Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 489863 11 pageshttpdxdoiorg1011552013489863

Research ArticleThe Effect of Magnetic Field and Initial Stress onFractional Order Generalized Thermoelastic Half-Space

Sunita Deswal Sandeep Singh Sheoran and Kapil Kumar Kalkal

Department of Mathematics G J University of Science and Technology Haryana Hisar 125001 India

Correspondence should be addressed to Kapil Kumar Kalkal kapilkalkal gjurediffmailcom

Received 26 December 2012 Revised 5 March 2013 Accepted 12 March 2013

Academic Editor Petr Ekel

Copyright copy 2013 Sunita Deswal et al 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 aim of this paper is to study magneto-thermoelastic interactions in an initially stressed isotropic homogeneous half-space inthe context of fractional order theory of generalized thermoelasticity State space formulation with the Laplace transform techniqueis used to obtain the general solution and the resulting formulation is applied to the ramp type increase in thermal load and zerostress Solutions of the problem in the physical domain are obtained by using a numerical method of the Laplace inverse transformbased on the Fourier expansion technique and the expressions for the displacement temperature and stress inside the half-spaceare obtained Numerical computations are carried out for a particular material for illustrating the results Results obtained for thefield variables are displayed graphically Some comparisons have been shown in figures to present the effect of fractional parameterramp parameter magnetic field and initial stress on the field variables Some particular cases of special interest have been deducedfrom the present investigation

1 Introduction

Biot [1] developed the coupled theory of thermoelasticityto overcome the paradox inherent in the uncoupled theorythat elastic changes have no effect on temperature In thistheory the equations of elasticity and heat conduction arecoupled But it shares the defect of the uncoupled theory inwhich it predicts infinite speed of propagation for heat wavesGeneralized thermoelastic theories have been developedwiththe objective of removing the defect of coupled theory Atpresent mainly two differentmodels of generalized thermoe-lasticity are being extensively used one proposed by Lord andShulman [2] and the other proposed by Green and Lindsay[3] The Lord-Shulman theory only modified the Fourierheat conduction equation and suggested one relaxation timewhereas the Green-Lindsay theory modified both the energyequation and the equation of motion and suggested tworelaxation times Dhaliwal and Sherief [4] extended the Lord-Shulman theory to include anisotropic case Hetnarski andIgnaczak [5] presented a survey article of various represen-tative theories in the range of generalized thermoelasticityYoussef [6] studied two-dimensional generalized thermoelas-ticity problem for half-space subjected to ramp type heating

The theory of magneto-thermoelasticity has receivedattention of many researchers due to its application in widelydiverse fields such as geophysics for understanding theeffect of earthrsquos magnetic field on seismic waves damping ofacoustic waves emission of electromagnetic radiations fromnuclear devices optics and so forth The theory of magneto-thermoelasticity was introduced by Knopoff [7] and Chad-wick [8] and developed by Kaliski and Petykiewicz [9] Thetheoretical outline of the development of magneto-thermo-elasticity was discussed by Paria [10] Paria studied thepropagation of planemagneto-thermoelastic waves in an iso-tropic unbounded medium under the influence of a mag-netic field acting transversely to the direction of propagationNayfeh and Nemat-Nasser [11] studied the propagation ofplane waves in a solid under the influence of electro-mag-netic field Sherief and Ezzat [12] discussed a thermal shockproblem in magneto-thermoelasticity with thermal relaxa-tion Sherief and Helmy [13] illustrated a two-dimensionalhalf-space problem in the context of electromagneto-thermo-elasticity theory subjected to a nonuniform thermal shockEzzat and Youssef [14] constructed a problem on genera-lized magneto-thermoelasticity in a perfectly conductingmedium Baksi et al [15] examined magneto-thermoelastic

2 Journal of Mathematics

problem with thermal relaxation and heat source in three-dimensional infinite rotating elastic medium

The development of initial stress in the medium is dueto many reasons such as the process of quenching resultingfrom difference of temperatures slow process of creepdifferential external forces and gravity variationsThe earth issupposed to be under high initial stressThe researchers haveshownmuch interest to study the effect of these stresses on thepropagation of waves Biot [16] solved the dynamic problemof elastic medium under initial stress Chattopadhyay et al[17] studied the reflection of elastic waves under initial stressat a free surface Montanaro [18] studied the isotropic linearthermoelasticity with hydrostatic initial stress by using Biotrsquoslinearization of the constitutive law for stress Othman andSong [19] investigated the reflection of plane waves from anelastic solid half-space under hydrostatic initial stress withoutenergy dissipation Singh [20] explored the effect of hydro-static initial stress on waves in a thermoelastic half-space

The theory of fractional derivative and integral was estab-lished in the second half of nineteenth centuryThefirst appli-cation of fractional derivative was given by Abel who appliedfractional calculus in the solution of an integral equationthat arises in the formulation of tautochrone problem In therecent years fractional calculus has been applied successfullyin various areas to modify many existing models of physicalprocesses such as heat conduction diffusion viscoelasticitywave propagation and electronics Caputo and Mainardi [2122] and Caputo [23] have established the relation betweenfractional derivative and theory of linear viscoelasticity Thegeneralization of the concept of derivative and integral to anoninteger order has been subjected to several approachesand some various alternative definitions of fractional deriva-tives appeared in [24ndash27] One can refer to Podlubny [28]for a survey of applications of fractional calculus Povstenko[29] has proposed a quasistatic uncoupled theory of ther-moelasticity based on fractional heat conduction equationSherief et al [30] introduced a newmodel of thermoelasticityusing fractional calculus proved a uniqueness theorem andderived a reciprocity relation and a variational principle Inthis model heat conduction equation takes the form as

119902119894+ 1205910

120597120572

120597119905120572119902119894= minus119896119894119895120579119895 (1)

where 119902119894are the components of the heat flux vector 120579 is the

temperature 1205910is the thermal relaxation time parameter 119896

119894119895is

thermal conductivity tensor and 120572 is a fractional parametersuch that 0 lt 120572 le 1 The above heat conduction equationreduces to the Maxwell-Cattaneo law in the limiting casewhen 120572 rarr 1 It should be mentioned here that the Maxwell-Cattaneo law has been employed by Lord and Shulman [2] todevelop first generalized theory of thermoelasticity Youssef[31] constructed another model of thermoelasticity in thecontext of a new consideration of heat conduction with afractional order and proved the uniqueness theorem In thismodel Youssef described different cases of conductivity 0 lt

120572 lt 1 corresponds toweak conductivity120572 = 1 corresponds tonormal conductivity and 1 lt 120572 le 2 corresponds to supercon-ductivity Ezzat [32 33] established a model of fractional heat

conduction equation by using the new Taylor series expan-sion of time-fractional order developed by Jumarie [34] El-Karamany and Ezzat [35] introduced two general modelsof fractional heat conduction law for a nonhomogeneousanisotropic elastic solid Uniqueness and reciprocal theoremsare proved and the convolutional variational principle isestablished and used to prove a uniqueness theorem with norestriction on the elasticity or thermal conductivity tensorsexcept symmetry conditions The two-temperature dynamiccoupled Lord-Shulman and fractional coupled thermoelas-ticity theories result as limit cases For fractional thermoe-lasticity not involving two-temperatures El-Karamany andEzzat [36] established the uniqueness reciprocal theoremsand convolution variational principle The dynamic coupledand the Green-Naghdi thermoelasticity theories result aslimit casesThe reciprocity relation in case of quiescent initialstate is found to be independent of the order of differintegra-tion [35 36] Fractional order theory of a perfect conductingthermoelastic medium not involving two temperatures wasinvestigated by El-Karamany and Ezzat [37] Kothari andMukhopadhyay [38] studied a half-space problem underfractional order theory of thermoelasticity and analyzed theeffect of the fractional order parameter on the field variables

In the present paper we study the effect of magnetic fieldand initial stress under fractional order theory of thermo-elasticity proposed by Sherief et al [30] We employ a statespace approach developed by Bahar and Hetnarski [39] onthe formulation The Laplace transform technique is used toobtain the general solution The inverse Laplace transform iscarried out using a numerical inversion method developedby Honig and Hirdes [40] Finally the effect of fractionalparameter ramp parameter magnetic field and initial stresson field variables is displayed graphically

2 Governing Equations

The governing equations in the context of fractional ordertheory of generalized thermoelasticity with initial stress andmagnetic field for isotropic and homogeneous elastic med-ium are considered as

(i) the equation of motion

120588119894= 120590119895119894119895

+ 119865119894 (2)

where = 1205830119869 times

(ii) heat conduction equation

119896120579119894119894= 120588119888119864(1 + 120591

0

120597120572

120597119905120572)120597120579

120597119905+ 1205731198790(1 + 120591

0

120597120572

120597119905120572)120597119906119894119894

120597119905 (3)

(iii) constitutive relations

120590119894119895= minus119901 (120575

119894119895+ 120596119894119895) + 2120583119890

119894119895+ 120582119890120575

119894119895minus 120573120579120575

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

120596119894119895=1

2(119906119895119894minus 119906119894119895)

(4)

Journal of Mathematics 3

We take the linearized Maxwell equations governing theelectromagnetic field for a perfectly conducting medium as

curl ℎ = 119869 + 1205760

120597

120597119905

curl = minus1205830

120597ℎ

120597119905

= minus1205830(120597ℎ

120597119905times )

div ℎ = 0

(5)

where 119906119894are the components of displacement vector 120579 =

119879 minus 1198790 119879 is the absolute temperature 119879

0is the reference

temperature assumed to obey the inequality |1205791198790| ≪ 1 120591

0

is the thermal relaxation time 120590119894119895are the components of the

stress tensor 119890119894119895are the components of strain tensor 120575

119894119895is the

Kronecker delta function 119890 is the cubical dilation 120588 is thedensity of the medium 119888

119864is the specific heat 119896 is the thermal

conductivity 120573 = (3120582 + 2120583)120572119905 120572119905is the coefficient of linear

thermal expansion 120582 and 120583 are the lame constants 119901 is theinitial stress 119865

119894are the components of Lorentzrsquos body force

vector 1205830is the magnetic permeability 120576

0is the electric

permittivity is the applied magnetic field ℎ is the inducedmagnetic field is the induced electric field and 119869 is thecurrent density vector

3 Problem Formulation

We consider a perfectly conducting isotropic homogeneousand fractional order generalized thermoelastic half-spacewith hydrostatic initial stress subjected to a constantmagneticfield (0119867

0 0) which produces an induced magnetic field

ℎ(0 ℎ2 0) and induced electric field (0 0 119864

3) We assume

one-dimensional motion for which all the field quantities arefunctions of 119909 and 119905

The displacement components take the form119906119909= 119906 (119909 119905) 119906

119910= 119906119911= 0 (6)

The strain component becomes

119890 = 119890119909119909

=120597119906

120597119909 (7)

The components of magnetic field vectors are119867119909= 0 119867

119910= 1198670 119867

119911= 0 (8)

The electric intensity vector is parallel to current densityvector 119869 Hence components of and 119869 are given as

119864119909= 119864119910= 0 119864

119911= 1198643

119869119909= 119869119910= 0 119869

119911= 1198693

(9)

Now the Maxwell equation (5) provides the followingresults

1198643= minus12058301198670

120597119906

120597119905 ℎ

2= minus1198670119890 119869

3= 120576012058301198670

1205972119906

1205972119905

(10)

Using (8) and (10) into the relation = 1205830119869 times we

obtain

119865119909= minus12057601205832

01198672

0 119865

119910= 119865119911= 0 (11)

The governing equations for one-dimensional case be-come

120588 =120597120590

120597119909+ 119865119909 (12)

1198961205972120579

1205971199092= 120588119888119864(1 + 120591

0

120597120572

120597119905120572)120597120579

120597119905+ 1205731198790(1 + 120591

0

120597120572

120597119905120572)120597119890

120597119905 (13)

120590 = 120590119909119909

= minus119901 + (120582 + 2120583) 119890 minus 120573120579 (14)

Now we will use the following nondimensional variables

1199091015840= 11988801205780119909 119906

1015840= 11988801205780119906

1199051015840= 1198882

01205780119905 120591

1015840

0= 1198882

012057801205910

1205791015840=

120579

1198790

1205901015840=

120590

120582 + 2120583 119901

1015840=

119901

1205731198790

(15)

where

1205780=120588119888119864

119896 119888

2

0=120582 + 2120583

120588 (16)

Expressing (12)ndash(14) in terms of the nondimensionalvariables given by (15) and dropping the prime sign forconvenience we have the following forms

1205972120590

1205971199092= 1205721119890

1205972120579

1205971199092= (1 + 120591

lowast

0

120597120572

120597119905120572) ( 120579 + 120573

1119890)

120590 = 119890 minus 1205761120579 minus 1205751

(17)

where

1205721= 1 +

12057601205832

01198672

0

120588 120573

1=

120573

120588119888119864

1205761=

1205731198790

120582 + 2120583

1205751= 1205761119901 120591

lowast

0=

1205910

(119888201205780)1minus120572

(18)

These equations will be supplemented with appropriateboundary conditions relevant to the particular applicationunder consideration

Taking the Laplace transform of (17) by using homoge-neous initial conditions defined and denoted as

119891 (119904) = int

infin

0

119890minus119904119905119891 (119905) 119889119905 119904 gt 0 (19)

4 Journal of Mathematics

we obtain

1205972120590

1205971199092= 12057211199042119890 (20)

1205972120579

1205971199092= (119904 + 120591

lowast

0119904120572+1

) (120579 + 1205731119890) (21)

120590 = 119890 minus 1205761120579 minus

1205751

119904 (22)

Eliminating the value of 119890 from (20) and (21) by using(22) we obtain

1205972120579

1205971199092= 1198711120579 + 1198712120590 + 1198713

1205972120590

1205971199092= 1198721120579 +119872

2120590 +119872

3

(23)

where

1198711= (1 + 120573

11205761) (119904 + 120591

lowast

0119904120572+1

) 1198712= 1205731(119904 + 120591

lowast

0119904120572+1

)

1198713= 12057311205751(1 + 120591

lowast

0119904120572)

1198721= 120572112057611199042 119872

2= 12057211199042 119872

3= 12057211205751119904

(24)

4 State-Space Formulation

Now choosing the temperature of heat conduction 120579 and thestress component 120590 in 119909-direction as state variables one canwrite (23) in matrix form as

1198892

1198891199092119881 (119909 119904) = 119860 (119904) 119881 (119909 119904) + 119862 (119909 119904) (25)

where

119860 (119904) = [1198711

1198712

11987211198722

] 119881 (119909 119904) = [120579 (119909 119904)

120590 (119909 119904)]

119862 (119909 119904) = [1198713

1198723

]

(26)

The formal solution of (25) can be written in the form

119881 (119909 119904) = exp [minusradic119860 (119904)119909] [119881 (0 119904) + 119862 (0 119904) 119860minus1(119904)]

minus 119862 (119909 119904) 119860minus1(119904)

(27)

where

119881 (0 119904) = [120579 (0 119904)

120590 (0 119904)] = [

1205790

1205900

] (28)

In the above solution we have cancelled the part ofexponential having positive power to get bounded solutionfor large 119909

Nowwewill use theCayley-Hamilton theorem to find theform of the matrix exp[minusradic119860(119904)119909] and for this we proceedas follows

The characteristic equation of the matrix119860(119904) is obtainedas

1205822minus (1198711+1198722) 120582 + (119871

11198722minus11987211198712) = 0 (29)

where the roots of (29) namely 1205821 and 120582

2satisfy the fol-

lowing relations

1205821+ 1205822= 1198711+1198722

12058211205822= 11987111198722minus 11987121198721

(30)

Now we write the spectral decomposition of matrix 119860(119904)as

119860 (119904) = 1205821119864 + 120582

2119865 (31)

where 119864 and 119865 are called the projectors of119860(119904) and satisfy thefollowing conditions

119864 + 119865 = 119868 (32a)

119864119865 = 119865119864 = 119874 (32b)

1198642= 119864 119865

2= 119865 (32c)

Then we have

radic119860 (119904) = radic1205821119864 + radic120582

2119865 (33)

where

119864 = minus1

1205821minus 1205822

[1205822minus 1198711

minus1198712

minus1198721

1205822minus1198722

]

119865 =1

1205821minus 1205822

[1205821minus 1198711

minus1198712

minus1198721

1205821minus1198722

]

119868 = [1 0

0 1] 119874 = [

0 0

0 0]

(34)

Finally we get

119861 (119904) = radic119860 (119904) =1

radic1205821+ radic1205822

[radic12058211205822+ 1198711

1198712

1198721

radic12058211205822+1198722

]

(35)

The characteristic equation of matrix 119861(119904) can be writtenas

1198962minus 119896 (radic120582

1+ radic1205822) + radic120582

1radic1205822= 0 (36)

where the roots of (36) namely 1198961and 1198962 can be written as

1198961= radic120582

1 119896

2= radic120582

2 (37)

Now the Taylor series expansion of exp[minusradic119860(119904)119909] yields

exp [minusradic119860 (119904)119909] = exp [minus119861 (119904) 119909] =infin

sum

119899=0

[minus119861 (119904) 119909]119899

119899 (38)

Journal of Mathematics 5

Using the Cayley-Hamilton theorem we can express 1198612and higher orders of matrix 119861 in terms of 119868 and 119861 where 119868is the identity matrix of second order Bahar and Hetnarski[39]

Therefore the infinite series in (38) can be expressed as

119871 (119909 119904) = exp [minus119861 (119904) 119909] = 1198870119868 + 1198871119861 (119904) (39)

where 1198870and 1198871are coefficients depending on 119909 and 119904 By the

Cayley-Hamilton theorem the characteristic roots 1198961and 1198962

of the matrix 119861must satisfy (39) so we have

exp [minus1198961119909] = 119887

0+ 11988711198961

exp [minus1198962119909] = 119887

0+ 11988711198962

(40)

By solving the above system of equations and using (37)we get

1198870=radic1205821119890minusradic1205822119909 minus radic120582

2119890minusradic1205821119909

radic1205821minus radic1205822

1198871=119890minusradic1205821119909 minus 119890

minusradic1205822119909

radic1205821minus radic1205822

(41)

Plugging the values of 1198870and 1198871in (39) we have

exp [minus119861 (119904) 119909] = 119871 (119909 119904) = [119897119894119895] 119894 119895 = 1 2 (42)

where the entries 119897119894119895(119909 119904) are given as

11989711=(1205821minus 1198711) 119890minusradic1205822119909 minus (120582

2minus 1198711) 119890minusradic1205821119909

1205821minus 1205822

11989712= 1198712(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989721= 1198721(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989722=(1205821minus1198722) 119890minusradic1205822119909 minus (120582

2minus1198722) 119890minusradic1205821119909

1205821minus 1205822

(43)

The solution of (25) can be written in the following form

119881 (119909 119904) = [119897119894119895] [119881 (0 119904) + 119860

minus1(119904) 119862 (0 119904)] minus 119860

minus1(119904) 119862 (119909 119904)

(44)

Substituting the values of119881(119909 119904)119860minus1(119904) and 119862(119909 119904) into(44) and performing the necessary matrix operations weobtain the values of 120579(119909 119904) and 120590(119909 119904) as

120579 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205822119909

minus (1205822minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205821119909]

minus 1205781

(45)

120590 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205822119909

minus (1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781)

times119890minusradic1205821119909] minus 120578

2

(46)

where

1205781=11987221198713minus 11987121198723

11987111198722minus11987211198712

1205782=11987111198723minus11987211198713

11987111198722minus11987211198712

(47)

Now considering (12) along with (15) and the Laplacetransform the displacement component is evaluated as

119906 (119909 119904) =1

12057211199042

120597120590

120597119909 (48)

Substituting (46) into (48) we get

119906 (119909 119904)

=1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205900+1205782)minus1198721(1205790+1205781) 119890minusradic1205822119909]

(49)

5 Application

Problem the ramp type boundary temperature of an elastichalf-space

We consider a homogeneous isotropic thermoelastic solidoccupying the half-space 119909 ge 0 The boundary of half-space119909 = 0 is affected by ramp type heating In mathematicalnotations the boundary conditions can be denoted as

120590 (0 119905) = 1205900= minus119901

120579 (0 119905) = 1205790= 120579lowastℎ (119905)

(50)

6 Journal of Mathematics

where 120579lowast is constant temperature and ℎ(119905) is defined as

ℎ (119905) =

0 119905 le 0

119905

1199050

0 lt 119905 le 1199050

1 119905 gt 1199050

(51)

where 1199050is ramping parameter

We employ nondimensional variables given in (15) on(50) and taking the Laplace transform we get

120590 (0 119904) = 1205900= minus

1205751

119904

120579 (0 119904) = 1205790= 120579lowast(1 minus 1198901199041199050

11990501199042

)

(52)

Substituting the values of 1205900and 120579

0from (52) into (45)-

(46) and (49) we find

120579 (119909 119904) =1

1205821minus 1205822

times [(1205821minus 1198711) (120579lowast120578lowast+ 1205781) minus 1198712(1205782minus1205751

119904)

times119890minusradic1205822119909 minus (120582

2minus 1198711) (120579lowast120578lowast+ 1205781)

minus1198712(1205782minus1205751

119904) 119890minusradic1205821119909] minus 120578

1

120590 (119909 119904) =1

1205821minus 1205822

times [(1205821minus1198722) (1205782minus1205751

119904) minus119872

1(120579lowast120578lowast+ 1205781)

times 119890minusradic1205822119909 minus (120582

2minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909]

minus 1205782

119906 (119909 119904) =1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205822119909]

(53)

where

120578lowast=1 minus 119890minus1199041199050

11990501199042

(54)

6 Limiting Cases

(1) The mathematical expressions for the field variablesstudied in the context of generalized magneto-ther-moelasticity theory with initial stress under appliedboundary condition can be obtained by applying 120572 =

1 in (13)(2) Neglecting initial stress effect by substituting 119901 = 0

in (14) and 120578 = 1 in (58) we obtain the expressionsof field variables under fractional order generalizedmagneto-thermoelasticity

(3) The expressions for studied fields in the context of thefractional order generalized thermoelasticity theorywith initial stress can be deduced by setting 119867

0= 0

in (12)

7 Numerical Inversion ofthe Laplace Transforms

We will now outline the numerical inversion method usedto find the solution in the physical domain The inversionformula of the Laplace transform is defined as

119891 (119905) =1

2120587120580int

119888+120580infin

119888minus120580infin

119890119904119905119891 (119904) 119889119904 (55)

where 119891(119904) is the Laplace transform of function 119891(119905)In order to invert the Laplace transforms in the equations

given in Section 5 we apply a numerical inversion methodbased on the Fourier series expansion explained byHonig andHirdes [40] In this method the inverse transform119891(119905) of theLaplace transform 119891(119904) is approximated by the relation as

119891 (119905) =119890119888119905

1199051

[1

2119891 (119888) + Re

119873

sum

119896=1

1198901205801198961205871199051199051119891(119888 +

120580119896120587

1199051

)]

0 le 119905 lt 21199051

(56)

where 119873 is a sufficiently large integer representing thenumber of terms in the truncated Fourier series chosen suchthat

119890119888119905 Re [1198901205801198731205871199051199051119891(119888 +

120580119873120587

1199051

)] le 1205761 (57)

where 1205761is a prescribed small positive value that corresponds

to the degree of accuracy to be achieved and 119888 is a positiveconstant and must be greater than the real parts of all thesingularities of 119891(119904) The optimal choice of 119888 was obtainedaccording to the criteria described by Honig and Hirdes [40]

8 Numerical Results and Discussions

To illustrate and compare the theoretical results obtainedin the Section 5 we now present some numerical results

Journal of Mathematics 7

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

119906

120572 = 05

120572 = 1

Figure 1 Displacement distribution 119906 for different values of 120572 at119905 = 01

which depict the variations of displacement temperatureand stress component The material chosen for the purposeof numerical evaluations is copper for which we take thefollowing values of the different physical constants

119864 = 369 times 1010 kgmminus1 sminus2 120576

0= (10

minus936120587) Fmminus1 119896 =

386Wmminus1 Kminus1 1198790= 293K 120588 = 8954 kgmminus3 120591

0= 002 s

V = 033 120572119905= 178 times 10

minus5 Kminus1 119888119864= 3831 Jkgminus1 Kminus1 119867

0=

(1074120587)Amminus1 120583

0= 412058710

minus7Hmminus1The general Lame constants 120582 and 120583 are given as

120582 =119864V

120578 (1 + V) (1 minus 2V) 120583 =

119864

2120578 (1 + V) (58)

where 120578 is the initial stress parameter 119864 is Youngrsquos modulusand V is Poisson ratio For isotropic elastic medium with noinitial stress we take 120578 = 1

The computations are carried out for 119905 = 01 1199050= 05

120572 = 05 and 120579lowast= 1 The numerical technique outlined in

previous section was used to invert the Laplace transform in(53) providing the displacement temperature and stress dis-tributions in the physical domainThe results are representedgraphically for different positions of 119909

Figures 1 2 and 3 exhibit the space variations of the fieldquantities in the context of fractional order theory of therm-oelasticity with magnetic field and initial stress for differentvalues of fractional parameter 120572

Figure 1 displays the variations of displacement compo-nent 119906 for different values of 120572 and it is noticed that themagnitude of displacement component 119906 decreases with theincrease in the value of fractional parameter 120572 In both thecases (ie 120572 = 05 and 120572 = 10) the displacement componentattains maximum value at the boundary of half-space andthen continuously decreases to zero Hence displacementcomponent 119906 has similar trend for both the values of 120572

minus05

0

05

1

15

2

25

0 04 08 12 16 2119909

120579

120572 = 05

120572 = 1

Figure 2 Temperature distribution 120579 for different values of 120572 at 119905 =01

0

05

1

15

2

25

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

120590

120572 = 05

120572 = 1

119909

Figure 3 Stress distribution 120590 for different values of 120572 at 119905 = 01

Figure 2 depicts the variations of temperature 120579 withdistance 119909 for different values of 120572 and it is noticed that inboth the cases (ie 120572 = 05 and 120572 = 10) maximum value of 120579is 2 which is on the boundary of half-space We observe fromthe figure that the difference is negligible in the beginningandwith the increase in 119909 the difference ismuch pronouncedup to 119909 le 14 both the series approach to zero The trends ofboth the series are alike only up to 119909 = 08

Figure 3 shows the variations of stress component 120590 withdistance119909 for different values of120572 It is evident from the figurethat both the series have similar trend that is first increasesto a maximum value and then decreases to a minimumvalue The value of 120590 increases with the increase of fractionalparameter when 0 le 119909 le 08 while the trend of change is

8 Journal of Mathematics

minus03

minus02

minus01

0

01

02

03

04

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

119906

Figure 4 Displacement distribution 119906 at 120572 = 05 119905 = 01

minus05

0

05

1

15

2

25

0 04 08 12 16 2

ISMTMTIST

119909

120579

Figure 5 Temperature distribution 120579 at 120572 = 05 119905 = 01

adverse when 08 le 119909 le 6 The difference is significant in therange 0 le 119909 le 28

Figures 4 5 and 6 depict the variations of displacementcomponent temperature and stress component for three dif-ferent cases defined as (a) ISMT (initially stressed magneto-thermoelasticity) 120578 = 25 119901 = 1 (b) MT (magneto-thermoelasticity) 120578 = 1 119901 = 0 and (c) IST (initially stressedthermoelasticity) 120578 = 25 119901 = 1 119867

0= 0 In all the above

cases (a) (b) and (c) the value of fractional order parameteris taken to be 120572 = 05

Figure 4 shows the variations of displacement componentunder the three cases discussed in (a) (b) and (c) It isobserved that the displacement component in case of ISMTand IST has maximum positive values but in case of MT

0

02

04

06

08

1

12

14

16

18

2

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

120590

Figure 6 Stress distribution 120590 at 120572 = 05 119905 = 01

the displacement component has maximum negative valueon the boundary of half-space The trend for displacementcomponent in case of ISMT and IST is same that is both theseries are continuously decreasing to zero but in case of MTthe series rapidly approaches to zero Figure 4 also exhibitsthat in the absence of magnetic field (IST) and initial stress(MT) decreases the value of displacement It is also observedthat as compared to ISMT the difference is significant in caseof IST but much pronounced in case of MT

Figure 5 displays the variations of temperature under theISMT MT and IST theories It is observed that the trend ofthe series in case of ISMT MT and IST is similar and thedifference is significant It is found that the absence of initialstress (MT) increases the value of temperature componentbut the absence of magnetic field (IST) decreases the value oftemperature as compared to the general case (ISMT)

Figure 6 exhibits the variations of stress under ISMTMT and IST theories We found that the behavior of stresscomponent in all the three cases is alike It also observedthat as compared to ISMT theory the stress component haslarge values in IST theory but has small values in MT theoryThe difference is significant in both IST and MT theoriescompared with ISMT theory but the difference is muchpronounced in MT theory as compared to IST theory

Figures 7 8 and 9 exhibit the variations of displacementcomponent temperature and stress component in the con-text of the fractional order theory of thermoelasticity withmagnetic field and initial stress for different values of rampparameterThe values of ramp parameter are taken as 01 03and 05 and the value of fractional order parameter 120572 is takento be 05 for all the three cases

Figure 7 demonstrates that as we increase the value oframp parameter then the value of displacement componentalso increases It is also observed that trend for all the cases is

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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

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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

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

2 Journal of Mathematics

problem with thermal relaxation and heat source in three-dimensional infinite rotating elastic medium

The development of initial stress in the medium is dueto many reasons such as the process of quenching resultingfrom difference of temperatures slow process of creepdifferential external forces and gravity variationsThe earth issupposed to be under high initial stressThe researchers haveshownmuch interest to study the effect of these stresses on thepropagation of waves Biot [16] solved the dynamic problemof elastic medium under initial stress Chattopadhyay et al[17] studied the reflection of elastic waves under initial stressat a free surface Montanaro [18] studied the isotropic linearthermoelasticity with hydrostatic initial stress by using Biotrsquoslinearization of the constitutive law for stress Othman andSong [19] investigated the reflection of plane waves from anelastic solid half-space under hydrostatic initial stress withoutenergy dissipation Singh [20] explored the effect of hydro-static initial stress on waves in a thermoelastic half-space

The theory of fractional derivative and integral was estab-lished in the second half of nineteenth centuryThefirst appli-cation of fractional derivative was given by Abel who appliedfractional calculus in the solution of an integral equationthat arises in the formulation of tautochrone problem In therecent years fractional calculus has been applied successfullyin various areas to modify many existing models of physicalprocesses such as heat conduction diffusion viscoelasticitywave propagation and electronics Caputo and Mainardi [2122] and Caputo [23] have established the relation betweenfractional derivative and theory of linear viscoelasticity Thegeneralization of the concept of derivative and integral to anoninteger order has been subjected to several approachesand some various alternative definitions of fractional deriva-tives appeared in [24ndash27] One can refer to Podlubny [28]for a survey of applications of fractional calculus Povstenko[29] has proposed a quasistatic uncoupled theory of ther-moelasticity based on fractional heat conduction equationSherief et al [30] introduced a newmodel of thermoelasticityusing fractional calculus proved a uniqueness theorem andderived a reciprocity relation and a variational principle Inthis model heat conduction equation takes the form as

119902119894+ 1205910

120597120572

120597119905120572119902119894= minus119896119894119895120579119895 (1)

where 119902119894are the components of the heat flux vector 120579 is the

temperature 1205910is the thermal relaxation time parameter 119896

119894119895is

thermal conductivity tensor and 120572 is a fractional parametersuch that 0 lt 120572 le 1 The above heat conduction equationreduces to the Maxwell-Cattaneo law in the limiting casewhen 120572 rarr 1 It should be mentioned here that the Maxwell-Cattaneo law has been employed by Lord and Shulman [2] todevelop first generalized theory of thermoelasticity Youssef[31] constructed another model of thermoelasticity in thecontext of a new consideration of heat conduction with afractional order and proved the uniqueness theorem In thismodel Youssef described different cases of conductivity 0 lt

120572 lt 1 corresponds toweak conductivity120572 = 1 corresponds tonormal conductivity and 1 lt 120572 le 2 corresponds to supercon-ductivity Ezzat [32 33] established a model of fractional heat

conduction equation by using the new Taylor series expan-sion of time-fractional order developed by Jumarie [34] El-Karamany and Ezzat [35] introduced two general modelsof fractional heat conduction law for a nonhomogeneousanisotropic elastic solid Uniqueness and reciprocal theoremsare proved and the convolutional variational principle isestablished and used to prove a uniqueness theorem with norestriction on the elasticity or thermal conductivity tensorsexcept symmetry conditions The two-temperature dynamiccoupled Lord-Shulman and fractional coupled thermoelas-ticity theories result as limit cases For fractional thermoe-lasticity not involving two-temperatures El-Karamany andEzzat [36] established the uniqueness reciprocal theoremsand convolution variational principle The dynamic coupledand the Green-Naghdi thermoelasticity theories result aslimit casesThe reciprocity relation in case of quiescent initialstate is found to be independent of the order of differintegra-tion [35 36] Fractional order theory of a perfect conductingthermoelastic medium not involving two temperatures wasinvestigated by El-Karamany and Ezzat [37] Kothari andMukhopadhyay [38] studied a half-space problem underfractional order theory of thermoelasticity and analyzed theeffect of the fractional order parameter on the field variables

In the present paper we study the effect of magnetic fieldand initial stress under fractional order theory of thermo-elasticity proposed by Sherief et al [30] We employ a statespace approach developed by Bahar and Hetnarski [39] onthe formulation The Laplace transform technique is used toobtain the general solution The inverse Laplace transform iscarried out using a numerical inversion method developedby Honig and Hirdes [40] Finally the effect of fractionalparameter ramp parameter magnetic field and initial stresson field variables is displayed graphically

2 Governing Equations

The governing equations in the context of fractional ordertheory of generalized thermoelasticity with initial stress andmagnetic field for isotropic and homogeneous elastic med-ium are considered as

(i) the equation of motion

120588119894= 120590119895119894119895

+ 119865119894 (2)

where = 1205830119869 times

(ii) heat conduction equation

119896120579119894119894= 120588119888119864(1 + 120591

0

120597120572

120597119905120572)120597120579

120597119905+ 1205731198790(1 + 120591

0

120597120572

120597119905120572)120597119906119894119894

120597119905 (3)

(iii) constitutive relations

120590119894119895= minus119901 (120575

119894119895+ 120596119894119895) + 2120583119890

119894119895+ 120582119890120575

119894119895minus 120573120579120575

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

120596119894119895=1

2(119906119895119894minus 119906119894119895)

(4)

Journal of Mathematics 3

We take the linearized Maxwell equations governing theelectromagnetic field for a perfectly conducting medium as

curl ℎ = 119869 + 1205760

120597

120597119905

curl = minus1205830

120597ℎ

120597119905

= minus1205830(120597ℎ

120597119905times )

div ℎ = 0

(5)

where 119906119894are the components of displacement vector 120579 =

119879 minus 1198790 119879 is the absolute temperature 119879

0is the reference

temperature assumed to obey the inequality |1205791198790| ≪ 1 120591

0

is the thermal relaxation time 120590119894119895are the components of the

stress tensor 119890119894119895are the components of strain tensor 120575

119894119895is the

Kronecker delta function 119890 is the cubical dilation 120588 is thedensity of the medium 119888

119864is the specific heat 119896 is the thermal

conductivity 120573 = (3120582 + 2120583)120572119905 120572119905is the coefficient of linear

thermal expansion 120582 and 120583 are the lame constants 119901 is theinitial stress 119865

119894are the components of Lorentzrsquos body force

vector 1205830is the magnetic permeability 120576

0is the electric

permittivity is the applied magnetic field ℎ is the inducedmagnetic field is the induced electric field and 119869 is thecurrent density vector

3 Problem Formulation

We consider a perfectly conducting isotropic homogeneousand fractional order generalized thermoelastic half-spacewith hydrostatic initial stress subjected to a constantmagneticfield (0119867

0 0) which produces an induced magnetic field

ℎ(0 ℎ2 0) and induced electric field (0 0 119864

3) We assume

one-dimensional motion for which all the field quantities arefunctions of 119909 and 119905

The displacement components take the form119906119909= 119906 (119909 119905) 119906

119910= 119906119911= 0 (6)

The strain component becomes

119890 = 119890119909119909

=120597119906

120597119909 (7)

The components of magnetic field vectors are119867119909= 0 119867

119910= 1198670 119867

119911= 0 (8)

The electric intensity vector is parallel to current densityvector 119869 Hence components of and 119869 are given as

119864119909= 119864119910= 0 119864

119911= 1198643

119869119909= 119869119910= 0 119869

119911= 1198693

(9)

Now the Maxwell equation (5) provides the followingresults

1198643= minus12058301198670

120597119906

120597119905 ℎ

2= minus1198670119890 119869

3= 120576012058301198670

1205972119906

1205972119905

(10)

Using (8) and (10) into the relation = 1205830119869 times we

obtain

119865119909= minus12057601205832

01198672

0 119865

119910= 119865119911= 0 (11)

The governing equations for one-dimensional case be-come

120588 =120597120590

120597119909+ 119865119909 (12)

1198961205972120579

1205971199092= 120588119888119864(1 + 120591

0

120597120572

120597119905120572)120597120579

120597119905+ 1205731198790(1 + 120591

0

120597120572

120597119905120572)120597119890

120597119905 (13)

120590 = 120590119909119909

= minus119901 + (120582 + 2120583) 119890 minus 120573120579 (14)

Now we will use the following nondimensional variables

1199091015840= 11988801205780119909 119906

1015840= 11988801205780119906

1199051015840= 1198882

01205780119905 120591

1015840

0= 1198882

012057801205910

1205791015840=

120579

1198790

1205901015840=

120590

120582 + 2120583 119901

1015840=

119901

1205731198790

(15)

where

1205780=120588119888119864

119896 119888

2

0=120582 + 2120583

120588 (16)

Expressing (12)ndash(14) in terms of the nondimensionalvariables given by (15) and dropping the prime sign forconvenience we have the following forms

1205972120590

1205971199092= 1205721119890

1205972120579

1205971199092= (1 + 120591

lowast

0

120597120572

120597119905120572) ( 120579 + 120573

1119890)

120590 = 119890 minus 1205761120579 minus 1205751

(17)

where

1205721= 1 +

12057601205832

01198672

0

120588 120573

1=

120573

120588119888119864

1205761=

1205731198790

120582 + 2120583

1205751= 1205761119901 120591

lowast

0=

1205910

(119888201205780)1minus120572

(18)

These equations will be supplemented with appropriateboundary conditions relevant to the particular applicationunder consideration

Taking the Laplace transform of (17) by using homoge-neous initial conditions defined and denoted as

119891 (119904) = int

infin

0

119890minus119904119905119891 (119905) 119889119905 119904 gt 0 (19)

4 Journal of Mathematics

we obtain

1205972120590

1205971199092= 12057211199042119890 (20)

1205972120579

1205971199092= (119904 + 120591

lowast

0119904120572+1

) (120579 + 1205731119890) (21)

120590 = 119890 minus 1205761120579 minus

1205751

119904 (22)

Eliminating the value of 119890 from (20) and (21) by using(22) we obtain

1205972120579

1205971199092= 1198711120579 + 1198712120590 + 1198713

1205972120590

1205971199092= 1198721120579 +119872

2120590 +119872

3

(23)

where

1198711= (1 + 120573

11205761) (119904 + 120591

lowast

0119904120572+1

) 1198712= 1205731(119904 + 120591

lowast

0119904120572+1

)

1198713= 12057311205751(1 + 120591

lowast

0119904120572)

1198721= 120572112057611199042 119872

2= 12057211199042 119872

3= 12057211205751119904

(24)

4 State-Space Formulation

Now choosing the temperature of heat conduction 120579 and thestress component 120590 in 119909-direction as state variables one canwrite (23) in matrix form as

1198892

1198891199092119881 (119909 119904) = 119860 (119904) 119881 (119909 119904) + 119862 (119909 119904) (25)

where

119860 (119904) = [1198711

1198712

11987211198722

] 119881 (119909 119904) = [120579 (119909 119904)

120590 (119909 119904)]

119862 (119909 119904) = [1198713

1198723

]

(26)

The formal solution of (25) can be written in the form

119881 (119909 119904) = exp [minusradic119860 (119904)119909] [119881 (0 119904) + 119862 (0 119904) 119860minus1(119904)]

minus 119862 (119909 119904) 119860minus1(119904)

(27)

where

119881 (0 119904) = [120579 (0 119904)

120590 (0 119904)] = [

1205790

1205900

] (28)

In the above solution we have cancelled the part ofexponential having positive power to get bounded solutionfor large 119909

Nowwewill use theCayley-Hamilton theorem to find theform of the matrix exp[minusradic119860(119904)119909] and for this we proceedas follows

The characteristic equation of the matrix119860(119904) is obtainedas

1205822minus (1198711+1198722) 120582 + (119871

11198722minus11987211198712) = 0 (29)

where the roots of (29) namely 1205821 and 120582

2satisfy the fol-

lowing relations

1205821+ 1205822= 1198711+1198722

12058211205822= 11987111198722minus 11987121198721

(30)

Now we write the spectral decomposition of matrix 119860(119904)as

119860 (119904) = 1205821119864 + 120582

2119865 (31)

where 119864 and 119865 are called the projectors of119860(119904) and satisfy thefollowing conditions

119864 + 119865 = 119868 (32a)

119864119865 = 119865119864 = 119874 (32b)

1198642= 119864 119865

2= 119865 (32c)

Then we have

radic119860 (119904) = radic1205821119864 + radic120582

2119865 (33)

where

119864 = minus1

1205821minus 1205822

[1205822minus 1198711

minus1198712

minus1198721

1205822minus1198722

]

119865 =1

1205821minus 1205822

[1205821minus 1198711

minus1198712

minus1198721

1205821minus1198722

]

119868 = [1 0

0 1] 119874 = [

0 0

0 0]

(34)

Finally we get

119861 (119904) = radic119860 (119904) =1

radic1205821+ radic1205822

[radic12058211205822+ 1198711

1198712

1198721

radic12058211205822+1198722

]

(35)

The characteristic equation of matrix 119861(119904) can be writtenas

1198962minus 119896 (radic120582

1+ radic1205822) + radic120582

1radic1205822= 0 (36)

where the roots of (36) namely 1198961and 1198962 can be written as

1198961= radic120582

1 119896

2= radic120582

2 (37)

Now the Taylor series expansion of exp[minusradic119860(119904)119909] yields

exp [minusradic119860 (119904)119909] = exp [minus119861 (119904) 119909] =infin

sum

119899=0

[minus119861 (119904) 119909]119899

119899 (38)

Journal of Mathematics 5

Using the Cayley-Hamilton theorem we can express 1198612and higher orders of matrix 119861 in terms of 119868 and 119861 where 119868is the identity matrix of second order Bahar and Hetnarski[39]

Therefore the infinite series in (38) can be expressed as

119871 (119909 119904) = exp [minus119861 (119904) 119909] = 1198870119868 + 1198871119861 (119904) (39)

where 1198870and 1198871are coefficients depending on 119909 and 119904 By the

Cayley-Hamilton theorem the characteristic roots 1198961and 1198962

of the matrix 119861must satisfy (39) so we have

exp [minus1198961119909] = 119887

0+ 11988711198961

exp [minus1198962119909] = 119887

0+ 11988711198962

(40)

By solving the above system of equations and using (37)we get

1198870=radic1205821119890minusradic1205822119909 minus radic120582

2119890minusradic1205821119909

radic1205821minus radic1205822

1198871=119890minusradic1205821119909 minus 119890

minusradic1205822119909

radic1205821minus radic1205822

(41)

Plugging the values of 1198870and 1198871in (39) we have

exp [minus119861 (119904) 119909] = 119871 (119909 119904) = [119897119894119895] 119894 119895 = 1 2 (42)

where the entries 119897119894119895(119909 119904) are given as

11989711=(1205821minus 1198711) 119890minusradic1205822119909 minus (120582

2minus 1198711) 119890minusradic1205821119909

1205821minus 1205822

11989712= 1198712(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989721= 1198721(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989722=(1205821minus1198722) 119890minusradic1205822119909 minus (120582

2minus1198722) 119890minusradic1205821119909

1205821minus 1205822

(43)

The solution of (25) can be written in the following form

119881 (119909 119904) = [119897119894119895] [119881 (0 119904) + 119860

minus1(119904) 119862 (0 119904)] minus 119860

minus1(119904) 119862 (119909 119904)

(44)

Substituting the values of119881(119909 119904)119860minus1(119904) and 119862(119909 119904) into(44) and performing the necessary matrix operations weobtain the values of 120579(119909 119904) and 120590(119909 119904) as

120579 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205822119909

minus (1205822minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205821119909]

minus 1205781

(45)

120590 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205822119909

minus (1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781)

times119890minusradic1205821119909] minus 120578

2

(46)

where

1205781=11987221198713minus 11987121198723

11987111198722minus11987211198712

1205782=11987111198723minus11987211198713

11987111198722minus11987211198712

(47)

Now considering (12) along with (15) and the Laplacetransform the displacement component is evaluated as

119906 (119909 119904) =1

12057211199042

120597120590

120597119909 (48)

Substituting (46) into (48) we get

119906 (119909 119904)

=1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205900+1205782)minus1198721(1205790+1205781) 119890minusradic1205822119909]

(49)

5 Application

Problem the ramp type boundary temperature of an elastichalf-space

We consider a homogeneous isotropic thermoelastic solidoccupying the half-space 119909 ge 0 The boundary of half-space119909 = 0 is affected by ramp type heating In mathematicalnotations the boundary conditions can be denoted as

120590 (0 119905) = 1205900= minus119901

120579 (0 119905) = 1205790= 120579lowastℎ (119905)

(50)

6 Journal of Mathematics

where 120579lowast is constant temperature and ℎ(119905) is defined as

ℎ (119905) =

0 119905 le 0

119905

1199050

0 lt 119905 le 1199050

1 119905 gt 1199050

(51)

where 1199050is ramping parameter

We employ nondimensional variables given in (15) on(50) and taking the Laplace transform we get

120590 (0 119904) = 1205900= minus

1205751

119904

120579 (0 119904) = 1205790= 120579lowast(1 minus 1198901199041199050

11990501199042

)

(52)

Substituting the values of 1205900and 120579

0from (52) into (45)-

(46) and (49) we find

120579 (119909 119904) =1

1205821minus 1205822

times [(1205821minus 1198711) (120579lowast120578lowast+ 1205781) minus 1198712(1205782minus1205751

119904)

times119890minusradic1205822119909 minus (120582

2minus 1198711) (120579lowast120578lowast+ 1205781)

minus1198712(1205782minus1205751

119904) 119890minusradic1205821119909] minus 120578

1

120590 (119909 119904) =1

1205821minus 1205822

times [(1205821minus1198722) (1205782minus1205751

119904) minus119872

1(120579lowast120578lowast+ 1205781)

times 119890minusradic1205822119909 minus (120582

2minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909]

minus 1205782

119906 (119909 119904) =1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205822119909]

(53)

where

120578lowast=1 minus 119890minus1199041199050

11990501199042

(54)

6 Limiting Cases

(1) The mathematical expressions for the field variablesstudied in the context of generalized magneto-ther-moelasticity theory with initial stress under appliedboundary condition can be obtained by applying 120572 =

1 in (13)(2) Neglecting initial stress effect by substituting 119901 = 0

in (14) and 120578 = 1 in (58) we obtain the expressionsof field variables under fractional order generalizedmagneto-thermoelasticity

(3) The expressions for studied fields in the context of thefractional order generalized thermoelasticity theorywith initial stress can be deduced by setting 119867

0= 0

in (12)

7 Numerical Inversion ofthe Laplace Transforms

We will now outline the numerical inversion method usedto find the solution in the physical domain The inversionformula of the Laplace transform is defined as

119891 (119905) =1

2120587120580int

119888+120580infin

119888minus120580infin

119890119904119905119891 (119904) 119889119904 (55)

where 119891(119904) is the Laplace transform of function 119891(119905)In order to invert the Laplace transforms in the equations

given in Section 5 we apply a numerical inversion methodbased on the Fourier series expansion explained byHonig andHirdes [40] In this method the inverse transform119891(119905) of theLaplace transform 119891(119904) is approximated by the relation as

119891 (119905) =119890119888119905

1199051

[1

2119891 (119888) + Re

119873

sum

119896=1

1198901205801198961205871199051199051119891(119888 +

120580119896120587

1199051

)]

0 le 119905 lt 21199051

(56)

where 119873 is a sufficiently large integer representing thenumber of terms in the truncated Fourier series chosen suchthat

119890119888119905 Re [1198901205801198731205871199051199051119891(119888 +

120580119873120587

1199051

)] le 1205761 (57)

where 1205761is a prescribed small positive value that corresponds

to the degree of accuracy to be achieved and 119888 is a positiveconstant and must be greater than the real parts of all thesingularities of 119891(119904) The optimal choice of 119888 was obtainedaccording to the criteria described by Honig and Hirdes [40]

8 Numerical Results and Discussions

To illustrate and compare the theoretical results obtainedin the Section 5 we now present some numerical results

Journal of Mathematics 7

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

119906

120572 = 05

120572 = 1

Figure 1 Displacement distribution 119906 for different values of 120572 at119905 = 01

which depict the variations of displacement temperatureand stress component The material chosen for the purposeof numerical evaluations is copper for which we take thefollowing values of the different physical constants

119864 = 369 times 1010 kgmminus1 sminus2 120576

0= (10

minus936120587) Fmminus1 119896 =

386Wmminus1 Kminus1 1198790= 293K 120588 = 8954 kgmminus3 120591

0= 002 s

V = 033 120572119905= 178 times 10

minus5 Kminus1 119888119864= 3831 Jkgminus1 Kminus1 119867

0=

(1074120587)Amminus1 120583

0= 412058710

minus7Hmminus1The general Lame constants 120582 and 120583 are given as

120582 =119864V

120578 (1 + V) (1 minus 2V) 120583 =

119864

2120578 (1 + V) (58)

where 120578 is the initial stress parameter 119864 is Youngrsquos modulusand V is Poisson ratio For isotropic elastic medium with noinitial stress we take 120578 = 1

The computations are carried out for 119905 = 01 1199050= 05

120572 = 05 and 120579lowast= 1 The numerical technique outlined in

previous section was used to invert the Laplace transform in(53) providing the displacement temperature and stress dis-tributions in the physical domainThe results are representedgraphically for different positions of 119909

Figures 1 2 and 3 exhibit the space variations of the fieldquantities in the context of fractional order theory of therm-oelasticity with magnetic field and initial stress for differentvalues of fractional parameter 120572

Figure 1 displays the variations of displacement compo-nent 119906 for different values of 120572 and it is noticed that themagnitude of displacement component 119906 decreases with theincrease in the value of fractional parameter 120572 In both thecases (ie 120572 = 05 and 120572 = 10) the displacement componentattains maximum value at the boundary of half-space andthen continuously decreases to zero Hence displacementcomponent 119906 has similar trend for both the values of 120572

minus05

0

05

1

15

2

25

0 04 08 12 16 2119909

120579

120572 = 05

120572 = 1

Figure 2 Temperature distribution 120579 for different values of 120572 at 119905 =01

0

05

1

15

2

25

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

120590

120572 = 05

120572 = 1

119909

Figure 3 Stress distribution 120590 for different values of 120572 at 119905 = 01

Figure 2 depicts the variations of temperature 120579 withdistance 119909 for different values of 120572 and it is noticed that inboth the cases (ie 120572 = 05 and 120572 = 10) maximum value of 120579is 2 which is on the boundary of half-space We observe fromthe figure that the difference is negligible in the beginningandwith the increase in 119909 the difference ismuch pronouncedup to 119909 le 14 both the series approach to zero The trends ofboth the series are alike only up to 119909 = 08

Figure 3 shows the variations of stress component 120590 withdistance119909 for different values of120572 It is evident from the figurethat both the series have similar trend that is first increasesto a maximum value and then decreases to a minimumvalue The value of 120590 increases with the increase of fractionalparameter when 0 le 119909 le 08 while the trend of change is

8 Journal of Mathematics

minus03

minus02

minus01

0

01

02

03

04

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

119906

Figure 4 Displacement distribution 119906 at 120572 = 05 119905 = 01

minus05

0

05

1

15

2

25

0 04 08 12 16 2

ISMTMTIST

119909

120579

Figure 5 Temperature distribution 120579 at 120572 = 05 119905 = 01

adverse when 08 le 119909 le 6 The difference is significant in therange 0 le 119909 le 28

Figures 4 5 and 6 depict the variations of displacementcomponent temperature and stress component for three dif-ferent cases defined as (a) ISMT (initially stressed magneto-thermoelasticity) 120578 = 25 119901 = 1 (b) MT (magneto-thermoelasticity) 120578 = 1 119901 = 0 and (c) IST (initially stressedthermoelasticity) 120578 = 25 119901 = 1 119867

0= 0 In all the above

cases (a) (b) and (c) the value of fractional order parameteris taken to be 120572 = 05

Figure 4 shows the variations of displacement componentunder the three cases discussed in (a) (b) and (c) It isobserved that the displacement component in case of ISMTand IST has maximum positive values but in case of MT

0

02

04

06

08

1

12

14

16

18

2

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

120590

Figure 6 Stress distribution 120590 at 120572 = 05 119905 = 01

the displacement component has maximum negative valueon the boundary of half-space The trend for displacementcomponent in case of ISMT and IST is same that is both theseries are continuously decreasing to zero but in case of MTthe series rapidly approaches to zero Figure 4 also exhibitsthat in the absence of magnetic field (IST) and initial stress(MT) decreases the value of displacement It is also observedthat as compared to ISMT the difference is significant in caseof IST but much pronounced in case of MT

Figure 5 displays the variations of temperature under theISMT MT and IST theories It is observed that the trend ofthe series in case of ISMT MT and IST is similar and thedifference is significant It is found that the absence of initialstress (MT) increases the value of temperature componentbut the absence of magnetic field (IST) decreases the value oftemperature as compared to the general case (ISMT)

Figure 6 exhibits the variations of stress under ISMTMT and IST theories We found that the behavior of stresscomponent in all the three cases is alike It also observedthat as compared to ISMT theory the stress component haslarge values in IST theory but has small values in MT theoryThe difference is significant in both IST and MT theoriescompared with ISMT theory but the difference is muchpronounced in MT theory as compared to IST theory

Figures 7 8 and 9 exhibit the variations of displacementcomponent temperature and stress component in the con-text of the fractional order theory of thermoelasticity withmagnetic field and initial stress for different values of rampparameterThe values of ramp parameter are taken as 01 03and 05 and the value of fractional order parameter 120572 is takento be 05 for all the three cases

Figure 7 demonstrates that as we increase the value oframp parameter then the value of displacement componentalso increases It is also observed that trend for all the cases is

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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

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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 3

We take the linearized Maxwell equations governing theelectromagnetic field for a perfectly conducting medium as

curl ℎ = 119869 + 1205760

120597

120597119905

curl = minus1205830

120597ℎ

120597119905

= minus1205830(120597ℎ

120597119905times )

div ℎ = 0

(5)

where 119906119894are the components of displacement vector 120579 =

119879 minus 1198790 119879 is the absolute temperature 119879

0is the reference

temperature assumed to obey the inequality |1205791198790| ≪ 1 120591

0

is the thermal relaxation time 120590119894119895are the components of the

stress tensor 119890119894119895are the components of strain tensor 120575

119894119895is the

Kronecker delta function 119890 is the cubical dilation 120588 is thedensity of the medium 119888

119864is the specific heat 119896 is the thermal

conductivity 120573 = (3120582 + 2120583)120572119905 120572119905is the coefficient of linear

thermal expansion 120582 and 120583 are the lame constants 119901 is theinitial stress 119865

119894are the components of Lorentzrsquos body force

vector 1205830is the magnetic permeability 120576

0is the electric

permittivity is the applied magnetic field ℎ is the inducedmagnetic field is the induced electric field and 119869 is thecurrent density vector

3 Problem Formulation

We consider a perfectly conducting isotropic homogeneousand fractional order generalized thermoelastic half-spacewith hydrostatic initial stress subjected to a constantmagneticfield (0119867

0 0) which produces an induced magnetic field

ℎ(0 ℎ2 0) and induced electric field (0 0 119864

3) We assume

one-dimensional motion for which all the field quantities arefunctions of 119909 and 119905

The displacement components take the form119906119909= 119906 (119909 119905) 119906

119910= 119906119911= 0 (6)

The strain component becomes

119890 = 119890119909119909

=120597119906

120597119909 (7)

The components of magnetic field vectors are119867119909= 0 119867

119910= 1198670 119867

119911= 0 (8)

The electric intensity vector is parallel to current densityvector 119869 Hence components of and 119869 are given as

119864119909= 119864119910= 0 119864

119911= 1198643

119869119909= 119869119910= 0 119869

119911= 1198693

(9)

Now the Maxwell equation (5) provides the followingresults

1198643= minus12058301198670

120597119906

120597119905 ℎ

2= minus1198670119890 119869

3= 120576012058301198670

1205972119906

1205972119905

(10)

Using (8) and (10) into the relation = 1205830119869 times we

obtain

119865119909= minus12057601205832

01198672

0 119865

119910= 119865119911= 0 (11)

The governing equations for one-dimensional case be-come

120588 =120597120590

120597119909+ 119865119909 (12)

1198961205972120579

1205971199092= 120588119888119864(1 + 120591

0

120597120572

120597119905120572)120597120579

120597119905+ 1205731198790(1 + 120591

0

120597120572

120597119905120572)120597119890

120597119905 (13)

120590 = 120590119909119909

= minus119901 + (120582 + 2120583) 119890 minus 120573120579 (14)

Now we will use the following nondimensional variables

1199091015840= 11988801205780119909 119906

1015840= 11988801205780119906

1199051015840= 1198882

01205780119905 120591

1015840

0= 1198882

012057801205910

1205791015840=

120579

1198790

1205901015840=

120590

120582 + 2120583 119901

1015840=

119901

1205731198790

(15)

where

1205780=120588119888119864

119896 119888

2

0=120582 + 2120583

120588 (16)

Expressing (12)ndash(14) in terms of the nondimensionalvariables given by (15) and dropping the prime sign forconvenience we have the following forms

1205972120590

1205971199092= 1205721119890

1205972120579

1205971199092= (1 + 120591

lowast

0

120597120572

120597119905120572) ( 120579 + 120573

1119890)

120590 = 119890 minus 1205761120579 minus 1205751

(17)

where

1205721= 1 +

12057601205832

01198672

0

120588 120573

1=

120573

120588119888119864

1205761=

1205731198790

120582 + 2120583

1205751= 1205761119901 120591

lowast

0=

1205910

(119888201205780)1minus120572

(18)

These equations will be supplemented with appropriateboundary conditions relevant to the particular applicationunder consideration

Taking the Laplace transform of (17) by using homoge-neous initial conditions defined and denoted as

119891 (119904) = int

infin

0

119890minus119904119905119891 (119905) 119889119905 119904 gt 0 (19)

4 Journal of Mathematics

we obtain

1205972120590

1205971199092= 12057211199042119890 (20)

1205972120579

1205971199092= (119904 + 120591

lowast

0119904120572+1

) (120579 + 1205731119890) (21)

120590 = 119890 minus 1205761120579 minus

1205751

119904 (22)

Eliminating the value of 119890 from (20) and (21) by using(22) we obtain

1205972120579

1205971199092= 1198711120579 + 1198712120590 + 1198713

1205972120590

1205971199092= 1198721120579 +119872

2120590 +119872

3

(23)

where

1198711= (1 + 120573

11205761) (119904 + 120591

lowast

0119904120572+1

) 1198712= 1205731(119904 + 120591

lowast

0119904120572+1

)

1198713= 12057311205751(1 + 120591

lowast

0119904120572)

1198721= 120572112057611199042 119872

2= 12057211199042 119872

3= 12057211205751119904

(24)

4 State-Space Formulation

Now choosing the temperature of heat conduction 120579 and thestress component 120590 in 119909-direction as state variables one canwrite (23) in matrix form as

1198892

1198891199092119881 (119909 119904) = 119860 (119904) 119881 (119909 119904) + 119862 (119909 119904) (25)

where

119860 (119904) = [1198711

1198712

11987211198722

] 119881 (119909 119904) = [120579 (119909 119904)

120590 (119909 119904)]

119862 (119909 119904) = [1198713

1198723

]

(26)

The formal solution of (25) can be written in the form

119881 (119909 119904) = exp [minusradic119860 (119904)119909] [119881 (0 119904) + 119862 (0 119904) 119860minus1(119904)]

minus 119862 (119909 119904) 119860minus1(119904)

(27)

where

119881 (0 119904) = [120579 (0 119904)

120590 (0 119904)] = [

1205790

1205900

] (28)

In the above solution we have cancelled the part ofexponential having positive power to get bounded solutionfor large 119909

Nowwewill use theCayley-Hamilton theorem to find theform of the matrix exp[minusradic119860(119904)119909] and for this we proceedas follows

The characteristic equation of the matrix119860(119904) is obtainedas

1205822minus (1198711+1198722) 120582 + (119871

11198722minus11987211198712) = 0 (29)

where the roots of (29) namely 1205821 and 120582

2satisfy the fol-

lowing relations

1205821+ 1205822= 1198711+1198722

12058211205822= 11987111198722minus 11987121198721

(30)

Now we write the spectral decomposition of matrix 119860(119904)as

119860 (119904) = 1205821119864 + 120582

2119865 (31)

where 119864 and 119865 are called the projectors of119860(119904) and satisfy thefollowing conditions

119864 + 119865 = 119868 (32a)

119864119865 = 119865119864 = 119874 (32b)

1198642= 119864 119865

2= 119865 (32c)

Then we have

radic119860 (119904) = radic1205821119864 + radic120582

2119865 (33)

where

119864 = minus1

1205821minus 1205822

[1205822minus 1198711

minus1198712

minus1198721

1205822minus1198722

]

119865 =1

1205821minus 1205822

[1205821minus 1198711

minus1198712

minus1198721

1205821minus1198722

]

119868 = [1 0

0 1] 119874 = [

0 0

0 0]

(34)

Finally we get

119861 (119904) = radic119860 (119904) =1

radic1205821+ radic1205822

[radic12058211205822+ 1198711

1198712

1198721

radic12058211205822+1198722

]

(35)

The characteristic equation of matrix 119861(119904) can be writtenas

1198962minus 119896 (radic120582

1+ radic1205822) + radic120582

1radic1205822= 0 (36)

where the roots of (36) namely 1198961and 1198962 can be written as

1198961= radic120582

1 119896

2= radic120582

2 (37)

Now the Taylor series expansion of exp[minusradic119860(119904)119909] yields

exp [minusradic119860 (119904)119909] = exp [minus119861 (119904) 119909] =infin

sum

119899=0

[minus119861 (119904) 119909]119899

119899 (38)

Journal of Mathematics 5

Using the Cayley-Hamilton theorem we can express 1198612and higher orders of matrix 119861 in terms of 119868 and 119861 where 119868is the identity matrix of second order Bahar and Hetnarski[39]

Therefore the infinite series in (38) can be expressed as

119871 (119909 119904) = exp [minus119861 (119904) 119909] = 1198870119868 + 1198871119861 (119904) (39)

where 1198870and 1198871are coefficients depending on 119909 and 119904 By the

Cayley-Hamilton theorem the characteristic roots 1198961and 1198962

of the matrix 119861must satisfy (39) so we have

exp [minus1198961119909] = 119887

0+ 11988711198961

exp [minus1198962119909] = 119887

0+ 11988711198962

(40)

By solving the above system of equations and using (37)we get

1198870=radic1205821119890minusradic1205822119909 minus radic120582

2119890minusradic1205821119909

radic1205821minus radic1205822

1198871=119890minusradic1205821119909 minus 119890

minusradic1205822119909

radic1205821minus radic1205822

(41)

Plugging the values of 1198870and 1198871in (39) we have

exp [minus119861 (119904) 119909] = 119871 (119909 119904) = [119897119894119895] 119894 119895 = 1 2 (42)

where the entries 119897119894119895(119909 119904) are given as

11989711=(1205821minus 1198711) 119890minusradic1205822119909 minus (120582

2minus 1198711) 119890minusradic1205821119909

1205821minus 1205822

11989712= 1198712(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989721= 1198721(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989722=(1205821minus1198722) 119890minusradic1205822119909 minus (120582

2minus1198722) 119890minusradic1205821119909

1205821minus 1205822

(43)

The solution of (25) can be written in the following form

119881 (119909 119904) = [119897119894119895] [119881 (0 119904) + 119860

minus1(119904) 119862 (0 119904)] minus 119860

minus1(119904) 119862 (119909 119904)

(44)

Substituting the values of119881(119909 119904)119860minus1(119904) and 119862(119909 119904) into(44) and performing the necessary matrix operations weobtain the values of 120579(119909 119904) and 120590(119909 119904) as

120579 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205822119909

minus (1205822minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205821119909]

minus 1205781

(45)

120590 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205822119909

minus (1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781)

times119890minusradic1205821119909] minus 120578

2

(46)

where

1205781=11987221198713minus 11987121198723

11987111198722minus11987211198712

1205782=11987111198723minus11987211198713

11987111198722minus11987211198712

(47)

Now considering (12) along with (15) and the Laplacetransform the displacement component is evaluated as

119906 (119909 119904) =1

12057211199042

120597120590

120597119909 (48)

Substituting (46) into (48) we get

119906 (119909 119904)

=1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205900+1205782)minus1198721(1205790+1205781) 119890minusradic1205822119909]

(49)

5 Application

Problem the ramp type boundary temperature of an elastichalf-space

We consider a homogeneous isotropic thermoelastic solidoccupying the half-space 119909 ge 0 The boundary of half-space119909 = 0 is affected by ramp type heating In mathematicalnotations the boundary conditions can be denoted as

120590 (0 119905) = 1205900= minus119901

120579 (0 119905) = 1205790= 120579lowastℎ (119905)

(50)

6 Journal of Mathematics

where 120579lowast is constant temperature and ℎ(119905) is defined as

ℎ (119905) =

0 119905 le 0

119905

1199050

0 lt 119905 le 1199050

1 119905 gt 1199050

(51)

where 1199050is ramping parameter

We employ nondimensional variables given in (15) on(50) and taking the Laplace transform we get

120590 (0 119904) = 1205900= minus

1205751

119904

120579 (0 119904) = 1205790= 120579lowast(1 minus 1198901199041199050

11990501199042

)

(52)

Substituting the values of 1205900and 120579

0from (52) into (45)-

(46) and (49) we find

120579 (119909 119904) =1

1205821minus 1205822

times [(1205821minus 1198711) (120579lowast120578lowast+ 1205781) minus 1198712(1205782minus1205751

119904)

times119890minusradic1205822119909 minus (120582

2minus 1198711) (120579lowast120578lowast+ 1205781)

minus1198712(1205782minus1205751

119904) 119890minusradic1205821119909] minus 120578

1

120590 (119909 119904) =1

1205821minus 1205822

times [(1205821minus1198722) (1205782minus1205751

119904) minus119872

1(120579lowast120578lowast+ 1205781)

times 119890minusradic1205822119909 minus (120582

2minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909]

minus 1205782

119906 (119909 119904) =1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205822119909]

(53)

where

120578lowast=1 minus 119890minus1199041199050

11990501199042

(54)

6 Limiting Cases

(1) The mathematical expressions for the field variablesstudied in the context of generalized magneto-ther-moelasticity theory with initial stress under appliedboundary condition can be obtained by applying 120572 =

1 in (13)(2) Neglecting initial stress effect by substituting 119901 = 0

in (14) and 120578 = 1 in (58) we obtain the expressionsof field variables under fractional order generalizedmagneto-thermoelasticity

(3) The expressions for studied fields in the context of thefractional order generalized thermoelasticity theorywith initial stress can be deduced by setting 119867

0= 0

in (12)

7 Numerical Inversion ofthe Laplace Transforms

We will now outline the numerical inversion method usedto find the solution in the physical domain The inversionformula of the Laplace transform is defined as

119891 (119905) =1

2120587120580int

119888+120580infin

119888minus120580infin

119890119904119905119891 (119904) 119889119904 (55)

where 119891(119904) is the Laplace transform of function 119891(119905)In order to invert the Laplace transforms in the equations

given in Section 5 we apply a numerical inversion methodbased on the Fourier series expansion explained byHonig andHirdes [40] In this method the inverse transform119891(119905) of theLaplace transform 119891(119904) is approximated by the relation as

119891 (119905) =119890119888119905

1199051

[1

2119891 (119888) + Re

119873

sum

119896=1

1198901205801198961205871199051199051119891(119888 +

120580119896120587

1199051

)]

0 le 119905 lt 21199051

(56)

where 119873 is a sufficiently large integer representing thenumber of terms in the truncated Fourier series chosen suchthat

119890119888119905 Re [1198901205801198731205871199051199051119891(119888 +

120580119873120587

1199051

)] le 1205761 (57)

where 1205761is a prescribed small positive value that corresponds

to the degree of accuracy to be achieved and 119888 is a positiveconstant and must be greater than the real parts of all thesingularities of 119891(119904) The optimal choice of 119888 was obtainedaccording to the criteria described by Honig and Hirdes [40]

8 Numerical Results and Discussions

To illustrate and compare the theoretical results obtainedin the Section 5 we now present some numerical results

Journal of Mathematics 7

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

119906

120572 = 05

120572 = 1

Figure 1 Displacement distribution 119906 for different values of 120572 at119905 = 01

which depict the variations of displacement temperatureand stress component The material chosen for the purposeof numerical evaluations is copper for which we take thefollowing values of the different physical constants

119864 = 369 times 1010 kgmminus1 sminus2 120576

0= (10

minus936120587) Fmminus1 119896 =

386Wmminus1 Kminus1 1198790= 293K 120588 = 8954 kgmminus3 120591

0= 002 s

V = 033 120572119905= 178 times 10

minus5 Kminus1 119888119864= 3831 Jkgminus1 Kminus1 119867

0=

(1074120587)Amminus1 120583

0= 412058710

minus7Hmminus1The general Lame constants 120582 and 120583 are given as

120582 =119864V

120578 (1 + V) (1 minus 2V) 120583 =

119864

2120578 (1 + V) (58)

where 120578 is the initial stress parameter 119864 is Youngrsquos modulusand V is Poisson ratio For isotropic elastic medium with noinitial stress we take 120578 = 1

The computations are carried out for 119905 = 01 1199050= 05

120572 = 05 and 120579lowast= 1 The numerical technique outlined in

previous section was used to invert the Laplace transform in(53) providing the displacement temperature and stress dis-tributions in the physical domainThe results are representedgraphically for different positions of 119909

Figures 1 2 and 3 exhibit the space variations of the fieldquantities in the context of fractional order theory of therm-oelasticity with magnetic field and initial stress for differentvalues of fractional parameter 120572

Figure 1 displays the variations of displacement compo-nent 119906 for different values of 120572 and it is noticed that themagnitude of displacement component 119906 decreases with theincrease in the value of fractional parameter 120572 In both thecases (ie 120572 = 05 and 120572 = 10) the displacement componentattains maximum value at the boundary of half-space andthen continuously decreases to zero Hence displacementcomponent 119906 has similar trend for both the values of 120572

minus05

0

05

1

15

2

25

0 04 08 12 16 2119909

120579

120572 = 05

120572 = 1

Figure 2 Temperature distribution 120579 for different values of 120572 at 119905 =01

0

05

1

15

2

25

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

120590

120572 = 05

120572 = 1

119909

Figure 3 Stress distribution 120590 for different values of 120572 at 119905 = 01

Figure 2 depicts the variations of temperature 120579 withdistance 119909 for different values of 120572 and it is noticed that inboth the cases (ie 120572 = 05 and 120572 = 10) maximum value of 120579is 2 which is on the boundary of half-space We observe fromthe figure that the difference is negligible in the beginningandwith the increase in 119909 the difference ismuch pronouncedup to 119909 le 14 both the series approach to zero The trends ofboth the series are alike only up to 119909 = 08

Figure 3 shows the variations of stress component 120590 withdistance119909 for different values of120572 It is evident from the figurethat both the series have similar trend that is first increasesto a maximum value and then decreases to a minimumvalue The value of 120590 increases with the increase of fractionalparameter when 0 le 119909 le 08 while the trend of change is

8 Journal of Mathematics

minus03

minus02

minus01

0

01

02

03

04

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

119906

Figure 4 Displacement distribution 119906 at 120572 = 05 119905 = 01

minus05

0

05

1

15

2

25

0 04 08 12 16 2

ISMTMTIST

119909

120579

Figure 5 Temperature distribution 120579 at 120572 = 05 119905 = 01

adverse when 08 le 119909 le 6 The difference is significant in therange 0 le 119909 le 28

Figures 4 5 and 6 depict the variations of displacementcomponent temperature and stress component for three dif-ferent cases defined as (a) ISMT (initially stressed magneto-thermoelasticity) 120578 = 25 119901 = 1 (b) MT (magneto-thermoelasticity) 120578 = 1 119901 = 0 and (c) IST (initially stressedthermoelasticity) 120578 = 25 119901 = 1 119867

0= 0 In all the above

cases (a) (b) and (c) the value of fractional order parameteris taken to be 120572 = 05

Figure 4 shows the variations of displacement componentunder the three cases discussed in (a) (b) and (c) It isobserved that the displacement component in case of ISMTand IST has maximum positive values but in case of MT

0

02

04

06

08

1

12

14

16

18

2

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

120590

Figure 6 Stress distribution 120590 at 120572 = 05 119905 = 01

the displacement component has maximum negative valueon the boundary of half-space The trend for displacementcomponent in case of ISMT and IST is same that is both theseries are continuously decreasing to zero but in case of MTthe series rapidly approaches to zero Figure 4 also exhibitsthat in the absence of magnetic field (IST) and initial stress(MT) decreases the value of displacement It is also observedthat as compared to ISMT the difference is significant in caseof IST but much pronounced in case of MT

Figure 5 displays the variations of temperature under theISMT MT and IST theories It is observed that the trend ofthe series in case of ISMT MT and IST is similar and thedifference is significant It is found that the absence of initialstress (MT) increases the value of temperature componentbut the absence of magnetic field (IST) decreases the value oftemperature as compared to the general case (ISMT)

Figure 6 exhibits the variations of stress under ISMTMT and IST theories We found that the behavior of stresscomponent in all the three cases is alike It also observedthat as compared to ISMT theory the stress component haslarge values in IST theory but has small values in MT theoryThe difference is significant in both IST and MT theoriescompared with ISMT theory but the difference is muchpronounced in MT theory as compared to IST theory

Figures 7 8 and 9 exhibit the variations of displacementcomponent temperature and stress component in the con-text of the fractional order theory of thermoelasticity withmagnetic field and initial stress for different values of rampparameterThe values of ramp parameter are taken as 01 03and 05 and the value of fractional order parameter 120572 is takento be 05 for all the three cases

Figure 7 demonstrates that as we increase the value oframp parameter then the value of displacement componentalso increases It is also observed that trend for all the cases is

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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

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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

4 Journal of Mathematics

we obtain

1205972120590

1205971199092= 12057211199042119890 (20)

1205972120579

1205971199092= (119904 + 120591

lowast

0119904120572+1

) (120579 + 1205731119890) (21)

120590 = 119890 minus 1205761120579 minus

1205751

119904 (22)

Eliminating the value of 119890 from (20) and (21) by using(22) we obtain

1205972120579

1205971199092= 1198711120579 + 1198712120590 + 1198713

1205972120590

1205971199092= 1198721120579 +119872

2120590 +119872

3

(23)

where

1198711= (1 + 120573

11205761) (119904 + 120591

lowast

0119904120572+1

) 1198712= 1205731(119904 + 120591

lowast

0119904120572+1

)

1198713= 12057311205751(1 + 120591

lowast

0119904120572)

1198721= 120572112057611199042 119872

2= 12057211199042 119872

3= 12057211205751119904

(24)

4 State-Space Formulation

Now choosing the temperature of heat conduction 120579 and thestress component 120590 in 119909-direction as state variables one canwrite (23) in matrix form as

1198892

1198891199092119881 (119909 119904) = 119860 (119904) 119881 (119909 119904) + 119862 (119909 119904) (25)

where

119860 (119904) = [1198711

1198712

11987211198722

] 119881 (119909 119904) = [120579 (119909 119904)

120590 (119909 119904)]

119862 (119909 119904) = [1198713

1198723

]

(26)

The formal solution of (25) can be written in the form

119881 (119909 119904) = exp [minusradic119860 (119904)119909] [119881 (0 119904) + 119862 (0 119904) 119860minus1(119904)]

minus 119862 (119909 119904) 119860minus1(119904)

(27)

where

119881 (0 119904) = [120579 (0 119904)

120590 (0 119904)] = [

1205790

1205900

] (28)

In the above solution we have cancelled the part ofexponential having positive power to get bounded solutionfor large 119909

Nowwewill use theCayley-Hamilton theorem to find theform of the matrix exp[minusradic119860(119904)119909] and for this we proceedas follows

The characteristic equation of the matrix119860(119904) is obtainedas

1205822minus (1198711+1198722) 120582 + (119871

11198722minus11987211198712) = 0 (29)

where the roots of (29) namely 1205821 and 120582

2satisfy the fol-

lowing relations

1205821+ 1205822= 1198711+1198722

12058211205822= 11987111198722minus 11987121198721

(30)

Now we write the spectral decomposition of matrix 119860(119904)as

119860 (119904) = 1205821119864 + 120582

2119865 (31)

where 119864 and 119865 are called the projectors of119860(119904) and satisfy thefollowing conditions

119864 + 119865 = 119868 (32a)

119864119865 = 119865119864 = 119874 (32b)

1198642= 119864 119865

2= 119865 (32c)

Then we have

radic119860 (119904) = radic1205821119864 + radic120582

2119865 (33)

where

119864 = minus1

1205821minus 1205822

[1205822minus 1198711

minus1198712

minus1198721

1205822minus1198722

]

119865 =1

1205821minus 1205822

[1205821minus 1198711

minus1198712

minus1198721

1205821minus1198722

]

119868 = [1 0

0 1] 119874 = [

0 0

0 0]

(34)

Finally we get

119861 (119904) = radic119860 (119904) =1

radic1205821+ radic1205822

[radic12058211205822+ 1198711

1198712

1198721

radic12058211205822+1198722

]

(35)

The characteristic equation of matrix 119861(119904) can be writtenas

1198962minus 119896 (radic120582

1+ radic1205822) + radic120582

1radic1205822= 0 (36)

where the roots of (36) namely 1198961and 1198962 can be written as

1198961= radic120582

1 119896

2= radic120582

2 (37)

Now the Taylor series expansion of exp[minusradic119860(119904)119909] yields

exp [minusradic119860 (119904)119909] = exp [minus119861 (119904) 119909] =infin

sum

119899=0

[minus119861 (119904) 119909]119899

119899 (38)

Journal of Mathematics 5

Using the Cayley-Hamilton theorem we can express 1198612and higher orders of matrix 119861 in terms of 119868 and 119861 where 119868is the identity matrix of second order Bahar and Hetnarski[39]

Therefore the infinite series in (38) can be expressed as

119871 (119909 119904) = exp [minus119861 (119904) 119909] = 1198870119868 + 1198871119861 (119904) (39)

where 1198870and 1198871are coefficients depending on 119909 and 119904 By the

Cayley-Hamilton theorem the characteristic roots 1198961and 1198962

of the matrix 119861must satisfy (39) so we have

exp [minus1198961119909] = 119887

0+ 11988711198961

exp [minus1198962119909] = 119887

0+ 11988711198962

(40)

By solving the above system of equations and using (37)we get

1198870=radic1205821119890minusradic1205822119909 minus radic120582

2119890minusradic1205821119909

radic1205821minus radic1205822

1198871=119890minusradic1205821119909 minus 119890

minusradic1205822119909

radic1205821minus radic1205822

(41)

Plugging the values of 1198870and 1198871in (39) we have

exp [minus119861 (119904) 119909] = 119871 (119909 119904) = [119897119894119895] 119894 119895 = 1 2 (42)

where the entries 119897119894119895(119909 119904) are given as

11989711=(1205821minus 1198711) 119890minusradic1205822119909 minus (120582

2minus 1198711) 119890minusradic1205821119909

1205821minus 1205822

11989712= 1198712(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989721= 1198721(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989722=(1205821minus1198722) 119890minusradic1205822119909 minus (120582

2minus1198722) 119890minusradic1205821119909

1205821minus 1205822

(43)

The solution of (25) can be written in the following form

119881 (119909 119904) = [119897119894119895] [119881 (0 119904) + 119860

minus1(119904) 119862 (0 119904)] minus 119860

minus1(119904) 119862 (119909 119904)

(44)

Substituting the values of119881(119909 119904)119860minus1(119904) and 119862(119909 119904) into(44) and performing the necessary matrix operations weobtain the values of 120579(119909 119904) and 120590(119909 119904) as

120579 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205822119909

minus (1205822minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205821119909]

minus 1205781

(45)

120590 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205822119909

minus (1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781)

times119890minusradic1205821119909] minus 120578

2

(46)

where

1205781=11987221198713minus 11987121198723

11987111198722minus11987211198712

1205782=11987111198723minus11987211198713

11987111198722minus11987211198712

(47)

Now considering (12) along with (15) and the Laplacetransform the displacement component is evaluated as

119906 (119909 119904) =1

12057211199042

120597120590

120597119909 (48)

Substituting (46) into (48) we get

119906 (119909 119904)

=1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205900+1205782)minus1198721(1205790+1205781) 119890minusradic1205822119909]

(49)

5 Application

Problem the ramp type boundary temperature of an elastichalf-space

We consider a homogeneous isotropic thermoelastic solidoccupying the half-space 119909 ge 0 The boundary of half-space119909 = 0 is affected by ramp type heating In mathematicalnotations the boundary conditions can be denoted as

120590 (0 119905) = 1205900= minus119901

120579 (0 119905) = 1205790= 120579lowastℎ (119905)

(50)

6 Journal of Mathematics

where 120579lowast is constant temperature and ℎ(119905) is defined as

ℎ (119905) =

0 119905 le 0

119905

1199050

0 lt 119905 le 1199050

1 119905 gt 1199050

(51)

where 1199050is ramping parameter

We employ nondimensional variables given in (15) on(50) and taking the Laplace transform we get

120590 (0 119904) = 1205900= minus

1205751

119904

120579 (0 119904) = 1205790= 120579lowast(1 minus 1198901199041199050

11990501199042

)

(52)

Substituting the values of 1205900and 120579

0from (52) into (45)-

(46) and (49) we find

120579 (119909 119904) =1

1205821minus 1205822

times [(1205821minus 1198711) (120579lowast120578lowast+ 1205781) minus 1198712(1205782minus1205751

119904)

times119890minusradic1205822119909 minus (120582

2minus 1198711) (120579lowast120578lowast+ 1205781)

minus1198712(1205782minus1205751

119904) 119890minusradic1205821119909] minus 120578

1

120590 (119909 119904) =1

1205821minus 1205822

times [(1205821minus1198722) (1205782minus1205751

119904) minus119872

1(120579lowast120578lowast+ 1205781)

times 119890minusradic1205822119909 minus (120582

2minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909]

minus 1205782

119906 (119909 119904) =1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205822119909]

(53)

where

120578lowast=1 minus 119890minus1199041199050

11990501199042

(54)

6 Limiting Cases

(1) The mathematical expressions for the field variablesstudied in the context of generalized magneto-ther-moelasticity theory with initial stress under appliedboundary condition can be obtained by applying 120572 =

1 in (13)(2) Neglecting initial stress effect by substituting 119901 = 0

in (14) and 120578 = 1 in (58) we obtain the expressionsof field variables under fractional order generalizedmagneto-thermoelasticity

(3) The expressions for studied fields in the context of thefractional order generalized thermoelasticity theorywith initial stress can be deduced by setting 119867

0= 0

in (12)

7 Numerical Inversion ofthe Laplace Transforms

We will now outline the numerical inversion method usedto find the solution in the physical domain The inversionformula of the Laplace transform is defined as

119891 (119905) =1

2120587120580int

119888+120580infin

119888minus120580infin

119890119904119905119891 (119904) 119889119904 (55)

where 119891(119904) is the Laplace transform of function 119891(119905)In order to invert the Laplace transforms in the equations

given in Section 5 we apply a numerical inversion methodbased on the Fourier series expansion explained byHonig andHirdes [40] In this method the inverse transform119891(119905) of theLaplace transform 119891(119904) is approximated by the relation as

119891 (119905) =119890119888119905

1199051

[1

2119891 (119888) + Re

119873

sum

119896=1

1198901205801198961205871199051199051119891(119888 +

120580119896120587

1199051

)]

0 le 119905 lt 21199051

(56)

where 119873 is a sufficiently large integer representing thenumber of terms in the truncated Fourier series chosen suchthat

119890119888119905 Re [1198901205801198731205871199051199051119891(119888 +

120580119873120587

1199051

)] le 1205761 (57)

where 1205761is a prescribed small positive value that corresponds

to the degree of accuracy to be achieved and 119888 is a positiveconstant and must be greater than the real parts of all thesingularities of 119891(119904) The optimal choice of 119888 was obtainedaccording to the criteria described by Honig and Hirdes [40]

8 Numerical Results and Discussions

To illustrate and compare the theoretical results obtainedin the Section 5 we now present some numerical results

Journal of Mathematics 7

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

119906

120572 = 05

120572 = 1

Figure 1 Displacement distribution 119906 for different values of 120572 at119905 = 01

which depict the variations of displacement temperatureand stress component The material chosen for the purposeof numerical evaluations is copper for which we take thefollowing values of the different physical constants

119864 = 369 times 1010 kgmminus1 sminus2 120576

0= (10

minus936120587) Fmminus1 119896 =

386Wmminus1 Kminus1 1198790= 293K 120588 = 8954 kgmminus3 120591

0= 002 s

V = 033 120572119905= 178 times 10

minus5 Kminus1 119888119864= 3831 Jkgminus1 Kminus1 119867

0=

(1074120587)Amminus1 120583

0= 412058710

minus7Hmminus1The general Lame constants 120582 and 120583 are given as

120582 =119864V

120578 (1 + V) (1 minus 2V) 120583 =

119864

2120578 (1 + V) (58)

where 120578 is the initial stress parameter 119864 is Youngrsquos modulusand V is Poisson ratio For isotropic elastic medium with noinitial stress we take 120578 = 1

The computations are carried out for 119905 = 01 1199050= 05

120572 = 05 and 120579lowast= 1 The numerical technique outlined in

previous section was used to invert the Laplace transform in(53) providing the displacement temperature and stress dis-tributions in the physical domainThe results are representedgraphically for different positions of 119909

Figures 1 2 and 3 exhibit the space variations of the fieldquantities in the context of fractional order theory of therm-oelasticity with magnetic field and initial stress for differentvalues of fractional parameter 120572

Figure 1 displays the variations of displacement compo-nent 119906 for different values of 120572 and it is noticed that themagnitude of displacement component 119906 decreases with theincrease in the value of fractional parameter 120572 In both thecases (ie 120572 = 05 and 120572 = 10) the displacement componentattains maximum value at the boundary of half-space andthen continuously decreases to zero Hence displacementcomponent 119906 has similar trend for both the values of 120572

minus05

0

05

1

15

2

25

0 04 08 12 16 2119909

120579

120572 = 05

120572 = 1

Figure 2 Temperature distribution 120579 for different values of 120572 at 119905 =01

0

05

1

15

2

25

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

120590

120572 = 05

120572 = 1

119909

Figure 3 Stress distribution 120590 for different values of 120572 at 119905 = 01

Figure 2 depicts the variations of temperature 120579 withdistance 119909 for different values of 120572 and it is noticed that inboth the cases (ie 120572 = 05 and 120572 = 10) maximum value of 120579is 2 which is on the boundary of half-space We observe fromthe figure that the difference is negligible in the beginningandwith the increase in 119909 the difference ismuch pronouncedup to 119909 le 14 both the series approach to zero The trends ofboth the series are alike only up to 119909 = 08

Figure 3 shows the variations of stress component 120590 withdistance119909 for different values of120572 It is evident from the figurethat both the series have similar trend that is first increasesto a maximum value and then decreases to a minimumvalue The value of 120590 increases with the increase of fractionalparameter when 0 le 119909 le 08 while the trend of change is

8 Journal of Mathematics

minus03

minus02

minus01

0

01

02

03

04

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

119906

Figure 4 Displacement distribution 119906 at 120572 = 05 119905 = 01

minus05

0

05

1

15

2

25

0 04 08 12 16 2

ISMTMTIST

119909

120579

Figure 5 Temperature distribution 120579 at 120572 = 05 119905 = 01

adverse when 08 le 119909 le 6 The difference is significant in therange 0 le 119909 le 28

Figures 4 5 and 6 depict the variations of displacementcomponent temperature and stress component for three dif-ferent cases defined as (a) ISMT (initially stressed magneto-thermoelasticity) 120578 = 25 119901 = 1 (b) MT (magneto-thermoelasticity) 120578 = 1 119901 = 0 and (c) IST (initially stressedthermoelasticity) 120578 = 25 119901 = 1 119867

0= 0 In all the above

cases (a) (b) and (c) the value of fractional order parameteris taken to be 120572 = 05

Figure 4 shows the variations of displacement componentunder the three cases discussed in (a) (b) and (c) It isobserved that the displacement component in case of ISMTand IST has maximum positive values but in case of MT

0

02

04

06

08

1

12

14

16

18

2

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

120590

Figure 6 Stress distribution 120590 at 120572 = 05 119905 = 01

the displacement component has maximum negative valueon the boundary of half-space The trend for displacementcomponent in case of ISMT and IST is same that is both theseries are continuously decreasing to zero but in case of MTthe series rapidly approaches to zero Figure 4 also exhibitsthat in the absence of magnetic field (IST) and initial stress(MT) decreases the value of displacement It is also observedthat as compared to ISMT the difference is significant in caseof IST but much pronounced in case of MT

Figure 5 displays the variations of temperature under theISMT MT and IST theories It is observed that the trend ofthe series in case of ISMT MT and IST is similar and thedifference is significant It is found that the absence of initialstress (MT) increases the value of temperature componentbut the absence of magnetic field (IST) decreases the value oftemperature as compared to the general case (ISMT)

Figure 6 exhibits the variations of stress under ISMTMT and IST theories We found that the behavior of stresscomponent in all the three cases is alike It also observedthat as compared to ISMT theory the stress component haslarge values in IST theory but has small values in MT theoryThe difference is significant in both IST and MT theoriescompared with ISMT theory but the difference is muchpronounced in MT theory as compared to IST theory

Figures 7 8 and 9 exhibit the variations of displacementcomponent temperature and stress component in the con-text of the fractional order theory of thermoelasticity withmagnetic field and initial stress for different values of rampparameterThe values of ramp parameter are taken as 01 03and 05 and the value of fractional order parameter 120572 is takento be 05 for all the three cases

Figure 7 demonstrates that as we increase the value oframp parameter then the value of displacement componentalso increases It is also observed that trend for all the cases is

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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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

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 5

Using the Cayley-Hamilton theorem we can express 1198612and higher orders of matrix 119861 in terms of 119868 and 119861 where 119868is the identity matrix of second order Bahar and Hetnarski[39]

Therefore the infinite series in (38) can be expressed as

119871 (119909 119904) = exp [minus119861 (119904) 119909] = 1198870119868 + 1198871119861 (119904) (39)

where 1198870and 1198871are coefficients depending on 119909 and 119904 By the

Cayley-Hamilton theorem the characteristic roots 1198961and 1198962

of the matrix 119861must satisfy (39) so we have

exp [minus1198961119909] = 119887

0+ 11988711198961

exp [minus1198962119909] = 119887

0+ 11988711198962

(40)

By solving the above system of equations and using (37)we get

1198870=radic1205821119890minusradic1205822119909 minus radic120582

2119890minusradic1205821119909

radic1205821minus radic1205822

1198871=119890minusradic1205821119909 minus 119890

minusradic1205822119909

radic1205821minus radic1205822

(41)

Plugging the values of 1198870and 1198871in (39) we have

exp [minus119861 (119904) 119909] = 119871 (119909 119904) = [119897119894119895] 119894 119895 = 1 2 (42)

where the entries 119897119894119895(119909 119904) are given as

11989711=(1205821minus 1198711) 119890minusradic1205822119909 minus (120582

2minus 1198711) 119890minusradic1205821119909

1205821minus 1205822

11989712= 1198712(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989721= 1198721(119890minusradic1205821119909 minus 119890

minusradic1205822119909

1205821minus 1205822

)

11989722=(1205821minus1198722) 119890minusradic1205822119909 minus (120582

2minus1198722) 119890minusradic1205821119909

1205821minus 1205822

(43)

The solution of (25) can be written in the following form

119881 (119909 119904) = [119897119894119895] [119881 (0 119904) + 119860

minus1(119904) 119862 (0 119904)] minus 119860

minus1(119904) 119862 (119909 119904)

(44)

Substituting the values of119881(119909 119904)119860minus1(119904) and 119862(119909 119904) into(44) and performing the necessary matrix operations weobtain the values of 120579(119909 119904) and 120590(119909 119904) as

120579 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205822119909

minus (1205822minus 1198711) (1205790+ 1205781) minus 1198712(1205900+ 1205782) 119890minusradic1205821119909]

minus 1205781

(45)

120590 (119909 119904) =1

1205821minus 1205822

times [ (1205821minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205822119909

minus (1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781)

times119890minusradic1205821119909] minus 120578

2

(46)

where

1205781=11987221198713minus 11987121198723

11987111198722minus11987211198712

1205782=11987111198723minus11987211198713

11987111198722minus11987211198712

(47)

Now considering (12) along with (15) and the Laplacetransform the displacement component is evaluated as

119906 (119909 119904) =1

12057211199042

120597120590

120597119909 (48)

Substituting (46) into (48) we get

119906 (119909 119904)

=1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205900+ 1205782) minus 119872

1(1205790+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205900+1205782)minus1198721(1205790+1205781) 119890minusradic1205822119909]

(49)

5 Application

Problem the ramp type boundary temperature of an elastichalf-space

We consider a homogeneous isotropic thermoelastic solidoccupying the half-space 119909 ge 0 The boundary of half-space119909 = 0 is affected by ramp type heating In mathematicalnotations the boundary conditions can be denoted as

120590 (0 119905) = 1205900= minus119901

120579 (0 119905) = 1205790= 120579lowastℎ (119905)

(50)

6 Journal of Mathematics

where 120579lowast is constant temperature and ℎ(119905) is defined as

ℎ (119905) =

0 119905 le 0

119905

1199050

0 lt 119905 le 1199050

1 119905 gt 1199050

(51)

where 1199050is ramping parameter

We employ nondimensional variables given in (15) on(50) and taking the Laplace transform we get

120590 (0 119904) = 1205900= minus

1205751

119904

120579 (0 119904) = 1205790= 120579lowast(1 minus 1198901199041199050

11990501199042

)

(52)

Substituting the values of 1205900and 120579

0from (52) into (45)-

(46) and (49) we find

120579 (119909 119904) =1

1205821minus 1205822

times [(1205821minus 1198711) (120579lowast120578lowast+ 1205781) minus 1198712(1205782minus1205751

119904)

times119890minusradic1205822119909 minus (120582

2minus 1198711) (120579lowast120578lowast+ 1205781)

minus1198712(1205782minus1205751

119904) 119890minusradic1205821119909] minus 120578

1

120590 (119909 119904) =1

1205821minus 1205822

times [(1205821minus1198722) (1205782minus1205751

119904) minus119872

1(120579lowast120578lowast+ 1205781)

times 119890minusradic1205822119909 minus (120582

2minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909]

minus 1205782

119906 (119909 119904) =1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205822119909]

(53)

where

120578lowast=1 minus 119890minus1199041199050

11990501199042

(54)

6 Limiting Cases

(1) The mathematical expressions for the field variablesstudied in the context of generalized magneto-ther-moelasticity theory with initial stress under appliedboundary condition can be obtained by applying 120572 =

1 in (13)(2) Neglecting initial stress effect by substituting 119901 = 0

in (14) and 120578 = 1 in (58) we obtain the expressionsof field variables under fractional order generalizedmagneto-thermoelasticity

(3) The expressions for studied fields in the context of thefractional order generalized thermoelasticity theorywith initial stress can be deduced by setting 119867

0= 0

in (12)

7 Numerical Inversion ofthe Laplace Transforms

We will now outline the numerical inversion method usedto find the solution in the physical domain The inversionformula of the Laplace transform is defined as

119891 (119905) =1

2120587120580int

119888+120580infin

119888minus120580infin

119890119904119905119891 (119904) 119889119904 (55)

where 119891(119904) is the Laplace transform of function 119891(119905)In order to invert the Laplace transforms in the equations

given in Section 5 we apply a numerical inversion methodbased on the Fourier series expansion explained byHonig andHirdes [40] In this method the inverse transform119891(119905) of theLaplace transform 119891(119904) is approximated by the relation as

119891 (119905) =119890119888119905

1199051

[1

2119891 (119888) + Re

119873

sum

119896=1

1198901205801198961205871199051199051119891(119888 +

120580119896120587

1199051

)]

0 le 119905 lt 21199051

(56)

where 119873 is a sufficiently large integer representing thenumber of terms in the truncated Fourier series chosen suchthat

119890119888119905 Re [1198901205801198731205871199051199051119891(119888 +

120580119873120587

1199051

)] le 1205761 (57)

where 1205761is a prescribed small positive value that corresponds

to the degree of accuracy to be achieved and 119888 is a positiveconstant and must be greater than the real parts of all thesingularities of 119891(119904) The optimal choice of 119888 was obtainedaccording to the criteria described by Honig and Hirdes [40]

8 Numerical Results and Discussions

To illustrate and compare the theoretical results obtainedin the Section 5 we now present some numerical results

Journal of Mathematics 7

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

119906

120572 = 05

120572 = 1

Figure 1 Displacement distribution 119906 for different values of 120572 at119905 = 01

which depict the variations of displacement temperatureand stress component The material chosen for the purposeof numerical evaluations is copper for which we take thefollowing values of the different physical constants

119864 = 369 times 1010 kgmminus1 sminus2 120576

0= (10

minus936120587) Fmminus1 119896 =

386Wmminus1 Kminus1 1198790= 293K 120588 = 8954 kgmminus3 120591

0= 002 s

V = 033 120572119905= 178 times 10

minus5 Kminus1 119888119864= 3831 Jkgminus1 Kminus1 119867

0=

(1074120587)Amminus1 120583

0= 412058710

minus7Hmminus1The general Lame constants 120582 and 120583 are given as

120582 =119864V

120578 (1 + V) (1 minus 2V) 120583 =

119864

2120578 (1 + V) (58)

where 120578 is the initial stress parameter 119864 is Youngrsquos modulusand V is Poisson ratio For isotropic elastic medium with noinitial stress we take 120578 = 1

The computations are carried out for 119905 = 01 1199050= 05

120572 = 05 and 120579lowast= 1 The numerical technique outlined in

previous section was used to invert the Laplace transform in(53) providing the displacement temperature and stress dis-tributions in the physical domainThe results are representedgraphically for different positions of 119909

Figures 1 2 and 3 exhibit the space variations of the fieldquantities in the context of fractional order theory of therm-oelasticity with magnetic field and initial stress for differentvalues of fractional parameter 120572

Figure 1 displays the variations of displacement compo-nent 119906 for different values of 120572 and it is noticed that themagnitude of displacement component 119906 decreases with theincrease in the value of fractional parameter 120572 In both thecases (ie 120572 = 05 and 120572 = 10) the displacement componentattains maximum value at the boundary of half-space andthen continuously decreases to zero Hence displacementcomponent 119906 has similar trend for both the values of 120572

minus05

0

05

1

15

2

25

0 04 08 12 16 2119909

120579

120572 = 05

120572 = 1

Figure 2 Temperature distribution 120579 for different values of 120572 at 119905 =01

0

05

1

15

2

25

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

120590

120572 = 05

120572 = 1

119909

Figure 3 Stress distribution 120590 for different values of 120572 at 119905 = 01

Figure 2 depicts the variations of temperature 120579 withdistance 119909 for different values of 120572 and it is noticed that inboth the cases (ie 120572 = 05 and 120572 = 10) maximum value of 120579is 2 which is on the boundary of half-space We observe fromthe figure that the difference is negligible in the beginningandwith the increase in 119909 the difference ismuch pronouncedup to 119909 le 14 both the series approach to zero The trends ofboth the series are alike only up to 119909 = 08

Figure 3 shows the variations of stress component 120590 withdistance119909 for different values of120572 It is evident from the figurethat both the series have similar trend that is first increasesto a maximum value and then decreases to a minimumvalue The value of 120590 increases with the increase of fractionalparameter when 0 le 119909 le 08 while the trend of change is

8 Journal of Mathematics

minus03

minus02

minus01

0

01

02

03

04

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

119906

Figure 4 Displacement distribution 119906 at 120572 = 05 119905 = 01

minus05

0

05

1

15

2

25

0 04 08 12 16 2

ISMTMTIST

119909

120579

Figure 5 Temperature distribution 120579 at 120572 = 05 119905 = 01

adverse when 08 le 119909 le 6 The difference is significant in therange 0 le 119909 le 28

Figures 4 5 and 6 depict the variations of displacementcomponent temperature and stress component for three dif-ferent cases defined as (a) ISMT (initially stressed magneto-thermoelasticity) 120578 = 25 119901 = 1 (b) MT (magneto-thermoelasticity) 120578 = 1 119901 = 0 and (c) IST (initially stressedthermoelasticity) 120578 = 25 119901 = 1 119867

0= 0 In all the above

cases (a) (b) and (c) the value of fractional order parameteris taken to be 120572 = 05

Figure 4 shows the variations of displacement componentunder the three cases discussed in (a) (b) and (c) It isobserved that the displacement component in case of ISMTand IST has maximum positive values but in case of MT

0

02

04

06

08

1

12

14

16

18

2

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

120590

Figure 6 Stress distribution 120590 at 120572 = 05 119905 = 01

the displacement component has maximum negative valueon the boundary of half-space The trend for displacementcomponent in case of ISMT and IST is same that is both theseries are continuously decreasing to zero but in case of MTthe series rapidly approaches to zero Figure 4 also exhibitsthat in the absence of magnetic field (IST) and initial stress(MT) decreases the value of displacement It is also observedthat as compared to ISMT the difference is significant in caseof IST but much pronounced in case of MT

Figure 5 displays the variations of temperature under theISMT MT and IST theories It is observed that the trend ofthe series in case of ISMT MT and IST is similar and thedifference is significant It is found that the absence of initialstress (MT) increases the value of temperature componentbut the absence of magnetic field (IST) decreases the value oftemperature as compared to the general case (ISMT)

Figure 6 exhibits the variations of stress under ISMTMT and IST theories We found that the behavior of stresscomponent in all the three cases is alike It also observedthat as compared to ISMT theory the stress component haslarge values in IST theory but has small values in MT theoryThe difference is significant in both IST and MT theoriescompared with ISMT theory but the difference is muchpronounced in MT theory as compared to IST theory

Figures 7 8 and 9 exhibit the variations of displacementcomponent temperature and stress component in the con-text of the fractional order theory of thermoelasticity withmagnetic field and initial stress for different values of rampparameterThe values of ramp parameter are taken as 01 03and 05 and the value of fractional order parameter 120572 is takento be 05 for all the three cases

Figure 7 demonstrates that as we increase the value oframp parameter then the value of displacement componentalso increases It is also observed that trend for all the cases is

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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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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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

6 Journal of Mathematics

where 120579lowast is constant temperature and ℎ(119905) is defined as

ℎ (119905) =

0 119905 le 0

119905

1199050

0 lt 119905 le 1199050

1 119905 gt 1199050

(51)

where 1199050is ramping parameter

We employ nondimensional variables given in (15) on(50) and taking the Laplace transform we get

120590 (0 119904) = 1205900= minus

1205751

119904

120579 (0 119904) = 1205790= 120579lowast(1 minus 1198901199041199050

11990501199042

)

(52)

Substituting the values of 1205900and 120579

0from (52) into (45)-

(46) and (49) we find

120579 (119909 119904) =1

1205821minus 1205822

times [(1205821minus 1198711) (120579lowast120578lowast+ 1205781) minus 1198712(1205782minus1205751

119904)

times119890minusradic1205822119909 minus (120582

2minus 1198711) (120579lowast120578lowast+ 1205781)

minus1198712(1205782minus1205751

119904) 119890minusradic1205821119909] minus 120578

1

120590 (119909 119904) =1

1205821minus 1205822

times [(1205821minus1198722) (1205782minus1205751

119904) minus119872

1(120579lowast120578lowast+ 1205781)

times 119890minusradic1205822119909 minus (120582

2minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909]

minus 1205782

119906 (119909 119904) =1

12057211199042 (1205821minus 1205822)

times [radic1205821(1205822minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205821119909

minus radic1205822(1205821minus1198722) (1205782minus1205751

119904)

minus1198721(120579lowast120578lowast+ 1205781) 119890minusradic1205822119909]

(53)

where

120578lowast=1 minus 119890minus1199041199050

11990501199042

(54)

6 Limiting Cases

(1) The mathematical expressions for the field variablesstudied in the context of generalized magneto-ther-moelasticity theory with initial stress under appliedboundary condition can be obtained by applying 120572 =

1 in (13)(2) Neglecting initial stress effect by substituting 119901 = 0

in (14) and 120578 = 1 in (58) we obtain the expressionsof field variables under fractional order generalizedmagneto-thermoelasticity

(3) The expressions for studied fields in the context of thefractional order generalized thermoelasticity theorywith initial stress can be deduced by setting 119867

0= 0

in (12)

7 Numerical Inversion ofthe Laplace Transforms

We will now outline the numerical inversion method usedto find the solution in the physical domain The inversionformula of the Laplace transform is defined as

119891 (119905) =1

2120587120580int

119888+120580infin

119888minus120580infin

119890119904119905119891 (119904) 119889119904 (55)

where 119891(119904) is the Laplace transform of function 119891(119905)In order to invert the Laplace transforms in the equations

given in Section 5 we apply a numerical inversion methodbased on the Fourier series expansion explained byHonig andHirdes [40] In this method the inverse transform119891(119905) of theLaplace transform 119891(119904) is approximated by the relation as

119891 (119905) =119890119888119905

1199051

[1

2119891 (119888) + Re

119873

sum

119896=1

1198901205801198961205871199051199051119891(119888 +

120580119896120587

1199051

)]

0 le 119905 lt 21199051

(56)

where 119873 is a sufficiently large integer representing thenumber of terms in the truncated Fourier series chosen suchthat

119890119888119905 Re [1198901205801198731205871199051199051119891(119888 +

120580119873120587

1199051

)] le 1205761 (57)

where 1205761is a prescribed small positive value that corresponds

to the degree of accuracy to be achieved and 119888 is a positiveconstant and must be greater than the real parts of all thesingularities of 119891(119904) The optimal choice of 119888 was obtainedaccording to the criteria described by Honig and Hirdes [40]

8 Numerical Results and Discussions

To illustrate and compare the theoretical results obtainedin the Section 5 we now present some numerical results

Journal of Mathematics 7

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

119906

120572 = 05

120572 = 1

Figure 1 Displacement distribution 119906 for different values of 120572 at119905 = 01

which depict the variations of displacement temperatureand stress component The material chosen for the purposeof numerical evaluations is copper for which we take thefollowing values of the different physical constants

119864 = 369 times 1010 kgmminus1 sminus2 120576

0= (10

minus936120587) Fmminus1 119896 =

386Wmminus1 Kminus1 1198790= 293K 120588 = 8954 kgmminus3 120591

0= 002 s

V = 033 120572119905= 178 times 10

minus5 Kminus1 119888119864= 3831 Jkgminus1 Kminus1 119867

0=

(1074120587)Amminus1 120583

0= 412058710

minus7Hmminus1The general Lame constants 120582 and 120583 are given as

120582 =119864V

120578 (1 + V) (1 minus 2V) 120583 =

119864

2120578 (1 + V) (58)

where 120578 is the initial stress parameter 119864 is Youngrsquos modulusand V is Poisson ratio For isotropic elastic medium with noinitial stress we take 120578 = 1

The computations are carried out for 119905 = 01 1199050= 05

120572 = 05 and 120579lowast= 1 The numerical technique outlined in

previous section was used to invert the Laplace transform in(53) providing the displacement temperature and stress dis-tributions in the physical domainThe results are representedgraphically for different positions of 119909

Figures 1 2 and 3 exhibit the space variations of the fieldquantities in the context of fractional order theory of therm-oelasticity with magnetic field and initial stress for differentvalues of fractional parameter 120572

Figure 1 displays the variations of displacement compo-nent 119906 for different values of 120572 and it is noticed that themagnitude of displacement component 119906 decreases with theincrease in the value of fractional parameter 120572 In both thecases (ie 120572 = 05 and 120572 = 10) the displacement componentattains maximum value at the boundary of half-space andthen continuously decreases to zero Hence displacementcomponent 119906 has similar trend for both the values of 120572

minus05

0

05

1

15

2

25

0 04 08 12 16 2119909

120579

120572 = 05

120572 = 1

Figure 2 Temperature distribution 120579 for different values of 120572 at 119905 =01

0

05

1

15

2

25

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

120590

120572 = 05

120572 = 1

119909

Figure 3 Stress distribution 120590 for different values of 120572 at 119905 = 01

Figure 2 depicts the variations of temperature 120579 withdistance 119909 for different values of 120572 and it is noticed that inboth the cases (ie 120572 = 05 and 120572 = 10) maximum value of 120579is 2 which is on the boundary of half-space We observe fromthe figure that the difference is negligible in the beginningandwith the increase in 119909 the difference ismuch pronouncedup to 119909 le 14 both the series approach to zero The trends ofboth the series are alike only up to 119909 = 08

Figure 3 shows the variations of stress component 120590 withdistance119909 for different values of120572 It is evident from the figurethat both the series have similar trend that is first increasesto a maximum value and then decreases to a minimumvalue The value of 120590 increases with the increase of fractionalparameter when 0 le 119909 le 08 while the trend of change is

8 Journal of Mathematics

minus03

minus02

minus01

0

01

02

03

04

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

119906

Figure 4 Displacement distribution 119906 at 120572 = 05 119905 = 01

minus05

0

05

1

15

2

25

0 04 08 12 16 2

ISMTMTIST

119909

120579

Figure 5 Temperature distribution 120579 at 120572 = 05 119905 = 01

adverse when 08 le 119909 le 6 The difference is significant in therange 0 le 119909 le 28

Figures 4 5 and 6 depict the variations of displacementcomponent temperature and stress component for three dif-ferent cases defined as (a) ISMT (initially stressed magneto-thermoelasticity) 120578 = 25 119901 = 1 (b) MT (magneto-thermoelasticity) 120578 = 1 119901 = 0 and (c) IST (initially stressedthermoelasticity) 120578 = 25 119901 = 1 119867

0= 0 In all the above

cases (a) (b) and (c) the value of fractional order parameteris taken to be 120572 = 05

Figure 4 shows the variations of displacement componentunder the three cases discussed in (a) (b) and (c) It isobserved that the displacement component in case of ISMTand IST has maximum positive values but in case of MT

0

02

04

06

08

1

12

14

16

18

2

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

120590

Figure 6 Stress distribution 120590 at 120572 = 05 119905 = 01

the displacement component has maximum negative valueon the boundary of half-space The trend for displacementcomponent in case of ISMT and IST is same that is both theseries are continuously decreasing to zero but in case of MTthe series rapidly approaches to zero Figure 4 also exhibitsthat in the absence of magnetic field (IST) and initial stress(MT) decreases the value of displacement It is also observedthat as compared to ISMT the difference is significant in caseof IST but much pronounced in case of MT

Figure 5 displays the variations of temperature under theISMT MT and IST theories It is observed that the trend ofthe series in case of ISMT MT and IST is similar and thedifference is significant It is found that the absence of initialstress (MT) increases the value of temperature componentbut the absence of magnetic field (IST) decreases the value oftemperature as compared to the general case (ISMT)

Figure 6 exhibits the variations of stress under ISMTMT and IST theories We found that the behavior of stresscomponent in all the three cases is alike It also observedthat as compared to ISMT theory the stress component haslarge values in IST theory but has small values in MT theoryThe difference is significant in both IST and MT theoriescompared with ISMT theory but the difference is muchpronounced in MT theory as compared to IST theory

Figures 7 8 and 9 exhibit the variations of displacementcomponent temperature and stress component in the con-text of the fractional order theory of thermoelasticity withmagnetic field and initial stress for different values of rampparameterThe values of ramp parameter are taken as 01 03and 05 and the value of fractional order parameter 120572 is takento be 05 for all the three cases

Figure 7 demonstrates that as we increase the value oframp parameter then the value of displacement componentalso increases It is also observed that trend for all the cases is

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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

Journal of Mathematics 7

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

119906

120572 = 05

120572 = 1

Figure 1 Displacement distribution 119906 for different values of 120572 at119905 = 01

which depict the variations of displacement temperatureand stress component The material chosen for the purposeof numerical evaluations is copper for which we take thefollowing values of the different physical constants

119864 = 369 times 1010 kgmminus1 sminus2 120576

0= (10

minus936120587) Fmminus1 119896 =

386Wmminus1 Kminus1 1198790= 293K 120588 = 8954 kgmminus3 120591

0= 002 s

V = 033 120572119905= 178 times 10

minus5 Kminus1 119888119864= 3831 Jkgminus1 Kminus1 119867

0=

(1074120587)Amminus1 120583

0= 412058710

minus7Hmminus1The general Lame constants 120582 and 120583 are given as

120582 =119864V

120578 (1 + V) (1 minus 2V) 120583 =

119864

2120578 (1 + V) (58)

where 120578 is the initial stress parameter 119864 is Youngrsquos modulusand V is Poisson ratio For isotropic elastic medium with noinitial stress we take 120578 = 1

The computations are carried out for 119905 = 01 1199050= 05

120572 = 05 and 120579lowast= 1 The numerical technique outlined in

previous section was used to invert the Laplace transform in(53) providing the displacement temperature and stress dis-tributions in the physical domainThe results are representedgraphically for different positions of 119909

Figures 1 2 and 3 exhibit the space variations of the fieldquantities in the context of fractional order theory of therm-oelasticity with magnetic field and initial stress for differentvalues of fractional parameter 120572

Figure 1 displays the variations of displacement compo-nent 119906 for different values of 120572 and it is noticed that themagnitude of displacement component 119906 decreases with theincrease in the value of fractional parameter 120572 In both thecases (ie 120572 = 05 and 120572 = 10) the displacement componentattains maximum value at the boundary of half-space andthen continuously decreases to zero Hence displacementcomponent 119906 has similar trend for both the values of 120572

minus05

0

05

1

15

2

25

0 04 08 12 16 2119909

120579

120572 = 05

120572 = 1

Figure 2 Temperature distribution 120579 for different values of 120572 at 119905 =01

0

05

1

15

2

25

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

120590

120572 = 05

120572 = 1

119909

Figure 3 Stress distribution 120590 for different values of 120572 at 119905 = 01

Figure 2 depicts the variations of temperature 120579 withdistance 119909 for different values of 120572 and it is noticed that inboth the cases (ie 120572 = 05 and 120572 = 10) maximum value of 120579is 2 which is on the boundary of half-space We observe fromthe figure that the difference is negligible in the beginningandwith the increase in 119909 the difference ismuch pronouncedup to 119909 le 14 both the series approach to zero The trends ofboth the series are alike only up to 119909 = 08

Figure 3 shows the variations of stress component 120590 withdistance119909 for different values of120572 It is evident from the figurethat both the series have similar trend that is first increasesto a maximum value and then decreases to a minimumvalue The value of 120590 increases with the increase of fractionalparameter when 0 le 119909 le 08 while the trend of change is

8 Journal of Mathematics

minus03

minus02

minus01

0

01

02

03

04

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

119906

Figure 4 Displacement distribution 119906 at 120572 = 05 119905 = 01

minus05

0

05

1

15

2

25

0 04 08 12 16 2

ISMTMTIST

119909

120579

Figure 5 Temperature distribution 120579 at 120572 = 05 119905 = 01

adverse when 08 le 119909 le 6 The difference is significant in therange 0 le 119909 le 28

Figures 4 5 and 6 depict the variations of displacementcomponent temperature and stress component for three dif-ferent cases defined as (a) ISMT (initially stressed magneto-thermoelasticity) 120578 = 25 119901 = 1 (b) MT (magneto-thermoelasticity) 120578 = 1 119901 = 0 and (c) IST (initially stressedthermoelasticity) 120578 = 25 119901 = 1 119867

0= 0 In all the above

cases (a) (b) and (c) the value of fractional order parameteris taken to be 120572 = 05

Figure 4 shows the variations of displacement componentunder the three cases discussed in (a) (b) and (c) It isobserved that the displacement component in case of ISMTand IST has maximum positive values but in case of MT

0

02

04

06

08

1

12

14

16

18

2

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

120590

Figure 6 Stress distribution 120590 at 120572 = 05 119905 = 01

the displacement component has maximum negative valueon the boundary of half-space The trend for displacementcomponent in case of ISMT and IST is same that is both theseries are continuously decreasing to zero but in case of MTthe series rapidly approaches to zero Figure 4 also exhibitsthat in the absence of magnetic field (IST) and initial stress(MT) decreases the value of displacement It is also observedthat as compared to ISMT the difference is significant in caseof IST but much pronounced in case of MT

Figure 5 displays the variations of temperature under theISMT MT and IST theories It is observed that the trend ofthe series in case of ISMT MT and IST is similar and thedifference is significant It is found that the absence of initialstress (MT) increases the value of temperature componentbut the absence of magnetic field (IST) decreases the value oftemperature as compared to the general case (ISMT)

Figure 6 exhibits the variations of stress under ISMTMT and IST theories We found that the behavior of stresscomponent in all the three cases is alike It also observedthat as compared to ISMT theory the stress component haslarge values in IST theory but has small values in MT theoryThe difference is significant in both IST and MT theoriescompared with ISMT theory but the difference is muchpronounced in MT theory as compared to IST theory

Figures 7 8 and 9 exhibit the variations of displacementcomponent temperature and stress component in the con-text of the fractional order theory of thermoelasticity withmagnetic field and initial stress for different values of rampparameterThe values of ramp parameter are taken as 01 03and 05 and the value of fractional order parameter 120572 is takento be 05 for all the three cases

Figure 7 demonstrates that as we increase the value oframp parameter then the value of displacement componentalso increases It is also observed that trend for all the cases is

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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 Journal of Mathematics

minus03

minus02

minus01

0

01

02

03

04

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

119906

Figure 4 Displacement distribution 119906 at 120572 = 05 119905 = 01

minus05

0

05

1

15

2

25

0 04 08 12 16 2

ISMTMTIST

119909

120579

Figure 5 Temperature distribution 120579 at 120572 = 05 119905 = 01

adverse when 08 le 119909 le 6 The difference is significant in therange 0 le 119909 le 28

Figures 4 5 and 6 depict the variations of displacementcomponent temperature and stress component for three dif-ferent cases defined as (a) ISMT (initially stressed magneto-thermoelasticity) 120578 = 25 119901 = 1 (b) MT (magneto-thermoelasticity) 120578 = 1 119901 = 0 and (c) IST (initially stressedthermoelasticity) 120578 = 25 119901 = 1 119867

0= 0 In all the above

cases (a) (b) and (c) the value of fractional order parameteris taken to be 120572 = 05

Figure 4 shows the variations of displacement componentunder the three cases discussed in (a) (b) and (c) It isobserved that the displacement component in case of ISMTand IST has maximum positive values but in case of MT

0

02

04

06

08

1

12

14

16

18

2

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

ISMTMTIST

119909

120590

Figure 6 Stress distribution 120590 at 120572 = 05 119905 = 01

the displacement component has maximum negative valueon the boundary of half-space The trend for displacementcomponent in case of ISMT and IST is same that is both theseries are continuously decreasing to zero but in case of MTthe series rapidly approaches to zero Figure 4 also exhibitsthat in the absence of magnetic field (IST) and initial stress(MT) decreases the value of displacement It is also observedthat as compared to ISMT the difference is significant in caseof IST but much pronounced in case of MT

Figure 5 displays the variations of temperature under theISMT MT and IST theories It is observed that the trend ofthe series in case of ISMT MT and IST is similar and thedifference is significant It is found that the absence of initialstress (MT) increases the value of temperature componentbut the absence of magnetic field (IST) decreases the value oftemperature as compared to the general case (ISMT)

Figure 6 exhibits the variations of stress under ISMTMT and IST theories We found that the behavior of stresscomponent in all the three cases is alike It also observedthat as compared to ISMT theory the stress component haslarge values in IST theory but has small values in MT theoryThe difference is significant in both IST and MT theoriescompared with ISMT theory but the difference is muchpronounced in MT theory as compared to IST theory

Figures 7 8 and 9 exhibit the variations of displacementcomponent temperature and stress component in the con-text of the fractional order theory of thermoelasticity withmagnetic field and initial stress for different values of rampparameterThe values of ramp parameter are taken as 01 03and 05 and the value of fractional order parameter 120572 is takento be 05 for all the three cases

Figure 7 demonstrates that as we increase the value oframp parameter then the value of displacement componentalso increases It is also observed that trend for all the cases is

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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

Journal of Mathematics 9

0

005

01

015

02

025

03

035

04

045

05

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6

Ramp parameter 01Ramp parameter 03Ramp parameter 05

119909

119906

Figure 7 Displacement distribution 119906 for different values of rampparameter at 120572 = 05 119905 = 01

minus2

0

2

4

6

8

10

12

14

0 04 08 12 16 2

120579

119909

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 8 Temperature distribution 120579 for different values of rampparameter at 120572 = 05 119905 = 01

similar and effect of change in ramp parameter is significantin first two cases and much pronounced in last two cases

Figure 8 depicts that as we increase the value of rampparameter then the value of the temperature rapidly de-creases It is also observed that trend for the series in all thethree cases is similar and the effect of change in ramp para-meter is much pronounced in the initial range of distance0 le 119909 le 10

Figure 9 displays that as we increase the value of rampparameter then the value of the stress component decreases

0

1

2

3

4

5

6

7

8

9

10

0 04 08 12 16 2 24 28 32 36 4 44 48 52 56 6119909

120590

Ramp parameter 01Ramp parameter 03Ramp parameter 05

Figure 9 Stress distribution 120590 for different values of ramp parame-ter at 120572 = 05 119905 = 01

It is also observed that trend for the series in all the threecases is similar (ie all the series have same initial value andfirst increases to a maximum value then decreases to a mini-mum value) and the effect of change in ramp parameter ismuch pronounced It is also apparent from the figure that aswe increase the value of ramp parameter from 01 to 03 thevalues of stress component decrease rapidly as compared toincrease in the value of ramp parameter from 03 to 05

9 Summary

We consider a perfectly conducting elastic homogeneoushalf-space in the context of fractional order generalized ther-moelasticity theory with magnetic field and initial stress Themethod of thematrix exponential which constitutes the basisof the state space approach of modern theory is applied tothe nondimensional equationsThe importance of state spaceapproach is recognized in the fields where the time behaviourof physical process is of interest The state space approach ismore general than the classical Laplace and Fourier transformtechnique Consequently state space is applicable to allsystems that can be analyzed by integral transform in timeand also is applicable to many systems for which transformtheory breaks down [41] The potential function approach isoften used to solve problems of thermoelasticity theoryThishowever has several disadvantages as outlined in [39] Thesemay be summarized in the fact that the boundary conditionsfor physical problems are related directly to the physicalquantities under consideration not to the potential functionsSecondly more stringent assumptions must be made onthe behaviour of potential functions than on the actualphysical quantities Last of all it was found thatmany integralrepresentations of physical quantities are convergent in theclassical sense while their potential function representations

10 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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 Journal of Mathematics

only converge in the mean All these reasons have led manyauthors to avoid the use of potential functions Among thealternatives is the state space formulation This approachenables one to use themethodology ofmodern control theoryin solving problems of thermoelasticity

The main conclusions due to the influence of magneticfield initial stress fractional parameter and ramp parametercan be summarized as follows

(1) the phenomenon of finite speed of propagation is pre-served for all the field variables except for stress com-ponent due to the presence of initial stress and allresults are in agreement with the generalized theoryof thermoelasticity

(2) the fractional parameter has a significant effect onall the studied fields The displacement componentdecreases with the increase in the value of the frac-tional parameter

(3) the thermodynamic temperature first decreases thenincreases but the stress component first increasesthen decreases to a minimum value with the increasein the value of fractional parameter

(4) the effect of magnetic field is silent as compared toinitial stress on studied fields Also all the field vari-ables for the cases ISMT MT and IST behave alikeexcept the displacement component for MT

(5) in the context of fractional order generalized ther-moelasticity theory with magnetic field and initialstress the increase in the value of ramp parameterincreases the magnitude of displacement componentbut decreases the magnitude of temperature distribu-tion and stress component

(6) displacement component 119906 temperature distribution120579 and stress component 120590 show almost similar pat-tern for different values of fractional parameter 120572 andramp parameter 119905

0

Acknowledgment

One of the authors Sandeep Singh Sheoran is thankful to UG C New Delhi for the financial support Vide Letter no F17-112008 (SA-1)

References

[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 pp 240ndash253 1956

[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967

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

[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980

[5] R B Hetnarski and J Ignaczak ldquoGeneralized thermoelasticityrdquoJournal of Thermal Stresses vol 22 no 4-5 pp 451ndash476 1999

[6] H M Youssef ldquoTwo-dimensional generalized thermoelasticityproblem for a half-space subjected to ramp-type heatingrdquoEuropean Journal of Mechanics ASolids vol 25 no 5 pp 745ndash763 2006

[7] L Knopoff ldquoThe interaction between elastic wave motion and amagnetic field in electrical conductorsrdquo Journal of GeophysicalResearch vol 60 pp 441ndash456 1955

[8] P Chadwick ldquoElastic wave propagation in a magnetic fieldrdquo inProceedings of the International Congress of Applied Mechanicspp 143ndash153 Brussels Belgium 1957

[9] S Kaliski and J Petykiewicz ldquoEquation of motion coupled withthe field of temperature in a magnetic field involving mech-anical and electrical relaxation for anisotropic bodiesrdquo Proceed-ings of Vibration Problems vol 4 pp 1ndash12 1959

[10] G Paria ldquoOn magneto-thermo-elastic plane wavesrdquo Proceed-ings of the Cambridge Philosophical Society vol 58 pp 527ndash5311962

[11] A H Nayfeh and S Nemat-Nasser ldquoElectromagneto-thermo-elastic plane waves in solids with thermal relaxationrdquo Journal ofApplied Mechanics Transactions ASME vol 39 no 1 pp 108ndash113 1972

[12] H H Sherief and M A Ezzat ldquoA thermal-shock problemin magneto-thermoelasticity with thermal relaxationrdquo Interna-tional Journal of Solids and Structures vol 33 no 30 pp 4449ndash4459 1996

[13] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002

[14] M A Ezzat and H M Youssef ldquoGeneralized magneto-therm-oelasticity in a perfectly conducting mediumrdquo InternationalJournal of Solids and Structures vol 42 no 24-25 pp 6319ndash6334 2005

[15] A Baksi R K Bera and L Debnath ldquoA study of magneto-thermoelastic problems with thermal relaxation and heatsources in a three-dimensional infinite rotating elastic med-iumrdquo International Journal of Engineering Science vol 43 no19-20 pp 1419ndash1434 2005

[16] M A Biot Mechanics of Incremental Deformation John Wileyamp Sons New York NY USA 1965

[17] A Chattopadhyay S Bose and M Chakraborty ldquoReflectionof elastic waves under initial stress at a free surface P and SVmotionrdquo Journal of the Acoustical Society of America vol 72 no1 pp 255ndash263 1982

[18] A Montanaro ldquoOn singular surfaces in isotropic linear ther-moelasticity with initial stressrdquo Journal of the Acoustical Societyof America vol 106 no 3 pp 1586ndash1588 1999

[19] M I A Othman and Y Song ldquoReflection of plane wavesfrom an elastic solid half-space under hydrostatic initial stresswithout energy dissipationrdquo International Journal of Solids andStructures vol 44 no 17 pp 5651ndash5664 2007

[20] B Singh ldquoEffect of hydrostatic initial stresses on waves ina thermoelastic solid half-spacerdquo Applied Mathematics andComputation vol 198 no 2 pp 494ndash505 2008

[21] M Caputo and F Mainardi ldquoA new dissipation model based onmemory mechanismrdquo Pure and Applied Geophysics vol 91 no1 pp 134ndash147 1971

[22] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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

Journal of Mathematics 11

[23] M Caputo ldquoVibrations of an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 no 3 pp 897ndash904 1974

[24] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional ordersrdquo in Fractals andFractional Calculus in ContinuumMechanics vol 378 pp 223ndash276 Springer Vienna Austria 1997

[25] R Hilfer Application of Fraction Calculus in Physics WorldScientific Publishing Singapore 2000

[26] J Ignaczak and M Ostoja-Starzewski Thermoelasticity withFinite Wave Speeds Oxford University Press Oxford UK 2010

[27] L Debnath and D Bhatta Integral Transforms and Their Appli-cations Chapman amp HallCRC Taylor amp Francis Group Lon-don UK 2nd edition 2007

[28] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[29] Y Z Povstenko ldquoFractional heat conduction equation and asso-ciated thermal stressrdquo Journal of Thermal Stresses vol 28 no 1pp 83ndash102 2005

[30] HH Sherief AMA El-Sayed andAMAbdEl-Latief ldquoFrac-tional order theory of thermoelasticityrdquo International Journal ofSolids and Structures vol 47 no 2 pp 269ndash275 2010

[31] HMYoussef ldquoTheory of fractional order generalized thermoe-lasticityrdquo Journal of Heat Transfer vol 132 no 6 pp 1ndash7 2010

[32] M A Ezzat ldquoThermoelectric MHD non-Newtonian fluid withfractional derivative heat transferrdquo Physica B vol 405 no 19pp 4188ndash4194 2010

[33] M A Ezzat ldquoMagneto-thermoelasticity with thermoelectricproperties and fractional derivative heat transferrdquoPhysica B vol406 no 1 pp 30ndash35 2011

[34] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[35] A S El-Karamany and M A Ezzat ldquoConvolutional variationalprinciple reciprocal and uniqueness theorems in linear frac-tional two-temperature thermoelasticityrdquo Journal of ThermalStresses vol 34 no 3 pp 264ndash284 2011

[36] A S El-Karamany andM A Ezzat ldquoOn fractional thermoelas-ticityrdquo Mathematics and Mechanics of Solids vol 16 no 3 pp334ndash346 2011

[37] A S El-Karamany and M A Ezzat ldquoFractional order theory ofa perfect conducting thermoelastic mediumrdquoCanadian Journalof Physics vol 89 no 3 pp 311ndash318 2011

[38] S Kothari and S Mukhopadhyay ldquoA problem on elastic halfspace under fractional order theory of thermoelasticityrdquo Journalof Thermal Stresses vol 34 no 7 pp 724ndash739 2011

[39] L Y Bahar and R B Hetnarski ldquoState space approach to therm-oelasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash147 1978

[40] G Honig and U Hirdes ldquoAmethod for the numerical inversionof Laplace transformsrdquo Journal of Computational and AppliedMathematics vol 10 no 1 pp 113ndash132 1984

[41] D Wiberg Theory and Problems of State Space and Linear Sys-tems Schaumrsquos Outline Series in Engineering Mc Graw-HillNew York NY USA 1971

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