A STUDY OF GREEN’S FUNCTIONS FOR TWO-DIMENSIONAL …

A STUDY OF GREEN’S FUNCTIONS FOR TWO-DIMENSIONAL

PROBLEM IN ORTHOTROPIC MAGNETOTHERMOELASTIC

MEDIA WITH MASS DIFFUSION

Rajneesh Kumar*, Vijay Chawla**

Department of Mathematics, Kurukshetra University, Kurukshetra, 136119, Haryana, India

*e-mail: [email protected]

**e-mail: [email protected] Abstract. The present investigation deals with the study of Green’s functions for two-dimensional problem in orthotropic magnetothermoelastic media with mass diffusion. After applying the dimensionless quantities and using the operator theory, two-dimensional general solution in orthotropic magnetothermoelastic diffusion media is derived. On the basis of general solution, the Green’s functions for a steady line on the surface of a semi-infinite orthotropic magnetothermoelastic diffusion material are constructed by four newly introduced harmonic functions. The components of displacement, stress, temperature distribution and mass concentration are expressed in terms of elementary functions. From the present investigation, some special cases of interest are also deduced and compared with the previous results obtained. The resulting quantities are computed numerically for semi-infinite magneto thermoelastic material and presented graphically to depict the effect of magnetic. 1. Introduction Fundamental solutions or Green’s functions play an important role in both applied and theoretical studied on the physics of solids. Fundamental solutions can be used to construct many analytical solutions solving boundary value problems of practical problems when boundary conditions are imposed. They are essential in boundary element method (BEM) as well as the study of cracks, defects and inclusion. Many researchers have been investigated the Green’s function for elastic solid in isotropic and anisotropic elastic media, notable among them are Freedholm [1], Lifshitz and Rezentsveig [2], Elliott [3], Kröner [4], Synge [5] , Lejcek [6], Pan and Chou [7], and Pan and Yuan [8].

When thermal effects are considered, Sharma [9] investigated the fundamental solution for transversely isotropic thermoelastic material in an integral form. Chen et al. [10] derived the three dimensional general solution for transversely isotropic thermoelastic materials. Hou et al. [11, 12] investigated the Green’s function for two and three-dimensional problem for a steady Point heat source in the interior of a semi-infinite thermoelastic materials. Also, Hou et.al [13] investigated the two dimensional general solutions and fundamental solutions for orthotropic thermoelastic materials.

The theory of magnetothermoelasticity is concerned with the interacting effects of the applied magnetic field on the elastic and thermoelastic deformation of a solid body. This theory has drawn the attention of many researchers because of its extensive uses in diverse fields, such as geophysics for understanding the effect of Earth’s magnetic field on seismic waves, damping of acoustic waves in a magnetic field. Kolaski and Nowacki [14] studied the magnetothermoelastic disturbance in a perfectly conducting elastic half-space in contact with

Materials Physics and Mechanics 15 (2012) 78-95 Received: September 29, 2012

© 2012, Institute of Problems of Mechanical Engineering

vacuum due to applied thermal disturbance on the plane boundary. Othman and Song [15] investigated Reflection of magnetothermoelastic waves with two relaxation times. Hou et al. [16] investigated the general solution and fundamental solution for orthotropic magnetothermoelastic materials.

Diffusion is defined as the spontaneous movement of the particles from a high concentration region to the low concentration region and it occurs in response to a concentration region and it occurs in response to a concentration gradient expressed as the change in the concentration due to change in position. Thermal diffusion utilizes the transfer of heat across a thin liquid or gas to accomplish isotope separation. Today, thermal remains a practical process to separate isotopes of noble gases (e.g. xexon) and other light isotopes (e.g. carbon) for research purpose.

Nowacki [17-20] developed the theory of thermoelastic diffusion by using coupled thermoelastic model. Sherief et al. [21] developed the generalized theory of thermoelastic diffusion with one relaxation time which allows finite speeds of propagation of waves. When diffusion effects are considered, Kumar and Chawla [22] investigated the fundamental solution in orthotropic thermoelastic diffusion material. Kumar and Chawla [23] studied the Green’s functions for two-dimensional problem in orthotropic thermoelastic diffusion material. Kumar and Chawla [24] derived the three-dimensional fundamental solution in transversely isotropic thermoelastic diffusion media. However, the important fundamental solution for two-dimensional problem in magnetothermoelastic material with mass diffusion has not been discussed so far in the literature.

The Green’s functions for two-dimensional problem in orthotropic magnetothermoelastic diffusion medium are investigated in this paper. Based on the two-dimensional general solution of orthotropic magnetothermoelastic diffusion media, the Green’s functions for a steady line heat source on the surface of a semi-infinite magnetothermoelastic diffusion material are obtained by four newly introduced harmonic functions. From the present investigation, some special cases of interest are also deduced. 2. Basic equations Following Ezzat [25], the simplified linear equations of electrodynamics of slowly moving medium for a homogeneous and perfectly conducting elastic solid are given by

0 ,curlt

E

h J (1)

0 ,curlt

h

E (2)

0 0 ,t

u

E H (3)

0,div h (4)

where 0H is the external applied magnetic field intensity vector, h is the induced magnetic

field vector, E is the induced electric field vector, J is the current density vector, u is the displacement vector, 0 and 0 are the magnetic and electric permeabilities respectively.

The above equations (1)-(4) are supplemented by equations of motion and constitutive relations in the theory of generalized thermoelastic diffusion, taking into account the Lorentz force (Eringen [26]).

79A study of Green’s functions for two-dimensional problem...

(i) Constitutive relations:

,Cbac ijijkmijkmij (5)

(ii) Equations of motion:

,,,, iijijjijjkmijkm uFCbTaec (6)

(iii) Equation of heat conduction:

,,00 ijijijijE KaCaC (7)

(iv) Equation of mass diffusion:

.,, **,

* CabCb ijijijijijkmkmij (8)

Here, )( ijmkjikmkmijijkm cccc are elastic parameters, ),( jiij aa )( jiij bb are, respectively,

the tensors of thermal and diffusion modules, is the density and EC is the specific heat at

constant strain, ba, are, respectively, coefficients describing the measure of thermoelastic

diffusion effects and diffusion effects, 0T is the reference temperature assumed to be such that

,10

T

T( ), ( )ij ji ij jiK K and

2,, ijji

ij

uu denote the components of thermal

conductivity, stress and strain tensor respectively, ),,,( tzyxT is the temperature change from

the reference temperature 0T and C is the mass concentration, iu are components of

displacement vector, )( **jiij are diffusion parameters, iF are components of Lorentz force.

In the above equations symbol (“,”) followed by a suffix denotes differentiation with respect to spatial coordinate and a superposed dot (“.”) denotes the derivative with respect to time respectively.

3. Formulation of the problem We consider homogenous orthotropic magnetothermoelastic diffusion medium. Let us take Oxyz as the frame of reference in Cartesian coordinates, the origin O being any point on the plane boundary.

For two-dimensional problem, we assume the displacement vector, temperature change and mass concentration are, respectively, of the form

( ,0, ),u wu ),,,( tzx ),,( tzxC , (9)

and Lorentz force is taken in the form (for two dimensional problem):

),(2

2

002001 t

u

x

eHF

(10a)

)(2

2

002003 t

w

z

eHF

, (10b)

80 Rajneesh Kumar, Vijay Chawla

where

.z

w

x

ue

Moreover, we are discussing static problem

.0

t

T

t

C

t

w

t

u (11)

We define the dimensionless quantities as:

*1

1 11 11

1( , , ) ( , , , ), ( , ) ( , ),x z u w x z u w T C a T b C

v c

1 1*

1 0 11 1 1

, ,ijij

a vH H

a c K

where

,121 bv

,

1

11*1 K

ac (12)

and 1b is the tensor of diffusion modules and 1K is the component of thermal conductivity.

Equations (5)-(8) for orthotropic materials, with the aid of Eqs. (9)-(12), after suppressing the primes, yields:

,02

32

2

22

2

1

Tx

Cx

wzx

uzx

(13)

0212

2

42

2

2

2

3

Tz

Cz

wzx

uzx

, (14)

,02

2

32

2

Tz

Tx

(15)

2 2 2 2 2 2 2 2

* * * * * * * *1 3 2 4 5 6 7 82 2 2 2 2 2 2 2

0,q q u q q w q q C q q Tx x z z x z x z x z

(16) where

20 0

111

1 ,H

c

2 * 22 3 4 55 13 55 0 0 1 33 0 0

11

1( , , ) , , ,c c c H c H

c 3

11

,b

b


32

1

,a

a

* ** ** * * *3 3 11 1

3 1 2 1 3 3 4 1 31 11 11

, ( , ) , , ( ) , ,K

q q b b q q b bK c c

* *

* * * * * * * *1 15 6 1 3 7 8 1 3

1 1

( , ) , , ( , ) , .b a

q q q qb a

The equations (13)-(16) can be written as

.0,,, tTCwuD (17) where D is differential operator matrix given by

2 2 2

1 2 32 2

2 2 2

3 2 4 1 22 2

2 2 2 2 2 2 2 2* * * * * * * *1 3 2 4 5 6 7 82 2 2 2 2 2 2 2

2 2

32 20 0 0

x z x z x x

x z x z z z

q q q q q q q qx x z z x z x x x x

x z

.

(18)

Equation (17) is a homogeneous set of differential equations in , , ,u w C T . The general solution by the operator theory as follows

1 2 3 4, , , , ( 1, 2, 3, 4)i i i iu A F w A F C A F T A F i (19)

where ijA are algebraic cofactors of the matrix D, of which the determinant is

,2

2

32

2

6

6*

24

6*

42

6*

6

6*

zxx

dzx

czx

bz

aD (20)

where

* * * * * * * * * * * *2 1 4 4 6 1 1 4 4 6 2 2 6 4 5 2 1 2 3 6 7 4 2 1 3 4( ), ( ) ( ) ( ) ( ) ,a q q b q q q q q q q q

* * * * * * * * * * * *

1 1 2 4 5 2 1 6 2 5 2 3 5 1 1 3 2 4 1 2 7 2 1 5 1( ) ( ) ( ) , ( ).c q q q q q q q q q d q q

The function F in equation (19) satisfies the following homogeneous equation:

.0FD (21)


It can be seen that if 1, 2, 3i are taken in equation (19), three general solution are obtained in which 0T . These solutions are identical to those without thermal fact and are not discussed here. Therefore if 4i should be taken in equation (19), the following solution is obtained

,4

4

122

2

14

2

1 x

F

zr

xzq

xpu

,4

4

222

2

24

2

2 z

F

zr

xzq

xpw

,6

6

342

6

324

6

36

6

3 Fx

lxz

rxz

qz

pC

,6

6*

42

6*

24

6*

6

6* F

xd

xzc

xzb

zaT

(22)

* * * * * * * * * *

1 7 5 2 1 2 1 7 5 2 2 6 8 4 7 8 8 1 2( ) , ( ) ( ) ( ) ,p q q q q q q q q q q q

,)()()( *

414*6

*4

*8

*64

*82

*62

*8121 qqqqqqqqr

* * * * * * *

2 3 5 7 1 2 1 1 1 7 2 5 2 1 1 8 2 6( ) ( ) ( ), ( ),p q q q q q r q q

),()()()( 12

*7

*6

*83

*52

*7122

*6

*8112 qqqqqqqq ,)( 2

*84

*423 qqp

* * * * * * * *

3 1 4 8 2 4 2 2 8 4 7 2 2 2 3 8 7 2 2 4 3 4( ) ( ) ( ) ( ) ,q q q q q q q q q

* * * * * * * *

3 1 2 8 4 7 2 2 7 3 7 2 2 1 3 2 1 2 7 4 1( ) ( ) ( ) ( ) ,r q q q q q q q q ).( *1

*7123 qql

Equations (21) can be rewritten as

2 24

2 21

0,j j

Fx z

(23)

where

14

3

,j j

Kz s z s

K

and ( 1, 2, 3)js j are three roots (with positive real part) of the following algebraic

equation:

.0*2*4*6* dscsbsa (24)


As known from the generalized Almansi theorem (Ding et al. [10]), the function F can be expressed in terms of four harmonic functions: (i) 4321 FFFFF for distinct ( 1, 2, 3, 4)js j ; (25a)

(ii) 4321 zFFFFF for 1 2 3 4s s s s ; (25b)

(iii) 4

2321 FzzFFFF for 1 2 3 4s s s s ; (25c)

(iv) 4

33

221 FzFzzFFF for 1 2 3 4s s s s . (25d)

Here ( 1, 2,3, 4)jF j satisfies the following harmonic equation:

).4,3,2,1(02

2

2

2

jF

zxj

j

(26)

The general solution for the case of distinct roots, can be derived as follows

4

15

5

2

4

14

5

1 ,,j j

jjj

j j

jj z

Fpsw

zx

Fpu .,

6 4

164

46

44

4

163

jj j

jj z

FpT

z

FpC (27)

In the similar way general solution for the other three cases can be derived. Equation (23) can be further simplified by taking

.4

4

1 jj

jj z

Fp

(28)

Using the formula (23) in equation (22) gives

4

1

,j

j

ux

4

11

,jj j

j j

w s Pz

24

2 21

,jj

j j

C Pz

24

434 2

1 4

,j

T Pz

(29)

where

.,, 144434132121 ppPppPppP jjjjjj (30)

The function j satisfies the harmonic equations:

2 2

2 20 1, 2, 3, 4.j

j

jx z

(31)

Making use of Eqs. (9), (11) and (12) in equation (1) and after suppressing the primes, with the aid of Eq. (29), we obtain:


,4

12

2

213112

11

j j

jjjjjxx z

PfPfPsff

(32a)

,4

12

2

233212

12

j j

jjjjjzz z

PhPhPshf

(32b)

,)1(4

1

2

14

j j

jjjzx zx

sPh

(32c)

where

,0333231 PPP (33a)

and

.,,,,,1

),,,,,( 551

113

1

113331311

01432121

c

b

cb

a

caccc

Tahhhhff (33b)

Substituting the values of zzxx , and zx from Eq. (32) in equations (6)-(7), with the aid of

Eqs. (9), (11) and (12), gives

,)1( 21421311

221 jjjjjj sPhPfPfPsff

),1( 1423321

212 jjjjj PhPhPhPshf

.01 3

23 jj Ps (34)

The general solution (32) with the help of Eq. (34) can be simplified as

2 2 24 4 42

1 1 12 21 1 1

, , ,j j jxx j j zz j zx j j

j j jj j j

s w w s wz z x z

(35)

where

.)1( 33232

11214212132

211

1 hPhPshPfPhs

fPfPfsPfw jjjjj

j

jjjjj

(36)

4. Green’s functions for a steady line heat source in a semi infinite orthotropic

magnetothermoelastic diffusion material As shown in Fig. 1 we consider a semi-infinite orthotropic magnetothermoelastic diffusion material 0z . A linear heat source H is applied at the line ),0( h in two dimensional


Cartesian coordinate ),( zx and the surface 0z is free, impermeable boundary and thermally insulated. The general solution given by equations (29) and (35) is derived in this section.

For future reference, following notations are introduced:

,,, kjjkkkjj hzzhshzsz

).4,3,2,1,(,,, 2222 kjzxrhzzzxr jkjkkjjkjkjk (37)

b O b

hhhhhhhhh

),0( h a

H

Fig. 1. Geometry of the problem.

Green’s functions in the semi-infinite plane are assumed of the following form:

2 2 11 3( ) log tan ( )

2 2j j jj jj jjjj

xA z x r xz

z

4

2 2 1

1

1 3( ) log tan ( ) ,

2 2jk jk jk jkk jk

xA z x r xz

z

(38)

where jA and ( , 1, 2, 3, 4)jkA j k are twenty constant to be determined.

The boundary conditions on the surface )0( z are in the form of

.0,0,0

z

T

z

Czxzz (39)

Substituting the equation (38) in equations (29) and (35) gives the expressions for components of displacement, mass concentration, temperature distribution and stress components as follows:

4

1

4

1

4

1

11 ,tan)1(logtan)1(logj j k jk

jkjkjkjj

jjjjj z

xzrxA

z

xzrxAu (40a)

1a

2a

z

x

H ),0( h


4

1

4

1

4

1

11

11 ,tan)1(logtan)1(log

j j k jkjkjkjkjj

jjjjjjjjj z

xxrzAPs

z

xxrzAPsw

(40b)

4

1

4

1

4

122 ,loglog

j j kjkjkjjjjj rAPrAPC

(40c)

,loglog4

1443444434

k

kk rAPrAPT (40d)

4

1

4

1

4

11

21

2 ,loglog

j jjk

kjkjjjjjjjxx rAwsrAws (40e)

4

1

4

1

4

111 ,loglog

j jjk

kjkjjjjjzz rAwrAw (40f)

4

1

4

1

4

1

11

11 .tantan

j j k jkjkjj

jjjjjzx z

xAws

z

xAws (40g)

Considering the continuity on plane hz for w and zx gives the following expressions

,01

4

1

jjj

j APs (41)

.01

4

1

jjj

j Aws (42)

Substituting jw1 from equation (36) in equation (42) yields

.0)1( 14

4

1

jjj

j APhs (43)

By virtue of equation (41), equation (43) can be simplified to

.04

1

jj

j As (44)


When the mechanical, concentration and thermal equilibrium for a rectangle of 21 aza

)0( 21 aha and bxb are considered (Fig. 1), three equations can be obtained:

,0),(),(),(),(2

1

12

dzzbzbdxaxax

a

a

zxzx

b

b

zzzz (45a)

,),(),(),(),(2

1

123 Hdzzbx

Tzb

x

Tdxax

z

Tax

z

Ta

a

b

b

(45b)

.0),(),(),(),(2

1

12

dzzbx

Czb

x

Cdxax

z

Cax

z

Ca

a

b

b

(45c)

Some useful integrals are given as follows:

),(tan)1(loglog 1

jjjjjjjj z

xzrxdxr (46a)

),(tan)1(loglog 1

jkjkjkjk z

xzrxdxr (46b)

,tantan4

1 4

14

44

14344

k kk z

xA

z

xAPsdx

z

T (46c)

,tantan4

1 4

14

44

14

4

34

k kz

xA

z

xA

s

Pdz

x

T (46d)

,tantan4

1

12

12

k jkjjjk

jjjjj z

xPsA

z

xPsAdx

z

C (46e)

.tantan4

1

12

12

k jkj

j

jk

jjj

j

j

z

xP

s

A

z

xP

s

Adz

x

C (46f)

It is noticed that the integrals (46d) and (46f) are not continuous at hz , thus following expression should be used

,2

1

2

1

dzx

Tdz

x

Tdz

x

Ta

h

h

a

a

a

(47a)


.2

1

2

1

dzx

Cdz

x

Cdz

x

Ca

h

h

a

a

a

(47b)

Substituting equations (40f) and (40g) into equation (45a) and using the integrals (46a) and (46b), we obtain

,02

4

1

4

111

4

11

IAwIAw

kjk

jjj

jj (48)

where

22

11

1 11 (log 1) tan ( ) log tan ( ) 0,

z ax bz a x b

jj jj jj jjjj jjz a x bx b z a

x xI x r z x r z

z z

(49a)

2

2

11

1 12 (log 1) tan ( ) log tan ( ) 0.

z ax bz a x b

jk jk jk jkjk jkz a x bx b z a

x xI x r z x r z

z z

(49b)

Equations (49a) and (49b) show that the equations (45a) and (48) are satisfied automatically. Substituting the value of C from equation (40c) into equation (45c) and using the integrals (46e), (46f) and (47b), we obtain

,04

1

4

12

*4

12

j kjkjj

jjjj APrrAP (50)

where

22

11

2 1 1 1tan ( ) tan ( ) tan ( )

z h z ax bz a x b x b

j jjj jj jjz a x b x bx b z a z h

x x xr s

z z z

2 1 1

2 1

2( 1) tan tan 2jj j j j

b bs

s a s h s a s h

and

2

1

12

1

1* )(tan)(tan

az

az

bx

bxjk

bx

bx

az

azjkj z

x

z

xr


2 1 1

2 1

2( 1) tan tan .jj j j j

b bs

s a s h s a s h

Substituting equation (40d) into equation (45b) with the aid of 314 / KKs and integrals

(46c) and (46d) and (47a), yields

,1334

6

4

1454

KKP

HIAIA

kk

(51)

where

22

11

1 1 15

44 44 44

tan ( ) tan ( ) tan ( ) 2 ,

z h z ax bz a x b x b

z a x b x bx b z a z h

x x xI

z z z

(52a)

.0)(tan)(tan2

1

2

1

4

1

4

16

bx

bx

az

azk

az

az

bx

bxk z

x

z

xI (52b)

From equations (51) and (52), we obtain

.2 1334

4KKP

HA

(53)

Equation (37) at the surface 0z gives

,,,0 kjkkkj hzhshz

2222 ,, kjkkjkkjk hxrhzhxr . (54)

Substituting equations (40c), (40d), (40f) and (40g) into equation (39) with the aid of

314 / KKs and equation (54), yields

,04

111

kj

kkkjjj AwsAws (55)

4

1 11

0,j j k kjk

w A w A

(56)

4 44 0,A A 4 0,kA (57)


,04

12

22

kkjkkjj APsAP (58)

where 1, 2, 3, 4, 1, 2, 3.j k

We have determined the twenty constants jA and ( , 1, 2, 3, 4)jkA j k from twenty

equations including equations (41), (44), (50), (53), (55), (56), (57) and (58) by the method of Crammer rule.

5. Special cases

(I) In case of negligible magnetic effect Eqs. (40a)-(40g) are reduced to

4

1

4

1

4

1

11 ,tan)1(logtan)1(logj j k jk

jkjkjkjj

jjjjj z

xzrxA

z

xzrxAu (59a)

4

1

4

1

4

1

11

11 ,tan)1(logtan)1(log

j j k jkjkjkjkjj

jjjjjjjjj z

xxrzAPs

z

xxrzAPsw

(59b)

4

1

4

1

4

122 ,loglog

j j kjkjkjjjjj rAPrAPC

(59c)

,loglog4

1443444434

k

kk rAPrAPT (59d)

4

1

4

1

4

11

21

2 ,loglog

j jjk

kjkjjjjjjjxx rAwsrAws (59e)

4

1

4

1

4

111 ,loglog

j jjk

kjkjjjjjzz rAwrAw (59f)

4 4 4

1 11 1

1 1 1

tan tanzx j j j j j jkj j kjj jk

x xs w A s w A

z z

, (59g)

which are similar to the results as those obtained by Kumar and Chawla [22].

(II) In the absence of magnetic and diffusion effect

Eqs. (40a)-(40g) are reduced to

,tan)1(log3

1

1

j jjjj z

xzrxAu (60a)


,tan)1(log3

1

11

j jjjjjj z

xxrzAPsw (60b)

4 34 4log ,T A P r (60c)

3

21

1

log ,xx j j j jj

s w A r

(60d)

3

11

log ,zz j j jj

w A r

(60e)

3

11

1

tan .zx j j jj j

xs w A

z

(60f)

The above results are similar to the results obtained by Hou et al. [13]. 6. Numerical results and discussion For the purpose of numerical computation, we take the following values of the relevant parameters as:

10 -1 -211c 18.78 10 Kg m s , 10 -1 -2

13c 8.0 10 Kg m s , 10 -1 -233c 10.2 10 Kg m s ,

10 -1 -255c 10.06 10 Kg m s , 3

0 0.293 10T K , 5 11 2.98 10 K , 5 1

3 2.4 10 ,K 4 3 1

1 1.1 10c m Kg , 3 1 11 0.12 10K W m K , 3 1 1

3 0.33 10K W m K , 4 2 2 11.4 10a m s K , 5 1 5 29 10b Kg m s , * 8 3

1 0.95 10 m s Kg , * 8 33 0.90 10 m s Kg , 0 00.38, 1H

1 11 1 13 3 3 13 1 33 3 1 11 1 13 3 3 13 1 33 3, , , .c c c ca c c a c c b c c b c c

Figures 2-5 depict the variation of horizontal displacement )(u , vertical displacement

)(w , temperature distribution )(T and mass concentration )(C w.r.t .x The solid line and dotted line correspond to thermoelastic diffusion (TD) and centre symbol on these lines correspond to magnetothermoelastic diffusion (MTD).

Figure 2 depicts the variation of horizontal displacement u with x and it indicates that the values of u increases for all values of x in both cases TD and MTD. It is noticed that the values of u in case of TD remain more (in comparison with MTD). Figure 3 shows that the values of vertical displacement w slightly decrease for smaller values of ,x whereas for higher values of x the values of w oscillates. It is evident that the values of w in case of MTD remain more for smaller values of x although for higher values of x reverse behavior occurs. Figure 4 exhibits the variation of temperature distribution T with x and it indicates that the values of T slightly increases for smaller values of x whereas for higher values of x , the values of T increases monotonically. It is noticed that the values of T in case of MTD remain more (in comparison with MTD) for higher values of .x Figure 5 shows that the values of mass concentration C slightly decrease for smaller values of ,x although for higher values of x the values of C increase. It is evident that the values of C in case of TD remain more (in comparison with MTD) for higher values of .x


0 2 4 6

Fig.2 variation of horizontal displacement w.r.t. x

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ho

rizo

nta

l dis

pla

cem

en

t

TD (z=5)

TD (z=10)

MTD (z=5)

MTD (z=10)

Fig. 2. Variation of horizontal displacement w.r.t. x.

0 2 4 6

Fig.3 variation of vertical displacement w.r.t.x

0

5

10

15

20

vert

ica

l dis

pla

cem

en

t

TD (z=5)

TD (z=10)

MTD (z=5)

MTD (z=10)

Fig. 3. Variation of vertical displacement w.r.t. x.

x

x


0 2 4 6

Fig.4 variation of temperature distribution w.r.t. x

0

2

4

6

8

10

Te

mp

era

ture

dis

trib

utio

n

TD (z=5)

TD (z=10)

MTD (z=5)

MTD (z=10)

Fig. 4. Variation of temperature distribution w.r.t. x.

0 2 4 6

Fig.5 variation of mass concentration w.r.t. x

0.4

0.8

1.2

1.6

Ma

ss c

on

cen

tra

tion

TD (z=5)

TD (z=10)

MTD (z=5)

MTD (z=10)

Fig. 5. Variation of mass concentration w.r.t. x.

x

x


7. Concluding remarks The Green’s functions for two-dimensional in orthotropic magnetothermoelastic diffusion material have been derived. By virtue of the two-dimensional general solution of orthotropic magnetothermoelastic diffusion material, the Green functions for a steady line heat source on the surface of a semi-infinite plane are obtained by four newly introduced harmonic functions

( 1, 2, 3, 4)j j . The general expression for components of displacement, stress, mass

concentration and temperature change are expressed in terms of elementary functions. Since all the components are expressed in terms of elementary functions, it is convenient to use them. From the present investigation, some special cases of interest are also deduced and compared with the previous results obtained. The components of displacement, mass concentration and temperature distribution are computed numerically and depicted graphically to depict the effect of magnetic. References [1] I. Freedholm // Acta Mathem. 23 (1900) 1. [2] I.M. Lifshitz, L.N. Rozentsveig // Zhurnal Eksperimental’noi i Teioretical Fiziki, 17

(1947) 783. [3] H.A. Elliott // Proc. Cambridge Philos. Soc. 44 (1948) 522. [4] E. Kröner // Z. Phys. 136 (1953) 402. [5] J.L. Synge, The Hypercircle in Mathematical Physics (Cambridge University Press,

London, UK, 1957). [6] L. Lejček // Czech. J. Phys. B 19 (1969) 799. [7] Y.C. Pan, T.W. Chou // J. Appl. Mech. 43 (1976) 608. [8] E. Pan, F.G. Yuan // Int. J. Solid Struct. 37 (2000) 5329. [9] B. Sharma // J. Appl. Mech. 23 (1958) 86. [10] H.J. Ding // Int. J. Solid Struct. 33 (1996) 2283. [11] P.F. Hou, A.Y.T. Leung, C.P. Chen // Z. Angew. Math. Mech. 1 (2008) 33. [12] P.F. Hou, L. Wang, T. Yi // Appl. Math. Model. 33 (2009) 1674. [13] P.F. Hou, H. Sha, C.P. Chen // Engineer. Anal. Bound. Elem. 45 (2008) 392. [14] S. Kaloski, W. Nowacki // Bull. Acad. Polon Sci. Sr.Sci.Techn. 18 (1970) 155. [15] M.I.A. Othman, Y. Song // Appl. Math. Model. 32 (2008) 483. [16] P.F. Hou, T. Yi, L. Wang // J. Thermal Stress. 31 (2008) 807. [17] W. Nowacki // Bulletin of Polish Academy of Sciences. Science and Tech. 22 (1974) 55. [18] W. Nowacki // Bulletin of Polish Academy of Sciences. Science and Tech. 22 (1974) 129. [19] W. Nowacki // Bulletin of Polish Academy of Sciences. Science and Tech. 22 (1974) 275. [20] W. Nowacki // Proc. Vib. Prob. 15 (1974) 105. [21] H.H. Sherief, H. Saleh // Int. J. of Solid and Struct. 42 (2005) 4484. [22] R. Kumar, V. Chawla // Int. Commun. Heat Mass Trans. 38 (2011) 456. [23] R. Kumar, V. Chawla // Engineering Analysis with Boundary Elements 36 (2012) 1272. [24] R. Kumar, V. Chawla // Theoretical and Applied Mechanics 39 (2012) 165. [25] M.A. Ezzat // Int. J. Eng. Sci. 35 (1997) 741. [26] A.C. Eringen, Foundations and Solids, Microcontinuum Field Theories (Springer-

Verlag, New York, 1999).


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A STUDY OF GREEN’S FUNCTIONS FOR TWO-DIMENSIONAL …