effect of magnetic field on transient wave in ...agrasen university, baddi, solan– 174103 (h. p.)...

International Journal of Computational and Applied Mathematics.

ISSN 1819-4966 Volume 12, Number 2 (2017), pp. 343-364

© Research India Publications

http://www.ripublication.com

Effect of Magnetic Field on Transient Wave in

Viscothermoelastic Half Space

D.K. Sharma1*, Inder Parkash1, S.S. Dhaliwal2, V. Walia3 and S. Chandel4

1Department of Mathematics, School of Basic and Applied Sciences, Maharaja Agrasen University, Baddi, Solan– 174103 (H. P.) India.

2Department of Mathematics, Sant Longowal Institute of Engineering and Technology Sangrur– 148106 (Punjab), India.

3Department of Mathematics, Govt. Degree College Tissa, District Chamba—176316, (H. P.) India.

4Department of Physics, Govt. Post Graduate College Bilaspur- 174001, (HP) India

Abstract

In this paper there is a study of generalized homogeneous isotropic magneto

viscothermoelastic solid half space have taken for investigation. By applying a

combination of Laplace and Fourier transform techniques on differential

equations in the context of generalized theories of thermoelasticity to obtain

equations in transformed domain. The disturbance due to strip load in a

homogeneous isotropic magneto viscothermoelastic half space has taken under

consideration. The numerical inverse Laplace transform technique have been

applied to obtain the exact closed algebraic expressions for the displacements,

temperatures, stresses and perturbation of electric and magnetic fields as a

function of time and horizontal distances, which are valid for all epicentral

distances. The displacements, stresses, temperatures and perturbation of

electric and magnetic fields are obtained in the physical domain, have been

computed numerically in MATLAB software tools and presented graphically.

A complete and comprehensive analysis of the results valid in different

theories of magneto viscothermoelasticity has also been made.

Keywords: Strip load; Transient wave; Relaxation time; Inverse Laplace

transform technique.

mailto:[email protected]

344 D.K. Sharma, Inder Parkash, S.S. Dhaliwal, V. Walia and S. Chandel

INTRODUCTION

Electric field is a region of space in which at every point a force would be felt by unit

charge when placed at that point in a region. Electric fields can be produced by

electric charges or by moving magnetic fields. The increasing interest in the theory of

thermoelasticity has been felt due to engineering applications in the magnetic storage

elements, structural elements and corresponding measurement techniques of

viscothemroelasticity. Love [1] took such type of problems in the context of classical

and non classical theories of thermoelasticity. Dhaliwal and Singh [2] and Graff [3]

have given more attention to such type of problems. Theoretical study on

development of magneto thermoelastic waves were carried out in detail by Paria [4].

Boit [5] firstly introduced the concept of classical coupled thermoelasticity. The

theory of elasticity, coupled thermoelasticity, generalized thermoelasticity and waves

in solids of thermoelasticity are well established in Nowacki [6]. The governing

equations in classical dynamic coupled thermoelasticity theory (CT), wave type

equation of motion and equation of heat conduction are of diffusion type. The energy

equation extends to infinity, implying that if a homogenous isotropic and anisotropic

elastic medium is subjected to mechanical and thermal disturbances, the effect of

thermal variations can be observed an infinite distance from the source of disturbance,

which is physically impossible. To remove this drawback, some researchers, such as

Lord and Shulman (LS) [7] modified Fourier law of heat conduction and constitutive

relations so as to get a hyperbolic equation that admits a finite speed of thermal

signals for heat conduction equation. To take into account the coupling effect between

temperature and strain fields, this work includes time needed for the acceleration of

heat flow for isotropic and anisotropic materials. The (LS) theory contains heat flux

and time derivative act as time relaxation parameter. The theory of thermoelasticity

has also been modified by Green and Lindsay (GL) [8] to have proved that second

sound effects, which are short, lived. This theory is based on modification on entropy

production inequality and developed temperature rate dependent theory introduces

two time relaxation parameters, one for mechanical and other for thermal one, which

do not violate classical Fourier law of heat conduction and predicts finite speed of

heat propagation. Chand et al. [9] presented an investigation on the distribution of

deformation stresses and magnetic field in a uniformly rotating homogeneous

isotropic, thermally conducting, elastic half space. Sharma and Thakur [10] studied

the propagation of plane wave homogenous isotropic conducting plate, rotating

normal to its faces about angular velocity in the context of generalized magneto

Effect of Magnetic Field on Transient Wave in Viscothermoelastic Half Space 345

thermoelasticity. Arani and Amir [11] studied the semi analytical solution of magneto

thermoelastic problem in functionally graded hollow rotating disc. Sherief and Ezzat

[12] studied thermal shock problem in magneto thermoelasticity with one relaxation

time parameter. Ezzat et al. [13] investigated the equations of magneto

thermoelasticity for conducting isotropic media in the context of generalized

thermoelasticity. Nayfeh and Nasser [14] used the Cagniard–De Hoop method to

study the transient behavior of thermoelastic waves in a solid half space. In 1960

Bland [15] had been established the theory of linear viscoelasticity. Several

mathematical models have been studied by Biot [16] and Hunter et al. [17] to have

accommodated the energy dissipation in vibrating and non vibrating solids. It was

observed that internal friction produces attenuation, dispersion and thermoelastic

damping. Roychoudhuri and Mukhopadhyay [18] studied infinite generalized

thermoviscoelastic solid of Kelvin-Voigt type, when the entire medium rotates with a

uniform angular velocity with relaxation times. Othman et al. [19] produced the exact

expressions for the temperature distributions, thermal stresses and the displacement

components in the context of (GL) theory of thermoelasticity in a thick plate subjected

to a time–dependent heat source on each face and a heated punch moving across the

surface of a thermoviscoelastic half–space. Sharma [20] studied interactions in an

infinite Kelvin–Voigt type viscoelastic model on the plate to obtain amplitudes of

temperature and displacements in the context of coupled thermoelasticity. Godara et

al. [21] studied two-phase medium consisting of a homogeneous, isotropic, perfectly

elastic half-space in smooth contact with a homogeneous, orthotropic, perfectly elastic

half-space.

There is no problem of dynamical generalized magneto viscothermoelasticity, which

is solved completely to obtain the results in a closed form by using the different types

of methods which are available in the literature. This motivated the authors to

investigate the response of homogeneous magneto viscothermoelastic half space by

using the inverse Laplace and Fourier transformation techniques. The present paper is

aimed at the investigation of the disturbance due to strip load in a homogeneous

isotropic magneto viscothermoelastic solid half space by using a combination of

Laplace and Fourier transforms in the context of generalized theories of LS and G L

of thermoelasticity. The numerical inverse Laplace transform integrals have been

evaluated to obtain the exact closed algebraic expressions for the field functions.

Computer simulated numerical results have been presented graphically for the carbon

steel material.


FORMULATION OF PROBLEM

We consider wave motion of homogeneous isotropic magneto viscothermoelastic and

elastic solid half space, at initial temperature 0T and initially magnetic field 0H .

The nature of material is taken as viscoelastic, described by the Kelvin – Voigt model

of linear viscoelasticity described in Roychoudhuri and Mukhopadhyay [18].

Introducing initial magnetic field ),0,0( 0H in viscothermoelastic cartesian co-

ordinate system ),,( zyx with origin at any point O of the plane boundary

0z and axisz pointing normally to half space which is represented by 0z .

The wave motion is caused by a strip load which is applied along y–axis at plane

boundary of undisturbed magneto viscothermoelastic solid half space which implies

that y–component of displacement vector vanishes everywhere and remaining

quantities are independent of y–coordinate shown in Fig. 1. The basic equations of

homogenous, isotropic, magneto viscothermoelastic body are described in Dhaliwal

and Singh [2] and Paria [4].

STRAIN–DISPLACEMENT RELATION

)3,2,1,(,2

,,

ji

uue ijji

ij (1)

-a a

Mechanical

Load

x

y

z Fig.1. Geometry of the Problem

Poling


MAXWELL EQUATIONS

00 hHuehe .,)( 00, (2)

STRESS–STRAIN–TEMPERATURE RELATIONS

F ijkijijkkij TtTee )(2 21

*** (3)

where F is Lorenz force whose value BJF is the one constitutive equation

for magnetic body force. J is electric current density and B is magnetic induction.

The displacement vector ),,(),,( wvutzx u and temperature

change ),,( tzxT and total magnetic field H in linear generalized magneto

viscothermoelasticity, in the absence of body forces and heat sources satisfy the

governing differential equations described by Sharma and Thakur [10]:

EQUATION OF MOTION

u-uu.euu*

0 )()(.)( 12

2

0

2

0

2*** TtT k2

0000 HHHH (4)

EQUATION OF HEAT CONDUCTION

)1k utTTtTCTK e 0

*

01

2 .()( u (5)

where

,,,,)( 02

2

2

22

00 hzx

HHHBHueJ 0

0

;1*,),0,(,),,(,),0,0( 03132100

teehhhH ehH (6)

T

Ttt)23(

,23,1*,1* 10001

,

Here collective previous equations constitute a complete system of generalized

magneto viscothermoelasticity equations for the medium with a perfect electrical

conductivity; 0H is initial magnetic field, hHH 0 is total magnetic field.

),0,( 31 eee and ),,( 321 hhhh are perturbation of electric and magnetic

fields, )3,2,1,(and jieijij are the stress and strain components;

** and are Lame’s viscoelastic constants for viscoelastic solid; and * are the

thermoelastic and viscothermoelastic coupling constants; 10 , are viscoelastic

relaxation times, , are Lame parameters, T is the coefficient of linear thermal


expansion, is mass density, eC is the specific heat at constant strain, K is

coefficient of thermal conductivity, 0t and 1t are the thermal relaxation times; is

the electrical conductivity, 0 is magnetic permeability, and 2,1; iik is

Kronecker delta, which describes Lord-Shulman (LS) theory for 1k and Green-

Lindsay (GL) theory for 2k . The superposed dots represent time differentiation.

BOUNDARY CONDITIONS

When boundary surface of half space is subjected to mechanical strip load, the

boundary conditions on the surface at 0z are

tax

taxtfzz

;0

;0 (7a)

,0xz (7b)

0*, THT z (7c)

0,0,0 231 hhRRihv

H

H (7d)

In this paper we consider three types of loads on the plane boundary for which f t

is as defined below

loadperiodicfor;cos

loadimpactfor;)(

loadcontinousfor;)(

)(

tttH

tf (8)

where 1i is imaginary number defined in equation (7d), 0 is constant of

normal stress, 2

1

2

0

cH

RH

, is resonant frequency 0and is permeability

of the free space. *H is the surface heat transfer coefficient of the medium. Here

* 0H corresponds to thermally insulated conditions and *H refers to

isothermal boundary conditions. According to Nayfeh and Nasser [14], the condition

at infinity requires the solution to be bounded as z becomes large. Finally, the

initial conditions are such that the medium is at rest for 0t .

SOLUTION OF THE PROBLEM

To remove the complexity of the problem and convenience of analysis, we define

following non–dimensional quantities:


;,,,, *

00

*

11

0

1

**

1

*

Tucutt

cxx i

i

;,,,,,2

10

*

100

0

*

01

*

1 cC

Hce

eHh

htttt He

Vj

jj

j

;,,,,)2( 2

1

2

22*

00

01

2

000

2

cc

TTT

cHR

CT

He

T

(9)

)(2, 01

2

00

0

Tij

ij

where

21 ,

2 cc are longitudinal, shear wave velocities and

KCe )2(*

is characteristic frequency of the medium, T is thermo-mechanical

coupling constant. V is thermoelastic deformation and H electromagnetic

coupling constant, HR is magnetic pressure number.

Substituting the non–dimensional quantities of equations (9) in equations (1) to (5) via

equations (6) we obtained following equations:

1210

131

2

0

2

331

2

110

),(1

,)()1(,1,1

TtTt

uRu

wt

ut

ut

kH

H

(10)

3210

330111

2

131

2

0

2

),(1

,1,1,)()1(

TtTt

w

wt

wt

ut

k

(11)

),,(),,(1)( 31103100

2 wutwut

TtTT kT

(12)

01 3

2

1

2

vvReRv

t H

H

TVH

H (13)

0,0,0 1,233,31,13,11,3 hehhee H (14)

0,0)( 3,213,11,3 vhehhu VTHHVT (15)

Here comma notation is used to denote spatial derivatives and primes have been

suppressed for convenience.


METHODS USED FOR TRANSFORMATION

Laplace and Fourier transforms are defined by the mathematical relations

),,(),,()(0

pzxfdttzxfetfL pt

(16)

),,(ˆ),,()( pzqfdttzxfepfF xqi

(17)

On applying these transforms in system of equations (10) to (15) we obtain

0ˆˆ)1(ˆ1

*

0

*22*

0

2*

1

2

TiqpwiqDuRpqD

H

H (18)

0ˆˆ))((ˆ)1( 1

*

0

2*

1

22*

0

*2 TDpwpqDuDiq (19)

0ˆ))((ˆˆ0

222

0

3*

00

3*

0 TpqDwDpuiqp TT (20)

3

22 ˆ1

ˆ)( eRp

vQDTVH

H

(21)

,0ˆˆ13 eDeiq (22)

0ˆˆ23 hiqe H (23)

,0ˆˆ31 hDhiq (24)

,0)ˆˆ(ˆ13 hDhiqup HVT (25)

0ˆˆˆ21 vphDe VTH (26)

where *

2

1

2

01*

11

1*

00

1*

00

1

1,,,

pppp (27)

,,,, 120

1

010

1

00

1*

00

1

kk tptptpp

*

1

2*

1

2

22

pRqQH

H ;

FORMAL SOLUTION TO OBTAIN FIELD FUNCTIONS

A formal solution upon using boundary conditions is then obtained as:

dqxiqzmAAAAALhhTwu kk

kkkkk )exp(),,,,(2

1),,,,(

3

154321

1

31 (28)

dqxiqzmAAAALeehv kk

kkkk )exp(),,,(2

1),,,(

5

49876

1

312 (29)


where 1L denotes the inverse Laplace transform, and

3,2,1and9.......3,2,1; jiAij are given below

/)()()()(ˆ

2

*

2333

*

322

*

011 SHmiqVmSHmiqVm

pfFA (30)

/)()()()(ˆ

3

*

3111

*

133

*

012 SHmiqVmSHmiqVm

pfFA (31)

/)()()()(ˆ

1

*

1222

*

211

*

013 SHmiqVmSHmiqVm

pfFA (32)

3,2,1;12 kAVA kkk (33)

3,2,1;13 kASA kkk (34)

3

11224 3,2,1;

kk

k

k

H

VTk kA

qmmpA (35)

3

11225 3,2,1;

1

kk

kH

VTk kA

qmipqA (36)

5

46564 ;1

mmAA (37)

)(;

22

55

2

47522

4

474 Qmm

mpAQm

mpAH

VT

H

VT (38)

)(;

122

55

48522

4

84 QmmmpA

QmpA VTVT (39)

)(;

22

55

49522

4

94 QmmmqipA

QmiqpA VTVT (40)

,ˆ2where 0

*

0 afF )( 210

222

3

2

2

2

1 FFpqmmm ;

*

1

2*

1

21320

222

1

2

3

2

3

2

2

2

2

2

1

1)(

H

HRpFFFpqmmmmmm;

4

*

1

2

2

130

222

3

2

2

2

1 )( qpqFFpqmmm;

;*

0

10

*

0

4

1

pF T


;)1(1

*

1

2

2

2

22*

*

1

2*

0

*

0

2*

1

2

*

1

2*

1

2

4*

02

2

qRpqpqFH

H

;*

1

*

0

2*

0

2

*

1

*

0

2

2

*

1

2

2

*

0

24

3

pRqRppqpqqFH

H

H

H

;

loadperiodicfor;

loadimpactfor;1

loadcontinousfor;1

)(ˆ

22

pp

ppf

(41)

;

))()(

)()1(2*

010

422*

1

22*

0

222

*

010

4

0

222*2

kTkok

Tkkk mppqpmppqm

ppqmpiqmV

(42)

;)1()(

1

*

0

*2

1

*

0

2

22*

1

22*

0

pqiV

mppqpmpS k

k

kk (43)

;

)()()()(

)()()()(

)()()()(

1

*

1222

*

211

*

3

1

*

1333

*

311

*

2

2

*

2333

*

322

*

1

SHmiqVmSHmiqVmDSHmiqVmSHmiqVmDSHmiqVmSHmiqVmD

;3,2,1;2)21()21( 1

*

0

*

1

2*

0

2*

1

2* kSpVmiqD kkkk (44)

;2

4,

2

4 2

5

2

4

nllmmllm

;)(2

1,)(

2

1 222222

H

H

H

H RliqqQlnR

liqqQlm (45)

;)(3

4)(;

2

4 222222

32

2 qQqQMMGG

l

This is to be noticed that ),( 31 hh are coupled with displacement and temperature

components ),,( Twu and ),,( 312 eeh are coupled with displacement

component )( v .

In order to have the solution which is bounded as z , we require that the roots

)3,2,1;( kmk must have positive real parts i.e. )3,2,1,0)((Re kmk see

the ref. Nayfeh and Nasser [14]. The stresses can be computed by using the stress

strain temperature relations from equation (3). The results for coupled theory of

magneto viscothermoelasticity can be obtained by taking )0( 10 tt in relevant


results, in case of magneto viscoelasticity can be obtained by taking )0( 10 ttT

and in case of magneto elasticity can be obtained by taking )0( **

010 ttT

and 0*

0

*

1

*

0 .

A formal solution obtained for stresses by using boundary conditions:

dqxiqzmpq

qafBDL kk

kkzz )exp(sinˆ2

2

10

3

1

*1 (46)

dqxiqzmBiqVmL kkk

kkxz )exp()(2

1 3

1

1 (47)

On applying boundary conditions in equations (46) and (47) to determine the

unknown parameters, we get:

0*

0

* FBD kk (48a)

0)(( kkk BiqVm (48b)

Here *

0F have been defined above.

INVERSE TRANSFORM TECHNIQUE

To calculate analytical solutions of inverse Laplace transform, it is very difficult to

find displacements, temperatures, stresses and strains in Laplace transform domain.

So we have to solve with the help of numerical computations by (Honig and Hirdes

[22]). The outline of the numerical procedure to solve the problem is as follow:

0

),,(),,(),,( dpeztxfztxfLzpxf pt

i

i

pt dpezpxfi

zpxfLztxf ),,(2

1),,(),,( 1 (49)

where is an arbitrary real number greater than all the real parts of all singularities

of ).,,( zpxf We take Ryiyp ,; . By Honig and Hirdes [22] both the

integrals exist for Rap )(Re and the function f is locally integrable, there exist

,and00 Rat such that 0),,( ttallforkeztxf at and for all

),0( t , there is a neighbourhood in which ),,( ztxf is bounded variation.

The resulting integral can be written as:

0

),)(,(

),)(,(2

),,(

dyezyixfe

dyezyixfeztxf

tyt

tyt

(50)


For fixed values of ,and, zxq the function inside the braces in equation (50) can

be considered as Laplace transform )( ph of some function )(th . Expanding the

function we suppose ),,(),,( ztxfeztxh t in a Fourier series in the

interval ]2,0[ T , we obtain the approximate formula

DEztxfztxf ),,(),,( (51)

where Ttaaztxfk

k 20;2

),,(1

0

(52)

z

Tikxf

Ttik

Ttak ,,expReexp (53)

and DE the discretisation error, can be made arbitrarily small by choosing large

enough.

Because the infinite series in equation (52) can only be summed up to a finite number

N of terms, the approximate value of ),,( ztxf will be

Ttaaztxfk

kN 20;2

),,(1

0

(54)

Proceeding in this way we evaluate a truncational error TE that must be added to the

discretisation error to produce the total approximate error. The ‘Korrector’ method is

used to reduce the discretisation error and algorithmε is used to reduce the

truncational error and hence to accelerate the convergence. To evaluate the function

),,( ztxf we use ‘Korrector’ method formula:

DEztTxftztxfztxf ),)2(,()2exp(),,(),,( (55)

where DD EE . The approximate value of ),,( ztxf becomes

),)2(,()2exp(),,(),,( ztTxftztxfztxf NNNk (56)

where N is an integer such that .NN We shall now describe the

algorithmε that is used to accelerate the convergence of the series in equation (54).

Let N be an odd number and let

m

kkm ap

1

, be the sequence of partial sum of

equation (54). We define the sequenceε by

...,3,2,1,,1

and,0,1,

1,1,1,1,0

mnpmnmn

mnmnmmm .

It can be shown (Honig and Hirdes [22]) that the sequence

1,1,51,31,1 ,...,,, n converges to 2/0)),,( aEztxf D faster than the


sequence of partial sum ..,.3,2,1, mpm . The actual procedure used to invert the

Laplace Transform consists of using equation (56) together with the

algorithmε (Honig and Hirdes [22]).

NUMERICAL RESULTS AND DISCUSSION

To represent and illustrating the analytical results for displacements, temperatures,

stresses and perturbation of magnetic and electrical fields which are obtained in the

previous sections in the contexts of generalized theories of magneto

viscothermoelasticity, we present some numerical results. The material is chosen for

the purpose of numerical evaluation is carbon steel for which data is given below:

Table 1: Physical data for carbon steel material.

S. No. Coefficient Units Value References

1. 3mkg 3109.7 [23, 24]

2. 0T K 1.293 [23, 24]

3. 2Nm 10103.9 [24, 25]

4. 2Nm 10104.8 [24, 25]

5. eC 11 degkgJ 2104.6 [24, 25]

6. K 11 KmW 50 [25, 26]

7. T 1deg 6102.13 [25, 26]

8. 10 s 2100.6 [25, 26]

9. H 1Am 0.1 [24, 25]

10 0 1Hm 6103.1 [25, 26]

Here the values of thermal relaxation time to have been estimated from the equation

(2.5) of Chandrasekharaiah [27] and 1t is taken proportional to 0t . The convergence

analysis of numerical have been carried out through the Cauchy’s general principle of

convergence i.e. ;1 nn ff being arbitrary small number to be selected at

random in order to achieve the desire level of accuracy.


Fig. 2. (a) Horizontal displacement )(u versus epicentral distance )(x .

Fig. 2. (b) Horizontal displacement )(u versus non dimensional time )(t .

Fig. 3. (a) Vertical displacement )(w versus epicentral distance )(x .


Fig. 3. (b) Vertical displacement )(w versus non dimensional time )(t .

In this paper Figs. 2 to 9 have been presented graphically for thermally insulated

boundary conditions. Comparison has been given for coupled thermoelasticity (CT)

and generalized thermoelasticity (GT) for impact and continuous loads. Figs. 2(a) and

2(b) have been presented for horizontal displacement )(u versus epicentral distance

)(x and non–dimensional time )(t for impact and continuous loading. It can be

inferred from these figures that initially the variation of horizontal

displacements )(u is large and as we move from the centre point towards epicentral

distance )(x and non–dimensional time )(t the vibrations go on decreasing and die

out. Figs. 3(a) and 3(b) have been plotted for vertical displacements )(w versus

epicentral distance )(x and non–dimensional time )(t for impact and continuous

loading. It is revealed from these figures that initially the variations are very low

below mean position, then rises and with increase in value of )(x and non–

dimensional time )(t the variations go on decreasing and die out.


Fig. 4. (a) Temperature change )(T versus epicentral distance )(x .

Fig. 4. (b) Temperature change )(T versus non dimensional time )(t .

Figs. 4(a) and 4(b) have been plotted for temperatures )(T versus epicentral distance

)(x and non–dimensional time )(t for impact and continuous loading. It is noticed

that the variation of temperatures )(T in Fig. 4(a) achieved maximum variation at

5.1x then become asymptotic with increasing value of )(x . But the Fig. 4(b)

shows the high variation in small time and with increase in value of )(t the variation

of vibrations go on deceasing and die out. This is to be observed from above

discussion that in case of generalized themoelasticity (GT), the behavior is large as

compared to coupled thermoelasticity (CT). Figs. 5 and 6 have been plotted for

perturbation of magnetic fields )(and)( 31 hh versus epicentral distance )(x for

impact and continuous loading. It can be inferred for these figures that there is a trend

of sinusoidal wave type vibrations.


Fig. 7 has been presented for horizontal displacement )(u , vertical

displacement )(w , temperature )(T and perturbation of magnetic fields

)(and)( 31 hh versus resonant frequencies )( for periodical loading. It is noticed

from this figure that variation of displacements and temperature is linear until

)1( after that increase in resonant frequencies the transient behavior of variations

die out at )1000( and in case of perturbation of magnetic fields )(and)( 31 hh ,

the variation is linear until )01.0( and with increase in value of the variation

is die out. From the trends of variations of Fig. 7, it is noticed that with large value of

)( , the variations of displacements )(and)( wu , temperature and perturbation of

magnetic fields )(and)( 31 hh becomes low. On the other hand, with increasing

values of )( , the variation of displacements, temperature and perturbation of

magnetic fields are high. This is due to the effect of resonance.

Figs. 8 shows the comparison of normal stress )( zz versus epicentral

distance )(x with and without magnetic field )0 h H(H respectively for coupled

thermoelasticity (CT) and gereralized thermoelasticity (GT) have been plotted for

thermally insulated boundary conditions for impact and continuous loads. It can be

inferred from Figs. 8 that the vibrations initially rises, achieved maximum variation at

)25.0( x and with increase in value of )(x the vibrations go on decreasing and die

out. Here the vibrations in presence of magnetic field, the variations are low, while in

absence of magnetic field the variations are high.

Fig. 5. Perturbation of magnetic field )( 1h versus epicentral distance )(x .


Fig. 6. Perturbation of magnetic field )( 3h versus epicentral distance )(x .

Fig. 7. Variation of displacements, temperature and perturbation of magnetic fields

)(and)( 31 hh versus resonant frequencies )( .

Fig. 8. Normal stress distribution )( zz versus epicentral distance )(x .


Fig. 9. Shear stress distribution )( xz versus epicentral distance )(x .

Figs. 9 shows the comparison of shear stresses )( xz versus distance )(x with and

without magnetic field )0 h H(H respectively have been presented for impact and

continuous loads. It is noticed from this Fig. 9 that the variation of vibrations is

initially decreasing and achieved maximum variation )25.0( x with increase in

epicentral distance )(x the variations increases and die out with increasing the value

of )(x . Here the variation is meager in the presence of magnetic field, where as the

variations are large without magnetic field. This shows that effect of magnetic field.

The effect of perturbation on electric fields has found negligible in case of

temperature, viscous effects and strip load.

CONCLUDING REMARKS

This paper studied the magneto viscothermoelastic half space in forced vibrations of

mechanical strip loadings in thermally insulated and isothermal boundary conditions.

The analytical and numerical results permit some concluding remarks:

1. The problem has been investigated with the help of non classical theories of

thermoelasticity based on Lord Shulman (LS) and Green Lindsay (GL).

2. The values of physical functions converge to zero with increase in distance

)( x and in the context of generalized theories of thermoelasticity and all

functions are continuous.

3. The problem have been analyzed and investigated with the help of three types

of loads i.e. impact loading, continuous loading and periodic loading.


Perturbation of electric fields is independent of temperature, loading and

viscous effects.

4. Effect of periodic loading has been shown for resonant frequencies. The Fig. 7

clearly shows that with increase in resonant frequencies, the variation of

displacements, temperature and perturbation of magnetic fields die out. As the

resonant frequencies are low, higher is the variation and with increase in

resonant frequencies, lower the variations.

5. In case of stresses, the variation is high without magnetic field and low with

magnetic field. Perturbation of magnetic fields is dependent on temperature

and loading. As the perturbation of magnetic field increases the variation of

stresses decreases. This clearly indicates the effect of magnetic field on

stresses.

6. The present theoretical results may provide interesting information and

mathematical foundation for working on the subject, because the increasing

interest in the theory of thermoelasticity has many engineering applications

such as magnetic storage elements, structural elements and corresponding

measurement techniques of magneto viscothermoelasticity which enrich this

work.

7. Perturbation of magnetic field is used to reduce high variation of stress

analysis. Study may also find useful and wide applications in the design and

construction of sensors and other acoustic waves in addition to possible bio

industries.

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