Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material
For 2D composite material geometrically composed %presentedby an interior of ellipse $E_1$ (with semi-axes $a_1$, $b_1$) and anouter elliptical ring with an outer ellipse $E_2$ (with semi-axes$a_2$, $b_2$), where $E_1$ and $E_2$ are confocal ellipses; it isdetermined in an analytic form the $x$-comp...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552691419316224 |
|---|---|
| author | Gryshchuk, S. V. Gryshchuk, S. V. |
| author_facet | Gryshchuk, S. V. Gryshchuk, S. V. |
| author_institution_txt_mv | [
{
"author": "S. V. Gryshchuk",
"institution": null
}
] |
| author_sort | Gryshchuk, S. V. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-23T12:10:05Z |
| description | For 2D composite material geometrically composed %presentedby an interior of ellipse $E_1$ (with semi-axes $a_1$, $b_1$) and anouter elliptical ring with an outer ellipse $E_2$ (with semi-axes$a_2$, $b_2$), where $E_1$ and $E_2$ are confocal ellipses; it isdetermined in an analytic form the $x$-component of the effectiveconductivity tensor as a sum of convergent power series withcoefficients depending on conductivities of the components of thecomposite material and geometrical characteristics $a_1$, $b_1$ and$a_2$, $b_2$. |
| first_indexed | 2026-08-04T01:03:27Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2015, том 12, № 3, 121–132
УДК 517.9 + 532.546
S.V. Gryshchuk
(Institute of Mathematics of NAS of Ukraine, Kyiv)
Effective Conductivity of 2D
Ellipse – Elliptical Ring
Composite Material
gryshchuk@imath.kiev.ua
For 2D composite material geometrically composed by an interior of ellipse
E1 (with semi-axes a1, b1) and an outer elliptical ring with an outer
ellipse E2 (with semi-axes a2, b2), where E1 and E2 are confocal ellipses;
it is determined in an analytic form the x-component of the effective
conductivity tensor as a sum of convergent power series with coefficients
depending on conductivities of the components of the composite material
and geometrical characteristics a1, b1 and a2, b2.
Для двовимiрного композицiйного матерiалу, що геометрично утворе-
ний внутрiшнiстю елiпса E1 з пiвосями a1, b1 та прилеглим елiптич-
ним кiльцем з зовнiшнiм елiпсом E2 з пiвосями a2, b2, а E1 та E2 ма-
ють спiльнi фокуси, одержано аналiтичний вираз x-компоненти тен-
зору ефективної провiдностi у виглядi суми збiжного числового ряду,
коефiцiєнти якого залежать вiд провiдностей компонентiв композиту
та геометричних характеристик a1, b1 та a2, b2.
Для плоского композиционного материала, геометрически образован-
ного внутренностью эллипса E1 с полуосями a1, b1 и прилежащим эл-
липтическим кольцом с внешним эллипсом E2 с полуосями a2, b2, а
E1 и E2 имеют общие фокусы, получено аналитическое выражение
x-компоненты тензора эффективной проводимости в виде суммы схо-
дящего числового ряда, коэффициенты которого зависят от проводи-
мостей компонент композита и геометрических характеристик a1, b1 и
a2, b2.
c⃝ Institute of Mathematics, 2015
122 S. V. Gryshchuk
Keywords: 2D composites, ellipse – elliptical ring composite, heat
conduction, R-linear problem, effective conductivity
MSC: 74A40, 74G10, 30E25
1. Introduction. Circular type 2D mechanical structures are
subject of long investigations. In the pioneering work [8] (see also [9])
it had been found a complete solution of the problem of torsion and
bending of elastic cylindric body reinforced by parallel cylindric beems Sk,
k = 1, . . . ,m, of different materials. The author showed that this problem
is equivalent to certain mathematical problem, which can be formulated
in the following form.
Problem M. Let Sk, k = 0, 1, . . . ,m, are simple connected domains of
the complex plane C with smooth boundaries C0, C1, . . . , Cm, such that the
contour C0 embraces any contour Ck, k = 1, . . . ,m. Let S := S0 \∪m
k=1Sk.
The problem is to find a function φ which harmonic in S, continuous in
cl(S0) and such that its normal derivative has a given jump on each contour
Ck, k = 0, 1, . . . ,m.
Here and anywhere cl(G) denote a closure of any domain G ⊂ C.
In [8], it was proved that the problem is equivalent to a system of
m+ 1 Fredholm integral equations of the second kind which has solutions
under some solvability conditions and the solution is unique up to an
additive constant. The main aim of this article is to calculate some
elastic characteristics of this elastic cylindric body reinforced by parallel
cylindric beams. One of these is, so-called, elastic torsional rigidity, which,
mathematically, is a real valued functional depending on the solution of
the latter problem.
Although, the solution of the problem M is a solution of a system of
Fredholm integral equations of the second kind, from the practical point of
view we have evident difficulties to get its explicit expressions. Therefore,
for the practice it is interesting any partial case of problem M which allows
to find a method for its solving in an explicit form. For example, in case
when m = k = 1 and S0 and S are disks (or what is equivalent S′
0 and
S′ are disks with common origin ) the problem M, is solved by the direct
method based on expansions in Taylor’s and Laurent series of a function
f(z), which is analytic in S′
1 ∪ S′, S′ = S′
0 \ cl (S′
1). The function f(z) is
such that its real part Ref equals to an unknown required function φ in
S′
1 ∪ S′. By using this solution the elastic torsional rigidity is calculated
in an explicit form in [11]. The case when m = k = 1 and S0 and S are
Effective Conductivity of Ellipse – Elliptical Ring Composite . . . 123
confocal ellipses has been considered in [10]. The proposed method in this
paper is based on the reduction of Problem M for confocal ellipses to
similar boundary value problem but for two bounded rings, therefore, the
method of solution of the latter problem based on conformal technique,
which maps our ellipses, by using an analytic function of Zhukovskii type,
to disks and after using Laurent expansions.
See also [5], where the method of functional equation (cf., [6]) is
developed to derive analytical approximate formulae for the effective
conductivity tensor of the two-dimensional composites with elliptical
inclusions which number is greater then two.
The aim of present paper is to solve a problem similar to Problem
M when m = k = 1 and S0 and S are bounded by confocal ellipses
E1 = C1 and E2 = C0, describing the 2D composite material with
the elliptical inclusion, geometrically represented by a simply connected
domain bounded by the ellipse E1, and the matrix, geometrically
represented by the elliptical annulus S bounded by two ellipses E1 and
E2.
On the base of this solution, we calculate an analog of the
elastic torsional rigidity, which is a x-component of, so-called, Effective
Conductivity Tensor (see, e.g., [2, 4]). More precisely, we consider the
problem of determination of the temperature distribution under perfect
contact condition in the above described inhomogeneous media loaded by
a simple heat flow. Note, that this problem is equivalent to the R-linear
conjugation problem on the complex plane (see [6, p. 45]). Note, that for a
case of disk–ring composite material in [3] it was delivered for x-component
of the effective conductivity tensor its exactly expression in terms of radius
r of the internal disk and so-called contrast parameter ρ introduced by
Bergmann [1] in the form of geometrical progression with respect to the
powers of r2.
2. Definitions and formulation of the problem. Let C
be the field of complex numbers and the complex plane, R be the field of
real numbers. For any ξ from C or R denote by |ξ| its modulus. For any
z ∈ C denote by Re z its real part, and by (η, θ), 0 < η < ∞, 0 ≤ θ ≤ 2π
its polar coordinates, thus, z = η exp{iθ}.
Let Ω be an open or closed domain in C and let τ : Ω −→ R be a
real valued function, then Ck(Ω), k ∈ 1, 2, . . . , denotes a class of functions
having continuous partial derivatives up to k-th order. If τ ∈ C2(Ω) then
∆τ(x+ iy) := ∂2τ
∂x2 + ∂2τ
∂y2 .
124 S. V. Gryshchuk
Let G and Go, G $ Go, are simply connected domains in C. For any
function u : Go −→ R by u1 and u2 we mean its restrictions, respectively,
to G and G̃ := Go \ cl(G), i.e., u1 := u|G и u1 := u|G̃. Denote by E1 and
E2, respectively, boundaries of G and Go, i.e, E1 = ∂G, E2 = ∂Go. Thus,
∂G̃ = E1 ∪ E2.
Problem A for the system of domains (G, G̃) is a problem on finding a
pair of real-valued functions u = (u1, u2), where u1 ∈ C2(G) ∩ C1(cl(G))
and u2 ∈ C2(G̃ ) ∩ C1(G̃ ), satisfying the following system of equations
∆u(z) = 0 ∀z ∈ G ∪ G̃,
u2(t) = −Re t ∀t ∈ E2,
u1(t) = u2(t) ∀t ∈ E1,
λ1
∂u1
∂n (t) = λ2
∂u2
∂n (t) ∀t ∈ E1,
(1)
where λ1 and λ2 are positive constants such, that the Bergmann’s contrast
parameter (see [1]) ρ := (λ1 + λ2)
−1(λ1 − λ2) is different from zero (it is
well known, that −1 < ρ < 1); u1(t) and u2(t) denote, the limiting on
E1 values of, respectively, u1(z), z ∈ G, and u2(z), z ∈ G̃; ∂u1
∂n and ∂u2
∂n
denote, respectively, the outward normal derivative of u1 with respect to
the boundary E1 of the domain G and the inward normal derivative of u2
with respect to the connected component E1 of the boundary ∂G̃ of G̃.
Note, that Problem (1) (cf., e.g., [6, p. 45] with λ2 = 1) can be reduced
to the following R-linear conjugation problem: to find two functions ϕ :
G ∪ G̃ −→ C, ϕ0 : C ∪ {∞} −→ C, ϕ0(∞) = 0, such that ϕ is analytic in
G ∪ G̃, ϕ0 is analytic in extGo := {z ∈ C : z ∈ cl(Go)} and continuous in
cl(extGo), functions ϕ and ϕ0 satisfy the following conjugation conditions
ϕ+(t) = ϕ−(t)− ρϕ−(t) ∀t ∈ E1, ϕ−(t) = ϕ0(t)− ϕ0(t)− t ∀t ∈ E2,
(2)
where ϕ+(t) and ϕ+(t), t ∈ E1, is boundary values of ϕ(z), accordingly,
from G, G̃, u+ iv := u− iv for u, v ∈ R.
In [7], it has been considered the problem of disturbance of a complex
potential after insertion of a foreign inclusion in the form of a two-
phase confocal elliptical annulus into a homogeneous medium. There were
investigated the cases of an arbitrary distribution of singularities.
Our aim is to calculate an x-component of the effective conductivity
tensor ( see, e.g. [4]), i.e. a quantity λxeff , satisfying the following equality
F xλxeff = λ1
∫
G
∂ u1
∂x
dx dy + λ2
∫
G̃
∂ u2
∂x
dx dy, (3)
Effective Conductivity of Ellipse – Elliptical Ring Composite . . . 125
where u(x, y) = (u1(x, y), u2(x, y)) ≡ (u1(x+ iy), u2(x+ iy)) is a solution
of Problem A for the system of domains (G, G̃), F x is a complete flux in
x-direction.
For any ϵ > 0 by Dϵ denote a disk {ζ ∈ C : |ζ| < ϵ}, γϵ means a circle
{ζ ∈ C : |ζ| = ϵ}. For a pair of positive real numbers r1, r2, r1 < r2, denote
by Ar1,r2 the set {ζ ∈ C : r1 < |ζ| < r2}.
In [3], a quantity (3) is calculated in an exactly form for the system of
domains (Dr, Ar,1), 0 < r < 1, as a functional depending of r, analytically
expressed as a sum of geometrical progression with respect to powers of
r2.
Assume further, that a boundary ∂Go of a domain Go is an ellipse E2
with semi-axes a2, b2, a2 > b2, ∂G is an ellipse E1 with semi-axes a1, b1,
a1 > b1, moreover, ellipses E1 and E2 have common focuses F1 = −c and
F2 = c, where c =
√
a22 − b22 =
√
a21 − b21. Then a2 > a1, b2 > b1, 2c is the
focus distance, G̃ is an elliptical ring bounded by ellipses E1 and E2.
3. Аuxiliary A–problem. Denote by
n = a1 + b1, m =
c
a1 + b1
< 1, R =
a2 + b2
a1 + b1
> 1. (4)
Consider a mapping ω : C \ {ζ ∈ C : |ζ| ≤ m} −→ C \ [F1, F2],
[F1, F2] := {x : −c ≤ x ≤ c} defined by
z = x+ iy =: ω(ζ) ≡ n
2
(
ζ +
m2
ζ
)
. (5)
Obviously, that the mapping (5) is an analytic function, thus, it is one-
to-one correspondence and the following formulae hold
x = x(η, θ) ≡ n
2
(
η +
m2
η
)
cos θ, y = y(η, θ) ≡ n
2
(
η − m2
η
)
sin θ, (6)
where ζ = η exp{iθ}.
If a point ζ = η exp{iθ} runs through the circle γm, then ω(ζ) twice
runs through the segment [F1, F2] and the following equality holds
ω (m exp{iθ}) = ω (m exp{−iθ}) , 0 ≤ θ ≤ 2π. (7)
Valid the following equalities
ω (Am,1) = G \ [F1F2], ω(γ1) = E1, ω(A1,R) = G̃, ω(γR) = E2. (8)
126 S. V. Gryshchuk
Let f(z) be analytic in G ∪ G̃ function such, that Ref(z) = u(z) is a
solution of the system (1). Then the function g(ζ) := f(ω(ζ)) is analytic
in Am,1 ∪ A1,R and a system (1) for a function u(ζ) ≡ v(ζ) := Re g(ζ),
taking in account equalities (6) and (8), transforms to the form
∆v(ζ) = 0 ∀ζ ∈ Am,1 ∪A1,R,
v2(ζ) = −n
2
(
R+ m2
R
)
cos θ ∀ζ = R exp{iθ} ∈ γR,
v1(ζ) = v2(ζ) ∀ζ ∈ γ1,
λ1
∂v1(η exp{iθ})
∂η
∣∣∣
η=1
= λ2
∂v2(η exp{iθ})
∂η
∣∣∣
η=1
∀θ ∈ [0, 2π].
(9)
A problem on finding a function v = (v1, v2), where v1 ∈ C2(Am,1) ∩
C1(cl(Am,1)) and v2 ∈ C2(A1,R) ∩ C1(cl(A1,R)), which satisfy the system
of equations (9) call an auxiliary A–problem.
Using analyticity of g1 := g|Am,1
and g2 := g|A1,R
, respectively, in Am,1
and A1,R, we have the following expansions in the Laurents’s series
g1(ζ) = a′0 + ib′0 +
∞∑
k=−∞, k ̸=0
(a′k + ib′k)ζ
k ∀ζ ∈ Am,1, (10)
g2(ζ) = a′′0 + ib′′0 +
∞∑
k=−∞, k ̸=0
(a′′k + ib′′k)ζ
k ∀ζ ∈ A1,R, (11)
where a′k, a
′′
k , b′k, b
′′
k , k = ±1,±2, . . . , denote unknown real numbers. Going
to real parts in (10) and (11), obtain formulas
v1(ζ) = a′0 +
∞∑
k=1
(
a′kη
k + a′−kη
−k
)
cos(kθ)−
−
∞∑
k=1
(
b′kη
k − b′−kη
−k
)
sin(kθ) ∀ζ = η exp{iθ} ∈ Am,1, (12)
v2(ζ) = a′′0 +
∞∑
k=1
(
a′′kη
k + a′′−kη
−k
)
cos(kθ)−
−
∞∑
k=1
(
b′′kη
k − b′′−kη
−k
)
sin(kθ) ∀ζ = η exp{iθ} ∈ A1,R. (13)
Effective Conductivity of Ellipse – Elliptical Ring Composite . . . 127
Since ω maps exterior extDm of the disk Dm in the complex plane of
the variable ζ to the complex plane of the variable z with a cut along
the segment [F1, F2], then the problem (1) is equivalent to auxiliary A–
problem if and only if when the function (10) has coinciding limiting
values on different sides of the cut γm, which, by virtue of the formula
(7), is equivalent to the validity of the following equality g1 (m exp{iθ}) =
g1 (m exp{−iθ}) , 0 ≤ θ ≤ 2π, which in turn leads equalities
a′−k = m2ka′k, b′−k = m2kb′k, k = 1, 2, . . . . (14)
Substituting (13) with η = R to the second condition in (9), obtain
a′′0 = 0, a′′−1 = −n
2
(
R2 +m2
)
−R2a′′1 , (15)
a′′−k = −R2ka′′k , k = 2, 3, . . . , b′′−k = R2kb′′k , k = 1, 2, . . . . (16)
Substituting equalities (12) — (15) with η = 1 to the third condition in
(9), obtain
a′0 = 0, a′1 =
1−R2
1 +m2
a′′1 − n
2
R2 +m2
1 +m2
, (17)
a′k =
1−R2k
1 +m2k
a′′k , k = 2, 3, . . . , b′k =
1−R2k
1−m2k
b′′k , k = 1, 2, . . . . (18)
Now the equality (12) by using the relations (14), (17) and (18), turns
to the form
v1(ζ) =
(
1−R2
1 +m2
a′′1 − n
2
R2 +m2
1 +m2
)(
η +
m2
η
)
cos θ+
+
∞∑
k=2
1−R2k
1 +m2k
a′′k
(
ηk +
m2k
ηk
)
cos kθ−
−
∞∑
k=1
1−R2k
1−m2k
b′′k
(
ηk − m2k
ηk
)
sin kθ ∀ζ = η exp{iθ} ∈ Am,1. (19)
Taking into account (15) and (16) we rewrite (13) in the form
v2(ζ) =
(
a′′1η −
(n
2
(R2 +m2) +R2a′′1
)
η−1
)
cos θ+
128 S. V. Gryshchuk
+
∞∑
k=2
a′′k
(
ηk − R2k
ηk
)
cos kθ−
−
∞∑
k=1
b′′k
(
ηk − R2k
ηk
)
sin kθ ∀ζ = η exp{iθ} ∈ A1,R. (20)
Substituting delivered equalities (19) and (20) to the fourth condition
in (9), obtain, after division on λ1 + λ2, the following relations((
1 +m2R2
)
ρ−m2 −R2
)
a′′1 =
n
2
(R2 +m2)(1−m2ρ), (21)
((
1 +m2kR2k
)
ρ−m2k −R2k
)
a′′k = 0, k = 2, 3, . . . , (22)((
1−m2kR2k
)
ρ+m2k −R2k
)
b′′k = 0, k = 1, 2, . . . . (23)
Since
ρ < 1, (1 +m2kR2k)−1(m2k +R2k) > 1, k = 1, 2, . . . , (24)
and the modulus of the expression (1−m2kR2k)−1(R2k −m2k) is greater
then one for all natural k by virtue of inequalities m < 1 and R > 1, then
relations (21) — (23) implies equalities
a′′1 =
n
2
(
R2 +m2
) (
1−m2ρ
)
(1 +m2R2)ρ−m2 −R2
, a′′k = b′′k−1 = 0, k = 2, 3, . . . . (25)
Substituting now (25) in (15) and (16), obtain equalities
a′′−1 = −n
2
(R2 +m2)(ρ−m2)
(1 +m2R2)ρ−m2 −R2
, a′′−k = 0, k = 2, 3, . . . . (26)
Substituting expressions for a′′1 and b′′k from (25) to the (17) and (16),
obtain
a′1 =
n
2
(R2 +m2)(1− ρ)
(1 +m2R2)ρ−m2 −R2
, b′′−k = 0, k = 1, 2, . . . . (27)
Substituting expression for a′1 from (27) to the (14) with k = 1, obtain
a′−1 =
n
2
m2(R2 +m2)(1− ρ)
(1 +m2R2)ρ−m2 −R2
. (28)
Effective Conductivity of Ellipse – Elliptical Ring Composite . . . 129
Using (25) rewrite formulae (18) and (14) in the form a′k =
= a′−k = 0, k = 2, 3, . . . , b′k = b′−k = 0, k = 1, 2, . . . .
Thus, substituting delivered relations to the formulas (12) and (13),
obtain a solution of the auxiliary A–problem with additional condition
(14) in the form
v1(ζ) = (1− ρ)ψ(ρ)
n
2
(
η +
m2
η
)
cos θ ∀ζ = η exp{iθ} ∈ Am,1, (29)
v2(ζ) = ψ(ρ)
n
2
(
η +
m2
η
−
(
m2η + η−1
)
ρ
)
cos θ ∀ζ = η exp{iθ} ∈ A1,R,
(30)
where ψ(ρ) := (R2 +m2)
(
(1 +m2R2)ρ−m2 −R2
)−1.
4. Solution of A–problem for a system of domains
bounded by confocal ellipses. Now summarizing results of the
latter section we can state the following theorem.
Theorem 1. Let A–Problem posed to the system of domains (G, G̃)
bounded by confocal ellipses E1 (inner) and E2 (outer), accordingly, with
semi-axes a1, b1, a1 > b1, and a2, b2, a2 > b2, relations (4) hold.
Then A–Problem for the system of domains (G, G̃) is equivalent to the
auxiliary A–problem if and only if when fulfilled a condition (14), where
v1 = Re g1 and g1 is expressed by the formula (10). Then these problems
are uniquely solvable and the following formulas hold
u(z) = v(ζ), z = x+ iy ≡ ω(ζ). (31)
A general solution v = (v1, v2) of the auxiliary A–problem is expressed
by the formulae (29) and (30). Furthermore, the boundary values for any
θ ∈ [0, 2π] have the form
v1(exp{iθ}) = v2(exp{iθ}) ≡ vγ1(θ) :=
n
2
(1− ρ)ψ(ρ)
(
1 +m2
)
cos θ, (32)
v2(R exp{iθ}) ≡ vγR
(θ) :=
n
2
ψ(ρ)
(
R+
m2
R
−
(
m2R+
1
R
)
ρ
)
cos θ.
(33)
A general solution of A–Problem for the system of domains (G, G̃) in
coordinates (x, y) has a form
u1(z) = ψ(ρ)(1− ρ)x ∀z = x+ iy ∈ G, (34)
130 S. V. Gryshchuk
u2(z) = ψ(ρ) (x− x′ρ) ∀z = x+ iy ∈ G̃, (35)
where x′ = Reω(ζ−1), z = ω(ζ).
5. Effective conductivity. In the considered case x-component
F x of the complete flux can be normalized as follows F x = −πa2b2 (see
[6]).
Theorem 2. Under assumptions of Theorem 1 x-component of the
effective conductivity tensor is expressed by the formula
λxeff = λ2 P1(ρ)
R2 +m2
m2 +R2 − ρ(1 +m2R2)
, (36)
where P1(ρ) :=
n2
4 a2b2
(
R2 − m4
R2 +
(
1−m4 −m2R2 + m2
R2
)
ρ
)
.
Furthermore, the function (3) is the sum of the following convergent
series
λxeff = λ2 P1(ρ)
(
1 +
∞∑
k=1
ρk
(
1 +m2R2
m2 +R2
)k
)
. (37)
Proof. By the Green’s formula
F xλxeff = λ1
∮
E1
u1(x, y) dy + λ2
(∮
E2
−
∮
E1
)
u2(x, y) dy =
=
∮
E1
(λ1u1(x, y)− λ2u2(x, y)) dy + λ2
∮
E2
u2(x, y) dy,
where in contour integrals the orientation is assumed to be opposite to
the clockwise direction. Doing a change of variables (6) with using of the
relations (8), (32) and (33), obtain
F xλxeff = (λ1 − λ2)
2π∫
0
vγ1(θ) dy(1, θ) + λ2
2π∫
0
vγR(θ) dy(R, θ) =
=
n2
4
ψ(ρ)
2π∫
0
(cos θ)
2
dθ
(
(1− ρ)(λ1 − λ2)(1−m4) +
+ λ2
(
R2 − m4
R2
)
− λ2ρ
(
R− m2
R
)(
m2R+
1
R
))
.
Effective Conductivity of Ellipse – Elliptical Ring Composite . . . 131
Substituting equalities (1 − ρ)(λ1 − λ2) = 2λ2ρ,
∫ 2π
0
(cos θ)2 dθ = π to
the last formula, and doing elementary transformations with using of the
expressions of ψ(ρ) and F x, obtain the required equality (36).
Then the equality (37) follows from the obtained formula after
substitution to it the following series expansion
R2 +m2
m2 +R2 − ρ(1 +m2R2)
= 1 +
∞∑
k=1
ρk
(
1 +m2R2
m2 +R2
)k
, (38)
which is valid due to the inequalities (24) with k = 1. The proof is
completed.
The author is grateful to Prof. S.A. Rogosin for the attention to the
work.
References
[1] Bergmann D.J. The dielectric constants of a composite material – a problem
in classical physics // Int. Journ.Phys. Rep. C. — 1978. — 43. — P. 377 –
407.
[2] Cherkaev A., Pruss A. Effective conductivity of spiral and other radial
symmetric assemblages // Int. Journ. Mechanics of Materials. — 2013. —
65. — P. 103 – 109.
[3] Gryshchuk S., Rogosin S. Effective Conductivity of 2D Disk - Ring
Composite Material // Int. Journ. Mathematical Modelling and Analysis. —
2013. — 18, No. 3. — P. 386 – 394,
[4] Milton G.W. The Theory of Composites. — Cambridge: Cambridge
University Press, 2004.
[5] Mityushev V. Conductivity of a two-dimensional composite containing
elliptical inclusions // Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. —
2009. — 465, No. 2110. — P. 2991 – 3010.
[6] Mityushev V.V., Rogosin S.V. Constructive methods to linear and non-
linear boundary value problems of the analytic function. Theory and
applications. — Chapman & Hall / CRC, Monographs and Surveys in Pure
and Applied Mathematics, 2000.
[7] Fadeev A.V. Solution of a problem of R-linear conjugation for confocal
elliptical annulus in the class of piecewise meromorphic functions // Russian
Mathematics. — 2013. — 57, No. 6. — P. 39 – 52
132 S. V. Gryshchuk
[8] Muskhelishvili N.I. On the problem of torsion and bending of elastic beams
composed of different materials // Izvestiya Akad. Nauk SSSR., Ser. Mat.
and Nat. Sci. — 1932. — No.7. — P. 907 – 945 (in Russian).
[9] Muskhelishvili N.I. Some basic problems of the mathematical theory of
elasticity. Fundamental equations, plane theory of elasticity, torsion and
bending., English transl. from the 4th Russian edition (R. M. Radok). —
Leiden: Noordhoff International Publishing, 1977.
[10] Vekua I.N., A.K. Rukhadze Torsion and transversal bending of the beam
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Matematika i Mechanika (Leningrad). — 1933. — No. 1. — P. 167 – 178 (in
Russian).
[11] Vekua I.N., Rukhadze A.K. The problem of the torsion of a circular cylinder
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|
| id | oai:trim.imath.kiev.ua:article-172 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:03:27Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/03/8952953428d6b5eab24a5fbbaca85803.pdf |
| spelling | oai:trim.imath.kiev.ua:article-1722018-01-23T12:10:05Z Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material Gryshchuk, S. V. Gryshchuk, S. V. For 2D composite material geometrically composed %presentedby an interior of ellipse $E_1$ (with semi-axes $a_1$, $b_1$) and anouter elliptical ring with an outer ellipse $E_2$ (with semi-axes$a_2$, $b_2$), where $E_1$ and $E_2$ are confocal ellipses; it isdetermined in an analytic form the $x$-component of the effectiveconductivity tensor as a sum of convergent power series withcoefficients depending on conductivities of the components of thecomposite material and geometrical characteristics $a_1$, $b_1$ and$a_2$, $b_2$. Інститут математики НАН України 2015-04-23 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/172 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 3 (2015): Analysis and Applications; 121–132 Сборник Трудов Института математики НАН Украины; Том 12 № 3 (2015): Анализ и приложения; 121–132 Збірник Праць Інституту математики НАН України; Том 12 № 3 (2015): Аналіз та застосування; 121–132 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/172/140 Авторське право (c) 2015 Праці Інституту математики НАН України |
| spellingShingle | Gryshchuk, S. V. Gryshchuk, S. V. Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material |
| title | Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material |
| title_full | Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material |
| title_fullStr | Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material |
| title_full_unstemmed | Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material |
| title_short | Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material |
| title_sort | effective conductivity of 2d ellipse – elliptical ring composite material |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/172 |
| work_keys_str_mv | AT gryshchuksv effectiveconductivityof2dellipseellipticalringcompositematerial AT gryshchuksv effectiveconductivityof2dellipseellipticalringcompositematerial |