Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 284646 11 pageshttpdxdoiorg1011552013284646

Research ArticleAn Analytical Solution for Effect of Magnetic Field andInitial Stress on an Infinite Generalized ThermoelasticRotating Nonhomogeneous Diffusion Medium

S R Mahmoud12

1 Mathematics Department Science Faculty King Abdulaziz University Jeddah 21589 Saudi Arabia2Mathematics Department Science Faculty Sohag University Sohag 82524 Egypt

Correspondence should be addressed to S R Mahmoud samsam73yahoocom

Received 12 July 2013 Revised 11 October 2013 Accepted 19 October 2013

Academic Editor Hasan Ali Yurtsever

Copyright copy 2013 S R Mahmoud This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problem of generalized magneto-thermoelastic diffusion in an infinite rotating nonhomogeneity medium subjected to certainboundary conditions is studied The chemical potential is also assumed to be a known function of time at the boundary of thecavity The analytical expressions for the displacements stresses temperature concentration and chemical potential are obtainedComparison was made between the results obtained in the presence and absence of diffusion The results indicate that the effect ofnonhomogeneity rotation magnetic field relaxation time and diffusion is very pronounced

1 Introduction

Diffusion can be defined as the spontaneous migration ofsubstances from regions of high concentration to regions oflow concentration There is now a great deal of interest inthe study of this phenomenon due to its many applicationsin geophysics and industrial applications Thermodiffusionin the solids is one of the transport processes which hasgreat practical importance Thermodiffusion in an elasticsolid is due to the coupling of the fields of temperature massdiffusion and that of strain This mater has attracted theattention of many researchers such as [1ndash5] Wave propaga-tion in rotating and nonhomogeneous media was studied byAbd-Alla et al [6ndash8] The extended thermoelasticity theoryintroducing one relaxation time in the thermoelastic processwas proposed by Lord and Shulman [9] In this theory amodified law of heat conduction including both the heat fluxand its time derivative replaces conventional Fourierrsquos lawThe heat equation associated with this is a hyperbolic oneand hence automatically eliminates the paradox of infinitespeeds of propagation inherent in the coupled theory ofthermoelasticity This theory was extended by Dhaliwal andSherief [10] to include the anisotropic case Abd-Alla and

Mahmoud [11] investigated themagneto-thermoelastic prob-lem in rotating nonhomogeneous orthotropic hollow cylin-der under the hyperbolic heat conduction model Mahmoud[12] investigated wave propagation in cylindrical poroelasticdry bones

Kumar andDevi [13] studied deformation in porous ther-moelastic material with temperature dependent propertiesOthman et al [14] presented the study of the two-dimensionalproblems of generalized thermoelasticity with one relaxationtime with the modulus of elasticity being dependent on thereference temperature for nonrotating and rotating mediumrespectively Kumar and Gupta [15] investigated deformationdue to inclined load in an orthotropic micropolar thermoe-lastic medium with two relaxation times The temperature-rate dependent theory of thermoelasticity which takes intoaccount two relaxation times was developed by Green andLindsay [16] Abd-Alla et al [17 18] investigated radialvibrations in a nonhomogeneous orthotropic elastic mediumsubjected to rotation and gravity field Sherief et al [19]developed the generalized theory of thermoelastic diffusionwith one relaxation time which allows the finite speedof propagation waves Sherief and Saleh [20] investigatedthe problem of a thermoelastic half-space in the context

2 Abstract and Applied Analysis

of the theory of generalized thermoelastic diffusion withone relaxation time The reflection of SV waves from thefree surface of an elastic solid in generalized thermoelasticdiffusion was discussed by Singh [21] Kumar and Kansal[22] discussed the propagation of Lambwaves in transverselyisotropic thermoelastic diffusive plates Thermomechanicalresponse of generalized thermoelastic diffusion with onerelaxation time due to time harmonic sources was discussedby Ram et al [23] Aouadi [24] examined the thermoe-lastic diffusion problem for an infinite elastic body withspherical cavity Abd-Alla and Mahmoud [25] investigatedanalytical solution of wave propagation in nonhomogeneousorthotropic rotating elastic media Othman et al [26] dis-cussed the effect of diffusion in a two-dimensional problemofgeneralized thermoelasticity with Green-Naghdi theory Xiaet al [27] studied the influence of diffusion on generalizedthermoelastic problems of infinite body with a cylindricalcavity Deswal and Kalkal [28] studied the two-dimensionalgeneralized electromagneto-thermoviscoelastic problem fora half-space with diffusion Abd-Alla and Abo-Dahab [29]found the time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energydissipation Mahmoud [30] discussed influence of rotationand generalized magnetothermoelastic on Rayleigh waves ina granular medium under effect of initial stress and gravityfield Abd-Alla et al [31 32] studied the generalizedmagneto-thermoelastic Rayleigh waves in a granular medium underthe influence of a gravity field and initial stress

In the present investigation the temperature displace-ments stresses diffusion and concentration as well as chem-ical potential are obtained in the physical domain using theharmonic vibrations Also study of the interaction betweenthe processes of elasticity nonhomogeneity rotation mag-netic field initial stress heat and diffusion in an infiniteelastic solidwith a spherical cavity in the context of the theoryof generalized thermoelastic diffusion is presented

2 Formulation of the Problem

Consider a perfect electric conductor and linearizedMaxwellequations governing the electromagnetic field in the absenceof the displacement current (SI) in the form as in Kraus[33] Applying an initial magnetic field vector

119867(0 01198670) in

spherical coordinates (119903 120579 120601) = (119906(119903 119905) 0 0) One willconsider a nonhomogeneous isotropic medium occupyingthe region 119886 le 119903 le 119887 where a is the radius of the sphericalcavity The strain tensor has the following components

119890119894119895=

1

2

(119906119894119895+ 119906119895119894) (1a)

120596119894119895=

1

2

(119906119895119894minus 119906119894119895) (1b)

119890119896119896

=

1

119903

2

120597

120597119903

(119903

2119906) 119894 119895 119896 = 1 2 3 (1c)

119890119903119903=

120597119906

120597119903

119890120579120579

= 119890120601120601

=

119906

119903

(1d)

The cubical dilatation is given by (1119903

2)(120597120597119903)(119903

2119906)

where the nonvanishing displacement component is theradial one 119906

119903= 119906(119903 119905) The elastic medium is rotating

uniformly with an angular velocity Ω = Ω 119899 where 119899 is a unit

vector representing the direction of the axis of rotation Thedisplacement equation of motion in the rotating frame hastwo additional terms

Ω times (

Ω times ) which is the centripetalacceleration due to time varying motion only and 2

Ω times

is the Coriolis acceleration where Ω = (0 Ω 0) Following

Sherief rsquos theory of generalized thermoelastic diffusion [19]and Sherief and Saleh [20] one is going to study an isotropicnonhomogeneous elastic medium which suffers thermalshock Due to spherical symmetry the stress-displacement-temperature-diffusion relation or constitutive equations aregiven by

120590119903119903= (2120583 + 119875

1)

120597119906

120597119903

+ (120582+1198751)

1

119903

2

120597

120597119903

(119903

2119906)

minus 1205731(120579 minus 120579

0) minus 1205732119862

(2a)

120590120601120601

= 120590120579120579

= (2120583 + 1198751)

119906

119903

+ (120582 + 1198751)

1

119903

2

120597

120597119903

(119903

2119906)

minus 1205731(120579 minus 120579

0) minus 1205732119862

(2b)

The chemical-displacement-temperature-diffusion rela-tion is given by

119875 = minus1205732

1

119903

2

120597

120597119903

(119903

2119906) + 119887119862 minus 119888 (120579 minus 120579

0) (3)

The governing equation for an isotropic nonhomoge-neous elastic solid with generalized magneto-thermoelasticdiffusion under effect of rotation is given by

120597 (1119903

2) (120597120597119903) (119903

2119906)

120597119903

minus

1205732

ℎ0

120597119862

120597119903

minus

1205731

ℎ0

120597 (120579 minus 1205790)

120597119903

+ 119865119903

=

120588

ℎ0

[

120597

2119906

120597119905

2minus Ω

2119906 minus 2Ω

120597119906

120597119905

]

(4)

where 120582 and 120583 are Lamersquos elastic constants 120575119894119895is Kroneckerrsquos

delta 1198751is the initial stress 120588 is the density of the medium

and 119865 is defined as Lorentzrsquos force which may be written as

119865 = 120583

119890(

119869 times

119867) = (120583119890119867

2

0

120597

120597119903

(

120597119906

120597119903

+

2119906

119903

) 0 0) (5)

where 120583119890is themagnetic permeability

119867 is themagnetic fieldvector

119869 is the electric current density is the displacementvector and 119905 is the time

Equation of heat conduction is given by

119870nabla

2120579 = (

120597

120597119905

+ 120591

120597

2

120597119905

2)(120588119888V120579 + 120579

01205731

1

119903

2

120597

120597119903

(119903

2119906) + 119888120579

0119862)

(6)

where the Laplacian operator nabla2 is given by nabla2 = (120597

2120597119903

2) +

(2119903)(120597120597119903) and ℎ0

= 2120583 + 120582 + (11987512) + 120583

119890119867

2

0 ℎ0is the

Abstract and Applied Analysis 3

coefficient of linear diffusion expansion 119870 is the thermalconductivity 120579 is the absolute temperature 120579

0is the initial

uniform temperature and |(120579 minus 1205790)1205790| ≪ 1

Equation of conservation of mass diffusion may bewritten as

1198631205732nabla

2 1

119903

2

120597

120597119903

(119903

2119906) + 119863119888nabla

2120579 + (

120597

120597119905

+ 120591

120597

2

120597119905

2)119862 = 119863119887nabla

2119862

(7)

where 120591 is the diffusion relaxation time 1205910is the thermal

relaxation time 120572119905is the coefficient of linear thermal expan-

sion and 1205790is constant where 120573

1= (3120582 + 2120583)120572

119905 1205732

=

(3120582 + 2120583)120572119888 120590119894119895are the components of the stress tensor

120591119894119895are the components of stress tensor 119887 and 119888 are the

measures of thermodiffusion and diffusive effects 119862 is theconcentration 119862V is the specific heat at constant strain 119863is the diffusive coefficient and 119890

119894119895are the components of

the strain tensor The thermal relaxation time 1205910will ensure

that the heat conduction equation will predict finite speedof heat propagation The diffusion relaxation time 120591 whichwill ensure the equation satisfied by the concentration 119862 willalso predict finite speed of propagation of matter from onemedium to the other

3 Dimensionless Quantities

Introduce the following nondimensional parameters

119903

lowast= 11988811205780119903 119906

lowast= 11988811205780119906 119879 =

1205731(120579 minus 120579

0)

ℎ0

119862

lowast=

1205732119862

ℎ0

Ω

lowast=

Ω

119888

2

11205780

120590

lowast

119894119895=

120590119894119895

ℎ0

119875

lowast=

119875

1205732

119905

lowast= 119888

2

11205780119905 120591

lowast

0= 119888

2

112057801205910

120591

lowast= 119888

2

11205780120591 120578

0=

120588119888V

119870

119888

2

1=

ℎ0

120588

(8)

The elastic constants 120582 120583 and the density 120588 of nonhomoge-neous material in form [32] are as follows

120582 = 119903

21198981205820 120583 = 119903

21198981205830 120583

ℎ= 119903

21198981205830

120588 = 119903

21198981205880 119901

lowast= 119901

lowast

0119903

2119898

(9)

Using the above non-dimensional parameters and (9) in(10)ndash(14) the non-dimensional system becomes

120597 (1119903

2) (120597120597119903) (119903

2119906)

120597119903

minus

120597119862

120597119903

minus

120597119879

120597119903

=

120597

2119906

120597119905

2minus Ω

2119906 minus 2Ω

120597119906

120597119905

(10)

nabla

2119879 = (

120597

120597119905

+ 1205910

120597

2

120597119905

2)(119879 + 120576

1

1

119903

2

120597

120597119903

(119903

2119906) + 120576

2119862) (11)

nabla

2 1

119903

2

120597

120597119903

(119903

2119906) + ℎ

4nabla

2119879 + ℎ6(

120597

120597119905

+ 120591

120597

2

120597119905

2)119862 = ℎ

5nabla

2119862

(12)

120590119903119903= ℎ1

120597119906

120597119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13a)

120590120579120579

= ℎ1

119906

119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13b)

120590120601120601

= ℎ1

119906

119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13c)

119875 = minus

1

119903

2

120597

120597119903

(119903

2119906) + ℎ

3119862 minus ℎ4119879 (14)

where

1205761=

120573

2

11198790119898

ℎ0120588119888V

1205762=

12057311198881198790ℎ0(119898 + 2)

1205732

ℎ1=

2120583

ℎ0

ℎ2=

120582

ℎ0

ℎ3=

119898119887ℎ0

120573

2

2

ℎ4=

119888ℎ0

12057311205732

ℎ5=

119863119887ℎ0

1205732

ℎ6=

2119898+ℎ0

120573

2

21198631205780

(15)

4 Boundary Conditions

The nonhomogeneous initial conditions are supplementedby the following boundary conditions The cavity surface istraction free

120590119903119903(119903 119905) + 120591

119903119903(119903 119905) = 0 119903 = 119886 (16a)

The cavity surface is subjected to a thermal shock

119879 (119886 119905) = 1198790119867(119905) (16b)

where119867(119905) is the Heaviside unit step function The chemicalpotential is also assumed to be a known function of time atthe cavity surface

119875 (119886 119905) = 1198750119867(119905) 119875

0is real constant (16c)

The displacement function is as follows

119906 (119903 119905) = 0 119903 = 119886 (16d)

5 Solution of the Problem

In this section one obtains the analytical solution of theproblem for a spherical region with boundary conditionsby taking the harmonic vibrations One assumes that thesolution of (10)ndash(12) as follows

119862 (119903 119905) = 119862

1015840(119903) 119890

119894120596119905 119879 (119903 119905) = 119879

1015840(119903) 119890

119894120596119905 (17a)

119906 (119903 119905) = 119906

1015840(119903) 119890

119894120596119905sdot 119890 (119903 119905) = 119864

1015840(119903) 119890

119894120596119905 (17b)

4 Abstract and Applied Analysis

10 12 14 16 18 20 22 24 26 28 30r

Chem

ical

pot

entia

l400300200100

0

minus600minus700

m = 04

m = 08

m = 15

minus100minus200minus300minus400minus500

t = 03 Ω = 12 1205910 = 04

(a)

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

10 12 14 16 18 20 22 24 26 28 30r

Chem

ical

pot

entia

l

400300200100

0

minus600minus700

minus100minus200minus300minus400minus500

(b)

Figure 1 Variation of chemical potential 119875 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

The c

once

ntra

tion

3E minus 015

4E minus 015

2E minus 015

1E minus 015

0E + 000

minus1E minus 015

minus2E minus 015

minus3E minus 015

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

10 12 14 16 18 20 22 24 26 28 30rTh

e con

cent

ratio

n

t = 03 m = 08 1205910 = 04

Ω = 06

Ω = 12

Ω = 18

3E minus 015

2E minus 015

1E minus 015

0E + 000

minus1E minus 015

minus2E minus 015

minus3E minus 015

(b)

Figure 2 Variation of concentration 119862 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

where 119890 = (120597119906120597119903) + (2119906119903) = (1119903

2)(120597120597119903)(119903

2119906)

Substituting (17a) and (17b) into (10)ndash(12) yields

120597119864

1015840

120597119903

minus

120597119862

1015840

120597119903

minus

120597119879

1015840

120597119903

= 120573119906

1015840

(18)

nabla

2119879

1015840= 1198961(119879

1015840+ 1205761119864

1015840+ 1205762119862

1015840) (19)

nabla

2119864

1015840+ ℎ4nabla

2119879

1015840+ ℎ61198962119866 = ℎ

5nabla

2119862

1015840 (20)

Applying the operator Laplacian operator nabla2 to (18) weobtain

(nabla

2minus 120573)119864

1015840= nabla

2119862

1015840+ nabla

2119879

1015840 (21)

From (19)ndash(21) we obtain

(nabla

6+ 1198871nabla

4+ 1198872nabla

2+ 1198873) (119864

1015840 119879

1015840 119862

1015840) = 0 (22)

where

1198871=

minus1

(ℎ5minus 1)

times [1198962ℎ6+ 1198961(ℎ5+ 1205762ℎ4) + 120573ℎ

5+ 1198961(1205761ℎ5+ 1205762)]

1198872=

1

(ℎ5minus 1)

times [11989611198962ℎ6+ 120573 (119896

2ℎ6+ 1198961(ℎ5+ 1205762ℎ4)) + 119896

111989621205761ℎ6]

1198873=

minus12057311989611198962ℎ6

(ℎ5minus 1)

1198961= 119894120596 (1 + 119894120596120591

0)

1198962= 119894120596 (1 + 119894120596120591) 120573 = minus (120596

2+ Ω

2+ 2119894120596Ω)

(23)

Abstract and Applied Analysis 5

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

(a)

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

The t

empe

ratu

re

(b)

Figure 3 Variation of temperature 120579 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus2010 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus20 10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 4 Variation of displacement 119906 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Equation (22) can be factorized as

(nabla

2+ 119876

2

1) (nabla

2+ 119876

2

2) (nabla

2+ 119876

2

3) (119864

1015840 119879

1015840 119862

1015840) = 0 (24)

where 119876

2

1 11987622 and 119876

2

3are the roots of the characteristic

equation

119876

6+ 1198871119876

4+ 1198872119876

2+ 1198873= 0 (25)

The solution of (24) which is bounded at infinity is given by

119879

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861119895 (120596)11987012

(119876119895119903)

119864

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903)

119862

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903)

(26)

where1198611198951198611015840119895 and11986110158401015840

119895are parameters depending only on120596 and

11987012

is the modified spherical Bessel function of the second

6 Abstract and Applied AnalysisRa

dial

stre

ss10

5

0

minus5

minus10

minus20

minus15

minus25

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

100000

80000

60000

40000

20000

0

Radi

al st

ress

minus20000

minus40000

minus60000

minus80000

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 5 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Tang

entia

l stre

ss

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

Tang

entia

l stre

ss

20

40

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 6 Variation of tangential stress 120590120601120601

with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

kind of order 12 Compatibility between (26) along with (19)and (20) will give rise to

119861

1015840

119895(120596) =

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895(120596)

119861

10158401015840

119895(120596) =

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895 (120596)

(27)

Substituting (26) into (17a) and (17b) we obtain

119879 (119903 119905) =

1

radic119903

3

sum

119895=1

119861119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (28)

119890 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (29)

119862 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (30)

Abstract and Applied Analysis 7

10 12 14 16 18 20 22 24 26 28 30r

The d

ispla

cem

ent

600

500

400

300

200

100

0

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 7 Variation of displacement 119906 with radius 119903 (thermoelasticnonhomogeneity medium)

100

12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

minus1000minus500

minus1500minus2000minus2500minus3000minus3500minus4000minus4500

Figure 8 Variation of temperature 120579 with radius 119903 (thermoelasticnonhomogeneity medium)

Integrating both sides of (29) from 119903 to infinity and assumingthat 119906(119903 119905) vanishes at infinity we obtain

119906 (119903 119905) =

1

radic119903

3

sum

119895=1

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 119861119895 (120596)11987032

(119876119895119903) 119890

119894120596119905

(31)

Radi

al st

ress

0

minus200

minus400

minus600

minus800

minus1000

minus1200

minus1400

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 9 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic

nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

0

200

400

600

800

1000

1200

1400

1600Ta

ngen

tial s

tress

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 10 Variation of tangential stress 120590120601120601

with radius 119903 (thermoe-lastic nonhomogeneity medium)

From (13a) (13b) and (13c)-(14) we get

120590119903119903=

1

radic119903

3

sum

119895=1

119861119895(120596)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times

[

[

(ℎ1+ℎ2) minus

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minusℎ61198962)

(119876

2

119895minus1198961) (ℎ5119876

2

119895minusℎ61198962)minus12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119903) minus

ℎ1

119903

11987032

(119876119895119903)

119890

119894120596119905

8 Abstract and Applied Analysis

120590120601120601

= 120590120579120579

=

1

radic119903

3

sum

119895=1

119861119895(120596)

times

ℎ1

119903

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 11987032

(119876119895119903)

+[

[

ℎ2

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus 1 minus

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

]

]

times 11987012

(119876119895119903)

119890

119894120596119905

119875 =

1

radic119903

3

sum

119895=1

119861119895 (120596)

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ ℎ3

12057611198961ℎ4119876

2

119895+119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119903) 119890

119894120596119905

(32)

Using the boundary conditions we get

1198611(120596) = (minus119873

2[radic11988612057901198823minus 11987012

(1198861198763) 1199010]

+1198733[radic11988612057901198822minus 11987012

(1198861198762) 1199010]) times (119872)

minus1

1198612(120596) = (radic119886119873

1[12057901198823minus 11987012

(1198861198763) 1199010]

+radic1198861198733[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

1198613 (120596) = (radic119886119873

1[11987012

(1198861198762) 1199010minus 12057901198822]

minusradic1198861198732[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

119873119895=

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times[

[

(ℎ1+ ℎ2)

minus

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119886) minus [

ℎ1

119886

+ 120583119890119867

2

120601(

3

119886

2minus 119876

2

119894)]

times 11987032

(119876119895119886)

119882119895=

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ℎ3

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119886)

119872 = 1198731[119882311987012

(1198861198762) minus 119882

211987012

(1198861198763)]

minus 1198732[119882311987012

(1198861198761) minus 119882

111987012

(1198861198761)]

+1198733[119882211987012

(1198861198761) minus 119882

111987012

(1198861198762)]

119895 = 1 2 3

(33)

6 Particular Case

If we neglect the initial stress and diffusion effects byeliminating (3) and (8) and putting 119875

1= 1205732= 119862 = 0 in (4)

and (6) we get (119879 119890) 119906(119903 119905) 120590119903119903 120590120601120601 and 120590

120579120579

(119879 119890) = 119860119890

minus120582119903+119894120596119905+ 119861119890

minus120582119903+119894120596119905 (34)

where

1205781= minus (ℓ

1+ 120573) 120578

2= ℓ1(120573 minus 120576

1)

(120582

2

1 120582

2

2) =

1

2

[1205781plusmn radic120578

2

1minus 41205782]

119906 (119903 119905) = minus [

119903

2120582

2

1+ 2119903120582

1+ 2

119903

2120582

3

1

119860 (120596) 119890

minus1205821119903

+

119903

2120582

2

2+21199031205822+2

119903

2120582

3

2

119861 (120596) 119890

minus1205822119903+119862 (120596)] 119890

119894120596119905

120590119903119903=[1205721(

4 + 41199031205821+ 2119903

2120582

2

1+ 119903

3120582

3

1

119903

3120582

3

1

+(1205722minus 1))119860 (120596) 119890

minus1205821119903

+ 1205721(

4 + 41199031205822+ 2119903

2120582

2

2+ 119903

3120582

3

2

119903

3120582

3

2

+ (1205722minus 1))

times 119861 (120596) 119890

minus1205822119903] 119890

119894120596119905

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

of the theory of generalized thermoelastic diffusion withone relaxation time The reflection of SV waves from thefree surface of an elastic solid in generalized thermoelasticdiffusion was discussed by Singh [21] Kumar and Kansal[22] discussed the propagation of Lambwaves in transverselyisotropic thermoelastic diffusive plates Thermomechanicalresponse of generalized thermoelastic diffusion with onerelaxation time due to time harmonic sources was discussedby Ram et al [23] Aouadi [24] examined the thermoe-lastic diffusion problem for an infinite elastic body withspherical cavity Abd-Alla and Mahmoud [25] investigatedanalytical solution of wave propagation in nonhomogeneousorthotropic rotating elastic media Othman et al [26] dis-cussed the effect of diffusion in a two-dimensional problemofgeneralized thermoelasticity with Green-Naghdi theory Xiaet al [27] studied the influence of diffusion on generalizedthermoelastic problems of infinite body with a cylindricalcavity Deswal and Kalkal [28] studied the two-dimensionalgeneralized electromagneto-thermoviscoelastic problem fora half-space with diffusion Abd-Alla and Abo-Dahab [29]found the time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energydissipation Mahmoud [30] discussed influence of rotationand generalized magnetothermoelastic on Rayleigh waves ina granular medium under effect of initial stress and gravityfield Abd-Alla et al [31 32] studied the generalizedmagneto-thermoelastic Rayleigh waves in a granular medium underthe influence of a gravity field and initial stress

In the present investigation the temperature displace-ments stresses diffusion and concentration as well as chem-ical potential are obtained in the physical domain using theharmonic vibrations Also study of the interaction betweenthe processes of elasticity nonhomogeneity rotation mag-netic field initial stress heat and diffusion in an infiniteelastic solidwith a spherical cavity in the context of the theoryof generalized thermoelastic diffusion is presented

2 Formulation of the Problem

Consider a perfect electric conductor and linearizedMaxwellequations governing the electromagnetic field in the absenceof the displacement current (SI) in the form as in Kraus[33] Applying an initial magnetic field vector

119867(0 01198670) in

spherical coordinates (119903 120579 120601) = (119906(119903 119905) 0 0) One willconsider a nonhomogeneous isotropic medium occupyingthe region 119886 le 119903 le 119887 where a is the radius of the sphericalcavity The strain tensor has the following components

119890119894119895=

1

2

(119906119894119895+ 119906119895119894) (1a)

120596119894119895=

1

2

(119906119895119894minus 119906119894119895) (1b)

119890119896119896

=

1

119903

2

120597

120597119903

(119903

2119906) 119894 119895 119896 = 1 2 3 (1c)

119890119903119903=

120597119906

120597119903

119890120579120579

= 119890120601120601

=

119906

119903

(1d)

The cubical dilatation is given by (1119903

2)(120597120597119903)(119903

2119906)

where the nonvanishing displacement component is theradial one 119906

119903= 119906(119903 119905) The elastic medium is rotating

uniformly with an angular velocity Ω = Ω 119899 where 119899 is a unit

vector representing the direction of the axis of rotation Thedisplacement equation of motion in the rotating frame hastwo additional terms

Ω times (

Ω times ) which is the centripetalacceleration due to time varying motion only and 2

Ω times

is the Coriolis acceleration where Ω = (0 Ω 0) Following

Sherief rsquos theory of generalized thermoelastic diffusion [19]and Sherief and Saleh [20] one is going to study an isotropicnonhomogeneous elastic medium which suffers thermalshock Due to spherical symmetry the stress-displacement-temperature-diffusion relation or constitutive equations aregiven by

120590119903119903= (2120583 + 119875

1)

120597119906

120597119903

+ (120582+1198751)

1

119903

2

120597

120597119903

(119903

2119906)

minus 1205731(120579 minus 120579

0) minus 1205732119862

(2a)

120590120601120601

= 120590120579120579

= (2120583 + 1198751)

119906

119903

+ (120582 + 1198751)

1

119903

2

120597

120597119903

(119903

2119906)

minus 1205731(120579 minus 120579

0) minus 1205732119862

(2b)

The chemical-displacement-temperature-diffusion rela-tion is given by

119875 = minus1205732

1

119903

2

120597

120597119903

(119903

2119906) + 119887119862 minus 119888 (120579 minus 120579

0) (3)

The governing equation for an isotropic nonhomoge-neous elastic solid with generalized magneto-thermoelasticdiffusion under effect of rotation is given by

120597 (1119903

2) (120597120597119903) (119903

2119906)

120597119903

minus

1205732

ℎ0

120597119862

120597119903

minus

1205731

ℎ0

120597 (120579 minus 1205790)

120597119903

+ 119865119903

=

120588

ℎ0

[

120597

2119906

120597119905

2minus Ω

2119906 minus 2Ω

120597119906

120597119905

]

(4)

where 120582 and 120583 are Lamersquos elastic constants 120575119894119895is Kroneckerrsquos

delta 1198751is the initial stress 120588 is the density of the medium

and 119865 is defined as Lorentzrsquos force which may be written as

119865 = 120583

119890(

119869 times

119867) = (120583119890119867

2

0

120597

120597119903

(

120597119906

120597119903

+

2119906

119903

) 0 0) (5)

where 120583119890is themagnetic permeability

119867 is themagnetic fieldvector

119869 is the electric current density is the displacementvector and 119905 is the time

Equation of heat conduction is given by

119870nabla

2120579 = (

120597

120597119905

+ 120591

120597

2

120597119905

2)(120588119888V120579 + 120579

01205731

1

119903

2

120597

120597119903

(119903

2119906) + 119888120579

0119862)

(6)

where the Laplacian operator nabla2 is given by nabla2 = (120597

2120597119903

2) +

(2119903)(120597120597119903) and ℎ0

= 2120583 + 120582 + (11987512) + 120583

119890119867

2

0 ℎ0is the

Abstract and Applied Analysis 3

coefficient of linear diffusion expansion 119870 is the thermalconductivity 120579 is the absolute temperature 120579

0is the initial

uniform temperature and |(120579 minus 1205790)1205790| ≪ 1

Equation of conservation of mass diffusion may bewritten as

1198631205732nabla

2 1

119903

2

120597

120597119903

(119903

2119906) + 119863119888nabla

2120579 + (

120597

120597119905

+ 120591

120597

2

120597119905

2)119862 = 119863119887nabla

2119862

(7)

where 120591 is the diffusion relaxation time 1205910is the thermal

relaxation time 120572119905is the coefficient of linear thermal expan-

sion and 1205790is constant where 120573

1= (3120582 + 2120583)120572

119905 1205732

=

(3120582 + 2120583)120572119888 120590119894119895are the components of the stress tensor

120591119894119895are the components of stress tensor 119887 and 119888 are the

measures of thermodiffusion and diffusive effects 119862 is theconcentration 119862V is the specific heat at constant strain 119863is the diffusive coefficient and 119890

119894119895are the components of

the strain tensor The thermal relaxation time 1205910will ensure

that the heat conduction equation will predict finite speedof heat propagation The diffusion relaxation time 120591 whichwill ensure the equation satisfied by the concentration 119862 willalso predict finite speed of propagation of matter from onemedium to the other

3 Dimensionless Quantities

Introduce the following nondimensional parameters

119903

lowast= 11988811205780119903 119906

lowast= 11988811205780119906 119879 =

1205731(120579 minus 120579

0)

ℎ0

119862

lowast=

1205732119862

ℎ0

Ω

lowast=

Ω

119888

2

11205780

120590

lowast

119894119895=

120590119894119895

ℎ0

119875

lowast=

119875

1205732

119905

lowast= 119888

2

11205780119905 120591

lowast

0= 119888

2

112057801205910

120591

lowast= 119888

2

11205780120591 120578

0=

120588119888V

119870

119888

2

1=

ℎ0

120588

(8)

The elastic constants 120582 120583 and the density 120588 of nonhomoge-neous material in form [32] are as follows

120582 = 119903

21198981205820 120583 = 119903

21198981205830 120583

ℎ= 119903

21198981205830

120588 = 119903

21198981205880 119901

lowast= 119901

lowast

0119903

2119898

(9)

Using the above non-dimensional parameters and (9) in(10)ndash(14) the non-dimensional system becomes

120597 (1119903

2) (120597120597119903) (119903

2119906)

120597119903

minus

120597119862

120597119903

minus

120597119879

120597119903

=

120597

2119906

120597119905

2minus Ω

2119906 minus 2Ω

120597119906

120597119905

(10)

nabla

2119879 = (

120597

120597119905

+ 1205910

120597

2

120597119905

2)(119879 + 120576

1

1

119903

2

120597

120597119903

(119903

2119906) + 120576

2119862) (11)

nabla

2 1

119903

2

120597

120597119903

(119903

2119906) + ℎ

4nabla

2119879 + ℎ6(

120597

120597119905

+ 120591

120597

2

120597119905

2)119862 = ℎ

5nabla

2119862

(12)

120590119903119903= ℎ1

120597119906

120597119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13a)

120590120579120579

= ℎ1

119906

119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13b)

120590120601120601

= ℎ1

119906

119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13c)

119875 = minus

1

119903

2

120597

120597119903

(119903

2119906) + ℎ

3119862 minus ℎ4119879 (14)

where

1205761=

120573

2

11198790119898

ℎ0120588119888V

1205762=

12057311198881198790ℎ0(119898 + 2)

1205732

ℎ1=

2120583

ℎ0

ℎ2=

120582

ℎ0

ℎ3=

119898119887ℎ0

120573

2

2

ℎ4=

119888ℎ0

12057311205732

ℎ5=

119863119887ℎ0

1205732

ℎ6=

2119898+ℎ0

120573

2

21198631205780

(15)

4 Boundary Conditions

The nonhomogeneous initial conditions are supplementedby the following boundary conditions The cavity surface istraction free

120590119903119903(119903 119905) + 120591

119903119903(119903 119905) = 0 119903 = 119886 (16a)

The cavity surface is subjected to a thermal shock

119879 (119886 119905) = 1198790119867(119905) (16b)

where119867(119905) is the Heaviside unit step function The chemicalpotential is also assumed to be a known function of time atthe cavity surface

119875 (119886 119905) = 1198750119867(119905) 119875

0is real constant (16c)

The displacement function is as follows

119906 (119903 119905) = 0 119903 = 119886 (16d)

5 Solution of the Problem

In this section one obtains the analytical solution of theproblem for a spherical region with boundary conditionsby taking the harmonic vibrations One assumes that thesolution of (10)ndash(12) as follows

119862 (119903 119905) = 119862

1015840(119903) 119890

119894120596119905 119879 (119903 119905) = 119879

1015840(119903) 119890

119894120596119905 (17a)

119906 (119903 119905) = 119906

1015840(119903) 119890

119894120596119905sdot 119890 (119903 119905) = 119864

1015840(119903) 119890

119894120596119905 (17b)

4 Abstract and Applied Analysis

10 12 14 16 18 20 22 24 26 28 30r

Chem

ical

pot

entia

l400300200100

0

minus600minus700

m = 04

m = 08

m = 15

minus100minus200minus300minus400minus500

t = 03 Ω = 12 1205910 = 04

(a)

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

10 12 14 16 18 20 22 24 26 28 30r

Chem

ical

pot

entia

l

400300200100

0

minus600minus700

minus100minus200minus300minus400minus500

(b)

Figure 1 Variation of chemical potential 119875 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

The c

once

ntra

tion

3E minus 015

4E minus 015

2E minus 015

1E minus 015

0E + 000

minus1E minus 015

minus2E minus 015

minus3E minus 015

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

10 12 14 16 18 20 22 24 26 28 30rTh

e con

cent

ratio

n

t = 03 m = 08 1205910 = 04

Ω = 06

Ω = 12

Ω = 18

3E minus 015

2E minus 015

1E minus 015

0E + 000

minus1E minus 015

minus2E minus 015

minus3E minus 015

(b)

Figure 2 Variation of concentration 119862 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

where 119890 = (120597119906120597119903) + (2119906119903) = (1119903

2)(120597120597119903)(119903

2119906)

Substituting (17a) and (17b) into (10)ndash(12) yields

120597119864

1015840

120597119903

minus

120597119862

1015840

120597119903

minus

120597119879

1015840

120597119903

= 120573119906

1015840

(18)

nabla

2119879

1015840= 1198961(119879

1015840+ 1205761119864

1015840+ 1205762119862

1015840) (19)

nabla

2119864

1015840+ ℎ4nabla

2119879

1015840+ ℎ61198962119866 = ℎ

5nabla

2119862

1015840 (20)

Applying the operator Laplacian operator nabla2 to (18) weobtain

(nabla

2minus 120573)119864

1015840= nabla

2119862

1015840+ nabla

2119879

1015840 (21)

From (19)ndash(21) we obtain

(nabla

6+ 1198871nabla

4+ 1198872nabla

2+ 1198873) (119864

1015840 119879

1015840 119862

1015840) = 0 (22)

where

1198871=

minus1

(ℎ5minus 1)

times [1198962ℎ6+ 1198961(ℎ5+ 1205762ℎ4) + 120573ℎ

5+ 1198961(1205761ℎ5+ 1205762)]

1198872=

1

(ℎ5minus 1)

times [11989611198962ℎ6+ 120573 (119896

2ℎ6+ 1198961(ℎ5+ 1205762ℎ4)) + 119896

111989621205761ℎ6]

1198873=

minus12057311989611198962ℎ6

(ℎ5minus 1)

1198961= 119894120596 (1 + 119894120596120591

0)

1198962= 119894120596 (1 + 119894120596120591) 120573 = minus (120596

2+ Ω

2+ 2119894120596Ω)

(23)

Abstract and Applied Analysis 5

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

(a)

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

The t

empe

ratu

re

(b)

Figure 3 Variation of temperature 120579 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus2010 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus20 10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 4 Variation of displacement 119906 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Equation (22) can be factorized as

(nabla

2+ 119876

2

1) (nabla

2+ 119876

2

2) (nabla

2+ 119876

2

3) (119864

1015840 119879

1015840 119862

1015840) = 0 (24)

where 119876

2

1 11987622 and 119876

2

3are the roots of the characteristic

equation

119876

6+ 1198871119876

4+ 1198872119876

2+ 1198873= 0 (25)

The solution of (24) which is bounded at infinity is given by

119879

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861119895 (120596)11987012

(119876119895119903)

119864

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903)

119862

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903)

(26)

where1198611198951198611015840119895 and11986110158401015840

119895are parameters depending only on120596 and

11987012

is the modified spherical Bessel function of the second

6 Abstract and Applied AnalysisRa

dial

stre

ss10

5

0

minus5

minus10

minus20

minus15

minus25

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

100000

80000

60000

40000

20000

0

Radi

al st

ress

minus20000

minus40000

minus60000

minus80000

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 5 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Tang

entia

l stre

ss

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

Tang

entia

l stre

ss

20

40

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 6 Variation of tangential stress 120590120601120601

with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

kind of order 12 Compatibility between (26) along with (19)and (20) will give rise to

119861

1015840

119895(120596) =

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895(120596)

119861

10158401015840

119895(120596) =

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895 (120596)

(27)

Substituting (26) into (17a) and (17b) we obtain

119879 (119903 119905) =

1

radic119903

3

sum

119895=1

119861119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (28)

119890 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (29)

119862 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (30)

Abstract and Applied Analysis 7

10 12 14 16 18 20 22 24 26 28 30r

The d

ispla

cem

ent

600

500

400

300

200

100

0

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 7 Variation of displacement 119906 with radius 119903 (thermoelasticnonhomogeneity medium)

100

12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

minus1000minus500

minus1500minus2000minus2500minus3000minus3500minus4000minus4500

Figure 8 Variation of temperature 120579 with radius 119903 (thermoelasticnonhomogeneity medium)

Integrating both sides of (29) from 119903 to infinity and assumingthat 119906(119903 119905) vanishes at infinity we obtain

119906 (119903 119905) =

1

radic119903

3

sum

119895=1

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 119861119895 (120596)11987032

(119876119895119903) 119890

119894120596119905

(31)

Radi

al st

ress

0

minus200

minus400

minus600

minus800

minus1000

minus1200

minus1400

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 9 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic

nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

0

200

400

600

800

1000

1200

1400

1600Ta

ngen

tial s

tress

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 10 Variation of tangential stress 120590120601120601

with radius 119903 (thermoe-lastic nonhomogeneity medium)

From (13a) (13b) and (13c)-(14) we get

120590119903119903=

1

radic119903

3

sum

119895=1

119861119895(120596)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times

[

[

(ℎ1+ℎ2) minus

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minusℎ61198962)

(119876

2

119895minus1198961) (ℎ5119876

2

119895minusℎ61198962)minus12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119903) minus

ℎ1

119903

11987032

(119876119895119903)

119890

119894120596119905

8 Abstract and Applied Analysis

120590120601120601

= 120590120579120579

=

1

radic119903

3

sum

119895=1

119861119895(120596)

times

ℎ1

119903

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 11987032

(119876119895119903)

+[

[

ℎ2

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus 1 minus

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

]

]

times 11987012

(119876119895119903)

119890

119894120596119905

119875 =

1

radic119903

3

sum

119895=1

119861119895 (120596)

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ ℎ3

12057611198961ℎ4119876

2

119895+119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119903) 119890

119894120596119905

(32)

Using the boundary conditions we get

1198611(120596) = (minus119873

2[radic11988612057901198823minus 11987012

(1198861198763) 1199010]

+1198733[radic11988612057901198822minus 11987012

(1198861198762) 1199010]) times (119872)

minus1

1198612(120596) = (radic119886119873

1[12057901198823minus 11987012

(1198861198763) 1199010]

+radic1198861198733[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

1198613 (120596) = (radic119886119873

1[11987012

(1198861198762) 1199010minus 12057901198822]

minusradic1198861198732[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

119873119895=

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times[

[

(ℎ1+ ℎ2)

minus

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119886) minus [

ℎ1

119886

+ 120583119890119867

2

120601(

3

119886

2minus 119876

2

119894)]

times 11987032

(119876119895119886)

119882119895=

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ℎ3

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119886)

119872 = 1198731[119882311987012

(1198861198762) minus 119882

211987012

(1198861198763)]

minus 1198732[119882311987012

(1198861198761) minus 119882

111987012

(1198861198761)]

+1198733[119882211987012

(1198861198761) minus 119882

111987012

(1198861198762)]

119895 = 1 2 3

(33)

6 Particular Case

If we neglect the initial stress and diffusion effects byeliminating (3) and (8) and putting 119875

1= 1205732= 119862 = 0 in (4)

and (6) we get (119879 119890) 119906(119903 119905) 120590119903119903 120590120601120601 and 120590

120579120579

(119879 119890) = 119860119890

minus120582119903+119894120596119905+ 119861119890

minus120582119903+119894120596119905 (34)

where

1205781= minus (ℓ

1+ 120573) 120578

2= ℓ1(120573 minus 120576

1)

(120582

2

1 120582

2

2) =

1

2

[1205781plusmn radic120578

2

1minus 41205782]

119906 (119903 119905) = minus [

119903

2120582

2

1+ 2119903120582

1+ 2

119903

2120582

3

1

119860 (120596) 119890

minus1205821119903

+

119903

2120582

2

2+21199031205822+2

119903

2120582

3

2

119861 (120596) 119890

minus1205822119903+119862 (120596)] 119890

119894120596119905

120590119903119903=[1205721(

4 + 41199031205821+ 2119903

2120582

2

1+ 119903

3120582

3

1

119903

3120582

3

1

+(1205722minus 1))119860 (120596) 119890

minus1205821119903

+ 1205721(

4 + 41199031205822+ 2119903

2120582

2

2+ 119903

3120582

3

2

119903

3120582

3

2

+ (1205722minus 1))

times 119861 (120596) 119890

minus1205822119903] 119890

119894120596119905

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

coefficient of linear diffusion expansion 119870 is the thermalconductivity 120579 is the absolute temperature 120579

0is the initial

uniform temperature and |(120579 minus 1205790)1205790| ≪ 1

Equation of conservation of mass diffusion may bewritten as

1198631205732nabla

2 1

119903

2

120597

120597119903

(119903

2119906) + 119863119888nabla

2120579 + (

120597

120597119905

+ 120591

120597

2

120597119905

2)119862 = 119863119887nabla

2119862

(7)

where 120591 is the diffusion relaxation time 1205910is the thermal

relaxation time 120572119905is the coefficient of linear thermal expan-

sion and 1205790is constant where 120573

1= (3120582 + 2120583)120572

119905 1205732

=

(3120582 + 2120583)120572119888 120590119894119895are the components of the stress tensor

120591119894119895are the components of stress tensor 119887 and 119888 are the

measures of thermodiffusion and diffusive effects 119862 is theconcentration 119862V is the specific heat at constant strain 119863is the diffusive coefficient and 119890

119894119895are the components of

the strain tensor The thermal relaxation time 1205910will ensure

that the heat conduction equation will predict finite speedof heat propagation The diffusion relaxation time 120591 whichwill ensure the equation satisfied by the concentration 119862 willalso predict finite speed of propagation of matter from onemedium to the other

3 Dimensionless Quantities

Introduce the following nondimensional parameters

119903

lowast= 11988811205780119903 119906

lowast= 11988811205780119906 119879 =

1205731(120579 minus 120579

0)

ℎ0

119862

lowast=

1205732119862

ℎ0

Ω

lowast=

Ω

119888

2

11205780

120590

lowast

119894119895=

120590119894119895

ℎ0

119875

lowast=

119875

1205732

119905

lowast= 119888

2

11205780119905 120591

lowast

0= 119888

2

112057801205910

120591

lowast= 119888

2

11205780120591 120578

0=

120588119888V

119870

119888

2

1=

ℎ0

120588

(8)

The elastic constants 120582 120583 and the density 120588 of nonhomoge-neous material in form [32] are as follows

120582 = 119903

21198981205820 120583 = 119903

21198981205830 120583

ℎ= 119903

21198981205830

120588 = 119903

21198981205880 119901

lowast= 119901

lowast

0119903

2119898

(9)

Using the above non-dimensional parameters and (9) in(10)ndash(14) the non-dimensional system becomes

120597 (1119903

2) (120597120597119903) (119903

2119906)

120597119903

minus

120597119862

120597119903

minus

120597119879

120597119903

=

120597

2119906

120597119905

2minus Ω

2119906 minus 2Ω

120597119906

120597119905

(10)

nabla

2119879 = (

120597

120597119905

+ 1205910

120597

2

120597119905

2)(119879 + 120576

1

1

119903

2

120597

120597119903

(119903

2119906) + 120576

2119862) (11)

nabla

2 1

119903

2

120597

120597119903

(119903

2119906) + ℎ

4nabla

2119879 + ℎ6(

120597

120597119905

+ 120591

120597

2

120597119905

2)119862 = ℎ

5nabla

2119862

(12)

120590119903119903= ℎ1

120597119906

120597119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13a)

120590120579120579

= ℎ1

119906

119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13b)

120590120601120601

= ℎ1

119906

119903

+ ℎ2

1

119903

2

120597

120597119903

(119903

2119906) minus 119879 minus 119862 (13c)

119875 = minus

1

119903

2

120597

120597119903

(119903

2119906) + ℎ

3119862 minus ℎ4119879 (14)

where

1205761=

120573

2

11198790119898

ℎ0120588119888V

1205762=

12057311198881198790ℎ0(119898 + 2)

1205732

ℎ1=

2120583

ℎ0

ℎ2=

120582

ℎ0

ℎ3=

119898119887ℎ0

120573

2

2

ℎ4=

119888ℎ0

12057311205732

ℎ5=

119863119887ℎ0

1205732

ℎ6=

2119898+ℎ0

120573

2

21198631205780

(15)

4 Boundary Conditions

The nonhomogeneous initial conditions are supplementedby the following boundary conditions The cavity surface istraction free

120590119903119903(119903 119905) + 120591

119903119903(119903 119905) = 0 119903 = 119886 (16a)

The cavity surface is subjected to a thermal shock

119879 (119886 119905) = 1198790119867(119905) (16b)

where119867(119905) is the Heaviside unit step function The chemicalpotential is also assumed to be a known function of time atthe cavity surface

119875 (119886 119905) = 1198750119867(119905) 119875

0is real constant (16c)

The displacement function is as follows

119906 (119903 119905) = 0 119903 = 119886 (16d)

5 Solution of the Problem

In this section one obtains the analytical solution of theproblem for a spherical region with boundary conditionsby taking the harmonic vibrations One assumes that thesolution of (10)ndash(12) as follows

119862 (119903 119905) = 119862

1015840(119903) 119890

119894120596119905 119879 (119903 119905) = 119879

1015840(119903) 119890

119894120596119905 (17a)

119906 (119903 119905) = 119906

1015840(119903) 119890

119894120596119905sdot 119890 (119903 119905) = 119864

1015840(119903) 119890

119894120596119905 (17b)

4 Abstract and Applied Analysis

10 12 14 16 18 20 22 24 26 28 30r

Chem

ical

pot

entia

l400300200100

0

minus600minus700

m = 04

m = 08

m = 15

minus100minus200minus300minus400minus500

t = 03 Ω = 12 1205910 = 04

(a)

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

10 12 14 16 18 20 22 24 26 28 30r

Chem

ical

pot

entia

l

400300200100

0

minus600minus700

minus100minus200minus300minus400minus500

(b)

Figure 1 Variation of chemical potential 119875 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

The c

once

ntra

tion

3E minus 015

4E minus 015

2E minus 015

1E minus 015

0E + 000

minus1E minus 015

minus2E minus 015

minus3E minus 015

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

10 12 14 16 18 20 22 24 26 28 30rTh

e con

cent

ratio

n

t = 03 m = 08 1205910 = 04

Ω = 06

Ω = 12

Ω = 18

3E minus 015

2E minus 015

1E minus 015

0E + 000

minus1E minus 015

minus2E minus 015

minus3E minus 015

(b)

Figure 2 Variation of concentration 119862 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

where 119890 = (120597119906120597119903) + (2119906119903) = (1119903

2)(120597120597119903)(119903

2119906)

Substituting (17a) and (17b) into (10)ndash(12) yields

120597119864

1015840

120597119903

minus

120597119862

1015840

120597119903

minus

120597119879

1015840

120597119903

= 120573119906

1015840

(18)

nabla

2119879

1015840= 1198961(119879

1015840+ 1205761119864

1015840+ 1205762119862

1015840) (19)

nabla

2119864

1015840+ ℎ4nabla

2119879

1015840+ ℎ61198962119866 = ℎ

5nabla

2119862

1015840 (20)

Applying the operator Laplacian operator nabla2 to (18) weobtain

(nabla

2minus 120573)119864

1015840= nabla

2119862

1015840+ nabla

2119879

1015840 (21)

From (19)ndash(21) we obtain

(nabla

6+ 1198871nabla

4+ 1198872nabla

2+ 1198873) (119864

1015840 119879

1015840 119862

1015840) = 0 (22)

where

1198871=

minus1

(ℎ5minus 1)

times [1198962ℎ6+ 1198961(ℎ5+ 1205762ℎ4) + 120573ℎ

5+ 1198961(1205761ℎ5+ 1205762)]

1198872=

1

(ℎ5minus 1)

times [11989611198962ℎ6+ 120573 (119896

2ℎ6+ 1198961(ℎ5+ 1205762ℎ4)) + 119896

111989621205761ℎ6]

1198873=

minus12057311989611198962ℎ6

(ℎ5minus 1)

1198961= 119894120596 (1 + 119894120596120591

0)

1198962= 119894120596 (1 + 119894120596120591) 120573 = minus (120596

2+ Ω

2+ 2119894120596Ω)

(23)

Abstract and Applied Analysis 5

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

(a)

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

The t

empe

ratu

re

(b)

Figure 3 Variation of temperature 120579 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus2010 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus20 10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 4 Variation of displacement 119906 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Equation (22) can be factorized as

(nabla

2+ 119876

2

1) (nabla

2+ 119876

2

2) (nabla

2+ 119876

2

3) (119864

1015840 119879

1015840 119862

1015840) = 0 (24)

where 119876

2

1 11987622 and 119876

2

3are the roots of the characteristic

equation

119876

6+ 1198871119876

4+ 1198872119876

2+ 1198873= 0 (25)

The solution of (24) which is bounded at infinity is given by

119879

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861119895 (120596)11987012

(119876119895119903)

119864

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903)

119862

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903)

(26)

where1198611198951198611015840119895 and11986110158401015840

119895are parameters depending only on120596 and

11987012

is the modified spherical Bessel function of the second

6 Abstract and Applied AnalysisRa

dial

stre

ss10

5

0

minus5

minus10

minus20

minus15

minus25

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

100000

80000

60000

40000

20000

0

Radi

al st

ress

minus20000

minus40000

minus60000

minus80000

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 5 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Tang

entia

l stre

ss

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

Tang

entia

l stre

ss

20

40

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 6 Variation of tangential stress 120590120601120601

with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

kind of order 12 Compatibility between (26) along with (19)and (20) will give rise to

119861

1015840

119895(120596) =

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895(120596)

119861

10158401015840

119895(120596) =

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895 (120596)

(27)

Substituting (26) into (17a) and (17b) we obtain

119879 (119903 119905) =

1

radic119903

3

sum

119895=1

119861119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (28)

119890 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (29)

119862 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (30)

Abstract and Applied Analysis 7

10 12 14 16 18 20 22 24 26 28 30r

The d

ispla

cem

ent

600

500

400

300

200

100

0

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 7 Variation of displacement 119906 with radius 119903 (thermoelasticnonhomogeneity medium)

100

12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

minus1000minus500

minus1500minus2000minus2500minus3000minus3500minus4000minus4500

Figure 8 Variation of temperature 120579 with radius 119903 (thermoelasticnonhomogeneity medium)

Integrating both sides of (29) from 119903 to infinity and assumingthat 119906(119903 119905) vanishes at infinity we obtain

119906 (119903 119905) =

1

radic119903

3

sum

119895=1

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 119861119895 (120596)11987032

(119876119895119903) 119890

119894120596119905

(31)

Radi

al st

ress

0

minus200

minus400

minus600

minus800

minus1000

minus1200

minus1400

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 9 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic

nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

0

200

400

600

800

1000

1200

1400

1600Ta

ngen

tial s

tress

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 10 Variation of tangential stress 120590120601120601

with radius 119903 (thermoe-lastic nonhomogeneity medium)

From (13a) (13b) and (13c)-(14) we get

120590119903119903=

1

radic119903

3

sum

119895=1

119861119895(120596)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times

[

[

(ℎ1+ℎ2) minus

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minusℎ61198962)

(119876

2

119895minus1198961) (ℎ5119876

2

119895minusℎ61198962)minus12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119903) minus

ℎ1

119903

11987032

(119876119895119903)

119890

119894120596119905

8 Abstract and Applied Analysis

120590120601120601

= 120590120579120579

=

1

radic119903

3

sum

119895=1

119861119895(120596)

times

ℎ1

119903

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 11987032

(119876119895119903)

+[

[

ℎ2

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus 1 minus

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

]

]

times 11987012

(119876119895119903)

119890

119894120596119905

119875 =

1

radic119903

3

sum

119895=1

119861119895 (120596)

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ ℎ3

12057611198961ℎ4119876

2

119895+119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119903) 119890

119894120596119905

(32)

Using the boundary conditions we get

1198611(120596) = (minus119873

2[radic11988612057901198823minus 11987012

(1198861198763) 1199010]

+1198733[radic11988612057901198822minus 11987012

(1198861198762) 1199010]) times (119872)

minus1

1198612(120596) = (radic119886119873

1[12057901198823minus 11987012

(1198861198763) 1199010]

+radic1198861198733[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

1198613 (120596) = (radic119886119873

1[11987012

(1198861198762) 1199010minus 12057901198822]

minusradic1198861198732[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

119873119895=

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times[

[

(ℎ1+ ℎ2)

minus

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119886) minus [

ℎ1

119886

+ 120583119890119867

2

120601(

3

119886

2minus 119876

2

119894)]

times 11987032

(119876119895119886)

119882119895=

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ℎ3

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119886)

119872 = 1198731[119882311987012

(1198861198762) minus 119882

211987012

(1198861198763)]

minus 1198732[119882311987012

(1198861198761) minus 119882

111987012

(1198861198761)]

+1198733[119882211987012

(1198861198761) minus 119882

111987012

(1198861198762)]

119895 = 1 2 3

(33)

6 Particular Case

If we neglect the initial stress and diffusion effects byeliminating (3) and (8) and putting 119875

1= 1205732= 119862 = 0 in (4)

and (6) we get (119879 119890) 119906(119903 119905) 120590119903119903 120590120601120601 and 120590

120579120579

(119879 119890) = 119860119890

minus120582119903+119894120596119905+ 119861119890

minus120582119903+119894120596119905 (34)

where

1205781= minus (ℓ

1+ 120573) 120578

2= ℓ1(120573 minus 120576

1)

(120582

2

1 120582

2

2) =

1

2

[1205781plusmn radic120578

2

1minus 41205782]

119906 (119903 119905) = minus [

119903

2120582

2

1+ 2119903120582

1+ 2

119903

2120582

3

1

119860 (120596) 119890

minus1205821119903

+

119903

2120582

2

2+21199031205822+2

119903

2120582

3

2

119861 (120596) 119890

minus1205822119903+119862 (120596)] 119890

119894120596119905

120590119903119903=[1205721(

4 + 41199031205821+ 2119903

2120582

2

1+ 119903

3120582

3

1

119903

3120582

3

1

+(1205722minus 1))119860 (120596) 119890

minus1205821119903

+ 1205721(

4 + 41199031205822+ 2119903

2120582

2

2+ 119903

3120582

3

2

119903

3120582

3

2

+ (1205722minus 1))

times 119861 (120596) 119890

minus1205822119903] 119890

119894120596119905

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

10 12 14 16 18 20 22 24 26 28 30r

Chem

ical

pot

entia

l400300200100

0

minus600minus700

m = 04

m = 08

m = 15

minus100minus200minus300minus400minus500

t = 03 Ω = 12 1205910 = 04

(a)

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

10 12 14 16 18 20 22 24 26 28 30r

Chem

ical

pot

entia

l

400300200100

0

minus600minus700

minus100minus200minus300minus400minus500

(b)

Figure 1 Variation of chemical potential 119875 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

The c

once

ntra

tion

3E minus 015

4E minus 015

2E minus 015

1E minus 015

0E + 000

minus1E minus 015

minus2E minus 015

minus3E minus 015

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

10 12 14 16 18 20 22 24 26 28 30rTh

e con

cent

ratio

n

t = 03 m = 08 1205910 = 04

Ω = 06

Ω = 12

Ω = 18

3E minus 015

2E minus 015

1E minus 015

0E + 000

minus1E minus 015

minus2E minus 015

minus3E minus 015

(b)

Figure 2 Variation of concentration 119862 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

where 119890 = (120597119906120597119903) + (2119906119903) = (1119903

2)(120597120597119903)(119903

2119906)

Substituting (17a) and (17b) into (10)ndash(12) yields

120597119864

1015840

120597119903

minus

120597119862

1015840

120597119903

minus

120597119879

1015840

120597119903

= 120573119906

1015840

(18)

nabla

2119879

1015840= 1198961(119879

1015840+ 1205761119864

1015840+ 1205762119862

1015840) (19)

nabla

2119864

1015840+ ℎ4nabla

2119879

1015840+ ℎ61198962119866 = ℎ

5nabla

2119862

1015840 (20)

Applying the operator Laplacian operator nabla2 to (18) weobtain

(nabla

2minus 120573)119864

1015840= nabla

2119862

1015840+ nabla

2119879

1015840 (21)

From (19)ndash(21) we obtain

(nabla

6+ 1198871nabla

4+ 1198872nabla

2+ 1198873) (119864

1015840 119879

1015840 119862

1015840) = 0 (22)

where

1198871=

minus1

(ℎ5minus 1)

times [1198962ℎ6+ 1198961(ℎ5+ 1205762ℎ4) + 120573ℎ

5+ 1198961(1205761ℎ5+ 1205762)]

1198872=

1

(ℎ5minus 1)

times [11989611198962ℎ6+ 120573 (119896

2ℎ6+ 1198961(ℎ5+ 1205762ℎ4)) + 119896

111989621205761ℎ6]

1198873=

minus12057311989611198962ℎ6

(ℎ5minus 1)

1198961= 119894120596 (1 + 119894120596120591

0)

1198962= 119894120596 (1 + 119894120596120591) 120573 = minus (120596

2+ Ω

2+ 2119894120596Ω)

(23)

Abstract and Applied Analysis 5

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

(a)

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

The t

empe

ratu

re

(b)

Figure 3 Variation of temperature 120579 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus2010 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus20 10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 4 Variation of displacement 119906 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Equation (22) can be factorized as

(nabla

2+ 119876

2

1) (nabla

2+ 119876

2

2) (nabla

2+ 119876

2

3) (119864

1015840 119879

1015840 119862

1015840) = 0 (24)

where 119876

2

1 11987622 and 119876

2

3are the roots of the characteristic

equation

119876

6+ 1198871119876

4+ 1198872119876

2+ 1198873= 0 (25)

The solution of (24) which is bounded at infinity is given by

119879

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861119895 (120596)11987012

(119876119895119903)

119864

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903)

119862

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903)

(26)

where1198611198951198611015840119895 and11986110158401015840

119895are parameters depending only on120596 and

11987012

is the modified spherical Bessel function of the second

6 Abstract and Applied AnalysisRa

dial

stre

ss10

5

0

minus5

minus10

minus20

minus15

minus25

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

100000

80000

60000

40000

20000

0

Radi

al st

ress

minus20000

minus40000

minus60000

minus80000

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 5 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Tang

entia

l stre

ss

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

Tang

entia

l stre

ss

20

40

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 6 Variation of tangential stress 120590120601120601

with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

kind of order 12 Compatibility between (26) along with (19)and (20) will give rise to

119861

1015840

119895(120596) =

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895(120596)

119861

10158401015840

119895(120596) =

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895 (120596)

(27)

Substituting (26) into (17a) and (17b) we obtain

119879 (119903 119905) =

1

radic119903

3

sum

119895=1

119861119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (28)

119890 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (29)

119862 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (30)

Abstract and Applied Analysis 7

10 12 14 16 18 20 22 24 26 28 30r

The d

ispla

cem

ent

600

500

400

300

200

100

0

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 7 Variation of displacement 119906 with radius 119903 (thermoelasticnonhomogeneity medium)

100

12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

minus1000minus500

minus1500minus2000minus2500minus3000minus3500minus4000minus4500

Figure 8 Variation of temperature 120579 with radius 119903 (thermoelasticnonhomogeneity medium)

Integrating both sides of (29) from 119903 to infinity and assumingthat 119906(119903 119905) vanishes at infinity we obtain

119906 (119903 119905) =

1

radic119903

3

sum

119895=1

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 119861119895 (120596)11987032

(119876119895119903) 119890

119894120596119905

(31)

Radi

al st

ress

0

minus200

minus400

minus600

minus800

minus1000

minus1200

minus1400

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 9 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic

nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

0

200

400

600

800

1000

1200

1400

1600Ta

ngen

tial s

tress

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 10 Variation of tangential stress 120590120601120601

with radius 119903 (thermoe-lastic nonhomogeneity medium)

From (13a) (13b) and (13c)-(14) we get

120590119903119903=

1

radic119903

3

sum

119895=1

119861119895(120596)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times

[

[

(ℎ1+ℎ2) minus

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minusℎ61198962)

(119876

2

119895minus1198961) (ℎ5119876

2

119895minusℎ61198962)minus12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119903) minus

ℎ1

119903

11987032

(119876119895119903)

119890

119894120596119905

8 Abstract and Applied Analysis

120590120601120601

= 120590120579120579

=

1

radic119903

3

sum

119895=1

119861119895(120596)

times

ℎ1

119903

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 11987032

(119876119895119903)

+[

[

ℎ2

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus 1 minus

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

]

]

times 11987012

(119876119895119903)

119890

119894120596119905

119875 =

1

radic119903

3

sum

119895=1

119861119895 (120596)

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ ℎ3

12057611198961ℎ4119876

2

119895+119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119903) 119890

119894120596119905

(32)

Using the boundary conditions we get

1198611(120596) = (minus119873

2[radic11988612057901198823minus 11987012

(1198861198763) 1199010]

+1198733[radic11988612057901198822minus 11987012

(1198861198762) 1199010]) times (119872)

minus1

1198612(120596) = (radic119886119873

1[12057901198823minus 11987012

(1198861198763) 1199010]

+radic1198861198733[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

1198613 (120596) = (radic119886119873

1[11987012

(1198861198762) 1199010minus 12057901198822]

minusradic1198861198732[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

119873119895=

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times[

[

(ℎ1+ ℎ2)

minus

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119886) minus [

ℎ1

119886

+ 120583119890119867

2

120601(

3

119886

2minus 119876

2

119894)]

times 11987032

(119876119895119886)

119882119895=

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ℎ3

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119886)

119872 = 1198731[119882311987012

(1198861198762) minus 119882

211987012

(1198861198763)]

minus 1198732[119882311987012

(1198861198761) minus 119882

111987012

(1198861198761)]

+1198733[119882211987012

(1198861198761) minus 119882

111987012

(1198861198762)]

119895 = 1 2 3

(33)

6 Particular Case

If we neglect the initial stress and diffusion effects byeliminating (3) and (8) and putting 119875

1= 1205732= 119862 = 0 in (4)

and (6) we get (119879 119890) 119906(119903 119905) 120590119903119903 120590120601120601 and 120590

120579120579

(119879 119890) = 119860119890

minus120582119903+119894120596119905+ 119861119890

minus120582119903+119894120596119905 (34)

where

1205781= minus (ℓ

1+ 120573) 120578

2= ℓ1(120573 minus 120576

1)

(120582

2

1 120582

2

2) =

1

2

[1205781plusmn radic120578

2

1minus 41205782]

119906 (119903 119905) = minus [

119903

2120582

2

1+ 2119903120582

1+ 2

119903

2120582

3

1

119860 (120596) 119890

minus1205821119903

+

119903

2120582

2

2+21199031205822+2

119903

2120582

3

2

119861 (120596) 119890

minus1205822119903+119862 (120596)] 119890

119894120596119905

120590119903119903=[1205721(

4 + 41199031205821+ 2119903

2120582

2

1+ 119903

3120582

3

1

119903

3120582

3

1

+(1205722minus 1))119860 (120596) 119890

minus1205821119903

+ 1205721(

4 + 41199031205822+ 2119903

2120582

2

2+ 119903

3120582

3

2

119903

3120582

3

2

+ (1205722minus 1))

times 119861 (120596) 119890

minus1205822119903] 119890

119894120596119905

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

(a)

60

50

40

30

20

10

0

minus10

minus20

minus30

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

The t

empe

ratu

re

(b)

Figure 3 Variation of temperature 120579 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus2010 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

The d

ispla

cem

ent

4020

0

minus40minus60minus80minus100minus120minus140minus160minus180minus200minus220minus240minus260

minus20 10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 4 Variation of displacement 119906 with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Equation (22) can be factorized as

(nabla

2+ 119876

2

1) (nabla

2+ 119876

2

2) (nabla

2+ 119876

2

3) (119864

1015840 119879

1015840 119862

1015840) = 0 (24)

where 119876

2

1 11987622 and 119876

2

3are the roots of the characteristic

equation

119876

6+ 1198871119876

4+ 1198872119876

2+ 1198873= 0 (25)

The solution of (24) which is bounded at infinity is given by

119879

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861119895 (120596)11987012

(119876119895119903)

119864

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903)

119862

1015840(119903 120596) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903)

(26)

where1198611198951198611015840119895 and11986110158401015840

119895are parameters depending only on120596 and

11987012

is the modified spherical Bessel function of the second

6 Abstract and Applied AnalysisRa

dial

stre

ss10

5

0

minus5

minus10

minus20

minus15

minus25

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

100000

80000

60000

40000

20000

0

Radi

al st

ress

minus20000

minus40000

minus60000

minus80000

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 5 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Tang

entia

l stre

ss

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

Tang

entia

l stre

ss

20

40

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 6 Variation of tangential stress 120590120601120601

with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

kind of order 12 Compatibility between (26) along with (19)and (20) will give rise to

119861

1015840

119895(120596) =

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895(120596)

119861

10158401015840

119895(120596) =

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895 (120596)

(27)

Substituting (26) into (17a) and (17b) we obtain

119879 (119903 119905) =

1

radic119903

3

sum

119895=1

119861119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (28)

119890 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (29)

119862 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (30)

Abstract and Applied Analysis 7

10 12 14 16 18 20 22 24 26 28 30r

The d

ispla

cem

ent

600

500

400

300

200

100

0

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 7 Variation of displacement 119906 with radius 119903 (thermoelasticnonhomogeneity medium)

100

12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

minus1000minus500

minus1500minus2000minus2500minus3000minus3500minus4000minus4500

Figure 8 Variation of temperature 120579 with radius 119903 (thermoelasticnonhomogeneity medium)

Integrating both sides of (29) from 119903 to infinity and assumingthat 119906(119903 119905) vanishes at infinity we obtain

119906 (119903 119905) =

1

radic119903

3

sum

119895=1

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 119861119895 (120596)11987032

(119876119895119903) 119890

119894120596119905

(31)

Radi

al st

ress

0

minus200

minus400

minus600

minus800

minus1000

minus1200

minus1400

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 9 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic

nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

0

200

400

600

800

1000

1200

1400

1600Ta

ngen

tial s

tress

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 10 Variation of tangential stress 120590120601120601

with radius 119903 (thermoe-lastic nonhomogeneity medium)

From (13a) (13b) and (13c)-(14) we get

120590119903119903=

1

radic119903

3

sum

119895=1

119861119895(120596)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times

[

[

(ℎ1+ℎ2) minus

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minusℎ61198962)

(119876

2

119895minus1198961) (ℎ5119876

2

119895minusℎ61198962)minus12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119903) minus

ℎ1

119903

11987032

(119876119895119903)

119890

119894120596119905

8 Abstract and Applied Analysis

120590120601120601

= 120590120579120579

=

1

radic119903

3

sum

119895=1

119861119895(120596)

times

ℎ1

119903

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 11987032

(119876119895119903)

+[

[

ℎ2

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus 1 minus

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

]

]

times 11987012

(119876119895119903)

119890

119894120596119905

119875 =

1

radic119903

3

sum

119895=1

119861119895 (120596)

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ ℎ3

12057611198961ℎ4119876

2

119895+119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119903) 119890

119894120596119905

(32)

Using the boundary conditions we get

1198611(120596) = (minus119873

2[radic11988612057901198823minus 11987012

(1198861198763) 1199010]

+1198733[radic11988612057901198822minus 11987012

(1198861198762) 1199010]) times (119872)

minus1

1198612(120596) = (radic119886119873

1[12057901198823minus 11987012

(1198861198763) 1199010]

+radic1198861198733[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

1198613 (120596) = (radic119886119873

1[11987012

(1198861198762) 1199010minus 12057901198822]

minusradic1198861198732[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

119873119895=

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times[

[

(ℎ1+ ℎ2)

minus

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119886) minus [

ℎ1

119886

+ 120583119890119867

2

120601(

3

119886

2minus 119876

2

119894)]

times 11987032

(119876119895119886)

119882119895=

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ℎ3

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119886)

119872 = 1198731[119882311987012

(1198861198762) minus 119882

211987012

(1198861198763)]

minus 1198732[119882311987012

(1198861198761) minus 119882

111987012

(1198861198761)]

+1198733[119882211987012

(1198861198761) minus 119882

111987012

(1198861198762)]

119895 = 1 2 3

(33)

6 Particular Case

If we neglect the initial stress and diffusion effects byeliminating (3) and (8) and putting 119875

1= 1205732= 119862 = 0 in (4)

and (6) we get (119879 119890) 119906(119903 119905) 120590119903119903 120590120601120601 and 120590

120579120579

(119879 119890) = 119860119890

minus120582119903+119894120596119905+ 119861119890

minus120582119903+119894120596119905 (34)

where

1205781= minus (ℓ

1+ 120573) 120578

2= ℓ1(120573 minus 120576

1)

(120582

2

1 120582

2

2) =

1

2

[1205781plusmn radic120578

2

1minus 41205782]

119906 (119903 119905) = minus [

119903

2120582

2

1+ 2119903120582

1+ 2

119903

2120582

3

1

119860 (120596) 119890

minus1205821119903

+

119903

2120582

2

2+21199031205822+2

119903

2120582

3

2

119861 (120596) 119890

minus1205822119903+119862 (120596)] 119890

119894120596119905

120590119903119903=[1205721(

4 + 41199031205821+ 2119903

2120582

2

1+ 119903

3120582

3

1

119903

3120582

3

1

+(1205722minus 1))119860 (120596) 119890

minus1205821119903

+ 1205721(

4 + 41199031205822+ 2119903

2120582

2

2+ 119903

3120582

3

2

119903

3120582

3

2

+ (1205722minus 1))

times 119861 (120596) 119890

minus1205822119903] 119890

119894120596119905

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

6 Abstract and Applied AnalysisRa

dial

stre

ss10

5

0

minus5

minus10

minus20

minus15

minus25

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

100000

80000

60000

40000

20000

0

Radi

al st

ress

minus20000

minus40000

minus60000

minus80000

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 5 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

Tang

entia

l stre

ss

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30

r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

(a)

Tang

entia

l stre

ss

20

40

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

10 12 14 16 18 20 22 24 26 28 30r

t = 03 m = 08 1205910 = 04

m = 06

m = 12

m = 18

(b)

Figure 6 Variation of tangential stress 120590120601120601

with radius 119903 (thermoelastic diffusion nonhomogeneity medium)

kind of order 12 Compatibility between (26) along with (19)and (20) will give rise to

119861

1015840

119895(120596) =

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895(120596)

119861

10158401015840

119895(120596) =

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

119861119895 (120596)

(27)

Substituting (26) into (17a) and (17b) we obtain

119879 (119903 119905) =

1

radic119903

3

sum

119895=1

119861119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (28)

119890 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

1015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (29)

119862 (119903 119905) =

1

radic119903

3

sum

119895=1

119861

10158401015840

119895(120596)11987012

(119876119895119903) 119890

119894120596119905 (30)

Abstract and Applied Analysis 7

10 12 14 16 18 20 22 24 26 28 30r

The d

ispla

cem

ent

600

500

400

300

200

100

0

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 7 Variation of displacement 119906 with radius 119903 (thermoelasticnonhomogeneity medium)

100

12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

minus1000minus500

minus1500minus2000minus2500minus3000minus3500minus4000minus4500

Figure 8 Variation of temperature 120579 with radius 119903 (thermoelasticnonhomogeneity medium)

Integrating both sides of (29) from 119903 to infinity and assumingthat 119906(119903 119905) vanishes at infinity we obtain

119906 (119903 119905) =

1

radic119903

3

sum

119895=1

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 119861119895 (120596)11987032

(119876119895119903) 119890

119894120596119905

(31)

Radi

al st

ress

0

minus200

minus400

minus600

minus800

minus1000

minus1200

minus1400

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 9 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic

nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

0

200

400

600

800

1000

1200

1400

1600Ta

ngen

tial s

tress

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 10 Variation of tangential stress 120590120601120601

with radius 119903 (thermoe-lastic nonhomogeneity medium)

From (13a) (13b) and (13c)-(14) we get

120590119903119903=

1

radic119903

3

sum

119895=1

119861119895(120596)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times

[

[

(ℎ1+ℎ2) minus

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minusℎ61198962)

(119876

2

119895minus1198961) (ℎ5119876

2

119895minusℎ61198962)minus12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119903) minus

ℎ1

119903

11987032

(119876119895119903)

119890

119894120596119905

8 Abstract and Applied Analysis

120590120601120601

= 120590120579120579

=

1

radic119903

3

sum

119895=1

119861119895(120596)

times

ℎ1

119903

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 11987032

(119876119895119903)

+[

[

ℎ2

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus 1 minus

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

]

]

times 11987012

(119876119895119903)

119890

119894120596119905

119875 =

1

radic119903

3

sum

119895=1

119861119895 (120596)

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ ℎ3

12057611198961ℎ4119876

2

119895+119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119903) 119890

119894120596119905

(32)

Using the boundary conditions we get

1198611(120596) = (minus119873

2[radic11988612057901198823minus 11987012

(1198861198763) 1199010]

+1198733[radic11988612057901198822minus 11987012

(1198861198762) 1199010]) times (119872)

minus1

1198612(120596) = (radic119886119873

1[12057901198823minus 11987012

(1198861198763) 1199010]

+radic1198861198733[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

1198613 (120596) = (radic119886119873

1[11987012

(1198861198762) 1199010minus 12057901198822]

minusradic1198861198732[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

119873119895=

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times[

[

(ℎ1+ ℎ2)

minus

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119886) minus [

ℎ1

119886

+ 120583119890119867

2

120601(

3

119886

2minus 119876

2

119894)]

times 11987032

(119876119895119886)

119882119895=

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ℎ3

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119886)

119872 = 1198731[119882311987012

(1198861198762) minus 119882

211987012

(1198861198763)]

minus 1198732[119882311987012

(1198861198761) minus 119882

111987012

(1198861198761)]

+1198733[119882211987012

(1198861198761) minus 119882

111987012

(1198861198762)]

119895 = 1 2 3

(33)

6 Particular Case

If we neglect the initial stress and diffusion effects byeliminating (3) and (8) and putting 119875

1= 1205732= 119862 = 0 in (4)

and (6) we get (119879 119890) 119906(119903 119905) 120590119903119903 120590120601120601 and 120590

120579120579

(119879 119890) = 119860119890

minus120582119903+119894120596119905+ 119861119890

minus120582119903+119894120596119905 (34)

where

1205781= minus (ℓ

1+ 120573) 120578

2= ℓ1(120573 minus 120576

1)

(120582

2

1 120582

2

2) =

1

2

[1205781plusmn radic120578

2

1minus 41205782]

119906 (119903 119905) = minus [

119903

2120582

2

1+ 2119903120582

1+ 2

119903

2120582

3

1

119860 (120596) 119890

minus1205821119903

+

119903

2120582

2

2+21199031205822+2

119903

2120582

3

2

119861 (120596) 119890

minus1205822119903+119862 (120596)] 119890

119894120596119905

120590119903119903=[1205721(

4 + 41199031205821+ 2119903

2120582

2

1+ 119903

3120582

3

1

119903

3120582

3

1

+(1205722minus 1))119860 (120596) 119890

minus1205821119903

+ 1205721(

4 + 41199031205822+ 2119903

2120582

2

2+ 119903

3120582

3

2

119903

3120582

3

2

+ (1205722minus 1))

times 119861 (120596) 119890

minus1205822119903] 119890

119894120596119905

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

10 12 14 16 18 20 22 24 26 28 30r

The d

ispla

cem

ent

600

500

400

300

200

100

0

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 7 Variation of displacement 119906 with radius 119903 (thermoelasticnonhomogeneity medium)

100

12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

The t

empe

ratu

re

minus1000minus500

minus1500minus2000minus2500minus3000minus3500minus4000minus4500

Figure 8 Variation of temperature 120579 with radius 119903 (thermoelasticnonhomogeneity medium)

Integrating both sides of (29) from 119903 to infinity and assumingthat 119906(119903 119905) vanishes at infinity we obtain

119906 (119903 119905) =

1

radic119903

3

sum

119895=1

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 119861119895 (120596)11987032

(119876119895119903) 119890

119894120596119905

(31)

Radi

al st

ress

0

minus200

minus400

minus600

minus800

minus1000

minus1200

minus1400

10 12 14 16 18 20 22 24 26 28 30r

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 9 Variation of radial stress 120590119903119903with radius 119903 (thermoelastic

nonhomogeneity medium)

10 12 14 16 18 20 22 24 26 28 30r

0

200

400

600

800

1000

1200

1400

1600Ta

ngen

tial s

tress

t = 03 Ω = 12 1205910 = 04m = 04

m = 08

m = 15

Figure 10 Variation of tangential stress 120590120601120601

with radius 119903 (thermoe-lastic nonhomogeneity medium)

From (13a) (13b) and (13c)-(14) we get

120590119903119903=

1

radic119903

3

sum

119895=1

119861119895(120596)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times

[

[

(ℎ1+ℎ2) minus

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minusℎ61198962)

(119876

2

119895minus1198961) (ℎ5119876

2

119895minusℎ61198962)minus12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119903) minus

ℎ1

119903

11987032

(119876119895119903)

119890

119894120596119905

8 Abstract and Applied Analysis

120590120601120601

= 120590120579120579

=

1

radic119903

3

sum

119895=1

119861119895(120596)

times

ℎ1

119903

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 11987032

(119876119895119903)

+[

[

ℎ2

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus 1 minus

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

]

]

times 11987012

(119876119895119903)

119890

119894120596119905

119875 =

1

radic119903

3

sum

119895=1

119861119895 (120596)

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ ℎ3

12057611198961ℎ4119876

2

119895+119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119903) 119890

119894120596119905

(32)

Using the boundary conditions we get

1198611(120596) = (minus119873

2[radic11988612057901198823minus 11987012

(1198861198763) 1199010]

+1198733[radic11988612057901198822minus 11987012

(1198861198762) 1199010]) times (119872)

minus1

1198612(120596) = (radic119886119873

1[12057901198823minus 11987012

(1198861198763) 1199010]

+radic1198861198733[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

1198613 (120596) = (radic119886119873

1[11987012

(1198861198762) 1199010minus 12057901198822]

minusradic1198861198732[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

119873119895=

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times[

[

(ℎ1+ ℎ2)

minus

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119886) minus [

ℎ1

119886

+ 120583119890119867

2

120601(

3

119886

2minus 119876

2

119894)]

times 11987032

(119876119895119886)

119882119895=

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ℎ3

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119886)

119872 = 1198731[119882311987012

(1198861198762) minus 119882

211987012

(1198861198763)]

minus 1198732[119882311987012

(1198861198761) minus 119882

111987012

(1198861198761)]

+1198733[119882211987012

(1198861198761) minus 119882

111987012

(1198861198762)]

119895 = 1 2 3

(33)

6 Particular Case

If we neglect the initial stress and diffusion effects byeliminating (3) and (8) and putting 119875

1= 1205732= 119862 = 0 in (4)

and (6) we get (119879 119890) 119906(119903 119905) 120590119903119903 120590120601120601 and 120590

120579120579

(119879 119890) = 119860119890

minus120582119903+119894120596119905+ 119861119890

minus120582119903+119894120596119905 (34)

where

1205781= minus (ℓ

1+ 120573) 120578

2= ℓ1(120573 minus 120576

1)

(120582

2

1 120582

2

2) =

1

2

[1205781plusmn radic120578

2

1minus 41205782]

119906 (119903 119905) = minus [

119903

2120582

2

1+ 2119903120582

1+ 2

119903

2120582

3

1

119860 (120596) 119890

minus1205821119903

+

119903

2120582

2

2+21199031205822+2

119903

2120582

3

2

119861 (120596) 119890

minus1205822119903+119862 (120596)] 119890

119894120596119905

120590119903119903=[1205721(

4 + 41199031205821+ 2119903

2120582

2

1+ 119903

3120582

3

1

119903

3120582

3

1

+(1205722minus 1))119860 (120596) 119890

minus1205821119903

+ 1205721(

4 + 41199031205822+ 2119903

2120582

2

2+ 119903

3120582

3

2

119903

3120582

3

2

+ (1205722minus 1))

times 119861 (120596) 119890

minus1205822119903] 119890

119894120596119905

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

120590120601120601

= 120590120579120579

=

1

radic119903

3

sum

119895=1

119861119895(120596)

times

ℎ1

119903

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times 11987032

(119876119895119903)

+[

[

ℎ2

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus 1 minus

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

]

]

times 11987012

(119876119895119903)

119890

119894120596119905

119875 =

1

radic119903

3

sum

119895=1

119861119895 (120596)

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ ℎ3

12057611198961ℎ4119876

2

119895+119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119903) 119890

119894120596119905

(32)

Using the boundary conditions we get

1198611(120596) = (minus119873

2[radic11988612057901198823minus 11987012

(1198861198763) 1199010]

+1198733[radic11988612057901198822minus 11987012

(1198861198762) 1199010]) times (119872)

minus1

1198612(120596) = (radic119886119873

1[12057901198823minus 11987012

(1198861198763) 1199010]

+radic1198861198733[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

1198613 (120596) = (radic119886119873

1[11987012

(1198861198762) 1199010minus 12057901198822]

minusradic1198861198732[11987012

(1198861198761) 1199010minus 12057901198821]) times (119872)

minus1

119873119895=

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

times[

[

(ℎ1+ ℎ2)

minus

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

]

]

times 11987012

(119876119895119886) minus [

ℎ1

119886

+ 120583119890119867

2

120601(

3

119886

2minus 119876

2

119894)]

times 11987032

(119876119895119886)

119882119895=

minus

(119876

2

119895minus 1198961) (ℎ5119876

2

119895minus ℎ61198962) minus 12057621198961ℎ4119876

2

119895

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

+ℎ3

12057611198961ℎ4119876

2

119895+ 119876

2

119895(119876

2

119895minus 1198961)

119876

2

11989512057621198961+ 11989611205761(ℎ5119876

2

119895minus ℎ61198962)

minus ℎ4

times 11987012

(119876119895119886)

119872 = 1198731[119882311987012

(1198861198762) minus 119882

211987012

(1198861198763)]

minus 1198732[119882311987012

(1198861198761) minus 119882

111987012

(1198861198761)]

+1198733[119882211987012

(1198861198761) minus 119882

111987012

(1198861198762)]

119895 = 1 2 3

(33)

6 Particular Case

If we neglect the initial stress and diffusion effects byeliminating (3) and (8) and putting 119875

1= 1205732= 119862 = 0 in (4)

and (6) we get (119879 119890) 119906(119903 119905) 120590119903119903 120590120601120601 and 120590

120579120579

(119879 119890) = 119860119890

minus120582119903+119894120596119905+ 119861119890

minus120582119903+119894120596119905 (34)

where

1205781= minus (ℓ

1+ 120573) 120578

2= ℓ1(120573 minus 120576

1)

(120582

2

1 120582

2

2) =

1

2

[1205781plusmn radic120578

2

1minus 41205782]

119906 (119903 119905) = minus [

119903

2120582

2

1+ 2119903120582

1+ 2

119903

2120582

3

1

119860 (120596) 119890

minus1205821119903

+

119903

2120582

2

2+21199031205822+2

119903

2120582

3

2

119861 (120596) 119890

minus1205822119903+119862 (120596)] 119890

119894120596119905

120590119903119903=[1205721(

4 + 41199031205821+ 2119903

2120582

2

1+ 119903

3120582

3

1

119903

3120582

3

1

+(1205722minus 1))119860 (120596) 119890

minus1205821119903

+ 1205721(

4 + 41199031205822+ 2119903

2120582

2

2+ 119903

3120582

3

2

119903

3120582

3

2

+ (1205722minus 1))

times 119861 (120596) 119890

minus1205822119903] 119890

119894120596119905

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

120590120601120601

= 120590120579120579

= minus (1205721(

119903

2120582

2

1+ 2119903120582

1+ 2

119903

3120582

3

1

) minus (1205722minus 1))

times 119860 (120596) 119890

minus1205821119903

+ [1205721(

119903

2120582

2

2+ 2119903120582

2+ 2

119903

3120582

3

2

) minus (1205722minus 1)]

times 119861 (120596) 119890

minus1205822119903 119890

119894120596119905

(35)Using the boundary conditions we obtain

119860 (120596) =

minusℎ21198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886 119861 (120596) =

ℎ11198790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

119862 (120596) =

(ℎ2ℎ3minus ℎ1ℎ4) 1205790

ℎ1119890

minus1205822119886minus ℎ2119890

minus1205821119886

ℎ1= 1205721(

4 + 41198861205821+ 2119886

2120582

2

1+ 119886

3120582

3

1

119886

3120582

3

1

+ (1205722minus 1 minus 120583

119890119867

2

1206011205821)) 119890

minus1205821119886

ℎ2= 1205721(

4 + 41198861205822+ 2119886

2120582

2

2+ 119886

3120582

3

2

119886

3120582

3

2

+ (1205722minus 1 minus 120583

119890119867

2

1206011205822)) 119890

minus1205822119886

ℎ3= (

119886

2120582

2

1+ 2119886120582

1+ 2

119886

2120582

3

1

) 119890

minus1205821119886

ℎ4= (

119886

2120582

2

2+ 2119886120582

2+ 2

119886

2120582

3

2

) 119890

minus1205822119886

(36)

7 Numerical Results and Discussion

For the purposes of numerical evaluations The coppermaterial was chosen The constants of the problem given byAouadi [24] Sokolnikoff [34] andThomas [35] are

120583 = 386 times 10

10 kgms3 120582 = 776 times 10

10 kgms3

120588 = 8954 kgm3 119888V = 3831 Jkg sdot K

120572119905= 178 times 10

minus5 Kminus1 120572119888= 198 times 10

minus4m3kg

119896 = 386WmK 119863 = 085 times 10

8 kg sdot sm3

1198790= 293K 119888 = 12 times 10

4m2s2K

119887 = 09 times 10

6m5s2kg 1205780= 888673 sm2

(37)

Using the above values we get 120583119890= 1 120579

0= 1 119875

0= 1

119886 = 2 120596 = 95 119867 = 07 times 10

minus5 1205910= 01 and 120591 = 02

The values of radial displacement 119906 temperature distribution120579 concentration 119862 stresses 120590

119903119903 120590120601120601 and chemical potential

distribution 119875 for thermoelastic diffusion and thermoelastic-ity are studied for force thermal source and chemical potentialsource The output is plotted in Figures 1ndash10 Figure 1 showsthat the values of chemical potential distribution 119875 haveoscillatory behavior with diffusion in the whole range ofradius 119903 The effects of nonhomogeneity 119898 rotation Ω time119905 and relaxation time 120591

0on chemical potential distribution

is shifting from the positive into the negative gradually withthe radius 119903 Figure 2 shows that the value of concentrationdistribution 119862 has oscillatory behavior for diffusion in thewhole range of radius 119903 under the effects of nonhomogeneityrotation and relaxation time while it is decreasing withan increase of nonhomogeneity 119898 In these figures it isclear that the distribution has a nonzero value only in thebounded region of space for 119905 = 015where the infinite speedof propagation is inherent The effects of nonhomogeneityrotation Ω time 119905 and relaxation time 120591

0on concentration

distribution is shifting from the positive into the negativegraduallyThis indicates that the equations are satisfied by theconcentration 119862 which predict a finite speed of propagationof matter from first medium to another one Figure 3 showsthat the value of temperature distribution 120579 has an oscillatorybehavior for thermoelastic diffusion in the whole range ofthe radius 119903 while the solution is notably different inside thesphere This is due to the fact that the thermal waves in thecoupled theory travel with an infinite speed of propagationas opposed to finite speed in the generalized case The effectsof nonhomogeneity rotationΩ time 119905 and relaxation time 120591

0

on temperature distribution shift from the positive into thenegative gradually This indicates that the heat propagates asa wave with finite velocity Figure 4 shows that the value ofradial displacement 119906 has oscillatory behavior with diffusionin the whole range of radius 119903 These figures indicate that themedium along 119903 undergoes expansion deformation due to thethermal shock while the other one shows the compressivedeformation The effect of nonhomogeneity rotation Ω andrelaxation time 120591

0on radial displacement becomes large

Increasing the nonhomogeneity the radial displacement isshifted upward from negative values to positive values Ata given instant the radial displacement is finite which isdue to the effect of nonhomogeneity rotation time andrelaxation time Figures 5 and 6 show the variations of theradial stress 120590

119903119903and tangential stress 120590

120601120601with respect to the

radius 119903 respectivelyThe values of radial stress and tangentialstress are increased and decreased due to the diffusion in anonuniform behavior for all values of the radius 119903 For thevalues of 120590

119903119903and 120590120601120601 depicting the effect of nonhomogeneity

diffusion rotation and relaxation time it is shown that theradial stress is compressive in its nature

Figure 7 shows the values of radial displacement 119906 inthermoelastic medium without diffusion This figure indi-cates clearly that the radial displacement at the cavity surfacetends to zero which agrees with the boundary conditionsprescribed This coincides with the mechanical boundarycondition of the cavity in case of fixed surface Figure 8 showsthe values of temperature distribution 120579 without diffusionin the whole range of radius 119903 It was found that the values

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

of 120579 under effect of nonhomogeneity and rotation Ω areincrease with an increase of nonhomogeneity and rotationΩ but are decreasing with the increase of the values of 119898Figures 9 and 10 show the values of radial stress 120590

119903119903and the

tangential stress 120590120601120601

without diffusion in the whole rangeof radius 119903 respectively It was found that the values of 120590

119903119903

under the effects of nonhomogeneity 119898 and rotation Ω areincreasing with an increase of the values of nonhomogeneity119898 and rotation Ω while the values of 120590

119903119903are decreasing

with an increase of nonhomogeneity 119898 while the tangentialstress 120590

120601120601is decreasing with the increase of the values of

nonhomogeneity 119898 and rotation Ω but the values of 120590120601120601

are increasing with an increase of 119898 Due to the compli-cated nature of the governing equations of the generalizedmagneto-thermoelastic diffusion theory the done works inthis field are unfortunately limited The method used in thisstudy provides quite a success in dealing with such problemsThis method gives exact solutions in the elastic mediumwithout any restrictions on the actual physical quantities thatappear in the governing equations of the considered problem

8 Conclusions

The results presented in this paper will be very helpfulfor researchers concerned with material science designersof new materials and low-temperature physicists as wellas for those working on the development of a theory ofhyperbolic propagation of hyperbolic thermodiffusion Studyof the phenomenon of nonhomogeneity rotation magneticfield and diffusion is also used to improve the conditions ofoil extractions It was found that for values of rotation andnonhomogeneity the coupled theory and the generalizationgive close resultsThe case is quite different whenwe considersmall value of rotation and nonhomogeneity ComparingFigures 1ndash6 in case of thermoelastic diffusion medium withthe Figures 7ndash10 in case of thermoelastic medium it wasfound that119906120590

119903119903120590120601120601119862 and119875have the same behavior in both

media But with the passage of nonhomogeneity and rotationthe numerical values of 119906 120590

119903119903 120590120601120601 119862 and 119875 in thermoelastic

diffusionmedium are large in comparison with those in ther-moelastic medium due to the influences of nonhomogeneitymagnetic field rotation and mass diffusion

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledge with thanks DSR technical and financialsupport

References

[1] D Motreanu and M Sofonea ldquoEvolutionary variational ine-qualities arising in quasistatic frictional contact problems forelastic materialsrdquo Abstract and Applied Analysis vol 4 no 4pp 255ndash279 1999

[2] T M Atanackovic S Konjik L Oparnica and D ZoricaldquoThermodynamical restrictions and wave propagation for a

class of fractional order viscoelastic rodsrdquo Abstract and AppliedAnalysis vol 2011 Article ID 975694 32 pages 2011

[3] R Fares G P Panasenko and R Stavre ldquoA viscous fluid flowthrough a thin channel withmixed rigid-elastic boundary vari-ational and asymptotic analysisrdquo Abstract and Applied Analysisvol 2012 Article ID 152743 47 pages 2012

[4] M Marin R P Agarwal and S R Mahmoud ldquoModelinga microstretch thermoelastic body with two temperaturesrdquoAbstract and Applied Analysis vol 2013 Article ID 583464 7pages 2013

[5] M N M Allam A M Zenkour and E R Elazab ldquoThe rotat-ing inhomogeneous elastic cylinders of variable-thickness anddensityrdquoAppliedMathematics amp Information Sciences vol 2 no3 pp 237ndash257 2008

[6] A M Abd-Alla S R Mahmoud and N A Al-Shehri ldquoEffectof the rotation on a non-homogeneous infinite cylinder of or-thotropicmaterialrdquoAppliedMathematics and Computation vol217 no 22 pp 8914ndash8922 2011

[7] A M Abd-Alla S R Mahmoud and S M Abo-Dahab ldquoOnproblem of transient coupled thermoelasticity of an annularfinrdquoMeccanica vol 47 no 5 pp 1295ndash1306 2012

[8] A M Abd-Alla S R Mahmoud and A M El-Naggar ldquoAna-lytical solution of electro-mechanical wave propagation in longbonesrdquo Applied Mathematics and Computation vol 119 no 1pp 77ndash98 2001

[9] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermo-elasticityrdquo Journal of theMechanics and Physics of Solidsvol 7 pp 71ndash75 1967

[10] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol33 no 1 pp 1ndash8 1980

[11] A M Abd-Alla and S R Mahmoud ldquoMagneto-thermoelasticproblem in rotating non-homogeneous orthotropic hollow cyl-inder under the hyperbolic heat conductionmodelrdquoMeccanicavol 45 no 4 pp 451ndash462 2010

[12] S R Mahmoud ldquoWave propagation in cylindrical poroelasticdry bonesrdquo Applied Mathematics amp Information Sciences vol 4no 2 pp 209ndash226 2010

[13] R Kumar and S Devi ldquoDeformation in porous thermoelas-tic material with temperature dependent propertiesrdquo AppliedMathematics amp Information Sciences vol 5 no 1 pp 132ndash1472011

[14] M I A Othman ldquoLord-Shulman theory under the dependenceof themodulus of elasticity on the reference temperature in two-dimensional generalized thermoelasticityrdquo Journal of ThermalStresses vol 25 no 11 pp 1027ndash1045 2009

[15] R Kumar and R R Gupta ldquoDeformation due to inclined loadin an orthotropic micropolar thermoelastic medium with tworelaxation timesrdquo Applied Mathematics amp Information Sciencesvol 4 no 3 pp 413ndash428 2010

[16] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972

[17] AMAbd-Alla G A Yahya and S RMahmoud ldquoRadial vibra-tions in a non-homogeneous orthotropic elastic hollow spheresubjected to rotationrdquo Journal of Computational andTheoreticalNanoscience vol 10 no 2 pp 455ndash463 2013

[18] AMAbd-Alla SMAbo-Dahab S RMahmoud andTAAl-Thamalia ldquoInfluence of the rotation and gravity field on stonelywaves in a non-homogeneous orthotropic elastic mediumrdquoJournal of Computational and Theoretical Nanoscience vol 10no 2 pp 297ndash305 2013

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

[19] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004

[20] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005

[21] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006

[22] R Kumar and T Kansal ldquoPropagation of Lamb waves in trans-versely isotropic thermoelastic diffusive platerdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5890ndash5913 2008

[23] P Ram N Sharma and R Kumar ldquoThermomechanical re-sponse of generalized thermoelastic diffusion with one relax-ation time due to time harmonic sourcesrdquo International Journalof Thermal Sciences vol 47 no 3 pp 315ndash323 2008

[24] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007

[25] A M Abd-Alla and S R Mahmoud ldquoAnalytical solution ofwave propagation in a non-homogeneous orthotropic rotatingelastic mediardquo Journal of Mechanical Science and Technologyvol 26 no 3 pp 917ndash926 2012

[26] M I A Othman S Y Atwa and R M Farouk ldquoThe effect ofdiffusion on two-dimensional problem of generalized thermoe-lasticity with Green-Naghdi theoryrdquo International Communica-tions inHeat andMass Transfer vol 36 no 8 pp 857ndash864 2009

[27] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009

[28] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011

[29] AMAbd-Alla and SMAbo-Dahab ldquoTime-harmonic sourcesin a generalized magneto-thermo-viscoelastic continuum withand without energy dissipationrdquo Applied Mathematical Mod-elling vol 33 no 5 pp 2388ndash2402 2009

[30] S R Mahmoud ldquoInfluence of rotation and generalized magne-to-thermoelastic on rayleighwaves in a granularmediumundereffect of initial stress and gravity fieldrdquoMeccanica vol 47 no 7pp 1561ndash1579 2012

[31] A M Abd-Alla S M Abo-Dahab H A Hammad and SR Mahmoud ldquoOn generalizedmagneto-thermoelastic rayleighwaves in a granular medium under the influence of a gravityfield and initial stressrdquo Journal of Vibration and Control vol 17no 1 pp 115ndash128 2011

[32] A M Abd-Alla and S R Mahmoud ldquoOn problem of radialvibrations in non-homogeneity isotropic cylinder under influ-ence of initial stress andmagnetic fieldrdquo Journal of Vibration andControl vol 19 no 9 pp 1283ndash1293 2013

[33] J D Kraus Electromagnetics Mc Graw-Hill New York NYUSA 1984

[34] I S SokolnikoffMathematical Theory of Elasticity Dover NewYork NY USA 1946

[35] L Thomas Fundamentals of Heat Transfer Prentice-Hall Eng-lewood Cliffs NJ USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of