Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating
Досліджено задачу про двовимірне магнітне мікрополярне узагальнене термопружне середовище, у якому присутній комбінований ефект струму Холла і яке нагрівається за законом пилоподібного типу. У середовищі існує сильне магнітне поперечне поле, яке є перпендикулярним до площини деформування, і прийнято...
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nasplib_isofts_kiev_ua-123456789-1005852025-02-09T23:30:31Z Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating Zakaria M. Досліджено задачу про двовимірне магнітне мікрополярне узагальнене термопружне середовище, у якому присутній комбінований ефект струму Холла і яке нагрівається за законом пилоподібного типу. У середовищі існує сильне магнітне поперечне поле, яке є перпендикулярним до площини деформування, і прийнято припущення про ігнорування наведеного електричного поля. Закон Ома модифіковано за допомогою включення двох доданків, один з яких є векторним добутком густини струму на початкове магнітне поле і другий є векторним добутком швидкості на початкове магнітне поле. Застосовано методи перетворення Лапласа і експоненціального перетворення Фур’є з метою зведення основних диференціальних рівнянь з частинними похідними до звичайних диференціальних рівнянь, які розв’язано точно. Проведено порівняння з роботами, опублікованими раніше, і результати оцінені як такі, що перебувають у добрій узгодженості. Отримано розподіли температури, зміщення, напруження, мікроповоротів і густини струму. Числові значення цих функцій представлено графічно. The problem of two dimensional magnetic micropolar generalized thermoelastic medium, in the presence of the combined effect of Hall currents subjected to ramptype heating, is investigated. The medium is permeated by a strong transverse magnetic field imposed perpendicularly to the displacement plane on the assumption of the induced electric field is neglected. Ohm’s law is modified by the inclusion of two terms, one for the cross product of the current density with the initial magnetic field and the second for the cross product of the velocity with the initial magnetic field. Laplace and exponential Fourier transform techniques are employed to transform the governing partial differential equations to ODE which were solved exactly. Comparisons with previously published work have been conducted and the results are found to be in good agreement. The distributions of the temperature, the displacement, the stress, the microrotation and the current density are obtained. The numerical values of these functions are represented graphically. 2014 Article Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating / M. Zakaria // Прикладная механика. — 2014. — Т. 50, № 1. — С. 130-144. — Бібліогр.: 25 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/100585 en Прикладная механика application/pdf Інститут механіки ім. С.П. Тимошенка НАН України |
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Досліджено задачу про двовимірне магнітне мікрополярне узагальнене термопружне середовище, у якому присутній комбінований ефект струму Холла і яке нагрівається за законом пилоподібного типу. У середовищі існує сильне магнітне поперечне поле, яке є перпендикулярним до площини деформування, і прийнято припущення про ігнорування наведеного електричного поля. Закон Ома модифіковано за допомогою включення двох доданків, один з яких є векторним добутком густини струму на початкове магнітне поле і другий є векторним добутком швидкості на початкове магнітне поле. Застосовано методи перетворення Лапласа і експоненціального перетворення Фур’є з метою зведення основних диференціальних рівнянь з частинними похідними до звичайних диференціальних рівнянь, які розв’язано точно. Проведено порівняння з роботами, опублікованими раніше, і результати оцінені як такі, що перебувають у добрій узгодженості. Отримано розподіли температури, зміщення, напруження, мікроповоротів і густини струму. Числові значення цих функцій представлено графічно. |
| format |
Article |
| author |
Zakaria M. |
| spellingShingle |
Zakaria M. Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating Прикладная механика |
| author_facet |
Zakaria M. |
| author_sort |
Zakaria M. |
| title |
Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating |
| title_short |
Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating |
| title_full |
Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating |
| title_fullStr |
Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating |
| title_full_unstemmed |
Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating |
| title_sort |
effect of hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating |
| publisher |
Інститут механіки ім. С.П. Тимошенка НАН України |
| publishDate |
2014 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/100585 |
| citation_txt |
Effect of Hall current on generalized magneto-thermoelasticity micro-polar solid subjected to ramp-type heating / M. Zakaria // Прикладная механика. — 2014. — Т. 50, № 1. — С. 130-144. — Бібліогр.: 25 назв. — англ. |
| series |
Прикладная механика |
| work_keys_str_mv |
AT zakariam effectofhallcurrentongeneralizedmagnetothermoelasticitymicropolarsolidsubjectedtoramptypeheating |
| first_indexed |
2025-12-01T18:02:30Z |
| last_indexed |
2025-12-01T18:02:30Z |
| _version_ |
1850329953201029120 |
| fulltext |
2014 ПРИКЛАДНАЯ МЕХАНИКА Том 50, № 1
130 ISSN0032–8243. Прикл. механика, 2014, 50, № 1
M . Z a k a r i a
EFFECT OF HALL CURRENT ON GENERALIZED
MAGNETO-THERMOELASTICITY MICROPOLAR SOLID
SUBJECTED TO RAMP-TYPE HEATING
Alexandria University,
21526 Alexandria, Egypt; e-mail: zakariandm@yahoo.com
Abstract. The problem of two dimensional magnetic micropolar generalized thermoe-
lastic medium, in the presence of the combined effect of Hall currents subjected to ramp-
type heating, is investigated. The medium is permeated by a strong transverse magnetic field
imposed perpendicularly to the displacement plane on the assumption of the induced electric
field is neglected. Ohm’s law is modified by the inclusion of two terms, one for the cross
product of the current density with the initial magnetic field and the second for the cross
product of the velocity with the initial magnetic field. Laplace and exponential Fourier
transform techniques are employed to transform the governing partial differential equations
to ODE which were solved exactly. Comparisons with previously published work have been
conducted and the results are found to be in good agreement. The distributions of the tem-
perature, the displacement, the stress, the microrotation and the current density are obtained.
The numerical values of these functions are represented graphically.
Keywords: Hall current, micropolar, thermoelasticity, ramp-type heating.
1. Introduction.
The linear theory of elasticity is of paramount importance in the stress analysis of steel,
which is the commonest engineering structural material. To a lesser extent, linear elasticity
describes the mechanical behavior of the other common solid materials, e.g. concrete, wood
and coal. However, the theory does not apply to the behavior of many of the new synthetic
materials of the elastomer and polymer type, e.g. polymethyl-methacrylate (Perspex), poly-
ethylene and polyvinyl chloride. The linear theory of micropolar elasticity is adequate to
represent the behavior of such materials. For ultrasonic waves, i.e. for the case of elastic
vibrations characterized by high frequencies and small wavelengths, the influence of the
body microstructure becomes significant. This influence of microstructure results in the de-
velopment of new type of waves, not found in the classical theory of elasticity. Metals,
polymers, composites, soils, rocks, concrete are typical media with microstructures. More
generally, most of the natural and manmade materials including engineering, geological and
biological media possess a microstructure. Eringen and Suhubi[1] and Suhubi and Eringen
[2] developed the nonlinear theory of micro-elastic solids. Later Eringen [3 – 5] developed a
theory for the special class of micro-elastic materials and called it the «linear theory of micropo-
lar elasticity». Under this theory, solids can undergo macro-deformations and macro-rotations.
The classical uncoupled theory of thermoelasticity predicts two phenomena not com-
patible with physical observations. First, the equation of heat conduction of this theory does
not contain any elastic terms, second, the heat equation is of a parabolic type, predicting
infinite speeds of propagation for heat waves.
Biot [6] introduced the theory of coupled thermoelasticity to overcome the first short-
coming. The governing equations for this theory are coupled, eliminating the first paradox
of the classical theory. However, both theories share the second shortcoming since the heat
equation for the coupled theory is also parabolic.
131
Two generalizations to the coupled theory were introduced. The first is due to Lord and
Shulman [7], who obtained a wave-type heat equation by postulating a new law of heat con-
duction to replace the classical Fourier’s law. Since the heat equation of this theory is of the
wave-type, it automatically ensures finite speeds of propagation for heat and elastic waves.
The remaining governing equations for this theory, namely, the equations of motion and
constitutive relations, remain the same as those for the coupled and the uncoupled theories.
The second generalization is known as the theory of thermoelasticity with two relaxation
times, or the theory of temperature-rate-dependent thermoelasticity, and was proposed by
Green and Lindsay [8]. It is based on a form of the entropy inequality proposed by Green
and Laws [9]. It does not violate Fourier’s law of heat conduction when the body under con-
sideration has a center of symmetry, and it is valid for both isotropic and anisotropic bodies.
Ezzat and Youssef [10] studied the generalized magneto-thermoelasticity in a perfectly con-
ducting medium.
The foundations of magnetoelasticity were presented by Knopoff [11] and Chadwick
[12] and developed by Kaliski and Petykiewicz [13]. An increasing attention is being de-
voted to the interaction between magnetic field and strain field in a thermoelastic solid due
to its many applications in the fields of geophysics, plasma physics and related topics. In all
papers quoted above it was assumed that the interactions between the two fields take place
by means of the Lorentz forces appearing in the equations of motion and by means of a term
entering Ohm’s law and describing the electric field produced by the velocity of a material
particle, moving in a magnetic field.
Many authors have considered the propagation of electromagneto-thermoelastic waves
in an electrically and thermally conducting solid. Paria [14] discussed the propagation of
plane magneto-thermoelastic waves in an isotropic unbounded medium under the influence
of a uniform thermal field and with a magnetic field acting transversely to the direction of
the propagation. Paria used the classical Fourier law of heat conduction, and neglected the
electric displacement. Wilson [15] extended Paria’s results by introducing a component of
the magnetic field parallel to the direction of the propagation. A comprehensive review of
the earlier contributions to the subject can be found in Paria [16]. Among the authors who
considered the generalized magneto-thermoelastic equations are Nayfeh and Namat – Nas-
ser [17] who studied the propagation of plane waves in a solid under the influence of an
electromagnetic field. They obtained the governing equations in the general case and the
solution for some particular cases. Choudhuri [18] extended these results to rotating media.
Sherief and Ezzat [19] solved a thermal shock half-space problem using asymptotic expan-
sions. Lately, Rajneesh Kumar and Rupender [20] studied the effect of rotation in magneto-
micropolar thermoelastic medium due to mechanical and thermal sources, Rajneesh Kumara
and ManjeetSingh [21] solved the effect of rotation and imperfection on reflection and
transmission of plane waves in anisotropic generalized thermoelastic media, R. Kumar,
Praveen Ailawalia [22] studied moving load response in micropolar thermoelastic medium
without energy dissipation possessing cubic symmetry, few attempts have been made to
solve two-dimensional problems in this field Moncef Aouadi [23] introduced temperature
dependence of an elastic modulus in generalized linear micropolar thermoelasticity.
When the magnetic field is very strong, the conductivity will be a tensor and the effect
of Hall current cannot be neglected. The conductivity normal to the magnetic field is re-
duced due to the free spiraling of electrons and ions about the magnetic lines of force before
suffering collisions and a current is induced in a direction normal to both the electric and
magnetic fields. This phenomenon is called the Hall effect. In all of the above investiga-
tions, the effects of Hall current have not been considered.
The objective of this paper is to consider two dimensional micropolar generalized mag-
neto-thermoelasticity medium, in the presence of a uniform strong magnetic field acts in y-
direction taken into consideration the Hall current effects. This new model is applied to gen-
eralization, Lord – Shulman theory and Green – Lindsay theory, as well as to the coupled
theory and solved by using Laplace – Fourier transform technique. The formulas of tem-
perature, displacement, stresses, the microrotation and the current density are obtained. The
ramp-type heating application is employed to our problem to get the solution in the com-
plete form. The considered variables are presented graphically and comparisons and discus-
sions are made.
132
2. Basic equations.
We shall consider a generalized magneto-thermoelasticity micropolar solid occupying
the region , , ,x o y z . The governing equations of motion for
homogenous, isotropic and elastic solid, when the Hall current effect is taken into account
consists of:
I. The equations of motion taking into account the Lorentz force have the form
, , ;j ij o ijr j r i ttJ H u (1)
, ,= ,ijr jr ji j i ttm j (2)
where ij is the force stress tensor, ui is the components of the displacement tensor, i is
the components of the microrotation tensor, ijm is the couple stress tensor, is the density,
j is the microinertia, ijr is the permutation symbol, µo is the magnetic permeability, rH is
the intensity tensor of magnetic field and rJ is the conduction current density.
The above equations are supplemented by the modified Ohm’s law for media with finite
conductivity and including the Hall current effect
, ,o
i o i ijr j t r i k k
e
J E u H J H
en
(3)
where 2( ),o e e en e t m is the electrical conductivity, e is the charge of an electron, en is
the electron number density, et is the electron collision time, em is the mass of the electron
and iE is the intensity tensor of the electric field.
II. The constitutive equation
, , , , 1 ,( ) ( ) ( ) ;ij r r ij i j j i j i ijr r t iju u u k u T T (4)
, , , ,ij r r ij i j j im (5)
where , are Lame’s constants, , , , and k are micropolar constants, T is the absolute
temperature of the medium, is the constant with dimension of time, called relaxation
time, 1 is the material constant given by 1 (3 2 ) ,TK and T being a coefficient
of linear thermal expansion.
III. The generalized equation of heat condition
, , 1 , ,( ) ( ),jj E t o t o i it o o i ittK T c T T T u n u (6)
where K is thermal conductivity, Ec is the specific heat at constant temperature, oT is the
uniform temperature, o is an other relaxation time and on is a non-dimensional constant.
For L S theory 0 1, 0, 0,on and for G L theory 0 = 0, > 0, 0.on
Substituting equations (4), and (5) into equation (5), and (6) we get
, , , 1 , , ,( ) ( ) ( ) ;j ji i jj ijr r j i it o rji r j i ttu k u k T T J H u (7)
, , , ,( ) 2 .j ji i jj ijr r j i i ttk u k j (8)
3. Formulation and solution of the problem.
The rectangular Cartesian system , , x y z having origin on the surface 0x with
x -axis vertical into the medium is introduction. In the following, we restrict our analysis
parallel to xz -plane with 0, ( , 0, ),ix u u w and (0, , 0).i A uniform very strong
magnetic field of strength Ho is assumed to be applied in the positive y-direction, we also
133
assume that 0.E Under these assumptions the modified Ohm’s law gives 0yJ every-
where in the medium. The current density components xJ and zJ are given by
2 ;
1
o o o
x
H u w
J m
m t t
(9)
2 ,
1
o o o
z
H u w
J m
m t t
(10)
where ( )e em t is the Hall parameter and ( )e o o ee µ H m is the electron frequency.
For simplifications we shall use the following non-dimensional variables:
* * * * * *
1 1
2
* * * * *
2
1 1 1
; ; ; ; ; ;
; ; ; ; ; ,
o o o o o o
o o o o
o o o o
ij o o o o
o ij ij ij
o o o o o o o
c c
x x z z u u w w t t
c c T T
B cT
T m m M
T T c T c T
where 2 ; ,o o Ec K c k and M is the Hartmann number or magnetic parameter.
The system of equations (6) – (8), using relations (9), and (10), takes the following form
– dropping the asterisks for convenience:
2
2
2 2 2 2 21 ;
1o o o
u e k k T M u w
u m
t c x c c z t x m t t
2
2
2 2 2 2 21 ;
1o o o
w e k k T M u w
w m
t c z c c x t z m t t
2
2
2 ( 2 ) ;
j u w
k k
t z x
(11)
22 2
2 1
2 2 .o
o o o
o
TT u w
T n
t t K t t x z
( 2 ) 1 ;xx
u w
k T
x z t
( 2 ) 1 ;zz
w u
k T
z x t
(12)
1 ;yy
u w
T
x z t
( ) ;xz
w u
k
x z
;xym
x
.zym
z
Introducing potential functions ( , , ),x z t and ( , , )x z t defined by
2 2; ; ; .
u w
u w e
x z z x z x
(13)
Substituting Eq. (13) into Eqs. (11) we obtain
134
2
2
2 2 2 2
2
1 ;
1 1o
k M M m
T
c t m t t m t
2
2
2 2 2 2 2 ;
1 1o o
k M k M m
c t m t c m t
2
2 2
2 2 ;
j
k k
t
(14)
22 2
2 21
2 2 .o
o o
o
T
T n
t t K t t
Taking Laplace transform defined by the relation
0
( ) ( ) ,stf s e f t dt
of both sides of
(12) and (14) using the homogenous initial conditions, we obtain
2 2
2 2 2
2
1 ;
1 1o
o
k sM M ms
s s T
c m m
2 2
2 2 2 2 ;
1 1o
o o
k Ms k M ms
s
c m c m
2 2 22 ;
j
s k k
2
2 211 1 ;o
o o o
o
T
s s T s n s
K
2 1 ;xx
u w
k s T
x z
(15)
2 1 .zz
w u
k s T
z x
1 ;yy
u w
s T
x z
(16)
;xz
w u
k
x z
(17)
;xym
x
.zym
z
(18)
Now we use the Fourier transform with respect to the space variable ,z defined by
* 1
( ) ( ) .
2
iqzf q e f z dy
Taking the Fourier transform of both sides of (15) – (17)
2 * * *
1 2 3( ) ;D T (19)
2 * * *
4 5 6( ) ;D (20)
135
2 * 2 2 *
7( ) ( ) ;D k D q (21)
2 * 2 2 *
8 9( ) ( ) ;D T D q (22)
* * * *( 2 ) (1 ) ;xx k Du iq w s T
* * * *( 2 ) (1 ) ;zz iq k w Du s T
* * * *( ) (1 ) ;yy Du iqw s T (23)
* * *.xz Dw iq k u
* *;xym D (24)
* *,zym i q (25)
where
2 2
2 2
1 22
(1 )
; ;
2 1 2
o o
o
c s c sMs
s q
k m k
2
3 2 ;
( 2 )(1 )
oMms c
k m
2
2 2
4 2 ;
1
o
o
c sM
s q
k m
2
5 6 2; ;
( )(1 )
os c Mmsk
k k m
2
2 2 2 1
7 8 92 ; (1 ) ; (1 )o
o o o
o
Tj
s k q s s q s n s
K
and the operator D means the derivative with respect to x .
On Eqs. (19) – (22) and after some simplification, we arrive at the following eighth or-
der differential equation satisfied by * * * *, , and :T
8 6 4 2 * * * *( , , , ) 0,D AD BD CD F T (26)
where
1 2 9 4 5 7 8;A k
2 2
1 8 2 9 4 7 5 1 8 2 9 4 7 5 ;B q q k
2 2
4 7 5 1 8 2 9 1 8 2 9 4 7 5( )( ) ( )( );C q q k
2 2
1 8 2 9 4 7 5( )( ).F q q
The solution of the Eq. (26) satisfying the radiation conditions that * * *, , and *T
tend to zero as x tends to infinity can be written as
4
* * * *
1 2 3
1
( , , , )( , , ) (1, , , ) ,jk x
j j j j
j
T x q s A e
(27)
where
2 2 2 2
2 9 1 4 8 1 1
1 2
3 4 8
( ) ( )( )
( 1, 2, 3, 4);
( )j
k q k k
j
k
136
2 2 2 2 2 2 2
4 2 9 8 1 96
2 32 2
3 5 8 5 4 8
( ) ( ) ( )( ) ( )
;
( ) ( )
j j j j j
j j
j
k k q k k k q
k k
аnd 1, 2, 3, 4jk i are the characteristic roots of the characteristic equation (21) which is
8 6 4 2 0.k Ak Bk Ck F
Eq. (13) together with Eq. (27), immediately give
4
*
1
1
( , , ) ( ) ;jk x
j j j
j
u x q s iq k A e
(28)
4
*
1
1
( , , ) ( ) .jk x
j j j
j
w x q s iq k A e
(29)
In order to get the components of stresses, substituting from equations (27) – (29) into
equations (23) we get
4
* 2 2
1 3
1
( ) (2 )( ) (1 ) ;jk x
xx j j j j j j
j
k q k k iq k s A e
(30)
4
* 2 2
1 3
1
(2 )( ) ( ) (1 ) ;jk x
zz j j j j j
j
iq k iq k k q s A e
(31)
4
* 2 2
3
1
( ) (1 ) ;jk x
yy j j j
j
k q s A e
(32)
4
*
1 1
1
( )( ) ( ) .jk x
xz j j j j j j
j
iq k iq k k iq k A e
(33)
Moreover, the components of couple stresses can be obtained in a similar manner by
substituting from (27) into (24) and (25) we get
4
*
2
1
;jk x
xy j j j
j
m k A e
(34)
4
*
2
1
.jk x
zy j j
j
m i q A e
(35)
To obtain the components of current density, substituting from equations (28) and (29)
into equations (9) and (10) and after taking Laplace – Fourier transform we get
4
12
1
;
1
jk x
x j j j j
j
Ms
J iqm k mk iq A e
m
(36)
4
12
1
.
1
jk x
z j j j j
j
Ms
J iq mk miq k A e
m
(37)
We shall now go to the applications of the problem to evaluate the unknown parameters
, 1, 2, 3, 4.jA g
4. Application.
We consider a micropolar thermoelastic material occupying the semi-space region
, , ,x o y z , let the surface of is traction free and subjected to
ramp-type heating,
137
a- Thermal boundary condition.
The boundary of the half-space is affected by ramp-type heating, which depends on the
coordinate z and the time t of the form
(0, , ) ( ) ( ),T z t G t F z (38)
where F(z) is an arbitrary function of z and ( )G t is a function defined as Misra et al., [25]
1
1
0 0;
( ) ( ) 0 0;
0;
o
t
G t T t t t
T t
(39)
where to indicates the length of time to rise the heat and T1 is constant, this means that the
boundary of the half-space, which is initially at rest and has a fixed temperature ,oT is sud-
denly raised to a temperature equal to the function ( ) ( )G t F z and maintained at this tem-
perature from then on.
Applying the Laplace and Fourier transforms to both sides of Eq. (38), we obtain
* *
1 2
0
(1 )
(0, , ) ( ) ( ) ( ) .
oste
T q s G s F q G s T
t s
(40)
b- Mechanical boundary conditions.
Under the assumption that the surface of Ω is traction free we can get the following
conditions:
(0, , ) (0, , ) 0.xx xz xyz t z t m (41)
Applying the Laplace and Fourier transforms, we get
* * *(0, , ) (0, , ) 0.xx xz xyz t z t m (42)
By substituting from equations (40) and (41) into equations (27), (30), (33) and (35), the
boundary conditions may be reduced to
4
*
3
1
( ) ( ) ,j j
j
G s F q A
(43)
4
2 2
1 3
1
0 ( ) (2 )( ) (1 ) ;j j j j j j
j
k q k k iq k s A
(44)
4
1 1
1
0 ( )( ) ( ) ;j j j j j j
j
iq k iq k k iq k A
(45)
4
2
1
0 .j j
j
A
(46)
By solving this system of equations for ( 1, 2, 3, 4),jA j we can complete the solution
of our problem.
5. Particular cases.
I. If 0,m in Eqs. (27) – (37), we obtain the components of displacements and tresses
in magneto-micropolar generalized thermoelastic without Hall current effect.
II. Taking Hartmann number 0,M in Eqs. (27) – (37), we obtain the components of
displacements and tresses in micropolar generalized thermoelastic.
138
III. Taking 0 1, 0, 0on in Eqs. (27) – (37), we obtain the corresponding expres-
sions for magneto-micropolar thermoelastic for –L S theory.
IV. Taking 0 0, 0, 0on in Eqs. (27) – (37), we obtain the corresponding expres-
sions for magneto-micropolar thermoelastic for G L theory.
V. Taking 0 0, 0, 0on we obtain the corresponding expressions for magneto-
micropolar coupled thermoelastic medium for C T theory.
6. Inversion of the transforms.
To obtain the solution of the problem in the physical domain ( , , ),x y t we have to in-
vert the iterated transforms in Eqs. (27) – (37). These expressions can be formally expressed
as functions of x and the parameter of the Fourier and Laplace transforms q and s of the
form *( , , )f x q s .
First, we invert the Fourier transform using the inversion formula given previously. This
gives the Laplace transform expression ( , , )f x q s of the function ( , , )f x y t as
1 1
( , , ) ( , , ) cos( ) sin( ) ,
2 2
iqy
e of x q t e f x q s dq qy f i qy f dq
where ef and of denote the even and the odd parts of the function *( , , )f x q s respectively.
We shall outline the numerical inversion method used to find the solution in the physi-
cal domain. For fixed values of , ,x y and q the function inside braces in the last integral can
be considered as a Laplace transform ( )g s of some function ( ).g t
The inversion formula for the Laplace transform can be written as
1
( ) ( ) ,
2
c i
st
c i
g t e g s ds
i
where c is an arbitrary real number greater than all the parts of the singularities ( )g s . Taking
,s c iy the above integral takes the form
( ) ( ) .
2
ct
iyte
g t e g c iy dy
Expanding the function ( ) exp( ) ( )h t ct g t in a Fourier series in the interval [0, 2L],
we obtain the approximate formula
( ) ( ) ,Dg t g t E
where
1
1
( ) for 0 2
2 o k
k
g t c c t L
(47)
and
Re ,
ct
ik t L
k
e
c e g c i k t L
L
(48)
,DE the discretization error, can be made arbitrary small by choosing c large enough.
Since the infinite series in Eq. (47) can be summed up to finite number N of terms, the
approximate value of ( )g t becomes
1
1
( ) for 0 2 .
2
N
N o k
k
g t c c t L
(49)
139
Using the above formula to evaluate ( ),g t we introduce a truncation error ET which,
must be added to the discretization error to produce the total approximation error.
Two methods are used to reduce the total error. First the «Korrecktur» method is used to
reduce the discretization error. Next, the e algorithm is used to reduce the truncation error
and hence to accelerate convergence.
The Korrecktur method uses the following formula to evaluate the function ( ) :g t
2( ) ( ) (2 ) ,cL
Dg t g t e g L t E
where the discretization error D DE E . Thus, the approximate value of ( )g t becomes
2( ) ( ) (2 );cL
NK N Ng t g t e g L t (50)
'N is an integer such that ' .N N
We shall now describe the e -algorithm that is used to accelerate the convergence of the
series in Eq. (49).
Let 2 1N q where q is a natural number, and let
1
m
m k
k
s c
be the sequence of partial sums of Eq. (49). We define the e-sequence by
, 1,0; 0o m m and 1, 1
1,
, 1 ,
1
( 1, 2, 3, . . . )p m
p m
p m p m
p
it can be shown that the sequence 1,1 3,1 5,1 ,1, , , ... , ,N converges to ( , , ) – 2of x y t ED c
faster than the sequence of partial sums ( 1, 2, 3, ...).ms m
The actual procedure used to invert the Laplace transform consists of using Eq. (50) to-
gether with the -algorithm. The values of c and L are chosen according to the criteria
outlined Honig and Hirdes [24].
7. Numerical results and discussion.
The analysis is conducted for a magnesium crystal-like material. Following reference
[25], the values of physical constants are
10 2 10 2 10 2 9,4 10 Nm ; 4,0 10 Nm ; 1,0 10 Nm ;k
3 3 9 15 21,74 10 gm/cm ; 0,779 10 N; 0,2 10 cm ;j
3 3 * 2 1 1 11,04 10 kgm ; 1,7 10 Jm s deg ; 298 K;E oc K T
6 21 1 3,68 10 Nm deg ; 0,02; 0,05,o on
we consider the following electric constants for our problem
5 2 8 9,36 10 Col /Cal.cm.sec; 10 Col/cm.sec.o oH
The computations are carried out for the non-dimensional time 0,2,t strip width
30,9 10a and on the surface plane 0.y The distribution of non-dimensional tempera-
ture ,T non-dimensional normal displacement u , non-dimensional transverse displacement
,w non-dimensional micro-rotation , non-dimensional normal force stress ,xx non-
dimensional shearing stresses ,xz non-dimensional couple stress ,xym non-dimensional
normal conduction current density field xJ and non-dimensional transverse conduction cur-
140
rent density field ,zJ with non-dimensional distance x have been shown in Figs. (1 – 9).
Numerical analysis has been carried out by taking x range from 0 to 3. These figures are
obtained to illustrate the relative influence of the parameter ,ot magnetic field parameter
(Hartmann number) M and Hall parameter m.
The solid line represents magneto-micropolar generalized thermoelastic with Hall current
effect MMGTH when 0,2, 0,6,M m and large dashes line represents magneto-micropolar
generalized thermoelastic without Hall current effect MMGT when 0,2, 0,M m while
small dotes line represents micropolar generalized thermoelastic MGT for 0, 0.M m
The field quantities, temperature, displacement, force stress, couple stress, and current
density fields do not depend only on the state and space variables ,t x and ,z but also de-
pend on Hall parameter .m It has been observed that, the finite Hall parameter m plays a
vital role on the development of temperature, displacement force stress, couple stress, and
current density fields. The numerical values for the field temperature are computed for a
wide range of values of finite pulse rise-time ot in the two situations ot t and ot t .
Fig. 1exhibits the space variation of temperature at 0,2t for different values of to and
m for the threemodels; magneto-micropolar generalized thermoelastic with Hall current
effect MMGTH , magneto-micropolar generalized thermoelastic without Hall current effect
MMGT and micropolar generalized thermoelastic MGT in which we observe when ot t in
some range of x we have great different values of temperature for the three models in com-
paring to the case ot t the three models. From this figure we see that the all curves start
from two values one at 0,52 when 0,1ot and the other 0,32 when 0,1ot then decreases
till it tends to zero at 3,x we observe that from this figure the temperature T increases in
the case MMGTH in comparison to curve MGT, while decreases in comparison to curves
MMGT.
Fig. 1. Variation of temperature distribution T with distance .x
The effect of Hall parameter m on the normal displacement u against x is displayed in
Fig. 2. It is noticed from the graphic representation of the normal displacement is started
from point 0,123x for all the cases and then increases up to the point 0,7x then de-
creases till it tends to zero at 3.x From this figure we observe that an increase in Hall
parameter m leads to a decrease in the displacement u distribution because the aim of this
latter parameter is to produce finite speed of displacement .u
141
Fig. 2. Variation of normal displacement u with distance .x
Fig. 3 illustrates the influence of the Hall parameter m and Hartmann number M on the
transverse displacement w in the medium of generalized magneto-thermoelasticity micropolar
solid. Application of magnetic field to MMGTH gives rise to a resistive type force called the
Lorentz force. This force has the tendency to slow down the transverse displacement w in the
medium. Also, the effects on w become more as the strength of the magnetic field increases,
also we record values higher in case MMGTH as compared to the values for case MMGT. It
is observed from the decreases in the transverse displacement we presented in Fig. 3.
Fig. 3. Variation of transverse displacement w with distance .x
Fig. 4 shows that micro-rotation φ profiles across the medium for different values of the
M and m parameters. The models MMGTH, MMGT and MMGT are representation corre-
sponding to = 0,6; 1,5; 0,6; 0M m M m and 0M m respectively. It is observed
that an increase in Hall parameter m leads to increase in the micro-rotation distribution.
These behaviors are clear from Fig. 4.
142
Fig. 4. Variation of micro-rotation with distance .x
Figs. 5, 6 describe the variation of the normal stresses xx and one of the shearing
stresses ,xz the influence of the Hall parameter m affects on the normal stresses by de-
creasing their values and on the shearing stresses by increasing their values up to the point
0,26x then opposite the direction of increasing.
Fig. 5. Variation of normal stresses xx with distance .x
Fig. 6. Variation of shearing stresses xz with distance .x
143
Fig. 7 shows that the Variation of couple stress yzm increase in the case MMGT in
comparison to curve MMGT. The figure indicates that yzm records values higher in case
MMGTH as compared to the values of case MGT.
Fig. 7. Variation of couple stresses xym with distance .x
Figs. 8, 9 describe the variation of two components of density component xJ and zJ
respectively; it is evident that the values of both fields are decreased in the MMGTH as
compared to the values of the cases MMGT and MGT.
Fig. 8. Variation of normal conduction current density field xJ with distance .x
Fig. 9. Variation of transverse conduction current density field zJ with distance .x
144
Р Е ЗЮМ Е . Досліджено задачу про двовимірне магнітне мікрополярне узагальнене термопружне
середовище, у якому присутній комбінований ефект струму Холла і яке нагрівається за законом пило-
подібного типу. У середовищі існує сильне магнітне поперечне поле, яке є перпендикулярним до пло-
щини деформування, і прийнято припущення про ігнорування наведеного електричного поля. Закон
Ома модифіковано за допомогою включення двох доданків, один з яких є векторним добутком густини
струму на початкове магнітне поле і другий є векторним добутком швидкості на початкове магнітне
поле. Застосовано методи перетворення Лапласа і експоненціального перетворення Фур’є з метою зве-
дення основних диференціальних рівнянь з частинними похідними до звичайних диференціальних рів-
нянь, які розв’язано точно. Проведено порівняння з роботами, опублікованими раніше, і результати
оцінені як такі, що перебувають у добрій узгодженості. Отримано розподіли температури, зміщення,
напруження, мікроповоротів і густини струму. Числові значення цих функцій представлено графічно.
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From the Editorial Board: The article corresponds completely to submitted manuscript.
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