On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation
In the present paper, the linear theory of micropolar thermoelasticity without energy dissipation is considered. This work is organized as follows: Section 2 is devoted to basic equations for micropolar thermoelastic materials, supposed to be isotropic and homogeneous, and to assumptions on constitu...
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| author | Giordano, P. Svanadze, M. Zampoli, V. Жордано, П. Сванадзе, М. Замполі, В. |
| author_facet | Giordano, P. Svanadze, M. Zampoli, V. Жордано, П. Сванадзе, М. Замполі, В. |
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| datestamp_date | 2020-03-18T19:53:10Z |
| description | In the present paper, the linear theory of micropolar thermoelasticity without energy dissipation is considered. This work is organized as follows: Section 2 is devoted to basic equations for micropolar thermoelastic materials, supposed to be isotropic and homogeneous, and to assumptions on constitutive constants. In Section 3, some theorems related to representations of a general solution are studied. |
| first_indexed | 2026-03-24T02:41:47Z |
| format | Article |
| fulltext |
UDC 539.3
M. Svanadze (Ilia Chavchavadze State Univ., Tbilisi, Georgia),
P. Giordano, V. Zampoli (Univ. Salerno, Italy)
ON THE REPRESENTATIONS OF GENERAL SOLUTION
IN THE THEORY OF MICROPOLAR THERMOELASTICITY
WITHOUT ENERGY DISSIPATION
ЗОБРАЖЕННЯ ЗАГАЛЬНОГО РОЗВ’ЯЗКУ
В ТЕОРIЇ МIКРОПОЛЯРНОЇ ТЕРМОЕЛАСТИЧНОСТI
БЕЗ РОЗСIЮВАННЯ ЕНЕРГIЇ
In the present paper, the linear theory of micropolar thermoelasticity without energy dissipation is con-
sidered. This work is articulated as follows. Section 2 regards the basic equations for micropolar
thermoelastic materials, supposed isotropic and homogeneous, and the assumptions on the constitutive
constants. In Section 3 some theorems connected with the representations of general solution are studied.
Розглянуто лiнiйну теорiю мiкрополярної термоеластичностi без розсiювання енергiї. Статтю побу-
довано таким чином. Другий пункт присвячено базовим рiвнянням для мiкрополярних термоелас-
тичних матерiалiв, якi вважаються iзотропними та однорiдними, та припущенням щодо основних
констант. У третьому пунктi доведено деякi теореми про зображення загального розв’язку.
1. Introduction. In [1], Eringen established the theory of micropolar thermoelasticity.
In recent years there has been very much written on the subject of this theory. The basis
results and extensive review of works on the theory of micropolar thermoelasticity can
be found in the books of Eringen [2] and Nowacki [3].
In [4], Boschi and Iesan extended a generalized theory of micropolar thermoelasticity
that permits the transmission of heat as thermal waves at finite speed. Recently, Green
and Naghdi [5] introduced a theory of thermoelasticity without energy dissipation. In
[6], Ciarletta presented a linear theory of micropolar thermoelasticity without energy
dissipation. This theory permits the transmission of heat as thermal waves at finite
speed, and the heat flow does not involve energy dissipation.
Contemporaly treatment of the various boundary-value problems on the elasticity
theory usually begins with the representation of a general solution of field equations in
terms of elementary (harmonic, biharmonic, metaharmonic and etс.) functions. In
the classical theory of elasticity the Boussinesq – Somiliana – Galerkin, Boussinesq –
Papkovitch – Neuber, Green – Lamé and Cauchy – Kovalevski – Somiliana solutions are
well known (see Gurtin [7], Kupradze and al. [8], Nowacki [9]). An excellent review of
the history of these solutions is given in Gurtin [7].
The representations of Galerkin-type solutions in the theory of micropolar thermo-
elasticity without energy dissipation, in the theory of thermoelastic materials with voids,
and in the dynamical theory of binary mixture consisting of a gas and an elastic solid are
established by Ciarletta [6, 10, 11]. In the theories of binary mixtures of elastic solids
and fluid-saturated porous media the representations of general solutions are presented
by Basheleishvili [12], Svanadze [13], and Svanadze and de Boer [14].
In this article the linear theory of isotropic and homogeneous micropolar thermoelastic
materials without energy dissipation [6] is considered. The representations of general
solution of the system of steady oscillations in terms of metaharmonic functions are
obtained.
c© M. SVANADZE, P. GIORDANO, V. ZAMPOLI, 2007
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10 1391
1392 M. SVANADZE, P. GIORDANO, V. ZAMPOLI
2. Basic equations. Let x = (x1, x2, x3) be the point of the Euclidean three-
dimensional space E3, and let t denote the time variable. We consider a linear mi-
cropolar thermoelastic material which occupies the region Ω of E3. The system of
linearized equations of motion in the theory of micropolar thermoelasticity without
energy dissipation for isotropic elastic solids can be written as [6]
(µ+ κ)∆ũ + (λ+ µ) grad div ũ + κ curl ϕ̃−m grad θ̃ = ρ(¨̃u−G′),
γ∆ϕ̃ + (α+ β) grad div ϕ̃ + κ curl ũ− 2κϕ̃ = ρ1
¨̃ϕ− ρG′′,
k0∆θ̃ − aT0
¨̃
θ −mT0 div ¨̃u = −ρṠ,
(1)
where ũ = (ũ1, ũ2, ũ3) is the displacement vector, ϕ̃ = (ϕ̃1, ϕ̃2, ϕ̃3) is the microrotation
vector, θ̃ is the temperature measured from the constant absolute temperature T0 (T0 >
> 0); λ, µ, κ, m, α, β, γ, a, k0 are constitutive coefficients, ρ (ρ > 0) is the reference
mass density, ρ1 (ρ1 > 0) is a coefficient of inertia, G′ is the body force density, G′′ is
the body couple density, and S is the heat source density [6]; ∆ is the Laplacian, and
dot denotes differentiation with respect to t: ˙̃u =
∂ũ
∂t
, ¨̃u =
∂2ũ
∂t2
.
If the body forces G′, G′′ and the heat source density S are assumed to be absent,
and the displacement vector ũ, the microrotation vector ϕ̃ and the temperature θ̃ are
postulated to have a harmonic time variation, that is
ũ(x, t) = Re
[
u(x)e−iωt
]
, ϕ̃(x, t) = Re
[
ϕ(x)e−iωt
]
, θ̃(x, t) = Re
[
θ(x)e−iωt
]
,
then from the system of equations of motion (1) we obtain the following system of
equations of steady oscillations (steady vibrations):
(µ+ κ)∆u + (λ+ µ) grad div u + κ curlϕ−m grad θ + ρω2u = 0,
γ∆ϕ + (α+ β) grad div ϕ + κ curlu + µ1ϕ = 0,
k0∆θ + a0θ +m0 div u = 0,
(2)
where µ1 = ρ1ω
2 − 2κ, a0 = aT0ω
2, m0 = mT0ω
2, and ω is the oscillation frequency
(ω > 0).
Throughout this article, it is assumed that all functions are continuous and differenti-
able up to the required order on Ω. We assume that the constitutive coefficients satisfy
the conditions [6]
3λ+ 2µ+ κ > 0, 2µ+ κ > 0, κ > 0, k0 > 0,
3α+ β + γ > 0, γ ± β > 0, a > 0.
In this article the representations of general solution of system (2) in terms of
metaharmonic functions are obtained.
3. Representations of general solution. We consider separately two possible cases:
ω 6= ω0 and ω = ω0, where ω0 =
√
2κ
ρ1
.
1. Let ω 6= ω0. In the sequel we use the following lemmas.
Lemma 1. If (u,ϕ, θ) is a solution of system (2), then
u = − 1
ρω2
[
grad(µ0 div u−mθ)− (µ+ κ) curl curlu + κ curlϕ
]
,
ϕ = − γ0
µ1
grad div ϕ +
1
µ1
(γ curl curlϕ− κ curlu).
(3)
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
ON THE REPRESENTATIONS OF GENERAL SOLUTION IN THE THEORY ... 1393
From system (2) directly follows (3).
Lemma 2. If (u,ϕ, θ) is a solution of system (2), then
(∆ + k2
1)(∆ + k2
2) div u = 0, (∆ + k2
3)(∆ + k2
4) curlu = 0, (4)
(∆ + k2
5) div ϕ = 0, (∆ + k2
3)(∆ + k2
4) curl ϕ = 0, (5)
(∆ + k2
1)(∆ + k2
2)θ = 0, (6)
where k2
1, k
2
2 and k2
3, k
2
4 are the roots of the equations (with respect to ξ)
µ0k0ξ
2 − (ρk0 + aT0µ0 +m2T0)ω2ξ + aT0ρω
4 = 0
and
γ(µ+ κ)ξ2 − [ρω2γ + µ1(µ+ κ) + κ2]ξ + ρω2µ1 = 0,
respectively, µ0 = λ+ 2µ+ κ, k2
5 =
µ1
γ0
, γ0 = α+ β + γ.
Proof. Applying the operator div to Eq. (2)2 we obtain Eq. (5)1. Applying the
operator div to Eq. (2)1 and taking into account Eq. (2)3 we get
(µ0∆ + ρω2) div u−m∆θ = 0,
m0 div u + (k0∆ + a0)θ = 0.
(7)
From system (7) we have
[µ0κ0∆2 + (ρκ0 + aT0µ0 +m2T0)ω2∆ + aT0ρω
4] div u = 0,
[µ0κ0∆2 + (ρκ0 + aT0µ0 +m2T0)ω2∆ + aT0ρω
4]θ = 0.
(8)
On the basis of (8) we obtain Eqs. (4)1 and (6).
Applying the operators (γ∆ + µ1) curl and curl to Eqs. (2)1 and (2)2, respectively,
we get
(γ∆ + µ1)[(µ+ κ)∆ + ρω2] curlu + κ(γ∆ + µ1) curl curl ϕ = 0,
(γ∆ + µ1) curl ϕ + κ curl curlu = 0.
(9)
Taking into account Eq. (9)2 and equality curl curlu = grad div u−∆u from (9)1 we
have {
(γ∆ + µ1)[(µ+ κ)∆ + ρω2] + κ2∆
}
curlu = 0. (10)
Obviously, from Eq. (10) we obtain Eq. (4)2. In the same way from Eqs. (2)1 and (2)2
we get Eq. (5)2.
Remark 1. It is easily seen that
i) k2
1 > 0, k2
2 > 0, k2
1 6= k2
2;
ii) k2
3 > 0, k2
4 > 0, k2
3 6= k2
4, k
2
5 > 0 for ω > ω0, k
2
3 > 0, k2
4 < 0, k2
5 < 0 for
ω < ω0;
iii) µ0k
2
2 − ρω2 6= 0, (µ+ κ)k2
4 − ρω2 6= 0.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
1394 M. SVANADZE, P. GIORDANO, V. ZAMPOLI
In the following we use the notations
α1 = k0k
2
1 − a0, α2 = −m, α3 = −m0k
2
1, α4 = µ0k
2
2 − ρω2,
β1 = γk2
3 − µ1, β2 = κk2
4, β3 = κ, β4 = (µ+ κ)k2
4 − ρω2,
λ−1
1 = α2α3k
2
2 − α1α4k
2
1 = k0k
2
1(k
2
2 − k2
1)α4.
(11)
It is obvious that
µ0αjk
2
j+2 +mαj+2 = ρω2αj ,
[(µ+ κ)βj − κβj+2]k2
j+2 = ρω2βj ,
γβj+2k
2
j+2 − κβj = µ1βj+2,
(k0k
2
j − a0)αj+2 +mαjk
2
j = 0, j = 1, 2.
(12)
Let
ψ1 = λ1(∆ + k2
2)(µ0 div u−mθ),
ψ2 = −λ0k0k
2
1(∆ + k2
1)θ,
ψ3 = − γ0
µ1
div ϕ.
(13)
On the basis of Eqs. (4)1, (5)1 and (6) we have
(∆ + k2
j )ψj = 0, j = 1, 2, 3. (14)
On the other hand, by virtue of (7) and (11), from (13) follows that
ψ1 = λ1(α4 div u + α2k
2
2θ), ψ2 = −λ1(α3 div u + α1k
2
1θ). (15)
From Eqs. (13)3 and (15) we get
div u = −(α1k
2
1ψ1 + α2k
2
2ψ2), div ϕ = −µ1
γ0
ψ3,
θ = α3ψ1 + α4ψ2.
(16)
We introduce the notation
w1 = (w11, w12, w13) =
1
β3k2
3(k
2
4 − k2
3)
(∆ + k2
4) curl ϕ,
w2 = (w21, w22, w23) =
1
β4k2
4(k
2
3 − k2
4)
(∆ + k2
3) curl ϕ.
(17)
Taking into account Eqs. (4)2 and (5)3, from (17) we have
(∆ + kj+2)wj = 0, div wj = 0, j = 1, 2, (18)
and
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
ON THE REPRESENTATIONS OF GENERAL SOLUTION IN THE THEORY ... 1395
curlϕ = β3k
2
3w1 + β4k
2
4w2. (19)
By virtue of Eq. (12)3 and (19) from (9)2 we get
curl curlu = − 1
β3
(γ∆ + µ1) curl ϕ =
1
β3
[
β1β3k
2
3w1 + β4k
2
4(γk
2
4 − µ1)w2
]
=
= β1k
2
3w1 + β2k
2
4w2. (20)
From Eq. (9)1 it follows that
curlu = − 1
ρω2
[(µ+ κ)∆ curlu + κ curl curlϕ] =
= − 1
ρω2
[−(µ+ κ) curl curl curlu + κ curl curlϕ]. (21)
On the basis of Eqs. (19) and (20) from (21) we obtain
curlu =
= − 1
ρω2
[
− (µ+ κ) curl(β1k
2
3w1 + β2k
2
4w2)− κ curl(β3k
2
3w1 + β4k
2
4w2)
]
=
=
1
ρω2
2∑
j=1
[
(µ+ κ)βj − κβj+2
]
k2
j+2 curlwj . (22)
In view of (12)2 from Eq. (22) we have
curlu = curl(β1w1 + β2w2). (23)
Theorem 1. If ω 6= ω0 and (u,ϕ, θ) is a solution of system (2), then
u = grad(α1ψ1 + α2ψ2) + β1w1 + β2w2,
ϕ = gradψ3 + curl(β3w1 + β4w2),
θ = α3ψ1 + α4ψ2,
(24)
where ψ1, ψ2, ψ3 are metaharmonic functions and w1, w2 are metaharmonic vectors
satisfy Eqs. (14) and (18), respectively; αj and βj , j = 1, 2, 3, 4, are defined by Eqs. (11).
Proof. Let (u,ϕ, θ) be a solution of system (2). Taking into account Eqs. (16), (19)
and (20), from Eq. (3)1 we have
u = − 1
ρω2
{
grad
[
− µ0(α1k
2
1ψ1 + α2k
2
2ψ2)−m(α3ψ1 + α4ψ2)
]
−
−(µ+ κ)(β1k
2
3w1 + β2k
2
4w2) + κ(β3k
2
3w1 + β4k
2
4w2)
}
=
=
1
ρω2
2∑
j=1
{
(µ0αjk
2
j +mαj+2) gradψj +
[
(µ+ κ)βj − κβj+2
]
k2
j+2wj
}
. (25)
In view of Eqs. (12) from (25) we obtain Eq. (24)1.
On the basis of Eqs. (16)2, (19) and (23) from (3)2 we get
ϕ = gradψ3 +
1
µ1
curl
2∑
j=1
(γβj+2k
2
j+2 − κβj)wj . (26)
By virtue of (12)3 from Eq. (26) we have Eq. (24)2.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
1396 M. SVANADZE, P. GIORDANO, V. ZAMPOLI
Theorem 2. If ω 6= ω0 and u, ϕ and θ given by Eqs. (24), where ψ1, ψ2, ψ3
and w1, w2 satisfy Eqs. (14) and (18), respectively, then (u,ϕ, θ) is the solution of
system (2) in Ω.
Proof. From Eqs. (24) we get
∆u = −
2∑
j=1
[
αjk
2
j gradψj + βjk
2
j+2wj
]
,
grad div u = −
2∑
j=1
αjk
2
j gradψj ,
curlϕ =
2∑
j=1
βj+2k
2
j+2wj .
(27)
Taking into account Eqs. (12), (24)3 and (27) we have
(µ+ κ)∆u + (λ+ µ) grad div u + κ curlϕ−m grad θ + ρω2u =
= −
2∑
j=1
(
µ0αjk
2
j +mαj+2 − ρω2αj
)
gradψj+
+
2∑
j=1
{[
(µ+ κ)k2
j+2 − ρω2
]
βj − κβj+2k
2
j+2
}
wj = 0.
On the other hand from Eqs. (24) follows that
∆ϕ = −k2
5 gradψ3 −
2∑
j=1
βj+2k
2
j+2 curlψj ,
grad div ϕ = −k2
5 gradψ3,
curlu = curl(β1w1 + β2w2).
(28)
By virtue of Eqs. (12), (28) we get
γ∆ϕ + (α+ β) grad div ϕ + κ curlu + µ1ϕ+
= (−γ0k
2
5 + µ1) gradψ3 +
2∑
j=1
(γβj+2k
2
j+2 − κβj − µ1βj+2) curlwj = 0.
Similarly, in view of (12) and
∆θ = −α3k
2
1ψ1 − α4k
2
2ψ2,
div u = −α1k
2
1ψ1 − α2k
2
2ψ2
we obtain
k0∆θ + a0θ +m0 div u =
2∑
j=1
[
(k0k
2
j − a0)αj+2 +m0αjk
2
j
]
ψj = 0.
Hence, we have Eq. (2)3.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
ON THE REPRESENTATIONS OF GENERAL SOLUTION IN THE THEORY ... 1397
Thus, the general solution (u,ϕ, θ) (vector with seven components) of system of
homogeneous equations (4) in terms of nine metaharmonic functions ψj , w1j and w2j ,
j = 1, 2, 3, is obtained.
2. Let ω = ω0. Obviously, µ1 = k4 = k5 = 0, k2
3 =
ρω2
0γ + κ2
γ(µ+ κ)
. From system (2)
we obtain Eqs. (3)1, (4)1, (6), and
∆ div ϕ = 0, ∆(∆ + k2
3) curlu = 0, quad∆(∆ + k2
3) curl ϕ = 0. (29)
We introduce the notations
β∗1 = γk2
3, β∗2 = − κ
ρω2
0
, λ∗ = α+ β − γ + 2κβ∗2 , µ∗ = γ − κβ∗2 . (30)
Theorem 3. If ω = ω0 and (u,ϕ, θ) is a solution of system (2), then
u = grad(α1ψ1 + α2ψ2) + β∗1v + β∗2 curlu0,
ϕ = u0 + κ curlv,
θ = α3ψ1 + α4ψ2,
(31)
where ψ1, ψ2 satisfy Eq. (14), the vectors v = (v1, v2, v3) and u0 are solutions of
following equations:
(∆ + k2
3)v = 0, div v = 0, (32)
and
µ∗∆u0 + (λ∗ + µ∗) grad div u∗
0 = 0; (33)
respectively, α1, α2, α3, α4 are defined by (12).
Proof. Let
v = − 1
κk4
3
∆ curlϕ, u0 = ϕ− κ curlv. (34)
Taking into account Eq. (29)3 from (34)1 we have (32).
On the basis of Eqs. (11), (13) and (31) from (3)1 we obtain Eq. (31)1. By virtue of
Eqs. (31)1 and (34) we get
µ∗∆u0 + (λ∗ + µ∗) grad div u0 =
= (γ − κβ∗2)∆u0 + (α+ β + κβ∗2) grad div u0 + (β∗1 − γk2
3)κ curlv =
= γ(∆u0 − κk2
3 curlv) + (α+ β) grad div u0+
+κ(β∗1 curlv + β∗2 curl curlu0). (35)
In view of Eq. (2)2 from (35) we have Eq. (33).
Obviously, from (34)2 it follows Eq. (31)2.
Thus, if ω = ω0, then the general solution of system (2) is presented by 5 metaharmonic
functions ψ1, ψ2, v1, v2, v3 and by vector function u0, that is solution of the homogeneous
equilibrium equation of the classical theory of elasticity with Lamé constants λ∗ and µ∗.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
1398 M. SVANADZE, P. GIORDANO, V. ZAMPOLI
Remark 2. As in classical theory of elasticity [7, 8], by virtue of Theorems 1 to 3
it is possible to construct the solutions of boundary-value problems in the linear theory
of micropolar thermoelasticity without energy dissipation.
1. Eringen A. C. Foundations of micropolar thermoelasticity // Int. Cent. Mech. Stud. Course and Lect.
– 1970. – № 23.
2. Eringen A. C. Microcontinuum field theories I: foundations and solids. – New York etc.: Springer,
1999.
3. Nowacki W. Theory of asymmetric elasticity. – Oxford: Pergamon, 1986.
4. Boschi E., Iesan D. A generalized theory of linear micropolar thermoelasticity // Meccanica. – 1973.
– 7. – P. 154 – 157.
5. Green A. E., Naghdi P. M. Thermoelasticity without energy dissipation // J. Elast. – 1993. – 31. –
P. 189 – 209.
6. Ciarletta M. A theory of micropolar thermoelasticity without energy dissipation // J. Thermal
Stresses. – 1999. – 22. – P. 581 – 594.
7. Gurtin M. E. The linear theory of elasticity // Handb. Physik / Ed. C. Trusdell. – Berlin: Springer,
1972. – Vol. VIa/2. – P. 1 – 295.
8. Kupradze V. D., Gegelia T. G., Basheleishvili M. O., Burchuladze T. B. Three-dimensional problems
of the mathematical theory of elasticity and thermoelasticity. – Amsterdam etc.: North-Holland,
1979.
9. Nowacki W. Dynamic problems of thermoelasticity. – Leyden: Noordhoff Int. Publ., 1975.
10. Ciarletta M. A solution of Galerkin type in the theory of thermoelastic materials with voids // J.
Thermal Stresses. – 1991. – 14. – P. 409 – 417.
11. Ciarletta M. General theorems and fundamental solutions in the dynamical theory of mixtures // J.
Elast. – 1995. – 39. – P. 229 – 246.
12. Basheleishvili M. Applications of analogues of general Kolosov – Muskhelishvili representation in
the theory of elastic mixtures // Georgian Math. J. – 1996. – 6. – P. 1 – 18.
13. Svanadze M. Representation of the general solution of the equation of steady oscillations of two-
component elastic mixtures // Int. Appl. Mech. – 1993. – 29. – P. 22 – 29.
14. Svanadze M., de Boer R. Representations of solutions in the theory of fluid-saturated porous media
// Quart. J. Mech. and Appl. Math. – 2005. – 58. – P. 551 – 562.
Received 14.06.2005,
after revision — 31.10.2006
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
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| id | umjimathkievua-article-3397 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:41:47Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b9/9efbc2cffaa6c060e6417e20d7a2abb9.pdf |
| spelling | umjimathkievua-article-33972020-03-18T19:53:10Z On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation Зображення загального розв'язку в теорії мікрополярної термоеластичності без розсіювання енерггї Giordano, P. Svanadze, M. Zampoli, V. Жордано, П. Сванадзе, М. Замполі, В. In the present paper, the linear theory of micropolar thermoelasticity without energy dissipation is considered. This work is organized as follows: Section 2 is devoted to basic equations for micropolar thermoelastic materials, supposed to be isotropic and homogeneous, and to assumptions on constitutive constants. In Section 3, some theorems related to representations of a general solution are studied. Розглянуто лінійну тєорію мікрополярної термоеластичності без розсіювання енергії. Статтю побудовано таким чином. Другий пункт присвячено базовим рівнянням для мікрополярних термоеластичних матеріалів, які вважаються ізотропними та однорідними, та припущенням щодо основних констант. У третьому пункті доведено деякі теореми про зображення загального розв'язку. Institute of Mathematics, NAS of Ukraine 2007-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3397 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 10 (2007); 1391–1398 Український математичний журнал; Том 59 № 10 (2007); 1391–1398 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3397/3537 https://umj.imath.kiev.ua/index.php/umj/article/view/3397/3538 Copyright (c) 2007 Giordano P.; Svanadze M.; Zampoli V. |
| spellingShingle | Giordano, P. Svanadze, M. Zampoli, V. Жордано, П. Сванадзе, М. Замполі, В. On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation |
| title | On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation |
| title_alt | Зображення загального розв'язку в теорії мікрополярної термоеластичності без розсіювання енерггї |
| title_full | On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation |
| title_fullStr | On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation |
| title_full_unstemmed | On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation |
| title_short | On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation |
| title_sort | on representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3397 |
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