Adv. Theor. Appl. Mech., Vol. 5, 2012, no. 2, 69 - 92

The Influence of Diffusion on Generalized Magneto-

Thermo-Viscoelastic Problem of a Homogenous

Isotropic Material

F. S. Bayones

Mathematics Department, Faculty of Science Umm Al-Qura University, P. O. Box 10109, Makkah, Saudi Arabia

F.S.Bayones@hotmail.com

Abstract The present paper is aimed at studying the effects of viscosity and diffusion on

generalized magneto- thermoelastic interactions in an isotropic with a spherical cavity. A harmonic function techniques are used. The thermal stresses, temperature and displacement have been obtained. These expressions are calculated numerically for a Copper material and depicted graphically. Keywords: Viscoelastic; Generalized magneto thermo; Isotropic; Diffusion; Spherical 1-Introduction

The diffusion can be defined as the spontaneous migration of substances from regions of high concentration to regions of low concentration . Thermodiffusion in the olids is one of the transport processes that has great practical importance . Thermodiffusion in an elastic solid is due to the coupling of the fields of temperature, mass diffusion and that of strain . Nowacki[1-3] developed the theory of thermoelastic diffusion . In this theory , the coupled thermoelastic model is used which implies infinite speeds of propagation of thermoelastic waves. Sherief et al.[4] developed the theory of generalized thermoelastic diffusion that predicts finite speeds of propagation for thermoelastic and diffusive waves. Sherief and Saleh [5] worked on a problem of a thermoelastic half-space with a permeating substance, in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion . Recently , Xia et al.[6] studied the dynamic response of an infinite body with a cylindrical cavity whose surface

70 F. S. Bayones suffers thermal shock using finite element method. Effect of rotation in a generalized magneto - thermo - viscoelastic media discussed by Abd-Alla et. [7] . Lord and Shulman [8,9] ,Green and Lindsay [10] explained two important generalized theories of thermoelasticity that have become the center of the interest of recent research in this area. Some problems of thermoelasticity or generalized thermoelasticity are investigated by Abd-Alla [11,12] Abd-Alla et al.[13,14] and Elnagar and Abd-Alla [15] respectively . . Roychoudhuri and Mukhopadhyay [16] studied the effect of rotation and relaxation times on plane waves in generalized thermo-viscoelasticity. Roychoudhuri and Banerjee [17] investigated the magneto-thermoelastic interactions in an infinite viscoelastic cylinder of temperature rate dependent material subjected to a periodic loading . Spherically symmetric thermo-viscoelastic waves in a viscoelastic medium with a spherical cavity discussed by Banerjee , et al. [18] . 2- Formulation of the problem

Let us consider the medium is a perfect electric conductor and the linearized Maxwell equations governing the electromagnetic field , in the absence of the displacement current (SI) in the form as in Roychoudhuri and Mukhopadhyay [16] and Kraus [19] :

⎟⎟⎠

⎞⎜⎜⎝

⎛×

∂∂

−=

=

=

=∂∂

−

=

Ht

UE

Ediv

hdiv

Ecurlth

hcurlJ

e

e

μ

μ

0

,0

,

,

(2.1)

where h is the perturbed magnetic field over the primary magnetic field , E is the electric intensity , J is the electric current density , eμ is the magnetic

permeability, H is the constant primary magnetic field and U is the displacement vector .

Applying an initial magnetic field vector ),0,0( φHH in spherical coordinate ),,( φθr to Eq.(2.1) we have

)0,,0(),,0,0(),0,0),,((tuHEHHHtruUU e ∂∂

−−=== φφ μ , (2.2a)

Influence of diffusion 71

⎟⎠⎞

⎜⎝⎛ +∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−=

ru

ruHh

rh

J 2,0,,0 φφφ , (2.2b)

)0,,0( 2EE = . (2.2c) The elastic medium is rotating uniformly with an angular velocity nΩ=Ω

where n is a unit vector representing the direction of the axis of rotation. The displacement equation of motion in the rotating frame has two additional term centripetal acceleration, )( u×Ω×Ω is the centripetal acceleration due to time

varying motion only and •

×Ω u2 is the Coriolis acceleration, and )0,,0( Ω=Ω . Following , (see [4,5]), the governing equations for an isotropic,

homogeneous elastic solid with generalized magneto-thermo- viscoelastic diffusion under effect of rotation are : (i)The Maxwell's electro-magnetic stress tensor ijτ is given by

( . ) , , 1, 2,3.ij e i j j i ijH h H h H h i jτ μ δ⎡ ⎤= + − =⎣ ⎦uur r

(2.3) (ii) The equation of motion:

{ } ,)]2()([)( ,2,1,, jrimimijjmjjim uuufCTuu•••

×Ω+×Ω×Ω+=+−−++ ρβτβτμλτμτ (2.4)

where rf is defined as Lorentz's force, Kraus [19], which may be written as

)2()( 2ru

ru

rHHJf eer +∂

∂∂∂

=×= φμμ ,

(iii) the equation of heat conduction :

)()( 01022

1, CcTeTTcttKT kkmvii ++∂

∂+

∂∂

= βτρτ , (2.5)

(iv) the equation of mass diffusion:

iiiiiikkm DbCCttDcTeD ,2

2

,,2 =⎟⎠

⎞⎜⎝

⎛∂∂

+∂∂

++ τβτ , (2.6)

(v) the constitutive equations: [ ]))((2 201 CTTee kkijijmij ββλδμτσ −−−+= , (2.7)

)( 02 TTcbCeP kkm −−+−= βτ , (2.8) where ρ is the mass density of the material, K is thermal conductivity,

tαμλβ )23(1 += and cαμλβ )23(2 += , tα and cα are , respectively, the coefficients of linear thermal and diffusion expansion , vc is specific heat at constant strain , μλ , are Lame' elastic constants, T is the absolute temperature of the medium , 0T is the reference uniform temperature of the body chosen such that 1/)( 00

72 F. S. Bayones the heat conduction equation will predict finite speed of heat propagation, and τ is the diffusion relaxation time, which will ensure that the equation satisfied by the concentration C will also predict finite speed of propagation of matter from one medium to the other, and D is the diffusive coefficient. The constants c and b are the measures of thermodiffusion effects and diffusive effects, respectively, ijσ are the components of the stress tensor, ije are the components of the strain tensor, and p is the chemical potential.

The non-vanishing displacement component is the radial one ),( truur = , is

.3,2,1,,,)(21

,, =Δ=+= jieuue jjijjiij (2.9)

Then, the strain tensor has the following components

ruee

ruerr ==∂∂

= φφθθ, . (2.10)

The cubical dilatation is thus given by:

( )urrrr

urue 2

2

12∂∂

=+∂∂

= . (2.11)

Due to spherical symmetry, Eqs. (2.4)-(2.8), take the form

⎥⎦

⎤⎢⎣

⎡∂∂

Ω−Ω−∂∂

=∂∂

−∂∂

−∂∂

tuu

tu

rrC

re

mm 22

2

2

120 ρθβτβτα , (2.12)

)()( 01022

12 CcTeTc

ttTK kkmv ++∂

∂+

∂∂

=∇ βτθρτ , (2.13)

CDbCtt

TDceD m2

2

222

2 ∇=⎟⎠

⎞⎜⎝

⎛∂∂

+∂∂

+∇+∇ τβτ , (2.14)

⎟⎠⎞−−+

∂∂

⎜⎝⎛= Ce

ru

mrr 212 βθβλμτσ , (2.15)

⎟⎠⎞−−+⎜

⎝⎛== Ce

ru

m 212 βθβλμτσσ θθφφ , (2.16)

θβτ cbCeP m −+−= 2 , (2.17) where

rrr ∂∂

+∂∂

=∇2

2

22 , )2(2

ru

ru

rHf er +∂

∂∂∂

= φμ 2

0 )2( φμμλτα Hem ++= . (2.18)

3- Dimensionless quantities

Now we introduce the following non – dimensional variables in Eqs.

(2.12)-(2.17):

Influence of diffusion 73

( ),,,,

,1,,

,,,)(

,,

021

*10

21

*100

21

*00

21

*

*2**2*

*

2

*

0

*

021

*

0

2*

0

0101

*01

*

τηττηττητη

βτα

σσ

ηαβτ

αβτ

θηη

ccctct

urrr

epp

cC

CTT

ucurcr

m

ijij

mm

====

∂∂

===

Ω=Ω=

−===

(3.1)

where ρα 02

1 =c and Kcv /0 ρη = , we have

*

***2

2*

*2

**

*

*

*

2tuu

tu

rrC

re

∂∂

Ω−Ω−∂∂

=∂∂

−∂∂

−∂∂ θ (3.2)

)()( *20*

10

2

2*

2*1*

2 Cett

m εαεατθτθ ++

∂∂

+∂∂

=∇ , (3.3)

*230

*2*

2*

*202

10*22 CC

tte mm ∇=⎟

⎠

⎞⎜⎝

⎛∂∂

+∂∂

+∇+∇ ταατααθαατ , (3.4)

**

0*

*

0

2 Ceru mm

rr −−+∂∂

= θαλτ

αμτσ , (3.5)

**

0*

*

0

2Ce

ru mm −−+== θ

αλτ

αμτ

σσ θθφφ , (3.6)

θββτ

αβτα

212

0*

22

20*

mm

cCbeP −+−= , (3.7)

where

.,1,,,2

30

22

221

12

012

02

11 β

αηβ

αββ

αβ

βερβε Db

DccT

cT

v

===== (3.8)

4- Boundary conditions

The homogeneous initial conditions are supplemented by the following boundary conditions : (1) The cavity surface is traction free

0),( =+ rrrr ta τσ . (4.1a) (2) The cavity surface is subjected to a thermal shock

).(),( 0 tHta θθ = (4.1b) (3) The chemical potential is also assumed to be a known function of time at the cavity surface

).(),( 0 tHptap = (4.1c) where 0θ and 0p are constants and H(t) is the Heaviside unit step function. (4) The displacement function

74 F. S. Bayones

.0),( =tau (4.1d) 5- Solution of the problem

Assuming a simple harmonic time dependent factor for all the quantities

and omitting the factor tie ω as following:

.)(),(,)(),(

,)(),(,)(),(***

**

*******

*******

titi

titi

erGtrCertre

ertrerUtruωω

ωω θ

=Ε=

Θ== (5.1)

Eqs. (2.12)-(2.14) become

Urr

Gr

β=∂Θ∂

−∂∂

−∂Ε∂

*** (5.2)

)( 4312 Gεε +Ε+Θ=Θ∇ l , (5.3)

GG 2542

32 ∇=+Θ∇+Ε∇ lll . (5.4)

By applying the operator **2*

22 2

rrr ∂∂

+∂∂

=∇ to Eq.(5.2) , we obtain

Θ∇+∇=Ε−∇ 222 )( Gβ , (5.5) From Eqs. (5.3)- (5.5) , we obtain

,0),,)(( 32

24

16 =ΘΕ+∇+∇+∇ Gaaa (5.6)

where

{ }541435433515

1 ])([)1(1

llllllll

βεεεε ++−++++−−

=a , (5.7)

{ })1())(()1(

134134514

52 εεβ ++++−= lllllll

a , (5.8)

)1( 541

3 −−

=l

llβa , (5.9)

).2(

,1,)2(,,

,,,,)1(),1(

*2*2

*0

*2**02

*041*

0

2*

3

*3

*0

52*

22*0

42*

1*0

3*

2*11

Ω+Ω+−=

+=++===

===+=+=

ωωβ

ωττμμλταεαεεατ

ε

ταα

ταα

ταα

ωτωωτω

φ

i

iH

iiii

memm

mmm

ll

llll

(5.10) The above system of equations can be factorized as

,0),,())()(( 2122

122

12 =ΘΕ+∇+∇+∇ Gkkk (5.11)

where k1, k2 and k3 are the roots of the characteristic equation .03

22

41

6 =+++ akakak (5.12)

Influence of diffusion 75

The solution of Eq.(5.11) , which is bounded at infinity , is given by

),()(1),( *2/13

1*

* rkKAr

r jjj

ωω ∑=

=Θ (5.13)

),()(1),( *2/13

1*

* rkKAr

r jjj

ωω ′=Ε ∑=

(5.14)

),()(1),( *2/13

1*

* rkKAr

rG jjj

ωω ′′= ∑=

(5.15)

where jj AA ′, and jA ′′ are parameters depending only on ω , and 2/1K is the

modified Bessel function of the second kind of order 1/2 . Compatibility between Eqs. (5.13)- (5.15) along with (5.3) and (5.4) will give rise to

)()(

))(()(

42

5312

41

23144

251

2

ωεε

εω j

jj

jjjj Akk

kkkA

llll

lllll

−+−−−

=′ , (5.16)

)()(

)()(

42

531142

1222

313 ωεε

εω j

jj

jjjj Akk

kkkA

llll

lll

−+−+

=′′ . (5.17)

Substituting from Eqs. (5.13)-(5.15) into (5.1), we obtain

,)()(1),( **2/13

1*

** tijj

jerkKA

rtr ωωθ ∑

=

= (5.18)

,)()(1),( **2/13

1*

*** tijj

jerkKA

rtre ωω′= ∑

=

(5.19)

.)()(1),( **2/13

1*

*** tijj

jerkKA

rtrC ωω′′= ∑

=

(5.20)

Integrating both sides of Eqs.(5.19) in the relation to the r* , we obtain

**2/3

42

5312

41

23144

251

23

1*

*** )()()(

))((1),( tijjjj

jjj

jerkKA

kkkkk

rtru ωω

εεεllll

lllll

−+−−−

= ∑=

. (5.21)

Also, from (3.5)-(3.7), we get

**2/3

1*2/12

31442

512

42

5312

4121

42

5312

41

23144

251

23

1*

)()())((

)()(

)())((

)(1

tijj

jjj

jj

jj

jjjj

jrr

erkKr

rkKkkk

kkmm

kkkkk

Ar

ωαε

εε

εεε

ωσ

⎪⎭

⎪⎬⎫

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−−

−+−+

⎪⎩

⎪⎨⎧

−+

−−−= ∑

=

lllll

llll

llll

lllll

(5.22)

76 F. S. Bayones

**2/1

42

531142

1222

313

42

5312

41

23144

251

2

2

*2/3

42

5312

41

23144

251

2

*1

3

1*

)()(

)(1

)())((

)()(

))(()(1

tij

jj

jjj

jj

jjj

jjj

jjjj

j

erkKkk

kkkkk

kkkm

rkKkk

kkkrm

Ar

ω

θθφφ

εεε

εεε

εεε

ωσσ

⎪⎭

⎪⎬⎫

⎥⎥⎦

⎤

−+

−+−−

−+

−−−

⎢⎢⎣

⎡+

−+

−−−

⎪⎩

⎪⎨⎧

== ∑=

llll

lll

llll

lllll

llll

lllll

(5

.23)

**2/14

42

531142

1222

3133

42

5312

41

23144

251

23

1*

)()(

)(

)())((

)(1

tij

jj

jjj

jj

jjjj

j

erkKmkk

kkkm

kkkkk

Ar

P

ω

εεε

εεε

ω

⎪⎭

⎪⎬⎫

−−+

−++

⎪⎩

⎪⎨⎧

−+

−−−−= ∑

=

llll

lll

llll

lllll

(5.24)

where

.,,,2

21*

*0

422

2*

*0

3*0

*

2*0

*

1 ββτα

βτα

αλτ

αμτ

mm

mm cmbmmm ====

By using the boundary conditions , we get

,])([])([)(

,])([])([)(

,])([])([)(

10012/1220022/113

10012/13032/13012

022/1203032/13021

MWpakKNaWpakKNaA

MWpakKNapakKWNaA

MpakKWaNpakKWaNA

θθω

θθω

θθω

−−−=

−+−=

−+−−=

(5.25)

⎪⎭

⎪⎬⎫

⎥⎦⎤⎟⎠⎞−⎜

⎝⎛+⎢⎣

⎡−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−−

−+−+

⎪⎩

⎪⎨⎧

−+

−−−=

)(3)())((

)()(

)())((

2/32

221

2/123144

251

2

42

5312

4121

42

5312

41

23144

251

2

akKka

Ham

akKkkk

kkmm

kkkkk

N

jiejjjj

jj

jj

jjjj

φμεεε

εεε

lllll

llll

llll

lllll

(5

.26)

)()(

)(

)())((

2/144

253114

2

1222

3133

42

5312

41

23144

251

2

akKmkk

kkkm

kkkkk

W

jjj

jjj

jj

jjjj

⎪⎭

⎪⎬⎫

−−+−+

+

⎪⎩

⎪⎨⎧

−+−−−

−=

llll

lll

llll

lllll

εεε

εεε

, j=1,2,3. (5.27)

Influence of diffusion 77

⎭⎬⎫−+

⎩⎨⎧ −−−=

)]()([

])()([])()([

22/1112/123

12/1112/13232/1222/131

akKWakKWN

akKWakKWNakKWakKWNM (5.28)

The discussion in thermoelastic diffusion medium is clear up from Figs. (1-

6) and in thermoelastic medium is clear up from Figs. (7-10):

-1.50E+00

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

0 0.5 1 1.5 2 2.5

-2.00E+00

-1.50E+00

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

0 0.5 1 1.5 2

-6.00E-01

-4.00E-01

-2.00E-010.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+001.20E+00

1.40E+00

1.60E+00

0 0.5 1 1.5 2 2.5

-2.00E+00

-1.00E+00

0.00E+00

1.00E+00

2.00E+00

3.00E+00

4.00E+00

0 0.5 1 1.5 2 2.5

Fig.(1): The displacement ),( tru in thermo viscoelastic diffusion medium

t

0=Ω __ _ ___ 2.0=Ω ________ 5.0=Ω ..........

1=r _ __ _ __ 5.1=r _______ 2=r ..............

u u

u

r

r

u

__ _ __ 01.00 =τ

1.00 =τ _______ 2.00 =τ..........

r

t=0 ___ _ __ t=1.4 ______

t=2 .............

78 F. S. Bayones

-1.20E+00

-1.00E+00

-8.00E-01

-6.00E-01

-4.00E-01

-2.00E-01

0.00E+000 0.5 1 1.5 2 2.5

-4.50E-01

-4.00E-01

-3.50E-01

-3.00E-01

-2.50E-01

-2.00E-01

-1.50E-01

-1.00E-01

-5.00E-02

0.00E+000 0.5 1 1.5 2 2.5

-4.50E-01

-4.00E-01

-3.50E-01

-3.00E-01

-2.50E-01

-2.00E-01

-1.50E-01

-1.00E-01

-5.00E-02

0.00E+000 0.5 1 1.5 2 2.5

-1.60E+00

-1.40E+00

-1.20E+00

-1.00E+00

-8.00E-01

-6.00E-01

-4.00E-01

-2.00E-01

0.00E+000 0.5 1 1.5 2 2.5

Fig.(2): The temperature ),( trθ in thermo viscoelastic diffusion medium

t=0 ___ _ __ t=1.4 ______

t=2 ...........

θ r

0=Ω _ __ __ 2.0=Ω _______ 5.0=Ω ..........

θ r

__ _ __ 01.00 =τ

1.00 =τ _______ 2.00 =τ ...........

θ

θ r 1=r _ __ _ __ 5.1=r _______ 2=r ..............

Influence of diffusion 79

0.00E+00

5.00E-18

1.00E-17

1.50E-17

2.00E-17

2.50E-17

3.00E-17

0 0.5 1 1.5 2 2.5

-5.00E-17

0.00E+00

5.00E-17

1.00E-16

1.50E-16

2.00E-16

2.50E-16

3.00E-16

0 0.5 1 1.5 2 2.5

-5.00E-17

0.00E+00

5.00E-17

1.00E-16

1.50E-16

2.00E-16

2.50E-16

3.00E-16

3.50E-16

0 0.5 1 1.5 2 2.5

0.00E+00

5.00E-17

1.00E-16

1.50E-16

2.00E-16

2.50E-16

3.00E-16

3.50E-16

4.00E-16

4.50E-16

5.00E-16

0 0.5 1 1.5 2 2.5

Fig.(3): The concentration ),( trC in thermo viscoelastic diffusion medium

C

t=0 ___ _ __ t=1.4 ______

t=2 ...........

__ _ __ 01.00 =τ

1.00 =τ _______ 2.00 =τ ............

r

0=Ω _ __ _ _ 2.0=Ω ________ 5.0=Ω ............

r

C

C C

1=r _ __ _ __ 5.1=r _______ 2=r .............

r t

80 F. S. Bayones

-2.00E+04

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

0 0.5 1 1.5 2 2.5

-2.00E+05

-1.50E+05

-1.00E+05

-5.00E+04

0.00E+00

5.00E+04

1.00E+05

1.50E+05

2.00E+05

0 0.5 1 1.5 2 2.5

-1.40E+05

-1.20E+05

-1.00E+05

-8.00E+04

-6.00E+04

-4.00E+04

-2.00E+04

0.00E+00

2.00E+04

4.00E+04

0 0.5 1 1.5 2 2.5

-5.00E+05

-4.00E+05

-3.00E+05

-2.00E+05

-1.00E+05

0.00E+00

1.00E+05

2.00E+05

0 0.5 1 1.5 2 2.5

Fig.(4): The stress rrσ in thermo viscoelastic diffusion medium

r

r

__ _ __ _ 01.00 =τ

1.00 =τ _______ 2.00 =τ .............

0=Ω __ _ __ 2.0=Ω ________ 5.0=Ω ............

t=0 ___ _ __ t=1.4 ______

t=2 ...........

1=r _ __ _ __ 5.1=r ______ 2=r .............

rrσ

t r

rrσ

rrσ

rrσ

Influence of diffusion 81

-4.00E-01

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

0 0.5 1 1.5 2 2.5

-1.50E+00

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

0 0.5 1 1.5 2 2.5

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

1.40E+00

1.60E+00

0 0.5 1 1.5 2 2.5

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

4.00E+00

4.50E+00

0 0.5 1 1.5 2 2.5

Fig.(5): The stress φφσ in thermo viscoelastic diffusion medium

φφσ φφσ

r

t=0 ___ _ __ t=1.4 ______

t=2 ...........

0=Ω _ __ __ 2.0=Ω _______ 5.0=Ω ..........

φφσ __ _ __ _ 01.00 =τ

1.00 =τ _______ 2.00 =τ .............

r

r

φφσ

1=r _ __ _ __ 5.1=r ______ 2=r .............

t

82 F. S. Bayones

0.00E+00

1.00E+00

2.00E+00

3.00E+00

4.00E+00

5.00E+00

6.00E+00

0 0.5 1 1.5 2 2.5

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

4.00E+00

4.50E+00

0 0.5 1 1.5 2 2.5

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

0 0.5 1 1.5 2 2.5

0.00E+00

1.00E+00

2.00E+00

3.00E+00

4.00E+00

5.00E+00

6.00E+00

0 0.5 1 1.5 2 2.5

Fig.(6): The chemical potential p in thermo viscoelastic diffusion medium

p p

p p

r r

r t

0=Ω _ __ __ 2.0=Ω _______ 5.0=Ω ..........

t=0 ___ _ __ t=1.4 ______

t=2 ...........

__ _ __ _ 01.00 =τ

1.00 =τ _______ 2.00 =τ .............

1=r _ __ _ __ 5.1=r ______ 2=r .............

Influence of diffusion 83 6- Particular case:

i- If we neglect the diffusion effect by eliminating Eqs.(2.6) and (2.8), and by putting 02 == cβ in Eqs. (2.4) , (2.5) and (2.8), we have

tirtir eBeAtre ωλωλ ωωθ +−+− += 21 )()(),)(,( , (6.1) where

[ ]2211222131211

421),(

,)(,)(

ηηηλλ

εβηβη

−±=

−=+−= ll (6.2)

,)()(22)(22),( 2132

22

22

2

31

21

21

2tirr eCeB

rrreA

rrrtru ωλλ ωω

λλλω

λλλ

⎥⎦

⎤⎢⎣

⎡+

+++

++−= −− (6.3)

(6.4)

,)()1(22

)()1(22

2

1

232

32

22

2

1

231

31

21

2

1

tir

r

eeBmr

rrm

eAmr

rrm

ωλ

λθθφφ

ωλλλ

ωλλλσσ

⎪⎭

⎪⎬⎫

⎥⎦⎤−−⎟

⎠⎞++

⎜⎝⎛

⎢⎣⎡+

⎥⎦⎤−−⎟

⎠⎞++

⎜⎝⎛

⎢⎣⎡

⎪⎩

⎪⎨⎧

−==

−

−

(6.4)

By using the boundary conditions , we have

,)()(

,)(,)(

12

1212

21

04132

21

01

21

02

aa

aaaa

ehehhhhhC

ehehhB

ehehhA

λλ

λλλλ

θω

θωθω

−−

−−−−

−−

=

−=

−−

= (6.5)

,)1(244

,)1(244

2

1

22

232

3

32

322

22

12

12

231

3

31

321

21

11

ae

ae

eHma

aaamh

eHma

aaamh

λφ

λφ

λμλ

λλλ

λμλ

λλλ

−

−

⎟⎠⎞−−+

+++⎜⎝⎛=

⎟⎠⎞−−+

+++⎜⎝⎛=

(6.6)

.22,22 21 32

22

22

2

431

21

21

2

3aa e

aaahe

aaah λλ

λλλ

λλλ −− ⎟

⎠⎞++

⎜⎝⎛=⎟

⎠⎞++

⎜⎝⎛= (6.7)

,)()1(244

)()1(244

2

1

232

3

32

322

22

1

231

3

31

321

21

1

tir

rrr

eeBmr

rrrm

eAmr

rrrm

ωλ

λ

ωλ

λλλ

ωλ

λλλσ

⎥⎥⎦

⎤⎟⎠⎞−+

+++⎜⎝⎛+

⎟⎠⎞−+

+++⎜⎝⎛

⎢⎢⎣

⎡=

−

−

84 F. S. Bayones

-1.00E+01

0.00E+00

1.00E+01

2.00E+01

3.00E+01

4.00E+01

5.00E+01

6.00E+01

0 0.5 1 1.5 2 2.5

-1.00E+01

0.00E+00

1.00E+01

2.00E+01

3.00E+01

4.00E+01

5.00E+01

0 0.5 1 1.5 2 2.5

-1.00E+02

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

5.00E+02

6.00E+02

7.00E+02

8.00E+02

9.00E+02

0 0.5 1 1.5 2 2.5

0.00E+00

2.00E+02

4.00E+02

6.00E+02

8.00E+02

1.00E+03

1.20E+03

1.40E+03

0 0.5 1 1.5 2 2.5

Fig.(7): The displacement ),( tru in thermo viscoelastic medium

u u

u u

r r

r t

0=Ω _ __ __ 2.0=Ω _______ 5.0=Ω ..........

t=0 ___ _ __ t=1.4 ______

t=2 ...........

__ _ __ _ 01.00 =τ

1.00 =τ _______ 2.00 =τ .............

1=r _ __ _ __ 5.1=r ______ 2=r .............

Influence of diffusion 85

-2.00E+03

-1.50E+03

-1.00E+03

-5.00E+02

0.00E+00

5.00E+02

0 0.5 1 1.5 2 2.5

-2.50E+03

-2.00E+03

-1.50E+03

-1.00E+03

-5.00E+02

0.00E+00

5.00E+02

0 0.5 1 1.5 2 2.5

-1.80E+03

-1.60E+03

-1.40E+03

-1.20E+03

-1.00E+03

-8.00E+02

-6.00E+02

-4.00E+02

-2.00E+02

0.00E+00

2.00E+02

0 0.5 1 1.5 2 2.5

-3.00E+03

-2.50E+03

-2.00E+03

-1.50E+03

-1.00E+03

-5.00E+02

0.00E+00

5.00E+02

0 0.5 1 1.5 2 2.5

Fig.(8): The temperature ),( trθ in thermo viscoelastic medium

θ

r r

r t

0=Ω _ __ __ 2.0=Ω _______ 5.0=Ω ..........

t=0 ___ _ __ t=1.4 ______

t=2 ...........

__ _ __ _ 01.00 =τ

1.00 =τ _______ 2.00 =τ .............

1=r _ __ _ __ 5.1=r ______ 2=r .............

θ

θ θ

86 F. S. Bayones

-1.40E+04

-1.20E+04

-1.00E+04

-8.00E+03

-6.00E+03

-4.00E+03

-2.00E+03

0.00E+00

2.00E+03

0 0.5 1 1.5 2 2.5

-1.20E+04

-1.00E+04

-8.00E+03

-6.00E+03

-4.00E+03

-2.00E+03

0.00E+00

2.00E+03

0 0.5 1 1.5 2 2.5

-9.00E+03

-8.00E+03

-7.00E+03

-6.00E+03

-5.00E+03

-4.00E+03

-3.00E+03

-2.00E+03

-1.00E+03

0.00E+00

1.00E+03

0 0.5 1 1.5 2 2.5

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

8.00E-01

9.00E-01

1.00E+00

0 0.5 1 1.5 2 2.5

Fig.(9): The stress rrσ in thermo viscoelastic medium

0=Ω _ __ __ 2.0=Ω _______ 5.0=Ω ..........

t=0 ___ _ __ t=1.4 ______

t=2 ...........

r=0 ___ _ __ r=1.4 ______

r=2 ...........

__ _ __ _ 01.00 =τ

1.00 =τ _______ 2.00 =τ .............

r r

r

t

rrσrrσ

rrσ rrσ

Influence of diffusion 87

-2.00E+01

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

1.60E+02

1.80E+02

0 0.5 1 1.5 2 2.5

-2.00E+01

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

1.60E+02

0 0.5 1 1.5 2 2.5

-2.00E+01

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 0.5 1 1.5 2 2.5

-7.00E-01

-6.00E-01

-5.00E-01

-4.00E-01

-3.00E-01

-2.00E-01

-1.00E-01

0.00E+000 0.5 1 1.5 2 2.5

Fig.(10): The stress φφσ in thermo viscoelastic medium

t=0 ___ _ __ t=1.4 ______

t=2 ...........

0=Ω _ __ __ 2.0=Ω _______ 5.0=Ω ..........

__ _ __ _ 01.00 =τ

1.00 =τ _______ 2.00 =τ .............

r=0 ___ _ __ r=1.4 ______

r=2 ...........

r r

r

t

r

φφσφφσ

φφσφφσ

88 F. S. Bayones 7- Numerical results and discussion

The Copper material was used chosen for purposes of numerical

evaluations. The constants of the problem given by [20] and [21], are

20

256224

038

3415

3310310

/73.8886,/109.0,/102.1

,293,/1085.0,/386

,/1098.1,1078.1,/1.383

,/8954,/1076.7,/1086.3

mskgsmbKsmc

KTmskgDmKWk

kgmKKkgJc

mkgmskgmskg

ctv

=×=×=

=×==

×=×==

=×=×=−−−

η

αα

ρλμ

Using these values, it was found that:

.2.0,2.0,01.0,107.0,5.9,2,1,1,2.0 105

00 ===×======− τττωθμ φHape

The numerical technique outlined above was used to obtain the temperature, radial displacement, radial stress and concentration as well as the chemical potential distribution inside the sphere. These distributions are shown in Figs. 1-10 respectively. For the sake of brevity some computational results are not being presented here. -In thermo - viscoelastic diffusion medium

Fig. (1) shows the variation of the radial displacement ),( tru under effects the change of time t , rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of ),( tru decreased with an increased of time , rotation and radius , while the values of ),( tru are take the same values (a slight change) at different values of mechanical relaxation time.

Fig. (2) shows the temperature distribution ),( trθ under effects the change of time t , rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of ),( trθ decreased and increased with an increased of time and radius respectively, while the values of ),( trθ are take the same values (a slight change) at different values of mechanical relaxation time and rotation.

Fig. (3) shows the concentration distribution ),( trC under effects the change of time t , rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of ),( trC decreased with an increased of time , radius and mechanical relaxation time, while the values of ),( trC are take the same values (a slight change) at different values of rotation.

Fig. (4) shows the radial stress stress rrσ under effects the change of time t , rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of rrσ decreased and increased with an increased of (time and rotation) and radius respectively, while the values of rrσ are take the same values (a slight change) at different values of mechanical relaxation time and rotation.

Fig. (5) shows the tangential stress φφσ under effects the change of time t ,

rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that

Influence of diffusion 89 the values of φφσ decreased with an increased of time , rotation and radius, while the values of φφσ are take the same values (a slight change) at different values of mechanical relaxation time and rotation.

Fig. (6) shows the chemical potential distribution p under effects the change of time t , rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of p decreased with an increased of time , rotation and radius, while the values of p are take the same values (a slight change) at different values of mechanical relaxation time and rotation. -In thermo viscoelastic medium

Fig. (7) shows the radial displacement ),( tru under effects the change of time t , rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of ),( tru decreased with an increased of time , rotation and radius, while the values of ),( tru are take the same values (a slight change) at different values of mechanical relaxation time and rotation.

Fig. (8) shows the temperature distribution ),( trθ under effects the change of time t , rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of ),( trθ decreased and increased with an increased of time , (rotation and radius) respectively , while the values of ),( trθ are take the same values (a slight change) at different values of mechanical relaxation time and rotation.

Fig. (9) shows the radial stress rrσ under effects the change of time t , rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of rrσ decreased and increased with an increased of (time and radius) and rotation respectively , while the values of rrσ are take the same values (a slight change) at different values of mechanical relaxation time and rotation.

Fig. (10) shows the tangential stress φφσ under effects the change of time t ,

rotation Ω , mechanical relaxation time 0τ and the radius r. It was found that the values of φφσ decreased and increased with an increased of (time and rotation) and radius respectively , while the values of φφσ are take the same values (a slight change) at different values of mechanical relaxation time and rotation.

8- Conclusions From the previous discussion in thermo viscoelastic diffusion medium, we

have the values of the displacement ),( tru , the concentration distribution ),( trC , the stress φφσ and the chemical potential distribution p decreased with increased values of time t , rotation Ω and the radius r , and take the same values (a slight

90 F. S. Bayones change) at different values of mechanical relaxation time 0τ . While the values of the temperature distribution ),( trθ , the stress rrσ take changes values increased , decreased or slight change with increased values of time t , rotation Ω , mechanical relaxation time 0τ and the radius r.

From the previous discussion in thermo viscoelastic medium, we have the values of the displacement ),( tru decreased with increased values of time t , rotation Ω and the radius r , and take the same values (a slight change) at different values of mechanical relaxation time 0τ . While the values of the temperature distribution ),( trθ , the stress rrσ , and the stress φφσ take changes values increased , decreased or slight change with increased values of time t , rotation Ω , mechanical relaxation time 0τ and the radius r.

Finally, the results obtained in this chapter should prove useful for research in material science, designers of new materials, low-temperature physicists, as well as for those working on the development of a theory of hyperbolic thermo diffusion. Cross effects of heat, magnetic field, rotation and mass diffusion exchange with the environment arising from and inside nuclear reactors influence their design and operations. Study of the phenomenon of diffusion is also used to improve the conditions of oil extraction.

References [1] Nowacki. W. 1974, Dynamical problems of thermodiffusion in solids Bull. Acad. Pol. Sci. Ser. Sci. Tech , 22 , pp. 55-64. [2] Nowacki . W. 1974 , Dynamical problems of thermodiffusion in solids II, Bull. Acad. Pol. Sci. Ser. Sci. Tech, 22 , pp.129-135. [3] Nowacki. W. 1974, Dynamical problems of thermo diffusion in solid III, Bull. Acad. Pol.Sci. Ser. Sci. Tech. 22 , pp. 266-275. [4] Sherief . H ., Hamza . F., Saleh . H . 2004 , The theory of generalized thermoelastic diffusion , Int. J. Eng. Sci, 42, pp. 591-608. [5] Sherief . H ., Saleh . H. 2005 , A half - space problem in the theory of generalized thermoelastic diffusion , Int. J. Solids Struct, 42 , pp. 4484- 4494. [6] Xia. R., Tian. X., Shen.Y. 2009,The influence of diffusion on generalized thrmoelastic problems of infinite body with a cylindrical cavity, Int. J. Eng. Sci. 47, pp. 669-679. [7] Abd-Alla . A. M., Bayones. F. S. 2011, Effect of rotation in a generalized magneto-thermo - viscoelastic media , Advanc Theoretical and Applied Mechanic . 4 , no. 1, pp. 15-42.

Influence of diffusion 91 [8] Lord, H.W., Shulman, Y.1967, A generalized dynamical theory of thermo- elasticity, J. Mech. Phys. Solids 7,pp. 71–75. [9] Lord, H. W ., Shulman . Y, 1967, A generalized thermodynamical theory of thermo-elasticity. Elasticity, J. Mech. Phys. Solids, 15 , pp. 299-309. [10] Green, A.E., Lindsay, A. 1972, Thermoelasticity, J. Elasticity 2, pp. 1–7. [11] Abd-Alla, A.M. 1995, Thermal stress in a transversely isotropic circular cylinder due to an instantaneous heat source, Appl. Math . Comput, 68, pp.113-124. [12] Abd-Alla , A. M . 1996, Transient response of inhomogeneous transversely isotropic thermoelastic cylinder to a dynamic input , J. Appl . Math .Comput, 74, pp. 1–13. [13] Abd-Alla , A. M. 1999, Propagation of Rayleigh waves in an elastic half-space of orthotropic material ,Applied Mathematics and Computation, Vol, 99, pp.61-619. [14] Abd-Alla, A. M., Abd-Alla , A.N., Zeidan, N . A. 2000 , Thermal stresses in a non-homogeneous orthotropic elastic multilayered cylinder , J .Thermal Stresses, 23 (5), pp. 413–428. [15] El-Naggar, A.M., Abd-Alla, A.M. 1987 , On a generalized thermoelastic problem in an infinite cylinder under initial stress, Earths, Moon Planets, 37, pp.213–223. [16] Roychoudhuri , S.K ., Mukhopadhyay , S . 2000, Effect of rotation and relaxation times on plane waves in generalized thermo– viscoelasticity, IJMMS, 23 (7), pp. 497–505. [17] Roychoudhuri, S. K ., Banerjee, S . 1998, Magneto-thermoelastic Interactions in an infinite viscoelastic cylinder of temperature rate dependent material subjected to a periodic loading, Int. J. Eng. Sci, 36 (5/6) , pp.635-643. [18] Banerjee, S., (Mukhopadhyay), Rychoudhuri, S. K. 1995 , Spherically symmetric thermo - viscoelastic waves in a viscoelastic medium with a spherical cavity, Comput . Math. Appl, 30, pp.91-98. [19] Kraus, J.D . 1984, Electromagnetics . Mc Graw-Hill, New York. [20] Singh, H., Sharma, J. N.1985 , Generalized thermoelastic waves in transversely isotropic media . J. Acoust . Soc . Am, 77, pp.1046-1053.

92 F. S. Bayones [21] Thomas . L .1956, Fundamentals of Heat Transfer . Prentice -Hall Inc , Englewood Cliffs, New Jersey. Received: November, 2011