The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion
The generalized magneto-thermoelasticity is developed. The formulation is done under two theories: the generalized thermoelasticity coupled theory and Lord – Shulman theory with one relaxation time. The normal mode analysis is used to obtain the expres- sions for 
 temperature, displacement...
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Інститут механіки ім. С.П. Тимошенка НАН України
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| Cite this: | The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion / M.I.A. Othman, R.M. Farouk, H.A. El Hamied // Прикладная механика. — 2013. — Т. 49, № 2. — С. 130-142. — Бібліогр.: 24 назв. — англ. |
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| author | Othman, M.I.A. Farouk, R.M. El Hamied, H.A. |
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| citation_txt | The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion / M.I.A. Othman, R.M. Farouk, H.A. El Hamied // Прикладная механика. — 2013. — Т. 49, № 2. — С. 130-142. — Бібліогр.: 24 назв. — англ. |
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| description | The generalized magneto-thermoelasticity is developed. The formulation is done under two theories: the generalized thermoelasticity coupled theory and Lord – Shulman theory with one relaxation time. The normal mode analysis is used to obtain the expres- sions for 
temperature, displacement components, the thermal stresses distributions and concentration of diffusion. The variations of the considered variables are represented graphically. A comparison is made with the results predicted by two theories in presence and absence of the magnetic field.
Розвинуто узагальнену теорію магнітотермопружності. Сформульовано задачу в
рамках двох підходів: теорії узагальненої зв’язаної термопружності та теорії Лорда – Шульмана з одним часом релаксації. Використано аналіз нормальної моди для отримання виразів для температури, компонентів зміщень, розподілу температурних напружень та концентрації дифузії. Зміну вказаних вище змінних представлено графічно. Порівняно результати в рамках обох теорій за умови наявності та відсутності магнітного поля.
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2013 ПРИКЛАДНАЯ МЕХАНИКА Том 49, № 2
130 ISSN0032–8243. Прикл. механика, 2013, 49, № 2
M . I . A . O t h m a n 1 , R . M . F a r o u k 2 , H . A . E l H a m i e d 3
THE EFFECT OF MAGNETIC FIELD AND THERMAL RELAXATION
ON 2-D PROBLEM OF GENERALIZED THERMOELASTIC DIFFUSION
1, 2, 3 Zagazig University, P.O. Box 44519, Zagazig, Egypt.
E-mail: m_i_othman@yahoo.com, rmfarouk1@yahoo.com, hemat7000@yahoo.com
Abstract. The generalized magneto-thermoelasticity is developed. The formulation is
done under two theories: the generalized thermoelasticity coupled theory and Lord – Shul-
man theory with one relaxation time. The normal mode analysis is used to obtain the expres-
sions for temperature, displacement components, the thermal stresses distributions and con-
centration of diffusion. The variations of the considered variables are represented graphi-
cally. A comparison is made with the results predicted by two theories in presence and ab-
sence of the magnetic field.
Key words: Generalized thermoelasticity, thermoelastic diffusion, Lord – Shulman,
electromagnetic field.
§1. Introduction.
The propagation of waves in thermoelastic materials has many applications in various
fields of science and technology, namely, atomic physics, industrial engineering, thermal
power plants, submarine structures, pressure vessel, aerospace, chemical pipe and metal-
lurgy. The importance of thermal stresses in causing structural damages and changes in
functioning of structure is well recognized whenever thermal stress environments are in-
volved. Therefore, the ability to predict electrodynamics stress induced by sudden thermal
loading in composite structures is essential for the proper and safe design and the knowledge
of its response during the last three decades, non classical theories of thermoelastic so-called
generalized thermoelasticity.
Biot [2] developed the coupled theory of thermoelastic to deal with a defect of the un-
coupled theory that mechanical causes have no effect on the temperature. However, this
theory shares a defect of the uncoupled theory in that it predicts infinite speeds of propaga-
tion for heat waves. Lord and Shulman (L – S) [7] introduced the theory of generalized ther-
moelasticity with one relaxation time for the special case of an isotropic body. This theory
was extended by Sherief [20] and Dhaliwal and Sherief [4] to include the anisotropic case.
In this theory, a modified law of heat conduction including both the heat flux and its time
derivative replaces the conventional Fourier's law. The heat equation associated with this
theory is hyperbolic and hence eliminates the paradox of infinite speeds of propagation in-
herent in both the uncoupled and coupled theories of thermoelasticity. For this theory,
Ignaczak [5] studied uniqueness of solution; Sherief [21] proved uniqueness and stability.
Othman [13] used L – S theory under the dependence of the modulus of elasticity on the
reference temperature in two dimensional generalized thermoelasticity. Othman [14] studied
the effect of rotation on plane waves in generalized thermoelasticity with two relaxation
times.
131
The theory of magneto-thermoelasticity is concerned with the influence of the magnetic
field on the elastic and thermoelastic deformations of a solid body. This theory has aroused
much interest in recent years, because of its application in various branches of science and
technology. Electro-magneto-thermoelasticity investigates the interaction between tempera-
ture, strain, stress and electromagnetic field in a solid elastic body. The origin of magneto-
elasticity as a separate discipline can be traced back to the work of Rikitake [19], who stud-
ied the excitation magneto-hydrodynamic waves by the seismic waves which pass through
the field core of the earth permeated by geomagnetic field. This in spired Cagniard [3] to
suggest the possible influence of the earth's magnetic field on the propagation of the seismic
waves and open a new field of research on the magneto-elastic waves in solids. A systematic
study in this area started with the important work of Knopoff [6] who developed the basic
equation governing the magneto-mechanical interactions in electrically conducting materials
and used them to study the influence of the terrestrial magnetic field on the propagation of
seismic waves in the interior of the earth. An excellent summary of the work done in mag-
neto-elasticity and magneto-thermoelasticity in the fifties and sixties of the present centuries
was gives by Paria [18]. Among the authors whom considered the generalized magneto-
thermoelastic equations are Nayfeh and Nasser [8] whom studied the propagation of plane
waves in a solid under the influence of an electromagnetic field, Agarwal [1] considered
thermoelastic and magneto-thermo-elastic plane wave propagation an infinity elastic me-
dium with two relaxation times, Othman [15] studied electro-magneto-thermoelastic plane
waves with thermal relaxation in a medium of prefect conductivity, Sherief and Helmy [23]
studied a two-dimensional problem in electro-magneto-thermoelasticity for a half-space,
whose surface in subjected to a non-uniform thermal shock and is stress free in the presence
of a transverse magnetic field, Othman et al. [16] investigated the effect of diffusion on two-
dimensional problem of generalized thermoelasticity with Green – Naghdi theory.
The thermo-diffusion in elastic solids is due to coupling of the fields of temperature,
mass diffusion and that of strain in addition to heat and mass exchange with environment.
Nowacki [9 – 12] developed the theory of thermoelastic diffusion by using coupled ther-
moelastic model. Diffusion can be defined as the random walk, of an ensemble of particles,
from regions of high concentration to regions of lower concentration. There is now a great
deal of interest in the study of this phenomenon, due to its many applications in geophysics
and industrial applications. In integrated circuit fabrication, diffusion is used to introduce
"do pants" in controlled amount into semiconductor substrate.
In particular, diffusion is used to form the base and emitter in bipolar transistors, form
integrated resistors, form the source drain regions in MOS transistors and dope ploy-silicon
gates in MOS transistors. In most of these applications, the concentration is calculated using
what is known as Fick’s law.
This is a simple law that does not take into consideration the mutual interaction between
the introduced substance and the medium into which it is introduced or the effect of the
temperature on this interaction. In this theory, the coupled thermoelastic model is used. This
implies infinite speeds of propagation of thermoelastic waves. Sherief et al. [22] developed
the theory of generalized thermoelastic diffusion that predicts finite speeds of propagation
for thermoelastic and diffusive waves. Sherief and Saleh [24] worked on a one dimensional
problem of a thermoelastic half-space with a permeating substance in contact with the
bounding plane in the context of the theory of generalized thermoelastic diffusion with one
relaxation time. Recently, Othman et al. [17] have studied the dependence of the modulus of
elasticity on reference temperature in the theory of generalized thermoelastic diffusion with
one relaxation time.
In this paper, we shall formulate the normal mode analysis of a two-dimensional prob-
lem of electro-magneto-thermoelasticity under Lord – Shulman theory in a perfectly con-
ducting medium.
The exact expressions for temperature distribution, thermal stress and displacement
components are obtained, and represented graphically in the presence and absence of the
magnetic field for the different theories.
132
§ 2. Formulation of the problem.
We consider an isotropic homogeneous, linear, thermally and perfectly conducting elas-
tic medium. The whole body is at constant temperature 0T and it is acted on throughout by
a constant magnetic field 00, ,0H H which is oriented towards the positive direction of
y-axis. We are being our consideration with the linearized equation of electromagnetism,
valid for slowly moving media [5]. We assume that all quantities are functions of the coor-
dinates ,x z and time t and independent of coordinate y. So the displacement vector will
have the components , ,xu u x z t and , ,zu w x z t . The electric intensity vector is
normal to both that the magnetic intensity and displacement vector. Thus, it has the
components
1 3, 0,x y zE E E E E , (2.1)
0curl h J E , (2.2)
0curl E h , (2.3)
0E u H , (2.4)
div 0h . (2.5)
The basic equations for electro-magneto-thermoelastic, in a homogeneous, in a homo-
geneous, isotropic solid without body forces, can be written as:
1) equation of motion
,i ij j iu F 0i i
F J H ; (2.6)
2) strain displacement equation relation
, ,
1
2ij i j j ie u u ; (2.7)
3) the constitutive law for the theory of generalized thermoelasticity
1 0 22ij ij ij kke e T T C , (2.8)
where, ijσ the components of stress tensor and , are lame's constants, is the density
and 1 2, are material constants given by 1 3 2 ,t t is the coefficient of lin-
ear thermal expansion and 2 3 2 ,c c is the coefficient of linear diffusion ex-
pansion;
4) heat conduction equation
2
, 0 1 0 02ii E kkK T C T e aT C
t t
, (2.9)
where K is the thermal conductivity, 0 is the thermal relaxation time, EC is the specific
heat at constant strain, a is a measure of thermo-diffusion effect, T is the absolute tempera-
ture, 0T is a reference temperature assumed to obey the inequality 0 0/ 1T T T ;
5) the equation of diffusion has the form
2 , , , 0kk ii ii iid e d aT C C dbC , (2.10)
133
where d is the diffusion coefficient, b is a measure diffusion effect, is the diffusion relaxa-
tion time and C is the concentration of diffusive material in the elastic.
We introduce the displacement potentials , ,x z t and , ,x z t through the relations
, ,x zu , , ,z xw (2.11)
( 2
, ,z xu w , 2
kke ). (2.12)
From Eqs. (4), (11) and (12) we have
0 0 ,0,E H w u , (2.13)
From Eqs. (3) and (13) we have
2
0 0kkh H e H , (2.14)
From Eqs. (2.2 ) and (2.13) we have
2
1 0 0 0 0 0 2
h u
F H H
x t
; (2.15)
2 0F ; (2.16)
2
3 0 0 0 0 0 2
h w
F H H
z t
. (2.17)
From Eq. (2.8), it follow that the components of stress tensor have the form
1 0 22xx xx kke e T T C ; (2.18)
1 0 22zz zz kke e T T C ; (2.19)
2xz xze ; (2.20)
1 0 2yy kke T T C ; (2.21)
0xy yz . (2.22)
The governing equations can be put in a more convenient form by using the following non-
dimensional variables:
1 0( , ) ( , )x z c x z ; 1 0, ,u w c u w ; 2
1 0t c t ; 2
2
1
C
C
c
;
1 0
2
1
T T
c
;
2
1
ij
ij
c
; 2
0 1 0 0c ; 2
1 0c ; 0
2
0 0 0 1
E
E
H c
; 0
0 0 0
h
h
H
;
2
1
2
c
0
EC
K
. (2.23)
Using Eq. (2.23) into Eqs. (2.6), (2.9) and (2.10) we have
134
2 2 2
0 0 0 01
e C h
u u A B u
x x x x
; (2.24)
2 2 2
0 0 0 01
e C h
w w A B w
z z z z
; (2.25)
2 2 2
2
0 0 1 02 2 2kke a C
t t tt t t
; (2.26)
2
2 2 2
1 2 32
0e C C
t t
, (2.27)
where
2
0
2
;c
2
0 2
0
1
;
c
2 2
0 0 0
0 2
0 1
;
H
A
c
2 2
0 0 0
0 ;
H
B
1 ;
Ec
2
1
1
2
;
E
a c
a
c
2
1
1
1 2
;
a c
2
1
2 2
2 0
;
c
d
2
1
3 2
2
b c
.
Using Eqs. (2.23) into Eqs. (2.18) – (2.21) we have
2
01 2xx
u w
C
x z
; (2.28)
2
01 2zz
w u
C
z x
; (2.29)
2
0xz
u w
z x
; (2.30)
2
01 2yy
u w
C
x z
. (2.31)
From Eqs. (2.11), (2.12) into Eqs. (2.24) – (2.27) we obtain
2
2
2
C
t
; (2.32)
2
0
2
0
2
2
2
1
0
b
t
; (2.33)
2 2 2
2 2
0 0 1 02 2 2
a C
t t tt t t
; (2.34)
2
4 2 2
1 2 32
0C C
t t
, (2.35)
135
where 0 01 A H , 01 B .
The initial conditions of the problem are taken to be homogeneous while the boundary
conditions are assumed to be
0 1, , ,zz zx z t f x t ; 0, , 0xz zx z t ; 0 0z
C
z
; 0 0zz
, (2.36)
where 1 ,f x t is known function of x and t.
The solution of considered physical variable can be decomposed in terms of normal
modes as the following:
* * * * *, , , , , , , , , , ( ) t ik x
ij ijC x z t C z e ; (2.37)
* * *2
1D s C ; (2.38)
*2 2 0D m ; (2.39)
2 * 2 2 * *
2 3 4D s s D k s C ; (2.40)
* * *
22 2 2 2 2
51 3 0D k D k s D C , (2.41)
where 1, 2,3, 4,5ns n and 2m are defined in the appendix.
Eqs. (2.38), (2.40) and (2.41) form a coupled system of * * *, ,C , while Eq. (2.39) un-
coupled, where /D d d z .
Eliminating * *and C between Eqs. (2.38), (2.40) and (2.41) we obtain
6 4 2 *
0 1 2 0D D D z . (2.42)
In a similar manner we arrive at,
6 4 2 * *
0 1 2 , 0D D D C z , (2.43)
Eq. (2.42) can be factorized as
2 2 2 2 2 2 *
1 2 3 0D k D k D k z ; (2.44)
2 2 * 0i iD k z , 1,2,3i , (2.45)
where 2 1, 2,3ik i are the roots of the following characteristic equation
6 4 2 *
0 1 2 0k k k z . (2.46)
The solution of Eq. (2.44)
3
1
* *
i
i
z
. (2.47)
The solution of Eq. (2.45) as z is given by
* ik z
i iG e . (2.48)
From Eq. (2.48) into Eq. (2.47)
3
*
1
k zi
i
i
z G e
. (2.49)
Similarly,
3
*
1
k zi
i
i
z G e
; (2.50)
136
3
*
1
k zi
i
i
C G e
. (2.51)
The solution of Eq. (2.39) has the form as z
* mzB e , (2.52)
where i, andi iG G G are some parameters.
Substituting from Eqs. (2.49) – (2.51) into Eqs. (2.38), (2.40) and (2.41) we get the fol-
lowing relation:
2 2 2
4 1 3
2
4 2
i i
i i
i
s k s s k k
G G
k s s
; (2.53)
1
2 2 2 2
2 3
2
4 2
i i i
i i
i
k s k s s k k
G G
k s s
. (2.54)
Substituting from Eqs. (2.53), (2.54) into Eqs. (2.50), (2.51) respectively, we obtain
*
3
1
ik z
i i
i
H G e
(2.55)
*
3
1
ik z
i i
i
C R G e
, (2.56)
where iH and iR are defined in the appendix.
Using Eqs. (2.37), (2.49) and (2.52) into Eq. (2.11) we can obtain the displacement
components * *,u w
3
*
1
ik z mz
i
i
u i k G e m B e
; (2.57)
3
*
1
i
i i
k z mz
i
w G k e i k B e
. (2.58)
Using Eq. (2.37) into Eqs. (2.28) – (2.31) we obtain
*
* * 2 * *
0(1 2 )xx
w
i k u C
z
; (2.59)
*
* 2 * * *
0(1 2 )zz
w
i k u C
z
; (2.60)
*
** 2
0
u
ik w
zxz
; (2.61)
*
* 2 * * *
0(1 2 )yy
w
i k u C
z
. (2.62)
From Eqs. (2.55) – (2.58) into Eqs. (2.59) – (2.62) we obtain the components of stress tensor
3
* 2 2 2 2
0 0
1
1 2 2ik z mz
xx i i i i
i
k k H R G e i m k B e
; (2.63)
3
* 2 2 2 2
0 0
1
1 2 2ik z mz
zz i i i i
i
k k H R G e i m k B e
; (2.64)
137
3
* 2 2 2
0
1
1 2 ik z
yy i i i i
i
k k H R G e
; (2.65)
3
* 2 2 2
0
1
2 ik z mz
xz i i
i
i k k G e m k B e
. (2.66)
In order to determine the parameters 1,2,3iG i substituting into the following boundary
conditions at 0z .
From Eq. (2.36) into Eqs. (2.55), (2.56), (2.64) and (2.66) we have
3
*
4 1
1
i i
i
A G A B f
; (2.67)
3
1
0
i
i ik G E B
; (2.68)
3
1
0
i
i iB G
; (2.69)
3
1
0
i
i iE G
, (2.70)
where ,i iA B and iE are defined in the appendix.
Solving Eqs. (2.67) – (2.70) we can obtain the parameters 1,2,3iG i and B by using
the mat lab program.
§3. Numerical results.
The copper material was chosen for purposes of numerical evaluations. The material
constants of the problem are thus given by SI units Thomas (1980);
0 293T K ; 38954 Kg m ; 0 0,02s ; 0,02s ; 383,1EC J KgK ;
5 101 21,78 10 ; 386 W mK ; 7,76 10 Kg ms ;t K K
10 3 42 3 33,86 10 Kg ms ; 8,950 10 Kg m ; 1,98 10 m Kg;c
8 4 63 2 2 5 20,85 10 Kgs m ; 1,2 10 m s K ; 0,9 10 m Kg s .d a b
Figs. 1 – 5 depict the influence of magnetic field on the temperature , the components
of displacement ,u w , the component of stress zz and the concentration C under (CD) and
(L – S) theories, when α 1 (i.e. 0H 0 ), α 1,8 (i.e. 8
0H 10 ).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
2.5
3
x 10
-3
z
CD
LS
=1.8
=1
Fig. 1. Variation of the temperature for 0 0 , 0 0 .
138
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
z
u
CD
LS
=1.8
=1
Fig. 2. Variation of the displacement component u for 0 0 , 0 0 .
We see from Fig. 1 that the magnetic field has decreasing effect on the temperature and
converges to zero with increase the distance z.
Figs. 2 and 3 show that the magnetic field has increasing effect on the components of
displacement ,u w and converges to zero with increase the distance z.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
z
w
CD
LS
=1.8
=1
Fig. 3. Variation of the displacement component w for 0 0 , 0 0 .
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.015
-0.01
-0.005
0
z
zz
CD
LS
=1.8
=1
Fig. 4. Variation of the stress component zz for 0 0 , 0 0 .
139
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.005
0.01
0.015
0.02
0.025
0.03
z
C
CD
LS
=1.8
=1
Fig. 5. Variation of the concentration C for 0 0 , 0 0 .
Fig. 4 depicts that the magnetic field has decreasing effect on the stress component zz
and converges to zero with increase the distance z. It is evident from Fig. 5 that the magnetic
field has decreasing effect on concentration C and converges to zero with increase the dis-
tance z.
In Figs. 1 – 10, we notice that all the curves converge to zero with increase the distance
z. Fig. 1, 4 and 5 indicate that the curves under Lord – Shulman theory are greater than those
of the coupled theory.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
2.5
3
3.5
x 10
-3
z
LS
t=0.1
t=0.3
t=0.5
Fig. 6. Variation of the temperature for 0 0 , 1.8 .
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
z
u
LS
t=0.3
t=0.1
t=0.5
Fig. 7. Variation of the displacement component u for 0 0 , 1.8 .
140
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
2.5
z
w
LS
t=0.1
t=0.3
t=0.5
Fig. 8. Variation of the displacement component w for 0 0 , 1.8 .
Figs. 2 and 3, show that the curves under the coupled theory are greater than those of
Lord – Shulman theory.
The variations of the temperature , the components of displacement ,u w the compo-
nent of stress zz and the concentration C with the distance z in the presence of magnetic
field for different values of time under (L – S) theory are shown in Figs. 6 – 10 at 1,8 ;
t = 0,1; t = 0,3 and t = 0,5. Fig. 6 shows that the temperature increases with increase t for
0 0.4x but for 0.4 x the temperature decrease with increase t and converges to zero
with increasing the distance z.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
5
10
15
20
25
30
z
zz
LS
t=0.1
t=0.3
t=0.5
Fig. 9. Variation of the stress component zz for 0 0 , 1.8 .
Fig. 10. Variation of the concentration C for 0 0 , 1.8 .
141
In Figs. 7 – 10 we notice that the components of displacement ,u w , the component of stress
zz and the concentration C are increasing with increase the time t and converges to zero
with increase the distance z.
§4. Conclusion.
In this paper, normal mode method is used to study the problem of the effect of mag-
netic field and thermal relaxation on two-dimensional problem of generalized thermoelastic
diffusion under Lord – Shulman theory.
We can obtain the following conclusions according to analysis above.
1. The values of distributions of all physical quantities converge to zero with increasing
the distance z.
2. The magnetic field has great influence on the distribution of physical quantities.
3. It is clear from Figs. 6 – 10 that the different values of the time play a significant role
in all the physical quantities except temperature.
4. All the physical quantities satisfy the boundary conditions and initial conditions.
5. The curves in the context of (CD) and (L – S) theories decrease exponentially with
increase z, this indicates that the thermoelastic diffusion waves are untenanted and non-
dispersive. Where purely thermoelastic diffusion waves undergo both attenuation and dis-
persion.
Appendix.
2 2
1s k ; 2 2
2 0s k ; 2
3 0s , 2
4 1 0s a ;
2 2
5 2 3s k ;
2
0
2
0
1 B
s
; 2m k s ;
2
50 2 4 3 1 3 2 1 3 4
3
1
2
1
k s s s s s s s s
;
42 2
5 5 51 2 1 3 3 2 3 1
3
1
2
1
k s k s s s k s s s s
–
2 2
1 4 1 4 1 32s k s k s s
;
4 42 2
5 52 3 2 1 2 4 1 3 1 1 4
3
1
1
s s k s k s s s k s s s s k
;
2 2 2
4 1 3
2
4 2
i i
i
i
s k s s k k
H
k s s
,
1
2 2 2 2
2 3
2
4 2
i i i
i
i
k s k s s k k
R
k s s
;
3
2 2 2
0
1
1 2i i i i
i
A k k H R
; 2
4 02A i m k ; 1 1 1B k H ; 2 2 2B k H ;
3 3 3B k H ; 1 1 1E k R ; 2 2 2E k R ; 3 3 3E k R and 2 2
2
i
E k m
k
.
Р Е ЗЮМ Е . Розвинуто узагальнену теорію магнітотермопружності. Сформульовано задачу в
рамках двох підходів: теорії узагальненої зв’язаної термопружності та теорії Лорда – Шульмана з
одним часом релаксації. Використано аналіз нормальної моди для отримання виразів для температу-
ри, компонентів зміщень, розподілу температурних напружень та концентрації дифузії. Зміну вказа-
них вище змінних представлено графічно. Порівняно результати в рамках обох теорій за умови наяв-
ності та відсутності магнітного поля.
142
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*From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 08.11.2010 Утверждена в печать 26.06.2012
|
| id | nasplib_isofts_kiev_ua-123456789-95504 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0032-8243 |
| language | English |
| last_indexed | 2025-12-07T16:37:46Z |
| publishDate | 2013 |
| publisher | Інститут механіки ім. С.П. Тимошенка НАН України |
| record_format | dspace |
| spelling | Othman, M.I.A. Farouk, R.M. El Hamied, H.A. 2016-02-27T10:01:27Z 2016-02-27T10:01:27Z 2013 The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion / M.I.A. Othman, R.M. Farouk, H.A. El Hamied // Прикладная механика. — 2013. — Т. 49, № 2. — С. 130-142. — Бібліогр.: 24 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/95504 The generalized magneto-thermoelasticity is developed. The formulation is done under two theories: the generalized thermoelasticity coupled theory and Lord – Shulman theory with one relaxation time. The normal mode analysis is used to obtain the expres- sions for 
 temperature, displacement components, the thermal stresses distributions and concentration of diffusion. The variations of the considered variables are represented graphically. A comparison is made with the results predicted by two theories in presence and absence of the magnetic field. Розвинуто узагальнену теорію магнітотермопружності. Сформульовано задачу в
 рамках двох підходів: теорії узагальненої зв’язаної термопружності та теорії Лорда – Шульмана з одним часом релаксації. Використано аналіз нормальної моди для отримання виразів для температури, компонентів зміщень, розподілу температурних напружень та концентрації дифузії. Зміну вказаних вище змінних представлено графічно. Порівняно результати в рамках обох теорій за умови наявності та відсутності магнітного поля. en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion Влияние магнитного поля и тепловой релаксации на двухмерную задачу обобщенной термоупругой диффузии Article published earlier |
| spellingShingle | The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion Othman, M.I.A. Farouk, R.M. El Hamied, H.A. |
| title | The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion |
| title_alt | Влияние магнитного поля и тепловой релаксации на двухмерную задачу обобщенной термоупругой диффузии |
| title_full | The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion |
| title_fullStr | The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion |
| title_full_unstemmed | The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion |
| title_short | The effect of magnetic field and thermal relaxation on 2-D problem of generalized thermoelastic diffusion |
| title_sort | effect of magnetic field and thermal relaxation on 2-d problem of generalized thermoelastic diffusion |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/95504 |
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