!
(3)
the connected domain obtained by removing of the inclusions from the cell Q(0,0) ;
cf. Fig. 1.
Let us consider the problem of determination of the effective conductivity of a
double periodic composite material with matrix
Dmatrix =
[
((D0 ∪ ∂Q(0,0) ) + m1 + ım2 )
(4)
m1 ,m2
and inclusions
Dinc =
N
[ [
(Dk + m1 + ım2 )
(5)
m1 ,m2 k=1
occupied by materials of conductivities λm > 0 and λi,k > 0, respectively. We
restrict our attention to the steady-state case of a potential field in each component
of the composite. This problem is equivalent to the determination of the potential
of the corresponding fields, i.e., a function u satisfying the Laplace equation
∆u(z) = 0,
z ∈ Dmatrix ∪ Dinc ,
(6)
which have to satisfy certain boundary conditions on all Tk , k = 1, 2, . . . , N . We
consider these conditions in the following form:
λm
λi,k
∂u+
∂u−
(t) = λi,k
(t),
∂n
∂n
∂u−
(t) + γk (u− (t) − u+ (t)) = 0,
∂n
(7)
t∈
[
m1 ,m2
Tk ,
(8)
where γk > 0 are the so-called resistance coefficients, the vector n = (n1 , n2 ) is the
outward unit normal vector to Tk , and
∂
∂
∂
= n1
+ n2
∂n
∂x
∂y
(9)
is the outward normal derivative,
u+ (t) :=
lim
z→t,z∈D0
u− (t) :=
u(z),
lim
z→t,z∈Dk
u(z), t ∈ Tk , k = 1, . . . , N.
(10)
We suppose additionally that the boundary functions u+ (t), u− (t) are continuously
differentiable on the corresponding curves. The conditions (7)–(8) form the socalled non-ideal contact conditions.
In addition, we assume that the external field is applied in the x-direction, and
u has a constant jump in the x-direction:
u(z + 1) = u(z) + 1,
u(z + ı) = u(z).
(11)
We introduce complex potentials ϕ(z) and ϕk (z) which are analytic in D0 and
Dk , and continuously differentiable in the closures of D0 and Dk , respectively, by
using the following relations
u(z) =

 ℜ (ϕ(z) + z), z ∈ D0 ,

2λm
λm +λi,k
(12)
ℜ ϕk (z), z ∈ Dk .
Thus, in particular, the function ϕ(z) is double periodic in the following sense:
ϕ(z + 1) = ϕ(z),
ϕ(z + ı) = ϕ(z).
(13)
The flux ∇u(x, y) can be represented via derivatives of the complex potentials:
ψ(z) :=
∂ϕ
∂z
ψk (z) :=
=
∂ϕk
∂z
∂u+
∂x
=
+
− ı ∂u
∂y − 1,
λm +λi,k
2λm
³
∂u−
∂x
z ∈ D0 ,
− ı ∂u
∂y
−
´
(14)
, z ∈ Dk .
Let us rewrite conditions (7)–(8) in terms of the complex potentials ϕ(z) and ϕk (z).
The boundary value of the normal derivative can be written in the form
∂u− (t)
2λm
= n · ∇u− (t) =
ℜ[(t − ak )(ϕk )′ (t)],
∂n
rk (λm + λi,k )
|t − ak | = rk . (15)
Let s be the natural parameter of the curve Tk and
∂
∂
∂
= −n2
+ n1
∂s
∂x
∂y
(16)
be the tangent derivative along Tk . Applying the Cauchy-Riemann equations
∂u±
∂v ±
=
,
∂x
∂y
∂u±
∂v ±
=−
,
∂y
∂x
(17)
where v is the imaginary part of the function ϕ(z), we obtain
∂u±
∂v ±
=
.
∂n
∂s
(18)
The equality (7) can be written as
λm
∂v −
∂v +
(t) = λi,k
(t),
∂s
∂s
|t − ak | = rk .
(19)
Integrating the last equality on s, we arrive at the relation
λm v + (t) = λi,k v − (t) + c,
(20)
where c is an arbitrary constant. We put c = 0 since the imaginary part of the
function ϕ is determined up to an additive constant which does not have impact
on the form of u. Using (12), we have
ℑϕ(t) = −ℑt +
2λi,k
ℑϕk (t),
λm + λi,k
|t − ak | = rk .
(21)
Using (15), we are able to write the equality (8) in the following form:
ℜϕ(t) =
2λi,k λm
2λm
ℜ[(t − ak )(ϕk )′ (t)] − ℜt +
ℜϕk .
rk γk (λm + λi,k )
λm + λi,k
(22)
By adding the relation (22) and (21) multiplied by ı, and by using ℜϕk =
r2
ϕk +ϕk
−ϕk
, ℑϕk = ϕk 2ı
, t − ak = t−ak , we obtain
2
k
ϕ(t) = ϕk (t) − ρk ϕk (t) + µk (t − ak )(ϕk )′ (t) + µk
rk2
(ϕk )′ (t) − t,
t − ak
|t − ak | = rk ,
(23)
λm (ρk +1)
2rk γk .
λi,k −λm
λi,k +λm
and µk =
where ρk =
Let us now differentiate (23). First, we note that
′
[ϕ(t)] = −
µ
rk
t − ak
¶2
[ϕ′ (t)]′
ϕ′ (t),
=−
µ
rk
t − ak
¶2
ψ ′ (t),
|t − ak | = rk .
(24)
This can be easily shown by representing the function ϕ in the form ϕ(z) =
∞
P
r2
αk (z − ak )l , |z − ak | ≤ rk , and by using the relation t = t−ak + ak on the
k
l=0
boundary |t − ak | = rk . Thus, after differentiating (23), we arrive at the following
R-linear boundary value problem ([12]) on each contour |t − ak | = rk ,
ψ(t) = (1+µk )ψk (t)+(ρk −µk )
µ
rk
t − ak
¶2
ψk (t)+µk (t−ak )ψk′ (t)−µk
rk4
ψ ′ (t)−1,
(t − ak )3 k
(25)
with k = 1, 2, . . . , N .
The formula for the effective conductivity tensor Λe of a composite material
represented by the cell Q(0,0) has the following form
Λe =
Ã
λxe λxy
e
y
λxy
e λe
!
.
The components of the tensor Λe of a composite material with N disjoint inclusions
are defined by the formulas (see [2])
λxe
= λm
N
Z
X
∂u
λi,k
dxdy +
∂x
Z
X
∂u
dxdy +
λi,k
∂y
k=1
D0
λxy
e = λm
N
D0
k=1
Z
∂u
dxdy,
∂x
(26)
Z
∂u
dxdy,
∂y
(27)
Dk
Dk
where u(x, y) is the solution to the problem (6)–(8).
Let us denote by ∂Dk+ (∂Dk− ) the boundary of the k-th disc, oriented in the
counter-clockwise (clockwise) direction, respectively. By Green’s formula we have
λxe
= λm
u dy +
Z
u+ dy + λm
+
k=1 −
∂Dk
∂Q(0,0)
= λm
N Z
X
¡
Z
¢
λm u+ − λi,k u− dy
N Z
X
¡
k=1 −
∂Dk
∂Q(0,0)
N
X
¢
(λi,k − λm )
u+ − u− dy +
k=1
Z
u− dy.(28)
Z
u− dx.
(29)
∂Dk+
Analogously,
λxy
e
= λm
Z
+
u dx +
k=1 −
∂Dk
∂Q(0,0)
= λm
Z
∂Q(0,0)
N Z
X
¡
+
u dx + λm
¢
λm u+ − λi,k u− dx
N Z
X
¡
k=1 −
∂Dk
+
u −u
−
¢
dx +
N
X
k=1
(λi,k − λm )
∂Dk+
By the relations (7) and (8) we have on the boundary of the k-th disc
u+ − u − =
λi,k ∂ u−
λm ∂ u +
=
.
γk ∂ n
γk ∂ n
At last, applying to each integral in the last sums of (28) and (29) Green’s formula
we obtain the following formula for effective conductivity in the considered case
λxe
−
ıλxy
e
Z
= λm
Q(0,0)
+
N
X
µ
∂ u+
∂ u+
−ı
∂x
∂y
(λi,k − λm )
k=1
Z µ
¶
dxdy + λm
Z
N
X
λm
k=1
∂ u−
∂ u−
−ı
∂x
∂y
Dk
¶
γk
∂ Dk−
µ
∂ u+
∂ u+
dy − ı
dx
∂n
∂n
dxdy.
¶
(30)
By using notation (14) and mean value theorem for analytic functions we can
rewrite the last formula in terms of functions ψ and ψk
λxe
−
ıλxy
e
= λm
Z
ψ(z)dxdy + (−ı)
N
X
λ2
m
k=1
Q(0,0)
γk
Z
∂ Dk−
N
X
∂ u+
dz + 2λm
ρk νk ψ(ak ),
∂n
k=1
(31)
where ρk =
λi,k −λm
(λi,k +λm ) ,
νk = πrk2 .
Remark 1 : When the external field is equal z then the first integral in (31) is
equal to the area of the unit cell. Thus, in the case of the ideal contact conditions
formula (31) coincides with Mityushev’s formula for effective conductivity of doubly
periodic composite materials with an array of circular cylindrical inclusions (see,
e.g. [12]).
3.
Functional equations in spaces of analytic functions
In order to determine the effective conductivity we have to find first the above introduced functions ψk , k = 1, 2, . . . , N . It means that we should solve the boundary
value problem (25). For this we use the so-called method of functional equations
(cf. [12]).
Notice that we have N contours Tk and N complex conjugation conditions on
each contour Tk but we need to find N + 1 functions ψ, ψ1 , . . . , ψN . This means
that we need one additional condition to close up the system. For this reason we
N
S
introduce a new function Φ, which is analytic in Q(0,0) and in
Dk , has the zero
k=1
jumps along each Tk , k = 1, 2, . . . , N , and is doubly periodic on C. Then, by using
Liouville’s theorem for doubly periodic functions, we have that Φ(z) = c (for some
constant c). Such consideration gives an additional condition on ψ, ψ1 , . . . , ψN .
We will now show that the just mentioned constant c turns out to be equal to
zero. Let us introduce the function Φ(z) by the following formulas:
Φ(z) =

 Φ(k) (z), |z − ak | ≤ rk ,

Φ(0) (z), z ∈ D0 ,
(32)
where
Φ(k) (z) =
(1 + µk )ψk (z) + µk (z − ak )ψk′ (z) − (ρk
N
X
X
− µk )
∗
Wm1 ,m2 ,m ψm (z)
m=1 m1 ,m2
+ µk
N
X
X
∗
′
′
Wm
ψm
(z) − 1, (33)
1 ,m2 ,m
m=1 m1 ,m2
Φ(0) (z) = ψ(z)−(ρk −µk )
N
X
X
Wm1 ,m2 ,m ψm (z)+µk
m=1 m1 ,m2
N
X
X
′
′
Wm
ψm
(z).
1 ,m2 ,m
m=1 m1 ,m2
(34)
Here,
Wm1 ,m2 ,k ψk (z) =
µ
′
Wm
ψk′ (z) =
1 ,m2 ,k
N
X
X
∗
rk
z − ak − m1 − ım2
¶2
ψk
rk4
ψ′
(z − ak − m1 − ım2 )3 k
Wm1 ,m2 ,m :=
X X
µ
µ
¶
rk2
+ ak ,
z − ak − m1 − ım2
¶
rk2
+ ak ,
z − ak − m1 − ım2
Wm1 ,m2 ,m +
The “prime” notation in
P
′
m1 ,m2
Wm1 ,m2 ,k .
(36)
(37)
m1 ,m2
m6=k m1 ,m2
m=1 m1 ,m2
X′
(35)
means that the summation occurs in all m1 and
m2 except at (m1 , m2 ) = (0, 0).
For each fixed k = 1, 2, . . . , N , we define the Banach space Ck of the continuous
functions on Tk with the usual norm k ψk k:= max |ψk (t)|, k = 1, 2, . . . , N , and
Tk
the closed subspace Ck+ ⊂ Ck of those functions ψk (z) which admit an analytic
continuation into Dk . Let Ck1 be the subspace of Ck consisting of continuously
differentiable functions on Tk endowed with the norm
k ψk k:= max |ψk (t)| + max |ψk′ (t)|,
Tk
Tk
k = 1, 2, . . . , N,
(38)
and consider also the closed subspace Ck1+ ⊂ Ck1 of functions which admit an analytic continuation into Dk . For each fixed k = 1, . . . , N , we also use the generalized
Hardy space H2 (Dk ) of analytic functions on Dk satisfying the condition
sup
0<r<rk
Z2π
0
|ψk (reıθ + ak )|2 dθ < ∞.
(39)
H2 (Dk ) is a Banach space under the following norm
kψk k2H2 (Dk )
:= sup
0<r<rk
Z2π
|ψk (reıθ + ak )|2 dθ.
(40)
0
Note that Ck1+ ⊂ Ck+ ⊂ H2 (Dk ).
We introduce a new function φ :
analytic in the disconnected domain
N
S
k=1
N
S
Dk → C, φ|Dk = ψk , which is piece-wise
Dk . We consider the Banach space H2 of
k=1
functions φ with “components” ψk ∈ H2 (Dk ) endowed with the norm kφk2H2 :=
kψ1 k2H2 (D1 ) + · · · + kψN k2H2 (DN ) .
Let
α(z) :=
rk2
+ ak .
z − ak − m1 − ım2
(41)
If m1 = m2 = 0, then α(z) becomes the symmetry with respect to the circle Tk .
If m1 + ım2 6= 0, then α(z) consists of the symmetry with respect to the circle
Tk + m1 + ım2 and translation by the vector −(m1 + ım2 ). Hence, α(z) with
m1 + ım2 6= 0 transforms the closed disk |z − ak | ≤ rk into another closed disk
which is compactly embedded into |z − ak | < rk . This means that α(z) possesses a
contraction property. This property allows us to assert that the operator Wm1 ,m2 ,k
′
is compact from Ck+ into Ck+ , and that Wm
is compact from Ck1+ into Ck1+ .
1 ,m2 ,k
Indeed, using the Cauchy integral formula, we have the following representations
ψk
ψk′
µ
rk2
+ ak
z − ak − m1 − ım2
µ
¶
rk2
+ ak
z − ak − m1 − ım2
¶
1
=
2πı
Z
Tk
1
=
2πı
Z
Tk
ψk (τ )dτ
τ−
rk2
z−ak −m1 −ım2
− ak
ψk′ (τ )dτ
τ−
rk2
z−ak −m1 −ım2
− ak
,
(42)
(43)
with continuous kernels as functions in (τ, z) ∈ Tk × Dk (due to the contraction
property), and thus gives the compactness of operators defined in (42) and (43).
Therefore, (35) and (36) are compositions of the compact operators arising in
(42) and (43), the operator of complex conjugation, and bounded operators of
multiplication by (rk /(z−ak −m1 −ım2 ))2 and rk4 /(z−ak −m1 −ım2 )3 , respectively.
It follows that the operators in (35) and (36) are compact.
4.
Analyticity of the functions and compactness of the operators
We show now that the function Φ(z) possesses all the above mentioned properties.
In particular, we justify here the analyticity of the involved functions, and the
compactness of the corresponding constructed operators.
Let us expand ψk (z) and ψk′ (z) into Taylor series
ψk (z) =
∞
X
ψk′ (z)
l
ψlk (z − ak ) ,
=
∞
X
ψlk l (z − ak )l−1
(44)
l=1
l=0
′
in order to sum up Wm1 ,m2 ,k ψk (z) and Wm
ψk′ (z) over all translations m1 +ım2
1 ,m2 ,k
and obtain double periodic functions.
Proposition 4.1:
(i) The series
P
Wj,k ψk (z) and
j
P
j
′ ψ ′ (z), where j = (m , m ) and k is
Wj,k
1
2
k
a fixed number, converge absolutely and uniformly in the perforated cell
D0 ∪ ∂D0 . They define functions which are analytic in D0 , continuous in
D0 ∪ ∂D0 and double periodic. These functions can be written in the form
X
∞
X
Wj,k ψk (z) =
j
X
2(l+1)
ψlk rk
El+2 (z − ak ),
(45)
l=0
′
Wj,k
ψk′ (z)
=
j
∞
X
2(l+1)
ψlk l rk
El+2 (z − ak ), z ∈ D0 ,
(46)
l=1
where Ep (z) is the elliptic P
Eisenstein function of order p (see [15]), defined
(z − m1 − ım2 )−p .
by the formula Ep (z) :=
m1 ,m2
(ii) The series
X′
Wj,k ψk (z) :=
X
Wj,k ψk (z) −
j
j
µ
rk
z − ak
¶2
ψk
µ
rk2
+ ak
z − ak
¶
(47)
and
X′
′
ψk′ (z)
Wj,k
:=
X
′
ψk′ (z)
Wj,k
j
j
rk4
−
ψ′
(z − ak )3 k
µ
rk2
+ ak
z − ak
¶
(48)
define analytic functions in the unit cell Q(0,0) , and continuous in Q(0,0) ∪
∂Q(0,0) . These functions can be written in the form
X′
Wj,k ψk (z) =
j
X′
j
′
Wj,k
ψk′ (z) =
∞
X
2(l+1)
ψlk rk
σl+2 (z − ak ),
(49)
l=0
∞
X
2(l+1)
ψlk l rk
σl+2 (z − ak ),
(50)
l=1
respectively, where σl is the modified Eisenstein function defined by the
formula σl (z) := El (z) − z −l , l = 1, 2, . . . .
P′
(iii) The linear operator
Wj,k is compact in Ck+ , and the linear operator
j
P′ ′
Wj,k is compact in Ck1+ .
j
P ′ ′
Wj,k ψk (z). This statement
Proof : Let us prove proposition (i) for the series
j
P
in the case of the series
Wj,k ψk (z) can be proved analogously. Therefore, the
j
corresponding proof is omitted here.
Let {ej }∞
j=0 be a linear ordered set formed by the numbers m1 + ım2 and with
∞
P
′ ψ ′ (z), where the number M is
Wj,k
e0 = 0. In addition, consider the series
k
j=M
chosen as follows: |z − ak − ej | ≥ h > 1 for all j ≥ M and z − ak ∈ Q(0,0) . Using
the Taylor expansion (44) of the function ψk′ (z), we have
∞
X
′
Wj,k
ψk′ (z) =
j=M
∞ X
∞
X
2(l+1)
ψlk l rk
(z − ak − ej )−3 (z − ak − ej )−l+1 .
(51)
j=M l=1
The series V (z) =
∞
P
|z − ak − ej |−3 converges uniformly in D0 ∪ ∂D0 (cf. [7]).
j=M
Since |z − ak − ej |−l+1 ≤ h−l+1 , l = 1, 2, . . . , and rk < 1, we have
∞
∞ ¯
∞ X
¯
X
X
¯ ¯ −l
¯
¯
2(l+1)
¯ψlk ¯ l h , (52)
(z − ak − ej )−3 (z − ak − ej )−l+1 ¯ ≤ h G
¯ψlk l rk
j=M l=1
l=1
where G := max |V (z)|. Since ψk ∈ Ck1+ and h > 1, the series
D0 ∪∂D0
∞
P
|ψlk | l h−l is
l=1
convergent. Therefore, the series (51) and hence the series (46) converge absolutely
and uniformly in D0 ∪ ∂D0 .
The analyticity of the functions (45) and (46) in D0 , their continuity in D0 ∪
∂D0 and the doubly periodicity follow from the corresponding properties of the
Eisenstein functions Ep (z).
P
To prove proposition (ii) we exclude from the series
Wj,k ψk (z) and
j
P ′ ′
Wj,k ψk (z) an item containing the pole. Then it follows that the functions
j
P′ ′ ′
P′
Wj,k ψk (z) are analytic in the unit cell Q(0,0) and continuous
Wj,k ψk (z) and
j
j
in Q(0,0) ∪ ∂Q(0,0) . The relations (49) and (50) follow from (44) and the definition
of σl (z). Thus, proposition (ii) is proved.
Since a converging series of compactPoperators converges to a compact operator,
′
Wj,k is compact in Ck+ . Analogously, the
we conclude that the linear operator
j
P′ ′
Wj,k is a compact operator in Ck1+ .
¤
linear operator
j
5.
A consequence of the generalized Liouville theorem
According to (25) the jump of Φ(z) across Tk takes the value
∆k := Φ(0) (t) − Φ(k) (t) = ψ(t) − (ρk − µk )
µ
rk
t − ak
¶2
ψk (t) + µk
− (1 + µk )ψk (t) − µk (t − ak )ψk′ (z) + 1 = 0,
rk4
ψ ′ (t)
(t − ak )3 k
|t − ak | = rk , (53)
Using the generalized Liouville theorem for double periodic functions, we conclude
that Φ(z) ≡ c, where c is a constant. Let ψ and ψk be solutions of the system
Φ(z) = c. Then, we have
ψ(z) = ψ̃(z) + c, ∈ D0 ,
(54)
with some double periodic function ψ̃(z). Inserting the last equality in (14) and
then in (12), we obtain
u(z) = ℜ(ϕ̃(z) + cz + z),
z ∈ D0
(55)
with some double periodic function ϕ̃(z) which together with (11) yields c = 0.
Thus, we have Φ(z) ≡ 0. Writing Φ(z) ≡ 0 in |z − ak | ≤ rk , we obtain the following
system of linear functional equations
N
ρk − µk X X ∗
µk
′
Wj,m ψm (z)
(z − ak )ψk (z) +
ψk (z) = −
1 + µk
1 + µk
m=1 j
−
N
µk X X ∗ ′
1
′
Wj,m ψm
(z) +
,
1 + µk
1 + µk
k = 1, 2, . . . , N, (56)
m=1 j
with respect to ψk (z) ∈ Ck1+ . The corresponding function ψ(z) has the form
ψ(z) =
N
X
(ρk − µk )
N X
X
m=1 j
k=1
Wj,m ψm (z) −
N
X
µk
k=1
N X
X
′
′
(z).
ψm
Wj,m
(57)
m=1 j
The system (56) can be considered as an equation for the function Ψ(z) in the
space H2 :
N
ρ̃ − µ̃ X X ∗
µ̃
′
Wj,m Ψ(z)
(z − ã)Ψ (z) +
Ψ(z) = −
1 + µ̃
1 + µ̃
m=1 j
−
N
N
[
µ̃ X X ∗ ′
1
(Dk ∪ Tk ),
Wj,m Ψ′ (z) +
,z∈
1 + µ̃
1 + µ̃
m=1 j
(58)
k=1
where Ψ(z) = ψk (z) in |z − ak | ≤ rk for each k = 1, . . . , N , and µ̃ = µk , ρ̃ = ρk , ã =
ak , for k = 1, . . . , N .
6.
On the solvability of the problem and characterization of the solution
Let us introduce on C 1+ the operator E defined by the following formulas
Ef (z) = f (z) +
µ̃
(z − ã)f ′ (z)
1 + µ̃
|z − ak | ≤ rk ,
k = 1, . . . , N,
(59)
where µ̃ = µk , ã = ak for z ∈ Dk , k = 1, . . . , N .
Let us also introduce the operator Ak in Ck1+ by the following formula
Ak ψk (z) = (ρk −µk )
N X
X
∗
Wj,m ψm (z)−µk
m=1 j
N X
X
∗
′
′
Wj,m
ψm
(z),
|z−ak | ≤ rk ,
m=1 j
and the operator A = Ak , k = 1, . . . , N. The operator
T := I −
1
E −1 A
1 + µ̃
(60)
characterizes the equation (58). Indeed, the system of equations (56) can be rewritten in the form of the following operator equation in H2 :
EΨ =
1
AΨ + g,
1 + µ̃
(61)
1
, z ∈ Dk ,
1 + µk
(62)
where the function
g(z) ≡
is a constant in each domain Dk . Equation (61) is now equivalently rewritten as
Ψ=
1
E −1 AΨ + h
1 + µ̃
(63)
and as
TΨ = h
(64)
with h = E −1 g.
Thus, it is possible to characterize the solvability of equation (58) (or equation
(64)) by studying the kernel and cokernel of this operator T .
µk
> 0, k = 1, . . . , N , then the operator T defined in (60)
Proposition 6.1: If 1+µ
k
has a kernel and cokernel of finite dimension and these dimensions coincide.
µk
Proof : It is shown in [4] that for 1+µ
> 0 the operator E −1 is bounded in H2
k
and kE −1 k ≤ 1. Thus, the corresponding conclusions are valid in our case too.
Besides, if f ∈ Ck1+ then E −1 f ∈ Ck1+ (see also [4]).
By Proposition 4.1, (iii), the operator Ak is compact in Ck+ . Therefore, the opµk
erator E −1 A is also a compact operator in H2 for 1+µ
> 0 (as a multiplication of
k
compact and bounded operators).
As a consequence, the structure of the operator T (being the identity plus a
compact operator) ensures that T is a Fredholm operator with zero Fredholm
index. This means (by definition) that the operator has a closed image, and an
equal finite dimension of the kernel and the cokernel.
¤
Theorem 6.2 : For sufficiently small parameters |ρk | and µk > 0, k = 1, . . . , N ,
the equation (58) has a unique solution in H2 .
Proof : From Proposition 6.1, we know already that in the conditions of the theorem the kernel and cokernel of the operator T have equal finite dimension. In
addition, we ¯take into
account that E −1 is a contractive operator and that µk > 0
¯
¯ 1 ¯
also implies ¯ 1+µk ¯ < 1. Let us choose µk > 0 and |ρk | small enough such that
kAk k < 1. Then from (63) we conclude by the method of successive approximations that the functional equation (58) has a unique solution in H2 .
¤
Remark 1 : Let us note that we actually construct ψk elements which are continuously differentiable in |z − ak | ≤ rk . This occurs due to the fact that the function g defined by (62) is continuously differentiable in |z − ak | ≤ rk . In addition,
E −1 g ∈ Ck1 , and the operator E −1 A transforms H2 into Ck1 .
Note also that the convergence in Ck+ means uniform convergence which preserves
′
analyticity in the limit. The operators Wm1 ,m2 ,m and Wm
depend analytically
1 ,m2 ,m
2
on rk . Thus, we can look for ψk (z) in the form of a series expansion in rk2 :
∞
X
ψk (z) =
(s)
(65)
(s)
(66)
ψk (z) rk2s .
s=0
In addition,
ψk′ (z)
=
∞
X
[ψk (z)]′ rk2s .
s=0
Substituting (65) into (31), we can write the effective conductivity λe in the form
of a series in the concentration of inclusions νk :
λe = λm + 2λm
N
X
(k)
(k)
(k)
ρk νk [A0 + A1 νk + A2 νk2 + . . . ],
(67)
k=1
(k)
where the coefficients As
are defined by the equalities
A(k)
s =
1 (s)
ψ (ak ), s = 0, 1, 2, . . . .
πs k
(68)
In particular, in the case of circular inclusions of equal radii (rk = r, k =
1, . . . , N ) filled in by material of equal conductivity (λi,k = λi , k = 1, . . . , N ) this
formula has the form
λe = λm + 2λm ρν[A0 + A1 ν + A2 ν 2 + . . . ],
(69)
where ν = N πr2 is the common concentration of inclusions in the unit cell, and
the coefficients As are defined by the equalities
As =
N
X
1
(s)
ψk (ak ), s = 0, 1, 2, . . . .
π s N s+1
k=1
(70)
Using the representations (45)–(50), we are now able to write (56) in the form
∞
X
∞
(s)
ψk (z) rk2s = −
s=0
X (s)
µk
[ψk (z)]′ rk2s
(z − ak )
1 + µk
s=0
+
ρk − µk X
1 + µk
∞
∞ X
X
(s)
2(l+s+1)
El+2 (z − am )
ψlm rm
m6=k l=0 s=0
+
∞ ∞
ρk − µk X X (s) 2(l+s+1)
ψlk rk
σl+2 (z − ak )
1 + µk
l=0 s=0
−
∞ ∞
µk X X X (s) 2(l+s+1)
ψlm l rm
El+2 (z − am )
1 + µk
m6=k l=1 s=0
−
∞ ∞
1
µk X X (s) 2(l+s+1)
ψlk l rk
σl+2 (z − ak ) +
,
1 + µk
1 + µk
l=1 s=0
|z − ak | ≤ rk ,
k = 1, 2, . . . , N,
where each term in (65) in expanded into the Taylor series
(s)
ψk (z) =
∞
X
(s)
ψlk (z − ak )l .
(71)
l=0
Collecting the coefficients of the same power of rk2 , we obtain the following theorem:
Theorem 6.3 : Let u(z) be the solution of the problem (6)–(8) and let ∇u be the
flux defined in (14)

ψ(z) + 1, z ∈ D0 ∪ ∂D0 ,
∂u 
∂u
−ı
=
 2λm ψ (z), z ∈ D ∪ T .
∂x
∂y
k
k
λm +λi,k k
(72)
Here, z = x+ıy and ψ(z) and ψk (z) are given by (57) and (65), respectively, where
(p)
ψk (z) are defined by the following recurrence relations:
(0)
µk
1
(0)
(z − ak )[ψk (z)]′ +
1 + µk
1 + µk
(1)
i
µk
ρk − µk h X (0)
(0)
(1)
ψ0m E2 (z − am ) + ψ0k σ2 (z − ak )
(z − ak )[ψk (z)]′ +
1 + µk
1 + µk
ψk (z) = −
ψk (z) = −
m6=k
..
.
(p)
ψk (z) = −
µk
ρk − µk h X (p−1)
(p)
ψ0m E2 (z − am )
(z − ak )[ψk (z)]′ +
1 + µk
1 + µk
m6=k
(p−1)
+ψ0k
σ2 (z − ak ) +
+··· +
X
X
(p−2)
(p−2)
ψ1m E3 (z − am ) + ψ1k
σ3 (z − ak )
m6=k
i
(0)
(0)
ψp−1m Ep+1 (z − am ) + ψp−1k σp+1 (z − ak )
m6=k
−
µk h X (p−2)
(p−2)
ψ1m E3 (z − am ) + ψ1k σ3 (z − ak )
1 + µk
m6=k
+
X
(p−3)
(p−3)
2 ψ2m E4 (z − am ) + 2 ψ2k
σ4 (z − ak )
m6=k
+··· +
X
m6=k
i
(0)
(0)
(p − 1) ψp−1m Ep+1 (z − am ) + (p − 1) ψp−1k σp+1 (z − ak ) ,
(73)
(s)
where p = 2, 3, . . . . Here, ψlm is the derivative of order l of the s-th approximation.
Note that each equation in the system of differential equations (73) has the form:
(p)
ψk (z) = −
µk
(p)
(z − ak )[ψk (z)]′ + q(z),
1 + µk
(74)
−
(p)
where q is a doubly periodic function. Substituting ψk (z) = (z − ak )
into (74), we obtain
(p)
ψk (z)
−
= (z − ak )
1+µk
µk
·
1 + µk
µk
Z
(z − ak )
1
µk
1+µk
µk
u(z)
¸
q(z) dz + cp ,
(75)
where cp is an arbitrary constant. Since we are looking for double periodic solutions,
−
1+µk
the term c (z − ak ) µk should disappear. It yields cp = 0.
Note also that if a value of µk is a rational number, then the integral on the
right-hand side of (75) can be represented through elementary functions. Finally,
(0)
(1)
note that the two first coefficients ψk (ak ), ψk (ak ) in the formula (68) can be
easily found. Namely, we have
(0)
ψk (ak ) =
1
,
1 + µk
(1)
ψk (ak ) =

ρk − µk 
(1 + µk )2
X
m6=k

E2 (ak − am ) + σ2 (0) . (76)
(p)
For a small concentration νk , it is enough to calculate several first terms ψk (ak )
(k)
for the coefficients Ap defined by (68). Formulas for the values of Ep (ak − am )
and σp (0) are presented, e.g., in [2].
Thus, we found a recursive algorithm to define the values ψk (ak ) and hence to
determine the effective conductivity in terms of basic parameters of the considered
model such as the conductivity of the matrix and inclusions, resistance coefficients,
radii and centers of the inclusions, and also the values of special Eisenstein functions.
Acknowledgements
This work is partially supported by Unidade de Investigação Matemática e
Aplicações of Universidade de Aveiro through the Portuguese Science Foundation
(FCT–Fundação para a Ciência e a Tecnologia), and by the Belarusian Fund for
Fundamental Scientific Research.
E. Pesetskaya is supported by the Portuguese Science Foundation through grant
number SFRH/BPD/34649/2007.
References
[1] N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging processes in periodic media, Nauka,
Moscow, 1984 (in Russian); English transl., Kluwer, Dordrecht/Boston/London, 1989.
[2] L. Berlyand and V. Mityushev, Generalized Clausius-Mossotti formula for random composite with
circular fibers, J. Statist. Phys. 102 (2001), pp. 115–145.
[3] L. Berlyand and V. Mityushev, Increase and decrease of the effective conductivity of two phase
composites due to plydisperisity, J. Statist. Phys. 118 (2005), pp. 481–509.
[4] P. Drygaś, A functional–differential equation in a class of analytic functions and its application,
Aequation Mathematicae 73 (2007), pp. 222–232.
[5] P. Drygaś, V. Mityushev Effective conductivity of arrays of a unidirectional cylinders with interfaciol
resistance, to appear.
[6] E.I. Grigolyuk and L.A. Filshtinskii, Periodic piece-wise homogeneous elastic structures, Nauka,
Moscow, 1992 (in Russian).
[7] A. Hurwitz and R. Courant, Vorlesungen uber allgemeine funktionentheorie und elliptische funktionen, Springer-Verlag, Berlin, 1964; Nauka, Moscow, 1968.
[8] V.V. Jikov, S.M. Kozlov, and O.A. Olejnik, Homogenization of Differential Operators and Integral
Functionals, Springer-Verlag, 1994.
[9] R. Lipton and B. Vernescu, Composites with imperfect interface, Proc. R. Soc. Lond. A 452, (1996),
329-358.
[10] V. Mityushev, Functional equations in a class of analytic functions and composite materials, Demonstr. Math. 30 (1997), pp. 63–70.
[11] V. Mityushev, Representative cell in mechanics of composites and generalized Eisenstein-Rayleigh
sums, Complex Var. Elliptic Equ. 51 (2006), pp. 1033–1045.
[12] V. Mityushev and S. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications, Chapman & Hall / CRC, Monographs and
Surveys in Pure and Applied Mathematics 108, 1999.
[13] E. Pesetskaya, Effective conductivity of composite materials with random positions of cylindrical
inclusions: finite number inclusions in the cell, Applicable Analysis 84 (2005), pp. 843–865.
[14] G.A. Vanin, Micromechanics of Composite Materials, Naukova Dumka, Kiev, 1985 (in Russian).
[15] A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Springer-Verlag, 1976.