research article effect of rotation for two-temperature...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 297274 13 pageshttpdxdoiorg1011552013297274

Research ArticleEffect of Rotation for Two-TemperatureGeneralized Thermoelasticity of Two-Dimensional underThermal Shock Problem

Kh Lotfy12 and Wafaa Hassan23

1 Department of Mathematics Faculty of Science PO Box 44519 Zagazig University Zagazig Egypt2 Department of Mathematics Faculty of Science and Arts Al-mithnab Qassim University PO Box 931Al-mithnab Buridah 51931 Saudi Arabia

3Mathematics and Physics Department Faculty of Engineering Port Said University Port Said Egypt

Correspondence should be addressed to Kh Lotfy khlotfy 1yahoocom

Received 6 July 2013 Accepted 3 September 2013

Academic Editor Franklin Mendivil

Copyright copy 2013 Kh Lotfy and W Hassan This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The theory of two-temperature generalized thermoelasticity based on the theory of Youssef is used to solve boundary value problemsof two-dimensional half-space The governing equations are solved using normal mode method under the purview of the Lord-Shulman (LS) and the classical dynamical coupled theory (CD) The general solution obtained is applied to a specific problem ofa half-space subjected to one type of heating the thermal shock type We study the influence of rotation on the total deformationof thermoelastic half-space and the interaction with each other under the influence of two temperature theory The material ishomogeneous isotropic elastic half-space The methodology applied here is use of the normal mode analysis techniques that areused to solve the resulting nondimensional coupled field equations for the two theories Numerical results for the displacementcomponents force stresses and temperature distribution are presented graphically and discussedThe conductive temperature thedynamical temperature the stress and the strain distributions are shown graphically with some comparisons

1 Introduction

The linear theory of elasticity is of paramount importancein the stress analysis of steel which is the commonest engi-neering structural material To a lesser extent linear elasticitydescribes themechanical behavior of the other common solidmaterials for example concrete wood and coal Howeverthe theory does not apply to the behavior of many of the newsynthetic materials of the clastomer and polymer type forexample polymethyl-methacrylate (Perspex) polyethyleneand polyvinyl chloride The linear theory of micropolar elas-ticity is adequate to represent the behavior of such materialsFor ultrasonic waves that is for the case of elastic vibrationscharacterized by high frequencies and small wavelengths theinfluence of the body microstructure becomes significantthis influence of microstructure results in the development ofnew type ofwaves that are not in the classical theory of elastic-ity Metals polymers composites solids rocks and concrete

are typical media with microstructures More generally mostof the natural andmanmadematerials including engineeringgeological and biological media possess a microstructure

The classical coupled thermoelasticity theory proposed byBiot [1] with the introduction of the strain-rate term in theFourier heat conduction equation leads to a parabolic-typeheat conduction equation called the diffusion equationThistheory predicts finite propagation speed for elastic waves butan infinite speed for thermal disturbance This is physicallyunrealistic To overcome such an absurdity generalized ther-moelasticity theories have been propounded by Lord andShulman [2] as well as Green and Lindsay [3] advocating theexistence of finite thermal wave speed in solids

These theories have been developed by introducing oneor two relaxation times in the thermoelastic process eitherby modifying Fourierrsquos heat conduction equation or by cor-recting the energy equation and Neuman-Duhamel relation

2 Mathematical Problems in Engineering

According to these generalized theories heat propagationcan be visualized as a wave phenomenon rather than adiffusion one in the literature it is usually referred to asthe second sound effect These two theories are structurallydifferent from one another and one cannot be obtained as aparticular case of the other Various problems characterizingthese theories have been investigated and has revealed someinteresting phenomena Brief reviews of this topic have beenreported by Chandrasekharaiah and Srinath [4] and Chan-drasekharaiah and Murthy [5] The interplay of the Maxwellelectromagnetic field with the motion of deformable solidsis largely being undertaken by many investigators owing tothe possibility of its application to geophysical problems andcertain topics in optics and acoustics Moreover the earth issubject to its ownmagnetic field and the material of the earthmay be electrically conducting Thus the magnetoelasticnature of the earthrsquos material may affect the propagationof waves Many authors have considered the propagationof electromagnetothermoelastic waves in an electrically andthermally conducting solid A comprehensive review of theearlier contributions to the subject can be found in thestudy by Puri [6] Among the authors who considered thegeneralized magnetothermoelastic equations are Nayfeh andNemat-Nasser [7] who studied the propagation of planewaves in a solid under the influence of an electromagneticfield They have obtained the governing equations in thegeneral case and the solution for some particular cases RoyChoudhuri and Mukhopdhyay [8] extended these resultsto rotating media Ezzat [9] has studied the problem ofgeneration of generalized magnetothermoelastic waves bythermal shock in a perfectly conducting half-space Ezzatet al [10] have established the model of two dimensionalequations of generalizedmagnetothermoelasticity In dealingwith classical or generalized thermoelastic problems in mostsituations the displacement potential function approach isused However Bahar and Hetnarski [11 12] outlined severaldisadvantages of the potential function approach These maybe summarized in the fact that the boundary and initialconditions of the problem are not related directly to thepotential function as it has no physical meaning explicitly

Secondly more stringent assumptions must be madeon the 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 representationsonly converge in the mean To get rid of these difficultiesBahar and Hetnarski [13] introduced the state space formu-lation in thermoelastic problems This state space approachhas been further developed in Sherief [14] to include theeffect of heat sources Sherief and Anwar [15] surveyed a two-dimensional thermal shock problem for a semi-infinite piezo-electric rod using state space approach Youssef and El-bary[16] put forward an analysis for a generalized thermoelasticinfinite layer problem under three theories using state spaceapproach State space formulation to the vibration of goldnanobeam in femtoseconds scale was done by Elsibai andYoussef [17]

The theory of heat conduction in a deformable bodyformulated by Chen and Gurtin [18] and Chen et al [19 20]

depends on two different temperatures the conductive tem-perature and the thermo dynamical temperature Chen et al[21] have suggested that the difference between these twotemperatures is proportional to heat supply In absence ofheat supply these two temperatures are identical for timeindependent situation However for time dependent casesparticularly for problems related to wave propagation thetwo temperatures are in general different regardless ofheat supply The two temperature thermoelasticity theoryhas gained much attention of the researchers in the recentyears The existence structural stability convergence andspatial behaviour of two temperature thermoelasticity havebeen provided by Quintanilla and Tien [22] Youssef [23] hasdeveloped a new model of generalized thermoelasticity thatdepends on two temperatures 119879 and 120601 where the differencebetween the two temperatures is proportional to heat supply120601119894119894

with a nonnegative constant 119886 (length2)Later on Youssef and Al-Lehaibi [24] Misra et al [25]

Singh [26] and Singh and Bala [27] investigated variousproblems on the basis of two temperature thermoelasticitywith relaxation times and showed that the obtained resultsare qualitatively different as compared to those in case of onetemperature thermoelasticity

Roy Choudhuri and Debnath [28 29] Othman [30 31]Othman and Singh [32] and Othman and Song [33] studiedthe effect of rotation in amicropolar-generalized thermoelas-tic and thermoviscoelasticity half-space under different the-ories The propagation of plane harmonic waves in a rotatingelastic medium without thermal field has been studied Itwas shown there that the rotation causes the elastic mediumto be dispersive and an isotropic These problems are basedon more realistic elastic model since earth moon and otherplants have angular velocity

Owing to the mathematical difficulties encountered intwo- and three-dimensional multifield coupled generalizedheat conduction problems the problems become too com-plicated to obtain an analytical solution Instead of analyticalmethods several authors have applied numerical methodssuch as finite difference method finite element method andboundary value method For solving such kind of problemsone can find several two-dimensional works based on thegeneralized thermoelasticity by using the normal modeanalysis in the literature Ezzat and Abd Elall [34] Othmanet al [35 36] Lotfy and Othman [37] Lotfy et al [38ndash40]and Sarkar and Lahiri [41] Using the normal mode analysistechnique we will get the solution in the Fourier transformeddomain actually To apply the normal mode analysis we haveto assume that all the relations are sufficiently smooth on thereal axis such that the normal mode analysis of all these func-tions exists The normal mode analysis [34ndash40] was used toobtain the exact expression for the temperature distributionthermal stresses and the displacement components

The present paper is intended to demonstrate the use ofthe normal mode analysis in analyzing the propagation ofthermoelastic waves in two temperature theory of thermoe-lasticity under the effect of rotation Normal mode analysisis employed to obtain the exact solution Numerical values ofthe field quantities are calculated by considering illustrative

Mathematical Problems in Engineering 3

examples and their variations with respect to space coordi-nate are displayed graphically and discussed under thermalshock problem

2 Formulation of the Problem

In the present paper the authors consider the problem ofa homogeneous isotropic elastic half-space (119909 ge 0) Thesurface of the half-space is subjected initially (119905 = 0) to athermal shock that is a function of 119910 and 119905 Thus all thequantities considered in this problem will be functions of thetime variable 119905 and coordinates 119909 119910 We can introduce theequations of the problem as follows

Theheat conduction equation takes the formYoussef [23]

119870120593119894119894

= (120597

120597119905+ 1205910

1205972

1205971199052)(120588119862

119864

119879 + 1205741198790

119906119894119895

) (1)

The constitutive equation takes the form

120590119894119895

= 120582119890119896119896

120575119894119895

+ 2120583119890119894119895

minus 120574119879120575119894119895

(2)

Since the medium is rotating uniformly with an angularvelocity Ω = Ω119899 where 119899 is a unit vector representing thedirection of the axis of rotationThe displacement equation ofmotion in the rotating frame of reference has two additionalterms centripetal acceleration Ω times (Ω times 119906) due to time-varying motion only and the Coriolis acceleration 2Ω times where 119906 is the dynamic displacement vector

The equations of motion in a rotating frame of referencein the context of generalized thermo elasticity are

x

z

Ω

yo

Geometry of the problem

120588 [119894

+ Ω times (Ω times 119906)119894

+ (2Ω times )119894

] = 120590119894119895119895

(119894 119895 = 1 2 3)

(3)

The relation between the heat conduction and the dynamicalheat takes the form

120593 minus 119879 = 119886120593119894119894

(4)

where 119886 gt 0 two-temperature parameter Youssef [23]Now we will suppose elastic and homogenous half-space

119909 ge 0which obey (1)ndash(4) and initially quiescent where all thestate functions are depend only on the dimension 119909 119910 andthe time 119905

The displacement components for one dimension medi-um have the form

119906119909

= 119906 (119909 119910 119905) 119906119910

= V (119909 119910 119905) 119906119911

= 0 (5)

The strain component takes the form

119890119894119895

=1

2(119906119894119895

+ 119906119895119894

) (6)

The heat conduction equation takes the form

119870(1205972

120593

1205971199092+

1205972

120593

1205971199102)

= (120597

120597119905+ 1205910

1205972

1205971199052)120588119862119864

119879

+ 1205741198790

(120597

120597119905+ 1205910

1205972

1205971199052)(

120597119906

120597119909+

120597V120597119910

)

(7)

The constitutive law equations can be written as

120590119909119909

= (2120583 + 120582)120597119906

120597119909+ 120582

120597V120597119910

minus 120574119879

120590119910119910

= (2120583 + 120582)120597V120597119910

+ 120582120597119906

120597119909minus 120574119879

120590119909119910

= 120583(120597119906

120597119910+

120597V120597119909

)

(8)

Using the summation convection From (8) we note that thethird equation ofmotion in (3) is identically satisfied and firsttwo equations become

120588(1205972

119906

1205971199052minus Ω2

119906 + 2ΩV) = 120583nabla2

119906 + (120583 + 120582)120597119890

120597119909minus 120574

120597119879

120597119909

120588(1205972V

1205971199052minus Ω2V minus 2Ω) = 120583nabla

2V + (120583 + 120582)120597119890

120597119910minus 120574

120597119879

120597119910

(9)

The relation between the heat conduction and dynamical heattakes the form

120593 minus 119879 = 119886(1205972

120593

1205971199092+

1205972

120593

1205971199102) (10)

To transform the above equations into nondimensionalforms we define the following non-dimensional variables

(1199091015840

1199101015840

1199061015840

V1015840) = 1198880

120578 (119909 119910 119906 V)

(1199051015840

1205911015840

0

1205921015840

0

) = 1198882

0

120578 (119905 1205910

1205920

)

(1205791015840

1205931015840

) =(119879 120593) minus 119879

0

1198790

(11)

1205901015840

119894119895

=

120590119894119895

2120583 + 120582 Ω

1015840

=Ω

1198882

0

120578 (12)

where 120578 = (120588119862119864

119870) 11986222

= (120583120588) and 1198622

0

= (2120583 + 120582)120588


Hence we have (dropping the dashed for convenience)

nabla2

120593 minus (1 + 1205910

120597

120597119905)

120597120579

120597119905minus 120576 (1 + 120591

0

120597

120597119905)

120597119890

120597119905= 0 (13)

120593 minus 120579 = 120573(1205972

120593

1205971199092+

1205972

120593

1205971199102) (14)

where 120576 = (120574120588119862119864

) and 120573 = 1198861205782

1198882

0

Assume the scalar potential functions Π(119909 119910 119905) and

120595(119909 119910 119905)defined by the relations in the nondimensional form

119906 =120597Π

120597119909+

120597120595

120597119910 V =

120597Π

120597119910minus

120597120595

120597119909 (15)

By using (15) and (12) in (9) we obtain

[nabla2

+ Ω2

minus1205972

1205971199052]Π + 2Ω

120597120595

120597119905minus 1198860

120579 = 0 (16)

(nabla2

minus 1198861

1205972

1205971199052+ 1198861

Ω2

)120595 minus 2Ω1198861

120597Π

120597119905= 0 (17)

where

1198861

=1205881198622

0

120583 119886

0

=1205741198790

1205881198622

0

(18)

The heat conduction equation (13) becomes

nabla2

120593 minus (1 + 1205910

120597

120597119905)

120597120579

120597119905minus 120576 (1 + 120591

0

120597

120597119905)

120597Π

120597119905= 0 (19)

3 Normal Mode Analysis

The solution of the physical variable can be decomposed interms of normal modes as the following way

[Π 120595 120593 120579 120590119894119895

] (119909 119910 119905)

= [Πlowast

120595lowast

119906lowast

(119909) 120593lowast

(119909) 120579lowast

(119909) 120590lowast

119894119895

(119909)]

times exp (120596119905 + 119894119887119910)

(20)

where 120596 is the (complex) time constant 119894 is the imaginary119887 be a wave number in the 119910-direction and Π

lowast 120595lowast 119906lowast(119909)120593lowast

(119909) 120579lowast(119909) and 120590lowast

119894119895

(119909) are the amplitude of the functionsBy using the normal mode defined in (20) (17)ndash(19) and (14)take the following forms

[1198632

minus 1198601

]Πlowast

+ 1198600

120595lowast

minus 1198602

120579lowast

= 0 (21)

(1198632

minus 1198604

) 120595lowast

minus 1198605

Π = 0 (22)

[1198632

minus 1198603

] 120593lowast

= minus120573lowast

120579lowast

(23)

(1198632

minus 1198872

) 120593 minus 119860120579lowast

minus 119861Πlowast

= 0 (24)

where1198600

= 2Ω120596119860 = 120596(1+1205961205910

) 119861 = 1205761198601198601

= 1198872

+1205962

minusΩ2

1198602

= 1198860

1198603

= (1205731198872

+ 1)120573 120573lowast = 1120573 1198604

= 1198872

+ 1198861

(1205962

minusΩ2

)1198605

= 1198861

1198600

and 119863 = 119889119889119909

Eliminating 120579lowast

(119909)Πlowast(119909) 120595lowast(119909) and 120593lowast

(119909) between (21)and (24) we obtain the partial differential equation satisfiedby 120579lowast

(119909)

[1198636

minus 1198641198634

+ 1198651198632

minus 119866]Πlowast

(119909) = 0 (25)

where 1198606

= (120573lowast

1198872

+ 1198603

119860)(120573lowast

+ 119860) 1198607

= minus119861(120573lowast

+ 119860)Since

119864 = 1198601

+ 1198604

+ 1198606

+ 1198602

1198607

119865 = 1198601

1198604

+ 1198600

1198605

+ 1198606

(1198601

+ 1198604

) + 1198602

1198607

(1198603

+ 1198604

)

119866 = 1198601

1198604

1198606

+ 1198602

1198603

1198604

1198607

+ 1198600

1198605

1198606

(26)

In a similar manner we get

[1198636

minus 1198641198634

+ 1198651198632

minus 119866] (120579lowast

120593lowast

120595lowast

) (119909) = 0 (27)

The above equation can be factorized

(1198632

minus 1198962

1

) (1198632

minus 1198962

2

) (1198632

minus 1198962

3

)Πlowast

(119909) = 0 (28)

where 1198962

119899

(119899 = 1 2 3) are the roots of the following charac-teristic equation

1198966

minus 1198641198964

+ 1198651198962

minus 119866 = 0 (29)

The solution of (28) which is bounded as 119909 rarr infin is given by

Πlowast

(119909) =

3

sum

119899=1

119872119899

(119887 120596) exp (minus119896119899

119909) (30)

Similarly

120579lowast

(119909) =

3

sum

119899=1

1198721015840

119899

(119887 120596) exp (minus119896119899

119909) (31)

120595lowast

(119909) =

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909) (32)

120593lowast

(119909) =

3

sum

119899=1

119872101584010158401015840

119899

(119887 120596) exp (minus119896119899

119909) (33)

Since

119906lowast

(119909) = 119863Πlowast

+ 119894119887120595lowast

(34)

Vlowast (119909) = 119894119887Πlowast

minus 119863120595lowast

(35)

119890lowast

(119909) = 119863119906lowast

+ 119894119887Vlowast (36)


Using (34) and (35) in order to obtain the amplitude of thedisplacement components 119906 and V which are bounded as119909 rarr infin then (34) and (35) become

119906lowast

(119909) = minus

3

sum

119899=1

119872119899

(119887 120596) 119896119899

119890minus119896119899119909

+ 119894119887

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909)

Vlowast (119909) = 119894119887

3

sum

119899=1

119872119899

(119887 120596) 119890minus119896119899119909

+

3

sum

119899=1

11987210158401015840

119899

(119887 120596) 119896119899

119890minus119896119899119909

(37)

where119872119899

1198721015840119899

11987210158401015840119899

and119872101584010158401015840

119899

are some parameters dependingon 120573 119887 and 120596

Substituting from (30)ndash(32) into (21)ndash(24) we have

1198721015840

119899

(119887 120596) = 1198671119899

119872119899

(119887 120596) 119899 = 1 2 3 (38)

11987210158401015840

119899

(119887 120596) = 1198672119899

119872119899

(119887 120596) 119899 = 1 2 3 (39)

119872101584010158401015840

119899

(119887 120596) = 1198673119899

119872119899

(119887 120596) 119899 = 1 2 3 (40)

where

1198671119899

=

1198607

(1198962

119899

minus 1198603

)

1198962119899

minus 1198606

119899 = 1 2 3 (41)

1198672119899

=1198605

(1198962119899

minus 1198604

) 119899 = 1 2 3 (42)

1198673119899

=1198607

120573lowast

1198606

minus 1198962119899

119899 = 1 2 3 (43)

Thus we have

120579lowast

(119909) =

3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120595lowast

(119909) =

3

sum

119899=1

1198672119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120593lowast

(119909) =

3

sum

119899=1

1198673119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

(44)

The stress components can be calculated by using (31) and(37) in (8) as follows

120590lowast

119909119909

=

3

sum

119899=1

ℎ119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119910119910

=

3

sum

119899=1

ℎ1015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119909119910

=

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

(45)

The displacements components can be reduced by using (39)in (37) and we get

119906lowast

(119909) =

3

sum

119899=1

(1198941198871198672119899

minus 119896119899

)119872119899

(119887 120596) 119890minus119896119899119909

Vlowast (119909) =

3

sum

119899=1

(119894119887 + 119896119899

1198672119899

)119872119899

(119887 120596) 119890minus119896119899119909

(46)

where

ℎ119899

= minus[119896119899

(1198941198871198672119899

minus 119896119899

) minus119894119887120582 (119894119887 + 119896

119899

1198672119899

)

2120583 + 120582+

1205741198790

1198671119899

2120583 + 120582]

ℎ1015840

119899

= [119894119887 (119896119899

1198672119899

+ 119894119887) minus120582119896119899

(1198941198871198672119899

minus 119896119899

)

2120583 + 120582minus

1205741198790

1198671119899

2120583 + 120582]

ℎ10158401015840

119899

=120583 [119894119887 (119894119887119867

2119899

minus 119896119899

) minus 119896119899

(119894119887 + 119896119899

1198672119899

)]

2120583 + 120582

(47)

The normal mode analysis is in fact to look for the solutionin Fourier transformed domain Assuming that all the fieldquantities are sufficiently smooth on the real line such thatnormal mode analysis of these functions exists

4 Application

41 Thermal Shock Problem In order to determine theconstants119872

119899

(119899 = 1 2 3) In the physical problemwe shouldsuppress the positive exponentials that are unbounded atinfinity The constants 119872

1

1198722

and 1198723

have to be chosensuch that the boundary conditions on the surface at 119909 = 0

take the form

(1) thermal boundary conditions that the surface of thehalf-space subjected to thermal shock

120579 (0 119910 119905) = 119891 (0 119910 119905) (48)

(2) mechanical boundary condition that surface of thehalf-space is traction-free

120590119909119909

(0 119910 119905) = 0 (49)


120590119909119910

(0 119910 119905) = 0 (50)

where 119891(0 119910 119905) is some given function in 119910 and 119905

Substituting from the expressions of the considered var-iables into the above boundary conditions (48)ndash(50) we


obtain the following equations satisfyed by the parametersafter some simple manipulations

3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) = 119891lowast

(119910 119905)

3

sum

119899=1

ℎ119899

119872119899

(119887 120573 120596) = 0

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120573 120596) = 0

(51)

Invoking the boundary conditions (51) at the surface 119909 = 0

of the plate we obtain a system of three equations Afterapplying the inverse of matrix method we have the valuesof the three constants 119872

119895

119895 = 1 2 3 Hence we obtain theexpressions of displacements temperature distribution andanother physical quantity of the plate

5 Numerical Results

In order to analyze the above problem numerically wenow consider a numerical example for which computationalresults are given The results depict the variation of temper-ature displacement and stress fields in the context of twotheories To study the effect of rotation and two temperatureon wave propagationThe copper material was chosen for thepurpose of numerical example The numerical constants (inSI unit) of the problem were taken as

120582 = 759 times 109Nm2 120583 = 386 times 10

10 kgms2

120588 = 8954 kgm3 1205910

= 002 s

120572 = minus128 times 109Nm2 120573 = 032 times 10

9Nm2

120578 = 888673ms2 120576 = 00168

120572119905

= 178 times 10minus5 Kminus1 119896 = 386Wmminus1Kminus1

119887 = 1 119862119864

= 3831 J (kgK)

1198790

= 293K 119891lowast

= 1 120596 = 1205960

+ 119894120585

1205960

= 2 120585 = 1

(52)

Since we have 120596 = 1205960

+ 119894120585 where 119894 is the imaginary unit119890120596119905

= 1198901205960119905

(cos 120585119905 + 119894 sin 120585119905) and for small value of time we cantake 120596 = 120596

0

(real)The computations were carried out for 119886 value of time

119905 = 01 The numerical technique outlined above wasused for the distribution of the real part of the thermaltemperature 120579 and 120601 the displacement 119906 V strain andthe stress (120590

119909119909

120590119910119910

120590119909119910

) distribution for the problem Thefield quantities temperature displacement components andstress components depend not only on space 119909 and time 119905 butalso on the thermal relaxation time 120591

0

Here all the variablesare taken in nondimensional forms

In the first group Figures 1(a) 1(b) 1(c) 1(d) 1(e) 1(f)1(g) and 1(h) the graph shows the two curves predictedby different theories of thermoelasticity In these figuresthe solid lines represent the solution in the Coupled theorythe dashed lines represent the solution in the generalizedLord and Shulman theory We notice that the results forthe temperature the displacement and stresses distributionwhen the relaxation time is including in the heat equation aredistinctly different from those when the relaxation time is notmentioned in heat equation because the thermal waves in theFourier theory of heat equation travel with an infinite speedof propagation as opposed to finite speed in the non-Fouriercase This demonstrates clearly the difference between thecoupled and the theory of thermoelasticity (LS)

The second group Figures 2(a) 2(b) 2(c) 2(d) 2(e) 2(f)2(g) and 2(h) show the comparison between the thermaltemperature 120579 and120601 displacement components 119906 V the forcestresses components 120590

119909119909

120590119910119910

and 120590119909119910

the case of differenttwo values of rotation and constant of two temperatureparameter (120573 = 1) under LS theory It should be noted(Figure 2(a)) in this problem It is clear from the graph that120579 sharp decreases to minimum value at the beginning whereit experiences smooth increases (with maximum positivegradient) Graph lines for both values of rotation showdifferent slopes In other words the temperature lines forΩ = 00 has the highest gradient when compared with thatof Ω = 02 in all ranges In addition all lines begin tocoincidewhen the horizontal distance119909 increases to reach thereference temperature of the solidThese results obey physicalreality for the behaviour of copper as a polycrystallinesolid Figure 2(b) the horizontal displacement 119906 despite thepeaks (for different values of rotation) the magnitude ofthe maximum displacement peak strongly depends on therotation It is also clear that the rate of change of 119906 decreaseswith increasing the rotation On the other hand Figure 2(c)shows atonable increase of the vertical displacement V nearthe beginning reachs minimum value and then reaching zerovalue at the infinity (state of particles equilibrium) whenΩ = 00 Figure 2(d) displays a comparison of the strain intwo cases which show the different behaviours when Ω =

00 and Ω = 02 we can say that significant difference inthe strain is noticed for different values of the rotation Inaddition all lines begin to coincide when the horizontaldistance119909 increases to reach zero at infinity In Figure 2(e) thehorizontal stresses 120590

119909119909

graph lines for both values of rotationshow different slopes In other words the 120590

119909119909

componentline for Ω = 00 has the highest gradient when comparedwith that of Ω = 02 In addition all lines begin to coincidewhen the horizontal distance 119909 is increased to reach zero aftertheir relaxations at infinity Variation ofΩ has a serious effecton both magnitudes of mechanical stresses These trendsobey elastic and thermoelastic properties of the solid underinvestigation Figure 2(f) shows that the stress component120590119910119910

takes the different behavior In other words the 120590119910119910

component line for Ω = 00 has the highest gradient whencompared with that of Ω = 02 Figure 2(g) shows thatthe stress component 120590

119909119910

satisfies the boundary condition itsharp decreases in the start and start increases (minimum) inthe context of theΩ = 02 but whenΩ = 00 take the different


0 2 4 6 8 10 12minus15

minus1

minus05

0

05

1

15

120579

x

(a) The thermodynamical heat distribution

0 2 4 6 8 10 12minus015

minus01

minus005

0

005

x

u

(b) The displacement distribution (119906)

minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x

(c) The displacement distribution (V)

minus02

minus01

0

01

02

0 2 4 6 8 10 12x

e

(d) The strain distribution (119890)

120590xx

0

005

01

015

minus0050 2 4 6 8 10 12

x

(e) The stress 120590119909119909

distribution

120590yy

0

002

004

006

minus002

minus004

minus006

minus0080 2 4 6 8 10 12

x

(f) The stress 120590119910119910

distribution

120590xy 0

001

002

003

minus002

minus001

minus0030 2 4 6 8 10 12

x

CDLS

(g) The stress 120590119909119910

distribution

120601 0

05

1

minus05

minus10 2 4 6 8 10 12

x

CDLS

(h) The conductive heat distribution

Figure 1The thermal temperature 120579 the displacement 119906 V the strain 119890 distribution the stresses distribution 120590119910119910

120590119909119909

and 120590119909119910

and the thermaltemperature 120601 distribution with 120573 = 01 and 119905 = 01 under CD and LS theories


minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

u


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

v


minus05

0

05

1

0 2 4 6 8 10 12x

e


0

01

02

03

0 2 4 6 8 10 12x

minus01

120590xx

(e) The stress 120590119909119909

distribution

0

01

02

03

04

0 2 4 6 8 10 12x

minus01

120590yy

(f) The stress 120590119910119910

distribution

minus002

minus004

Ω = 00

Ω = 02

0

002

004

006

008

0 2 4 6 8 10 12x

120590xy

(g) The stress 120590119909119910

distribution

Ω = 00

Ω = 02

0 2 4 6 8 10 12x

minus05

minus1

0

05

1

120601



120590119909119909

and 120590119909119910

and the thermaltemperature 120601 distribution with different values of rotation when 120573 = 01 and 119905 = 01


behaviour The lines for Ω = 00 has the highest gradientwhen compared with that of Ω = 02 These trends obeyelastic and thermoelastic properties of the solid Figure 2(h)displays the conductive temperature in which we observe thesignificant difference in the conductive temperature that isnoticed for the value of rotationThe conductive temperaturebegins from the positive values and then decreases to arrivethe minimum amplitudes in two cases ofΩ also move in thewave propagation beyond it falls again to try to retain zero atinfinity

The third group Figures 3(a) 3(b) 3(c) 3(d) 3(e) 3(f)3(g) and 3(h) show the comparison between the thermaltemperature 120579 and 120601 displacement components (119906 and V)strain distribution and the stress (120590

119909119909

120590119910119910

120590119909119910

) distributionthe case of different two values of two temperature parameterFor the value of 119910 namely 119910 = minus1 were substituted in per-forming the computation Figure 3(a) exhibits the space vari-ation of temperature distribution inwhichwe observe the fol-lowing Significant difference in the thermodynamical tem-perature is noticed for different value of the nondimensionaltwo-temperature parameter It should be noted (Figure 3(a))It is clear from the graph that 120579 has decreased to arrive theminimum value at the beginning in two cases 120573 = 01 and120573 = 02 (two temperature)The value of temperature quantityconverges to zero with increasing the distance 119909 and satisfiesthe boundary conditions at 119909 = 0 Also from this figure wecan see when 120573 = 01 and 120573 = 02move in the wave function

In Figure 3(b) the horizontal displacement 119906 we seethat the displacement component 119906 always starts from thenegative value when 120573 = 01 and 120573 = 02 and terminates atthe zero value beginswith increase (then smooth increases) toreach its maximum magnitude Beyond it 119906 falls again to tryto retain zero at infinity beyond reaching zero at the infinity(state of particles equilibrium)The displacements 119906 show thesame behaviours at different values of 120573 In Figure 3(c) thevertical displacement V we see that the displacement compo-nent V always starts from the positive value and terminates atthe zero value to reach the minimum value beyond reachingzero at infinity with increases of 119909 Figure 3(d) displays acomparison of the strain in the context of two cases whichshow the same behaviours when 120573 = 01 and 120573 = 02We can say that significant difference in the strain is noticedfor different values of the non-dimensional two-temperatureparameter In addition all lines begin to coincide when thehorizontal distance 119909 increases to reach zero at infinity Thestrain distribution is continuous smooth and moves in thewave function These trends obey elastic and thermoelasticproperties of the solid The stress component 120590

119909119909

reachcoincidence with zero value (Figure 3(e)) and reaches themaximum value in the beginning and smooth decreases thenconverges to zero with increasing the distance 119909 Figure 3(f)shows that the stress component 120590

119910119910

increases in the startand arrive to maximum in the context of the two values of 120573These trends obey elastic and thermoelastic properties of thesolid under investigation In Figure 3(g) the stress component120590119909119910

satisfies the boundary condition and starts from zero Itsharp decreases in the start to arrive the minimum and thenstart smooth increases to maximum when 120573 = 01 but sharpincreases in the start to arrive the maximum and then start

smooth decreases to minimum when 120573 = 02 Figure 3(h)displays the conductive temperature in which we observe thesignificant difference in the conductive temperature that isnoticed for the value of the non-dimensional two temperatureparameter 120573 where the case of 120573 = 01 and 120573 = 02 indicatesthe new case (two-temperature)The conductive temperaturebegins from the positive values and then decreases to arrivethe minimum amplitudes in two cases of 120573 also move in thewave propagation when 120573 = 01 and 120573 = 02 beyond it fallsagain to try to retain zero at infinity

The forth group Figures 4(a) 4(b) 4(c) 4(d) 4(e) 4(f)4(g) and 4(h) show the comparison between the temperature120579 the displacement components 119906 and the force stressescomponents 120590

119909119909

and 120590119909119910

and the case of different two valuesof time (namely 119905 = 01 and 119905 = 03) under Lord-Shulman(LS) theoryThis group shows the effect of time on the resultsand we found that the curve when 119905 = 01 is greater than thecurves when 119905 = 02 in all figures These results obey physicalreality for the behaviour of copper as a polycrystalline solid

6 Conclusions

(1) The curves of the physical quantitieswith (CD) theoryin most of figures are lower in comparison with thoseunder (LS) theory due to the relaxation times

(2) Analytical solutions based upon normal mode anal-ysis for thermoelastic problem in solids have beendeveloped and utilized

(3) The theory of two-temperature generalized thermoe-lasticity describes the behavior of the particles ofthe elastic body more real than the theory of one-temperature generalized thermoelasticity

(4) In the context of the theory of two-temperature thephysical functions are continuous

(5) The value of all the physical quantities converges tozero with an increase in distance 119909 and all functionsare continuous

(6) Deformation of a body depends on the nature offorced applied as well as the type of boundary con-ditions

(7) It is clear from all the figures that all the distributionsconsidered have nonzero value only in a boundedregion of the half-space Outside of this region thevalues vanish identically and this means that theregion has not felt thermal disturbance yet

(8) From the temperature distributions we have found awave type heat propagation with finite speeds in themedium The heat wave front moves forward with afinite speed in the medium with the passage of timewhich proves that the generalized thermoelasticitytheory with two temperature heat transfer is veryclose to the behavior of the elastic materials Thisis not the case for the CD theory where an infinitespeeds of thermal propagation can be found andhence all the considered physical quantities have anonzero (possibly very small) value for any point


minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910

and the thermaltemperature 120601 distribution with different values of two-temperature parameter at the constants Ω = 02 and 119905 = 01


minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910

and the thermaltemperature 120601 distribution with different values of time at the constants Ω = 02 and 120573 = 01


in the medium This indicates that the generalizedFourierrsquos heat conduction mechanism is completelydifferent from the classical Fourierrsquos law

Nomenclature

120582 120583 Counterparts of Lamersquos parameters119901 Initial pressure120578 Initial stress parameter119886 Two temperature parameter120572119905

Coefficient of linear thermal expansion120579 = 119879 minus 119879

0

Thermodynamical temperature120601 = 120601

0

minus 119879 Conductive temperature119879 Absolute temperature1198790

Temperature of the medium in its naturalstate assumed to be |(119879 minus 119879

0

)1198790

| lt 1

120590119894119895

Components of the stress tensor119906119894

Components of the displacement vector120588 Density of the medium119890119894119895

Components of the strain tensor119890 Cubical dilatation119862119864

Specific heat at constant strain119870 Thermal conductivity1205910

Thermal relaxation time1205830

Magnetic permeability1205760

Electric permittivity119865119894

Lorentz force120575119894119895

Kronecker delta function

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] D S Chandrasekharaiah and K S Srinath ldquoThermoelasticinteractions without energy dissipation due to a point heatsourcerdquo Journal of Elasticity vol 50 no 2 pp 97ndash108 1998

[5] D S Chandrasekharaiah andHNMurthy ldquoTemperature-rate-dependent thermoelastic interactions due to a line heat sourcerdquoActa Mechanica vol 89 no 1ndash4 pp 1ndash12 1991

[6] P Puri ldquoPlane waves in thermoelasticity and magneto-ther-moelasticityrdquo International Journal of Engineering Science vol10 no 5 pp 467ndash477 1972

[7] ANayfeh and SNemat-Nasser ldquoTransient thermoelastic wavesin a half-space with thermal relaxationrdquo Zeitschrift fur Ange-wandte Mathematik und Physik vol 23 no 1 pp 50ndash68 1972

[8] S K Roy Choudhuri and S Mukhopdhyay ldquoEffect of rota-tion and relaxation on plane waves in generalized thermo-viscoelasticityrdquo International Journal of Mathematics and Math-ematical Sciences vol 23 pp 479ndash505 2000

[9] M A Ezzat and M I A Othman ldquoElectromagneto-ther-moelastic plane waves with two relaxation times in a mediumof perfect conductivityrdquo International Journal of EngineeringScience vol 38 no 1 pp 107ndash120 2000

[10] M Ezzat M I A Othman and A S El-Karamany ldquoElec-tromagneto-thermoelastic plane waves with thermal relaxation

in a medium of perfect conductivityrdquo Journal of ThermalStresses vol 24 no 5 pp 411ndash432 2001

[11] L Y Bahar and R B Hetnarski ldquoState space approach tothermoelasticityrdquo in Proceedings of the 6th Canadian Congressof Applied Mechanics pp 17ndash18 University of British ColumbiaVancouver Canada 1977

[12] L Y Bahar and R B Hetnarski ldquoTransfer matrix approach tothermoelasticityrdquo in Proceedings of the 15th Midwest Mechan-ical Conference pp 161ndash163 University of Illinois at ChicagoChicago Ill USA 1977

[13] L Y Bahar and R Hetnarski ldquoState space approach to thermoe-lasticityrdquo Journal of Thermal Stresses vol 1 pp 135ndash145 1978

[14] H H Sherief ldquoState space formulation for generalized ther-moelasticity with one relaxation time including heat sourcesrdquoJournal of Thermal Stresses vol 16 no 2 pp 163ndash180 1993

[15] H Sherief and M Anwar ldquoTwo-dimensional generalized ther-moelasticity problem for an infinitely long cylinderrdquo Journal ofThermal Stresses vol 17 no 2 pp 227ndash217 1994

[16] H M Youssef and A A El-Bary ldquoMathematical model forthermal shock problem of a generalized thermoelastic layeredcomposite material with variable thermal conductivityrdquo Com-putational Methods in Science and Technology vol 12 no 2 pp165ndash171 2006

[17] K A Elsibai and H M Youssef ldquoState-space approach tovibration of gold nano-beam induced by ramp type heatingwithout energy dissipation in femtoseconds scalerdquo Journal ofThermal Stresses vol 34 no 3 pp 244ndash263 2011

[18] P J Chen and M E Gurtin ldquoOn a theory of heat conductioninvolving two temperaturesrdquo Zeitschrift fur Angewandte Mathe-matik und Physik vol 19 no 4 pp 614ndash627 1968

[19] P J Chen andW O Williams ldquoA note on non-simple heat con-ductionrdquo Zeitschrift fur Angewandte Mathematik und Physikvol 19 no 6 pp 969ndash970 1968

[20] P J ChenM E Gurtin andWOWilliams ldquoOn the thermody-namics of non-simple elastic materials with two temperaturesrdquoZeitschrift fur Angewandte Mathematik und Physik vol 20 no1 pp 107ndash112 1969

[21] J K Chen J E Beraun and C L Tham ldquoUltrafast thermoe-lasticity for short-pulse laser heatingrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 793ndash807 2004

[22] T Q Quintanilla and C L Tien ldquoHeat transfer mechanismduring short-pulse laser heating of metalsrdquo Journal of HeatTransfer vol 115 pp 835ndash841 1993

[23] H M Youssef ldquoTheory of two-temperature-generalized ther-moelasticityrdquo IMA Journal of Applied Mathematics vol 71 no3 pp 383ndash390 2006

[24] H M Youssef and E A Al-Lehaibi ldquoState-space approachof two-temperature generalized thermoelasticity of one-dimensional problemrdquo International Journal of Solids andStructures vol 44 no 5 pp 1550ndash1562 2007

[25] J C Misra S B Kar and S C Samanta ldquoEffects of mechanicaland thermal relaxations on the stresses in a heated viscoelasticcontinuum with a cylindrical holerdquo Transactions of the Cana-dian Society for Mechanical Engineering vol 11 no 3 pp 151ndash159 1987

[26] B Singh ldquoPropagation of Rayleigh wave in a two-temperaturegeneralized thermoelastic solid half-spacerdquo ISRN Geophysicsvol 2013 Article ID 857937 6 pages 2013

[27] B Singh and K Bala ldquoOn Rayleigh wave in two-temperaturegeneralized thermoelastic mediumwithout energy dissipationrdquoApplied Mathematics vol 4 no 1 pp 107ndash112 2013


[28] S K R Choudhuri and L Debnath ldquoMagneto-thermoelasticplane waves in a rotating mediardquo International Journal ofEngineering Science vol 21 pp 155ndash163 1983

[29] S K R Choudhuri and L Debnath ldquoMagneto-elastic planewaves in infinite rotating mediardquo Journal of Applied Mechanicsvol 50 pp 283ndash288 1983

[30] M I A Othman ldquoEffect of rotation on plane waves in general-ized thermo-elasticity with two relaxation timesrdquo InternationalJournal of Solids and Structures vol 41 no 11-12 pp 2939ndash29562004

[31] M I A Othman ldquoEffect of rotation and relaxation time on athermal shock problem for a half-space in generalized thermo-viscoelasticityrdquo Acta Mechanica vol 174 no 3-4 pp 129ndash1432005

[32] M I A Othman and B Singh ldquoThe effect of rotation ongeneralized micropolar thermoelasticity for a half-space underfive theoriesrdquo International Journal of Solids and Structures vol44 no 9 pp 2748ndash2762 2007

[33] M I A Othman and Y Song ldquoEffect of rotation on plane wavesof generalized electro-magneto-thermoviscoelasticity with tworelaxation timesrdquo Applied Mathematical Modelling vol 32 no5 pp 811ndash825 2008

[34] M A Ezzat and M Z Abd Elall ldquoGeneralized magneto-thermoelasticity with modified Ohmrsquos lawrdquo Mechanics ofAdvancedMaterials and Structures vol 17 no 1 pp 74ndash84 2010

[35] M I A Othman and Kh Lotfy ldquoOn the plane waves ofgeneralized thermo-microstretch elastic half-space under threetheoriesrdquo International Communications in Heat and MassTransfer vol 37 no 2 pp 192ndash200 2010

[36] M I A Othman Kh Lotfy and R M Farouk ldquoGeneralizedthermo-microstretch elastic medium with temperature depen-dent properties for different theoriesrdquo Engineering Analysis withBoundary Elements vol 34 no 3 pp 229ndash237 2010

[37] M Othman and Kh Lotfy ldquoThe effect of magnetic field androtation of the 2-D problem of a fiber-reinforced thermoelasticunder three theories with influence of gravityrdquo Mechanics ofMaterials vol 60 pp 120ndash143 2013

[38] Kh Lotfy and W Hassan ldquoA mode-I crack problem fortwo-dimensional problem of a fiber-reinforced thermoelasticwith normal mode analysisrdquo International Journal of PhysicalSciences vol 8 no 22 pp 1228ndash1245 2013

[39] Kh Lotfy and M Othman ldquoThe effect of rotation on planewaves in generalized thermo-microstretch elastic solid with onerelaxation time for a mode-I crack problemrdquo Chinese Physics Bvol 20 no 7 Article ID 074601 2011

[40] Kh Lotfy ldquoMode-I crack in a two-dimensional fibre-reinforcedgeneralized thermoelastic problemrdquo Chinese Physics B vol 21no 1 Article ID 014209 2012

[41] N Sarkar and A Lahiri ldquoA three-dimensional thermoelasticproblem for a half-space without energy dissipationrdquo Interna-tional Journal of Engineering Science vol 51 pp 310ndash325 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


According to these generalized theories heat propagationcan be visualized as a wave phenomenon rather than adiffusion one in the literature it is usually referred to asthe second sound effect These two theories are structurallydifferent from one another and one cannot be obtained as aparticular case of the other Various problems characterizingthese theories have been investigated and has revealed someinteresting phenomena Brief reviews of this topic have beenreported by Chandrasekharaiah and Srinath [4] and Chan-drasekharaiah and Murthy [5] The interplay of the Maxwellelectromagnetic field with the motion of deformable solidsis largely being undertaken by many investigators owing tothe possibility of its application to geophysical problems andcertain topics in optics and acoustics Moreover the earth issubject to its ownmagnetic field and the material of the earthmay be electrically conducting Thus the magnetoelasticnature of the earthrsquos material may affect the propagationof waves Many authors have considered the propagationof electromagnetothermoelastic waves in an electrically andthermally conducting solid A comprehensive review of theearlier contributions to the subject can be found in thestudy by Puri [6] Among the authors who considered thegeneralized magnetothermoelastic equations are Nayfeh andNemat-Nasser [7] who studied the propagation of planewaves in a solid under the influence of an electromagneticfield They have obtained the governing equations in thegeneral case and the solution for some particular cases RoyChoudhuri and Mukhopdhyay [8] extended these resultsto rotating media Ezzat [9] has studied the problem ofgeneration of generalized magnetothermoelastic waves bythermal shock in a perfectly conducting half-space Ezzatet al [10] have established the model of two dimensionalequations of generalizedmagnetothermoelasticity In dealingwith classical or generalized thermoelastic problems in mostsituations the displacement potential function approach isused However Bahar and Hetnarski [11 12] outlined severaldisadvantages of the potential function approach These maybe summarized in the fact that the boundary and initialconditions of the problem are not related directly to thepotential function as it has no physical meaning explicitly

Secondly more stringent assumptions must be madeon the 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 representationsonly converge in the mean To get rid of these difficultiesBahar and Hetnarski [13] introduced the state space formu-lation in thermoelastic problems This state space approachhas been further developed in Sherief [14] to include theeffect of heat sources Sherief and Anwar [15] surveyed a two-dimensional thermal shock problem for a semi-infinite piezo-electric rod using state space approach Youssef and El-bary[16] put forward an analysis for a generalized thermoelasticinfinite layer problem under three theories using state spaceapproach State space formulation to the vibration of goldnanobeam in femtoseconds scale was done by Elsibai andYoussef [17]

The theory of heat conduction in a deformable bodyformulated by Chen and Gurtin [18] and Chen et al [19 20]

depends on two different temperatures the conductive tem-perature and the thermo dynamical temperature Chen et al[21] have suggested that the difference between these twotemperatures is proportional to heat supply In absence ofheat supply these two temperatures are identical for timeindependent situation However for time dependent casesparticularly for problems related to wave propagation thetwo temperatures are in general different regardless ofheat supply The two temperature thermoelasticity theoryhas gained much attention of the researchers in the recentyears The existence structural stability convergence andspatial behaviour of two temperature thermoelasticity havebeen provided by Quintanilla and Tien [22] Youssef [23] hasdeveloped a new model of generalized thermoelasticity thatdepends on two temperatures 119879 and 120601 where the differencebetween the two temperatures is proportional to heat supply120601119894119894

with a nonnegative constant 119886 (length2)Later on Youssef and Al-Lehaibi [24] Misra et al [25]

Singh [26] and Singh and Bala [27] investigated variousproblems on the basis of two temperature thermoelasticitywith relaxation times and showed that the obtained resultsare qualitatively different as compared to those in case of onetemperature thermoelasticity

Roy Choudhuri and Debnath [28 29] Othman [30 31]Othman and Singh [32] and Othman and Song [33] studiedthe effect of rotation in amicropolar-generalized thermoelas-tic and thermoviscoelasticity half-space under different the-ories The propagation of plane harmonic waves in a rotatingelastic medium without thermal field has been studied Itwas shown there that the rotation causes the elastic mediumto be dispersive and an isotropic These problems are basedon more realistic elastic model since earth moon and otherplants have angular velocity

Owing to the mathematical difficulties encountered intwo- and three-dimensional multifield coupled generalizedheat conduction problems the problems become too com-plicated to obtain an analytical solution Instead of analyticalmethods several authors have applied numerical methodssuch as finite difference method finite element method andboundary value method For solving such kind of problemsone can find several two-dimensional works based on thegeneralized thermoelasticity by using the normal modeanalysis in the literature Ezzat and Abd Elall [34] Othmanet al [35 36] Lotfy and Othman [37] Lotfy et al [38ndash40]and Sarkar and Lahiri [41] Using the normal mode analysistechnique we will get the solution in the Fourier transformeddomain actually To apply the normal mode analysis we haveto assume that all the relations are sufficiently smooth on thereal axis such that the normal mode analysis of all these func-tions exists The normal mode analysis [34ndash40] was used toobtain the exact expression for the temperature distributionthermal stresses and the displacement components

The present paper is intended to demonstrate the use ofthe normal mode analysis in analyzing the propagation ofthermoelastic waves in two temperature theory of thermoe-lasticity under the effect of rotation Normal mode analysisis employed to obtain the exact solution Numerical values ofthe field quantities are calculated by considering illustrative






119870120593119894119894

= (120597

120597119905+ 1205910

1205972

1205971199052)(120588119862

119864

119879 + 1205741198790

119906119894119895

) (1)


120590119894119895

= 120582119890119896119896

120575119894119895

+ 2120583119890119894119895

minus 120574119879120575119894119895

(2)



x

z

Ω

yo


120588 [119894

+ Ω times (Ω times 119906)119894

+ (2Ω times )119894

] = 120590119894119895119895

(119894 119895 = 1 2 3)

(3)


120593 minus 119879 = 119886120593119894119894

(4)




119906119909

= 119906 (119909 119910 119905) 119906119910

= V (119909 119910 119905) 119906119911

= 0 (5)


119890119894119895

=1

2(119906119894119895

+ 119906119895119894

) (6)


119870(1205972

120593

1205971199092+

1205972

120593

1205971199102)

= (120597

120597119905+ 1205910

1205972

1205971199052)120588119862119864

119879

+ 1205741198790

(120597

120597119905+ 1205910

1205972

1205971199052)(

120597119906

120597119909+

120597V120597119910

)

(7)


120590119909119909

= (2120583 + 120582)120597119906

120597119909+ 120582

120597V120597119910

minus 120574119879

120590119910119910

= (2120583 + 120582)120597V120597119910

+ 120582120597119906

120597119909minus 120574119879

120590119909119910

= 120583(120597119906

120597119910+

120597V120597119909

)

(8)


120588(1205972

119906

1205971199052minus Ω2

119906 + 2ΩV) = 120583nabla2

119906 + (120583 + 120582)120597119890

120597119909minus 120574

120597119879

120597119909

120588(1205972V


2V + (120583 + 120582)120597119890

120597119910minus 120574

120597119879

120597119910

(9)


120593 minus 119879 = 119886(1205972

120593

1205971199092+

1205972

120593

1205971199102) (10)


(1199091015840

1199101015840

1199061015840

V1015840) = 1198880

120578 (119909 119910 119906 V)

(1199051015840

1205911015840

0

1205921015840

0

) = 1198882

0

120578 (119905 1205910

1205920

)

(1205791015840

1205931015840

) =(119879 120593) minus 119879

0

1198790

(11)

1205901015840

119894119895

=

120590119894119895

2120583 + 120582 Ω

1015840

=Ω

1198882

0

120578 (12)

where 120578 = (120588119862119864

119870) 11986222

= (120583120588) and 1198622

0

= (2120583 + 120582)120588



nabla2

120593 minus (1 + 1205910

120597

120597119905)

120597120579

120597119905minus 120576 (1 + 120591

0

120597

120597119905)

120597119890

120597119905= 0 (13)

120593 minus 120579 = 120573(1205972

120593

1205971199092+

1205972

120593

1205971199102) (14)

where 120576 = (120574120588119862119864

) and 120573 = 1198861205782

1198882

0



119906 =120597Π

120597119909+

120597120595

120597119910 V =

120597Π

120597119910minus

120597120595

120597119909 (15)


[nabla2

+ Ω2

minus1205972

1205971199052]Π + 2Ω

120597120595

120597119905minus 1198860

120579 = 0 (16)

(nabla2

minus 1198861

1205972

1205971199052+ 1198861

Ω2

)120595 minus 2Ω1198861

120597Π

120597119905= 0 (17)

where

1198861

=1205881198622

0

120583 119886

0

=1205741198790

1205881198622

0

(18)


nabla2

120593 minus (1 + 1205910

120597

120597119905)

120597120579

120597119905minus 120576 (1 + 120591

0

120597

120597119905)

120597Π

120597119905= 0 (19)



[Π 120595 120593 120579 120590119894119895

] (119909 119910 119905)

= [Πlowast

120595lowast

119906lowast

(119909) 120593lowast

(119909) 120579lowast

(119909) 120590lowast

119894119895

(119909)]

times exp (120596119905 + 119894119887119910)

(20)




119894119895


[1198632

minus 1198601

]Πlowast

+ 1198600

120595lowast

minus 1198602

120579lowast

= 0 (21)

(1198632

minus 1198604

) 120595lowast

minus 1198605

Π = 0 (22)

[1198632

minus 1198603

] 120593lowast

= minus120573lowast

120579lowast

(23)

(1198632

minus 1198872

) 120593 minus 119860120579lowast


= 0 (24)

where1198600

= 2Ω120596119860 = 120596(1+1205961205910

) 119861 = 1205761198601198601

= 1198872

+1205962

minusΩ2

1198602

= 1198860

1198603

= (1205731198872

+ 1)120573 120573lowast = 1120573 1198604

= 1198872

+ 1198861

(1205962

minusΩ2

)1198605

= 1198861

1198600

and 119863 = 119889119889119909




(119909)

[1198636

minus 1198641198634

+ 1198651198632


(119909) = 0 (25)

where 1198606

= (120573lowast

1198872

+ 1198603

119860)(120573lowast

+ 119860) 1198607


+ 119860)Since

119864 = 1198601

+ 1198604

+ 1198606

+ 1198602

1198607

119865 = 1198601

1198604

+ 1198600

1198605

+ 1198606

(1198601

+ 1198604

) + 1198602

1198607

(1198603

+ 1198604

)

119866 = 1198601

1198604

1198606

+ 1198602

1198603

1198604

1198607

+ 1198600

1198605

1198606

(26)


[1198636

minus 1198641198634

+ 1198651198632


120593lowast

120595lowast

) (119909) = 0 (27)


(1198632

minus 1198962

1

) (1198632

minus 1198962

2

) (1198632

minus 1198962

3

)Πlowast

(119909) = 0 (28)

where 1198962

119899


1198966

minus 1198641198964

+ 1198651198962

minus 119866 = 0 (29)


Πlowast

(119909) =

3

sum

119899=1

119872119899

(119887 120596) exp (minus119896119899

119909) (30)

Similarly

120579lowast

(119909) =

3

sum

119899=1

1198721015840

119899

(119887 120596) exp (minus119896119899

119909) (31)

120595lowast

(119909) =

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909) (32)

120593lowast

(119909) =

3

sum

119899=1

119872101584010158401015840

119899

(119887 120596) exp (minus119896119899

119909) (33)

Since

119906lowast

(119909) = 119863Πlowast

+ 119894119887120595lowast

(34)

Vlowast (119909) = 119894119887Πlowast


(35)

119890lowast

(119909) = 119863119906lowast

+ 119894119887Vlowast (36)



119906lowast

(119909) = minus

3

sum

119899=1

119872119899

(119887 120596) 119896119899

119890minus119896119899119909

+ 119894119887

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909)

Vlowast (119909) = 119894119887

3

sum

119899=1

119872119899

(119887 120596) 119890minus119896119899119909

+

3

sum

119899=1

11987210158401015840

119899

(119887 120596) 119896119899

119890minus119896119899119909

(37)

where119872119899

1198721015840119899

11987210158401015840119899

and119872101584010158401015840

119899



1198721015840

119899

(119887 120596) = 1198671119899

119872119899

(119887 120596) 119899 = 1 2 3 (38)

11987210158401015840

119899

(119887 120596) = 1198672119899

119872119899

(119887 120596) 119899 = 1 2 3 (39)

119872101584010158401015840

119899

(119887 120596) = 1198673119899

119872119899

(119887 120596) 119899 = 1 2 3 (40)

where

1198671119899

=

1198607

(1198962

119899

minus 1198603

)

1198962119899

minus 1198606

119899 = 1 2 3 (41)

1198672119899

=1198605

(1198962119899

minus 1198604

) 119899 = 1 2 3 (42)

1198673119899

=1198607

120573lowast

1198606

minus 1198962119899

119899 = 1 2 3 (43)

Thus we have

120579lowast

(119909) =

3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120595lowast

(119909) =

3

sum

119899=1

1198672119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120593lowast

(119909) =

3

sum

119899=1

1198673119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

(44)


120590lowast

119909119909

=

3

sum

119899=1

ℎ119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119910119910

=

3

sum

119899=1

ℎ1015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119909119910

=

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

(45)


119906lowast

(119909) =

3

sum

119899=1

(1198941198871198672119899

minus 119896119899

)119872119899

(119887 120596) 119890minus119896119899119909

Vlowast (119909) =

3

sum

119899=1

(119894119887 + 119896119899

1198672119899

)119872119899

(119887 120596) 119890minus119896119899119909

(46)

where

ℎ119899

= minus[119896119899

(1198941198871198672119899

minus 119896119899

) minus119894119887120582 (119894119887 + 119896

119899

1198672119899

)

2120583 + 120582+

1205741198790

1198671119899

2120583 + 120582]

ℎ1015840

119899

= [119894119887 (119896119899

1198672119899

+ 119894119887) minus120582119896119899

(1198941198871198672119899

minus 119896119899

)

2120583 + 120582minus

1205741198790

1198671119899

2120583 + 120582]

ℎ10158401015840

119899

=120583 [119894119887 (119894119887119867

2119899

minus 119896119899

) minus 119896119899

(119894119887 + 119896119899

1198672119899

)]

2120583 + 120582

(47)


4 Application


119899


1

1198722

and 1198723


take the form


120579 (0 119910 119905) = 119891 (0 119910 119905) (48)


120590119909119909

(0 119910 119905) = 0 (49)


120590119909119910

(0 119910 119905) = 0 (50)





3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) = 119891lowast

(119910 119905)

3

sum

119899=1

ℎ119899

119872119899

(119887 120573 120596) = 0

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120573 120596) = 0

(51)



119895


5 Numerical Results


120582 = 759 times 109Nm2 120583 = 386 times 10

10 kgms2

120588 = 8954 kgm3 1205910

= 002 s


9Nm2

120578 = 888673ms2 120576 = 00168

120572119905


119887 = 1 119862119864

= 3831 J (kgK)

1198790

= 293K 119891lowast

= 1 120596 = 1205960

+ 119894120585

1205960

= 2 120585 = 1

(52)

Since we have 120596 = 1205960


= 1198901205960119905


0



119909119909

120590119910119910

120590119909119910


0




119909119909

120590119910119910

and 120590119909119910



119909119909


119909119909




119909119910



0 2 4 6 8 10 12minus15

minus1

minus05

0

05

1

15

120579

x


0 2 4 6 8 10 12minus015

minus01

minus005

0

005

x

u


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus02

minus01

0

01

02

0 2 4 6 8 10 12x

e


120590xx

0

005

01

015

minus0050 2 4 6 8 10 12

x

(e) The stress 120590119909119909

distribution

120590yy

0

002

004

006

minus002

minus004

minus006

minus0080 2 4 6 8 10 12

x

(f) The stress 120590119910119910

distribution

120590xy 0

001

002

003

minus002

minus001

minus0030 2 4 6 8 10 12

x

CDLS

(g) The stress 120590119909119910

distribution

120601 0

05

1

minus05

minus10 2 4 6 8 10 12

x

CDLS



120590119909119909

and 120590119909119910



minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

u


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

v


minus05

0

05

1

0 2 4 6 8 10 12x

e


0

01

02

03

0 2 4 6 8 10 12x

minus01

120590xx

(e) The stress 120590119909119909

distribution

0

01

02

03

04

0 2 4 6 8 10 12x

minus01

120590yy

(f) The stress 120590119910119910

distribution

minus002

minus004

Ω = 00

Ω = 02

0

002

004

006

008

0 2 4 6 8 10 12x

120590xy

(g) The stress 120590119909119910

distribution

Ω = 00

Ω = 02

0 2 4 6 8 10 12x

minus05

minus1

0

05

1

120601



120590119909119909

and 120590119909119910





119909119909

120590119910119910

120590119909119910



119909119909


119910119910





119909119909

and 120590119909119910


6 Conclusions










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119870120593119894119894

= (120597

120597119905+ 1205910

1205972

1205971199052)(120588119862

119864

119879 + 1205741198790

119906119894119895

) (1)


120590119894119895

= 120582119890119896119896

120575119894119895

+ 2120583119890119894119895

minus 120574119879120575119894119895

(2)



x

z

Ω

yo


120588 [119894

+ Ω times (Ω times 119906)119894

+ (2Ω times )119894

] = 120590119894119895119895

(119894 119895 = 1 2 3)

(3)


120593 minus 119879 = 119886120593119894119894

(4)




119906119909

= 119906 (119909 119910 119905) 119906119910

= V (119909 119910 119905) 119906119911

= 0 (5)


119890119894119895

=1

2(119906119894119895

+ 119906119895119894

) (6)


119870(1205972

120593

1205971199092+

1205972

120593

1205971199102)

= (120597

120597119905+ 1205910

1205972

1205971199052)120588119862119864

119879

+ 1205741198790

(120597

120597119905+ 1205910

1205972

1205971199052)(

120597119906

120597119909+

120597V120597119910

)

(7)


120590119909119909

= (2120583 + 120582)120597119906

120597119909+ 120582

120597V120597119910

minus 120574119879

120590119910119910

= (2120583 + 120582)120597V120597119910

+ 120582120597119906

120597119909minus 120574119879

120590119909119910

= 120583(120597119906

120597119910+

120597V120597119909

)

(8)


120588(1205972

119906

1205971199052minus Ω2

119906 + 2ΩV) = 120583nabla2

119906 + (120583 + 120582)120597119890

120597119909minus 120574

120597119879

120597119909

120588(1205972V


2V + (120583 + 120582)120597119890

120597119910minus 120574

120597119879

120597119910

(9)


120593 minus 119879 = 119886(1205972

120593

1205971199092+

1205972

120593

1205971199102) (10)


(1199091015840

1199101015840

1199061015840

V1015840) = 1198880

120578 (119909 119910 119906 V)

(1199051015840

1205911015840

0

1205921015840

0

) = 1198882

0

120578 (119905 1205910

1205920

)

(1205791015840

1205931015840

) =(119879 120593) minus 119879

0

1198790

(11)

1205901015840

119894119895

=

120590119894119895

2120583 + 120582 Ω

1015840

=Ω

1198882

0

120578 (12)

where 120578 = (120588119862119864

119870) 11986222

= (120583120588) and 1198622

0

= (2120583 + 120582)120588



nabla2

120593 minus (1 + 1205910

120597

120597119905)

120597120579

120597119905minus 120576 (1 + 120591

0

120597

120597119905)

120597119890

120597119905= 0 (13)

120593 minus 120579 = 120573(1205972

120593

1205971199092+

1205972

120593

1205971199102) (14)

where 120576 = (120574120588119862119864

) and 120573 = 1198861205782

1198882

0



119906 =120597Π

120597119909+

120597120595

120597119910 V =

120597Π

120597119910minus

120597120595

120597119909 (15)


[nabla2

+ Ω2

minus1205972

1205971199052]Π + 2Ω

120597120595

120597119905minus 1198860

120579 = 0 (16)

(nabla2

minus 1198861

1205972

1205971199052+ 1198861

Ω2

)120595 minus 2Ω1198861

120597Π

120597119905= 0 (17)

where

1198861

=1205881198622

0

120583 119886

0

=1205741198790

1205881198622

0

(18)


nabla2

120593 minus (1 + 1205910

120597

120597119905)

120597120579

120597119905minus 120576 (1 + 120591

0

120597

120597119905)

120597Π

120597119905= 0 (19)



[Π 120595 120593 120579 120590119894119895

] (119909 119910 119905)

= [Πlowast

120595lowast

119906lowast

(119909) 120593lowast

(119909) 120579lowast

(119909) 120590lowast

119894119895

(119909)]

times exp (120596119905 + 119894119887119910)

(20)




119894119895


[1198632

minus 1198601

]Πlowast

+ 1198600

120595lowast

minus 1198602

120579lowast

= 0 (21)

(1198632

minus 1198604

) 120595lowast

minus 1198605

Π = 0 (22)

[1198632

minus 1198603

] 120593lowast

= minus120573lowast

120579lowast

(23)

(1198632

minus 1198872

) 120593 minus 119860120579lowast


= 0 (24)

where1198600

= 2Ω120596119860 = 120596(1+1205961205910

) 119861 = 1205761198601198601

= 1198872

+1205962

minusΩ2

1198602

= 1198860

1198603

= (1205731198872

+ 1)120573 120573lowast = 1120573 1198604

= 1198872

+ 1198861

(1205962

minusΩ2

)1198605

= 1198861

1198600

and 119863 = 119889119889119909




(119909)

[1198636

minus 1198641198634

+ 1198651198632


(119909) = 0 (25)

where 1198606

= (120573lowast

1198872

+ 1198603

119860)(120573lowast

+ 119860) 1198607


+ 119860)Since

119864 = 1198601

+ 1198604

+ 1198606

+ 1198602

1198607

119865 = 1198601

1198604

+ 1198600

1198605

+ 1198606

(1198601

+ 1198604

) + 1198602

1198607

(1198603

+ 1198604

)

119866 = 1198601

1198604

1198606

+ 1198602

1198603

1198604

1198607

+ 1198600

1198605

1198606

(26)


[1198636

minus 1198641198634

+ 1198651198632


120593lowast

120595lowast

) (119909) = 0 (27)


(1198632

minus 1198962

1

) (1198632

minus 1198962

2

) (1198632

minus 1198962

3

)Πlowast

(119909) = 0 (28)

where 1198962

119899


1198966

minus 1198641198964

+ 1198651198962

minus 119866 = 0 (29)


Πlowast

(119909) =

3

sum

119899=1

119872119899

(119887 120596) exp (minus119896119899

119909) (30)

Similarly

120579lowast

(119909) =

3

sum

119899=1

1198721015840

119899

(119887 120596) exp (minus119896119899

119909) (31)

120595lowast

(119909) =

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909) (32)

120593lowast

(119909) =

3

sum

119899=1

119872101584010158401015840

119899

(119887 120596) exp (minus119896119899

119909) (33)

Since

119906lowast

(119909) = 119863Πlowast

+ 119894119887120595lowast

(34)

Vlowast (119909) = 119894119887Πlowast


(35)

119890lowast

(119909) = 119863119906lowast

+ 119894119887Vlowast (36)



119906lowast

(119909) = minus

3

sum

119899=1

119872119899

(119887 120596) 119896119899

119890minus119896119899119909

+ 119894119887

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909)

Vlowast (119909) = 119894119887

3

sum

119899=1

119872119899

(119887 120596) 119890minus119896119899119909

+

3

sum

119899=1

11987210158401015840

119899

(119887 120596) 119896119899

119890minus119896119899119909

(37)

where119872119899

1198721015840119899

11987210158401015840119899

and119872101584010158401015840

119899



1198721015840

119899

(119887 120596) = 1198671119899

119872119899

(119887 120596) 119899 = 1 2 3 (38)

11987210158401015840

119899

(119887 120596) = 1198672119899

119872119899

(119887 120596) 119899 = 1 2 3 (39)

119872101584010158401015840

119899

(119887 120596) = 1198673119899

119872119899

(119887 120596) 119899 = 1 2 3 (40)

where

1198671119899

=

1198607

(1198962

119899

minus 1198603

)

1198962119899

minus 1198606

119899 = 1 2 3 (41)

1198672119899

=1198605

(1198962119899

minus 1198604

) 119899 = 1 2 3 (42)

1198673119899

=1198607

120573lowast

1198606

minus 1198962119899

119899 = 1 2 3 (43)

Thus we have

120579lowast

(119909) =

3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120595lowast

(119909) =

3

sum

119899=1

1198672119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120593lowast

(119909) =

3

sum

119899=1

1198673119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

(44)


120590lowast

119909119909

=

3

sum

119899=1

ℎ119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119910119910

=

3

sum

119899=1

ℎ1015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119909119910

=

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

(45)


119906lowast

(119909) =

3

sum

119899=1

(1198941198871198672119899

minus 119896119899

)119872119899

(119887 120596) 119890minus119896119899119909

Vlowast (119909) =

3

sum

119899=1

(119894119887 + 119896119899

1198672119899

)119872119899

(119887 120596) 119890minus119896119899119909

(46)

where

ℎ119899

= minus[119896119899

(1198941198871198672119899

minus 119896119899

) minus119894119887120582 (119894119887 + 119896

119899

1198672119899

)

2120583 + 120582+

1205741198790

1198671119899

2120583 + 120582]

ℎ1015840

119899

= [119894119887 (119896119899

1198672119899

+ 119894119887) minus120582119896119899

(1198941198871198672119899

minus 119896119899

)

2120583 + 120582minus

1205741198790

1198671119899

2120583 + 120582]

ℎ10158401015840

119899

=120583 [119894119887 (119894119887119867

2119899

minus 119896119899

) minus 119896119899

(119894119887 + 119896119899

1198672119899

)]

2120583 + 120582

(47)


4 Application


119899


1

1198722

and 1198723


take the form


120579 (0 119910 119905) = 119891 (0 119910 119905) (48)


120590119909119909

(0 119910 119905) = 0 (49)


120590119909119910

(0 119910 119905) = 0 (50)





3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) = 119891lowast

(119910 119905)

3

sum

119899=1

ℎ119899

119872119899

(119887 120573 120596) = 0

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120573 120596) = 0

(51)



119895


5 Numerical Results


120582 = 759 times 109Nm2 120583 = 386 times 10

10 kgms2

120588 = 8954 kgm3 1205910

= 002 s


9Nm2

120578 = 888673ms2 120576 = 00168

120572119905


119887 = 1 119862119864

= 3831 J (kgK)

1198790

= 293K 119891lowast

= 1 120596 = 1205960

+ 119894120585

1205960

= 2 120585 = 1

(52)

Since we have 120596 = 1205960


= 1198901205960119905


0



119909119909

120590119910119910

120590119909119910


0




119909119909

120590119910119910

and 120590119909119910



119909119909


119909119909




119909119910



0 2 4 6 8 10 12minus15

minus1

minus05

0

05

1

15

120579

x


0 2 4 6 8 10 12minus015

minus01

minus005

0

005

x

u


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus02

minus01

0

01

02

0 2 4 6 8 10 12x

e


120590xx

0

005

01

015

minus0050 2 4 6 8 10 12

x

(e) The stress 120590119909119909

distribution

120590yy

0

002

004

006

minus002

minus004

minus006

minus0080 2 4 6 8 10 12

x

(f) The stress 120590119910119910

distribution

120590xy 0

001

002

003

minus002

minus001

minus0030 2 4 6 8 10 12

x

CDLS

(g) The stress 120590119909119910

distribution

120601 0

05

1

minus05

minus10 2 4 6 8 10 12

x

CDLS



120590119909119909

and 120590119909119910



minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

u


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

v


minus05

0

05

1

0 2 4 6 8 10 12x

e


0

01

02

03

0 2 4 6 8 10 12x

minus01

120590xx

(e) The stress 120590119909119909

distribution

0

01

02

03

04

0 2 4 6 8 10 12x

minus01

120590yy

(f) The stress 120590119910119910

distribution

minus002

minus004

Ω = 00

Ω = 02

0

002

004

006

008

0 2 4 6 8 10 12x

120590xy

(g) The stress 120590119909119910

distribution

Ω = 00

Ω = 02

0 2 4 6 8 10 12x

minus05

minus1

0

05

1

120601



120590119909119909

and 120590119909119910





119909119909

120590119910119910

120590119909119910



119909119909


119910119910





119909119909

and 120590119909119910


6 Conclusions










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











nabla2

120593 minus (1 + 1205910

120597

120597119905)

120597120579

120597119905minus 120576 (1 + 120591

0

120597

120597119905)

120597119890

120597119905= 0 (13)

120593 minus 120579 = 120573(1205972

120593

1205971199092+

1205972

120593

1205971199102) (14)

where 120576 = (120574120588119862119864

) and 120573 = 1198861205782

1198882

0



119906 =120597Π

120597119909+

120597120595

120597119910 V =

120597Π

120597119910minus

120597120595

120597119909 (15)


[nabla2

+ Ω2

minus1205972

1205971199052]Π + 2Ω

120597120595

120597119905minus 1198860

120579 = 0 (16)

(nabla2

minus 1198861

1205972

1205971199052+ 1198861

Ω2

)120595 minus 2Ω1198861

120597Π

120597119905= 0 (17)

where

1198861

=1205881198622

0

120583 119886

0

=1205741198790

1205881198622

0

(18)


nabla2

120593 minus (1 + 1205910

120597

120597119905)

120597120579

120597119905minus 120576 (1 + 120591

0

120597

120597119905)

120597Π

120597119905= 0 (19)



[Π 120595 120593 120579 120590119894119895

] (119909 119910 119905)

= [Πlowast

120595lowast

119906lowast

(119909) 120593lowast

(119909) 120579lowast

(119909) 120590lowast

119894119895

(119909)]

times exp (120596119905 + 119894119887119910)

(20)




119894119895


[1198632

minus 1198601

]Πlowast

+ 1198600

120595lowast

minus 1198602

120579lowast

= 0 (21)

(1198632

minus 1198604

) 120595lowast

minus 1198605

Π = 0 (22)

[1198632

minus 1198603

] 120593lowast

= minus120573lowast

120579lowast

(23)

(1198632

minus 1198872

) 120593 minus 119860120579lowast


= 0 (24)

where1198600

= 2Ω120596119860 = 120596(1+1205961205910

) 119861 = 1205761198601198601

= 1198872

+1205962

minusΩ2

1198602

= 1198860

1198603

= (1205731198872

+ 1)120573 120573lowast = 1120573 1198604

= 1198872

+ 1198861

(1205962

minusΩ2

)1198605

= 1198861

1198600

and 119863 = 119889119889119909




(119909)

[1198636

minus 1198641198634

+ 1198651198632


(119909) = 0 (25)

where 1198606

= (120573lowast

1198872

+ 1198603

119860)(120573lowast

+ 119860) 1198607


+ 119860)Since

119864 = 1198601

+ 1198604

+ 1198606

+ 1198602

1198607

119865 = 1198601

1198604

+ 1198600

1198605

+ 1198606

(1198601

+ 1198604

) + 1198602

1198607

(1198603

+ 1198604

)

119866 = 1198601

1198604

1198606

+ 1198602

1198603

1198604

1198607

+ 1198600

1198605

1198606

(26)


[1198636

minus 1198641198634

+ 1198651198632


120593lowast

120595lowast

) (119909) = 0 (27)


(1198632

minus 1198962

1

) (1198632

minus 1198962

2

) (1198632

minus 1198962

3

)Πlowast

(119909) = 0 (28)

where 1198962

119899


1198966

minus 1198641198964

+ 1198651198962

minus 119866 = 0 (29)


Πlowast

(119909) =

3

sum

119899=1

119872119899

(119887 120596) exp (minus119896119899

119909) (30)

Similarly

120579lowast

(119909) =

3

sum

119899=1

1198721015840

119899

(119887 120596) exp (minus119896119899

119909) (31)

120595lowast

(119909) =

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909) (32)

120593lowast

(119909) =

3

sum

119899=1

119872101584010158401015840

119899

(119887 120596) exp (minus119896119899

119909) (33)

Since

119906lowast

(119909) = 119863Πlowast

+ 119894119887120595lowast

(34)

Vlowast (119909) = 119894119887Πlowast


(35)

119890lowast

(119909) = 119863119906lowast

+ 119894119887Vlowast (36)



119906lowast

(119909) = minus

3

sum

119899=1

119872119899

(119887 120596) 119896119899

119890minus119896119899119909

+ 119894119887

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909)

Vlowast (119909) = 119894119887

3

sum

119899=1

119872119899

(119887 120596) 119890minus119896119899119909

+

3

sum

119899=1

11987210158401015840

119899

(119887 120596) 119896119899

119890minus119896119899119909

(37)

where119872119899

1198721015840119899

11987210158401015840119899

and119872101584010158401015840

119899



1198721015840

119899

(119887 120596) = 1198671119899

119872119899

(119887 120596) 119899 = 1 2 3 (38)

11987210158401015840

119899

(119887 120596) = 1198672119899

119872119899

(119887 120596) 119899 = 1 2 3 (39)

119872101584010158401015840

119899

(119887 120596) = 1198673119899

119872119899

(119887 120596) 119899 = 1 2 3 (40)

where

1198671119899

=

1198607

(1198962

119899

minus 1198603

)

1198962119899

minus 1198606

119899 = 1 2 3 (41)

1198672119899

=1198605

(1198962119899

minus 1198604

) 119899 = 1 2 3 (42)

1198673119899

=1198607

120573lowast

1198606

minus 1198962119899

119899 = 1 2 3 (43)

Thus we have

120579lowast

(119909) =

3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120595lowast

(119909) =

3

sum

119899=1

1198672119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120593lowast

(119909) =

3

sum

119899=1

1198673119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

(44)


120590lowast

119909119909

=

3

sum

119899=1

ℎ119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119910119910

=

3

sum

119899=1

ℎ1015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119909119910

=

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

(45)


119906lowast

(119909) =

3

sum

119899=1

(1198941198871198672119899

minus 119896119899

)119872119899

(119887 120596) 119890minus119896119899119909

Vlowast (119909) =

3

sum

119899=1

(119894119887 + 119896119899

1198672119899

)119872119899

(119887 120596) 119890minus119896119899119909

(46)

where

ℎ119899

= minus[119896119899

(1198941198871198672119899

minus 119896119899

) minus119894119887120582 (119894119887 + 119896

119899

1198672119899

)

2120583 + 120582+

1205741198790

1198671119899

2120583 + 120582]

ℎ1015840

119899

= [119894119887 (119896119899

1198672119899

+ 119894119887) minus120582119896119899

(1198941198871198672119899

minus 119896119899

)

2120583 + 120582minus

1205741198790

1198671119899

2120583 + 120582]

ℎ10158401015840

119899

=120583 [119894119887 (119894119887119867

2119899

minus 119896119899

) minus 119896119899

(119894119887 + 119896119899

1198672119899

)]

2120583 + 120582

(47)


4 Application


119899


1

1198722

and 1198723


take the form


120579 (0 119910 119905) = 119891 (0 119910 119905) (48)


120590119909119909

(0 119910 119905) = 0 (49)


120590119909119910

(0 119910 119905) = 0 (50)





3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) = 119891lowast

(119910 119905)

3

sum

119899=1

ℎ119899

119872119899

(119887 120573 120596) = 0

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120573 120596) = 0

(51)



119895


5 Numerical Results


120582 = 759 times 109Nm2 120583 = 386 times 10

10 kgms2

120588 = 8954 kgm3 1205910

= 002 s


9Nm2

120578 = 888673ms2 120576 = 00168

120572119905


119887 = 1 119862119864

= 3831 J (kgK)

1198790

= 293K 119891lowast

= 1 120596 = 1205960

+ 119894120585

1205960

= 2 120585 = 1

(52)

Since we have 120596 = 1205960


= 1198901205960119905


0



119909119909

120590119910119910

120590119909119910


0




119909119909

120590119910119910

and 120590119909119910



119909119909


119909119909




119909119910



0 2 4 6 8 10 12minus15

minus1

minus05

0

05

1

15

120579

x


0 2 4 6 8 10 12minus015

minus01

minus005

0

005

x

u


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus02

minus01

0

01

02

0 2 4 6 8 10 12x

e


120590xx

0

005

01

015

minus0050 2 4 6 8 10 12

x

(e) The stress 120590119909119909

distribution

120590yy

0

002

004

006

minus002

minus004

minus006

minus0080 2 4 6 8 10 12

x

(f) The stress 120590119910119910

distribution

120590xy 0

001

002

003

minus002

minus001

minus0030 2 4 6 8 10 12

x

CDLS

(g) The stress 120590119909119910

distribution

120601 0

05

1

minus05

minus10 2 4 6 8 10 12

x

CDLS



120590119909119909

and 120590119909119910



minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

u


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

v


minus05

0

05

1

0 2 4 6 8 10 12x

e


0

01

02

03

0 2 4 6 8 10 12x

minus01

120590xx

(e) The stress 120590119909119909

distribution

0

01

02

03

04

0 2 4 6 8 10 12x

minus01

120590yy

(f) The stress 120590119910119910

distribution

minus002

minus004

Ω = 00

Ω = 02

0

002

004

006

008

0 2 4 6 8 10 12x

120590xy

(g) The stress 120590119909119910

distribution

Ω = 00

Ω = 02

0 2 4 6 8 10 12x

minus05

minus1

0

05

1

120601



120590119909119909

and 120590119909119910





119909119909

120590119910119910

120590119909119910



119909119909


119910119910





119909119909

and 120590119909119910


6 Conclusions










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906lowast

(119909) = minus

3

sum

119899=1

119872119899

(119887 120596) 119896119899

119890minus119896119899119909

+ 119894119887

3

sum

119899=1

11987210158401015840

119899

(119887 120596) exp (minus119896119899

119909)

Vlowast (119909) = 119894119887

3

sum

119899=1

119872119899

(119887 120596) 119890minus119896119899119909

+

3

sum

119899=1

11987210158401015840

119899

(119887 120596) 119896119899

119890minus119896119899119909

(37)

where119872119899

1198721015840119899

11987210158401015840119899

and119872101584010158401015840

119899



1198721015840

119899

(119887 120596) = 1198671119899

119872119899

(119887 120596) 119899 = 1 2 3 (38)

11987210158401015840

119899

(119887 120596) = 1198672119899

119872119899

(119887 120596) 119899 = 1 2 3 (39)

119872101584010158401015840

119899

(119887 120596) = 1198673119899

119872119899

(119887 120596) 119899 = 1 2 3 (40)

where

1198671119899

=

1198607

(1198962

119899

minus 1198603

)

1198962119899

minus 1198606

119899 = 1 2 3 (41)

1198672119899

=1198605

(1198962119899

minus 1198604

) 119899 = 1 2 3 (42)

1198673119899

=1198607

120573lowast

1198606

minus 1198962119899

119899 = 1 2 3 (43)

Thus we have

120579lowast

(119909) =

3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120595lowast

(119909) =

3

sum

119899=1

1198672119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

120593lowast

(119909) =

3

sum

119899=1

1198673119899

119872119899

(119887 120573lowast

120596) exp (minus119896119899

119909)

(44)


120590lowast

119909119909

=

3

sum

119899=1

ℎ119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119910119910

=

3

sum

119899=1

ℎ1015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

120590lowast

119909119910

=

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120596) exp (minus119896119899

119909)

(45)


119906lowast

(119909) =

3

sum

119899=1

(1198941198871198672119899

minus 119896119899

)119872119899

(119887 120596) 119890minus119896119899119909

Vlowast (119909) =

3

sum

119899=1

(119894119887 + 119896119899

1198672119899

)119872119899

(119887 120596) 119890minus119896119899119909

(46)

where

ℎ119899

= minus[119896119899

(1198941198871198672119899

minus 119896119899

) minus119894119887120582 (119894119887 + 119896

119899

1198672119899

)

2120583 + 120582+

1205741198790

1198671119899

2120583 + 120582]

ℎ1015840

119899

= [119894119887 (119896119899

1198672119899

+ 119894119887) minus120582119896119899

(1198941198871198672119899

minus 119896119899

)

2120583 + 120582minus

1205741198790

1198671119899

2120583 + 120582]

ℎ10158401015840

119899

=120583 [119894119887 (119894119887119867

2119899

minus 119896119899

) minus 119896119899

(119894119887 + 119896119899

1198672119899

)]

2120583 + 120582

(47)


4 Application


119899


1

1198722

and 1198723


take the form


120579 (0 119910 119905) = 119891 (0 119910 119905) (48)


120590119909119909

(0 119910 119905) = 0 (49)


120590119909119910

(0 119910 119905) = 0 (50)





3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) = 119891lowast

(119910 119905)

3

sum

119899=1

ℎ119899

119872119899

(119887 120573 120596) = 0

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120573 120596) = 0

(51)



119895


5 Numerical Results


120582 = 759 times 109Nm2 120583 = 386 times 10

10 kgms2

120588 = 8954 kgm3 1205910

= 002 s


9Nm2

120578 = 888673ms2 120576 = 00168

120572119905


119887 = 1 119862119864

= 3831 J (kgK)

1198790

= 293K 119891lowast

= 1 120596 = 1205960

+ 119894120585

1205960

= 2 120585 = 1

(52)

Since we have 120596 = 1205960


= 1198901205960119905


0



119909119909

120590119910119910

120590119909119910


0




119909119909

120590119910119910

and 120590119909119910



119909119909


119909119909




119909119910



0 2 4 6 8 10 12minus15

minus1

minus05

0

05

1

15

120579

x


0 2 4 6 8 10 12minus015

minus01

minus005

0

005

x

u


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus02

minus01

0

01

02

0 2 4 6 8 10 12x

e


120590xx

0

005

01

015

minus0050 2 4 6 8 10 12

x

(e) The stress 120590119909119909

distribution

120590yy

0

002

004

006

minus002

minus004

minus006

minus0080 2 4 6 8 10 12

x

(f) The stress 120590119910119910

distribution

120590xy 0

001

002

003

minus002

minus001

minus0030 2 4 6 8 10 12

x

CDLS

(g) The stress 120590119909119910

distribution

120601 0

05

1

minus05

minus10 2 4 6 8 10 12

x

CDLS



120590119909119909

and 120590119909119910



minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

u


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

v


minus05

0

05

1

0 2 4 6 8 10 12x

e


0

01

02

03

0 2 4 6 8 10 12x

minus01

120590xx

(e) The stress 120590119909119909

distribution

0

01

02

03

04

0 2 4 6 8 10 12x

minus01

120590yy

(f) The stress 120590119910119910

distribution

minus002

minus004

Ω = 00

Ω = 02

0

002

004

006

008

0 2 4 6 8 10 12x

120590xy

(g) The stress 120590119909119910

distribution

Ω = 00

Ω = 02

0 2 4 6 8 10 12x

minus05

minus1

0

05

1

120601



120590119909119909

and 120590119909119910





119909119909

120590119910119910

120590119909119910



119909119909


119910119910





119909119909

and 120590119909119910


6 Conclusions










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











3

sum

119899=1

1198671119899

119872119899

(119887 120573lowast

120596) = 119891lowast

(119910 119905)

3

sum

119899=1

ℎ119899

119872119899

(119887 120573 120596) = 0

3

sum

119899=1

ℎ10158401015840

119899

119872119899

(119887 120573 120596) = 0

(51)



119895


5 Numerical Results


120582 = 759 times 109Nm2 120583 = 386 times 10

10 kgms2

120588 = 8954 kgm3 1205910

= 002 s


9Nm2

120578 = 888673ms2 120576 = 00168

120572119905


119887 = 1 119862119864

= 3831 J (kgK)

1198790

= 293K 119891lowast

= 1 120596 = 1205960

+ 119894120585

1205960

= 2 120585 = 1

(52)

Since we have 120596 = 1205960


= 1198901205960119905


0



119909119909

120590119910119910

120590119909119910


0




119909119909

120590119910119910

and 120590119909119910



119909119909


119909119909




119909119910



0 2 4 6 8 10 12minus15

minus1

minus05

0

05

1

15

120579

x


0 2 4 6 8 10 12minus015

minus01

minus005

0

005

x

u


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus02

minus01

0

01

02

0 2 4 6 8 10 12x

e


120590xx

0

005

01

015

minus0050 2 4 6 8 10 12

x

(e) The stress 120590119909119909

distribution

120590yy

0

002

004

006

minus002

minus004

minus006

minus0080 2 4 6 8 10 12

x

(f) The stress 120590119910119910

distribution

120590xy 0

001

002

003

minus002

minus001

minus0030 2 4 6 8 10 12

x

CDLS

(g) The stress 120590119909119910

distribution

120601 0

05

1

minus05

minus10 2 4 6 8 10 12

x

CDLS



120590119909119909

and 120590119909119910



minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

u


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

v


minus05

0

05

1

0 2 4 6 8 10 12x

e


0

01

02

03

0 2 4 6 8 10 12x

minus01

120590xx

(e) The stress 120590119909119909

distribution

0

01

02

03

04

0 2 4 6 8 10 12x

minus01

120590yy

(f) The stress 120590119910119910

distribution

minus002

minus004

Ω = 00

Ω = 02

0

002

004

006

008

0 2 4 6 8 10 12x

120590xy

(g) The stress 120590119909119910

distribution

Ω = 00

Ω = 02

0 2 4 6 8 10 12x

minus05

minus1

0

05

1

120601



120590119909119909

and 120590119909119910





119909119909

120590119910119910

120590119909119910



119909119909


119910119910





119909119909

and 120590119909119910


6 Conclusions










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12minus15

minus1

minus05

0

05

1

15

120579

x


0 2 4 6 8 10 12minus015

minus01

minus005

0

005

x

u


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus02

minus01

0

01

02

0 2 4 6 8 10 12x

e


120590xx

0

005

01

015

minus0050 2 4 6 8 10 12

x

(e) The stress 120590119909119909

distribution

120590yy

0

002

004

006

minus002

minus004

minus006

minus0080 2 4 6 8 10 12

x

(f) The stress 120590119910119910

distribution

120590xy 0

001

002

003

minus002

minus001

minus0030 2 4 6 8 10 12

x

CDLS

(g) The stress 120590119909119910

distribution

120601 0

05

1

minus05

minus10 2 4 6 8 10 12

x

CDLS



120590119909119909

and 120590119909119910



minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

u


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

v


minus05

0

05

1

0 2 4 6 8 10 12x

e


0

01

02

03

0 2 4 6 8 10 12x

minus01

120590xx

(e) The stress 120590119909119909

distribution

0

01

02

03

04

0 2 4 6 8 10 12x

minus01

120590yy

(f) The stress 120590119910119910

distribution

minus002

minus004

Ω = 00

Ω = 02

0

002

004

006

008

0 2 4 6 8 10 12x

120590xy

(g) The stress 120590119909119910

distribution

Ω = 00

Ω = 02

0 2 4 6 8 10 12x

minus05

minus1

0

05

1

120601



120590119909119909

and 120590119909119910





119909119909

120590119910119910

120590119909119910



119909119909


119910119910





119909119909

and 120590119909119910


6 Conclusions










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

u


0

02

minus02

minus04

minus060 2 4 6 8 10 12

x

v


minus05

0

05

1

0 2 4 6 8 10 12x

e


0

01

02

03

0 2 4 6 8 10 12x

minus01

120590xx

(e) The stress 120590119909119909

distribution

0

01

02

03

04

0 2 4 6 8 10 12x

minus01

120590yy

(f) The stress 120590119910119910

distribution

minus002

minus004

Ω = 00

Ω = 02

0

002

004

006

008

0 2 4 6 8 10 12x

120590xy

(g) The stress 120590119909119910

distribution

Ω = 00

Ω = 02

0 2 4 6 8 10 12x

minus05

minus1

0

05

1

120601



120590119909119909

and 120590119909119910





119909119909

120590119910119910

120590119909119910



119909119909


119910119910





119909119909

and 120590119909119910


6 Conclusions










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119909119909

120590119910119910

120590119909119910



119909119909


119910119910





119909119909

and 120590119909119910


6 Conclusions










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus15

minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

u

0 2 4 6 8 10 12x


minus01

minus005

0

005

01

015

0 2 4 6 8 10 12x


minus015

minus01

minus005

0

005

01

015

02

e

0 2 4 6 8 10 12x


0

005

01

015

minus005

120590xx

0 2 4 6 8 10 12x

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus01

minus008

minus006

minus004

minus002

0

002

004

120590yy

(f) The stress 120590119910119910

distribution

120573 = 01

120573 = 02

0

002

004

minus006

minus004

minus002

120590xy

0 2 4 6 8 10 12x

(g) The stress 120590119909119910

distribution

120573 = 01

120573 = 02

0

05

1

minus1

minus05

120601

0 2 4 6 8 10 12x



120590119909119909

and 120590119909119910



minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus1

minus05

0

05

1

15

120579

0 2 4 6 8 10 12x


0 2 4 6 8 10 12x

minus015

minus01

minus005

0

005

u


0

002

004

006

008

01

0 2 4 6 8 10 12x

minus002


0 2 4 6 8 10 12x

minus02

minus015

minus01

minus005

0

005

01

015

e


0 2 4 6 8 10 12x

0

005

01

015

minus005

120590xx

(e) The stress 120590119909119909

distribution

0 2 4 6 8 10 12x

minus008

minus006

minus004

minus002

0

002

120590yy

(f) The stress 120590119910119910

distribution

0 2 4 6 8 10 12x

minus003

minus002

minus001

0

002

001

120590xy

t = 01

t = 03

(g) The stress 120590119909119910

distribution

0 2 4 6 8 10 12x

minus04

minus02

0

02

04

06

08

1

120601

t = 01

t = 03



120590119909119909

and 120590119909119910




Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Nomenclature



0


0



0

)1198790

| lt 1

120590119894119895








Lorentz force120575119894119895


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article effect of rotation for two-temperature...

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