The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer
An effect of three factors - the order of fractional derivative, the hydrostatic initial stress, and parameter of magnetic field – on the plane waves in the half-space made of fiber-reinforced material, that is described by the theory of generalized magnetothermoelasticity, is studied. The problem i...
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irk-123456789-1410492018-07-22T01:23:43Z The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer Sarkar, N. Atwa, S.Y. Othman, M.I.A. An effect of three factors - the order of fractional derivative, the hydrostatic initial stress, and parameter of magnetic field – on the plane waves in the half-space made of fiber-reinforced material, that is described by the theory of generalized magnetothermoelasticity, is studied. The problem is solved numerically using the normal mode analysis. The results correspond to the Lord-Shulman model and the model, that uses the fractional derivatives and are presented in the form of graphs. The findings show pronounced effect of mentioned three factors. The results are compared with the case, when the initial stress and magnetic field are absent. Вивчено вплив трьох факторів – порядку дробових похідних, початкового гідростатичного напруження, параметру магнітного поля – на плоскі хвилі в півпросторі з армованого волокнами матеріалу, який описується теорією узагальненої магнітотермопружності. Задача розв’язана чисельно за допомогою аналізу нормальних мод. Результати аналізу відповідають моделі Лорда – Шульмана і моделі, що описується за допомогою дробових похідних, і представлені у вигляді графіків. Отримані результати показують добре виражений вплив вказаних трьох факторів. Також ці результати порівнюються з результатами, що отримані для випадку відсутності початкового гідростатичного напруження і магнітного поля. 2016 Article The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer / N. Sarkar, S.Y. Atwa, M.I.A. Othman // Прикладная механика. — 2016. — Т. 52, № 2. — С. 126-143. — Бібліогр.: 34 назв. — англ. 0032-8243 http://dspace.nbuv.gov.ua/handle/123456789/141049 en Прикладная механика Інститут механіки ім. С.П. Тимошенка НАН України |
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An effect of three factors - the order of fractional derivative, the hydrostatic initial stress, and parameter of magnetic field – on the plane waves in the half-space made of fiber-reinforced material, that is described by the theory of generalized magnetothermoelasticity, is studied. The problem is solved numerically using the normal mode analysis. The results correspond to the Lord-Shulman model and the model, that uses the fractional derivatives and are presented in the form of graphs. The findings show pronounced effect of mentioned three factors. The results are compared with the case, when the initial stress and magnetic field are absent. |
| format |
Article |
| author |
Sarkar, N. Atwa, S.Y. Othman, M.I.A. |
| spellingShingle |
Sarkar, N. Atwa, S.Y. Othman, M.I.A. The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer Прикладная механика |
| author_facet |
Sarkar, N. Atwa, S.Y. Othman, M.I.A. |
| author_sort |
Sarkar, N. |
| title |
The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer |
| title_short |
The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer |
| title_full |
The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer |
| title_fullStr |
The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer |
| title_full_unstemmed |
The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer |
| title_sort |
effect of hydrostatic initial stress on the plane waves in a fiber-reinforced magneto-thermoelastic medium with fractional derivative heat transfer |
| publisher |
Інститут механіки ім. С.П. Тимошенка НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/141049 |
| citation_txt |
The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer / N. Sarkar, S.Y. Atwa, M.I.A. Othman // Прикладная механика. — 2016. — Т. 52, № 2. — С. 126-143. — Бібліогр.: 34 назв. — англ. |
| series |
Прикладная механика |
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2025-07-10T11:50:12Z |
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2025-07-10T11:50:12Z |
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2016 ПРИКЛАДНАЯ МЕХАНИКА Том 52, № 2
126 ISSN0032–8243. Прикл. механика, 2016, 52, №2
N . S a r k a r 1 , S . Y . A t w a 2 , M . I . A . O t h m a n 3
THE EFFECT OF HYDROSTATIC INITIAL STRESS ON THE PLANE WAVES
IN A FIBER-REINFORCED MAGNETO-THERMOELASTIC MEDIUM
WITH FRACTIONAL DERIVATIVE HEAT TRANSFER
1Department of Mathematics, The University of Burdwan, West Bengal, India.
2Higher Institute of Engineer, Department of Engineering Mathematics and Physics,
Shorouk Academy, Egypt.
3Department of Mathematics, Faculty of Science, Zagazig University, Egypt.
Corresponding author e-mail: nantu.math@gmail.com (N. Sarkar)
Abstract. An effect of three factors - the order of fractional derivative, the hydrostatic
initial stress, and parameter of magnetic field – on the plane waves in the half-space made of
fiber-reinforced material, that is described by the theory of generalized magneto-
thermoelasticity, is studied. The problem is solved numerically using the normal mode
analysis. The results correspond to the Lord-Shulman model and the model, that uses the
fractional derivatives and are presented in the form of graphs. The findings show pro-
nounced effect of mentioned three factors. The results are compared with the case, when the
initial stress and magnetic field are absent.
Key words: generalized magneto-thermoelasticity, fractional derivative, fiber rein-
forced material, hydrostatic initial stress, normal mode analysis.
1. Introduction.
During recent years, by applying the fractional calculus several interesting models have
been established successfully to study the physical processes particularly in the area of me-
chanics of solids, control theory, electricity, heat conduction, diffusion problems, visco-
elasticity etc. It has been verified/examined that the use of fractional order deriva-
tives/integrals lead to the formulation of certain physical problem which is more economical
and useful than the classical approach. There are some materials (e.g. porous materials, bio-
logical materials/polymers and colloids, glassy etc.) and physical situations (like low-
temperature, amorphous media and transient loading etc.) where the conventional coupled
dynamical theory (CD) [1] based on the classical Fourier’s law is unsuitable (see [2] for
details). In such cases, one needs to use a generalized thermoelastic (and more generally
thermo-viscoelastic) model based on an anomalous heat conduction theory involving frac-
tional time-derivatives; see Ignaczak & Ostoja-Starzewski [3]. Recently, fractional calculus
has also been employed in the area of thermoelasticity. Povstenko [4] has constructed a
quasi-static uncoupled thermoelasticity model based on the heat conduction equation with
fractional order time derivatives. He has used the Caputo fractional derivative (see [5] for
details) and obtained the stress components corresponding to the fundamental solution of a
Cauchy problem for the fractional order heat conduction equation in both the one-
dimensional and two-dimensional cases. In 2010, a new theory of generalized thermoelastic-
ity in the context of a new consideration of the heat conduction equation with fractional or-
der time derivatives has been proposed by Youssef [6].
127
The uniqueness of the solution has also been proved in the same work. Youssef & Al-
Lehaibi [7] have studied a problem on an elastic half space using this theory. Sherief et al.
[8] and Ezzat & Fayik [9] have also constructed some model in generalized thermoelasticity
by using fractional time-derivatives.
Fiber-reinforced composites are widely used in engineering structures, due to their supe-
riority over the structural materials in applications requiring high strength and stiffness in
lightweight components. A continuum model is used to explain the mechanical properties of
such materials. A reinforced concrete member should be designed for all conditions of
stresses that may occur and in accordance with the principles of mechanics. The characteris-
tic property of a reinforced concrete member is that its components, namely concrete and
steel, act together as a single unit as long as they remain in the elastic condition, i.e., the two
components are bound together so that there can be no relative displacement between them.
In the linear case, the associated constitutive relations, relating infinitesimal stress and
strain components, have five material constants. In the last three decades, the analysis of
stress and deformation of fiber-reinforced composite materials has been an important re-
search area of solid mechanics. Belfield et al. [10] has introduced the idea of continuous
self-reinforcement at every point of an elastic solid. Spencer [11], Pipkin [12] and Rogers
[13, 14] have done pioneering works on this subject. Fibers are assumed as an inherent ma-
terial property, rather than some form of inclusion in such models, see [11] for details. One
can find some work on transversely isotropic elasticity in the literatures [15-19].
The study of the magneto-thermoelastic interactions which deals with the interactions
among the strain, temperature and the electromagnetic field in an elastic solid is of great
practical importance due to its extensive uses in diverse field, such as geophysics (for under-
standing the effect of the Earth’s magnetic field on seismic waves), damping of acoustic
waves in a magnetic field, designing machine elements like heat exchangers, boiler tubes
where the temperature induced elastic deformation occurs, biomedical engineering (prob-
lems involving thermal stress), emissions of the electromagnetic radiations from nuclear
devices, development of a highly sensitive super conducting magnetometer, electrical power
engineering, plasma physics etc. [20, 21]. Many works in generalized magneto-
thermoelasticity can be found in the literatures Sarkar & Lahiri [22], Abbas et al. [23], Ezzat
& Youssef [24, 25], Youssef [26], Ezzat & Abd Elall [27] and Xion & Tian [28].
The aim of the present paper is to investigate the influences of fractional order, hydro-
static initial stress and the magnetic field on the plane waves in a fiber-reinforced general-
ized thermoelastic solid half-space. The problem has been solved numerically using the
normal mode analysis [23, 29 – 32]. Numerical results for the temperature, displacement
components and the stresses are represented graphically for the Lord-Shulman (LS) and
farctional order (FO) model of generalized thermoelasticity and analyze the results. The
graphical results indicate that the effect of fractional order, hydrostatic initial stress and
magnetic field on plane waves are very pronounced. Comparisons are made with the results
in the absence of the hydrostatic initial stress and the magnetic field.
2. Formulation of the Problem and Basic Equations.
We consider the problem of a fiber-reinforced generalized thermoelastic half-space
( 0x ). A magnetic field with a constant intensity 00,0,H H
acts parallel to the bound-
ary plane (taken as the direction of the z-axis). We begin our consideration with linearized
equations of electro-dynamics of a slowly moving medium [23]:
0J h E
, (1)
0E h
, (2)
0E u H
, (3)
128
0h
. (4)
The above equations are supplemented by the displacement equations of the theory of
elasticity, taking into consideration the Lorentz force to give
,ij j i iF u , (5)
0i i
F J H
. (6)
The constitutive relations for a fiber-reinforced linearly thermoelastic isotropic medium
with respect to the reinforcement direction a with an initial hydrostatic stress and without
body forces and heat sources are given by Lord & Shulman [33], Montanaro [34] and Singh
[16] as
0
2
2 ,
ij ij ij kk ij T ij k m km ij i j kk
L T i k ik j k k j k m km i j ij
P e e a a e a a e
a a e a a e a a e a a T T
(7)
, ,
1
,
2ij i j j ie u u (8)
, ,
1
2ij i j j iu u . (9)
The heat conduction equation with fractional derivative heat transfer heat transfer pro-
posed by Ezzat & Fayik [9] is
2 0
01 , 0 1,
! Ek T C T T e
t t
(10)
where 2 2 2
1 2 3 1 2 3( , , ), 1a a a a a a a
and
1
( , ) ( ,0) when 0,
( , )
( , ) : when 0 1,
( , )
when 1.
f x t f x
f x t
f x t I
tt
f x t
t
In the above definition, the Riemann–Liouville fractional integral operator I is defined
as
1
0
1
[ ( , )] : ( ) ( , ) ,
( )
t
I f x t t s f x s ds
where (...) is the well-known Gamma function. The comma notation is used for spatial
derivatives and superimposed dot represents time differentiation.
For plane strain deformation in the xy-plane, all the considered functions will be depend
on the time t and the coordinates x and y and the displacement vector u
will have the com-
ponents
( , , ), ( , , ), 0.x y zu u u x y t v u v x y t w u (11)
We choose the fiber-direction as (1,0,0)a
so that the preferred direction is the x-axis,
and Eq. (5)-(7) simplify, as given below,
129
02 4 2 ,xx L T
u v
P T T
x y
(12)
02 ,yy T
u v
P T T
x y
(13)
,
2 2xy L L
P v P u
x y
(14)
,
2 2yx L L
P v P u
x y
(15)
2 2 2
2
0 0 0 02 2
,x
u v u
F H
x yx t
(16)
2 2 2
2
0 0 0 02 2
,y
v u v
F H
x yy t
(17)
22 2 2 2
2 2
11 122 2 2 2
1 ,
2 2
H
H L H L
Ru P v P u T u
A R A R
x y xx y c t
(18)
22 2 2 2
2 2
22 122 2 2 2
1 .
2 2
H
H L H L
Rv P u P v T v
A R A R
x y yy x c t
(19)
To transform the above equations in non-dimensional forms, we will use the following
non-dimensional variables
2
1 0 1 0( ', ', ', ') ( , , , ), ( ', ') ( , ),x y u v c x y u v t c t
0
2 2
01 1 0
' , , ' , .ij E
ij
T T ch
h
H kc c
Using the above non-dimensional variables, Eqs. (12) – (15), (18), (19) and (10) take
the following forms (omitting the primes for convenience)
12 ,xx P
u v
R B
x y
(20)
1 22 ,yy P
u v
R B B
x y
(21)
3 3 ,xy P P
v u
B R B R
x y
(22)
3 3 ,yx P P
v u
B R B R
x y
(23)
2 2 2 2
1 1 3 1 3 22 2 2
1 ,P P
u v u u
M B B R M B R M
x y xx y t
(24)
2 2 2 2
2 1 1 3 1 3 22 2 2
,P P
v u v v
B M B B R M B R M
x y yy x t
(25)
2 0
01 ,
!
e
t tt
(26)
where
2 2
1 2 3 11 22 1 22 2
1 111
1
, , , , , , 1 , .
2
H H
L P
R R P
B B B A A M M R
A Ac c
130
3. Normal mode analysis.
The solution of the physical quantities can be decomposed in terms of normal modes in
the following form:
, , , , ( . . ) , , , , ( )expij iju v e x y t u v e x t imy , (27)
where *( )u x etc. are the amplitude of the function u(x, y, t) etc., i is the imaginary unit,
(complex) is the time constant and m is the wave number in the y-direction.
By using (27), we can obtain the following equations from (24)-(26) respectively:
2
41 45 46
2
54 52 53
2
64 62 63
( ) ( ) ( ) 0,
( ) ( ) ( ) 0,
( ) ( ) ( ) 0,
D C u x C Dv x C D x
C Du x D C v x C x
C Du x C v x D C x
(28)
where
2 2
3 2 1 3 1
41 45 46
1 1 1
1
, , ,
1 1 1
P P
m B R M im R B B M
C C C
M M M
2 2
2 1 2 1 3 1
52 53 54
3 3 3
, , ,P
P P P
m B M M im R B B Mim
C C C
B R B R B R
2 0
62 1 63 1 64 1 1
1
, , , 1 , .
! ( 1 )
n
t
n
t
C im C m C e t
n
Eliminating ( )v x and ( )x from Eqs. (28), we get after some simple computations
the following sixth-order ordinary differential equation satisfied by *( )u x
6 4 2
1 2 3 ( ) 0D g D g D g u x (29)
where
1 41 52 63 45 54 46 64 ,g C C C C C C C
2 41 52 53 62 46 54 62 41 63 52 63 45 54 63 46 52 64 45 53 64 ,g C C C C C C C C C C C C C C C C C C C C
3 41 53 62 41 52 63 .g C C C C C C
In a similar manner, we can show that ( )v x and ( )x satisfy the following equations
6 4 2
1 2 3 ( ), ( ) 0.D g D g D g v x x (30)
The general solution of Eq. (29) which is regular at x can be written as
3
1
( ) ( , )exp ,j j
j
u x R m k x
(31)
where ( 1,2,3)jk j , the roots (with positive real part) of the following characteristics equa-
tion
131
6 4 2
1 2 3 0,k g k g k g (32)
are given by
2 2 2
1 1 2 1 3 1
1 1 1
2 sin , [ 3 cos sin ] , [ 3 cos sin ] ,
3 3 3
k p q g k p q q g k p q q g
and
31
2 1 2 1 3
1 2 3
9 2 27sin
3 , , .
3 2
g g g gr
p g g q r
p
Similarly, the solutions for ( )v x and ( )x can be written as
3
'
1
( ) ( , )exp ,j j
j
v x R m k x
(33)
3
''
1
( ) ( , )exp .j j
j
x R m k x
(34)
Substituting from Eqs. (31), (33) and (34) into the Eqs. (28), we obtain the following re-
lations
' ''
1 2, ,j j j j j jR G R R G R (35)
2 2 2 2 2
53 41 46 54 52 41 45 54
1 22 2
45 53 46 52 45 53 46 52
, , 1,2,3.
j j j j j
j j
j j j j j j
C C k k C C k C C k k C C
G G j
k C C k C k C k C C k C k C
(36)
By using the relation (27), the solution for the Eqs. (24) – (26) can be written as
3
1
( , , ) exp ( , )exp ,j j
j
u x y t t imy R m k x
(37)
3
1
1
( , , ) exp ( , )exp ,j j j
j
v x y t t imy G R m k x
(38)
3
2
1
( , , ) exp ( , )exp ,j j j
j
x y t t imy G R m k x
(39)
Substituting from Eqs. (37) – (39) into Eqs. (20)-(23), we get the following expressions
for the stress components
3
1
1
( , , ) 2 exp exp ,xx P j j j
j
x y t R t imy M R k x
(40)
3
2
1
( , , ) 2 exp exp ,yy P j j j
j
x y t R t imy M R k x
(41)
3
3
1
( , , ) exp exp ,xy j j j
j
x y t t imy M R k x
(42)
3
4
1
( , , ) exp exp ,yx j j j
j
x y t t imy M R k x
(43)
132
where
1 1 2 2 1 2 2, ,j j ij j j j ij jM k imG B G M k B imG B G
3 1 3 3 4 1 3 3( ) ( ) , ( ) ( ) .j j j P P j j j P PM G k B R im B R M G k B R im B R
4. Application.
We consider the problem of a fiber-reinforced elastic half-space under hydrostatic initial
stress which fills the region defined as follows:
, , : 0 , , .x y z x y z
We apply the following boundary conditions for the present problem. The boundary
conditions at the plane surface x=0 subjected to an arbitrary normal force 1P are
1(0, , ) exp( ), (0, , ) 0, (0, , ) 0.xx xyy t P t imy y t y t (44)
Substituting the expressions of the variables considered into the above boundary condi-
tions, we obtain the following equations satisfied by the parameters ( 1,2,3)jR j
3
1
1
,j j P
j
M R R
(45)
3
3
1
0,j j
j
M R
(46)
3
2
1
0j j
j
G R
, (47)
where
12 exp( ) .P PR R t imy P
Solving Eqs. (45) – (47), we get the parameters ( 1,2,3)jR j with the following forms
respectively
23 32 22 33 21 33 23 31 22 31 21 32
1 2 3, , ,P P PR G M G M R G M G M R G M G M
R R R
(48)
where
11 23 32 22 33 12 21 33 23 31 13 22 31 21 32 .M G M G M M G M G M M G M G M
5. Particular cases.
(i) Isotropic generalized magneto-thermoelastic medium with hydrostatic initial
stress;
Substituting L T and 0 in Eqs. (37) – (43), we obtain the correspond-
ing expressions of the temperature, the displacements and the stress distribution in isotropic
generalized thermoelastic medium with hydrostatic initial stress and magnetic field.
(ii) Isotropic generalized thermoelastic medium with hydrostatic initial stress;
Substituting L T , 0 and 0HR in Eqs. (37) – (43), we obtain the cor-
responding expressions of all the physical quantities in isotropic generalized thermoelastic
medium with hydrostatic initial stress and without magnetic field.
(iii) Isotropic generalized magneto-thermoelastic medium without hydrostatic ini-
tial stress;
Substituting L T , 0 and 0PR in Eqs. (37) – (43), we obtain the
corresponding expressions of all the physical quantities in an isotropic generalized magneto-
thermoelastic medium without hydrostatic initial stress and with magnetic field.
133
(iv) Isotropic generalized thermoelastic medium without hydrostatic initial stress
and magnetic field;
Substituting L T , 0 , 0PR and 0HR in Eqs. (37) – (43), for the
above physical quantities in an isotropic generalized thermoelastic medium without hydro-
static initial stress and with magnetic field.
(v) Fiber-reinforced generalized magneto-thermoelastic medium;
Setting 0PR , the expressions in Eqs. (37) – (43) reduce to the case of a fiber-
reinforced generalized thermoelastic medium without hydrostatic initial stress and with
magnetic field.
(vi) Fiber-reinforced generalized thermoelastic medium;
Setting 0PR and 0HR , the expressions in Eqs. (37) – (43) reduce to the case of a
fiber-reinforced generalized thermoelastic medium without hydrostatic initial stress and
magnetic field.
6. Special cases of thermoelasticity theory
(i) Classical dynamical theory of thermoelasticity (CD-theory);
Setting 1 and 0 0 , the equations of the CD-theory can be obtained.
(ii) Lord-Shulman theory of thermoelasticity (LS-theory);
Setting 1 where 0 0 , the equations of the LS-theory can be obtained.
(ii) Fractional order theory of thermoelasticity (FO-theory);
In this case, we take 0.5 where 0 0 .
7. Numerical results.
With the analytical procedure presented earlier, we consider a numerical example for
which computational results are given. We use the following physical constants of a fiber-
reinforced thermoelastic solid to study the effect of the reinforcement, fractional parameter
and the magnetic field on the wave propagation:
4 -2 2 -2 3 -2 2 -2
3 -2 -3 -1 -1 -1 -1 -1
5 -2 4 -1 -1
0 0 0 0 0
5.65 10 N.m , 2.46 10 N.m , 5.66 10 N.m , 1.28 10 N.m ,
220.9 10 N.m , 2660kg.m , 0.3J.m . .K , 787J.kg .K ,
1.7810 N.m , 298K, 0.05s,H 10 A.m , 0.03F.m , 0.04H
T L
E
t
k s c
T
-1.m .
0 1 2 3 4 5 6 7 8 9 10
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
x
=1 FR
=0.5 FR
=1 NFR
=0.5 NFR
Fig. 1
Variation of the temperature distribution with x at 4 3
0 10 , 10pH R .
134
0 1 2 3 4 5 6 7 8 9 10
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
x
u
=1 FR
=0.5 FR
=1 NFR
=0.5 NFR
Fig. 2
Variation of the displacement distribution u with x at 4 3
0 10 , 10pH R .
0 1 2 3 4 5 6 7 8 9 10
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
x
v
=1 FR
=0.5 FR
=1 NFR
=0.5 NFR
Fig. 3
Variation of the displacement distribution v with x at 4 3
0 10 , 10pH R .
0 1 2 3 4 5 6 7 8 9 10
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
x
yy
=1 FR
=0.5 FR
=1 NFR
=0.5 NFR
Fig. 4
Variation of the stress distribution yy with x at 4 3
0 10 , 10pH R .
135
The other constants of the problem may be taken as 0.5, 3.6,m
0 0 1, 2.5, 10, 0.1, 100.i P P The computations are carried out on the sur-
face y=1.5 at time t=0.3. The distribution of the real part of the non-dimensional temperature
( ), the displacement components (u, v) and the stress component ( yy ) for the problem
considered are shown in figs. 1 – 16 for three different cases. In the first case, we are inves-
tigating how the non-dimensional temperature ( ), the displacement components (u, v) and
the stress component ( yy ) vary with different values of the fractional parameter
1.0 and 0.5 against x for the fiber-reinforced (FR) and non-fiber-reinforced (NFR)
elastic half-space when the initial hydrostatic stress and the magnetic field remain constant.
In the second case, we will show how the non-dimensional temperature ( ), the displace-
ments (u, v) and the stress ( yy ) vary with different values of the fractional parameter
1.0 and 0.5 in the presence 4
0( 10 )H and absence 0( 0)H of the magnetic field
against x for the fiber-reinforced elastic half-space when the initial hydrostatic stress re-
mains constant. The third case is investigating how the non-dimensional temperature , the
displacements u and v and the stress ( yy ) vary with different values of the fractional pa-
rameter 1.0 and 0.5 in the presence 3( 10 )pR and absence ( 0)pR of the initial
hydrostatic stress against x for the fiber-reinforced elastic half-space when the magnetic
field remains constant.
Figs. 1 – 4 depict the variety of the real part of the non-dimensional temperature ( ), the dis-
placement components (u, v) and the stress component ( yy ) for two different values of the frac-
tional parameter ( ) for the fiber-reinforced (FR) and non-fiber-reinforced (NFR) elastic half-
space. Fig. 1 and 4 show that the range of magnitude of the temperature ( ) and the stress ( yy )
are greater in the NFR thermoelastic medium for 1.0 . Fig. 2 exhibits that the normal dis-
placement u starts with a zero value and shows the oscillatory nature and converges to the zero
value rapidly with the increase of the distance x. Fig. 3 shows that the horizontal displacement v
starts with a positive initial value for the FR case but with a negative value for the NFR case and
vanishes identically with the increase of the distance x. It is also clearly depicted from figs. 2, 3
that the values of u and v are maximum in the FR elastic half-space for 0.5 . Figs. 5 – 8 exhibit
that as the value of x increases, the values of the non-dimensional temperature ( ), the displace-
ments (u, v) and the stress component ( yy ) approach rapidly to the zero value in the fiber-
reinforced elastic half-space without the effect of the magnetic field. It is also clearly depicted that
the values of all the physical quantities are maximum in the fiber-reinforced thermoelastic medium
for 0.5 when the effect of the magnetic field is present. Figs. 9 – 12 display the distribution of
the real part of the non-dimensional temperature ( ), the displacement components (u, v) and the
stress component ( yy ) for two different values of the fractional parameter in the presence and
absence of the hydrostatic initial stress for the FR elastic medium. The values of all the non-
dimensional physical quantities approach rapidly to the zero value in the fiber-reinforced elastic
half-space when the effect of the hydrostatic initial stress is absent. It can also be noted that the
values of all the physical quantities are maximum in the fiber-reinforced thermoelastic medium for
0.5 in the presence of the hydrostatic initial stress. Figs. 1, 5, 9 depict that the temperature ( )
is zero on the boundary surface x=0.0 for all values of y and t. The isothermal boundary condition
(44) on the surface x=0 of the half-space 0x is thus found to be satisfied numerically. This is
consistent with our theoretical result. Figs. 13 – 16 depict the three-dimensional distribution of the
real part of the non-dimensional temperature ( ), the displacement components (u, v) and the
stress component ( yy ) for two different values of the fractional parameter ( ) for the fiber-
reinforced elastic half-space in the presence of the hydrostatic initial stress and with the magnetic
field effect.
136
0 1 2 3 4 5 6 7 8
-0.02
-0.01
0
0.01
x
=1 H0=0
=0.5 H0=0
=1 H0=104
=0.5 H0=104
Fig. 5
Variation of the temperature distribution with x at 310pR .
0 1 2 3 4 5 6 7 8
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
x
u
=1 H0=0
=0.5 H0=0
=1 H0=104
=0.5 H0=104
Fig. 6
Variation of the displacement distribution u with x at 310pR .
0 1 2 3 4 5 6 7 8
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
x
v
=1 H0=0
=0.5 H0=0
=1 H0=104
=0.5 H0=104
Fig. 7
Variation of the displacement distribution v with x at 310pR .
137
0 1 2 3 4 5 6 7 8
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
x
yy
=1 H0=0
=0.5 H0=0
=1 H0=104
=0.5 H0=104
Fig. 8
Variation of the stress distribution yy with x at 310pR .
0 1 2 3 4 5 6 7 8
-0.02
-0.01
0
0.01
x
=1 Rp=0
=0.5 Rp=0
=1 Rp=103
=0.5 Rp=103
Fig. 9
Variation of the temperature distribution with x at 4
0 10H .
0 1 2 3 4 5 6 7 8
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
x
u
=1 Rp=0
=0.5 Rp=0
=1 Rp=103
=0.5 Rp=103
Fig. 10
Variation of the displacement distribution u with x at 4
0 10H .
138
0 1 2 3 4 5 6 7 8
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
x
v
=1 Rp=0
=0.5 Rp=0
=1 Rp=103
=0.5 Rp=103
Fig. 11
Variation of the displacement distribution v with x at 4
0 10H .
0 1 2 3 4 5 6 7 8
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
x
yy
=1 R
p
=0
=0.5 R
p
=0
=1 R
p
=103
=0.5 R
p
=103
Fig. 12
Variation of the stress distribution yy with x at 4
0 10H .
-2
-1
0
1
2
0
5
10
-3
-2
-1
0
1
2
y
=0.5 FR, Rp=103, H0=104
x
Fig. 13.
The three-dimensional temperature distribution with distance x and y.
139
-2
-1
0
1
2
0
5
10
-4
-2
0
2
4
6
8
y
=0.5 FR, Rp=103, H0=104
x
u
Fig. 14
The three-dimensional displacement distribution u with distance x and y.
-2
-1
0
1
2
0
5
10
-10
-5
0
5
y
=0.5 FR, R
p
=103, H0=104
x
v
Fig. 15
The three-dimensional displacement distribution v with distance x and y.
-2
-1
0
1
2
0
5
10
-0.2
-0.1
0
0.1
0.2
0.3
y
=0.5 FR, Rp=103, H0=104
x
yy
Fig. 16
The three-dimensional stress distribution yy with distance x and y.
140
Fig. 13 also clearly shows that the temperature starts with a zero value which satisfies the
boundary condition (44).
8. Concluding remarks.
According to the analysis above, we can conclude the following points:
1. The hydrostatic initial stress and the fractional parameter have a great effect on the
distribution of the field quantities. The presence of the magnetic field plays a significant role
in the field quantities.
2. It is clear from all the figures that all the distributions considered have a non-zero
value only in a bounded region of the fiber-reinforced elastic half-space. Outside of this re-
gion, the values vanish identically and this means that the region has not felt a thermal dis-
turbance yet.
3. The values of all the physical quantities converge to zero with increasing distance x.
4. All the physical quantities satisfy the boundary conditions.
5. Deformation of a body depends on the nature of the applied force as well as the type
of boundary conditions.
6. The method that was used in the present article is applicable to a wide range of prob-
lems in thermodynamics and thermoelasticity.
7. Analytical solutions based upon normal mode analysis of the thermoelastic problem
in solids have been developed and utilized.
8. From the temperature distributions, we have found wave type heat propagation with
finite speeds in the medium.
9. The results presented in this paper should prove useful for researchers in material sci-
ence, designers of new materials, low temperature physicists, as well as for those working
on the development of a theory of hyperbolic thermoelasticity with a fractional derivative
heat transfer. The introduction of the magnetic field and the fractional derivative heat trans-
fer to the generalized thermoelastic medium provides a more realistic model for these stud-
ies.
Р Е ЗЮМ Е . Вивчено вплив трьох факторів – порядку дробових похідних, початкового гідро-
статичного напруження, параметру магнітного поля – на плоскі хвилі в півпросторі з армованого
волокнами матеріалу, який описується теорією узагальненої магнітотермопружності. Задача
розв’язана чисельно за допомогою аналізу нормальних мод. Результати аналізу відповідають моделі
Лорда – Шульмана і моделі, що описується за допомогою дробових похідних, і представлені у вигля-
ді графіків. Отримані результати показують добре виражений вплив вказаних трьох факторів. Також
ці результати порівнюються з результатами, що отримані для випадку відсутності початкового гідро-
статичного напруження і магнітного поля.
141
Nomenclature
H
applied magnetic field vector
J
current density vector
E
induced electric field vector
h
induced magnetic field vector
0 electric permeability
0 magnetic permeability
, ,L T elastic parameters
, , ,L T reinforcement parameters
acceleration due to gravity
Ec specific heat of the solid at constant strain
0 the thermal relaxation time parameter
fractional parameter
ij components of the stress tensor
ije components of the strain tensor
iu components of the displacement vector u
kke =e, cubical dilatation
t time variable
x, y space variables
T absolute temperature
0T the temperature of the medium in it’s natural state, assumed to be such
that
0
1
T
3 2 T T
T coefficient of linear thermal expansion
k thermal conductivity
P the initial hydrostatic pressure
ij Kronecker delta
11A 2 4 2L T
12A
22A 2 T
2,HR 2
0 0H
1c 11A
2c
0 0
1
, speed of light
D
2
0
11 E
T
A c
d
dx
142
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*From the Editorial Board: The article corresponds completely to submitted manuscript.
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