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© 2020 IAU, Arak Branch. All rights reserved.
Journal of Solid Mechanics Vol. 12, No. 3 (2020) pp. 663-680
DOI: 10.22034/jsm.2020.1885944.1530
Three Dimensional Thermal Shock Problem in Magneto-Thermoelastic Orthotropic Medium
S. Biswas 1,*
, S.M. Abo-Dahab 2,3
1Department of Mathematics, University of North Bengal, Darjeeling, India
2Department of Mathematics, Faculty of Science, Taif University, Saudi Arabia
3Department of Mathematics, Faculty of Science, South Valley University, Egypt
Received 28 June 2020; accepted 25 August 2020
ABSTRACT
The paper is concerned with the study of magneto-
thermoelastic interactions in three dimensional thermoelastic
medium under the purview of three-phase-lag model of
generalized thermoelasticity. The medium under consideration
is assumed to be homogeneous orthotropic medium. The
fundamental equations of the three-dimensional problem of
generalized thermoelasticity are obtained as a vector-matrix
differential equation form by employing normal mode analysis
which is then solved by eigenvalue approach. Stresses and
displacements are presented graphically for different
thermoelastic models.
© 2020 IAU, Arak Branch. All rights reserved.
Keywords: Eigenvalue approach; Orthotropic medium; Three-
phase-lag model; Magnetic effect.
1 INTRODUCTION
ENERALIZED thermoelasticity theory is developed to overcome the paradox of infinite speed of thermal
wave inherent in the classical coupled thermoelasticity theory. Lord and Shulman [1] formulated the
generalized thermoelasticity theory by introducing one relaxation time which is known as LS model. Green and
Lindsay [2] introduced GL theory by incorporating two relaxation times. Later Green and Naghdi [3, 4, 5]
developed three models for generalized thermoelasticity of homogeneous isotropic materials, which are labeled as
G-N models I, II, III. Detailed information regarding these theories is available in [6, 7, 8]. Tzou [9] introduced two-
phase lags to both the heat flux vector and the temperature gradient and considered as constitutive equation to
describe the lagging behavior in the heat conduction in solids. Roy Choudhuri [10] has established a generalized
mathematical model of a coupled thermoelasticity theory that includes three-phase-lags in the heat flux vector, the
temperature gradient and in the thermal displacement gradient. The interplay of the Maxwell electromagnetic filed
with the motion of deformable solids is largely being undertaken by many investigators [11-14] owing to the
possibility of its application to geophysical problems and certain topics in optics and acoustics. El-Karamany and
Ezzat [15] considered thermal shock problem in generalized thermoelasticity under four theories. Sherief et al. [16]
discussed stochastic thermal shock in generalized thermoelasticity and Ezzat and Youssef [17] investigated three
______ *Corresponding author.
E-mail address: siddharthabsws957@gmail.com (S. Biswas).
G
664 S. Biswas and S.M. Abo-Dahab
© 2020 IAU, Arak Branch
dimensional thermal shock problem of generalized thermoelastic half-space. Kalkal and Deswal [18] considered the
effects of phase lags on three dimensional wave propagation with temperature dependent properties. El-Karamany
and Ezzat [19] discussed three-phase-lag linear micro polar thermoelasticity theory. Ezzat et al. [20] proposed
fractional order theory in thermoelastic solid with three-phase-lag heat transfer. Said and Othman [21] discussed the
Effects of gravitational and hydrostatic initial stress on a two-temperature fiber-reinforced thermoelastic medium for
three-phase-lag model. Lofty [22] studied two temperature generalized magneto-thermoelastic interactions in an
elastic medium under three theories. Sarkar and Lahiri [23] considered electro magneto-thermoelastic interactions in
an orthotropic slab with two thermal relaxation times. Das and Bhakta [24] proposed eigen function expansion
method to the solution of simultaneous equations and its application in mechanics. Ezzat [25] considered the
relaxation effects of the volume properties of electrically conducting viscoelastic material. Ezzat [26] discussed
fundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect conductor
cylindrical region. Ezzat et al. [27] studied electro-thermoelasticity theory with memory-dependent heat transfer.
The present article deals with a three dimensional electro-magneto-thermoelastic coupled problem for
homogeneous orthotropic thermally and electrically conducting solid whose surface is subjected to time dependent
thermal shock. The normal mode analysis and eigenvalue approach is used to solve the problem. Numerical results
for the displacements and thermal stress distribution are presented graphically for dual-phase lag (DPL) model,
Green-Naghdi type-III model (GN-III) and three-phase-lag model (TPL).
2 FORMULATION OF THE PROBLEM
The homogeneous orthotropic medium is supposed to be initially unstrained and unstressed. The basic equations of
linear magneto-thermoelasticity with three-phase-lag model are as follows:
The equations of motion
ij j ii
J B u , (1)
Maxwell’s equations (in absence of the displacement current and charge density)
curl H J , curl
BE
t
, div B 0,
eB H (2)
The modified Ohm’s law is
uJ E B
t
(3)
where H the total magnetic field vector x y zH H H , , , B magnetic inductance vector
x y zB B B , , ,
E electric field vector x y zE E E , , , e magnetic permeability of the medium, electric conductivity of
the medium, constant mass density, ij component of stress tensor, i j x y z, , , , u displacement vector
u v w , , . If we take x yH H H , ,0 , we get from Eqs. (2) and (3), after neglecting second order differentiation
of xH and yH ,
yx x
e
EH E
t y z
(4)
y x z
e
H E E
t z x
(5)
Three Dimensional Thermal Shock Problem …. 665
© 2020 IAU, Arak Branch
Eqs. (4) and (5) can be linearized by setting x xH H h 0
and y yH H h 0
where xh and
yh denote change in
the basic magnetic field H 0(called the perturbed field) and then neglecting product terms with
xh andyh .
After linearization, Eqs. (4) and (5) with the help of (2) become
x
v u wh H
y y z
0 (6)
y
u v wh H
x x z
0
(7)
The thermal stresses in an orthotropic infinite elastic solid subject to plane strain in three dimensions are
xx
u v wc c c T
x y z
11 12 13 1
yy
u v wc c c T
x y z
12 22 23 2
zz
u v wc c c T
x y z
13 23 33 3
yz
v wc
z y
44
xz
w uc
x z
55
xy
u vc
y x
66
(8)
where ijc are elastic constants of the orthotropic material and 1 2 3, , are thermal moduli along x y, and z axis
respectively.
From Eqs. (1), (6), (7) and (8), after neglecting higher order of small quantities, we get
e
u u u v w w v u v u wc c c c c c c H
x y x z x z y zx y z x x y y
Tu
x
2 2 2 2 2 2 2 2 2 2 22
11 66 55 12 66 13 55 02 2 2 2 2 2 2
1
(9)
e
v v v u w w v u v u wc c c c c c c H
x y y z x z y zx y z x x y y
Tv
y
2 2 2 2 2 2 2 2 2 2 22
66 22 44 12 66 23 44 02 2 2 2 2 2 2
2
(10)
e
w w w u v w v u v u wc c c c c c c H
x z y z x z x z y z y zx y z z z
Tw
z
2 2 2 2 2 2 2 2 2 2 22
55 44 33 13 55 23 44 02 2 2 2 2
3
(11)
where T is the temperature above reference temperature.
Equation of three-phase-lag model in orthotropic medium is
666 S. Biswas and S.M. Abo-Dahab
© 2020 IAU, Arak Branch
T T T
q
q e
T T T T TK K K K K
t t t t tx y z x y
T u v wK C T T
t t x y zz t
2 2 2 2 2* *
1 2 3 1 22 2 2 2 2
22 2*
3 0 1 2 32 2
1 1 1 1 1
1 12
(12)
where T 0 is the reference temperature, iK i 1,2,3 is the thermal conductivity tensor, iK i * 1,2,3 are the
material constant characteristic of the theory, q T , and are the phase lags for heat flux vector, temperature
gradient and thermal displacement gradient respectively.
Maxwell’s electromagnetic stress tensor ij is given by
ij i j j i ijH h H h H h
.
3 BOUNDARY CONDITIONS
We consider the case where the surface of the three dimensional orthotropic medium is subjected to a time
dependent thermal shock and the surface is traction free. Then we consider the case where the surface of medium is
rigidly fixed and a thermal shock is applied on it.
In order to determine the parameters, we need to consider the following boundary conditions at z 0 :
3.1 Case 1
Thermal boundary condition that the surface of the medium is subjected to a time dependent thermal shock
T x y t F t H a x H b y ( , ,0, )
where H denotes Heaviside function.
Mechanical boundary condition that the surface to the medium is traction free
zz zz yz yz xz xzx y t x y t x y t x y t x y t x y t , ,0, , ,0, , ,0, , ,0, , ,0, , ,0, 0
3.2 Case 2
Thermal boundary condition that the surface of the medium subjected to a time dependent thermal shock
T x y t F t H a x H b y ( , ,0, )
Mechanical boundary condition that the surface of the medium is rigidly fixed
u x y t v x y t w x y t , ,0, , ,0, , ,0, 0
4 SOLUTION OF THE PROBLEM
We take the solutions of the Eqs. (9)-(12) in the following form:
u v w T x y z t u v w T z i kx ly t , , , , , , , , , exp (13)
Three Dimensional Thermal Shock Problem …. 667
© 2020 IAU, Arak Branch
where k and l are wave number along x and y axis respectively and is angular frequency.
e e
e
d u dwc ik c c i H k l H k l c k c l u
dzdz
H k l c c kl v ik T
22 2 2 2 2 2 2
55 13 55 0 0 11 662
2 2 2
0 12 66 1 0
(14)
e e
e
d v dwc il c c i H k l H k l c c kl u
dzdz
H k l c l c k v il T
22 2 2 2
44 23 44 0 0 12 662
2 2 2 2 2 2
0 22 66 2 0
(15)
e e e
d w du dvc H c c ik i H k l c c il i H k l
dz dzdz
dTc k c l w
dz
22 2 2
33 0 13 55 0 23 44 02
2 2 2
55 44 3
2
0
(16)
e
d T dwK K i T ikT u ilT v K i k K il K k K l C T
dzdz
2* 2 2 2 2 2 * 2 * 2 2
3 2 3 1 0 3 0 1 0 2 1 1 2 1 2 1 220 (17)
where T
q
q
i
i
1 2 2
1,
12
q
q
i
i
2 2 2
1.
12
T u v, , and w must be bounded at infinity so as to satisfy the regularity condition at infinity. So we assume that
T u v, , and w as well as their derivatives vanish at infinity.
Eqs. (14)-(17) can now be written in the form of a vector matrix differential equation as follows:
dVAV
dz (18)
where
T
du dv dw dTV u v w T
dz dz dz dz
, , , , , , ,
The matrix A is I
AP Q
0
where
A A A
A A AP
A
A A A
51 52 54
61 62 64
73
81 82 84
0
0,
0 0 0
0
A
AQ
A A A
A
57
67
75 76 78
87
0 0 0
0 0 0,
0
0 0 0
0 0 0 0
0 0 0 00 ,
0 0 0 0
0 0 0 0
I
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
ec k c l H k lA
c
2 2 2 2 2 2
11 66 0
51
55
, ec c kl H k l
Ac
2 2 2
12 66 0
52
55
,
ikA
c
1
54
55
, ei H k l ik c c
Ac
2
0 13 55
57
55
, ec c kl H k l
Ac
2 2 2
12 66 0
61
44
,
ec k c l H k lA
c
2 2 2 2 2 2
66 22 0
62
44
,il
Ac
2
64
44
, ei H k l c c il
Ac
2
0 23 44
67
44
,
668 S. Biswas and S.M. Abo-Dahab
© 2020 IAU, Arak Branch
e
c k c lA
c H
2 2 2
55 44
73 2
33 0
,2
e
e
i H k l ik c cA
c H
2
0 13 55
75 2
33 0
,2
e
e
i H k l c c ilA
c H
2
0 23 44
76 2
33 0
,2
e
Ac H
3
78 2
33 0
,2
2
0 1
81 *
3 2 1 3
,ik T
AK i K
il T
AK i K
2
0 2
82 *
3 2 1 3
,
eK i k K il K k K l C
AK i K
2 2 * 2 * 2 2
1 1 2 1 2 1 2
84 *
3 2 1 3
,
TA
K i K
2
0 3
87 *
3 2 1 3
5 SOLUTION OF THE VECTOR MATRIX DIFFERENTIAL EQUATION
As for the solution of the Eq. (18), we follow the method of eigenvalue approach as in Das and Bhakta [24],The
characteristic equation of the matrix A takes the form
B B B B 8 6 4 2
1 2 3 4 0 (19)
where
1 78 87 67 76 57 75 81 62 73 84 ,B A A A A A A A A A A
2 81 54 81 57 78 82 64 82 67 78 51 78 87 62 78 87 75 54 87 87 64 76 62 73
51 62 51 73 51 67 76 52 61 52 67 75 57 61 76 75 62 84 51 84 73 84 62
84 67 76 84 57 75 ,
B A A A A A A A A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A A A A A A A A A A A
A A A A A A
3 81 54 62 81 54 73 81 54 67 76 81 57 62 78 81 57 64 76 52 64 81 81 52 67 78
82 64 73 82 51 64 82 54 61 82 51 67 78 82 57 61 78 82 57 64 75 82 54 75 67
75 52 64 87 75 62 54 87 78 87
B A A A A A A A A A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A A A A A A A A A A A A A
A A A A A A A A A A
51 62 78 52 61 67 76 61 54 87 76 87 51 64
51 62 73 52 61 73 84 62 73 84 51 62 51 84 73 84 51 67 76 84 52 61 52 67 75 84
84 57 61 76 84 57 75 62 ,
A A A A A A A A A A A A A A
A A A A A A A A A A A A A A A A A A A A A A A A A A
A A A A A A A A
4 73 51 62 84 64 82 52 61 84 64 81 54 61 82 62 81B A A A A A A A A A A A A A A A A
The roots of the characteristic Eq. (19) which are also the eigenvalues of the matrix A are of the form
1, 2 , 3 , 4 (20)
The right and left eigenvectors X and Y of the matrix A corresponding to the eigenvalue can be taken as
follows:
1 2 3 4 5 6 7 8, , , , , , ,T
X x x x x x x x x (21)
1 2 3 4 5 6 7 8, , , , , , ,Y y y y y y y y y (22)
where
4 2
1 57 78 54 54 62 54 73 52 64 54 67 76 52 67 78 57 64 76 57 62 78
52 64 73 54 62 73 ,
x A A A A A A A A A A A A A A A A A A A A A
A A A A A A
4 2
2 67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64 64 51 61 54
73 51 64 73 61 54 ,
x A A A A A A A A A A A A A A A A A A A A A
A A A A A A
4 2
3 78 51 78 62 78 64 76 54 75 51 62 78 51 64 76 61 54 76 52 61 78 52 64 75 54 62 75 ,x A A A A A A A A A A A A A A A A A A A A A A A A A A A
Three Dimensional Thermal Shock Problem …. 669
© 2020 IAU, Arak Branch
6 4 2
4 51 62 73 57 75 67 76 51 62 51 73 52 61 62 73 51 67 76
52 75 67 57 62 75 57 61 76 51 62 73 61 52 73
(
) ,
x A A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A A A
5 1,x x 6 2 ,x x
7 3 ,x x 8 4x x
and
7 5 3
1 62 73 84 57 75 76 67 87 78 62 78 67 82 64 62 87 78 62 73
62 84 84 73 76 87 64 67 76 84 75 52 67 75 62 57 75 57 84 75 54 87 55
52 67 84 52 64 87 62 57 84 62 54 87 8
(
)
y A A A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A
2 57 64 82 67 54 ,A A A A A
5 3
2 52 52 87 78 52 84 52 73 82 57 78 82 54 76 57 84 76 54 87 52 67 76
62 57 76 76 57 84 76 54 87 52 84 73 82 54 73 52 67 84 52 64 87 62 57 84
62 54 87 82 57 64 82 67 54
(
) (
),
y A A A A A A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A A A A A A A A A A A A
A A A A A A A A A
4 2
3 73 57 52 67 62 57 57 84 54 87 52 67 84 52 64 87 62 57 84 62 54 87
82 57 64 82 67 54
[ (
)],
y A A A A A A A A A A A A A A A A A A A A A A
A A A A A A
5 3
4 54 78 57 84 78 54 87 62 54 54 73 52 64 76 57 64 76 67 54
78 52 67 84 78 52 64 87 78 62 57 84 78 62 54 87 78 82 57 64 78 82 67 54 52 64 73 62 54 73 ,
y A A A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A
6 4 2
5 62 73 84 67 76 78 87 62 73 84 62 84 73 84 67 76 87 62 78
87 64 76 82 64 82 67 78 84 62 73 82 64 73
(
) ,
y A A A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A A
4 2
6 52 57 76 52 73 52 84 84 57 76 54 82 82 57 78 87 52 78 87 54 76
52 73 84 82 54 73 ,
y A A A A A A A A A A A A A A A A A A A A A
A A A A A A
5 3
7 57 67 52 57 62 84 57 54 87 52 67 64 52 64 87 62 57 84 62 54 87
82 57 64 82 67 54
(
),
y A A A A A A A A A A A A A A A A A A A A A
A A A A A A
4 2
8 54 57 78 54 73 54 62 64 52 57 62 78 52 67 78 54 76 67 57 64 76
54 62 73 52 64 73
y A A A A A A A A A A A A A A A A A A A A A
A A A A A A
For our further reference we shall use the following notations:
1
1 ,X X
1
2 ,X X
2
3 ,X X
2
4 ,X X
3
5 ,X X
3
6 ,X X
4
7 ,X X
4
8X X
(23)
and
1
1 ,Y Y
1
2 ,Y Y
2
3 ,Y Y
2
4 ,Y Y
3
5 ,Y Y
3
6 ,Y Y
4
7 ,Y Y
4
8Y Y
(24)
Assuming the regularity condition at infinity, the solution of the Eq. (18) can be written as:
1 2 1 2 4 2 3 6 3 4 8 4V A X exp z A X exp z A X exp z A X exp z
Now we get
44 2
57 78 54 54 62 54 73 52 64 54 67 76 52 67 78 57 64 76 57 62 78
1
52 64 73 54 62 73
n n n
n
n
u x , y , z ,t A [ A A A A A A A A A A A A A A A A A A A A A
A A A A A A ] exp z
(25)
670 S. Biswas and S.M. Abo-Dahab
© 2020 IAU, Arak Branch
44 2
67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64
1
64 51 61 54 73 51 64 73 61 54
n n n
n
n
v x , y , z ,t A [ A A A ( A A A A A A A A A A A A A A
A A A A ) A A A A A A ] exp z
(26)
45 3
78 51 78 62 78 64 76 54 75 51 62 78 51 64 76 61 54 76
1
52 61 78 52 64 75 54 62 75
n n n n
n
n
w x , y , z ,t A [ A A A A A A A A A ( A A A A A A A A A
A A A A A A A A A )] exp z
(27)
46 4 2
51 62 73 57 75 67 76 51 62 51 73 52 61 62 73
1
51 67 76 52 75 67 57 62 75 57 61 76 51 62 73 61 52 73
n n n n
n
n
T x , y , z ,t A [ A A A A A A A ( A A A A A A A A
A A A A A A A A A A A A ) A A A A A A ] exp z
(28)
The stress components are obtained as:
44 2
11 57 78 54 54 62 54 73 52 64 54 67 76 52 67 78 57 64 76 57 62 78 52 64 73 54 62 73
1
44 2
12 67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64 64
1
xx n n n
n
n n n
n
{ c ikA [ A A A A A A A A A A A A A A A A A A A A A A A A A A A ]
c ilA [ A A A ( A A A A A A A A A A A A A A A
51 61 54 73 51 64 73 61 54
45 3
13 78 51 78 62 78 64 76 54 75 51 62 78 51 64 76 61 54 76 52 61 78 52 64 75 54 62 75
1
46 4 2
1 51 62 73 57 75 67 76 5
1
n n n n n
n
n n n n
n
A A A ) A A A A A A ]
c A [ A A A A A A A A A ( A A A A A A A A A A A A A A A A A A )]
A [ A A A A A A A ( A
1 62 51 73 52 61 62 73 51 67 76 52 75 67 57 62 75 57 61 76
51 62 73 61 52 73 n
A A A A A A A A A A A A A A A A A A A )
A A A A A A ]} exp z i kx ly t
44 2
13 57 78 54 54 62 54 73 52 64 54 67 76 52 67 78 57 64 76 57 62 78 52 64 73 54 62 73
1
44 2
23 67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64 64
1
zz n n n
n
n n n
n
{ c ikA [ A A A A A A A A A A A A A A A A A A A A A A A A A A A ]
c A il [ A A A ( A A A A A A A A A A A A A A A
51 61 54 73 51 64 73 61 54
45 3
33 78 51 78 62 78 64 76 54 75 51 62 78 51 64 76 61 54 76 52 62 78 52 64 75 54 62 75
1
46 4 2
3 51 62 73 57 75 67 76 51 62
1
n n n n n
n
n n n n
n
A A A ) A A A A A A ]
c A [ A A A A A A A A A A A A A A A A A A A A A A A A A A A
A [ A A A A A A A ( A A
51 73 52 61 62 73 51 67 76 52 75 67 57 62 75 57 61 76
51 62 73 61 52 73 n
A A A A A A A A A A A A A A A A A A )
A A A A A A ]} exp z i kx ly t
44 2
44 67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64
1
45 3
64 51 61 54 73 51 64 73 61 54 78 51 78 62 78 64 76 54 75
1
51 62 78 51 64 76 61 54
yz n n n n
n
n n n
n
n
c { A [ A A A ( A A A A A A A A A A A A A A
A A A A ) A A A A A A ] A il [ A A A A A A A A A
( A A A A A A A A A
76 52 61 78 52 64 75 54 62 75 nA A A A A A A A A )]} exp z i kx ly t
45 3
55 78 51 78 62 78 64 76 54 75 51 62 78 51 64 76 61 54 76 52 61 78
1
44 2
52 64 75 54 62 75 57 78 54 54 62 54 73 52 64 54 67 76 52 67 78
1
57 64 7
xz n n n n
n
n n n n
n
c { ik A [ A A A A A A A A A ( A A A A A A A A A A A A
A A A A A A )] A [ A A A ( A A A A A A A A A A A A
A A A
6 57 62 78 52 64 73 54 62 73 nA A A ) A A A A A A ]} exp z i kx ly t
4 2
66 57 78 54 54 62 54 73 52 64 54 67 76 52 67 78 57 64 76 57 62 78
4 2
52 64 73 54 62 73 67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64
64 51 61 54 73 5
xy n n
n n
c [ il{ A A A A A A A A A A A A A A A A A A A A A
A A A A A A } ik{ A A A ( A A A A A A A A A A A A A A
A A A A ) A A
1 64 73 61 54 nA A A A }] exp z i kx ly t
The constants 1 2 3 4nA n , , , can be obtained by using boundary conditions.
For case 1 we obtain the constants as follows:
Three Dimensional Thermal Shock Problem …. 671
© 2020 IAU, Arak Branch
1
1
DA ,
D 2
2
DA ,
D 3
3
DA ,
D 4
4
DA
D
In which
1 2 3 4 4 3 3 2 4 4 2 4 2 3 3 2
2 1 3 4 4 3 3 1 4 4 1 4 1 3 3 1
3 1 2 4 4 2 2 1 4 4 1 4 1 2 2 1
4 1 2 3 3 2 2 1 3 3 1 3 1 2 2 1
D M N P Q P Q N P Q P Q N P Q P Q
M N P Q P Q N PQ P Q N PQ P Q
M N P Q P Q N PQ P Q N PQ P Q
M N P Q P Q N PQ P Q N PQ P Q
1D 1 2 3 4 4 3 3 2 4 4 2 4 2 3 3 2F N PQ P Q N P Q P Q N P Q PQ
2D 1 1 3 4 4 3 3 1 4 4 1 4 1 3 3 1F N PQ P Q N PQ P Q N PQ PQ
3D 1 1 2 4 4 2 2 1 4 4 1 4 1 2 2 1F N P Q P Q N PQ P Q N PQ P Q
4D 1 1 2 3 3 2 2 1 3 3 1 3 1 2 2 1F N P Q PQ N PQ PQ N PQ P Q
6 4 2
51 62 73 57 75 67 76 51 62 51 73 52 61 62 73
51 67 76 52 75 67 57 62 75 57 61 76 51 62 73 61 52 73
n n n nM A A A A A A A ( A A A A A A A A
A A A A A A A A A A A A ) A A A A A A
2 4 2
13 0 57 78 54 54 62 54 73 52 64 54 67 76 52 67 78 57 64 76 57 62 78 52 64 73 54 62 73
2 4 2
23 0 67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64 64 51
n e n n
e n n
N ik c H [ A A A A A A A A A A A A A A A A A A A A A A A A A A A ]
il c H [ A A A ( A A A A A A A A A A A A A A A A
61 54 73 51 64 73 61 54
2 5 3
33 0 78 51 78 62 78 64 76 54 75 51 62 78 51 64 76 61 54 76 52 62 78 52 64 75 54 62 75
3
e n n n n
n
A A ) A A A A A A ]
c H [ A A A A A A A A A A A A A A A A A A A A A A A A A A A ]
M
5 3
44 78 51 78 62 78 64 76 54 75 51 62 78 51 64 76 61 54 76 52 61 78
4 2
52 64 75 54 62 75 57 78 54 54 62 54 73 52 64 54 67 76 52 67 78
57 64 76 57 62 78 5
n n n n
n n n
P c { ik [ A A A A A A A A A ( A A A A A A A A A A A A
A A A A A A )] [ A A A ( A A A A A A A A A A A A
A A A A A A ) A
2 64 73 54 62 73A A A A A ]}
4 2
55 67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64
5 3
64 51 61 54 73 51 64 73 61 54 78 51 78 62 78 64 76 54 75
51 62 78 51 64 76 61 54 76 52 61 78 5
n n n n
n n
n
Q c { [ A A A ( A A A A A A A A A A A A A A
A A A A ) A A A A A A ] il [ A A A A A A A A A
( A A A A A A A A A A A A A
2 64 75 54 62 75A A A A A )]}
1F x , y ,t F t H a x H b y exp i kx ly t
For case 2 we obtain the constants as follows:
1
1A ,
2
2A ,
3
3A ,
4
4A
In which
1 2 3 4 4 3 3 2 4 4 2 4 2 3 3 2
2 1 3 4 4 3 3 1 4 4 1 4 1 3 3 1
3 1 2 4 4 2 2 1 4 4 1 4 1 2 2 1
4 1 2 3 3 2 2 1 3 3 1 3 1 2 2 1
M L R S R S L R S R S L R S R S
M L R S R S L R S R S L R S R S
M L R S R S L R S R S L R S R S
M L R S R S L R S R S L R S R S
1 1 2 3 4 4 3 3 2 4 4 2 4 2 3 3 2F L R S R S L R S R S L R S R S
2 1 1 3 4 4 3 3 1 4 4 1 4 1 3 3 1F L R S R S L R S R S L R S R S
3 1 1 2 4 4 2 2 1 4 4 1 4 1 2 2 1F L R S R S L R S R S L R S R S
672 S. Biswas and S.M. Abo-Dahab
© 2020 IAU, Arak Branch
4 1 1 2 3 3 2 2 1 3 3 1 3 1 2 2 1F L R S R S L R S R S L R S R S
4 2
57 78 54 54 62 54 73 52 64 54 67 76 52 67 78 57 64 76 57 62 78
52 64 73 54 62 73
n n nL A A A A A A A A A A A A A A A A A A A A A
A A A A A A
4 2
67 78 64 64 73 51 67 78 54 67 75 57 61 78 57 75 64 64 51
61 54 73 51 64 73 61 54
n n nR A A A ( A A A A A A A A A A A A A A A A
A A ) A A A A A A
5 3
78 51 78 62 78 64 76 54 75 51 62 78 51 64 76 61 54 76 52 61 78
52 64 75 54 62 75
n n n nS A A A A A A A A A ( A A A A A A A A A A A A
A A A A A A )
6 SPECIAL CASES
We discuss some special cases for different values of the parameters considered in the problem.
The above discussion converts for orthotropic medium without magnetic effect when we put
0 0H .
This problem reduces for isotropic medium with three-phase-lag model, if we take
11 22 33 13 23 12 44 1 2 3 1 2 3 1 2 32 * * * *c c c ,c c c ,c , ,K K K K ,K K K K
The study reduces to the case of orthotropic elasticity if we neglect the thermal parameters i.e.
* * *,K K K ,K K K 1 2 3 1 2 3 1 2 30 0 0
7 PARTICULAR CASES
From the general Eqs. (9)-(12), we now classify the problem into three classes for our further reference and for our
comparison of numerical computations of the results:
i. If q , 2 0q , 0 T , 0 * * *K K K 1 2 3 0 , then the problem reduces to the problem of Lord-Shulman
(LS) model.
ii. If q T 0 then the problem reduces to the problem of Green-Naghdi theory type-III (GN-III).
iii. If * * *K K K 1 2 3 0 then the problem reduces to the problem of dual-phase-lag model (DPL).
8 NUMERICAL RESULTS AND DISCUSSION
For numerical computations, we take the following values of the relevant parameters for cobalt material as follows:
2c . Nm , 11
11 3 071 10 2c . Nm , 11
12 1 650 10 2c . Nm , 11
13 1 027 10 2c . Nm , 11
23 1 027 10
2c . Nm , 11
22 3 071 10 2c . Nm , 11
33 3 581 10 2c . Nm , 11
44 1 510 10 2 1. Nm deg , 6
1 7 04 10
2 1. Nm deg , 6
2 7 04 10 2 1. Nm deg , 6
3 6 90 10 1 1K Wm deg , 1 69 1 1K Wm deg , 2 69
1 1K Wm deg , 3 69 * 1 1 1`K . Wm deg s , 1 13 1 * 1 1 1`K . Wm deg s , 2 15 4 * 1 1 1`K . Wm deg s , 3 15 4
3. Kgm , 37 14 10 1 1
eC . JKg deg , 381 4 T K,0 296 q s, 72 10 T . s , 71 5 10
v s. 81 10
We consider F t exp( dt ) 0 where 0 is a constant. Further for numerical purpose we take
Three Dimensional Thermal Shock Problem …. 673
© 2020 IAU, Arak Branch
1 1 1
e,d . ,a b m ,k . , . Hm , . Fm ,H Am . 5
0 0 010 0 1 1 1 2 1 2 1 2 10
In Figs. 1-12, displacements and stresses for traction free surface are presented. In Fig. 1, variation of u with
respect to z for three-phase-lag model is presented. It is observed that u decreases with the increase of magnetic
field. u is showing oscillatory behavior and converging towards zero with the increase of z.
Fig.1
Variation of u with respect to z.
In Fig. 2 and 3, variation of v and w with respect to z for three-phase-lag model is presented. It is noticed that v
and w decrease with the increase of magnetic field. Displacements are showing oscillatory behavior and converging
towards zero with the increase of z.
Fig.2
Variation of v with respect to z.
Fig.3
Variation of w with respect to z.
In Fig. 4, variation of xz with respect to z for three-phase-lag model is presented. It is noticed that
xz decrease
with the increase of magnetic field. Stress is showing oscillatory behavior and converging towards zero with the
increase of z.
674 S. Biswas and S.M. Abo-Dahab
© 2020 IAU, Arak Branch
Fig.4
Variation of xz with respect to z.
In Fig. 5, variation of yz with respect to z for three-phase-lag model is presented. It is noticed that
yz increase
with the increase of magnetic field. Stress is showing oscillatory behavior and converging towards zero with the
increase of z.
Fig.5
Variation of yz with respect to z.
In Fig. 6, variation of zz with respect to z for three-phase-lag model is presented. It is noticed that zz decrease
with the increase of magnetic field. Stress is showing oscillatory behavior and converging towards zero with the
increase of z.
Fig.6
Variation of zz with respect to z.
In Figs. 7 and 8, variation of u and v with respect to z for GN-III and DPL model is presented. It is noticed that u
and v increase with the increase of magnetic field. Displacements for DPL model are greater than displacements for
GN-III model. Displacements are showing oscillatory behavior and converging towards zero with the increase of z.
Three Dimensional Thermal Shock Problem …. 675
© 2020 IAU, Arak Branch
Fig.7
Comparison of u with respect to z for GN and DPL.
Fig.8
Comparison of v with respect to z for GN and DPL.
In Fig. 9, variation of w with respect to z for GN-III and DPL model is presented. It is noticed that w increase
with the increase of magnetic field. Displacement for DPL model is greater than displacement for GN-III model.
Displacement is converging towards zero with the increase of z.
Fig.9
Comparison of w with respect to z for GN and DPL.
In Fig.10, comparison of zz with respect to z for GN-III and DPL model is presented. It is noticed that zz
increase with the increase of magnetic field. Stress for DPL model is greater than stress for GN-III model. Stress is
oscillating and converging towards zero with the increase of z.
Fig.10
Comparison of zz with respect to z for GN and DPL.
676 S. Biswas and S.M. Abo-Dahab
© 2020 IAU, Arak Branch
In Fig. 11, comparison of xz with respect to z for GN-III and DPL model is presented. It is noticed that
xz
increase with the increase of magnetic field. Stress for DPL model is greater than stress for GN-III model. Stress is
oscillating and converging towards zero with the increase of z.
Fig.11
Comparison of xz with respect to z for GN and DPL.
In Fig. 12, comparison of yz with respect to z for GN-III and DPL model is presented. It is noticed that
yz
decrease with the increase of magnetic field. Stress for DPL model is greater than stress for GN-III model. Stress is
oscillating and converging towards zero with the increase of z.
Fig.12
Comparison of yz with respect to z for GN and DPL.
In Figs. 13-21, displacements and stresses for rigidly fixed surface are presented. In Fig. 13, variation of u with
respect to z for three-phase-lag model is presented. It is observed that u increases with the increase of magnetic field.
u is showing oscillatory behavior and converging towards zero with the increase of z.
Fig.13
Comparison of u with respect to z.
In Fig.14 and 15, variation of v and w with respect to z for three-phase-lag model is presented. It is noticed that
v and w increase with the increase of magnetic field. Displacements are showing oscillatory behavior and
converging towards zero with the increase of z.
Three Dimensional Thermal Shock Problem …. 677
© 2020 IAU, Arak Branch
Fig.14
Comparison of v with respect to z.
Fig.15
Comparison of w with respect to z.
In Fig. 16, variation of xz with respect to z for TPL model is presented. It is noticed that
xz increase with the
increase of magnetic field. Stress is oscillating and converging towards zero with the increase of z.
Fig.16
Comparison of xz with respect to z.
In Figs. 17 and 18, variation of u and v with respect to z for GN-III and DPL model is presented. It is noticed
that u and v increase with the increase of magnetic field. Displacements for DPL model are greater than
displacements for GN-III model. Displacements are showing oscillatory behavior and converging towards zero with
the increase of z.
Fig.17
Comparison of u with respect to z for GN and DPL.
678 S. Biswas and S.M. Abo-Dahab
© 2020 IAU, Arak Branch
Fig.18
Comparison of v with respect to z for GN and DPL.
In Fig. 19, variation of w with respect to z for GN-III and DPL model is presented. It is noticed that w increase
with the increase of magnetic field. Displacement for DPL model is greater than displacement for GN-III model.
Displacement is converging towards zero with the increase of z.
Fig.19
Comparison of w with respect to z for GN and DPL.
In Fig. 20, comparison of xz with respect to z for GN-III and DPL model is presented. It is noticed that
xz
increase with the increase of magnetic field. Stress for DPL model is greater than stress for GN-III model. Stress is
oscillating and converging towards zero with the increase of z.
Fig.20
Comparison of xz with respect to z for GN and DPL.
In Fig. 21, comparison of yz with respect to z for GN-III and DPL model is presented. It is noticed that yz
decrease with the increase of magnetic field. Stress for DPL model is greater than stress for GN-III model. Stress is
oscillating and converging towards zero with the increase of z.
Fig.21
Comparison of yz with respect to z for GN and DPL.
Three Dimensional Thermal Shock Problem …. 679
© 2020 IAU, Arak Branch
9 CONCLUSIONS
The present article provides a detail analysis of propagation of thermoelastic disturbances in an orthotropic medium
in the presence of a time dependent thermal shock. With the view of theoretical analysis and numerical computation,
we can conclude the following phenomena:
(i) This problem with three-phase-lag model is more general as other problem with different thermoelastic
models can be derived as a special case from this.
(ii) Displacements decrease with the increase of magnetic field for TPL model but displacements increase with
the increase of magnetic field for GN-III and DPL model.
(iii) Displacements and stresses are showing oscillatory behavior and converging towards zero with the increase
of distance.
(iv) Displacements and stresses are showing similar nature for both traction free and rigidly fixed surface.
(v) Displacements and stresses for DPL model are greater than displacements and stresses for GN-III model.
The results presented in this article may be useful for researchers who are working on material science,
mathematical physics and thermodynamics with low temperatures as well as on the development of the hyperbolic
thermoelasticity theory.
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