Three Dimensional Thermal Shock Problem in Magneto ...jsm.iau-arak.ac.ir/article_677284_317712f8e20be9749b2ec...ij component of stress tensor, i j x y z, , , , u displacement vector

© 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: [email protected] (S. Biswas).

G

mailto:[email protected]

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


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



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)



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

,



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



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 ,


A A A A A A 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

(

) ,


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

(

),


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


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)



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:



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



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




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






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



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.



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


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


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.



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.



In Fig. 11, comparison of xz with respect to z for GN-III and DPL model is presented. It is noticed that

xz



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


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.



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.



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



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


Fig.21

Comparison of yz with respect to z for GN and DPL.



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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Three Dimensional Thermal Shock Problem in Magneto ...jsm.iau-arak.ac.ir/article_677284_317712f8e20be9749b2ec...ij component of stress tensor, i j x y z, , , , u displacement vector