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
81A study of Green’s functions for two-dimensional problem...
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)
82 Rajneesh Kumar, Vijay Chawla
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)
83A study of Green’s functions for two-dimensional problem...
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:
84 Rajneesh Kumar, Vijay Chawla
,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
85A study of Green’s functions for two-dimensional problem...
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
86 Rajneesh Kumar, Vijay Chawla
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)
87A study of Green’s functions for two-dimensional problem...
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)
88 Rajneesh Kumar, Vijay Chawla
.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
89A study of Green’s functions for two-dimensional problem...
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)
90 Rajneesh Kumar, Vijay Chawla
,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)
91A study of Green’s functions for two-dimensional problem...
,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
92 Rajneesh Kumar, Vijay Chawla
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
93A study of Green’s functions for two-dimensional problem...
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
94 Rajneesh Kumar, Vijay Chawla
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).
95A study of Green’s functions for two-dimensional problem...