Viscoelasticity and Thermodiffusion in Electric Field
Basic equations for description and modelling of electric, thermal and diffusion processes in multicomponent viscoelactic structures under presence of the external electric field are presented. The corresponding constitutive relations are analyzed. Сформульовано вихідні рівняння модельного опису еле...
Збережено в:
| Опубліковано в: : | Фізико-математичне моделювання та інформаційні технології |
|---|---|
| Дата: | 2006 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України
2006
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/20973 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Viscoelasticity and Thermodiffusion in Electric Field / J. Jędrzejczyk-Kubik // Фіз.-мат. моделювання та інформ. технології. — 2006. — Вип. 3. — С. 84-90. — Бібліогр.: 11 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860258973253369856 |
|---|---|
| author | Jędrzejczyk-Kubik, J. |
| author_facet | Jędrzejczyk-Kubik, J. |
| citation_txt | Viscoelasticity and Thermodiffusion in Electric Field / J. Jędrzejczyk-Kubik // Фіз.-мат. моделювання та інформ. технології. — 2006. — Вип. 3. — С. 84-90. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Фізико-математичне моделювання та інформаційні технології |
| description | Basic equations for description and modelling of electric, thermal and diffusion processes in multicomponent viscoelactic structures under presence of the external electric field are presented. The corresponding constitutive relations are analyzed.
Сформульовано вихідні рівняння модельного опису електротермодифузійних процесів у n-компонентних в’язкопружних структурах за наявності зовнішнього електричного поля. Проведено аналіз отриманих визначальних співвідношень.
Сформулированы исходные уравнения модельного описания электротермодиффузионных процессов в n-компонентных вязкоупругих структурах при наличии внешнего электрического поля. Проведен анализ полученных определяющих соотношений.
|
| first_indexed | 2025-12-07T18:52:47Z |
| format | Article |
| fulltext |
Viscoelasticity and Thermodiffusion in Electric Field
Jadwiga Jędrzejczyk-Kubik
Doctor, Silesian University of Technology , Department of Theory of Structures, Chair of Theoretical Mechanics,
5 Akademicka st., PL-44-100, Gliwice, Poland, e-mail: jjedrzejczykkubik@poczta.onet.pl
Basic equations for description and modelling of electric, thermal and diffusion processes in
multicomponent viscoelactic structures under presence of the external electric field are presented.
The corresponding constitutive relations are analyzed.
Key words: multicomponent continua, constitutive equations, mechano-thermo-
diffusion, electric field, viscoelasticity
Introduction. We shall analyze a multicomponent continuous medium in which diffu-
sive transport takes place being caused by both chemical potential gradient and by the
action of electric field.
In the medium there is a skeleton component of the mass density ρ0 which is one
order higher than that for the remaining components ρ0 >> ρα, n,1=α . We shall treat
this skeleton, as a viscoelastic body. The medium analyzed is in an electric field which
generates ionization of the particular components of the medium and diffusive flow.
Thus, the sources of mass appear as a result of the ionization and recombination
processes. We shall also assume that the skeleton of the body is dielectric while the
remaining components are the ions which diffuse with respect to this skeleton.
The model of the medium assumed in this paper can serve, among others, for
describing the transport of electrolyte, electrodiffusion in a body with capillary-porous
structure, and in the description of electrochemical corrosion of reinforced concrete or
the mass transfer processes in the plastics [4, 10, 11].
The starting point of our consideration is the system of balances for multicompo-
nent mixture which is in the electric field. Within this range, we analyze the balance of
mass, momentum, energy and inequality of entropy increase.
The interaction of electric field generates Lorentz's force and electric polariza-
tion which appear consequently as an additional body forces in the balance of momen-
tum and energy. As the final result we obtain residual inequality which determines the
trend of evolution of the electrodiffusive process.
Taking into consideration the constitutive assumption we shall obtain the sought
physical equations describing the process of thermal-electrodiffusion in viscoelastic body.
УДК 393.3
84
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
2006, вип. 3, 84-90
85
1. Balances of the process
The equations of multicomponent simple mixture in the electromagnetic field are the
starting point of our consideration.
In these equations each component is cinematically equivalent, as opposed to
diffusion, in which the migration of components in relation to the solid )0( =α gene-
rally takes place; at the same time the mass density of the skeleton is one order higher
than that of the migrating component αρ , n,1=α .
The fields describing processes are as follows [1, 9]:
), tX(χx α α=
rrr is function of motion of the particle αX
r
;
),( tXχx
rrr
= is function of motion of the solid component in reference
configuration ( 3,1,);,( =χ= KktXx Kkk );
t
tXχv
∂
∂
=
αα
α ),(
vr
r
is velocity of particle αX
r
;
),()),,((),( 0
1
0 txv
t
ttx
t
tXv
rr
rrrrr
r
=
∂
χχ∂
=
∂
χ∂
=
−
is Eulerian velocity of the solid com-
ponent. (1)
The mean velocity of the mixture and diffusion velocities are defined respectively as
∑
=α
αα ≈ρ
ρ
=
n
iii vvw
0
01 , .3,1, =−= αα iwvu iii (2)
In the classical notations the balance laws have the forms:
• Conservation of mass
ααα
α
=ρ+
∂
ρ∂ Rv
t ii ,)( , (3)
where Rα is the mass supply of the constituent α. An index following a comma denotes
the partial derivation and repeated Latin indicates are summed but not Greek indices.
Let us sum up the components of the mixture considered. Then from (3) we obtain
0)( , =ρ+
∂
ρ∂
iiw
t
, ∑
α
α = 0R . (4)
If we introduce the mass concentration of the α-th constituent αc defined as
ρρ= αα /c , we will obtain another version of (3)
ααα −=ρ iijRc ,& . (5)
In the above equation, ααα ρ= ii uj denotes mass flux of the constituent α and
)()()( o
ro
o gradw
t
⋅+
∂
∂
= indicates material derivative following the motion wr .
Jadwiga Jędrzejczyk-Kubik
Viscoelasticity and Thermodiffusion in Electric Field
86
• Balance of momentum [1, 2]
nEeFvvv
t jijiiijjii ,1,)()( ,, =ασ+ρ+ρ+φ=ρ+ρ
∂
∂ ααααααααααα ;
0
,,
000
,
00000 )()( jijjjiiijjii PEFvvv
t
σ++ρ+φ=ρ+ρ
∂
∂ , (6)
where ααρ e is the charge density of α -th component, αφi is the momentum supply,
α
iF is body force density acting for the α -th constituent, ασij is partial stress tensor, iP
is electric polarization per unit volume and iE is electric field. The terms iEeααρ and
jji PE , on the right-hand side of this equation are the volume electric force.
After summing up we obtain [6, 7]
∑
α
αα +ρ+σ+ρ=ρ jjkklklkk PEEeFw ,,& . (7)
Here ∑
α
ααρ=ρ kk FF , ∑
α
ασ=σ klkl .
• Balance of energy [1, 2].
e
n
ii
n
jiijii hMjcMwqrU −−ρ+σ+−ρ=ρ ∑∑
=α
αα
=α
α
1
,
1
,, && . (8)
In the above equation U is specific internal energy, iq is the heat flux, r is the
heat supply, αM is chemical potential for the α -th constituent. The quantity eh is the
electric energy source [2] given by
∑
α
α πρ+= iiiie EEJh & , (9)
where iπ is polarization per unit mass ρ=π /ii P .
The complete set of balances of the process closes the inequality of the entropy
increase. This inequality proposed in [2, 8] is taken in the following form
0
,
2
, ≥
ρ
−−−ρ
T
rq
T
T
T
q
S i
iii& . (10)
2. Residual inequality
Making use of the balance of energy (8) and inequality of entropy (10) we shall obtain
the following residual inequality
+−ρ+σ+ρ−ρ ∑∑
=α
αα
=α
αα
n
ii
n
jiij MjcMwUST
1
,
1
, &&&
0
1
, ≥+πρ++ ∑
=α
α
n
iiiii
i EJEq
T
T
& . (11)
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
2006, вип. 3, 84-90
87
In further considerations it is convenient to make use of function
ρ−−= /kk PESTUA .
Then residual inequality has the form
+−ρ+σ+ρ−ρ− ∑∑
=α
αα
=α
αα
n
ii
n
jiij MjcMwAST
1
,
1
, &&&
0
1
' ≥+−+ ∑
=α
α
n
iiiii
i EJEPq
T
T & (12)
Although descriptions of the coupled flow phenomena of interest are naturally
posed in current configuration, numerical solutions of governing equations are more
conveniently carried out on a fixed Lagrangian reference configuration. It is, therefore
necessary to define the following material tensor field [3]
kiiLkKKL XJXT σ= ,, , kkKK qJXQ ,= ,
kkKK PJX ,=Π , αα = kkKK JJXJ , ,
LkKkKL xxC ,,= , KkkK xTT ,,, = ,
KkkK xEE ,= , KkxJ ,det= ,
αα = kkKK jJXj , , KkkK xMM ,,, = . (13)
The residual inequality in terms of the material field can be written as
−ρ++++ρ− ∑
=α
αα
n
KKKLKLE cMTQ
T
CTTSA
1
,0
1
2
1)( &&&&
0
1 1
, ≥+Π−− ∑ ∑
=α =α
ααα
n
KKKKKK EJEMj & , (14)
where klElLkKKLE XJXT σ= ,, , kkklklE EP+σ=σ .
Inequality (14) is the significant importance for defining physical equations of thermo-
diffusion.
3. Constitutive equations
Let us now define the process by history Λ
{ }
ααα
α
−=−=Θ
Θ=Λ
00 ,
)(),(),(),(
cccTT
scsEsEs KJK
T
.
Here 2)( KLKLKJ CE δ−= is the Lagrangian tensor deformation, Θ and c denotes the
increment of the temperature and of the concentration, respectively.
We make the following constitutive assumptions
[ ])();(
0
tstA
s
Λ−Λρ=ρ
∞
=
A (15)
Jadwiga Jędrzejczyk-Kubik
Viscoelasticity and Thermodiffusion in Electric Field
88
from which we obtain physical equations
[ ],)();(
0
E tstT
s
IJIJ Λ−Λ=
∞
=
J [ ])();(
0
tstS
s
Λ−Λ=
∞
=
G ,
[ ])();(
0
tst
s
IiI Λ−Λ=Π
∞
=
D , [ ])(
0
stTq
s
ii −∇=
∞
=
Q ,
[ ])(
0
stMj
s
ii −∇= α
∞
=
αα J , ∑
α
αα= ii jeJ . (16)
Then the present set of equations has the form partly close to the equations of
viscoelasticity and thermodiffusion.
Changes appear, however, when defining the flux of mass. In the simplest case,
the flux of ions is defined by equation
( ) αααα
∞
=
ααα −≈−∇= jij
s
ii MeDstMeJ ,
0
)(J , αα
α
α
α
α ∑∑ −== jijii MeDJJ , , (17)
or in an isotropic case
∑ ∑
α α
αααααααα −==−= iiiii MeDJJMeDJ ,, , . (18)
Confining ourselves to the linear problems, we approximate the functional Aρ
only by the linear and square functionals of the form [5]
−ητ
η∂
η∂
τ∂
τ∂
−τ−+
+τ
τ∂
τ∂
τ−+τ
τ∂
τ∂
τ−+
+τ
τ∂
τΘ∂
τ−−τ
τ∂
τ∂
τ−+ρ=ρ
∫ ∫
∫∑ ∫
∫∫
∞− ∞−
∞−α ∞−
α
∞−∞−
ddEEnttG
dEtLdctL
dtLdEtLAA
t
KLKL
t
IJKL
I
t
I
t
t
KL
t
KL
)()(),(
2
1
)()()()(
)()()()(
43
21
0
+ητ
η∂
ηΘ∂
τ∂
τ∂
η−τ−Φ− ∫ ∫
∞− ∞−
ddEtt
t t
IJ
IJ
)()(),(
∑ ∫ ∫
α
α
∞− ∞−
α −ητ
η∂
η∂
τ∂
τ∂
η−τ−ψ− ddcEtt
t t
IJ
IJ
)()(),(
+ητ
η∂
η∂
τ∂
τ∂
η−τ−− ∫ ∫
∞− ∞−
ddEEttA K
t t
IJ
IJK
)()(),(
−ητ
η∂
ηΘ∂
τ∂
τΘ∂
η−τ−− ∫ ∫
∞− ∞−
ddttm
t t )()(),(
2
1
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
2006, вип. 3, 84-90
89
−ητ
η∂
η∂
τ∂
τΘ∂
η−τ−−
+ητ
η∂
η∂
τ∂
τΘ∂
η−τ−−
∫ ∫
∑ ∫ ∫
∞− ∞−
α
α
∞− ∞−
α
ddEttR
ddcttl
K
t t
K
t t
)()(),(
)()(
),(
∑ ∫ ∫
∑ ∫ ∫
α ∞− ∞−
α
α
α
α
∞− ∞−
α
α
+ητ
η∂
η∂
τ∂
τ∂
η−τ−+
+ητ
η∂
η∂
τ∂
τ∂
η−τ−+
ddEcttC
ddccttn
K
t t
K
t t
)()(),(
)()(),(
2
1
)(0)()(),(
2
1 2ε+ητ
η∂
η∂
τ∂
τ∂
η−τ−+ ∫ ∫
∞− ∞−
ddEEttW L
t t
K
KL . (19)
Here ),,(),,(),,(),,(),,(),(),(),(),( 4321 ητητητψητΦητττττ α mAGLLLL IJKIJIJIJKLIIJ
),( ηταl , ),(),,(),,(),,( ητητητητ αα
KLII WnCR are the relaxation functions, which de-
termine physical properties of the material. These functions are continuous for
0,0 ≥η≥τ and disappear for 0<τ and 0<η .
Let us introduce the functional (19) into the inequality (16). After transformation
we obtain the following set of the constitutive equations for:
• stress tensor
−ττΘτ−Φ−τττ−+= ∫ ∫ dtdEtGLtT
t t
IJKLIJKLIJIJE )(),0()()0,()0()(
0 0
1 &&
τττ−Λ−τττ−ψ− ∫∑∫ α
α
α dEtdct K
t
IJK
t
IJ )(),0()(),0(
00
&& ; (20)
• entropy
+τττ−Φ+ττΘτ−+= ∫ ∫ dEtdtmLtS IJ
t t
IJ )()0,()()0,()0()(
0 0
2 &
τττ−+τττ−+ ∫∑∫ α
α
α dEtRdctl I
t
I
t
)(),0()(),0(
00
&& ; (21)
• chemical potential
−τττ−ψ+τττ−+= ∫ ∫ααα dEtdctnLtM IJ
t t
IJ )()0,()()0,()0()(
0 0
3 &&
τττ−+ττΘτ−− ∫∫ αα dEtCdtl I
t
I
t
)(),0()()0,(
00
&& ; (22)
Jadwiga Jędrzejczyk-Kubik
Viscoelasticity and Thermodiffusion in Electric Field
90
• electric polarization
−τττ−−τττ−+Π=Π ∫ ∫ dEtAdEtWLt JK
t t
IJKJIJII )()0,()()0,()0(4)(
0 0
4 &&
∑∫∫
α
αα τττ−+ττΘτ−− dctCdtR
tt
)()0,()()0,(
00
&& . (23)
The suggested set of equations is the simplest description of the diffusive flows
in the electric field.
References
[1] Bowen R. M. Diffusion model by implied by the theory of mixtures in rational thermody-
namics, ed. C. Truesdell. — New York: Springer-Verlag, 1984. — 578 p.
[2] Eringen A. C. A mixture theory of electromagnetism and superconductivity // Int. J.
Engng. Sci. — 1998. — Vol. 36, № 5/6. — P. 525-543.
[3] Eringen A. C., Maugin G. A. Electrodynamics of continua I, II. — New York: Springer-
Verlag, 1990. — 753 p.
[4] Hladik J. Physics of electrolytes. Transport processes in solid electrolytes and electrodes. —
London: Academic Press, 1972.
[5] Karnauhov V. G., Kirichok I. F. Elektrotermo-vyazkouprugost. — K.: Nauk. dumka,
1988. — 319 p. (in Russian).
[6] Kubik J. Thermodiffusion in Viscoelastic Solids // Studia Geotechnica et Mechanica —
1986. — Vol. 8, № 2. — P. 29-47.
[7] Kubik J. The correspondencr between equations of thermodiffusion and theory of mixtu-
res // Acta Mech. — 1987. — Vol. 70 — P. 51-56.
[8] Muller I, Ruggeri T. Extended Thermodynamics. — New York: Springer-Verlag, 1993.
[9] Wilmański K. Lagrangean Model of Two-Phase Porous Material // J. Non-Equilib. Ther-
modyn. — 1995. — Vol. 20. — P. 50-77.
[10] Zybura A. The degradation of reinforced concrete in corrosive conditions // Zeszyty
Nauk. Pol. Śl. — 1990. — № 1096. — P. 157. (in Polish).
[11] Бурак Я. Й. Вибрані праці. — Львів: ЦММ ІППММ ім. Я. С.Підстригача НАН
України, 2001. — 352 с.
В’язкопружність і термодифузія в електричному полі
Ядвіґа Єнджейчик-Кубік
Сформульовано вихідні рівняння модельного опису електротермодифузійних процесів у
n-компонентних в’язкопружних структурах за наявності зовнішнього електричного поля.
Проведено аналіз отриманих визначальних співвідношень.
Вязкоупругость и термодиффузия в электрическом поле
Ядвига Єнджейчык-Кубик
Сформулированы исходные уравнения модельного описания электротермодиффузионных
процессов в n-компонентных вязкоупругих структурах при наличии внешнего электричес-
кого поля. Проведен анализ полученных определяющих соотношений.
Отримано 15.02.06
|
| id | nasplib_isofts_kiev_ua-123456789-20973 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1816-1545 |
| language | English |
| last_indexed | 2025-12-07T18:52:47Z |
| publishDate | 2006 |
| publisher | Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України |
| record_format | dspace |
| spelling | Jędrzejczyk-Kubik, J. 2011-06-13T18:44:06Z 2011-06-13T18:44:06Z 2006 Viscoelasticity and Thermodiffusion in Electric Field / J. Jędrzejczyk-Kubik // Фіз.-мат. моделювання та інформ. технології. — 2006. — Вип. 3. — С. 84-90. — Бібліогр.: 11 назв. — англ. 1816-1545 https://nasplib.isofts.kiev.ua/handle/123456789/20973 393.3 Basic equations for description and modelling of electric, thermal and diffusion processes in multicomponent viscoelactic structures under presence of the external electric field are presented. The corresponding constitutive relations are analyzed. Сформульовано вихідні рівняння модельного опису електротермодифузійних процесів у n-компонентних в’язкопружних структурах за наявності зовнішнього електричного поля. Проведено аналіз отриманих визначальних співвідношень. Сформулированы исходные уравнения модельного описания электротермодиффузионных процессов в n-компонентных вязкоупругих структурах при наличии внешнего электрического поля. Проведен анализ полученных определяющих соотношений. en Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України Фізико-математичне моделювання та інформаційні технології Viscoelasticity and Thermodiffusion in Electric Field Вязкоупругость и термодиффузия в электрическом поле В’язкопружність і термодифузія в електричному полі Article published earlier |
| spellingShingle | Viscoelasticity and Thermodiffusion in Electric Field Jędrzejczyk-Kubik, J. |
| title | Viscoelasticity and Thermodiffusion in Electric Field |
| title_alt | Вязкоупругость и термодиффузия в электрическом поле В’язкопружність і термодифузія в електричному полі |
| title_full | Viscoelasticity and Thermodiffusion in Electric Field |
| title_fullStr | Viscoelasticity and Thermodiffusion in Electric Field |
| title_full_unstemmed | Viscoelasticity and Thermodiffusion in Electric Field |
| title_short | Viscoelasticity and Thermodiffusion in Electric Field |
| title_sort | viscoelasticity and thermodiffusion in electric field |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/20973 |
| work_keys_str_mv | AT jedrzejczykkubikj viscoelasticityandthermodiffusioninelectricfield AT jedrzejczykkubikj vâzkouprugostʹitermodiffuziâvélektričeskompole AT jedrzejczykkubikj vâzkopružnístʹítermodifuzíâvelektričnomupolí |